id	sid	tid	token	lemma	pos
eajbcs-667	1	1	finite	finite	PROPN
eajbcs-667	1	2	element	element	NOUN
eajbcs-667	1	3	method	method	NOUN
eajbcs-667	1	4	;	;	PUNCT
eajbcs-667	1	5	varational	varational	ADJ
eajbcs-667	1	6	formulation	formulation	NOUN
eajbcs-667	1	7	;	;	PUNCT
eajbcs-667	1	8	numerical	numerical	ADJ
eajbcs-667	1	9	integration	integration	NOUN
eajbcs-667	1	10	an	an	DET
eajbcs-667	1	11	advection	advection	NOUN
eajbcs-667	1	12	-	-	PUNCT
eajbcs-667	1	13	diffusion	diffusion	NOUN
eajbcs-667	1	14	equation	equation	NOUN
eajbcs-667	1	15	(	(	PUNCT
eajbcs-667	1	16	ade	ade	PROPN
eajbcs-667	1	17	)	)	PUNCT
eajbcs-667	1	18	is	be	AUX
eajbcs-667	1	19	a	a	DET
eajbcs-667	1	20	mathematical	mathematical	ADJ
eajbcs-667	1	21	model	model	NOUN
eajbcs-667	1	22	that	that	PRON
eajbcs-667	1	23	has	have	AUX
eajbcs-667	1	24	been	be	AUX
eajbcs-667	1	25	used	use	VERB
eajbcs-667	1	26	to	to	PART
eajbcs-667	1	27	model	model	VERB
eajbcs-667	1	28	the	the	DET
eajbcs-667	1	29	concentration	concentration	NOUN
eajbcs-667	1	30	of	of	ADP
eajbcs-667	1	31	pollutants	pollutant	NOUN
eajbcs-667	1	32	.	.	PUNCT
eajbcs-667	2	1	it	it	PRON
eajbcs-667	2	2	gives	give	VERB
eajbcs-667	2	3	the	the	DET
eajbcs-667	2	4	amount	amount	NOUN
eajbcs-667	2	5	of	of	ADP
eajbcs-667	2	6	pollutant	pollutant	ADJ
eajbcs-667	2	7	concentration	concentration	NOUN
eajbcs-667	2	8	fields	field	NOUN
eajbcs-667	2	9	a	a	DET
eajbcs-667	2	10	fter	fter	NOUN
eajbcs-667	3	1	i	i	PRON
eajbcs-667	3	2	nput	nput	NOUN
eajbcs-667	4	1	o	o	X
eajbcs-667	4	2	f	f	NOUN
eajbcs-667	5	1	t	t	NOUN
eajbcs-667	5	2	he	he	PRON
eajbcs-667	5	3	velocity	velocity	NOUN
eajbcs-667	5	4	data	datum	NOUN
eajbcs-667	5	5	from	from	ADP
eajbcs-667	5	6	the	the	DET
eajbcs-667	5	7	hydrodynamic	hydrodynamic	ADJ
eajbcs-667	5	8	model	model	NOUN
eajbcs-667	5	9	which	which	PRON
eajbcs-667	5	10	are	be	AUX
eajbcs-667	5	11	derived	derive	VERB
eajbcs-667	5	12	from	from	ADP
eajbcs-667	5	13	mass	mass	ADJ
eajbcs-667	5	14	balances	balance	NOUN
eajbcs-667	5	15	.	.	PUNCT
eajbcs-667	6	1	formally	formally	ADV
eajbcs-667	6	2	the	the	DET
eajbcs-667	6	3	ade	ade	NOUN
eajbcs-667	6	4	equation	equation	NOUN
eajbcs-667	6	5	is	be	AUX
eajbcs-667	6	6	given	give	VERB
eajbcs-667	6	7	by	by	ADP
eajbcs-667	6	8	ut	ut	PROPN
eajbcs-667	6	9	+	+	CCONJ
eajbcs-667	6	10	a∇u	a∇u	X
eajbcs-667	6	11	=	=	SYM
eajbcs-667	6	12	∇(d∇u	∇(d∇u	PROPN
eajbcs-667	6	13	)	)	PUNCT
eajbcs-667	7	1	+	+	SYM
eajbcs-667	7	2	f	f	X
eajbcs-667	7	3	,	,	PUNCT
eajbcs-667	7	4	(	(	PUNCT
eajbcs-667	7	5	1	1	X
eajbcs-667	7	6	)	)	PUNCT
eajbcs-667	7	7	where	where	SCONJ
eajbcs-667	7	8	u	u	NOUN
eajbcs-667	7	9	is	be	AUX
eajbcs-667	7	10	the	the	DET
eajbcs-667	7	11	concentration	concentration	NOUN
eajbcs-667	7	12	of	of	ADP
eajbcs-667	7	13	the	the	DET
eajbcs-667	7	14	pollutant	pollutant	NOUN
eajbcs-667	7	15	,	,	PUNCT
eajbcs-667	7	16	a	a	PRON
eajbcs-667	7	17	is	be	AUX
eajbcs-667	7	18	the	the	DET
eajbcs-667	7	19	velocity	velocity	NOUN
eajbcs-667	7	20	of	of	ADP
eajbcs-667	7	21	the	the	DET
eajbcs-667	7	22	considered	consider	VERB
eajbcs-667	7	23	particle	particle	NOUN
eajbcs-667	7	24	,	,	PUNCT
eajbcs-667	7	25	d	d	PROPN
eajbcs-667	7	26	the	the	DET
eajbcs-667	7	27	diffusion	diffusion	NOUN
eajbcs-667	7	28	coefficient	coefficient	NOUN
eajbcs-667	7	29	and	and	CCONJ
eajbcs-667	7	30	f	f	PROPN
eajbcs-667	7	31	defines	define	VERB
eajbcs-667	7	32	the	the	DET
eajbcs-667	7	33	sources	source	NOUN
eajbcs-667	7	34	and	and	CCONJ
eajbcs-667	7	35	sinks	sink	NOUN
eajbcs-667	7	36	due	due	ADJ
eajbcs-667	7	37	to	to	ADP
eajbcs-667	7	38	different	different	ADJ
eajbcs-667	7	39	processes	process	NOUN
eajbcs-667	7	40	.	.	PUNCT
eajbcs-667	8	1	east	east	ADJ
eajbcs-667	8	2	african	african	PROPN
eajbcs-667	8	3	journal	journal	PROPN
eajbcs-667	8	4	of	of	ADP
eajbcs-667	8	5	biophysical	biophysical	ADJ
eajbcs-667	8	6	and	and	CCONJ
eajbcs-667	8	7	computational	computational	ADJ
eajbcs-667	8	8	sciences	sciences	PROPN
eajbcs-667	8	9	journal	journal	PROPN
eajbcs-667	8	10	homepage	homepage	NOUN
eajbcs-667	8	11	:	:	PUNCT
eajbcs-667	8	12	https://journals.hu.edu.et/hu-journals/index.php/eajbcs	https://journals.hu.edu.et/hu-journals/index.php/eajbcs	VERB
eajbcs-667	8	13	college	college	NOUN
eajbcs-667	8	14	of	of	ADP
eajbcs-667	8	15	natural	natural	PROPN
eajbcs-667	8	16	&	&	CCONJ
eajbcs-667	8	17	com	com	PROPN
eajbcs-667	8	18	h	h	NOUN
eajbcs-667	9	1	pu	pu	PROPN
eajbcs-667	10	1	aw	aw	INTJ
eajbcs-667	10	2	tat	tat	NOUN
eajbcs-667	10	3	as	as	ADP
eajbcs-667	10	4	ion	ion	NOUN
eajbcs-667	10	5	sa	sa	NOUN
eajbcs-667	10	6	u	u	PROPN
eajbcs-667	10	7	al	al	PROPN
eajbcs-667	10	8	n	n	PROPN
eajbcs-667	10	9	sc	sc	PROPN
eajbcs-667	11	1	i	i	X
eajbcs-667	11	2	ve	ve	VERB
eajbcs-667	11	3	ie	ie	ADV
eajbcs-667	11	4	r	r	PROPN
eajbcs-667	11	5	nc	nc	PROPN
eajbcs-667	11	6	sit	sit	PROPN
eajbcs-667	11	7	e	e	PROPN
eajbcs-667	11	8	y	y	PROPN
eajbcs-667	11	9	s	s	PART
eajbcs-667	11	10	year	year	NOUN
eajbcs-667	11	11	2021	2021	NUM
eajbcs-667	11	12	volume	volume	NOUN
eajbcs-667	11	13	xx	xx	NUM
eajbcs-667	12	1	no	no	DET
eajbcs-667	12	2	xx	xx	NUM
eajbcs-667	12	3	keywords	keyword	NOUN
eajbcs-667	12	4	:	:	PUNCT
eajbcs-667	12	5	abstract	abstract	ADJ
eajbcs-667	12	6	*	*	PUNCT
eajbcs-667	12	7	corresponding	correspond	VERB
eajbcs-667	12	8	author	author	NOUN
eajbcs-667	12	9	:	:	PUNCT
eajbcs-667	12	10	email	email	NOUN
eajbcs-667	12	11	:	:	PUNCT
eajbcs-667	12	12	kassahunm@hu.edu.et	kassahunm@hu.edu.et	PROPN
eajbcs-667	12	13	+251918491415	+251918491415	PROPN
eajbcs-667	12	14	https://dx.doi.org/10.4314/eajbcs.v4i1.5s	https://dx.doi.org/10.4314/eajbcs.v4i1.5s	PROPN
eajbcs-667	12	15	east	east	PROPN
eajbcs-667	12	16	afr	afr	PROPN
eajbcs-667	12	17	.	.	PUNCT
eajbcs-667	13	1	j.	j.	PROPN
eajbcs-667	13	2	biophys	biophys	PROPN
eajbcs-667	13	3	.	.	PUNCT
eajbcs-667	14	1	comput	comput	NOUN
eajbcs-667	14	2	.	.	PUNCT
eajbcs-667	15	1	sci	sci	PROPN
eajbcs-667	15	2	.	.	PUNCT
eajbcs-667	15	3	(	(	PUNCT
eajbcs-667	15	4	2023	2023	NUM
eajbcs-667	15	5	)	)	PUNCT
eajbcs-667	15	6	,	,	PUNCT
eajbcs-667	15	7	vol	vol	NOUN
eajbcs-667	15	8	.	.	PROPN
eajbcs-667	15	9	4	4	NUM
eajbcs-667	15	10	,	,	PUNCT
eajbcs-667	15	11	issue	issue	NOUN
eajbcs-667	15	12	.	.	PUNCT
eajbcs-667	16	1	1	1	NUM
eajbcs-667	16	2	,	,	PUNCT
eajbcs-667	16	3	52	52	NUM
eajbcs-667	16	4	-	-	SYM
eajbcs-667	16	5	74	74	NUM
eajbcs-667	16	6	for	for	ADP
eajbcs-667	16	7	the	the	DET
eajbcs-667	16	8	vast	vast	ADJ
eajbcs-667	16	9	majority	majority	NOUN
eajbcs-667	16	10	of	of	ADP
eajbcs-667	16	11	geometries	geometry	NOUN
eajbcs-667	16	12	and	and	CCONJ
eajbcs-667	16	13	problems	problem	NOUN
eajbcs-667	16	14	,	,	PUNCT
eajbcs-667	16	15	eq	eq	NOUN
eajbcs-667	16	16	.	.	PROPN
eajbcs-667	16	17	1	1	NUM
eajbcs-667	16	18	can	can	AUX
eajbcs-667	16	19	not	not	PART
eajbcs-667	16	20	be	be	AUX
eajbcs-667	16	21	solved	solve	VERB
eajbcs-667	16	22	with	with	ADP
eajbcs-667	16	23	analytical	analytical	ADJ
eajbcs-667	16	24	methods	method	NOUN
eajbcs-667	16	25	,	,	PUNCT
eajbcs-667	16	26	and	and	CCONJ
eajbcs-667	16	27	an	an	DET
eajbcs-667	16	28	approximation	approximation	NOUN
eajbcs-667	16	29	of	of	ADP
eajbcs-667	16	30	the	the	DET
eajbcs-667	16	31	equations	equation	NOUN
eajbcs-667	16	32	can	can	AUX
eajbcs-667	16	33	be	be	AUX
eajbcs-667	16	34	constructed	construct	VERB
eajbcs-667	16	35	with	with	ADP
eajbcs-667	16	36	different	different	ADJ
eajbcs-667	16	37	types	type	NOUN
eajbcs-667	16	38	of	of	ADP
eajbcs-667	16	39	discretizations	discretization	NOUN
eajbcs-667	16	40	.	.	PUNCT
eajbcs-667	17	1	many	many	ADJ
eajbcs-667	17	2	numerical	numerical	ADJ
eajbcs-667	17	3	schemes	scheme	NOUN
eajbcs-667	17	4	have	have	AUX
eajbcs-667	17	5	been	be	AUX
eajbcs-667	17	6	implemented	implement	VERB
eajbcs-667	17	7	to	to	PART
eajbcs-667	17	8	approximately	approximately	ADV
eajbcs-667	17	9	solve	solve	VERB
eajbcs-667	17	10	the	the	DET
eajbcs-667	17	11	ade	ade	NOUN
eajbcs-667	17	12	(	(	PUNCT
eajbcs-667	17	13	lima	lima	PROPN
eajbcs-667	17	14	et	et	PROPN
eajbcs-667	17	15	al	al	PROPN
eajbcs-667	17	16	.	.	PROPN
eajbcs-667	17	17	,	,	PUNCT
eajbcs-667	17	18	2021	2021	NUM
eajbcs-667	17	19	;	;	PUNCT
eajbcs-667	17	20	mahmud	mahmud	PROPN
eajbcs-667	17	21	,	,	PUNCT
eajbcs-667	17	22	2012	2012	NUM
eajbcs-667	17	23	;	;	PUNCT
eajbcs-667	17	24	pochai	pochai	ADJ
eajbcs-667	17	25	and	and	CCONJ
eajbcs-667	17	26	deepana	deepana	ADJ
eajbcs-667	17	27	,	,	PUNCT
eajbcs-667	17	28	2011	2011	NUM
eajbcs-667	17	29	;	;	PUNCT
eajbcs-667	18	1	lian	lian	PROPN
eajbcs-667	18	2	et	et	PROPN
eajbcs-667	18	3	al	al	PROPN
eajbcs-667	18	4	.	.	PROPN
eajbcs-667	18	5	,	,	PUNCT
eajbcs-667	18	6	2016	2016	NUM
eajbcs-667	18	7	;	;	PUNCT
eajbcs-667	18	8	szymkiewicz	szymkiewicz	NOUN
eajbcs-667	18	9	and	and	CCONJ
eajbcs-667	18	10	gkakassahun	gkakassahun	PROPN
eajbcs-667	18	11	getnet	getnet	NOUN
eajbcs-667	18	12	mekonen	mekonen	PROPN
eajbcs-667	18	13	*	*	PROPN
eajbcs-667	18	14	,	,	PUNCT
eajbcs-667	18	15	zerihun	zerihun	PROPN
eajbcs-667	18	16	kinfe	kinfe	PROPN
eajbcs-667	18	17	birhanu	birhanu	PROPN
eajbcs-667	18	18	department	department	PROPN
eajbcs-667	18	19	of	of	ADP
eajbcs-667	18	20	mathematics	mathematics	PROPN
eajbcs-667	18	21	,	,	PUNCT
eajbcs-667	18	22	college	college	NOUN
eajbcs-667	18	23	of	of	ADP
eajbcs-667	18	24	natural	natural	ADJ
eajbcs-667	18	25	and	and	CCONJ
eajbcs-667	18	26	computational	computational	ADJ
eajbcs-667	18	27	sciences	science	NOUN
eajbcs-667	18	28	,	,	PUNCT
eajbcs-667	18	29	hawassa	hawassa	ADJ
eajbcs-667	18	30	university	university	NOUN
eajbcs-667	18	31	,	,	PUNCT
eajbcs-667	18	32	hawassa	hawassa	NOUN
eajbcs-667	18	33	,	,	PUNCT
eajbcs-667	18	34	ethiopia	ethiopia	PROPN
eajbcs-667	18	35	.	.	PUNCT
eajbcs-667	19	1	in	in	ADP
eajbcs-667	19	2	this	this	DET
eajbcs-667	19	3	paper	paper	NOUN
eajbcs-667	19	4	,	,	PUNCT
eajbcs-667	19	5	we	we	PRON
eajbcs-667	19	6	have	have	AUX
eajbcs-667	19	7	implemented	implement	VERB
eajbcs-667	19	8	the	the	DET
eajbcs-667	19	9	finite	finite	ADJ
eajbcs-667	19	10	element	element	NOUN
eajbcs-667	19	11	method	method	NOUN
eajbcs-667	19	12	for	for	ADP
eajbcs-667	19	13	the	the	DET
eajbcs-667	19	14	numerical	numerical	ADJ
eajbcs-667	19	15	solution	solution	NOUN
eajbcs-667	19	16	of	of	ADP
eajbcs-667	19	17	a	a	DET
eajbcs-667	19	18	boundary	boundary	ADJ
eajbcs-667	19	19	and	and	CCONJ
eajbcs-667	19	20	initial	initial	ADJ
eajbcs-667	19	21	value	value	NOUN
eajbcs-667	19	22	problems	problem	NOUN
eajbcs-667	19	23	,	,	PUNCT
eajbcs-667	19	24	mainly	mainly	ADV
eajbcs-667	19	25	on	on	ADP
eajbcs-667	19	26	solving	solve	VERB
eajbcs-667	19	27	the	the	DET
eajbcs-667	19	28	one	one	NUM
eajbcs-667	19	29	and	and	CCONJ
eajbcs-667	19	30	two	two	NUM
eajbcs-667	19	31	dimensional	dimensional	ADJ
eajbcs-667	19	32	advection	advection	NOUN
eajbcs-667	19	33	-	-	PUNCT
eajbcs-667	19	34	diffusion	diffusion	NOUN
eajbcs-667	19	35	equation	equation	NOUN
eajbcs-667	19	36	with	with	ADP
eajbcs-667	19	37	constant	constant	ADJ
eajbcs-667	19	38	parameters	parameter	NOUN
eajbcs-667	19	39	.	.	PUNCT
eajbcs-667	20	1	in	in	ADP
eajbcs-667	20	2	doing	do	VERB
eajbcs-667	20	3	so	so	ADV
eajbcs-667	20	4	,	,	PUNCT
eajbcs-667	20	5	the	the	DET
eajbcs-667	20	6	basic	basic	ADJ
eajbcs-667	20	7	idea	idea	NOUN
eajbcs-667	20	8	is	be	AUX
eajbcs-667	20	9	to	to	PART
eajbcs-667	20	10	first	first	ADV
eajbcs-667	20	11	rewrite	rewrite	VERB
eajbcs-667	20	12	the	the	DET
eajbcs-667	20	13	problem	problem	NOUN
eajbcs-667	20	14	as	as	ADP
eajbcs-667	20	15	a	a	DET
eajbcs-667	20	16	variational	variational	ADJ
eajbcs-667	20	17	equation	equation	NOUN
eajbcs-667	20	18	,	,	PUNCT
eajbcs-667	20	19	and	and	CCONJ
eajbcs-667	20	20	then	then	ADV
eajbcs-667	20	21	seek	seek	VERB
eajbcs-667	20	22	a	a	DET
eajbcs-667	20	23	solution	solution	NOUN
eajbcs-667	20	24	approximation	approximation	NOUN
eajbcs-667	20	25	from	from	ADP
eajbcs-667	20	26	the	the	DET
eajbcs-667	20	27	space	space	NOUN
eajbcs-667	20	28	of	of	ADP
eajbcs-667	20	29	continuous	continuous	ADJ
eajbcs-667	20	30	piece	piece	NOUN
eajbcs-667	20	31	-	-	PUNCT
eajbcs-667	20	32	wise	wise	ADJ
eajbcs-667	20	33	linear	linear	NOUN
eajbcs-667	20	34	’s	’s	NOUN
eajbcs-667	20	35	.	.	PUNCT
eajbcs-667	21	1	this	this	DET
eajbcs-667	21	2	discretization	discretization	NOUN
eajbcs-667	21	3	procedure	procedure	NOUN
eajbcs-667	21	4	results	result	NOUN
eajbcs-667	21	5	in	in	ADP
eajbcs-667	21	6	a	a	DET
eajbcs-667	21	7	linear	linear	ADJ
eajbcs-667	21	8	system	system	NOUN
eajbcs-667	21	9	that	that	PRON
eajbcs-667	21	10	can	can	AUX
eajbcs-667	21	11	be	be	AUX
eajbcs-667	21	12	solved	solve	VERB
eajbcs-667	21	13	by	by	ADP
eajbcs-667	21	14	using	use	VERB
eajbcs-667	21	15	a	a	DET
eajbcs-667	21	16	numerical	numerical	ADJ
eajbcs-667	21	17	algorithm	algorithm	NOUN
eajbcs-667	21	18	for	for	ADP
eajbcs-667	21	19	systems	system	NOUN
eajbcs-667	21	20	of	of	ADP
eajbcs-667	21	21	these	these	DET
eajbcs-667	21	22	equations	equation	NOUN
eajbcs-667	21	23	.	.	PUNCT
eajbcs-667	22	1	the	the	DET
eajbcs-667	22	2	techniques	technique	NOUN
eajbcs-667	22	3	are	be	AUX
eajbcs-667	22	4	based	base	VERB
eajbcs-667	22	5	on	on	ADP
eajbcs-667	22	6	the	the	DET
eajbcs-667	22	7	finite	finite	ADJ
eajbcs-667	22	8	element	element	NOUN
eajbcs-667	22	9	approximations	approximation	NOUN
eajbcs-667	22	10	using	use	VERB
eajbcs-667	22	11	galerkin	galerkin	PROPN
eajbcs-667	22	12	’s	’s	PART
eajbcs-667	22	13	method	method	NOUN
eajbcs-667	22	14	in	in	ADP
eajbcs-667	22	15	space	space	NOUN
eajbcs-667	22	16	resulting	result	VERB
eajbcs-667	22	17	system	system	NOUN
eajbcs-667	22	18	of	of	ADP
eajbcs-667	22	19	the	the	DET
eajbcs-667	22	20	first	first	ADJ
eajbcs-667	22	21	order	order	NOUN
eajbcs-667	22	22	ode	ode	NOUN
eajbcs-667	22	23	’s	’s	NOUN
eajbcs-667	22	24	and	and	CCONJ
eajbcs-667	22	25	then	then	ADV
eajbcs-667	22	26	solving	solve	VERB
eajbcs-667	22	27	this	this	DET
eajbcs-667	22	28	first	first	ADJ
eajbcs-667	22	29	order	order	NOUN
eajbcs-667	23	1	ode	ode	AUX
eajbcs-667	23	2	’s	’s	PART
eajbcs-667	23	3	using	use	VERB
eajbcs-667	23	4	backward	backward	ADJ
eajbcs-667	23	5	euler	euler	NOUN
eajbcs-667	23	6	descritization	descritization	NOUN
eajbcs-667	23	7	in	in	ADP
eajbcs-667	23	8	time	time	NOUN
eajbcs-667	23	9	.	.	PUNCT
eajbcs-667	24	1	for	for	ADP
eajbcs-667	24	2	the	the	DET
eajbcs-667	24	3	two	two	NUM
eajbcs-667	24	4	dimensional	dimensional	ADJ
eajbcs-667	24	5	problems	problem	NOUN
eajbcs-667	24	6	,	,	PUNCT
eajbcs-667	24	7	we	we	PRON
eajbcs-667	24	8	use	use	VERB
eajbcs-667	24	9	the	the	DET
eajbcs-667	24	10	ode	ode	PROPN
eajbcs-667	24	11	solver	solver	NOUN
eajbcs-667	24	12	ode15i	ode15i	X
eajbcs-667	24	13	to	to	PART
eajbcs-667	24	14	descritize	descritize	VERB
eajbcs-667	24	15	time	time	NOUN
eajbcs-667	24	16	.	.	PUNCT
eajbcs-667	25	1	the	the	DET
eajbcs-667	25	2	validity	validity	NOUN
eajbcs-667	25	3	of	of	ADP
eajbcs-667	25	4	the	the	DET
eajbcs-667	25	5	numerical	numerical	ADJ
eajbcs-667	25	6	model	model	NOUN
eajbcs-667	25	7	is	be	AUX
eajbcs-667	25	8	verified	verify	VERB
eajbcs-667	25	9	using	use	VERB
eajbcs-667	25	10	different	different	ADJ
eajbcs-667	25	11	test	test	NOUN
eajbcs-667	25	12	examples	example	NOUN
eajbcs-667	25	13	.	.	PUNCT
eajbcs-667	26	1	the	the	DET
eajbcs-667	26	2	computed	compute	VERB
eajbcs-667	26	3	results	result	NOUN
eajbcs-667	26	4	showed	show	VERB
eajbcs-667	26	5	that	that	SCONJ
eajbcs-667	26	6	the	the	DET
eajbcs-667	26	7	use	use	NOUN
eajbcs-667	26	8	of	of	ADP
eajbcs-667	26	9	the	the	DET
eajbcs-667	26	10	current	current	ADJ
eajbcs-667	26	11	method	method	NOUN
eajbcs-667	26	12	is	be	AUX
eajbcs-667	26	13	very	very	ADV
eajbcs-667	26	14	applicable	applicable	ADJ
eajbcs-667	26	15	for	for	ADP
eajbcs-667	26	16	the	the	DET
eajbcs-667	26	17	solution	solution	NOUN
eajbcs-667	26	18	of	of	ADP
eajbcs-667	26	19	the	the	DET
eajbcs-667	26	20	advection	advection	NOUN
eajbcs-667	26	21	-	-	PUNCT
eajbcs-667	26	22	diffusion	diffusion	NOUN
eajbcs-667	26	23	equation	equation	NOUN
eajbcs-667	26	24	.	.	PUNCT
eajbcs-667	27	1	numerical	numerical	ADJ
eajbcs-667	27	2	solutions	solution	NOUN
eajbcs-667	27	3	of	of	ADP
eajbcs-667	27	4	advection	advection	NOUN
eajbcs-667	27	5	diffusion	diffusion	NOUN
eajbcs-667	27	6	equations	equation	NOUN
eajbcs-667	27	7	using	use	VERB
eajbcs-667	27	8	finite	finite	PROPN
eajbcs-667	27	9	element	element	NOUN
eajbcs-667	27	10	method	method	NOUN
eajbcs-667	27	11	introduction	introduction	NOUN
eajbcs-667	27	12	research	research	NOUN
eajbcs-667	27	13	article	article	NOUN
eajbcs-667	27	14	equation	equation	NOUN
eajbcs-667	27	15	by	by	ADP
eajbcs-667	27	16	a	a	DET
eajbcs-667	27	17	test	test	NOUN
eajbcs-667	27	18	function	function	NOUN
eajbcs-667	27	19	and	and	CCONJ
eajbcs-667	27	20	integration	integration	NOUN
eajbcs-667	27	21	by	by	ADP
eajbcs-667	27	22	parts	part	NOUN
eajbcs-667	27	23	(	(	PUNCT
eajbcs-667	27	24	green	green	ADJ
eajbcs-667	27	25	-	-	PUNCT
eajbcs-667	27	26	gauss	gauss	NOUN
eajbcs-667	27	27	theorem	theorem	NOUN
eajbcs-667	27	28	)	)	PUNCT
eajbcs-667	27	29	to	to	PART
eajbcs-667	27	30	reduce	reduce	VERB
eajbcs-667	27	31	second	second	ADJ
eajbcs-667	27	32	order	order	NOUN
eajbcs-667	27	33	derivatives	derivative	NOUN
eajbcs-667	27	34	to	to	ADP
eajbcs-667	27	35	first	first	ADJ
eajbcs-667	27	36	order	order	NOUN
eajbcs-667	27	37	terms	term	NOUN
eajbcs-667	27	38	,	,	PUNCT
eajbcs-667	27	39	i.e.	i.e.	X
eajbcs-667	27	40	,	,	PUNCT
eajbcs-667	27	41	weak	weak	ADJ
eajbcs-667	27	42	formulation	formulation	NOUN
eajbcs-667	27	43	.	.	PUNCT
eajbcs-667	28	1	then	then	ADV
eajbcs-667	28	2	we	we	PRON
eajbcs-667	28	3	represent	represent	VERB
eajbcs-667	28	4	the	the	DET
eajbcs-667	28	5	approximate	approximate	ADJ
eajbcs-667	28	6	solution	solution	NOUN
eajbcs-667	28	7	by	by	ADP
eajbcs-667	28	8	the	the	DET
eajbcs-667	28	9	linear	linear	ADJ
eajbcs-667	28	10	combination	combination	NOUN
eajbcs-667	28	11	of	of	ADP
eajbcs-667	28	12	basis	basis	NOUN
eajbcs-667	28	13	functions	function	NOUN
eajbcs-667	28	14	,	,	PUNCT
eajbcs-667	28	15	by	by	ADP
eajbcs-667	28	16	constructing	construct	VERB
eajbcs-667	28	17	a	a	DET
eajbcs-667	28	18	set	set	NOUN
eajbcs-667	28	19	of	of	ADP
eajbcs-667	28	20	basis	basis	NOUN
eajbcs-667	28	21	functions	function	NOUN
eajbcs-667	28	22	based	base	VERB
eajbcs-667	28	23	on	on	ADP
eajbcs-667	28	24	the	the	DET
eajbcs-667	28	25	mesh	mesh	NOUN
eajbcs-667	28	26	of	of	ADP
eajbcs-667	28	27	our	our	PRON
eajbcs-667	28	28	domain	domain	NOUN
eajbcs-667	28	29	.	.	PUNCT
eajbcs-667	29	1	that	that	PRON
eajbcs-667	29	2	is	is	ADV
eajbcs-667	29	3	,	,	PUNCT
eajbcs-667	29	4	the	the	DET
eajbcs-667	29	5	solution	solution	NOUN
eajbcs-667	29	6	u	u	NOUN
eajbcs-667	29	7	can	can	AUX
eajbcs-667	29	8	be	be	AUX
eajbcs-667	29	9	approximated	approximate	VERB
eajbcs-667	29	10	by	by	ADP
eajbcs-667	29	11	a	a	DET
eajbcs-667	29	12	function	function	NOUN
eajbcs-667	29	13	uh	uh	INTJ
eajbcs-667	29	14	using	use	VERB
eajbcs-667	29	15	the	the	DET
eajbcs-667	29	16	linear	linear	ADJ
eajbcs-667	29	17	combinations	combination	NOUN
eajbcs-667	29	18	of	of	ADP
eajbcs-667	29	19	the	the	DET
eajbcs-667	29	20	basis	basis	NOUN
eajbcs-667	29	21	functions	function	NOUN
eajbcs-667	29	22	ϕi	ϕi	ADP
eajbcs-667	29	23	according	accord	VERB
eajbcs-667	29	24	to	to	ADP
eajbcs-667	29	25	the	the	DET
eajbcs-667	29	26	following	following	ADJ
eajbcs-667	29	27	expressions	expression	NOUN
eajbcs-667	29	28	:	:	PUNCT
eajbcs-667	29	29	u	u	NOUN
eajbcs-667	30	1	≈	≈	PROPN
eajbcs-667	30	2	uh	uh	INTJ
eajbcs-667	30	3	=	=	NOUN
eajbcs-667	30	4	∑	∑	NOUN
eajbcs-667	30	5	uiϕi	uiϕi	ADJ
eajbcs-667	30	6	.	.	PUNCT
eajbcs-667	31	1	(	(	PUNCT
eajbcs-667	31	2	2	2	X
eajbcs-667	31	3	)	)	PUNCT
eajbcs-667	31	4	and	and	CCONJ
eajbcs-667	31	5	we	we	PRON
eajbcs-667	31	6	solve	solve	VERB
eajbcs-667	31	7	the	the	DET
eajbcs-667	31	8	linear	linear	ADJ
eajbcs-667	31	9	system	system	NOUN
eajbcs-667	31	10	of	of	ADP
eajbcs-667	31	11	equations	equation	NOUN
eajbcs-667	31	12	to	to	PART
eajbcs-667	31	13	obtain	obtain	VERB
eajbcs-667	31	14	the	the	DET
eajbcs-667	31	15	approximate	approximate	ADJ
eajbcs-667	31	16	solution	solution	NOUN
eajbcs-667	31	17	.	.	PUNCT
eajbcs-667	32	1	to	to	PART
eajbcs-667	32	2	derive	derive	VERB
eajbcs-667	32	3	the	the	DET
eajbcs-667	32	4	advection	advection	NOUN
eajbcs-667	32	5	diffusion	diffusion	NOUN
eajbcs-667	32	6	equations	equation	NOUN
eajbcs-667	32	7	for	for	ADP
eajbcs-667	32	8	the	the	DET
eajbcs-667	32	9	application	application	NOUN
eajbcs-667	32	10	of	of	ADP
eajbcs-667	32	11	pollution	pollution	NOUN
eajbcs-667	32	12	models	model	NOUN
eajbcs-667	32	13	,	,	PUNCT
eajbcs-667	32	14	consider	consider	VERB
eajbcs-667	32	15	an	an	DET
eajbcs-667	32	16	elementary	elementary	ADJ
eajbcs-667	32	17	water	water	NOUN
eajbcs-667	32	18	body	body	NOUN
eajbcs-667	32	19	.	.	PUNCT
eajbcs-667	33	1	water	water	NOUN
eajbcs-667	33	2	quality	quality	NOUN
eajbcs-667	33	3	within	within	ADP
eajbcs-667	33	4	this	this	DET
eajbcs-667	33	5	body	body	NOUN
eajbcs-667	33	6	depends	depend	VERB
eajbcs-667	33	7	on	on	ADP
eajbcs-667	33	8	the	the	DET
eajbcs-667	33	9	polluting	pollute	VERB
eajbcs-667	33	10	substance	substance	NOUN
eajbcs-667	33	11	mass	mass	NOUN
eajbcs-667	33	12	present	present	ADJ
eajbcs-667	33	13	there	there	ADV
eajbcs-667	33	14	.	.	PUNCT
eajbcs-667	34	1	the	the	DET
eajbcs-667	34	2	water	water	NOUN
eajbcs-667	34	3	quality	quality	NOUN
eajbcs-667	34	4	models	model	NOUN
eajbcs-667	34	5	describe	describe	VERB
eajbcs-667	34	6	the	the	DET
eajbcs-667	34	7	change	change	NOUN
eajbcs-667	34	8	in	in	ADP
eajbcs-667	34	9	the	the	DET
eajbcs-667	34	10	mass	mass	NOUN
eajbcs-667	34	11	of	of	ADP
eajbcs-667	34	12	a	a	DET
eajbcs-667	34	13	polluting	pollute	VERB
eajbcs-667	34	14	substance	substance	NOUN
eajbcs-667	34	15	within	within	ADP
eajbcs-667	34	16	the	the	DET
eajbcs-667	34	17	water	water	NOUN
eajbcs-667	34	18	body	body	NOUN
eajbcs-667	34	19	.	.	PUNCT
eajbcs-667	35	1	the	the	DET
eajbcs-667	35	2	change	change	NOUN
eajbcs-667	35	3	is	be	AUX
eajbcs-667	35	4	calculated	calculate	VERB
eajbcs-667	35	5	as	as	ADP
eajbcs-667	35	6	the	the	DET
eajbcs-667	35	7	difference	difference	NOUN
eajbcs-667	35	8	between	between	ADP
eajbcs-667	35	9	mass	mass	NOUN
eajbcs-667	35	10	-	-	PUNCT
eajbcs-667	35	11	flows	flow	NOUN
eajbcs-667	35	12	(	(	PUNCT
eajbcs-667	35	13	mass	mass	ADJ
eajbcs-667	35	14	fluxes	flux	NOUN
eajbcs-667	35	15	)	)	PUNCT
eajbcs-667	35	16	entering	enter	VERB
eajbcs-667	35	17	and	and	CCONJ
eajbcs-667	35	18	leaving	leave	VERB
eajbcs-667	35	19	this	this	DET
eajbcs-667	35	20	water	water	NOUN
eajbcs-667	35	21	body	body	NOUN
eajbcs-667	35	22	,	,	PUNCT
eajbcs-667	35	23	considering	consider	VERB
eajbcs-667	35	24	also	also	ADV
eajbcs-667	35	25	the	the	DET
eajbcs-667	35	26	effects	effect	NOUN
eajbcs-667	35	27	of	of	ADP
eajbcs-667	35	28	internal	internal	ADJ
eajbcs-667	35	29	sources	source	NOUN
eajbcs-667	35	30	and	and	CCONJ
eajbcs-667	35	31	sinks	sink	NOUN
eajbcs-667	35	32	of	of	ADP
eajbcs-667	35	33	the	the	DET
eajbcs-667	35	34	substance	substance	NOUN
eajbcs-667	35	35	,	,	PUNCT
eajbcs-667	35	36	if	if	SCONJ
eajbcs-667	35	37	any	any	PRON
eajbcs-667	35	38	.	.	PUNCT
eajbcs-667	36	1	the	the	DET
eajbcs-667	36	2	mechanism	mechanism	NOUN
eajbcs-667	36	3	of	of	ADP
eajbcs-667	36	4	mass	mass	ADJ
eajbcs-667	36	5	transfer	transfer	NOUN
eajbcs-667	36	6	into	into	ADP
eajbcs-667	36	7	and	and	CCONJ
eajbcs-667	36	8	out	out	ADP
eajbcs-667	36	9	of	of	ADP
eajbcs-667	36	10	this	this	DET
eajbcs-667	36	11	water	water	NOUN
eajbcs-667	36	12	body	body	NOUN
eajbcs-667	36	13	includes	include	VERB
eajbcs-667	36	14	the	the	DET
eajbcs-667	36	15	following	follow	VERB
eajbcs-667	36	16	processes	process	NOUN
eajbcs-667	36	17	:	:	PUNCT
eajbcs-667	36	18	�	�	PROPN
eajbcs-667	36	19	mass	mass	NOUN
eajbcs-667	36	20	is	be	AUX
eajbcs-667	36	21	transported	transport	VERB
eajbcs-667	36	22	by	by	ADP
eajbcs-667	36	23	the	the	DET
eajbcs-667	36	24	flow	flow	NOUN
eajbcs-667	36	25	,	,	PUNCT
eajbcs-667	36	26	a	a	PRON
eajbcs-667	36	27	,	,	PUNCT
eajbcs-667	36	28	of	of	ADP
eajbcs-667	36	29	the	the	DET
eajbcs-667	36	30	velocity	velocity	NOUN
eajbcs-667	36	31	vector	vector	NOUN
eajbcs-667	36	32	.	.	PUNCT
eajbcs-667	37	1	this	this	DET
eajbcs-667	37	2	process	process	NOUN
eajbcs-667	37	3	is	be	AUX
eajbcs-667	37	4	termed	term	VERB
eajbcs-667	37	5	as	as	ADP
eajbcs-667	37	6	the	the	DET
eajbcs-667	37	7	advective	advective	ADJ
eajbcs-667	37	8	mass	mass	NOUN
eajbcs-667	37	9	transfer	transfer	NOUN
eajbcs-667	37	10	.	.	PUNCT
eajbcs-667	38	1	the	the	DET
eajbcs-667	38	2	transfer	transfer	NOUN
eajbcs-667	38	3	of	of	ADP
eajbcs-667	38	4	mass	mass	NOUN
eajbcs-667	38	5	,	,	PUNCT
eajbcs-667	38	6	that	that	PRON
eajbcs-667	38	7	is	be	AUX
eajbcs-667	38	8	the	the	DET
eajbcs-667	38	9	mass	mass	ADJ
eajbcs-667	38	10	flux	flux	NOUN
eajbcs-667	38	11	can	can	AUX
eajbcs-667	38	12	be	be	AUX
eajbcs-667	38	13	calculated	calculate	VERB
eajbcs-667	38	14	as	as	ADP
eajbcs-667	38	15	u	u	PROPN
eajbcs-667	38	16	×	×	NOUN
eajbcs-667	38	17	a	a	X
eajbcs-667	38	18	,	,	PUNCT
eajbcs-667	38	19	where	where	SCONJ
eajbcs-667	38	20	u	u	NOUN
eajbcs-667	38	21	is	be	AUX
eajbcs-667	38	22	the	the	DET
eajbcs-667	38	23	concentration	concentration	NOUN
eajbcs-667	38	24	of	of	ADP
eajbcs-667	38	25	the	the	DET
eajbcs-667	38	26	substance	substance	NOUN
eajbcs-667	38	27	in	in	ADP
eajbcs-667	38	28	the	the	DET
eajbcs-667	38	29	water	water	NOUN
eajbcs-667	38	30	.	.	PUNCT
eajbcs-667	39	1	�	�	PROPN
eajbcs-667	40	1	the	the	DET
eajbcs-667	40	2	dispersive	dispersive	ADJ
eajbcs-667	40	3	mass	mass	NOUN
eajbcs-667	40	4	transfer	transfer	NOUN
eajbcs-667	40	5	is	be	AUX
eajbcs-667	40	6	usually	usually	ADV
eajbcs-667	40	7	expressed	express	VERB
eajbcs-667	40	8	by	by	ADP
eajbcs-667	40	9	the	the	DET
eajbcs-667	40	10	law	law	NOUN
eajbcs-667	40	11	of	of	ADP
eajbcs-667	40	12	fix	fix	NOUN
eajbcs-667	40	13	which	which	PRON
eajbcs-667	40	14	states	state	VERB
eajbcs-667	40	15	that	that	SCONJ
eajbcs-667	40	16	the	the	DET
eajbcs-667	40	17	transport	transport	NOUN
eajbcs-667	40	18	of	of	ADP
eajbcs-667	40	19	east	east	PROPN
eajbcs-667	40	20	afr	afr	PROPN
eajbcs-667	40	21	.	.	PUNCT
eajbcs-667	41	1	j.	j.	PROPN
eajbcs-667	41	2	biophys	biophys	PROPN
eajbcs-667	41	3	.	.	PUNCT
eajbcs-667	42	1	comput	comput	NOUN
eajbcs-667	42	2	.	.	PUNCT
eajbcs-667	43	1	sci	sci	PROPN
eajbcs-667	43	2	.	.	PUNCT
eajbcs-667	43	3	(	(	PUNCT
eajbcs-667	43	4	2023	2023	NUM
eajbcs-667	43	5	)	)	PUNCT
eajbcs-667	43	6	,	,	PUNCT
eajbcs-667	43	7	vol	vol	NOUN
eajbcs-667	43	8	.	.	PROPN
eajbcs-667	43	9	4	4	NUM
eajbcs-667	43	10	,	,	PUNCT
eajbcs-667	43	11	no	no	INTJ
eajbcs-667	43	12	.	.	NOUN
eajbcs-667	43	13	1	1	NUM
eajbcs-667	43	14	,	,	PUNCT
eajbcs-667	43	15	52	52	NUM
eajbcs-667	43	16	-	-	SYM
eajbcs-667	43	17	74	74	NUM
eajbcs-667	43	18	52	52	NUM
eajbcs-667	43	19	here	here	ADV
eajbcs-667	43	20	,	,	PUNCT
eajbcs-667	43	21	ϕi	ϕi	ADP
eajbcs-667	43	22	denotes	denote	NOUN
eajbcs-667	43	23	the	the	DET
eajbcs-667	43	24	basis	basis	NOUN
eajbcs-667	43	25	functions	function	NOUN
eajbcs-667	43	26	and	and	CCONJ
eajbcs-667	43	27	ui	ui	NOUN
eajbcs-667	43	28	denotes	denote	VERB
eajbcs-667	43	29	the	the	DET
eajbcs-667	43	30	coefficients	coefficient	NOUN
eajbcs-667	43	31	of	of	ADP
eajbcs-667	43	32	the	the	DET
eajbcs-667	43	33	functions	function	NOUN
eajbcs-667	43	34	that	that	PRON
eajbcs-667	43	35	approximate	approximate	VERB
eajbcs-667	43	36	u	u	NOUN
eajbcs-667	43	37	with	with	ADP
eajbcs-667	43	38	uh	uh	INTJ
eajbcs-667	43	39	.	.	PUNCT
eajbcs-667	44	1	after	after	ADP
eajbcs-667	44	2	this	this	PRON
eajbcs-667	44	3	we	we	PRON
eajbcs-667	44	4	get	get	VERB
eajbcs-667	44	5	a	a	DET
eajbcs-667	44	6	system	system	NOUN
eajbcs-667	44	7	of	of	ADP
eajbcs-667	44	8	linear	linear	PROPN
eajbcs-667	44	9	equations	equation	NOUN
eajbcs-667	44	10	siorowski	siorowski	VERB
eajbcs-667	44	11	,	,	PUNCT
eajbcs-667	44	12	2021	2021	NUM
eajbcs-667	44	13	)	)	PUNCT
eajbcs-667	44	14	.	.	PUNCT
eajbcs-667	45	1	in	in	ADP
eajbcs-667	45	2	numerical	numerical	ADJ
eajbcs-667	45	3	method	method	NOUN
eajbcs-667	45	4	,	,	PUNCT
eajbcs-667	45	5	a	a	DET
eajbcs-667	45	6	discrete	discrete	ADJ
eajbcs-667	45	7	approximations	approximation	NOUN
eajbcs-667	45	8	for	for	ADP
eajbcs-667	45	9	the	the	DET
eajbcs-667	45	10	solution	solution	NOUN
eajbcs-667	45	11	is	be	AUX
eajbcs-667	45	12	computed	compute	VERB
eajbcs-667	45	13	by	by	ADP
eajbcs-667	45	14	descritizing	descritize	VERB
eajbcs-667	45	15	the	the	DET
eajbcs-667	45	16	given	give	VERB
eajbcs-667	45	17	domain	domain	NOUN
eajbcs-667	45	18	in	in	ADP
eajbcs-667	45	19	to	to	ADP
eajbcs-667	45	20	different	different	ADJ
eajbcs-667	45	21	sets	set	NOUN
eajbcs-667	45	22	of	of	ADP
eajbcs-667	45	23	sub	sub	NOUN
eajbcs-667	45	24	domains	domain	NOUN
eajbcs-667	45	25	.	.	PUNCT
eajbcs-667	46	1	in	in	ADP
eajbcs-667	46	2	this	this	DET
eajbcs-667	46	3	paper	paper	NOUN
eajbcs-667	46	4	we	we	PRON
eajbcs-667	46	5	were	be	AUX
eajbcs-667	46	6	focus	focus	VERB
eajbcs-667	46	7	on	on	ADP
eajbcs-667	46	8	finite	finite	ADJ
eajbcs-667	46	9	element	element	NOUN
eajbcs-667	46	10	method	method	NOUN
eajbcs-667	46	11	to	to	PART
eajbcs-667	46	12	solve	solve	VERB
eajbcs-667	46	13	the	the	DET
eajbcs-667	46	14	pde	pde	NOUN
eajbcs-667	46	15	given	give	VERB
eajbcs-667	46	16	in	in	ADP
eajbcs-667	46	17	eq	eq	ADJ
eajbcs-667	46	18	.	.	PROPN
eajbcs-667	47	1	1	1	X
eajbcs-667	47	2	.	.	X
eajbcs-667	47	3	we	we	PRON
eajbcs-667	47	4	will	will	AUX
eajbcs-667	47	5	implement	implement	VERB
eajbcs-667	47	6	the	the	DET
eajbcs-667	47	7	method	method	NOUN
eajbcs-667	47	8	for	for	ADP
eajbcs-667	47	9	one	one	NUM
eajbcs-667	47	10	and	and	CCONJ
eajbcs-667	47	11	two	two	NUM
eajbcs-667	47	12	dimensional	dimensional	ADJ
eajbcs-667	47	13	pde	pde	NOUN
eajbcs-667	47	14	’s	’s	NOUN
eajbcs-667	47	15	.	.	PUNCT
eajbcs-667	48	1	in	in	ADP
eajbcs-667	48	2	this	this	DET
eajbcs-667	48	3	method	method	NOUN
eajbcs-667	48	4	,	,	PUNCT
eajbcs-667	48	5	first	first	ADV
eajbcs-667	48	6	we	we	PRON
eajbcs-667	48	7	develop	develop	VERB
eajbcs-667	48	8	a	a	DET
eajbcs-667	48	9	weak	weak	ADJ
eajbcs-667	48	10	formulation	formulation	NOUN
eajbcs-667	48	11	,	,	PUNCT
eajbcs-667	48	12	from	from	ADP
eajbcs-667	48	13	which	which	PRON
eajbcs-667	48	14	we	we	PRON
eajbcs-667	48	15	derive	derive	VERB
eajbcs-667	48	16	the	the	DET
eajbcs-667	48	17	discretization	discretization	NOUN
eajbcs-667	48	18	by	by	ADP
eajbcs-667	48	19	multiplying	multiply	VERB
eajbcs-667	48	20	both	both	DET
eajbcs-667	48	21	sides	side	NOUN
eajbcs-667	48	22	of	of	ADP
eajbcs-667	48	23	the	the	DET
eajbcs-667	48	24	ade	ade	NOUN
eajbcs-667	48	25	the	the	DET
eajbcs-667	48	26	development	development	NOUN
eajbcs-667	48	27	of	of	ADP
eajbcs-667	48	28	finite	finite	PROPN
eajbcs-667	48	29	element	element	NOUN
eajbcs-667	48	30	method	method	NOUN
eajbcs-667	48	31	has	have	AUX
eajbcs-667	48	32	favored	favor	VERB
eajbcs-667	48	33	by	by	ADP
eajbcs-667	48	34	the	the	DET
eajbcs-667	48	35	progress	progress	NOUN
eajbcs-667	48	36	of	of	ADP
eajbcs-667	48	37	computer	computer	NOUN
eajbcs-667	48	38	technology	technology	NOUN
eajbcs-667	48	39	and	and	CCONJ
eajbcs-667	48	40	numerical	numerical	ADJ
eajbcs-667	48	41	calculus	calculus	NOUN
eajbcs-667	48	42	,	,	PUNCT
eajbcs-667	48	43	and	and	CCONJ
eajbcs-667	48	44	originally	originally	ADV
eajbcs-667	48	45	applied	apply	VERB
eajbcs-667	48	46	for	for	ADP
eajbcs-667	48	47	mechanical	mechanical	ADJ
eajbcs-667	48	48	structures	structure	NOUN
eajbcs-667	48	49	(	(	PUNCT
eajbcs-667	48	50	lima	lima	NOUN
eajbcs-667	48	51	et	et	PROPN
eajbcs-667	48	52	al	al	PROPN
eajbcs-667	48	53	.	.	PROPN
eajbcs-667	48	54	,	,	PUNCT
eajbcs-667	48	55	2021	2021	NUM
eajbcs-667	48	56	;	;	PUNCT
eajbcs-667	48	57	mahmud	mahmud	PROPN
eajbcs-667	48	58	,	,	PUNCT
eajbcs-667	48	59	2012	2012	NUM
eajbcs-667	48	60	;	;	PUNCT
eajbcs-667	48	61	donea	donea	PROPN
eajbcs-667	48	62	and	and	CCONJ
eajbcs-667	48	63	huerta	huerta	PROPN
eajbcs-667	48	64	,	,	PUNCT
eajbcs-667	48	65	2003	2003	NUM
eajbcs-667	48	66	)	)	PUNCT
eajbcs-667	48	67	.	.	PUNCT
eajbcs-667	49	1	several	several	ADJ
eajbcs-667	49	2	procedures	procedure	NOUN
eajbcs-667	49	3	have	have	AUX
eajbcs-667	49	4	been	be	AUX
eajbcs-667	49	5	tried	try	VERB
eajbcs-667	49	6	to	to	PART
eajbcs-667	49	7	interpret	interpret	VERB
eajbcs-667	49	8	separately	separately	ADV
eajbcs-667	49	9	the	the	DET
eajbcs-667	49	10	advection	advection	NOUN
eajbcs-667	49	11	and	and	CCONJ
eajbcs-667	49	12	diffusion	diffusion	NOUN
eajbcs-667	49	13	pollutant	pollutant	ADJ
eajbcs-667	49	14	transport	transport	NOUN
eajbcs-667	49	15	.	.	PUNCT
eajbcs-667	50	1	the	the	DET
eajbcs-667	50	2	fem	fem	NOUN
eajbcs-667	50	3	can	can	AUX
eajbcs-667	50	4	help	help	VERB
eajbcs-667	50	5	to	to	PART
eajbcs-667	50	6	face	face	VERB
eajbcs-667	50	7	more	more	ADJ
eajbcs-667	50	8	complex	complex	ADJ
eajbcs-667	50	9	problems	problem	NOUN
eajbcs-667	50	10	and	and	CCONJ
eajbcs-667	50	11	the	the	DET
eajbcs-667	50	12	privilege	privilege	NOUN
eajbcs-667	50	13	importance	importance	NOUN
eajbcs-667	50	14	of	of	ADP
eajbcs-667	50	15	the	the	DET
eajbcs-667	50	16	method	method	NOUN
eajbcs-667	50	17	is	be	AUX
eajbcs-667	50	18	that	that	SCONJ
eajbcs-667	50	19	it	it	PRON
eajbcs-667	50	20	can	can	AUX
eajbcs-667	50	21	be	be	AUX
eajbcs-667	50	22	adapted	adapt	VERB
eajbcs-667	50	23	to	to	ADP
eajbcs-667	50	24	complex	complex	ADJ
eajbcs-667	50	25	geometry	geometry	NOUN
eajbcs-667	50	26	domains	domain	NOUN
eajbcs-667	50	27	,	,	PUNCT
eajbcs-667	50	28	but	but	CCONJ
eajbcs-667	50	29	the	the	DET
eajbcs-667	50	30	element	element	NOUN
eajbcs-667	50	31	wise	wise	ADJ
eajbcs-667	50	32	intervals	interval	NOUN
eajbcs-667	50	33	can	can	AUX
eajbcs-667	50	34	assume	assume	VERB
eajbcs-667	50	35	any	any	DET
eajbcs-667	50	36	form	form	NOUN
eajbcs-667	50	37	of	of	ADP
eajbcs-667	50	38	size	size	NOUN
eajbcs-667	50	39	,	,	PUNCT
eajbcs-667	50	40	and	and	CCONJ
eajbcs-667	50	41	obviously	obviously	ADV
eajbcs-667	50	42	there	there	PRON
eajbcs-667	50	43	is	be	VERB
eajbcs-667	50	44	an	an	DET
eajbcs-667	50	45	expense	expense	NOUN
eajbcs-667	50	46	of	of	ADP
eajbcs-667	50	47	more	more	ADV
eajbcs-667	50	48	burdensome	burdensome	ADJ
eajbcs-667	50	49	calculations	calculation	NOUN
eajbcs-667	50	50	.	.	PUNCT
eajbcs-667	51	1	we	we	PRON
eajbcs-667	51	2	had	have	VERB
eajbcs-667	51	3	to	to	PART
eajbcs-667	51	4	do	do	VERB
eajbcs-667	51	5	the	the	DET
eajbcs-667	51	6	descritization	descritization	NOUN
eajbcs-667	51	7	process	process	NOUN
eajbcs-667	51	8	of	of	ADP
eajbcs-667	51	9	the	the	DET
eajbcs-667	51	10	finite	finite	ADJ
eajbcs-667	51	11	element	element	NOUN
eajbcs-667	51	12	method	method	NOUN
eajbcs-667	51	13	for	for	ADP
eajbcs-667	51	14	the	the	DET
eajbcs-667	51	15	1d	1d	NUM
eajbcs-667	51	16	and	and	CCONJ
eajbcs-667	51	17	2d	2d	NOUN
eajbcs-667	51	18	poisson	poisson	NOUN
eajbcs-667	51	19	equations	equation	NOUN
eajbcs-667	51	20	in	in	ADP
eajbcs-667	51	21	which	which	PRON
eajbcs-667	51	22	it	it	PRON
eajbcs-667	51	23	is	be	AUX
eajbcs-667	51	24	an	an	DET
eajbcs-667	51	25	auxiliary	auxiliary	ADJ
eajbcs-667	51	26	step	step	NOUN
eajbcs-667	51	27	in	in	ADP
eajbcs-667	51	28	solving	solve	VERB
eajbcs-667	51	29	the	the	DET
eajbcs-667	51	30	advection	advection	NOUN
eajbcs-667	51	31	diffusion	diffusion	NOUN
eajbcs-667	51	32	equation	equation	NOUN
eajbcs-667	51	33	with	with	ADP
eajbcs-667	51	34	the	the	DET
eajbcs-667	51	35	fem	fem	NOUN
eajbcs-667	51	36	.	.	PUNCT
eajbcs-667	52	1	usually	usually	ADV
eajbcs-667	52	2	the	the	DET
eajbcs-667	52	3	numerical	numerical	ADJ
eajbcs-667	52	4	solutions	solution	NOUN
eajbcs-667	52	5	of	of	ADP
eajbcs-667	52	6	pde	pde	NOUN
eajbcs-667	52	7	’	'	PUNCT
eajbcs-667	52	8	including	include	VERB
eajbcs-667	52	9	the	the	DET
eajbcs-667	52	10	equation	equation	NOUN
eajbcs-667	52	11	(	(	PUNCT
eajbcs-667	52	12	eq	eq	NOUN
eajbcs-667	52	13	.	.	NOUN
eajbcs-667	52	14	1	1	NUM
eajbcs-667	52	15	)	)	PUNCT
eajbcs-667	52	16	are	be	AUX
eajbcs-667	52	17	done	do	VERB
eajbcs-667	52	18	with	with	ADP
eajbcs-667	52	19	the	the	DET
eajbcs-667	52	20	finite	finite	ADJ
eajbcs-667	52	21	difference	difference	NOUN
eajbcs-667	52	22	method	method	NOUN
eajbcs-667	52	23	.	.	PUNCT
eajbcs-667	53	1	mathematicalmodel	mathematicalmodel	NOUN
eajbcs-667	53	2	formulation	formulation	NOUN
eajbcs-667	53	3	the	the	DET
eajbcs-667	53	4	substance	substance	NOUN
eajbcs-667	53	5	in	in	ADP
eajbcs-667	53	6	the	the	DET
eajbcs-667	53	7	direction	direction	NOUN
eajbcs-667	53	8	of	of	ADP
eajbcs-667	53	9	a	a	DET
eajbcs-667	53	10	space	space	NOUN
eajbcs-667	53	11	is	be	AUX
eajbcs-667	53	12	proportional	proportional	ADJ
eajbcs-667	53	13	to	to	ADP
eajbcs-667	53	14	the	the	DET
eajbcs-667	53	15	concentration	concentration	NOUN
eajbcs-667	53	16	of	of	ADP
eajbcs-667	53	17	this	this	DET
eajbcs-667	53	18	substance	substance	NOUN
eajbcs-667	53	19	in	in	ADP
eajbcs-667	53	20	that	that	DET
eajbcs-667	53	21	direction	direction	NOUN
eajbcs-667	53	22	and	and	CCONJ
eajbcs-667	53	23	the	the	DET
eajbcs-667	53	24	proportionality	proportionality	NOUN
eajbcs-667	53	25	factor	factor	NOUN
eajbcs-667	53	26	being	be	AUX
eajbcs-667	53	27	the	the	DET
eajbcs-667	53	28	coefficient	coefficient	NOUN
eajbcs-667	53	29	of	of	ADP
eajbcs-667	53	30	dispersion	dispersion	NOUN
eajbcs-667	53	31	,	,	PUNCT
eajbcs-667	53	32	d∇u	d∇u	PROPN
eajbcs-667	53	33	.	.	PUNCT
eajbcs-667	54	1	transport	transport	NOUN
eajbcs-667	54	2	by	by	ADP
eajbcs-667	54	3	advection	advection	NOUN
eajbcs-667	54	4	=	=	SYM
eajbcs-667	54	5	auda	auda	NOUN
eajbcs-667	54	6	,	,	PUNCT
eajbcs-667	54	7	transport	transport	NOUN
eajbcs-667	54	8	by	by	ADP
eajbcs-667	54	9	diffusion	diffusion	NOUN
eajbcs-667	54	10	=	=	SYM
eajbcs-667	54	11	dx	dx	PROPN
eajbcs-667	54	12	∂u	∂u	PROPN
eajbcs-667	54	13	∂x	∂x	PROPN
eajbcs-667	54	14	,	,	PUNCT
eajbcs-667	54	15	where	where	SCONJ
eajbcs-667	54	16	da	da	PROPN
eajbcs-667	54	17	is	be	AUX
eajbcs-667	54	18	an	an	DET
eajbcs-667	54	19	elemental	elemental	ADJ
eajbcs-667	54	20	cross	cross	ADJ
eajbcs-667	54	21	-	-	ADJ
eajbcs-667	54	22	sectional	sectional	ADJ
eajbcs-667	54	23	area	area	NOUN
eajbcs-667	54	24	of	of	ADP
eajbcs-667	54	25	the	the	DET
eajbcs-667	54	26	cubic	cubic	ADJ
eajbcs-667	54	27	element	element	NOUN
eajbcs-667	54	28	,	,	PUNCT
eajbcs-667	54	29	and	and	CCONJ
eajbcs-667	54	30	dx	dx	PROPN
eajbcs-667	54	31	is	be	AUX
eajbcs-667	54	32	the	the	DET
eajbcs-667	54	33	diffusion	diffusion	NOUN
eajbcs-667	54	34	coefficient	coefficient	NOUN
eajbcs-667	54	35	in	in	ADP
eajbcs-667	54	36	the	the	DET
eajbcs-667	54	37	x	x	NOUN
eajbcs-667	54	38	-	-	NOUN
eajbcs-667	54	39	direction	direction	NOUN
eajbcs-667	54	40	.	.	PUNCT
eajbcs-667	55	1	∂	∂	NUM
eajbcs-667	56	1	∂t	∂t	PROPN
eajbcs-667	56	2	u(x	u(x	PROPN
eajbcs-667	56	3	,	,	PUNCT
eajbcs-667	56	4	t	t	PROPN
eajbcs-667	56	5	)	)	PUNCT
eajbcs-667	56	6	+	+	NUM
eajbcs-667	56	7	∂	∂	NUM
eajbcs-667	56	8	∂x	∂x	PROPN
eajbcs-667	56	9	(	(	PUNCT
eajbcs-667	56	10	a(x	a(x	NOUN
eajbcs-667	56	11	,	,	PUNCT
eajbcs-667	56	12	t)u(x	t)u(x	NOUN
eajbcs-667	56	13	,	,	PUNCT
eajbcs-667	56	14	t	t	NOUN
eajbcs-667	56	15	)	)	PUNCT
eajbcs-667	56	16	)	)	PUNCT
eajbcs-667	56	17	=	=	SYM
eajbcs-667	56	18	∂	∂	NUM
eajbcs-667	56	19	∂x	∂x	PROPN
eajbcs-667	56	20	(	(	PUNCT
eajbcs-667	56	21	d(x	d(x	PROPN
eajbcs-667	56	22	,	,	PUNCT
eajbcs-667	56	23	t	t	PROPN
eajbcs-667	56	24	)	)	PUNCT
eajbcs-667	56	25	∂	∂	NUM
eajbcs-667	56	26	∂x	∂x	PROPN
eajbcs-667	56	27	u(x	u(x	PROPN
eajbcs-667	56	28	,	,	PUNCT
eajbcs-667	56	29	t	t	PROPN
eajbcs-667	56	30	)	)	PUNCT
eajbcs-667	56	31	)	)	PUNCT
eajbcs-667	57	1	+	+	CCONJ
eajbcs-667	57	2	f(t	f(t	NOUN
eajbcs-667	57	3	,	,	PUNCT
eajbcs-667	57	4	x	x	NOUN
eajbcs-667	57	5	)	)	PUNCT
eajbcs-667	57	6	.	.	PUNCT
eajbcs-667	58	1	(	(	PUNCT
eajbcs-667	58	2	3	3	X
eajbcs-667	58	3	)	)	PUNCT
eajbcs-667	58	4	space	space	NOUN
eajbcs-667	58	5	interval	interval	NOUN
eajbcs-667	59	1	ω	ω	X
eajbcs-667	60	1	⊂	⊂	PROPN
eajbcs-667	60	2	r	r	NOUN
eajbcs-667	60	3	with	with	ADP
eajbcs-667	60	4	time	time	NOUN
eajbcs-667	60	5	t	t	PROPN
eajbcs-667	60	6	≥	≥	NOUN
eajbcs-667	60	7	0	0	NUM
eajbcs-667	60	8	.	.	PUNCT
eajbcs-667	61	1	an	an	DET
eajbcs-667	61	2	initial	initial	ADJ
eajbcs-667	61	3	condition	condition	NOUN
eajbcs-667	61	4	u(x	u(x	NOUN
eajbcs-667	61	5	,	,	PUNCT
eajbcs-667	61	6	0	0	NUM
eajbcs-667	61	7	)	)	PUNCT
eajbcs-667	61	8	will	will	AUX
eajbcs-667	61	9	be	be	AUX
eajbcs-667	61	10	given	give	VERB
eajbcs-667	61	11	and	and	CCONJ
eajbcs-667	61	12	we	we	PRON
eajbcs-667	61	13	also	also	ADV
eajbcs-667	61	14	assume	assume	VERB
eajbcs-667	61	15	that	that	SCONJ
eajbcs-667	61	16	suitable	suitable	ADJ
eajbcs-667	61	17	boundary	boundary	ADJ
eajbcs-667	61	18	conditions	condition	NOUN
eajbcs-667	61	19	are	be	AUX
eajbcs-667	61	20	provided	provide	VERB
eajbcs-667	61	21	,	,	PUNCT
eajbcs-667	61	22	and	and	CCONJ
eajbcs-667	61	23	for	for	ADP
eajbcs-667	61	24	our	our	PRON
eajbcs-667	61	25	work	work	NOUN
eajbcs-667	61	26	we	we	PRON
eajbcs-667	61	27	consider	consider	VERB
eajbcs-667	61	28	the	the	DET
eajbcs-667	61	29	velocity	velocity	NOUN
eajbcs-667	61	30	field	field	NOUN
eajbcs-667	61	31	and	and	CCONJ
eajbcs-667	61	32	the	the	DET
eajbcs-667	61	33	diffusion	diffusion	NOUN
eajbcs-667	61	34	term	term	NOUN
eajbcs-667	61	35	as	as	ADP
eajbcs-667	61	36	constants	constant	NOUN
eajbcs-667	61	37	.	.	PUNCT
eajbcs-667	62	1	pressions	pression	NOUN
eajbcs-667	62	2	of	of	ADP
eajbcs-667	62	3	the	the	DET
eajbcs-667	62	4	advection	advection	NOUN
eajbcs-667	62	5	equation	equation	NOUN
eajbcs-667	62	6	a∂u	a∂u	NOUN
eajbcs-667	62	7	∂x	∂x	PROPN
eajbcs-667	62	8	which	which	PRON
eajbcs-667	62	9	has	have	VERB
eajbcs-667	62	10	a	a	DET
eajbcs-667	62	11	first	first	ADJ
eajbcs-667	62	12	-	-	PUNCT
eajbcs-667	62	13	order	order	NOUN
eajbcs-667	62	14	derivative	derivative	NOUN
eajbcs-667	62	15	,	,	PUNCT
eajbcs-667	62	16	and	and	CCONJ
eajbcs-667	62	17	the	the	DET
eajbcs-667	62	18	diffusion	diffusion	NOUN
eajbcs-667	62	19	equation	equation	NOUN
eajbcs-667	62	20	d	d	PROPN
eajbcs-667	62	21	∂2u	∂2u	PROPN
eajbcs-667	62	22	∂x2	∂x2	PROPN
eajbcs-667	62	23	that	that	PRON
eajbcs-667	62	24	has	have	VERB
eajbcs-667	62	25	a	a	DET
eajbcs-667	62	26	second	second	ADJ
eajbcs-667	62	27	-	-	PUNCT
eajbcs-667	62	28	order	order	NOUN
eajbcs-667	62	29	derivative	derivative	NOUN
eajbcs-667	62	30	.	.	PUNCT
eajbcs-667	63	1	east	east	PROPN
eajbcs-667	63	2	afr	afr	PROPN
eajbcs-667	63	3	.	.	PUNCT
eajbcs-667	64	1	j.	j.	PROPN
eajbcs-667	64	2	biophys	biophys	PROPN
eajbcs-667	64	3	.	.	PUNCT
eajbcs-667	65	1	comput	comput	NOUN
eajbcs-667	65	2	.	.	PUNCT
eajbcs-667	66	1	sci	sci	PROPN
eajbcs-667	66	2	.	.	PUNCT
eajbcs-667	66	3	(	(	PUNCT
eajbcs-667	66	4	2023	2023	NUM
eajbcs-667	66	5	)	)	PUNCT
eajbcs-667	66	6	,	,	PUNCT
eajbcs-667	66	7	vol	vol	NOUN
eajbcs-667	66	8	.	.	PROPN
eajbcs-667	66	9	4	4	NUM
eajbcs-667	66	10	,	,	PUNCT
eajbcs-667	66	11	no	no	INTJ
eajbcs-667	66	12	.	.	NOUN
eajbcs-667	66	13	1	1	NUM
eajbcs-667	66	14	,	,	PUNCT
eajbcs-667	66	15	52	52	NUM
eajbcs-667	66	16	-	-	SYM
eajbcs-667	66	17	74	74	NUM
eajbcs-667	66	18	52	52	NUM
eajbcs-667	66	19	both	both	CCONJ
eajbcs-667	66	20	advection	advection	NOUN
eajbcs-667	66	21	and	and	CCONJ
eajbcs-667	66	22	diffusion	diffusion	NOUN
eajbcs-667	66	23	move	move	VERB
eajbcs-667	66	24	a	a	DET
eajbcs-667	66	25	pollutant	pollutant	ADJ
eajbcs-667	66	26	material	material	NOUN
eajbcs-667	66	27	from	from	ADP
eajbcs-667	66	28	one	one	NUM
eajbcs-667	66	29	place	place	NOUN
eajbcs-667	66	30	to	to	ADP
eajbcs-667	66	31	another	another	PRON
eajbcs-667	66	32	,	,	PUNCT
eajbcs-667	66	33	but	but	CCONJ
eajbcs-667	66	34	each	each	PRON
eajbcs-667	66	35	accomplishes	accomplish	VERB
eajbcs-667	66	36	this	this	PRON
eajbcs-667	66	37	differently	differently	ADV
eajbcs-667	66	38	.	.	PUNCT
eajbcs-667	67	1	the	the	DET
eajbcs-667	67	2	essential	essential	ADJ
eajbcs-667	67	3	difference	difference	NOUN
eajbcs-667	67	4	of	of	ADP
eajbcs-667	67	5	the	the	DET
eajbcs-667	67	6	advection	advection	NOUN
eajbcs-667	67	7	and	and	CCONJ
eajbcs-667	67	8	diffusion	diffusion	NOUN
eajbcs-667	67	9	is	be	AUX
eajbcs-667	67	10	that	that	SCONJ
eajbcs-667	67	11	advection	advection	NOUN
eajbcs-667	67	12	moves	move	VERB
eajbcs-667	67	13	the	the	DET
eajbcs-667	67	14	pollutants	pollutant	NOUN
eajbcs-667	67	15	in	in	ADP
eajbcs-667	67	16	one	one	NUM
eajbcs-667	67	17	way	way	NOUN
eajbcs-667	67	18	(	(	PUNCT
eajbcs-667	67	19	downstream	downstream	ADJ
eajbcs-667	67	20	)	)	PUNCT
eajbcs-667	67	21	but	but	CCONJ
eajbcs-667	67	22	diffusion	diffusion	NOUN
eajbcs-667	67	23	goes	go	VERB
eajbcs-667	67	24	in	in	ADP
eajbcs-667	67	25	both	both	DET
eajbcs-667	67	26	ways	way	NOUN
eajbcs-667	67	27	(	(	PUNCT
eajbcs-667	67	28	regardless	regardless	ADV
eajbcs-667	67	29	of	of	ADP
eajbcs-667	67	30	a	a	DET
eajbcs-667	67	31	stream	stream	NOUN
eajbcs-667	67	32	direction	direction	NOUN
eajbcs-667	67	33	)	)	PUNCT
eajbcs-667	67	34	.	.	PUNCT
eajbcs-667	68	1	this	this	PRON
eajbcs-667	68	2	is	be	AUX
eajbcs-667	68	3	seen	see	VERB
eajbcs-667	68	4	in	in	ADP
eajbcs-667	68	5	the	the	DET
eajbcs-667	68	6	respective	respective	ADJ
eajbcs-667	68	7	mathematical	mathematical	ADJ
eajbcs-667	68	8	exquestions	exquestion	NOUN
eajbcs-667	68	9	are	be	AUX
eajbcs-667	68	10	arise	arise	ADJ
eajbcs-667	68	11	like	like	ADP
eajbcs-667	68	12	,	,	PUNCT
eajbcs-667	68	13	can	can	AUX
eajbcs-667	68	14	we	we	PRON
eajbcs-667	68	15	have	have	VERB
eajbcs-667	68	16	cases	case	NOUN
eajbcs-667	68	17	of	of	ADP
eajbcs-667	68	18	fast	fast	ADJ
eajbcs-667	68	19	advection	advection	NOUN
eajbcs-667	68	20	and	and	CCONJ
eajbcs-667	68	21	relatively	relatively	ADV
eajbcs-667	68	22	weak	weak	ADJ
eajbcs-667	68	23	diffusion	diffusion	NOUN
eajbcs-667	68	24	and	and	CCONJ
eajbcs-667	68	25	other	other	ADJ
eajbcs-667	68	26	cases	case	NOUN
eajbcs-667	68	27	of	of	ADP
eajbcs-667	68	28	negligible	negligible	ADJ
eajbcs-667	68	29	advection	advection	NOUN
eajbcs-667	68	30	and	and	CCONJ
eajbcs-667	68	31	fast	fast	ADJ
eajbcs-667	68	32	diffusion	diffusion	NOUN
eajbcs-667	68	33	?	?	PUNCT
eajbcs-667	69	1	to	to	PART
eajbcs-667	69	2	answer	answer	VERB
eajbcs-667	69	3	this	this	PRON
eajbcs-667	69	4	assuming	assume	VERB
eajbcs-667	69	5	that	that	SCONJ
eajbcs-667	69	6	the	the	DET
eajbcs-667	69	7	two	two	NUM
eajbcs-667	69	8	components	component	NOUN
eajbcs-667	69	9	(	(	PUNCT
eajbcs-667	69	10	advection	advection	NOUN
eajbcs-667	69	11	and	and	CCONJ
eajbcs-667	69	12	diffusion	diffusion	NOUN
eajbcs-667	69	13	)	)	PUNCT
eajbcs-667	69	14	may	may	AUX
eajbcs-667	69	15	be	be	AUX
eajbcs-667	69	16	superposed	superpose	VERB
eajbcs-667	69	17	,	,	PUNCT
eajbcs-667	69	18	the	the	DET
eajbcs-667	69	19	total	total	ADJ
eajbcs-667	69	20	amount	amount	NOUN
eajbcs-667	69	21	of	of	ADP
eajbcs-667	69	22	material	material	NOUN
eajbcs-667	69	23	transported	transport	VERB
eajbcs-667	69	24	parallel	parallel	NOUN
eajbcs-667	69	25	to	to	ADP
eajbcs-667	69	26	any	any	DET
eajbcs-667	69	27	given	give	VERB
eajbcs-667	69	28	direction	direction	NOUN
eajbcs-667	69	29	is	be	AUX
eajbcs-667	69	30	obtained	obtain	VERB
eajbcs-667	69	31	by	by	ADP
eajbcs-667	69	32	summing	sum	VERB
eajbcs-667	69	33	the	the	DET
eajbcs-667	69	34	advective	advective	ADJ
eajbcs-667	69	35	and	and	CCONJ
eajbcs-667	69	36	diffusive	diffusive	ADJ
eajbcs-667	69	37	transports	transport	NOUN
eajbcs-667	69	38	.	.	PUNCT
eajbcs-667	70	1	using	use	VERB
eajbcs-667	70	2	the	the	DET
eajbcs-667	70	3	mass	mass	NOUN
eajbcs-667	70	4	balance	balance	NOUN
eajbcs-667	70	5	approach	approach	NOUN
eajbcs-667	70	6	by	by	ADP
eajbcs-667	70	7	equating	equate	VERB
eajbcs-667	70	8	the	the	DET
eajbcs-667	70	9	difference	difference	NOUN
eajbcs-667	70	10	between	between	ADP
eajbcs-667	70	11	the	the	DET
eajbcs-667	70	12	mass	mass	NOUN
eajbcs-667	70	13	of	of	ADP
eajbcs-667	70	14	material	material	NOUN
eajbcs-667	70	15	entering	enter	VERB
eajbcs-667	70	16	a	a	DET
eajbcs-667	70	17	volume	volume	NOUN
eajbcs-667	70	18	element	element	NOUN
eajbcs-667	70	19	and	and	CCONJ
eajbcs-667	70	20	that	that	SCONJ
eajbcs-667	70	21	leaving	leave	VERB
eajbcs-667	70	22	the	the	DET
eajbcs-667	70	23	element	element	NOUN
eajbcs-667	70	24	(	(	PUNCT
eajbcs-667	70	25	i.e.	i.e.	X
eajbcs-667	70	26	,	,	PUNCT
eajbcs-667	70	27	net	net	ADJ
eajbcs-667	70	28	influx	influx	NOUN
eajbcs-667	70	29	of	of	ADP
eajbcs-667	70	30	mass	mass	NOUN
eajbcs-667	70	31	)	)	PUNCT
eajbcs-667	70	32	to	to	ADP
eajbcs-667	70	33	the	the	DET
eajbcs-667	70	34	rate	rate	NOUN
eajbcs-667	70	35	of	of	ADP
eajbcs-667	70	36	accumulation	accumulation	NOUN
eajbcs-667	70	37	of	of	ADP
eajbcs-667	70	38	mass	mass	NOUN
eajbcs-667	70	39	inside	inside	ADP
eajbcs-667	70	40	the	the	DET
eajbcs-667	70	41	volume	volume	NOUN
eajbcs-667	70	42	.	.	PUNCT
eajbcs-667	71	1	by	by	ADP
eajbcs-667	71	2	considering	consider	VERB
eajbcs-667	71	3	a	a	DET
eajbcs-667	71	4	volume	volume	NOUN
eajbcs-667	71	5	element	element	NOUN
eajbcs-667	71	6	of	of	ADP
eajbcs-667	71	7	porous	porous	ADJ
eajbcs-667	71	8	mediums	medium	NOUN
eajbcs-667	71	9	in	in	ADP
eajbcs-667	71	10	three	three	NUM
eajbcs-667	71	11	dimensional	dimensional	ADJ
eajbcs-667	71	12	cartesian	cartesian	ADJ
eajbcs-667	71	13	coordinates	coordinate	NOUN
eajbcs-667	71	14	the	the	DET
eajbcs-667	71	15	equations	equation	NOUN
eajbcs-667	71	16	are	be	AUX
eajbcs-667	71	17	derived	derive	VERB
eajbcs-667	71	18	(	(	PUNCT
eajbcs-667	71	19	bajellan	bajellan	NOUN
eajbcs-667	71	20	,	,	PUNCT
eajbcs-667	71	21	2015	2015	NUM
eajbcs-667	71	22	)	)	PUNCT
eajbcs-667	71	23	.	.	PUNCT
eajbcs-667	72	1	since	since	SCONJ
eajbcs-667	72	2	we	we	PRON
eajbcs-667	72	3	are	be	AUX
eajbcs-667	72	4	considering	consider	VERB
eajbcs-667	72	5	advection	advection	NOUN
eajbcs-667	72	6	and	and	CCONJ
eajbcs-667	72	7	diffusion	diffusion	NOUN
eajbcs-667	72	8	as	as	ADP
eajbcs-667	72	9	the	the	DET
eajbcs-667	72	10	two	two	NUM
eajbcs-667	72	11	modes	mode	NOUN
eajbcs-667	72	12	of	of	ADP
eajbcs-667	72	13	transport	transport	NOUN
eajbcs-667	72	14	of	of	ADP
eajbcs-667	72	15	a	a	DET
eajbcs-667	72	16	fluid	fluid	NOUN
eajbcs-667	72	17	within	within	ADP
eajbcs-667	72	18	the	the	DET
eajbcs-667	72	19	porous	porous	ADJ
eajbcs-667	72	20	medium	medium	NOUN
eajbcs-667	72	21	,	,	PUNCT
eajbcs-667	72	22	we	we	PRON
eajbcs-667	72	23	can	can	AUX
eajbcs-667	72	24	represent	represent	VERB
eajbcs-667	72	25	these	these	DET
eajbcs-667	72	26	two	two	NUM
eajbcs-667	72	27	transport	transport	NOUN
eajbcs-667	72	28	modes	mode	NOUN
eajbcs-667	72	29	in	in	ADP
eajbcs-667	72	30	the	the	DET
eajbcs-667	72	31	x	x	NOUN
eajbcs-667	72	32	-	-	NOUN
eajbcs-667	72	33	direction	direction	NOUN
eajbcs-667	72	34	mathematically	mathematically	ADV
eajbcs-667	72	35	as	as	SCONJ
eajbcs-667	72	36	:	:	PUNCT
eajbcs-667	72	37	the	the	DET
eajbcs-667	72	38	multidimensional	multidimensional	ADJ
eajbcs-667	72	39	advectiondiffusion	advectiondiffusion	NOUN
eajbcs-667	72	40	equation	equation	NOUN
eajbcs-667	72	41	is	be	AUX
eajbcs-667	72	42	used	use	VERB
eajbcs-667	72	43	for	for	ADP
eajbcs-667	72	44	analyzing	analyze	VERB
eajbcs-667	72	45	mixing	mix	VERB
eajbcs-667	72	46	problems	problem	NOUN
eajbcs-667	72	47	in	in	ADP
eajbcs-667	72	48	rivers	river	NOUN
eajbcs-667	72	49	.	.	PUNCT
eajbcs-667	73	1	one	one	NUM
eajbcs-667	73	2	of	of	ADP
eajbcs-667	73	3	the	the	DET
eajbcs-667	73	4	practical	practical	ADJ
eajbcs-667	73	5	difficulties	difficulty	NOUN
eajbcs-667	73	6	is	be	AUX
eajbcs-667	73	7	that	that	SCONJ
eajbcs-667	73	8	the	the	DET
eajbcs-667	73	9	equation	equation	NOUN
eajbcs-667	73	10	requires	require	VERB
eajbcs-667	73	11	some	some	DET
eajbcs-667	73	12	prior	prior	ADJ
eajbcs-667	73	13	information	information	NOUN
eajbcs-667	73	14	about	about	ADP
eajbcs-667	73	15	water	water	NOUN
eajbcs-667	73	16	depths	depth	NOUN
eajbcs-667	73	17	,	,	PUNCT
eajbcs-667	73	18	velocities	velocity	NOUN
eajbcs-667	73	19	,	,	PUNCT
eajbcs-667	73	20	and	and	CCONJ
eajbcs-667	73	21	diffusion	diffusion	NOUN
eajbcs-667	73	22	coefficients	coefficient	NOUN
eajbcs-667	73	23	,	,	PUNCT
eajbcs-667	73	24	which	which	PRON
eajbcs-667	73	25	could	could	AUX
eajbcs-667	73	26	not	not	PART
eajbcs-667	73	27	conveniently	conveniently	ADV
eajbcs-667	73	28	be	be	AUX
eajbcs-667	73	29	gathered	gather	VERB
eajbcs-667	73	30	in	in	ADP
eajbcs-667	73	31	field	field	NOUN
eajbcs-667	73	32	experiments	experiment	NOUN
eajbcs-667	73	33	.	.	PUNCT
eajbcs-667	74	1	in	in	ADP
eajbcs-667	74	2	some	some	DET
eajbcs-667	74	3	particular	particular	ADJ
eajbcs-667	74	4	mixing	mixing	NOUN
eajbcs-667	74	5	problems	problem	NOUN
eajbcs-667	74	6	,	,	PUNCT
eajbcs-667	74	7	however	however	ADV
eajbcs-667	74	8	,	,	PUNCT
eajbcs-667	74	9	some	some	PRON
eajbcs-667	74	10	of	of	ADP
eajbcs-667	74	11	the	the	DET
eajbcs-667	74	12	terms	term	NOUN
eajbcs-667	74	13	in	in	ADP
eajbcs-667	74	14	the	the	DET
eajbcs-667	74	15	multidimensional	multidimensional	ADJ
eajbcs-667	74	16	advection	advection	NOUN
eajbcs-667	74	17	-	-	PUNCT
eajbcs-667	74	18	diffusion	diffusion	NOUN
eajbcs-667	74	19	equation	equation	NOUN
eajbcs-667	74	20	are	be	AUX
eajbcs-667	74	21	negligibly	negligibly	ADV
eajbcs-667	74	22	small	small	ADJ
eajbcs-667	74	23	,	,	PUNCT
eajbcs-667	74	24	so	so	SCONJ
eajbcs-667	74	25	that	that	SCONJ
eajbcs-667	74	26	the	the	DET
eajbcs-667	74	27	problem	problem	NOUN
eajbcs-667	74	28	can	can	AUX
eajbcs-667	74	29	be	be	AUX
eajbcs-667	74	30	simplified	simplify	VERB
eajbcs-667	74	31	by	by	ADP
eajbcs-667	74	32	reducing	reduce	VERB
eajbcs-667	74	33	the	the	DET
eajbcs-667	74	34	model	model	NOUN
eajbcs-667	74	35	to	to	ADP
eajbcs-667	74	36	one	one	NUM
eajbcs-667	74	37	dimension	dimension	NOUN
eajbcs-667	74	38	(	(	PUNCT
eajbcs-667	74	39	lima	lima	NOUN
eajbcs-667	74	40	et	et	PROPN
eajbcs-667	74	41	al	al	PROPN
eajbcs-667	74	42	.	.	PROPN
eajbcs-667	74	43	,	,	PUNCT
eajbcs-667	74	44	2021	2021	NUM
eajbcs-667	74	45	;	;	PUNCT
eajbcs-667	74	46	hundsdorfer	hundsdorfer	NOUN
eajbcs-667	74	47	,	,	PUNCT
eajbcs-667	74	48	1996	1996	NUM
eajbcs-667	74	49	)	)	PUNCT
eajbcs-667	74	50	and	and	CCONJ
eajbcs-667	74	51	the	the	DET
eajbcs-667	74	52	one	one	NUM
eajbcs-667	74	53	dimensional	dimensional	ADJ
eajbcs-667	74	54	advection	advection	NOUN
eajbcs-667	74	55	diffusion	diffusion	NOUN
eajbcs-667	74	56	equation	equation	NOUN
eajbcs-667	74	57	is	be	AUX
eajbcs-667	74	58	given	give	VERB
eajbcs-667	74	59	in	in	ADP
eajbcs-667	74	60	eq	eq	ADJ
eajbcs-667	74	61	.	.	PROPN
eajbcs-667	75	1	3	3	X
eajbcs-667	75	2	.	.	PUNCT
eajbcs-667	76	1	the	the	DET
eajbcs-667	76	2	time	time	NOUN
eajbcs-667	76	3	-	-	PUNCT
eajbcs-667	76	4	derivative	derivative	ADJ
eajbcs-667	76	5	term	term	NOUN
eajbcs-667	76	6	expresses	express	VERB
eajbcs-667	76	7	accumulation	accumulation	NOUN
eajbcs-667	76	8	of	of	ADP
eajbcs-667	76	9	mass	mass	NOUN
eajbcs-667	76	10	at	at	ADP
eajbcs-667	76	11	a	a	DET
eajbcs-667	76	12	point	point	NOUN
eajbcs-667	76	13	in	in	ADP
eajbcs-667	76	14	space	space	NOUN
eajbcs-667	76	15	,	,	PUNCT
eajbcs-667	76	16	the	the	DET
eajbcs-667	76	17	advection	advection	NOUN
eajbcs-667	76	18	term	term	NOUN
eajbcs-667	76	19	a∇u	a∇u	DET
eajbcs-667	76	20	transport	transport	NOUN
eajbcs-667	76	21	of	of	ADP
eajbcs-667	76	22	mass	mass	NOUN
eajbcs-667	76	23	with	with	ADP
eajbcs-667	76	24	the	the	DET
eajbcs-667	76	25	flow	flow	NOUN
eajbcs-667	76	26	,	,	PUNCT
eajbcs-667	76	27	and	and	CCONJ
eajbcs-667	76	28	the	the	DET
eajbcs-667	76	29	diffusion	diffusion	NOUN
eajbcs-667	76	30	term	term	NOUN
eajbcs-667	76	31	d∇2u	d∇2u	PROPN
eajbcs-667	76	32	reflects	reflect	VERB
eajbcs-667	76	33	transport	transport	NOUN
eajbcs-667	76	34	of	of	ADP
eajbcs-667	76	35	mass	mass	NOUN
eajbcs-667	76	36	due	due	ADP
eajbcs-667	76	37	to	to	ADP
eajbcs-667	76	38	molecular	molecular	ADJ
eajbcs-667	76	39	diffusion	diffusion	NOUN
eajbcs-667	76	40	(	(	PUNCT
eajbcs-667	76	41	langtangen	langtangen	PROPN
eajbcs-667	76	42	,	,	PUNCT
eajbcs-667	76	43	1999	1999	NUM
eajbcs-667	76	44	)	)	PUNCT
eajbcs-667	76	45	.	.	PUNCT
eajbcs-667	77	1	we	we	PRON
eajbcs-667	77	2	shall	shall	AUX
eajbcs-667	77	3	consider	consider	VERB
eajbcs-667	77	4	the	the	DET
eajbcs-667	77	5	equation	equation	NOUN
eajbcs-667	77	6	in	in	ADP
eajbcs-667	77	7	the	the	DET
eajbcs-667	77	8	question	question	NOUN
eajbcs-667	77	9	,	,	PUNCT
eajbcs-667	77	10	we	we	PRON
eajbcs-667	77	11	must	must	AUX
eajbcs-667	77	12	compare	compare	VERB
eajbcs-667	77	13	the	the	DET
eajbcs-667	77	14	sizes	size	NOUN
eajbcs-667	77	15	of	of	ADP
eajbcs-667	77	16	the	the	DET
eajbcs-667	77	17	a∂u	a∂u	NOUN
eajbcs-667	77	18	∂x	∂x	PROPN
eajbcs-667	77	19	and	and	CCONJ
eajbcs-667	77	20	d	d	PROPN
eajbcs-667	77	21	∂2u	∂2u	ADJ
eajbcs-667	77	22	∂x2	∂x2	PROPN
eajbcs-667	77	23	terms	term	NOUN
eajbcs-667	77	24	to	to	ADP
eajbcs-667	77	25	each	each	DET
eajbcs-667	77	26	other	other	ADJ
eajbcs-667	77	27	,	,	PUNCT
eajbcs-667	77	28	and	and	CCONJ
eajbcs-667	77	29	this	this	PRON
eajbcs-667	77	30	is	be	AUX
eajbcs-667	77	31	accomplishes	accomplish	VERB
eajbcs-667	77	32	by	by	ADP
eajbcs-667	77	33	introducing	introduce	VERB
eajbcs-667	77	34	scales	scale	NOUN
eajbcs-667	77	35	.	.	PUNCT
eajbcs-667	78	1	using	use	VERB
eajbcs-667	78	2	these	these	DET
eajbcs-667	78	3	scales	scale	NOUN
eajbcs-667	78	4	,	,	PUNCT
eajbcs-667	78	5	we	we	PRON
eajbcs-667	78	6	can	can	AUX
eajbcs-667	78	7	derive	derive	VERB
eajbcs-667	78	8	estimates	estimate	NOUN
eajbcs-667	78	9	of	of	ADP
eajbcs-667	78	10	the	the	DET
eajbcs-667	78	11	sizes	size	NOUN
eajbcs-667	78	12	of	of	ADP
eajbcs-667	78	13	the	the	DET
eajbcs-667	78	14	different	different	ADJ
eajbcs-667	78	15	terms	term	NOUN
eajbcs-667	78	16	.	.	PUNCT
eajbcs-667	79	1	since	since	SCONJ
eajbcs-667	79	2	the	the	DET
eajbcs-667	79	3	derivative	derivative	ADJ
eajbcs-667	79	4	∂u	∂u	PROPN
eajbcs-667	79	5	∂x	∂x	PROPN
eajbcs-667	79	6	is	be	AUX
eajbcs-667	79	7	expressing	express	VERB
eajbcs-667	79	8	the	the	DET
eajbcs-667	79	9	difference	difference	NOUN
eajbcs-667	79	10	in	in	ADP
eajbcs-667	79	11	concentration	concentration	NOUN
eajbcs-667	79	12	over	over	ADP
eajbcs-667	79	13	a	a	DET
eajbcs-667	79	14	distance	distance	NOUN
eajbcs-667	79	15	of	of	ADP
eajbcs-667	79	16	infinitesimal	infinitesimal	ADJ
eajbcs-667	79	17	limit	limit	NOUN
eajbcs-667	79	18	,	,	PUNCT
eajbcs-667	79	19	we	we	PRON
eajbcs-667	79	20	can	can	AUX
eajbcs-667	79	21	estimate	estimate	VERB
eajbcs-667	79	22	it	it	PRON
eajbcs-667	79	23	to	to	PART
eajbcs-667	79	24	be	be	AUX
eajbcs-667	79	25	approximately	approximately	ADV
eajbcs-667	79	26	u	u	NOUN
eajbcs-667	79	27	x	x	NOUN
eajbcs-667	79	28	,	,	PUNCT
eajbcs-667	79	29	and	and	CCONJ
eajbcs-667	79	30	the	the	DET
eajbcs-667	79	31	advection	advection	NOUN
eajbcs-667	79	32	term	term	NOUN
eajbcs-667	79	33	scales	scale	NOUN
eajbcs-667	79	34	as	as	ADP
eajbcs-667	79	35	:	:	PUNCT
eajbcs-667	79	36	a	a	DET
eajbcs-667	79	37	∂u	∂u	PROPN
eajbcs-667	79	38	∂x	∂x	PROPN
eajbcs-667	79	39	∼	∼	NOUN
eajbcs-667	79	40	v	v	NOUN
eajbcs-667	79	41	u	u	NOUN
eajbcs-667	79	42	x	x	X
eajbcs-667	79	43	.	.	PUNCT
eajbcs-667	80	1	similarly	similarly	ADV
eajbcs-667	80	2	,	,	PUNCT
eajbcs-667	80	3	the	the	DET
eajbcs-667	80	4	second	second	ADJ
eajbcs-667	80	5	derivative	derivative	ADJ
eajbcs-667	80	6	∂2u	∂2u	ADJ
eajbcs-667	80	7	∂x2	∂x2	PROPN
eajbcs-667	80	8	represents	represent	VERB
eajbcs-667	80	9	the	the	DET
eajbcs-667	80	10	difference	difference	NOUN
eajbcs-667	80	11	of	of	ADP
eajbcs-667	80	12	a	a	DET
eajbcs-667	80	13	gradient	gradient	NOUN
eajbcs-667	80	14	over	over	ADP
eajbcs-667	80	15	a	a	DET
eajbcs-667	80	16	specified	specify	VERB
eajbcs-667	80	17	distance	distance	NOUN
eajbcs-667	80	18	and	and	CCONJ
eajbcs-667	80	19	is	be	AUX
eajbcs-667	80	20	estimated	estimate	VERB
eajbcs-667	80	21	at	at	ADP
eajbcs-667	80	22	(	(	PUNCT
eajbcs-667	80	23	u	u	NOUN
eajbcs-667	80	24	x	x	PROPN
eajbcs-667	80	25	)	)	PUNCT
eajbcs-667	80	26	x	x	X
eajbcs-667	81	1	=	=	PUNCT
eajbcs-667	81	2	u	u	NOUN
eajbcs-667	81	3	x2	x2	NOUN
eajbcs-667	81	4	,	,	PUNCT
eajbcs-667	81	5	and	and	CCONJ
eajbcs-667	81	6	the	the	DET
eajbcs-667	81	7	diffusion	diffusion	NOUN
eajbcs-667	81	8	term	term	NOUN
eajbcs-667	81	9	scales	scale	VERB
eajbcs-667	81	10	as	as	ADP
eajbcs-667	81	11	:	:	PUNCT
eajbcs-667	81	12	d	d	PROPN
eajbcs-667	81	13	∂2u	∂2u	ADJ
eajbcs-667	81	14	∂x2	∂x2	NOUN
eajbcs-667	81	15	∼	∼	NOUN
eajbcs-667	81	16	d	d	NOUN
eajbcs-667	81	17	u	u	NOUN
eajbcs-667	81	18	x2	x2	PROPN
eajbcs-667	81	19	.	.	PUNCT
eajbcs-667	82	1	equipped	equip	VERB
eajbcs-667	82	2	with	with	ADP
eajbcs-667	82	3	these	these	DET
eajbcs-667	82	4	estimates	estimate	NOUN
eajbcs-667	82	5	,	,	PUNCT
eajbcs-667	82	6	we	we	PRON
eajbcs-667	82	7	can	can	AUX
eajbcs-667	82	8	then	then	ADV
eajbcs-667	82	9	compare	compare	VERB
eajbcs-667	82	10	the	the	DET
eajbcs-667	82	11	two	two	NUM
eajbcs-667	82	12	processes	process	NOUN
eajbcs-667	82	13	by	by	ADP
eajbcs-667	82	14	forming	form	VERB
eajbcs-667	82	15	the	the	DET
eajbcs-667	82	16	ratio	ratio	NOUN
eajbcs-667	82	17	of	of	ADP
eajbcs-667	82	18	their	their	PRON
eajbcs-667	82	19	scales	scale	NOUN
eajbcs-667	82	20	:	:	PUNCT
eajbcs-667	82	21	advecton	advecton	ADJ
eajbcs-667	82	22	diffusion	diffusion	NOUN
eajbcs-667	82	23	=	=	PUNCT
eajbcs-667	83	1	v	v	NUM
eajbcs-667	83	2	u	u	NOUN
eajbcs-667	83	3	x	x	PROPN
eajbcs-667	83	4	d	d	X
eajbcs-667	83	5	u	u	NOUN
eajbcs-667	83	6	x2	x2	NOUN
eajbcs-667	83	7	=	=	X
eajbcs-667	83	8	v	v	PROPN
eajbcs-667	83	9	x	x	PROPN
eajbcs-667	83	10	d	d	NOUN
eajbcs-667	83	11	this	this	DET
eajbcs-667	83	12	ratio	ratio	NOUN
eajbcs-667	83	13	is	be	AUX
eajbcs-667	83	14	dimensionless	dimensionless	ADJ
eajbcs-667	83	15	and	and	CCONJ
eajbcs-667	83	16	traditionally	traditionally	ADV
eajbcs-667	83	17	,	,	PUNCT
eajbcs-667	83	18	it	it	PRON
eajbcs-667	83	19	is	be	AUX
eajbcs-667	83	20	called	call	VERB
eajbcs-667	83	21	the	the	DET
eajbcs-667	83	22	peclet	peclet	NOUN
eajbcs-667	83	23	number	number	NOUN
eajbcs-667	83	24	and	and	CCONJ
eajbcs-667	83	25	is	be	AUX
eajbcs-667	83	26	denoted	denote	VERB
eajbcs-667	83	27	by	by	ADP
eajbcs-667	83	28	pe	pe	NOUN
eajbcs-667	83	29	:	:	PUNCT
eajbcs-667	84	1	pe	pe	PROPN
eajbcs-667	84	2	=	=	SYM
eajbcs-667	84	3	v	v	PROPN
eajbcs-667	84	4	x	x	SYM
eajbcs-667	84	5	d	d	NOUN
eajbcs-667	84	6	.	.	PUNCT
eajbcs-667	85	1	if	if	SCONJ
eajbcs-667	85	2	pe	pe	PROPN
eajbcs-667	85	3	≪	≪	ADJ
eajbcs-667	85	4	1	1	NUM
eajbcs-667	85	5	(	(	PUNCT
eajbcs-667	85	6	if	if	SCONJ
eajbcs-667	85	7	pe	pe	X
eajbcs-667	85	8	<	<	X
eajbcs-667	85	9	0.1	0.1	NUM
eajbcs-667	85	10	):	):	PUNCT
eajbcs-667	85	11	the	the	DET
eajbcs-667	85	12	advection	advection	NOUN
eajbcs-667	85	13	term	term	NOUN
eajbcs-667	85	14	will	will	AUX
eajbcs-667	85	15	result	result	VERB
eajbcs-667	85	16	significantly	significantly	ADV
eajbcs-667	85	17	smaller	small	ADJ
eajbcs-667	85	18	than	than	ADP
eajbcs-667	85	19	the	the	DET
eajbcs-667	85	20	diffusion	diffusion	NOUN
eajbcs-667	85	21	term	term	NOUN
eajbcs-667	85	22	.	.	PUNCT
eajbcs-667	86	1	physically	physically	ADV
eajbcs-667	86	2	,	,	PUNCT
eajbcs-667	86	3	diffusion	diffusion	NOUN
eajbcs-667	86	4	dominates	dominate	NOUN
eajbcs-667	86	5	and	and	CCONJ
eajbcs-667	86	6	advection	advection	NOUN
eajbcs-667	86	7	is	be	AUX
eajbcs-667	86	8	negligible	negligible	ADJ
eajbcs-667	86	9	.	.	PUNCT
eajbcs-667	87	1	so	so	ADV
eajbcs-667	87	2	,	,	PUNCT
eajbcs-667	87	3	spreading	spread	VERB
eajbcs-667	87	4	occurs	occur	VERB
eajbcs-667	87	5	symmetrically	symmetrically	ADV
eajbcs-667	87	6	despite	despite	SCONJ
eajbcs-667	87	7	of	of	ADP
eajbcs-667	87	8	the	the	DET
eajbcs-667	87	9	flow	flow	NOUN
eajbcs-667	87	10	of	of	ADP
eajbcs-667	87	11	the	the	DET
eajbcs-667	87	12	directional	directional	ADJ
eajbcs-667	87	13	bias	bias	NOUN
eajbcs-667	87	14	.	.	PUNCT
eajbcs-667	88	1	if	if	SCONJ
eajbcs-667	88	2	we	we	PRON
eajbcs-667	88	3	wish	wish	VERB
eajbcs-667	88	4	to	to	PART
eajbcs-667	88	5	simplify	simplify	VERB
eajbcs-667	88	6	the	the	DET
eajbcs-667	88	7	problem	problem	NOUN
eajbcs-667	88	8	,	,	PUNCT
eajbcs-667	88	9	we	we	PRON
eajbcs-667	88	10	may	may	AUX
eajbcs-667	88	11	drop	drop	VERB
eajbcs-667	88	12	the	the	DET
eajbcs-667	88	13	a∂u	a∂u	NOUN
eajbcs-667	88	14	∂x	∂x	PROPN
eajbcs-667	88	15	term	term	NOUN
eajbcs-667	88	16	,	,	PUNCT
eajbcs-667	88	17	as	as	SCONJ
eajbcs-667	88	18	if	if	SCONJ
eajbcs-667	88	19	a	a	PRON
eajbcs-667	88	20	were	be	AUX
eajbcs-667	88	21	nil	nil	NOUN
eajbcs-667	88	22	(	(	PUNCT
eajbcs-667	88	23	no	no	DET
eajbcs-667	88	24	amount	amount	NOUN
eajbcs-667	88	25	at	at	ADV
eajbcs-667	88	26	all	all	ADV
eajbcs-667	88	27	)	)	PUNCT
eajbcs-667	88	28	.	.	PUNCT
eajbcs-667	89	1	the	the	DET
eajbcs-667	89	2	relative	relative	ADJ
eajbcs-667	89	3	error	error	NOUN
eajbcs-667	89	4	occurred	occur	VERB
eajbcs-667	89	5	in	in	ADP
eajbcs-667	89	6	the	the	DET
eajbcs-667	89	7	solution	solution	NOUN
eajbcs-667	89	8	is	be	AUX
eajbcs-667	89	9	expected	expect	VERB
eajbcs-667	89	10	to	to	PART
eajbcs-667	89	11	be	be	AUX
eajbcs-667	89	12	on	on	ADP
eajbcs-667	89	13	the	the	DET
eajbcs-667	89	14	order	order	NOUN
eajbcs-667	89	15	of	of	ADP
eajbcs-667	89	16	the	the	DET
eajbcs-667	89	17	peclet	peclet	NOUN
eajbcs-667	89	18	number	number	NOUN
eajbcs-667	89	19	,	,	PUNCT
eajbcs-667	89	20	and	and	CCONJ
eajbcs-667	89	21	the	the	DET
eajbcs-667	89	22	smaller	small	ADJ
eajbcs-667	89	23	pe	pe	NOUN
eajbcs-667	89	24	leads	lead	VERB
eajbcs-667	89	25	to	to	ADP
eajbcs-667	89	26	the	the	DET
eajbcs-667	89	27	smaller	small	ADJ
eajbcs-667	89	28	error	error	NOUN
eajbcs-667	89	29	.	.	PUNCT
eajbcs-667	90	1	the	the	DET
eajbcs-667	90	2	solutions	solution	NOUN
eajbcs-667	90	3	established	establish	VERB
eajbcs-667	90	4	with	with	ADP
eajbcs-667	90	5	diffusion	diffusion	NOUN
eajbcs-667	90	6	only	only	ADV
eajbcs-667	90	7	were	be	AUX
eajbcs-667	90	8	based	base	VERB
eajbcs-667	90	9	on	on	ADP
eajbcs-667	90	10	such	such	ADJ
eajbcs-667	90	11	simplification	simplification	NOUN
eajbcs-667	90	12	and	and	CCONJ
eajbcs-667	90	13	are	be	AUX
eajbcs-667	90	14	thus	thus	ADV
eajbcs-667	90	15	valid	valid	ADJ
eajbcs-667	90	16	as	as	ADV
eajbcs-667	90	17	long	long	ADV
eajbcs-667	90	18	as	as	ADP
eajbcs-667	90	19	pe	pe	X
eajbcs-667	90	20	≪	≪	ADJ
eajbcs-667	90	21	1	1	X
eajbcs-667	90	22	.	.	PUNCT
eajbcs-667	91	1	if	if	SCONJ
eajbcs-667	91	2	pe	pe	PROPN
eajbcs-667	91	3	≫	≫	NOUN
eajbcs-667	91	4	1	1	NUM
eajbcs-667	91	5	(	(	PUNCT
eajbcs-667	91	6	if	if	SCONJ
eajbcs-667	91	7	pe	pe	INTJ
eajbcs-667	91	8	>	>	X
eajbcs-667	91	9	10	10	NUM
eajbcs-667	91	10	):	):	PUNCT
eajbcs-667	91	11	the	the	DET
eajbcs-667	91	12	advection	advection	NOUN
eajbcs-667	91	13	term	term	NOUN
eajbcs-667	91	14	is	be	AUX
eajbcs-667	91	15	significantly	significantly	ADV
eajbcs-667	91	16	bigger	big	ADJ
eajbcs-667	91	17	than	than	ADP
eajbcs-667	91	18	the	the	DET
eajbcs-667	91	19	diffusion	diffusion	NOUN
eajbcs-667	91	20	term	term	NOUN
eajbcs-667	91	21	.	.	PUNCT
eajbcs-667	92	1	physically	physically	ADV
eajbcs-667	92	2	,	,	PUNCT
eajbcs-667	92	3	the	the	DET
eajbcs-667	92	4	diffusion	diffusion	NOUN
eajbcs-667	92	5	term	term	NOUN
eajbcs-667	92	6	is	be	AUX
eajbcs-667	92	7	negligible	negligible	ADJ
eajbcs-667	92	8	and	and	CCONJ
eajbcs-667	92	9	advection	advection	NOUN
eajbcs-667	92	10	dominates	dominate	VERB
eajbcs-667	92	11	,	,	PUNCT
eajbcs-667	92	12	and	and	CCONJ
eajbcs-667	92	13	spreading	spread	VERB
eajbcs-667	92	14	is	be	AUX
eajbcs-667	92	15	existent	existent	ADJ
eajbcs-667	92	16	,	,	PUNCT
eajbcs-667	92	17	with	with	SCONJ
eajbcs-667	92	18	the	the	DET
eajbcs-667	92	19	patch	patch	NOUN
eajbcs-667	92	20	(	(	PUNCT
eajbcs-667	92	21	small	small	ADJ
eajbcs-667	92	22	area	area	NOUN
eajbcs-667	92	23	)	)	PUNCT
eajbcs-667	92	24	of	of	ADP
eajbcs-667	92	25	pollutant	pollutant	ADJ
eajbcs-667	92	26	being	be	AUX
eajbcs-667	92	27	simply	simply	ADV
eajbcs-667	92	28	moved	move	VERB
eajbcs-667	92	29	along	along	ADV
eajbcs-667	92	30	by	by	ADP
eajbcs-667	92	31	the	the	DET
eajbcs-667	92	32	flow	flow	NOUN
eajbcs-667	92	33	.	.	PUNCT
eajbcs-667	93	1	if	if	SCONJ
eajbcs-667	93	2	we	we	PRON
eajbcs-667	93	3	wish	wish	VERB
eajbcs-667	93	4	to	to	PART
eajbcs-667	93	5	simplify	simplify	VERB
eajbcs-667	93	6	the	the	DET
eajbcs-667	93	7	problem	problem	NOUN
eajbcs-667	93	8	,	,	PUNCT
eajbcs-667	93	9	we	we	PRON
eajbcs-667	93	10	may	may	AUX
eajbcs-667	93	11	drop	drop	VERB
eajbcs-667	93	12	the	the	DET
eajbcs-667	93	13	d	d	PROPN
eajbcs-667	93	14	∂2u	∂2u	ADJ
eajbcs-667	93	15	∂x2	∂x2	PROPN
eajbcs-667	93	16	term	term	NOUN
eajbcs-667	93	17	,	,	PUNCT
eajbcs-667	93	18	as	as	SCONJ
eajbcs-667	93	19	if	if	SCONJ
eajbcs-667	93	20	d	d	PROPN
eajbcs-667	93	21	were	be	AUX
eajbcs-667	93	22	zero	zero	NUM
eajbcs-667	93	23	.	.	PUNCT
eajbcs-667	94	1	the	the	DET
eajbcs-667	94	2	relative	relative	ADJ
eajbcs-667	94	3	error	error	NOUN
eajbcs-667	94	4	occurred	occur	VERB
eajbcs-667	94	5	in	in	ADP
eajbcs-667	94	6	doing	do	VERB
eajbcs-667	94	7	the	the	DET
eajbcs-667	94	8	solution	solution	NOUN
eajbcs-667	94	9	is	be	AUX
eajbcs-667	94	10	expected	expect	VERB
eajbcs-667	94	11	as	as	ADP
eajbcs-667	94	12	the	the	DET
eajbcs-667	94	13	order	order	NOUN
eajbcs-667	94	14	of	of	ADP
eajbcs-667	94	15	the	the	DET
eajbcs-667	94	16	peclet	peclet	NOUN
eajbcs-667	94	17	number	number	NOUN
eajbcs-667	94	18	inverse	inverse	NOUN
eajbcs-667	94	19	(	(	PUNCT
eajbcs-667	94	20	1	1	NUM
eajbcs-667	94	21	/	/	SYM
eajbcs-667	94	22	p	p	NOUN
eajbcs-667	94	23	e	e	NOUN
eajbcs-667	94	24	)	)	PUNCT
eajbcs-667	94	25	,	,	PUNCT
eajbcs-667	94	26	and	and	CCONJ
eajbcs-667	94	27	the	the	DET
eajbcs-667	94	28	larger	large	ADJ
eajbcs-667	94	29	p	p	NOUN
eajbcs-667	94	30	e	e	NOUN
eajbcs-667	94	31	will	will	AUX
eajbcs-667	94	32	result	result	VERB
eajbcs-667	94	33	the	the	DET
eajbcs-667	94	34	smaller	small	ADJ
eajbcs-667	94	35	error	error	NOUN
eajbcs-667	94	36	.	.	PUNCT
eajbcs-667	95	1	east	east	PROPN
eajbcs-667	95	2	afr	afr	PROPN
eajbcs-667	95	3	.	.	PUNCT
eajbcs-667	96	1	j.	j.	PROPN
eajbcs-667	96	2	biophys	biophys	PROPN
eajbcs-667	96	3	.	.	PUNCT
eajbcs-667	97	1	comput	comput	NOUN
eajbcs-667	97	2	.	.	PUNCT
eajbcs-667	98	1	sci	sci	PROPN
eajbcs-667	98	2	.	.	PUNCT
eajbcs-667	98	3	(	(	PUNCT
eajbcs-667	98	4	2023	2023	NUM
eajbcs-667	98	5	)	)	PUNCT
eajbcs-667	98	6	,	,	PUNCT
eajbcs-667	98	7	vol	vol	NOUN
eajbcs-667	98	8	.	.	PROPN
eajbcs-667	98	9	4	4	NUM
eajbcs-667	98	10	,	,	PUNCT
eajbcs-667	98	11	no	no	INTJ
eajbcs-667	98	12	.	.	NOUN
eajbcs-667	98	13	1	1	NUM
eajbcs-667	98	14	,	,	PUNCT
eajbcs-667	98	15	52	52	NUM
eajbcs-667	98	16	-	-	SYM
eajbcs-667	98	17	74	74	NUM
eajbcs-667	98	18	55	55	NUM
eajbcs-667	98	19	if	if	SCONJ
eajbcs-667	98	20	p	p	NOUN
eajbcs-667	98	21	e	e	NOUN
eajbcs-667	98	22	∼	∼	NOUN
eajbcs-667	98	23	1	1	NUM
eajbcs-667	98	24	(	(	PUNCT
eajbcs-667	98	25	in	in	ADP
eajbcs-667	98	26	practice	practice	NOUN
eajbcs-667	98	27	,	,	PUNCT
eajbcs-667	98	28	if	if	SCONJ
eajbcs-667	98	29	0.1	0.1	NUM
eajbcs-667	98	30	<	<	X
eajbcs-667	98	31	p	p	X
eajbcs-667	98	32	e	e	X
eajbcs-667	98	33	<	<	X
eajbcs-667	98	34	10	10	NUM
eajbcs-667	98	35	):	):	PUNCT
eajbcs-667	98	36	the	the	DET
eajbcs-667	98	37	advection	advection	NOUN
eajbcs-667	98	38	and	and	CCONJ
eajbcs-667	98	39	diffusion	diffusion	NOUN
eajbcs-667	98	40	terms	term	NOUN
eajbcs-667	98	41	are	be	AUX
eajbcs-667	98	42	not	not	PART
eajbcs-667	98	43	significantly	significantly	ADV
eajbcs-667	98	44	different	different	ADJ
eajbcs-667	98	45	which	which	PRON
eajbcs-667	98	46	results	result	VERB
eajbcs-667	98	47	for	for	ADP
eajbcs-667	98	48	the	the	DET
eajbcs-667	98	49	non	non	ADJ
eajbcs-667	98	50	dominance	dominance	NOUN
eajbcs-667	98	51	of	of	ADP
eajbcs-667	98	52	the	the	DET
eajbcs-667	98	53	two	two	NUM
eajbcs-667	98	54	in	in	ADP
eajbcs-667	98	55	the	the	DET
eajbcs-667	98	56	process	process	NOUN
eajbcs-667	98	57	.	.	PUNCT
eajbcs-667	99	1	the	the	DET
eajbcs-667	99	2	full	full	ADJ
eajbcs-667	99	3	equation	equation	NOUN
eajbcs-667	99	4	must	must	AUX
eajbcs-667	99	5	be	be	AUX
eajbcs-667	99	6	utilized	utilize	VERB
eajbcs-667	99	7	as	as	SCONJ
eajbcs-667	99	8	there	there	PRON
eajbcs-667	99	9	will	will	AUX
eajbcs-667	99	10	no	no	DET
eajbcs-667	99	11	approximation	approximation	NOUN
eajbcs-667	99	12	to	to	ADP
eajbcs-667	99	13	the	the	DET
eajbcs-667	99	14	equation	equation	NOUN
eajbcs-667	99	15	will	will	AUX
eajbcs-667	99	16	be	be	AUX
eajbcs-667	99	17	justified	justify	VERB
eajbcs-667	99	18	.	.	PUNCT
eajbcs-667	100	1	variable	variable	ADJ
eajbcs-667	100	2	scale	scale	NOUN
eajbcs-667	100	3	choice	choice	NOUN
eajbcs-667	100	4	of	of	ADP
eajbcs-667	100	5	value	value	NOUN
eajbcs-667	100	6	u	u	NOUN
eajbcs-667	100	7	u	u	NOUN
eajbcs-667	100	8	the	the	DET
eajbcs-667	100	9	concentration	concentration	NOUN
eajbcs-667	100	10	value	value	NOUN
eajbcs-667	100	11	such	such	ADJ
eajbcs-667	100	12	as	as	ADP
eajbcs-667	100	13	initial	initial	ADJ
eajbcs-667	100	14	,	,	PUNCT
eajbcs-667	100	15	boundary	boundary	ADJ
eajbcs-667	100	16	,	,	PUNCT
eajbcs-667	100	17	or	or	CCONJ
eajbcs-667	100	18	average	average	ADJ
eajbcs-667	100	19	value	value	NOUN
eajbcs-667	100	20	a	a	DET
eajbcs-667	100	21	v	v	NOUN
eajbcs-667	100	22	the	the	DET
eajbcs-667	100	23	maximum	maximum	ADJ
eajbcs-667	100	24	velocity	velocity	NOUN
eajbcs-667	100	25	value	value	NOUN
eajbcs-667	100	26	x	x	X
eajbcs-667	100	27	x	x	X
eajbcs-667	100	28	approximate	approximate	ADJ
eajbcs-667	100	29	length	length	NOUN
eajbcs-667	100	30	of	of	ADP
eajbcs-667	100	31	the	the	DET
eajbcs-667	100	32	domain	domain	NOUN
eajbcs-667	100	33	or	or	CCONJ
eajbcs-667	100	34	size	size	NOUN
eajbcs-667	100	35	of	of	ADP
eajbcs-667	100	36	release	release	NOUN
eajbcs-667	100	37	location	location	NOUN
eajbcs-667	100	38	in	in	ADP
eajbcs-667	100	39	this	this	DET
eajbcs-667	100	40	paper	paper	NOUN
eajbcs-667	100	41	we	we	PRON
eajbcs-667	100	42	use	use	VERB
eajbcs-667	100	43	the	the	DET
eajbcs-667	100	44	finite	finite	ADJ
eajbcs-667	100	45	element	element	NOUN
eajbcs-667	100	46	method	method	NOUN
eajbcs-667	100	47	(	(	PUNCT
eajbcs-667	100	48	fem	fem	NOUN
eajbcs-667	100	49	)	)	PUNCT
eajbcs-667	100	50	to	to	PART
eajbcs-667	100	51	approximate	approximate	VERB
eajbcs-667	100	52	the	the	DET
eajbcs-667	100	53	solution	solution	NOUN
eajbcs-667	100	54	of	of	ADP
eajbcs-667	100	55	the	the	DET
eajbcs-667	100	56	advection	advection	NOUN
eajbcs-667	100	57	diffusion	diffusion	NOUN
eajbcs-667	100	58	equation	equation	NOUN
eajbcs-667	100	59	.	.	PUNCT
eajbcs-667	101	1	the	the	DET
eajbcs-667	101	2	method	method	NOUN
eajbcs-667	101	3	is	be	AUX
eajbcs-667	101	4	examined	examine	VERB
eajbcs-667	101	5	as	as	ADP
eajbcs-667	101	6	an	an	DET
eajbcs-667	101	7	emerging	emerge	VERB
eajbcs-667	101	8	tool	tool	NOUN
eajbcs-667	101	9	for	for	ADP
eajbcs-667	101	10	the	the	DET
eajbcs-667	101	11	approximate	approximate	ADJ
eajbcs-667	101	12	solution	solution	NOUN
eajbcs-667	101	13	of	of	ADP
eajbcs-667	101	14	differential	differential	ADJ
eajbcs-667	101	15	equations	equation	NOUN
eajbcs-667	101	16	describing	describe	VERB
eajbcs-667	101	17	different	different	ADJ
eajbcs-667	101	18	physical	physical	ADJ
eajbcs-667	101	19	processes	process	NOUN
eajbcs-667	101	20	(	(	PUNCT
eajbcs-667	101	21	yang	yang	PROPN
eajbcs-667	101	22	et	et	PROPN
eajbcs-667	101	23	al	al	PROPN
eajbcs-667	101	24	.	.	PROPN
eajbcs-667	101	25	,	,	PUNCT
eajbcs-667	101	26	2020	2020	NUM
eajbcs-667	101	27	)	)	PUNCT
eajbcs-667	101	28	.	.	PUNCT
eajbcs-667	102	1	it	it	PRON
eajbcs-667	102	2	is	be	AUX
eajbcs-667	102	3	based	base	VERB
eajbcs-667	102	4	on	on	ADP
eajbcs-667	102	5	the	the	DET
eajbcs-667	102	6	basic	basic	ADJ
eajbcs-667	102	7	finite	finite	NOUN
eajbcs-667	102	8	e	e	PROPN
eajbcs-667	102	9	lement	lement	NOUN
eajbcs-667	102	10	procedures	procedure	NOUN
eajbcs-667	102	11	,	,	PUNCT
eajbcs-667	102	12	those	those	PRON
eajbcs-667	102	13	are	be	AUX
eajbcs-667	102	14	:	:	PUNCT
eajbcs-667	102	15	the	the	DET
eajbcs-667	102	16	variation	variation	NOUN
eajbcs-667	102	17	form	form	VERB
eajbcs-667	102	18	fornumerical	fornumerical	ADJ
eajbcs-667	102	19	method	method	NOUN
eajbcs-667	102	20	mulation	mulation	NOUN
eajbcs-667	102	21	of	of	ADP
eajbcs-667	102	22	the	the	DET
eajbcs-667	102	23	problem	problem	NOUN
eajbcs-667	102	24	,	,	PUNCT
eajbcs-667	102	25	the	the	DET
eajbcs-667	102	26	discretization	discretization	NOUN
eajbcs-667	102	27	of	of	ADP
eajbcs-667	102	28	the	the	DET
eajbcs-667	102	29	formulation	formulation	NOUN
eajbcs-667	102	30	in	in	ADP
eajbcs-667	102	31	a	a	DET
eajbcs-667	102	32	finite	finite	ADJ
eajbcs-667	102	33	element	element	NOUN
eajbcs-667	102	34	,	,	PUNCT
eajbcs-667	102	35	and	and	CCONJ
eajbcs-667	102	36	the	the	DET
eajbcs-667	102	37	solution	solution	NOUN
eajbcs-667	102	38	of	of	ADP
eajbcs-667	102	39	the	the	DET
eajbcs-667	102	40	resulting	result	VERB
eajbcs-667	102	41	finite	finite	ADJ
eajbcs-667	102	42	element	element	NOUN
eajbcs-667	102	43	equations	equation	NOUN
eajbcs-667	102	44	.	.	PUNCT
eajbcs-667	103	1	fem	fem	PROPN
eajbcs-667	103	2	cuts	cut	VERB
eajbcs-667	103	3	a	a	DET
eajbcs-667	103	4	given	give	VERB
eajbcs-667	103	5	domain	domain	NOUN
eajbcs-667	103	6	into	into	ADP
eajbcs-667	103	7	several	several	ADJ
eajbcs-667	103	8	elements	element	NOUN
eajbcs-667	103	9	(	(	PUNCT
eajbcs-667	103	10	pieces	piece	NOUN
eajbcs-667	103	11	of	of	ADP
eajbcs-667	103	12	the	the	DET
eajbcs-667	103	13	domain	domain	NOUN
eajbcs-667	103	14	)	)	PUNCT
eajbcs-667	103	15	and	and	CCONJ
eajbcs-667	103	16	connected	connect	VERB
eajbcs-667	103	17	in	in	ADP
eajbcs-667	103	18	a	a	DET
eajbcs-667	103	19	finite	finite	ADJ
eajbcs-667	103	20	number	number	NOUN
eajbcs-667	103	21	of	of	ADP
eajbcs-667	103	22	nodal	nodal	NOUN
eajbcs-667	103	23	points	point	NOUN
eajbcs-667	103	24	.	.	PUNCT
eajbcs-667	104	1	finite	finite	PROPN
eajbcs-667	104	2	element	element	NOUN
eajbcs-667	104	3	implementation	implementation	NOUN
eajbcs-667	104	4	of	of	ADP
eajbcs-667	104	5	the	the	DET
eajbcs-667	104	6	1d	1d	NOUN
eajbcs-667	104	7	governing	governing	NOUN
eajbcs-667	104	8	equation	equation	NOUN
eajbcs-667	104	9	let	let	VERB
eajbcs-667	104	10	us	we	PRON
eajbcs-667	104	11	consider	consider	VERB
eajbcs-667	104	12	the	the	DET
eajbcs-667	104	13	one	one	NUM
eajbcs-667	104	14	dimensional	dimensional	ADJ
eajbcs-667	104	15	advection	advection	NOUN
eajbcs-667	104	16	-	-	PUNCT
eajbcs-667	104	17	diffusion	diffusion	NOUN
eajbcs-667	104	18	equation	equation	NOUN
eajbcs-667	104	19	given	give	VERB
eajbcs-667	104	20	by	by	ADP
eajbcs-667	104	21	:	:	PUNCT
eajbcs-667	104	22	∂u	∂u	PROPN
eajbcs-667	104	23	∂t	∂t	PROPN
eajbcs-667	104	24	+	+	CCONJ
eajbcs-667	104	25	a	a	DET
eajbcs-667	104	26	∂u	∂u	PROPN
eajbcs-667	104	27	∂x	∂x	PROPN
eajbcs-667	104	28	=	=	SYM
eajbcs-667	104	29	d	d	PROPN
eajbcs-667	104	30	∂2u	∂2u	X
eajbcs-667	104	31	∂x2	∂x2	PROPN
eajbcs-667	104	32	+	+	NOUN
eajbcs-667	104	33	f	f	PROPN
eajbcs-667	104	34	,	,	PUNCT
eajbcs-667	104	35	u(0	u(0	NOUN
eajbcs-667	104	36	)	)	PUNCT
eajbcs-667	104	37	=	=	SYM
eajbcs-667	105	1	u(1	u(1	PROPN
eajbcs-667	105	2	)	)	PUNCT
eajbcs-667	105	3	=	=	SYM
eajbcs-667	105	4	0	0	NUM
eajbcs-667	105	5	,	,	PUNCT
eajbcs-667	105	6	(	(	PUNCT
eajbcs-667	105	7	4	4	X
eajbcs-667	105	8	)	)	PUNCT
eajbcs-667	105	9	where	where	SCONJ
eajbcs-667	105	10	u	u	NOUN
eajbcs-667	105	11	is	be	AUX
eajbcs-667	105	12	the	the	DET
eajbcs-667	105	13	concentration	concentration	NOUN
eajbcs-667	105	14	of	of	ADP
eajbcs-667	105	15	the	the	DET
eajbcs-667	105	16	pollutant	pollutant	NOUN
eajbcs-667	105	17	,	,	PUNCT
eajbcs-667	105	18	a	a	PRON
eajbcs-667	105	19	is	be	AUX
eajbcs-667	105	20	the	the	DET
eajbcs-667	105	21	velocity	velocity	NOUN
eajbcs-667	105	22	,	,	PUNCT
eajbcs-667	105	23	f	f	PROPN
eajbcs-667	105	24	is	be	AUX
eajbcs-667	105	25	the	the	DET
eajbcs-667	105	26	source	source	NOUN
eajbcs-667	105	27	term	term	NOUN
eajbcs-667	105	28	,	,	PUNCT
eajbcs-667	105	29	and	and	CCONJ
eajbcs-667	105	30	d	d	NOUN
eajbcs-667	105	31	is	be	AUX
eajbcs-667	105	32	the	the	DET
eajbcs-667	105	33	diffusion	diffusion	NOUN
eajbcs-667	105	34	coefficient	coefficient	NOUN
eajbcs-667	105	35	,	,	PUNCT
eajbcs-667	105	36	with	with	SCONJ
eajbcs-667	105	37	all	all	DET
eajbcs-667	105	38	the	the	DET
eajbcs-667	105	39	three	three	NUM
eajbcs-667	105	40	variables	variable	NOUN
eajbcs-667	105	41	be	be	VERB
eajbcs-667	105	42	constants	constant	NOUN
eajbcs-667	105	43	.	.	PUNCT
eajbcs-667	106	1	implementation	implementation	NOUN
eajbcs-667	106	2	of	of	ADP
eajbcs-667	106	3	the	the	DET
eajbcs-667	106	4	1d	1d	NUM
eajbcs-667	106	5	advection	advection	NOUN
eajbcs-667	106	6	equation	equation	NOUN
eajbcs-667	106	7	let	let	VERB
eajbcs-667	106	8	we	we	PRON
eajbcs-667	106	9	first	first	ADV
eajbcs-667	106	10	consider	consider	VERB
eajbcs-667	106	11	only	only	ADV
eajbcs-667	106	12	an	an	DET
eajbcs-667	106	13	advection	advection	NOUN
eajbcs-667	106	14	equation	equation	NOUN
eajbcs-667	106	15	,	,	PUNCT
eajbcs-667	106	16	that	that	PRON
eajbcs-667	106	17	is	be	AUX
eajbcs-667	106	18	the	the	DET
eajbcs-667	106	19	diffusion	diffusion	NOUN
eajbcs-667	106	20	term	term	NOUN
eajbcs-667	106	21	does	do	AUX
eajbcs-667	106	22	not	not	PART
eajbcs-667	106	23	exist	exist	VERB
eajbcs-667	106	24	;	;	PUNCT
eajbcs-667	107	1	∂u	∂u	PROPN
eajbcs-667	107	2	∂t	∂t	PROPN
eajbcs-667	107	3	+	+	CCONJ
eajbcs-667	107	4	a	a	DET
eajbcs-667	107	5	∂u	∂u	PROPN
eajbcs-667	107	6	∂x	∂x	PROPN
eajbcs-667	107	7	=	=	SYM
eajbcs-667	107	8	0	0	PROPN
eajbcs-667	107	9	.	.	PUNCT
eajbcs-667	108	1	the	the	DET
eajbcs-667	108	2	first	first	ADJ
eajbcs-667	108	3	step	step	NOUN
eajbcs-667	108	4	is	be	AUX
eajbcs-667	108	5	constructing	construct	VERB
eajbcs-667	108	6	a	a	DET
eajbcs-667	108	7	variational	variational	ADJ
eajbcs-667	108	8	or	or	CCONJ
eajbcs-667	108	9	weak	weak	ADJ
eajbcs-667	108	10	formulation	formulation	NOUN
eajbcs-667	108	11	,	,	PUNCT
eajbcs-667	108	12	by	by	ADP
eajbcs-667	108	13	multiplying	multiply	VERB
eajbcs-667	108	14	both	both	DET
eajbcs-667	108	15	sides	side	NOUN
eajbcs-667	108	16	of	of	ADP
eajbcs-667	108	17	the	the	DET
eajbcs-667	108	18	differential	differential	ADJ
eajbcs-667	108	19	equation	equation	NOUN
eajbcs-667	108	20	by	by	ADP
eajbcs-667	108	21	a	a	DET
eajbcs-667	108	22	test	test	NOUN
eajbcs-667	108	23	function	function	NOUN
eajbcs-667	108	24	v(x	v(x	NOUN
eajbcs-667	108	25	)	)	PUNCT
eajbcs-667	108	26	satisfying	satisfy	VERB
eajbcs-667	108	27	the	the	DET
eajbcs-667	108	28	boundary	boundary	ADJ
eajbcs-667	108	29	conditions	condition	NOUN
eajbcs-667	108	30	(	(	PUNCT
eajbcs-667	108	31	bc	bc	PROPN
eajbcs-667	108	32	)	)	PUNCT
eajbcs-667	108	33	v(0	v(0	PROPN
eajbcs-667	108	34	)	)	PUNCT
eajbcs-667	109	1	=	=	SYM
eajbcs-667	109	2	0	0	NUM
eajbcs-667	109	3	,	,	PUNCT
eajbcs-667	109	4	v(1	v(1	ADJ
eajbcs-667	109	5	)	)	PUNCT
eajbcs-667	109	6	=	=	SYM
eajbcs-667	109	7	0	0	NUM
eajbcs-667	109	8	and	and	CCONJ
eajbcs-667	109	9	v	v	ADP
eajbcs-667	109	10	∈	∈	PROPN
eajbcs-667	109	11	h1	h1	NOUN
eajbcs-667	109	12	0	0	NUM
eajbcs-667	109	13	(	(	PUNCT
eajbcs-667	109	14	0	0	NUM
eajbcs-667	109	15	,	,	PUNCT
eajbcs-667	109	16	1	1	NUM
eajbcs-667	109	17	)	)	PUNCT
eajbcs-667	109	18	,	,	PUNCT
eajbcs-667	109	19	where	where	SCONJ
eajbcs-667	109	20	h1	h1	PROPN
eajbcs-667	109	21	0	0	SYM
eajbcs-667	109	22	(	(	PUNCT
eajbcs-667	109	23	0	0	NUM
eajbcs-667	109	24	,	,	PUNCT
eajbcs-667	109	25	1	1	NUM
eajbcs-667	109	26	)	)	PUNCT
eajbcs-667	109	27	is	be	AUX
eajbcs-667	109	28	the	the	DET
eajbcs-667	109	29	sobolev	sobolev	ADJ
eajbcs-667	109	30	space	space	NOUN
eajbcs-667	109	31	,	,	PUNCT
eajbcs-667	109	32	h1	h1	NOUN
eajbcs-667	109	33	0	0	NUM
eajbcs-667	109	34	(	(	PUNCT
eajbcs-667	109	35	0	0	NUM
eajbcs-667	109	36	,	,	PUNCT
eajbcs-667	109	37	1	1	NUM
eajbcs-667	109	38	)	)	PUNCT
eajbcs-667	109	39	=	=	PRON
eajbcs-667	109	40	{	{	PUNCT
eajbcs-667	109	41	v	v	NUM
eajbcs-667	109	42	∈	∈	PROPN
eajbcs-667	109	43	l2(0	l2(0	NOUN
eajbcs-667	109	44	,	,	PUNCT
eajbcs-667	109	45	1	1	NUM
eajbcs-667	109	46	)	)	PUNCT
eajbcs-667	109	47	;	;	PUNCT
eajbcs-667	109	48	v′	v′	PROPN
eajbcs-667	109	49	∈	∈	PROPN
eajbcs-667	109	50	l2(0	l2(0	NOUN
eajbcs-667	109	51	,	,	PUNCT
eajbcs-667	109	52	1	1	NUM
eajbcs-667	109	53	)	)	PUNCT
eajbcs-667	109	54	}	}	PUNCT
eajbcs-667	109	55	,	,	PUNCT
eajbcs-667	109	56	and	and	CCONJ
eajbcs-667	109	57	it	it	PRON
eajbcs-667	109	58	is	be	AUX
eajbcs-667	109	59	a	a	DET
eajbcs-667	109	60	function	function	NOUN
eajbcs-667	109	61	space	space	NOUN
eajbcs-667	109	62	where	where	SCONJ
eajbcs-667	109	63	all	all	DET
eajbcs-667	109	64	the	the	DET
eajbcs-667	109	65	functions	function	NOUN
eajbcs-667	109	66	are	be	AUX
eajbcs-667	109	67	bounded	bound	VERB
eajbcs-667	109	68	.	.	PUNCT
eajbcs-667	110	1	let	let	VERB
eajbcs-667	110	2	us	we	PRON
eajbcs-667	110	3	now	now	ADV
eajbcs-667	110	4	define	define	VERB
eajbcs-667	110	5	a	a	DET
eajbcs-667	110	6	sub	sub	NOUN
eajbcs-667	110	7	-	-	NOUN
eajbcs-667	110	8	space	space	NOUN
eajbcs-667	110	9	of	of	ADP
eajbcs-667	110	10	h	h	NOUN
eajbcs-667	110	11	where	where	SCONJ
eajbcs-667	110	12	we	we	PRON
eajbcs-667	110	13	can	can	AUX
eajbcs-667	110	14	find	find	VERB
eajbcs-667	110	15	our	our	PRON
eajbcs-667	110	16	solution	solution	NOUN
eajbcs-667	110	17	u.	u.	VERB
eajbcs-667	111	1	we	we	PRON
eajbcs-667	111	2	call	call	VERB
eajbcs-667	111	3	this	this	DET
eajbcs-667	111	4	v	v	NOUN
eajbcs-667	111	5	and	and	CCONJ
eajbcs-667	111	6	v	v	NOUN
eajbcs-667	111	7	=	=	PUNCT
eajbcs-667	111	8	{	{	PUNCT
eajbcs-667	111	9	v	v	NUM
eajbcs-667	111	10	∈	∈	PROPN
eajbcs-667	111	11	h(x	h(x	PROPN
eajbcs-667	111	12	)	)	PUNCT
eajbcs-667	111	13	:	:	PUNCT
eajbcs-667	112	1	v|∂ω	v|∂ω	NOUN
eajbcs-667	112	2	=	=	SYM
eajbcs-667	112	3	0	0	NUM
eajbcs-667	112	4	}	}	PUNCT
eajbcs-667	112	5	,	,	PUNCT
eajbcs-667	112	6	where	where	SCONJ
eajbcs-667	112	7	ω	ω	PROPN
eajbcs-667	112	8	is	be	AUX
eajbcs-667	112	9	our	our	PRON
eajbcs-667	112	10	domain	domain	NOUN
eajbcs-667	112	11	,	,	PUNCT
eajbcs-667	112	12	then	then	ADV
eajbcs-667	112	13	multiplying	multiply	VERB
eajbcs-667	112	14	and	and	CCONJ
eajbcs-667	112	15	integrating	integrate	VERB
eajbcs-667	112	16	both	both	DET
eajbcs-667	112	17	sides	side	NOUN
eajbcs-667	112	18	in	in	ADP
eajbcs-667	112	19	the	the	DET
eajbcs-667	112	20	domain	domain	NOUN
eajbcs-667	112	21	we	we	PRON
eajbcs-667	112	22	have	have	VERB
eajbcs-667	112	23	that	that	PRON
eajbcs-667	112	24	:	:	PUNCT
eajbcs-667	112	25	∂u	∂u	PROPN
eajbcs-667	112	26	∂t	∂t	PROPN
eajbcs-667	112	27	.v	.v	PROPN
eajbcs-667	113	1	+	+	CCONJ
eajbcs-667	113	2	a	a	DET
eajbcs-667	113	3	∂u	∂u	PROPN
eajbcs-667	113	4	∂x	∂x	PROPN
eajbcs-667	113	5	.v	.v	PUNCT
eajbcs-667	114	1	=	=	PUNCT
eajbcs-667	114	2	0	0	NUM
eajbcs-667	114	3	,	,	PUNCT
eajbcs-667	114	4	∫	∫	PROPN
eajbcs-667	114	5	1	1	NUM
eajbcs-667	114	6	0	0	NUM
eajbcs-667	114	7	(	(	PUNCT
eajbcs-667	114	8	∂u	∂u	PROPN
eajbcs-667	114	9	∂t	∂t	PROPN
eajbcs-667	114	10	.v	.v	PROPN
eajbcs-667	115	1	+	+	CCONJ
eajbcs-667	115	2	a	a	DET
eajbcs-667	115	3	∂u	∂u	PROPN
eajbcs-667	115	4	∂x	∂x	PROPN
eajbcs-667	115	5	.v	.v	NOUN
eajbcs-667	115	6	)	)	PUNCT
eajbcs-667	116	1	=	=	PUNCT
eajbcs-667	116	2	0	0	X
eajbcs-667	116	3	.	.	PUNCT
eajbcs-667	116	4	east	east	PROPN
eajbcs-667	116	5	afr	afr	PROPN
eajbcs-667	116	6	.	.	PUNCT
eajbcs-667	117	1	j.	j.	PROPN
eajbcs-667	117	2	biophys	biophys	PROPN
eajbcs-667	117	3	.	.	PUNCT
eajbcs-667	118	1	comput	comput	NOUN
eajbcs-667	118	2	.	.	PUNCT
eajbcs-667	119	1	sci	sci	PROPN
eajbcs-667	119	2	.	.	PUNCT
eajbcs-667	119	3	(	(	PUNCT
eajbcs-667	119	4	2023	2023	NUM
eajbcs-667	119	5	)	)	PUNCT
eajbcs-667	119	6	,	,	PUNCT
eajbcs-667	119	7	vol	vol	NOUN
eajbcs-667	119	8	.	.	PROPN
eajbcs-667	119	9	4	4	NUM
eajbcs-667	119	10	,	,	PUNCT
eajbcs-667	119	11	no	no	INTJ
eajbcs-667	119	12	.	.	NOUN
eajbcs-667	119	13	1	1	NUM
eajbcs-667	119	14	,	,	PUNCT
eajbcs-667	119	15	52	52	NUM
eajbcs-667	119	16	-	-	SYM
eajbcs-667	119	17	74	74	NUM
eajbcs-667	119	18	56	56	NUM
eajbcs-667	119	19	the	the	DET
eajbcs-667	119	20	fem	fem	NOUN
eajbcs-667	119	21	is	be	AUX
eajbcs-667	119	22	based	base	VERB
eajbcs-667	119	23	on	on	ADP
eajbcs-667	119	24	the	the	DET
eajbcs-667	119	25	integration	integration	NOUN
eajbcs-667	119	26	of	of	ADP
eajbcs-667	119	27	the	the	DET
eajbcs-667	119	28	terms	term	NOUN
eajbcs-667	119	29	in	in	ADP
eajbcs-667	119	30	the	the	DET
eajbcs-667	119	31	equation	equation	NOUN
eajbcs-667	119	32	to	to	PART
eajbcs-667	119	33	be	be	AUX
eajbcs-667	119	34	solved	solve	VERB
eajbcs-667	119	35	,	,	PUNCT
eajbcs-667	119	36	in	in	ADP
eajbcs-667	119	37	form	form	NOUN
eajbcs-667	119	38	of	of	ADP
eajbcs-667	119	39	point	point	NOUN
eajbcs-667	119	40	discretization	discretization	NOUN
eajbcs-667	119	41	schemes	scheme	NOUN
eajbcs-667	119	42	.	.	PUNCT
eajbcs-667	120	1	it	it	PRON
eajbcs-667	120	2	utilizes	utilize	VERB
eajbcs-667	120	3	the	the	DET
eajbcs-667	120	4	method	method	NOUN
eajbcs-667	120	5	of	of	ADP
eajbcs-667	120	6	weighted	weight	VERB
eajbcs-667	120	7	residuals	residual	NOUN
eajbcs-667	120	8	and	and	CCONJ
eajbcs-667	120	9	integration	integration	NOUN
eajbcs-667	120	10	by	by	ADP
eajbcs-667	120	11	parts	part	NOUN
eajbcs-667	120	12	(	(	PUNCT
eajbcs-667	120	13	green	green	ADJ
eajbcs-667	120	14	-	-	PUNCT
eajbcs-667	120	15	gauss	gauss	NOUN
eajbcs-667	120	16	theorem	theorem	NOUN
eajbcs-667	120	17	)	)	PUNCT
eajbcs-667	120	18	to	to	PART
eajbcs-667	120	19	reduce	reduce	VERB
eajbcs-667	120	20	second	second	ADJ
eajbcs-667	120	21	order	order	NOUN
eajbcs-667	120	22	derivatives	derivative	NOUN
eajbcs-667	120	23	to	to	ADP
eajbcs-667	120	24	first	first	ADJ
eajbcs-667	120	25	order	order	NOUN
eajbcs-667	120	26	terms	term	NOUN
eajbcs-667	120	27	.	.	PUNCT
eajbcs-667	121	1	the	the	DET
eajbcs-667	121	2	solution	solution	NOUN
eajbcs-667	121	3	domain	domain	NOUN
eajbcs-667	121	4	is	be	AUX
eajbcs-667	121	5	discretized	discretize	VERB
eajbcs-667	121	6	into	into	ADP
eajbcs-667	121	7	individual	individual	ADJ
eajbcs-667	121	8	elements	element	NOUN
eajbcs-667	121	9	and	and	CCONJ
eajbcs-667	121	10	these	these	DET
eajbcs-667	121	11	elements	element	NOUN
eajbcs-667	121	12	are	be	AUX
eajbcs-667	121	13	operated	operate	VERB
eajbcs-667	121	14	upon	upon	SCONJ
eajbcs-667	121	15	individually	individually	ADV
eajbcs-667	121	16	and	and	CCONJ
eajbcs-667	121	17	then	then	ADV
eajbcs-667	121	18	solved	solve	VERB
eajbcs-667	121	19	globally	globally	ADV
eajbcs-667	121	20	using	use	VERB
eajbcs-667	121	21	matrix	matrix	NOUN
eajbcs-667	121	22	solution	solution	NOUN
eajbcs-667	121	23	techniques	technique	NOUN
eajbcs-667	121	24	.	.	PUNCT
eajbcs-667	122	1	such	such	DET
eajbcs-667	122	2	a	a	DET
eajbcs-667	122	3	task	task	NOUN
eajbcs-667	122	4	could	could	AUX
eajbcs-667	122	5	be	be	AUX
eajbcs-667	122	6	done	do	VERB
eajbcs-667	122	7	automatically	automatically	ADV
eajbcs-667	122	8	by	by	ADP
eajbcs-667	122	9	a	a	DET
eajbcs-667	122	10	computer	computer	NOUN
eajbcs-667	122	11	,	,	PUNCT
eajbcs-667	122	12	but	but	CCONJ
eajbcs-667	122	13	it	it	PRON
eajbcs-667	122	14	necessitates	necessitate	VERB
eajbcs-667	122	15	an	an	DET
eajbcs-667	122	16	amount	amount	NOUN
eajbcs-667	122	17	of	of	ADP
eajbcs-667	122	18	mathematical	mathematical	ADJ
eajbcs-667	122	19	skill	skill	NOUN
eajbcs-667	122	20	that	that	PRON
eajbcs-667	122	21	to	to	ADP
eajbcs-667	122	22	day	day	NOUN
eajbcs-667	122	23	still	still	ADV
eajbcs-667	122	24	requires	require	VERB
eajbcs-667	122	25	human	human	ADJ
eajbcs-667	122	26	involvement	involvement	NOUN
eajbcs-667	122	27	,	,	PUNCT
eajbcs-667	122	28	(	(	PUNCT
eajbcs-667	122	29	brenner	brenner	PROPN
eajbcs-667	122	30	et	et	PROPN
eajbcs-667	122	31	al	al	PROPN
eajbcs-667	122	32	.	.	PROPN
eajbcs-667	122	33	,	,	PUNCT
eajbcs-667	122	34	2008	2008	NUM
eajbcs-667	122	35	)	)	PUNCT
eajbcs-667	122	36	.	.	PUNCT
eajbcs-667	123	1	the	the	DET
eajbcs-667	123	2	theories	theory	NOUN
eajbcs-667	123	3	of	of	ADP
eajbcs-667	123	4	finite	finite	ADJ
eajbcs-667	123	5	element	element	NOUN
eajbcs-667	123	6	methods	method	NOUN
eajbcs-667	123	7	provided	provide	VERB
eajbcs-667	123	8	the	the	DET
eajbcs-667	123	9	reasons	reason	NOUN
eajbcs-667	123	10	why	why	SCONJ
eajbcs-667	123	11	it	it	PRON
eajbcs-667	123	12	worked	work	VERB
eajbcs-667	123	13	well	well	ADV
eajbcs-667	123	14	for	for	ADP
eajbcs-667	123	15	the	the	DET
eajbcs-667	123	16	class	class	NOUN
eajbcs-667	123	17	boundary	boundary	NOUN
eajbcs-667	123	18	/	/	SYM
eajbcs-667	123	19	initial	initial	ADJ
eajbcs-667	123	20	value	value	NOUN
eajbcs-667	123	21	problems	problem	NOUN
eajbcs-667	123	22	(	(	PUNCT
eajbcs-667	123	23	lima	lima	NOUN
eajbcs-667	123	24	et	et	PROPN
eajbcs-667	123	25	al	al	PROPN
eajbcs-667	123	26	.	.	PROPN
eajbcs-667	123	27	,	,	PUNCT
eajbcs-667	123	28	2021	2021	NUM
eajbcs-667	123	29	;	;	PUNCT
eajbcs-667	123	30	ahsan	ahsan	ADJ
eajbcs-667	123	31	,	,	PUNCT
eajbcs-667	123	32	2012	2012	NUM
eajbcs-667	123	33	;	;	PUNCT
eajbcs-667	123	34	larson	larson	PROPN
eajbcs-667	123	35	and	and	CCONJ
eajbcs-667	123	36	bengzon	bengzon	PROPN
eajbcs-667	123	37	,	,	PUNCT
eajbcs-667	123	38	2010	2010	NUM
eajbcs-667	123	39	)	)	PUNCT
eajbcs-667	123	40	.	.	PUNCT
eajbcs-667	124	1	extension	extension	NOUN
eajbcs-667	124	2	of	of	ADP
eajbcs-667	124	3	the	the	DET
eajbcs-667	124	4	mathematical	mathematical	ADJ
eajbcs-667	124	5	basis	basis	NOUN
eajbcs-667	124	6	to	to	ADP
eajbcs-667	124	7	nonlinear	nonlinear	ADJ
eajbcs-667	124	8	and	and	CCONJ
eajbcs-667	124	9	non	non	ADJ
eajbcs-667	124	10	-	-	ADJ
eajbcs-667	124	11	structural	structural	ADJ
eajbcs-667	124	12	problems	problem	NOUN
eajbcs-667	124	13	was	be	AUX
eajbcs-667	124	14	achieved	achieve	VERB
eajbcs-667	124	15	through	through	ADP
eajbcs-667	124	16	the	the	DET
eajbcs-667	124	17	method	method	NOUN
eajbcs-667	124	18	of	of	ADP
eajbcs-667	124	19	weighted	weight	VERB
eajbcs-667	124	20	residuals	residual	NOUN
eajbcs-667	124	21	(	(	PUNCT
eajbcs-667	124	22	mwr	mwr	PROPN
eajbcs-667	124	23	)	)	PUNCT
eajbcs-667	124	24	,	,	PUNCT
eajbcs-667	124	25	originally	originally	ADV
eajbcs-667	124	26	conceived	conceive	VERB
eajbcs-667	124	27	by	by	ADP
eajbcs-667	124	28	galerkin	galerkin	PROPN
eajbcs-667	124	29	in	in	ADP
eajbcs-667	124	30	the	the	DET
eajbcs-667	124	31	early	early	ADJ
eajbcs-667	124	32	20th	20th	ADJ
eajbcs-667	124	33	century	century	NOUN
eajbcs-667	124	34	.	.	PUNCT
eajbcs-667	125	1	the	the	DET
eajbcs-667	125	2	basics	basic	NOUN
eajbcs-667	125	3	of	of	ADP
eajbcs-667	125	4	the	the	DET
eajbcs-667	125	5	method	method	NOUN
eajbcs-667	125	6	requires	require	VERB
eajbcs-667	125	7	multiplying	multiply	VERB
eajbcs-667	125	8	of	of	ADP
eajbcs-667	125	9	the	the	DET
eajbcs-667	125	10	governing	govern	VERB
eajbcs-667	125	11	differential	differential	ADJ
eajbcs-667	125	12	equation	equation	NOUN
eajbcs-667	125	13	by	by	ADP
eajbcs-667	125	14	a	a	DET
eajbcs-667	125	15	set	set	NOUN
eajbcs-667	125	16	of	of	ADP
eajbcs-667	125	17	predetermined	predetermine	VERB
eajbcs-667	125	18	weights	weight	NOUN
eajbcs-667	125	19	and	and	CCONJ
eajbcs-667	125	20	integrating	integrate	VERB
eajbcs-667	125	21	the	the	DET
eajbcs-667	125	22	resulting	result	VERB
eajbcs-667	125	23	product	product	NOUN
eajbcs-667	125	24	over	over	ADP
eajbcs-667	125	25	a	a	DET
eajbcs-667	125	26	region	region	NOUN
eajbcs-667	125	27	.	.	PUNCT
eajbcs-667	126	1	most	most	ADJ
eajbcs-667	126	2	of	of	ADP
eajbcs-667	126	3	the	the	DET
eajbcs-667	126	4	finite	finite	PROPN
eajbcs-667	126	5	element	element	NOUN
eajbcs-667	126	6	method	method	NOUN
eajbcs-667	126	7	uses	use	VERB
eajbcs-667	126	8	the	the	DET
eajbcs-667	126	9	galerkin	galerkin	PROPN
eajbcs-667	126	10	’s	’s	PART
eajbcs-667	126	11	method	method	NOUN
eajbcs-667	126	12	to	to	PART
eajbcs-667	126	13	establish	establish	VERB
eajbcs-667	126	14	the	the	DET
eajbcs-667	126	15	approximations	approximation	NOUN
eajbcs-667	126	16	of	of	ADP
eajbcs-667	126	17	the	the	DET
eajbcs-667	126	18	governing	govern	VERB
eajbcs-667	126	19	equations	equation	NOUN
eajbcs-667	126	20	,	,	PUNCT
eajbcs-667	126	21	(	(	PUNCT
eajbcs-667	126	22	aragonés	aragoné	NOUN
eajbcs-667	126	23	et	et	NOUN
eajbcs-667	126	24	al	al	PROPN
eajbcs-667	126	25	.	.	PROPN
eajbcs-667	126	26	,	,	PUNCT
eajbcs-667	126	27	2019	2019	NUM
eajbcs-667	126	28	;	;	PUNCT
eajbcs-667	126	29	lima	lima	PROPN
eajbcs-667	126	30	et	et	PROPN
eajbcs-667	126	31	al	al	PROPN
eajbcs-667	126	32	.	.	PROPN
eajbcs-667	126	33	,	,	PUNCT
eajbcs-667	126	34	2021	2021	NUM
eajbcs-667	126	35	;	;	PUNCT
eajbcs-667	126	36	ahsan	ahsan	PROPN
eajbcs-667	126	37	,	,	PUNCT
eajbcs-667	126	38	2012	2012	NUM
eajbcs-667	126	39	;	;	PUNCT
eajbcs-667	126	40	yang	yang	PROPN
eajbcs-667	126	41	et	et	PROPN
eajbcs-667	126	42	al	al	PROPN
eajbcs-667	126	43	.	.	PROPN
eajbcs-667	126	44	,	,	PUNCT
eajbcs-667	126	45	2020	2020	NUM
eajbcs-667	126	46	;	;	PUNCT
eajbcs-667	126	47	brenner	brenner	PROPN
eajbcs-667	126	48	et	et	PROPN
eajbcs-667	126	49	al	al	PROPN
eajbcs-667	126	50	.	.	PROPN
eajbcs-667	126	51	,	,	PUNCT
eajbcs-667	126	52	2008	2008	NUM
eajbcs-667	126	53	)	)	PUNCT
eajbcs-667	126	54	.	.	PUNCT
eajbcs-667	127	1	it	it	PRON
eajbcs-667	127	2	allows	allow	VERB
eajbcs-667	127	3	us	we	PRON
eajbcs-667	127	4	to	to	PART
eajbcs-667	127	5	convert	convert	VERB
eajbcs-667	127	6	a	a	DET
eajbcs-667	127	7	continuous	continuous	ADJ
eajbcs-667	127	8	form	form	NOUN
eajbcs-667	127	9	of	of	ADP
eajbcs-667	127	10	the	the	DET
eajbcs-667	127	11	problem	problem	NOUN
eajbcs-667	127	12	,	,	PUNCT
eajbcs-667	127	13	such	such	ADJ
eajbcs-667	127	14	as	as	ADP
eajbcs-667	127	15	the	the	DET
eajbcs-667	127	16	weak	weak	ADJ
eajbcs-667	127	17	formulation	formulation	NOUN
eajbcs-667	127	18	for	for	ADP
eajbcs-667	127	19	the	the	DET
eajbcs-667	127	20	partial	partial	ADJ
eajbcs-667	127	21	differential	differential	ADJ
eajbcs-667	127	22	equation	equation	NOUN
eajbcs-667	127	23	into	into	ADP
eajbcs-667	127	24	a	a	DET
eajbcs-667	127	25	discrete	discrete	ADJ
eajbcs-667	127	26	problem	problem	NOUN
eajbcs-667	127	27	that	that	PRON
eajbcs-667	127	28	may	may	AUX
eajbcs-667	127	29	be	be	AUX
eajbcs-667	127	30	solved	solve	VERB
eajbcs-667	127	31	numerically	numerically	ADV
eajbcs-667	127	32	.	.	PUNCT
eajbcs-667	128	1	that	that	SCONJ
eajbcs-667	128	2	is;∫	is;∫	PRON
eajbcs-667	128	3	1	1	NUM
eajbcs-667	128	4	0	0	NUM
eajbcs-667	128	5	∂u	∂u	PROPN
eajbcs-667	128	6	∂t	∂t	PROPN
eajbcs-667	128	7	.v	.v	PROPN
eajbcs-667	129	1	+	+	CCONJ
eajbcs-667	129	2	∫	∫	PROPN
eajbcs-667	129	3	1	1	NUM
eajbcs-667	129	4	0	0	NUM
eajbcs-667	129	5	a	a	DET
eajbcs-667	129	6	∂u	∂u	PROPN
eajbcs-667	129	7	∂x	∂x	PROPN
eajbcs-667	129	8	.v	.v	PUNCT
eajbcs-667	130	1	=	=	PUNCT
eajbcs-667	130	2	0	0	NUM
eajbcs-667	130	3	,	,	PUNCT
eajbcs-667	130	4	(	(	PUNCT
eajbcs-667	130	5	5	5	NUM
eajbcs-667	130	6	)	)	PUNCT
eajbcs-667	130	7	which	which	PRON
eajbcs-667	130	8	is	be	AUX
eajbcs-667	130	9	the	the	DET
eajbcs-667	130	10	weak	weak	ADJ
eajbcs-667	130	11	formulation	formulation	NOUN
eajbcs-667	130	12	of	of	ADP
eajbcs-667	130	13	the	the	DET
eajbcs-667	130	14	one	one	NUM
eajbcs-667	130	15	dimensional	dimensional	ADJ
eajbcs-667	130	16	advection	advection	NOUN
eajbcs-667	130	17	equation	equation	NOUN
eajbcs-667	130	18	.	.	PUNCT
eajbcs-667	131	1	advantages	advantage	NOUN
eajbcs-667	131	2	of	of	ADP
eajbcs-667	131	3	weak	weak	ADJ
eajbcs-667	131	4	form	form	NOUN
eajbcs-667	131	5	compared	compare	VERB
eajbcs-667	131	6	to	to	ADP
eajbcs-667	131	7	strong	strong	ADJ
eajbcs-667	131	8	form	form	NOUN
eajbcs-667	131	9	equation	equation	NOUN
eajbcs-667	131	10	5	5	NUM
eajbcs-667	131	11	is	be	AUX
eajbcs-667	131	12	the	the	DET
eajbcs-667	131	13	final	final	ADJ
eajbcs-667	131	14	weak	weak	ADJ
eajbcs-667	131	15	formulation	formulation	NOUN
eajbcs-667	131	16	.	.	PUNCT
eajbcs-667	132	1	it	it	PRON
eajbcs-667	132	2	is	be	AUX
eajbcs-667	132	3	equivalent	equivalent	ADJ
eajbcs-667	132	4	to	to	ADP
eajbcs-667	132	5	the	the	DET
eajbcs-667	132	6	strong	strong	ADJ
eajbcs-667	132	7	form	form	NOUN
eajbcs-667	132	8	,	,	PUNCT
eajbcs-667	132	9	since	since	SCONJ
eajbcs-667	132	10	we	we	PRON
eajbcs-667	132	11	can	can	AUX
eajbcs-667	132	12	reverse	reverse	VERB
eajbcs-667	132	13	all	all	DET
eajbcs-667	132	14	the	the	DET
eajbcs-667	132	15	steps	step	NOUN
eajbcs-667	132	16	,	,	PUNCT
eajbcs-667	132	17	and	and	CCONJ
eajbcs-667	132	18	get	get	VERB
eajbcs-667	132	19	back	back	ADV
eajbcs-667	132	20	to	to	ADP
eajbcs-667	132	21	the	the	DET
eajbcs-667	132	22	original	original	ADJ
eajbcs-667	132	23	equation	equation	NOUN
eajbcs-667	132	24	.	.	PUNCT
eajbcs-667	133	1	firstly	firstly	ADV
eajbcs-667	133	2	,	,	PUNCT
eajbcs-667	133	3	if	if	SCONJ
eajbcs-667	133	4	we	we	PRON
eajbcs-667	133	5	look	look	VERB
eajbcs-667	133	6	at	at	ADP
eajbcs-667	133	7	the	the	DET
eajbcs-667	133	8	strong	strong	ADJ
eajbcs-667	133	9	form	form	NOUN
eajbcs-667	133	10	,	,	PUNCT
eajbcs-667	133	11	we	we	PRON
eajbcs-667	133	12	have	have	VERB
eajbcs-667	133	13	two	two	NUM
eajbcs-667	133	14	separate	separate	ADJ
eajbcs-667	133	15	partial	partial	ADJ
eajbcs-667	133	16	derivatives	derivative	NOUN
eajbcs-667	133	17	of	of	ADP
eajbcs-667	133	18	u	u	NOUN
eajbcs-667	133	19	,	,	PUNCT
eajbcs-667	133	20	so	so	ADV
eajbcs-667	133	21	the	the	DET
eajbcs-667	133	22	strong	strong	ADJ
eajbcs-667	133	23	form	form	NOUN
eajbcs-667	133	24	requires	require	VERB
eajbcs-667	133	25	that	that	SCONJ
eajbcs-667	133	26	u	u	PRON
eajbcs-667	133	27	be	be	AUX
eajbcs-667	133	28	continuously	continuously	ADV
eajbcs-667	133	29	differentiable	differentiable	ADJ
eajbcs-667	133	30	until	until	ADP
eajbcs-667	133	31	at	at	ADP
eajbcs-667	133	32	least	least	ADJ
eajbcs-667	133	33	second	second	ADJ
eajbcs-667	133	34	partial	partial	ADJ
eajbcs-667	133	35	derivative	derivative	NOUN
eajbcs-667	133	36	.	.	PUNCT
eajbcs-667	134	1	our	our	PRON
eajbcs-667	134	2	new	new	ADJ
eajbcs-667	134	3	formulations	formulation	NOUN
eajbcs-667	134	4	has	have	AUX
eajbcs-667	134	5	lowered	lower	VERB
eajbcs-667	134	6	this	this	DET
eajbcs-667	134	7	requirement	requirement	NOUN
eajbcs-667	134	8	to	to	ADP
eajbcs-667	134	9	only	only	ADV
eajbcs-667	134	10	first	first	ADJ
eajbcs-667	134	11	partial	partial	ADJ
eajbcs-667	134	12	derivatives	derivative	NOUN
eajbcs-667	134	13	by	by	ADP
eajbcs-667	134	14	transforming	transform	VERB
eajbcs-667	134	15	one	one	NUM
eajbcs-667	134	16	of	of	ADP
eajbcs-667	134	17	the	the	DET
eajbcs-667	134	18	partial	partial	ADJ
eajbcs-667	134	19	derivatives	derivative	NOUN
eajbcs-667	134	20	onto	onto	ADP
eajbcs-667	134	21	the	the	DET
eajbcs-667	134	22	weightfunction	weightfunction	NOUN
eajbcs-667	134	23	v(x	v(x	PROPN
eajbcs-667	134	24	,	,	PUNCT
eajbcs-667	134	25	y	y	PROPN
eajbcs-667	134	26	)	)	PUNCT
eajbcs-667	134	27	.	.	PUNCT
eajbcs-667	135	1	this	this	PRON
eajbcs-667	135	2	is	be	AUX
eajbcs-667	135	3	the	the	DET
eajbcs-667	135	4	first	first	ADJ
eajbcs-667	135	5	big	big	ADJ
eajbcs-667	135	6	advantage	advantage	NOUN
eajbcs-667	135	7	of	of	ADP
eajbcs-667	135	8	a	a	DET
eajbcs-667	135	9	weak	weak	ADJ
eajbcs-667	135	10	formulation	formulation	NOUN
eajbcs-667	135	11	.	.	PUNCT
eajbcs-667	136	1	the	the	DET
eajbcs-667	136	2	subspace	subspace	NOUN
eajbcs-667	136	3	v	v	NOUN
eajbcs-667	136	4	is	be	AUX
eajbcs-667	136	5	not	not	PART
eajbcs-667	136	6	difficult	difficult	ADJ
eajbcs-667	136	7	to	to	PART
eajbcs-667	136	8	understand	understand	VERB
eajbcs-667	136	9	;	;	PUNCT
eajbcs-667	136	10	it	it	PRON
eajbcs-667	136	11	is	be	AUX
eajbcs-667	136	12	a	a	DET
eajbcs-667	136	13	subspace	subspace	NOUN
eajbcs-667	136	14	of	of	ADP
eajbcs-667	136	15	h	h	NOUN
eajbcs-667	136	16	because	because	SCONJ
eajbcs-667	136	17	our	our	PRON
eajbcs-667	136	18	weak	weak	ADJ
eajbcs-667	136	19	form	form	NOUN
eajbcs-667	136	20	requires	require	VERB
eajbcs-667	136	21	that	that	SCONJ
eajbcs-667	136	22	the	the	DET
eajbcs-667	136	23	functions	function	NOUN
eajbcs-667	136	24	are	be	AUX
eajbcs-667	136	25	in	in	ADP
eajbcs-667	136	26	h	h	NOUN
eajbcs-667	136	27	;	;	PUNCT
eajbcs-667	136	28	our	our	PRON
eajbcs-667	136	29	strong	strong	ADJ
eajbcs-667	136	30	form	form	NOUN
eajbcs-667	136	31	requires	require	VERB
eajbcs-667	136	32	that	that	SCONJ
eajbcs-667	136	33	u	u	PRON
eajbcs-667	136	34	be	be	VERB
eajbcs-667	136	35	0	0	NUM
eajbcs-667	136	36	along	along	ADP
eajbcs-667	136	37	the	the	DET
eajbcs-667	136	38	boundary	boundary	NOUN
eajbcs-667	136	39	,	,	PUNCT
eajbcs-667	136	40	so	so	CCONJ
eajbcs-667	136	41	v	v	NOUN
eajbcs-667	136	42	is	be	AUX
eajbcs-667	136	43	the	the	DET
eajbcs-667	136	44	subspace	subspace	NOUN
eajbcs-667	136	45	of	of	ADP
eajbcs-667	136	46	all	all	DET
eajbcs-667	136	47	function	function	NOUN
eajbcs-667	136	48	which	which	PRON
eajbcs-667	136	49	are	be	AUX
eajbcs-667	136	50	zero	zero	NUM
eajbcs-667	136	51	on	on	ADP
eajbcs-667	136	52	the	the	DET
eajbcs-667	136	53	boundary	boundary	NOUN
eajbcs-667	136	54	.	.	PUNCT
eajbcs-667	137	1	ih	ih	INTJ
eajbcs-667	137	2	,	,	PUNCT
eajbcs-667	137	3	i	i	PRON
eajbcs-667	137	4	=	=	NOUN
eajbcs-667	137	5	0	0	NUM
eajbcs-667	137	6	,	,	PUNCT
eajbcs-667	137	7	1	1	NUM
eajbcs-667	137	8	,	,	PUNCT
eajbcs-667	137	9	...	...	PUNCT
eajbcs-667	137	10	,	,	PUNCT
eajbcs-667	137	11	n	n	CCONJ
eajbcs-667	137	12	,	,	PUNCT
eajbcs-667	137	13	where	where	SCONJ
eajbcs-667	137	14	h	h	NOUN
eajbcs-667	137	15	=	=	SYM
eajbcs-667	137	16	1	1	NUM
eajbcs-667	137	17	n	n	NOUN
eajbcs-667	137	18	,	,	PUNCT
eajbcs-667	137	19	and	and	CCONJ
eajbcs-667	137	20	ϕi(xj	ϕi(xj	ADV
eajbcs-667	137	21	)	)	PUNCT
eajbcs-667	137	22	=	=	SYM
eajbcs-667	137	23	δij	δij	NOUN
eajbcs-667	137	24	,	,	PUNCT
eajbcs-667	137	25	i	i	PRON
eajbcs-667	137	26	,	,	PUNCT
eajbcs-667	137	27	j	j	PROPN
eajbcs-667	137	28	=	=	SYM
eajbcs-667	137	29	1	1	NUM
eajbcs-667	137	30	,	,	PUNCT
eajbcs-667	137	31	...	...	PUNCT
eajbcs-667	137	32	,	,	PUNCT
eajbcs-667	137	33	n−	n−	NOUN
eajbcs-667	137	34	1	1	NUM
eajbcs-667	137	35	,	,	PUNCT
eajbcs-667	137	36	(	(	PUNCT
eajbcs-667	137	37	6	6	NUM
eajbcs-667	137	38	)	)	PUNCT
eajbcs-667	137	39	where	where	SCONJ
eajbcs-667	137	40	δij	δij	NOUN
eajbcs-667	137	41	being	be	AUX
eajbcs-667	137	42	the	the	DET
eajbcs-667	137	43	kronecker	kronecker	NOUN
eajbcs-667	137	44	delta	delta	NOUN
eajbcs-667	137	45	.	.	PUNCT
eajbcs-667	138	1	the	the	DET
eajbcs-667	138	2	function	function	NOUN
eajbcs-667	138	3	ϕi	ϕi	ADP
eajbcs-667	138	4	is	be	AUX
eajbcs-667	138	5	therefore	therefore	ADV
eajbcs-667	138	6	piece	piece	NOUN
eajbcs-667	138	7	wise	wise	ADJ
eajbcs-667	138	8	linear	linear	ADV
eajbcs-667	138	9	and	and	CCONJ
eajbcs-667	138	10	are	be	AUX
eajbcs-667	138	11	fix	fix	VERB
eajbcs-667	138	12	with	with	ADP
eajbcs-667	138	13	one	one	NUM
eajbcs-667	138	14	node	node	NOUN
eajbcs-667	138	15	(	(	PUNCT
eajbcs-667	138	16	vertex	vertex	NOUN
eajbcs-667	138	17	)	)	PUNCT
eajbcs-667	138	18	and	and	CCONJ
eajbcs-667	138	19	figure	figure	VERB
eajbcs-667	138	20	1	1	NUM
eajbcs-667	138	21	:	:	PUNCT
eajbcs-667	138	22	the	the	DET
eajbcs-667	138	23	basis	basis	NOUN
eajbcs-667	138	24	(	(	PUNCT
eajbcs-667	138	25	hat	hat	NOUN
eajbcs-667	138	26	)	)	PUNCT
eajbcs-667	138	27	function	function	NOUN
eajbcs-667	138	28	ϕi	ϕi	ADP
eajbcs-667	138	29	associated	associate	VERB
eajbcs-667	138	30	to	to	PART
eajbcs-667	138	31	node	node	VERB
eajbcs-667	138	32	xj	xj	PROPN
eajbcs-667	138	33	,	,	PUNCT
eajbcs-667	138	34	in	in	ADP
eajbcs-667	138	35	this	this	DET
eajbcs-667	138	36	figure	figure	NOUN
eajbcs-667	138	37	φi	φi	ADV
eajbcs-667	138	38	on	on	ADP
eajbcs-667	138	39	a	a	DET
eajbcs-667	138	40	mesh	mesh	NOUN
eajbcs-667	138	41	.	.	PUNCT
eajbcs-667	139	1	also	also	ADV
eajbcs-667	139	2	shown	show	VERB
eajbcs-667	139	3	is	be	AUX
eajbcs-667	139	4	the	the	DET
eajbcs-667	139	5	half	half	ADJ
eajbcs-667	139	6	hat	hat	NOUN
eajbcs-667	139	7	ϕ0	ϕ0	NOUN
eajbcs-667	139	8	.	.	PUNCT
eajbcs-667	140	1	its	its	PRON
eajbcs-667	140	2	expression	expression	NOUN
eajbcs-667	140	3	is	be	AUX
eajbcs-667	140	4	given	give	VERB
eajbcs-667	140	5	by	by	ADP
eajbcs-667	140	6	ϕi(x	ϕi(x	PROPN
eajbcs-667	140	7	)	)	PUNCT
eajbcs-667	140	8	=	=	PUNCT
eajbcs-667	141	1			PROPN
eajbcs-667	141	2	x−xi−1	x−xi−1	PROPN
eajbcs-667	141	3	xi−xi−1	xi−xi−1	PROPN
eajbcs-667	141	4	,	,	PUNCT
eajbcs-667	141	5	if	if	SCONJ
eajbcs-667	141	6	xi−1	xi−1	PROPN
eajbcs-667	141	7	≤	≤	ADV
eajbcs-667	141	8	x	x	PUNCT
eajbcs-667	141	9	≤	≤	NUM
eajbcs-667	141	10	xi	xi	X
eajbcs-667	141	11	xi+1−x	xi+1−x	PUNCT
eajbcs-667	142	1	xi+1−xi	xi+1−xi	PROPN
eajbcs-667	142	2	,	,	PUNCT
eajbcs-667	142	3	if	if	SCONJ
eajbcs-667	142	4	xi	xi	ADP
eajbcs-667	142	5	≤	≤	NUM
eajbcs-667	142	6	x	x	X
eajbcs-667	142	7	≤	≤	NUM
eajbcs-667	142	8	xi+1	xi+1	PROPN
eajbcs-667	142	9	0	0	NUM
eajbcs-667	142	10	,	,	PUNCT
eajbcs-667	142	11	other	other	ADJ
eajbcs-667	142	12	wise	wise	ADJ
eajbcs-667	142	13	,	,	PUNCT
eajbcs-667	142	14	for	for	ADP
eajbcs-667	142	15	i	i	PROPN
eajbcs-667	142	16	=	=	SYM
eajbcs-667	142	17	1	1	NUM
eajbcs-667	142	18	,	,	PUNCT
eajbcs-667	142	19	2	2	NUM
eajbcs-667	142	20	,	,	PUNCT
eajbcs-667	142	21	.	.	PUNCT
eajbcs-667	142	22	.	.	PUNCT
eajbcs-667	142	23	.	.	PUNCT
eajbcs-667	143	1	,	,	PUNCT
eajbcs-667	143	2	n−	n−	NOUN
eajbcs-667	143	3	1	1	NUM
eajbcs-667	143	4	.	.	PUNCT
eajbcs-667	144	1	(	(	PUNCT
eajbcs-667	144	2	7	7	NUM
eajbcs-667	144	3	)	)	PUNCT
eajbcs-667	144	4	that	that	PRON
eajbcs-667	144	5	is	be	AUX
eajbcs-667	144	6	,	,	PUNCT
eajbcs-667	144	7	ϕi(xj	ϕi(xj	ADV
eajbcs-667	144	8	)	)	PUNCT
eajbcs-667	144	9	=	=	SYM
eajbcs-667	144	10	δij	δij	NOUN
eajbcs-667	144	11	=	=	SYM
eajbcs-667	144	12	{	{	PUNCT
eajbcs-667	144	13	1	1	NUM
eajbcs-667	144	14	,	,	PUNCT
eajbcs-667	144	15	if	if	SCONJ
eajbcs-667	144	16	i	i	PRON
eajbcs-667	144	17	=	=	SYM
eajbcs-667	144	18	j	j	PROPN
eajbcs-667	144	19	0	0	NUM
eajbcs-667	144	20	,	,	PUNCT
eajbcs-667	144	21	if	if	SCONJ
eajbcs-667	144	22	i	i	PRON
eajbcs-667	144	23	̸=	̸=	PROPN
eajbcs-667	144	24	j	j	PROPN
eajbcs-667	144	25	.	.	PUNCT
eajbcs-667	145	1	(	(	PUNCT
eajbcs-667	145	2	8)	8)	NUM
eajbcs-667	145	3	east	east	PROPN
eajbcs-667	145	4	afr	afr	PROPN
eajbcs-667	145	5	.	.	PUNCT
eajbcs-667	146	1	j.	j.	PROPN
eajbcs-667	146	2	biophys	biophys	PROPN
eajbcs-667	146	3	.	.	PUNCT
eajbcs-667	147	1	comput	comput	NOUN
eajbcs-667	147	2	.	.	PUNCT
eajbcs-667	148	1	sci	sci	PROPN
eajbcs-667	148	2	.	.	PUNCT
eajbcs-667	148	3	(	(	PUNCT
eajbcs-667	148	4	2023	2023	NUM
eajbcs-667	148	5	)	)	PUNCT
eajbcs-667	148	6	,	,	PUNCT
eajbcs-667	148	7	vol	vol	NOUN
eajbcs-667	148	8	.	.	PROPN
eajbcs-667	148	9	4	4	NUM
eajbcs-667	148	10	,	,	PUNCT
eajbcs-667	148	11	no	no	INTJ
eajbcs-667	148	12	.	.	NOUN
eajbcs-667	148	13	1	1	NUM
eajbcs-667	148	14	,	,	PUNCT
eajbcs-667	148	15	52	52	NUM
eajbcs-667	148	16	-	-	SYM
eajbcs-667	148	17	74	74	NUM
eajbcs-667	148	18	57	57	NUM
eajbcs-667	148	19	the	the	DET
eajbcs-667	148	20	next	next	ADJ
eajbcs-667	148	21	step	step	NOUN
eajbcs-667	148	22	is	be	AUX
eajbcs-667	148	23	to	to	PART
eajbcs-667	148	24	generate	generate	VERB
eajbcs-667	148	25	a	a	DET
eajbcs-667	148	26	mesh	mesh	NOUN
eajbcs-667	148	27	,	,	PUNCT
eajbcs-667	148	28	let	let	VERB
eajbcs-667	148	29	be	be	AUX
eajbcs-667	148	30	a	a	DET
eajbcs-667	148	31	uniform	uniform	ADJ
eajbcs-667	148	32	cartesian	cartesian	ADJ
eajbcs-667	148	33	mesh	mesh	NOUN
eajbcs-667	148	34	xi	xi	X
eajbcs-667	148	35	=	=	PUNCT
eajbcs-667	148	36	associate	associate	VERB
eajbcs-667	148	37	the	the	DET
eajbcs-667	148	38	value	value	NOUN
eajbcs-667	148	39	one	one	NUM
eajbcs-667	148	40	to	to	ADP
eajbcs-667	148	41	this	this	DET
eajbcs-667	148	42	node	node	NOUN
eajbcs-667	148	43	and	and	CCONJ
eajbcs-667	148	44	zero	zero	NUM
eajbcs-667	148	45	at	at	ADP
eajbcs-667	148	46	the	the	DET
eajbcs-667	148	47	remaining	remain	VERB
eajbcs-667	148	48	nodes	node	NOUN
eajbcs-667	148	49	of	of	ADP
eajbcs-667	148	50	the	the	DET
eajbcs-667	148	51	partition	partition	NOUN
eajbcs-667	148	52	(	(	PUNCT
eajbcs-667	148	53	see	see	VERB
eajbcs-667	148	54	fig	fig	NOUN
eajbcs-667	148	55	.	.	PUNCT
eajbcs-667	149	1	1	1	NUM
eajbcs-667	149	2	,	,	PUNCT
eajbcs-667	149	3	(	(	PUNCT
eajbcs-667	149	4	larson	larson	PROPN
eajbcs-667	149	5	and	and	CCONJ
eajbcs-667	149	6	bengzon	bengzon	PROPN
eajbcs-667	149	7	,	,	PUNCT
eajbcs-667	149	8	2010	2010	NUM
eajbcs-667	149	9	)	)	PUNCT
eajbcs-667	149	10	)	)	PUNCT
eajbcs-667	149	11	.	.	PUNCT
eajbcs-667	150	1	we	we	PRON
eajbcs-667	150	2	define	define	VERB
eajbcs-667	150	3	the	the	DET
eajbcs-667	150	4	intervals	interval	NOUN
eajbcs-667	150	5	as	as	ADP
eajbcs-667	150	6	[	[	X
eajbcs-667	150	7	xi−1	xi−1	PROPN
eajbcs-667	150	8	,	,	PUNCT
eajbcs-667	150	9	xi	xi	ADP
eajbcs-667	150	10	]	]	PUNCT
eajbcs-667	150	11	,	,	PUNCT
eajbcs-667	150	12	i	i	PRON
eajbcs-667	150	13	=	=	NOUN
eajbcs-667	150	14	1	1	NUM
eajbcs-667	150	15	,	,	PUNCT
eajbcs-667	150	16	2	2	NUM
eajbcs-667	150	17	,	,	PUNCT
eajbcs-667	150	18	...	...	PUNCT
eajbcs-667	150	19	,	,	PUNCT
eajbcs-667	150	20	n.	n.	NOUN
eajbcs-667	150	21	after	after	ADP
eajbcs-667	150	22	generating	generate	VERB
eajbcs-667	150	23	a	a	DET
eajbcs-667	150	24	mesh	mesh	NOUN
eajbcs-667	150	25	we	we	PRON
eajbcs-667	150	26	construct	construct	VERB
eajbcs-667	150	27	a	a	DET
eajbcs-667	150	28	set	set	NOUN
eajbcs-667	150	29	of	of	ADP
eajbcs-667	150	30	basis	basis	NOUN
eajbcs-667	150	31	functions	function	NOUN
eajbcs-667	150	32	based	base	VERB
eajbcs-667	150	33	on	on	ADP
eajbcs-667	150	34	the	the	DET
eajbcs-667	150	35	mesh	mesh	NOUN
eajbcs-667	150	36	for	for	ADP
eajbcs-667	150	37	each	each	DET
eajbcs-667	150	38	intervals	interval	NOUN
eajbcs-667	150	39	,	,	PUNCT
eajbcs-667	150	40	such	such	ADJ
eajbcs-667	150	41	as	as	ADP
eajbcs-667	150	42	the	the	DET
eajbcs-667	150	43	piece	piece	NOUN
eajbcs-667	150	44	wise	wise	ADJ
eajbcs-667	150	45	linear	linear	NOUN
eajbcs-667	150	46	functions	function	NOUN
eajbcs-667	150	47	for	for	ADP
eajbcs-667	150	48	i	i	PRON
eajbcs-667	150	49	=	=	NOUN
eajbcs-667	150	50	1	1	NUM
eajbcs-667	150	51	,	,	PUNCT
eajbcs-667	150	52	2	2	NUM
eajbcs-667	150	53	,	,	PUNCT
eajbcs-667	150	54	...	...	PUNCT
eajbcs-667	150	55	,	,	PUNCT
eajbcs-667	150	56	n−	n−	NOUN
eajbcs-667	150	57	1	1	NUM
eajbcs-667	150	58	.	.	PUNCT
eajbcs-667	151	1	the	the	DET
eajbcs-667	151	2	characteristic	characteristic	ADJ
eajbcs-667	151	3	basis	basis	NOUN
eajbcs-667	151	4	functions	function	NOUN
eajbcs-667	151	5	are	be	AUX
eajbcs-667	151	6	characterized	characterize	VERB
eajbcs-667	151	7	by	by	ADP
eajbcs-667	151	8	the	the	DET
eajbcs-667	151	9	following	follow	VERB
eajbcs-667	151	10	property	property	NOUN
eajbcs-667	151	11	,	,	PUNCT
eajbcs-667	151	12	(	(	PUNCT
eajbcs-667	151	13	quarteroni	quarteroni	NOUN
eajbcs-667	151	14	and	and	CCONJ
eajbcs-667	151	15	quarteroni	quarteroni	NOUN
eajbcs-667	151	16	,	,	PUNCT
eajbcs-667	151	17	2009	2009	NUM
eajbcs-667	151	18	)	)	PUNCT
eajbcs-667	151	19	uh(x	uh(x	X
eajbcs-667	151	20	)	)	PUNCT
eajbcs-667	152	1	=	=	PUNCT
eajbcs-667	152	2	n−1∑	n−1∑	NUM
eajbcs-667	152	3	j=1	j=1	PROPN
eajbcs-667	152	4	cjϕi(x	cjϕi(x	PROPN
eajbcs-667	152	5	)	)	PUNCT
eajbcs-667	152	6	,	,	PUNCT
eajbcs-667	152	7	(	(	PUNCT
eajbcs-667	152	8	9	9	X
eajbcs-667	152	9	)	)	PUNCT
eajbcs-667	152	10	where	where	SCONJ
eajbcs-667	152	11	the	the	DET
eajbcs-667	152	12	coefficients	coefficient	NOUN
eajbcs-667	152	13	cj	cj	NOUN
eajbcs-667	152	14	are	be	AUX
eajbcs-667	152	15	the	the	DET
eajbcs-667	152	16	unknowns	unknown	NOUN
eajbcs-667	152	17	to	to	PART
eajbcs-667	152	18	be	be	AUX
eajbcs-667	152	19	determined	determine	VERB
eajbcs-667	152	20	.	.	PUNCT
eajbcs-667	153	1	since	since	SCONJ
eajbcs-667	153	2	the	the	DET
eajbcs-667	153	3	hat	hat	NOUN
eajbcs-667	153	4	(	(	PUNCT
eajbcs-667	153	5	basis	basis	NOUN
eajbcs-667	153	6	)	)	PUNCT
eajbcs-667	153	7	functions	function	NOUN
eajbcs-667	153	8	are	be	AUX
eajbcs-667	153	9	piece	piece	NOUN
eajbcs-667	153	10	wise	wise	ADJ
eajbcs-667	153	11	linear	linear	NOUN
eajbcs-667	153	12	,	,	PUNCT
eajbcs-667	153	13	uh(x	uh(x	PRON
eajbcs-667	153	14	)	)	PUNCT
eajbcs-667	153	15	is	be	AUX
eajbcs-667	153	16	also	also	ADV
eajbcs-667	153	17	a	a	DET
eajbcs-667	153	18	piece	piece	NOUN
eajbcs-667	153	19	wise	wise	ADJ
eajbcs-667	153	20	linear	linear	NOUN
eajbcs-667	153	21	function	function	NOUN
eajbcs-667	153	22	,	,	PUNCT
eajbcs-667	153	23	although	although	SCONJ
eajbcs-667	153	24	this	this	PRON
eajbcs-667	153	25	is	be	AUX
eajbcs-667	153	26	not	not	PART
eajbcs-667	153	27	usually	usually	ADV
eajbcs-667	153	28	the	the	DET
eajbcs-667	153	29	case	case	NOUN
eajbcs-667	153	30	for	for	ADP
eajbcs-667	153	31	the	the	DET
eajbcs-667	153	32	true	true	ADJ
eajbcs-667	153	33	solution	solution	NOUN
eajbcs-667	153	34	u(x	u(x	NOUN
eajbcs-667	153	35	)	)	PUNCT
eajbcs-667	153	36	,	,	PUNCT
eajbcs-667	153	37	and	and	CCONJ
eajbcs-667	153	38	here	here	ADV
eajbcs-667	153	39	we	we	PRON
eajbcs-667	153	40	have	have	VERB
eajbcs-667	153	41	,	,	PUNCT
eajbcs-667	153	42	uh(xj	uh(xj	ADJ
eajbcs-667	153	43	)	)	PUNCT
eajbcs-667	153	44	=	=	SYM
eajbcs-667	154	1	n−1∑	n−1∑	PROPN
eajbcs-667	154	2	i=1	i=1	PROPN
eajbcs-667	154	3	cjϕi(xj	cjϕi(xj	ADV
eajbcs-667	154	4	)	)	PUNCT
eajbcs-667	155	1	=	=	SYM
eajbcs-667	155	2	cj	cj	NOUN
eajbcs-667	155	3	.	.	PUNCT
eajbcs-667	156	1	we	we	PRON
eajbcs-667	156	2	then	then	ADV
eajbcs-667	156	3	derive	derive	VERB
eajbcs-667	156	4	a	a	DET
eajbcs-667	156	5	linear	linear	ADJ
eajbcs-667	156	6	system	system	NOUN
eajbcs-667	156	7	of	of	ADP
eajbcs-667	156	8	equations	equation	NOUN
eajbcs-667	156	9	for	for	ADP
eajbcs-667	156	10	the	the	DET
eajbcs-667	156	11	coefficients	coefficient	NOUN
eajbcs-667	156	12	by	by	ADP
eajbcs-667	156	13	substituting	substitute	VERB
eajbcs-667	156	14	the	the	DET
eajbcs-667	156	15	approximate	approximate	ADJ
eajbcs-667	156	16	solution	solution	NOUN
eajbcs-667	156	17	uh(x	uh(x	NUM
eajbcs-667	156	18	)	)	PUNCT
eajbcs-667	156	19	for	for	ADP
eajbcs-667	156	20	the	the	DET
eajbcs-667	156	21	exact	exact	ADJ
eajbcs-667	156	22	solution	solution	NOUN
eajbcs-667	156	23	u(x	u(x	NOUN
eajbcs-667	156	24	)	)	PUNCT
eajbcs-667	156	25	in	in	ADP
eajbcs-667	156	26	the	the	DET
eajbcs-667	156	27	weak	weak	ADJ
eajbcs-667	156	28	form	form	NOUN
eajbcs-667	156	29	:	:	PUNCT
eajbcs-667	156	30	u(t	u(t	NOUN
eajbcs-667	156	31	,	,	PUNCT
eajbcs-667	156	32	x	x	NOUN
eajbcs-667	156	33	)	)	PUNCT
eajbcs-667	156	34	=	=	SYM
eajbcs-667	157	1	n−1∑	n−1∑	PROPN
eajbcs-667	157	2	i=1	i=1	PROPN
eajbcs-667	158	1	uiϕi(x	uiϕi(x	PROPN
eajbcs-667	158	2	)	)	PUNCT
eajbcs-667	158	3	,	,	PUNCT
eajbcs-667	158	4	(	(	PUNCT
eajbcs-667	158	5	10	10	NUM
eajbcs-667	158	6	)	)	PUNCT
eajbcs-667	158	7	v(t	v(t	NOUN
eajbcs-667	158	8	,	,	PUNCT
eajbcs-667	158	9	x	x	NOUN
eajbcs-667	158	10	)	)	PUNCT
eajbcs-667	158	11	=	=	SYM
eajbcs-667	158	12	n−1∑	n−1∑	NUM
eajbcs-667	158	13	j=1	j=1	NOUN
eajbcs-667	158	14	vjϕj(x	vjϕj(x	NOUN
eajbcs-667	158	15	)	)	PUNCT
eajbcs-667	158	16	.	.	PUNCT
eajbcs-667	159	1	(	(	PUNCT
eajbcs-667	159	2	11	11	NUM
eajbcs-667	159	3	)	)	PUNCT
eajbcs-667	159	4	now	now	ADV
eajbcs-667	159	5	substituting	substitute	VERB
eajbcs-667	159	6	eq.10	eq.10	NOUN
eajbcs-667	159	7	and	and	CCONJ
eajbcs-667	159	8	eq.11	eq.11	NUM
eajbcs-667	159	9	in	in	ADP
eajbcs-667	159	10	the	the	DET
eajbcs-667	159	11	weak	weak	ADJ
eajbcs-667	159	12	formulation	formulation	NOUN
eajbcs-667	159	13	of	of	ADP
eajbcs-667	159	14	the	the	DET
eajbcs-667	159	15	equation	equation	NOUN
eajbcs-667	159	16	eq.5	eq.5	PROPN
eajbcs-667	159	17	,	,	PUNCT
eajbcs-667	159	18	we	we	PRON
eajbcs-667	159	19	have	have	VERB
eajbcs-667	159	20	:	:	PUNCT
eajbcs-667	159	21	∫	∫	PROPN
eajbcs-667	159	22	1	1	NUM
eajbcs-667	159	23	0	0	NUM
eajbcs-667	159	24	∂	∂	NUM
eajbcs-667	160	1	∂t	∂t	PROPN
eajbcs-667	160	2	n−1∑	n−1∑	NUM
eajbcs-667	160	3	i=1	i=1	PROPN
eajbcs-667	160	4	uiϕi	uiϕi	ADJ
eajbcs-667	160	5	.	.	PUNCT
eajbcs-667	161	1	n−1∑	n−1∑	NUM
eajbcs-667	161	2	j=1	j=1	PROPN
eajbcs-667	161	3	vjϕj	vjϕj	NOUN
eajbcs-667	162	1	+	+	CCONJ
eajbcs-667	162	2	∫	∫	PROPN
eajbcs-667	162	3	1	1	NUM
eajbcs-667	162	4	0	0	NUM
eajbcs-667	162	5	a	a	DET
eajbcs-667	162	6	∂	∂	NOUN
eajbcs-667	162	7	∂x	∂x	PROPN
eajbcs-667	162	8	(	(	PUNCT
eajbcs-667	162	9	n−1∑	n−1∑	NUM
eajbcs-667	162	10	i=1	i=1	PROPN
eajbcs-667	162	11	uiϕi	uiϕi	ADJ
eajbcs-667	162	12	)	)	PUNCT
eajbcs-667	162	13	.	.	PUNCT
eajbcs-667	163	1	n−1∑	n−1∑	NUM
eajbcs-667	163	2	j=1	j=1	NOUN
eajbcs-667	163	3	vjϕj	vjϕj	NOUN
eajbcs-667	163	4	=	=	SYM
eajbcs-667	163	5	0	0	X
eajbcs-667	163	6	.	.	NOUN
eajbcs-667	163	7	which	which	PRON
eajbcs-667	163	8	then	then	ADV
eajbcs-667	163	9	implies	imply	VERB
eajbcs-667	163	10	,	,	PUNCT
eajbcs-667	163	11	n−1∑	n−1∑	NUM
eajbcs-667	163	12	j=1	j=1	PROPN
eajbcs-667	163	13	vj	vj	X
eajbcs-667	163	14	(	(	PUNCT
eajbcs-667	163	15	∂	∂	NUM
eajbcs-667	163	16	∂t	∂t	PROPN
eajbcs-667	163	17	n−1∑	n−1∑	NUM
eajbcs-667	163	18	i=1	i=1	PROPN
eajbcs-667	163	19	ui	ui	PROPN
eajbcs-667	163	20	∫	∫	PROPN
eajbcs-667	163	21	1	1	NUM
eajbcs-667	163	22	0	0	NUM
eajbcs-667	163	23	ϕi.ϕj	ϕi.ϕj	NOUN
eajbcs-667	163	24	+	+	CCONJ
eajbcs-667	163	25	a	a	DET
eajbcs-667	163	26	n−1∑	n−1∑	NUM
eajbcs-667	163	27	i=1	i=1	PROPN
eajbcs-667	163	28	ui	ui	PROPN
eajbcs-667	163	29	∫	∫	PROPN
eajbcs-667	163	30	1	1	NUM
eajbcs-667	163	31	0	0	X
eajbcs-667	163	32	ϕ′	ϕ′	PUNCT
eajbcs-667	164	1	i.ϕj	i.ϕj	INTJ
eajbcs-667	164	2	)	)	PUNCT
eajbcs-667	164	3	=	=	SYM
eajbcs-667	165	1	0	0	X
eajbcs-667	165	2	.	.	PUNCT
eajbcs-667	166	1	that	that	PRON
eajbcs-667	166	2	is	be	AUX
eajbcs-667	166	3	∂	∂	NUM
eajbcs-667	166	4	∂t	∂t	PROPN
eajbcs-667	166	5	n−1∑	n−1∑	NUM
eajbcs-667	166	6	i=1	i=1	PROPN
eajbcs-667	167	1	ui	ui	PROPN
eajbcs-667	168	1	∫	∫	PROPN
eajbcs-667	169	1	1	1	NUM
eajbcs-667	169	2	0	0	NUM
eajbcs-667	169	3	ϕi.ϕj	ϕi.ϕj	NOUN
eajbcs-667	169	4	+	+	CCONJ
eajbcs-667	169	5	a	a	DET
eajbcs-667	169	6	n−1∑	n−1∑	NUM
eajbcs-667	169	7	i=1	i=1	PROPN
eajbcs-667	169	8	ui	ui	PROPN
eajbcs-667	170	1	∫	∫	PROPN
eajbcs-667	170	2	1	1	NUM
eajbcs-667	170	3	0	0	X
eajbcs-667	170	4	ϕ′	ϕ′	PUNCT
eajbcs-667	171	1	i.ϕj	i.ϕj	INTJ
eajbcs-667	171	2	=	=	NOUN
eajbcs-667	171	3	0	0	X
eajbcs-667	171	4	.	.	PUNCT
eajbcs-667	172	1	in	in	ADP
eajbcs-667	172	2	a	a	DET
eajbcs-667	172	3	matrix	matrix	NOUN
eajbcs-667	172	4	form	form	NOUN
eajbcs-667	172	5	it	it	PRON
eajbcs-667	172	6	can	can	AUX
eajbcs-667	172	7	be	be	AUX
eajbcs-667	172	8	written	write	VERB
eajbcs-667	172	9	as	as	ADP
eajbcs-667	172	10	:	:	PUNCT
eajbcs-667	172	11	mu̇	mu̇	X
eajbcs-667	172	12	+	+	CCONJ
eajbcs-667	172	13	abu	abu	PROPN
eajbcs-667	172	14	=	=	SYM
eajbcs-667	172	15	0	0	PROPN
eajbcs-667	172	16	,	,	PUNCT
eajbcs-667	172	17	(	(	PUNCT
eajbcs-667	172	18	12	12	NUM
eajbcs-667	172	19	)	)	PUNCT
eajbcs-667	172	20	where	where	SCONJ
eajbcs-667	172	21	,	,	PUNCT
eajbcs-667	172	22	east	east	PROPN
eajbcs-667	172	23	afr	afr	PROPN
eajbcs-667	172	24	.	.	PUNCT
eajbcs-667	173	1	j.	j.	PROPN
eajbcs-667	173	2	biophys	biophys	PROPN
eajbcs-667	173	3	.	.	PUNCT
eajbcs-667	174	1	comput	comput	NOUN
eajbcs-667	174	2	.	.	PUNCT
eajbcs-667	175	1	sci	sci	PROPN
eajbcs-667	175	2	.	.	PUNCT
eajbcs-667	175	3	(	(	PUNCT
eajbcs-667	175	4	2023	2023	NUM
eajbcs-667	175	5	)	)	PUNCT
eajbcs-667	175	6	,	,	PUNCT
eajbcs-667	175	7	vol	vol	NOUN
eajbcs-667	175	8	.	.	PROPN
eajbcs-667	175	9	4	4	NUM
eajbcs-667	175	10	,	,	PUNCT
eajbcs-667	175	11	no	no	INTJ
eajbcs-667	175	12	.	.	NOUN
eajbcs-667	175	13	1	1	NUM
eajbcs-667	175	14	,	,	PUNCT
eajbcs-667	175	15	52	52	NUM
eajbcs-667	175	16	-	-	SYM
eajbcs-667	175	17	74	74	NUM
eajbcs-667	175	18	58	58	NUM
eajbcs-667	175	19	we	we	PRON
eajbcs-667	175	20	use	use	VERB
eajbcs-667	175	21	this	this	DET
eajbcs-667	175	22	hat	hat	NOUN
eajbcs-667	175	23	(	(	PUNCT
eajbcs-667	175	24	basis	basis	NOUN
eajbcs-667	175	25	)	)	PUNCT
eajbcs-667	175	26	functions	function	NOUN
eajbcs-667	175	27	through	through	ADP
eajbcs-667	175	28	out	out	ADP
eajbcs-667	175	29	the	the	DET
eajbcs-667	175	30	1d	1d	NUM
eajbcs-667	175	31	space	space	NOUN
eajbcs-667	175	32	of	of	ADP
eajbcs-667	175	33	this	this	DET
eajbcs-667	175	34	research	research	NOUN
eajbcs-667	175	35	using	use	VERB
eajbcs-667	175	36	equal	equal	ADJ
eajbcs-667	175	37	spaced	space	VERB
eajbcs-667	175	38	step	step	NOUN
eajbcs-667	175	39	size	size	NOUN
eajbcs-667	175	40	(	(	PUNCT
eajbcs-667	175	41	xi+1−xi	xi+1−xi	PROPN
eajbcs-667	175	42	=	=	SYM
eajbcs-667	175	43	h	h	NOUN
eajbcs-667	175	44	,	,	PUNCT
eajbcs-667	175	45	for	for	ADP
eajbcs-667	175	46	all	all	DET
eajbcs-667	175	47	i	i	PRON
eajbcs-667	175	48	)	)	PUNCT
eajbcs-667	175	49	.	.	PUNCT
eajbcs-667	176	1	we	we	PRON
eajbcs-667	176	2	say	say	VERB
eajbcs-667	176	3	that	that	SCONJ
eajbcs-667	176	4	the	the	DET
eajbcs-667	176	5	functions	function	NOUN
eajbcs-667	176	6	are	be	AUX
eajbcs-667	176	7	basis	basis	NOUN
eajbcs-667	176	8	for	for	ADP
eajbcs-667	176	9	the	the	DET
eajbcs-667	176	10	following	follow	VERB
eajbcs-667	176	11	reasons	reason	NOUN
eajbcs-667	176	12	.	.	PUNCT
eajbcs-667	177	1	if	if	SCONJ
eajbcs-667	177	2	we	we	PRON
eajbcs-667	177	3	want	want	VERB
eajbcs-667	177	4	to	to	PART
eajbcs-667	177	5	approximate	approximate	VERB
eajbcs-667	177	6	our	our	PRON
eajbcs-667	177	7	continuous	continuous	ADJ
eajbcs-667	177	8	function	function	NOUN
eajbcs-667	177	9	u	u	NOUN
eajbcs-667	177	10	with	with	ADP
eajbcs-667	177	11	a	a	DET
eajbcs-667	177	12	piece	piece	NOUN
eajbcs-667	177	13	wise	wise	ADJ
eajbcs-667	177	14	continuous	continuous	ADJ
eajbcs-667	177	15	linear	linear	PROPN
eajbcs-667	177	16	functhen	functhen	NOUN
eajbcs-667	177	17	let	let	VERB
eajbcs-667	177	18	the	the	DET
eajbcs-667	177	19	approximate	approximate	ADJ
eajbcs-667	177	20	solution	solution	NOUN
eajbcs-667	177	21	for	for	ADP
eajbcs-667	177	22	u	u	NOUN
eajbcs-667	177	23	be	be	AUX
eajbcs-667	177	24	given	give	VERB
eajbcs-667	177	25	by	by	ADP
eajbcs-667	177	26	a	a	DET
eajbcs-667	177	27	linear	linear	ADJ
eajbcs-667	177	28	combination	combination	NOUN
eajbcs-667	177	29	of	of	ADP
eajbcs-667	177	30	basis	basis	NOUN
eajbcs-667	177	31	functions	function	NOUN
eajbcs-667	177	32	ϕi	ϕi	ADP
eajbcs-667	177	33	=	=	NOUN
eajbcs-667	177	34	δij	δij	NOUN
eajbcs-667	177	35	,	,	PUNCT
eajbcs-667	177	36	as	as	SCONJ
eajbcs-667	177	37	given	give	VERB
eajbcs-667	177	38	in	in	ADP
eajbcs-667	177	39	eq	eq	NOUN
eajbcs-667	177	40	.	.	PROPN
eajbcs-667	177	41	10	10	NUM
eajbcs-667	177	42	,	,	PUNCT
eajbcs-667	177	43	and	and	CCONJ
eajbcs-667	177	44	also	also	ADV
eajbcs-667	177	45	for	for	ADP
eajbcs-667	177	46	v	v	NOUN
eajbcs-667	177	47	as	as	ADP
eajbcs-667	177	48	in	in	ADP
eajbcs-667	177	49	eq	eq	NOUN
eajbcs-667	177	50	.	.	PROPN
eajbcs-667	177	51	11	11	NUM
eajbcs-667	177	52	.	.	PUNCT
eajbcs-667	178	1	now	now	ADV
eajbcs-667	178	2	we	we	PRON
eajbcs-667	178	3	find	find	VERB
eajbcs-667	178	4	a	a	DET
eajbcs-667	178	5	finite	finite	ADJ
eajbcs-667	178	6	element	element	NOUN
eajbcs-667	178	7	solution	solution	NOUN
eajbcs-667	178	8	of	of	ADP
eajbcs-667	178	9	the	the	DET
eajbcs-667	178	10	discrete	discrete	ADJ
eajbcs-667	178	11	problem	problem	NOUN
eajbcs-667	178	12	by	by	ADP
eajbcs-667	178	13	using	use	VERB
eajbcs-667	178	14	the	the	DET
eajbcs-667	178	15	hat	hat	NOUN
eajbcs-667	178	16	functions	function	NOUN
eajbcs-667	178	17	ϕi(x	ϕi(x	NOUN
eajbcs-667	178	18	)	)	PUNCT
eajbcs-667	178	19	defined	define	VERB
eajbcs-667	178	20	in	in	ADP
eajbcs-667	178	21	eq	eq	NOUN
eajbcs-667	178	22	.	.	PROPN
eajbcs-667	179	1	7	7	X
eajbcs-667	179	2	.	.	X
eajbcs-667	179	3	for	for	ADP
eajbcs-667	179	4	the	the	DET
eajbcs-667	179	5	given	give	VERB
eajbcs-667	179	6	basis	basis	NOUN
eajbcs-667	179	7	function	function	NOUN
eajbcs-667	179	8	the	the	DET
eajbcs-667	179	9	approximation	approximation	NOUN
eajbcs-667	179	10	of	of	ADP
eajbcs-667	179	11	u	u	NOUN
eajbcs-667	179	12	and	and	CCONJ
eajbcs-667	179	13	v	v	NOUN
eajbcs-667	179	14	can	can	AUX
eajbcs-667	179	15	be	be	AUX
eajbcs-667	179	16	written	write	VERB
eajbcs-667	179	17	as	as	ADP
eajbcs-667	179	18	:	:	PUNCT
eajbcs-667	179	19	tion	tion	PROPN
eajbcs-667	179	20	u′	u′	PROPN
eajbcs-667	179	21	,	,	PUNCT
eajbcs-667	179	22	these	these	DET
eajbcs-667	179	23	functions	function	NOUN
eajbcs-667	179	24	are	be	AUX
eajbcs-667	179	25	what	what	PRON
eajbcs-667	179	26	we	we	PRON
eajbcs-667	179	27	need	need	VERB
eajbcs-667	179	28	.	.	PUNCT
eajbcs-667	180	1	these	these	DET
eajbcs-667	180	2	functions	function	NOUN
eajbcs-667	180	3	are	be	AUX
eajbcs-667	180	4	linearly	linearly	ADV
eajbcs-667	180	5	independent	independent	ADJ
eajbcs-667	180	6	of	of	ADP
eajbcs-667	180	7	each	each	DET
eajbcs-667	180	8	other	other	ADJ
eajbcs-667	180	9	;	;	PUNCT
eajbcs-667	180	10	it	it	PRON
eajbcs-667	180	11	is	be	AUX
eajbcs-667	180	12	not	not	PART
eajbcs-667	180	13	possible	possible	ADJ
eajbcs-667	180	14	to	to	PART
eajbcs-667	180	15	make	make	VERB
eajbcs-667	180	16	one	one	NUM
eajbcs-667	180	17	out	out	ADP
eajbcs-667	180	18	of	of	ADP
eajbcs-667	180	19	a	a	DET
eajbcs-667	180	20	combination	combination	NOUN
eajbcs-667	180	21	of	of	ADP
eajbcs-667	180	22	others	other	NOUN
eajbcs-667	180	23	.	.	PUNCT
eajbcs-667	181	1	for	for	ADP
eajbcs-667	181	2	example	example	NOUN
eajbcs-667	181	3	,	,	PUNCT
eajbcs-667	181	4	only	only	ADV
eajbcs-667	181	5	one	one	NUM
eajbcs-667	181	6	of	of	ADP
eajbcs-667	181	7	these	these	DET
eajbcs-667	181	8	functions	function	NOUN
eajbcs-667	181	9	,	,	PUNCT
eajbcs-667	181	10	ϕi	ϕi	ADP
eajbcs-667	181	11	,	,	PUNCT
eajbcs-667	181	12	is	be	AUX
eajbcs-667	181	13	non	non	ADJ
eajbcs-667	181	14	-	-	ADJ
eajbcs-667	181	15	zero	zero	NUM
eajbcs-667	181	16	(	(	PUNCT
eajbcs-667	181	17	equal	equal	ADJ
eajbcs-667	181	18	to	to	ADP
eajbcs-667	181	19	1	1	NUM
eajbcs-667	181	20	)	)	PUNCT
eajbcs-667	181	21	at	at	ADP
eajbcs-667	181	22	node	node	PROPN
eajbcs-667	181	23	i.	i.	PROPN
eajbcs-667	181	24	the	the	DET
eajbcs-667	181	25	next	next	ADJ
eajbcs-667	181	26	step	step	NOUN
eajbcs-667	181	27	in	in	ADP
eajbcs-667	181	28	approximating	approximate	VERB
eajbcs-667	181	29	a	a	DET
eajbcs-667	181	30	pde	pde	NOUN
eajbcs-667	181	31	with	with	ADP
eajbcs-667	181	32	fem	fem	NOUN
eajbcs-667	181	33	is	be	AUX
eajbcs-667	181	34	represent	represent	VERB
eajbcs-667	181	35	the	the	DET
eajbcs-667	181	36	approximate	approximate	ADJ
eajbcs-667	181	37	(	(	PUNCT
eajbcs-667	181	38	fe	fe	NOUN
eajbcs-667	181	39	)	)	PUNCT
eajbcs-667	181	40	solution	solution	NOUN
eajbcs-667	181	41	by	by	ADP
eajbcs-667	181	42	the	the	DET
eajbcs-667	181	43	linear	linear	ADJ
eajbcs-667	181	44	combination	combination	NOUN
eajbcs-667	181	45	of	of	ADP
eajbcs-667	181	46	such	such	ADJ
eajbcs-667	181	47	basis	basis	NOUN
eajbcs-667	181	48	functions	function	NOUN
eajbcs-667	181	49	,	,	PUNCT
eajbcs-667	181	50	(	(	PUNCT
eajbcs-667	181	51	quarteroni	quarteroni	NOUN
eajbcs-667	181	52	and	and	CCONJ
eajbcs-667	181	53	quarteroni	quarteroni	NOUN
eajbcs-667	181	54	,	,	PUNCT
eajbcs-667	181	55	2009	2009	NUM
eajbcs-667	181	56	)	)	PUNCT
eajbcs-667	181	57	as	as	SCONJ
eajbcs-667	181	58	�	�	PROPN
eajbcs-667	181	59	m	m	PROPN
eajbcs-667	181	60	is	be	AUX
eajbcs-667	181	61	the	the	DET
eajbcs-667	181	62	mass	mass	ADJ
eajbcs-667	181	63	matrix	matrix	NOUN
eajbcs-667	181	64	with	with	ADP
eajbcs-667	181	65	entries	entry	NOUN
eajbcs-667	181	66	:	:	PUNCT
eajbcs-667	181	67	mi	mi	PROPN
eajbcs-667	181	68	,	,	PUNCT
eajbcs-667	181	69	j	j	PROPN
eajbcs-667	182	1	=	=	SYM
eajbcs-667	182	2	∫	∫	PROPN
eajbcs-667	182	3	1	1	NUM
eajbcs-667	182	4	0	0	NUM
eajbcs-667	183	1	ϕi(x)ϕj(x)dx	ϕi(x)ϕj(x)dx	NOUN
eajbcs-667	183	2	.	.	PUNCT
eajbcs-667	184	1	�	�	PROPN
eajbcs-667	184	2	b	b	PROPN
eajbcs-667	184	3	is	be	AUX
eajbcs-667	184	4	a	a	DET
eajbcs-667	184	5	matrix	matrix	NOUN
eajbcs-667	184	6	with	with	ADP
eajbcs-667	184	7	entries	entry	NOUN
eajbcs-667	184	8	:	:	PUNCT
eajbcs-667	184	9	bij	bij	NOUN
eajbcs-667	184	10	=	=	SYM
eajbcs-667	185	1	∫	∫	PROPN
eajbcs-667	185	2	1	1	NUM
eajbcs-667	185	3	0	0	NUM
eajbcs-667	185	4	ϕ′	ϕ′	NOUN
eajbcs-667	186	1	i(x)ϕj(x)dx	i(x)ϕj(x)dx	NOUN
eajbcs-667	186	2	=	=	PUNCT
eajbcs-667	187	1			PROPN
eajbcs-667	187	2	0	0	NUM
eajbcs-667	187	3	,	,	PUNCT
eajbcs-667	187	4	if	if	SCONJ
eajbcs-667	187	5	i	i	PRON
eajbcs-667	187	6	=	=	SYM
eajbcs-667	187	7	j	j	PROPN
eajbcs-667	187	8	−1	−1	NOUN
eajbcs-667	187	9	2	2	NUM
eajbcs-667	187	10	,	,	PUNCT
eajbcs-667	187	11	if	if	SCONJ
eajbcs-667	187	12	i−	i−	PROPN
eajbcs-667	187	13	j	j	PROPN
eajbcs-667	187	14	=	=	SYM
eajbcs-667	187	15	1	1	NUM
eajbcs-667	187	16	1	1	NUM
eajbcs-667	187	17	2	2	NUM
eajbcs-667	187	18	,	,	PUNCT
eajbcs-667	187	19	if	if	SCONJ
eajbcs-667	187	20	j	j	PROPN
eajbcs-667	187	21	−	−	VERB
eajbcs-667	188	1	i	i	PRON
eajbcs-667	188	2	=	=	NOUN
eajbcs-667	188	3	1	1	NUM
eajbcs-667	188	4	0	0	NUM
eajbcs-667	188	5	,	,	PUNCT
eajbcs-667	188	6	other	other	ADJ
eajbcs-667	188	7	wise	wise	ADJ
eajbcs-667	188	8	.	.	PUNCT
eajbcs-667	189	1	(	(	PUNCT
eajbcs-667	189	2	13	13	NUM
eajbcs-667	189	3	)	)	PUNCT
eajbcs-667	189	4	m	m	PROPN
eajbcs-667	189	5	(	(	PUNCT
eajbcs-667	189	6	un+1	un+1	PROPN
eajbcs-667	189	7	−	−	PROPN
eajbcs-667	189	8	un	un	PROPN
eajbcs-667	189	9	∆t	∆t	PROPN
eajbcs-667	189	10	)	)	PUNCT
eajbcs-667	190	1	+	+	PUNCT
eajbcs-667	190	2	abun+1	abun+1	NOUN
eajbcs-667	190	3	=	=	SYM
eajbcs-667	190	4	0	0	X
eajbcs-667	190	5	.	.	PUNCT
eajbcs-667	191	1	here	here	ADV
eajbcs-667	191	2	un	un	PROPN
eajbcs-667	191	3	denotes	denote	VERB
eajbcs-667	191	4	u	u	NOUN
eajbcs-667	191	5	at	at	ADP
eajbcs-667	191	6	time	time	NOUN
eajbcs-667	191	7	t	t	PROPN
eajbcs-667	191	8	=	=	SYM
eajbcs-667	191	9	tn	tn	PROPN
eajbcs-667	191	10	=	=	SYM
eajbcs-667	191	11	∆tn	∆tn	NOUN
eajbcs-667	191	12	,	,	PUNCT
eajbcs-667	191	13	and	and	CCONJ
eajbcs-667	191	14	∆t	∆t	PROPN
eajbcs-667	191	15	is	be	AUX
eajbcs-667	191	16	the	the	DET
eajbcs-667	191	17	time	time	NOUN
eajbcs-667	191	18	step	step	NOUN
eajbcs-667	191	19	.	.	PUNCT
eajbcs-667	192	1	rearranging	rearrange	VERB
eajbcs-667	192	2	the	the	DET
eajbcs-667	192	3	terms	term	NOUN
eajbcs-667	192	4	we	we	PRON
eajbcs-667	192	5	obtain	obtain	VERB
eajbcs-667	192	6	the	the	DET
eajbcs-667	192	7	system	system	NOUN
eajbcs-667	192	8	:(	:(	PUNCT
eajbcs-667	192	9	m	m	PART
eajbcs-667	192	10	∆t	∆t	PROPN
eajbcs-667	192	11	+	+	CCONJ
eajbcs-667	192	12	ab	ab	PROPN
eajbcs-667	192	13	)	)	PUNCT
eajbcs-667	192	14	un+1	un+1	PROPN
eajbcs-667	192	15	=	=	SYM
eajbcs-667	192	16	1	1	NUM
eajbcs-667	192	17	∆t	∆t	PROPN
eajbcs-667	192	18	mun	mun	PROPN
eajbcs-667	192	19	,	,	PUNCT
eajbcs-667	192	20	n	n	NOUN
eajbcs-667	192	21	=	=	SYM
eajbcs-667	192	22	0	0	NUM
eajbcs-667	192	23	,	,	PUNCT
eajbcs-667	192	24	1	1	NUM
eajbcs-667	192	25	,	,	PUNCT
eajbcs-667	192	26	2	2	NUM
eajbcs-667	192	27	,	,	PUNCT
eajbcs-667	192	28	.	.	PUNCT
eajbcs-667	192	29	.	.	PUNCT
eajbcs-667	192	30	.	.	PUNCT
eajbcs-667	193	1	,	,	PUNCT
eajbcs-667	193	2	to	to	PART
eajbcs-667	193	3	be	be	AUX
eajbcs-667	193	4	solved	solve	VERB
eajbcs-667	193	5	for	for	ADP
eajbcs-667	193	6	un+1	un+1	NUM
eajbcs-667	193	7	by	by	ADP
eajbcs-667	193	8	using	use	VERB
eajbcs-667	193	9	initial	initial	ADJ
eajbcs-667	193	10	condition	condition	NOUN
eajbcs-667	193	11	for	for	ADP
eajbcs-667	193	12	u0	u0	ADJ
eajbcs-667	193	13	=	=	NOUN
eajbcs-667	193	14	u(x	u(x	PROPN
eajbcs-667	193	15	,	,	PUNCT
eajbcs-667	193	16	t	t	NOUN
eajbcs-667	193	17	=	=	SYM
eajbcs-667	193	18	0	0	NUM
eajbcs-667	193	19	)	)	PUNCT
eajbcs-667	193	20	.	.	PUNCT
eajbcs-667	194	1	assembly	assembly	NOUN
eajbcs-667	194	2	of	of	ADP
eajbcs-667	194	3	the	the	DET
eajbcs-667	194	4	mass	mass	ADJ
eajbcs-667	194	5	matrix	matrix	NOUN
eajbcs-667	194	6	m	m	VERB
eajbcs-667	194	7	in	in	ADP
eajbcs-667	194	8	1d	1d	NUM
eajbcs-667	194	9	∫	∫	PROPN
eajbcs-667	194	10	ω	ω	X
eajbcs-667	194	11	ϕiϕjdx	ϕiϕjdx	X
eajbcs-667	194	12	exactly	exactly	ADV
eajbcs-667	194	13	.	.	PUNCT
eajbcs-667	195	1	moreover	moreover	ADV
eajbcs-667	195	2	,	,	PUNCT
eajbcs-667	195	3	since	since	SCONJ
eajbcs-667	195	4	the	the	DET
eajbcs-667	195	5	hats	hat	NOUN
eajbcs-667	195	6	ϕi	ϕi	X
eajbcs-667	195	7	and	and	CCONJ
eajbcs-667	195	8	ϕj	ϕj	AUX
eajbcs-667	195	9	figure	figure	VERB
eajbcs-667	195	10	2	2	NUM
eajbcs-667	195	11	:	:	PUNCT
eajbcs-667	195	12	illustration	illustration	NOUN
eajbcs-667	195	13	of	of	ADP
eajbcs-667	195	14	the	the	DET
eajbcs-667	195	15	hat	hat	NOUN
eajbcs-667	195	16	functions	function	NOUN
eajbcs-667	195	17	ϕi−1	ϕi−1	PROPN
eajbcs-667	195	18	and	and	CCONJ
eajbcs-667	195	19	ϕi	ϕi	ADP
eajbcs-667	195	20	,	,	PUNCT
eajbcs-667	195	21	in	in	ADP
eajbcs-667	195	22	this	this	DET
eajbcs-667	195	23	figure	figure	NOUN
eajbcs-667	195	24	φ	φ	NOUN
eajbcs-667	195	25	,	,	PUNCT
eajbcs-667	195	26	and	and	CCONJ
eajbcs-667	195	27	their	their	PRON
eajbcs-667	195	28	support	support	NOUN
eajbcs-667	195	29	.	.	PUNCT
eajbcs-667	196	1	east	east	PROPN
eajbcs-667	196	2	afr	afr	PROPN
eajbcs-667	196	3	.	.	PUNCT
eajbcs-667	197	1	j.	j.	PROPN
eajbcs-667	197	2	biophys	biophys	PROPN
eajbcs-667	197	3	.	.	PUNCT
eajbcs-667	198	1	comput	comput	NOUN
eajbcs-667	198	2	.	.	PUNCT
eajbcs-667	199	1	sci	sci	PROPN
eajbcs-667	199	2	.	.	PUNCT
eajbcs-667	199	3	(	(	PUNCT
eajbcs-667	199	4	2023	2023	NUM
eajbcs-667	199	5	)	)	PUNCT
eajbcs-667	199	6	,	,	PUNCT
eajbcs-667	199	7	vol	vol	NOUN
eajbcs-667	199	8	.	.	PROPN
eajbcs-667	199	9	4	4	NUM
eajbcs-667	199	10	,	,	PUNCT
eajbcs-667	199	11	no	no	INTJ
eajbcs-667	199	12	.	.	NOUN
eajbcs-667	199	13	1	1	NUM
eajbcs-667	199	14	,	,	PUNCT
eajbcs-667	199	15	52	52	NUM
eajbcs-667	199	16	-	-	SYM
eajbcs-667	199	17	74	74	NUM
eajbcs-667	199	18	59	59	NUM
eajbcs-667	199	19	the	the	DET
eajbcs-667	199	20	next	next	ADJ
eajbcs-667	199	21	step	step	NOUN
eajbcs-667	199	22	is	be	AUX
eajbcs-667	199	23	to	to	PART
eajbcs-667	199	24	descritize	descritize	VERB
eajbcs-667	199	25	the	the	DET
eajbcs-667	199	26	sys	sys	PROPN
eajbcs-667	199	27	-	-	PROPN
eajbcs-667	199	28	tem	tem	PROPN
eajbcs-667	199	29	eq.12	eq.12	NOUN
eajbcs-667	199	30	in	in	ADP
eajbcs-667	199	31	time	time	NOUN
eajbcs-667	199	32	.	.	PUNCT
eajbcs-667	200	1	here	here	ADV
eajbcs-667	200	2	we	we	PRON
eajbcs-667	200	3	were	be	AUX
eajbcs-667	200	4	consider	consider	VERB
eajbcs-667	200	5	finite	finite	ADJ
eajbcs-667	200	6	difference	difference	NOUN
eajbcs-667	200	7	approximations	approximation	NOUN
eajbcs-667	200	8	specially	specially	ADV
eajbcs-667	200	9	the	the	DET
eajbcs-667	200	10	implicit	implicit	ADJ
eajbcs-667	200	11	euler	euler	NOUN
eajbcs-667	200	12	(	(	PUNCT
eajbcs-667	200	13	back	back	PROPN
eajbcs-667	200	14	ward	ward	PROPN
eajbcs-667	200	15	euler	euler	PROPN
eajbcs-667	200	16	)	)	PUNCT
eajbcs-667	200	17	method	method	NOUN
eajbcs-667	200	18	.	.	PUNCT
eajbcs-667	201	1	by	by	ADP
eajbcs-667	201	2	using	use	VERB
eajbcs-667	201	3	the	the	DET
eajbcs-667	201	4	back	back	ADJ
eajbcs-667	201	5	ward	ward	NOUN
eajbcs-667	201	6	euler	euler	NOUN
eajbcs-667	201	7	scheme	scheme	PROPN
eajbcs-667	201	8	,	,	PUNCT
eajbcs-667	201	9	the	the	DET
eajbcs-667	201	10	system	system	NOUN
eajbcs-667	201	11	eq.12	eq.12	AUX
eajbcs-667	201	12	results	result	VERB
eajbcs-667	201	13	the	the	DET
eajbcs-667	201	14	following	follow	VERB
eajbcs-667	201	15	system	system	NOUN
eajbcs-667	201	16	of	of	ADP
eajbcs-667	201	17	algebraic	algebraic	ADJ
eajbcs-667	201	18	equations	equation	NOUN
eajbcs-667	201	19	:	:	PUNCT
eajbcs-667	201	20	euler	euler	VERB
eajbcs-667	201	21	back	back	ADP
eajbcs-667	201	22	ward	ward	NOUN
eajbcs-667	201	23	represents	represent	VERB
eajbcs-667	201	24	an	an	DET
eajbcs-667	201	25	implicit	implicit	ADJ
eajbcs-667	201	26	scheme	scheme	NOUN
eajbcs-667	201	27	which	which	PRON
eajbcs-667	201	28	is	be	AUX
eajbcs-667	201	29	stable	stable	ADJ
eajbcs-667	201	30	for	for	ADP
eajbcs-667	201	31	all	all	DET
eajbcs-667	201	32	choices	choice	NOUN
eajbcs-667	201	33	of	of	ADP
eajbcs-667	201	34	∆t	∆t	PROPN
eajbcs-667	201	35	(	(	PUNCT
eajbcs-667	201	36	?	?	PUNCT
eajbcs-667	201	37	)	)	PUNCT
eajbcs-667	201	38	.	.	PUNCT
eajbcs-667	202	1	since	since	SCONJ
eajbcs-667	202	2	the	the	DET
eajbcs-667	202	3	scheme	scheme	NOUN
eajbcs-667	202	4	is	be	AUX
eajbcs-667	202	5	implicit	implicit	ADJ
eajbcs-667	202	6	,	,	PUNCT
eajbcs-667	202	7	we	we	PRON
eajbcs-667	202	8	have	have	VERB
eajbcs-667	202	9	to	to	PART
eajbcs-667	202	10	solve	solve	VERB
eajbcs-667	202	11	a	a	DET
eajbcs-667	202	12	system	system	NOUN
eajbcs-667	202	13	of	of	ADP
eajbcs-667	202	14	algebraic	algebraic	ADJ
eajbcs-667	202	15	equations	equation	NOUN
eajbcs-667	202	16	at	at	ADP
eajbcs-667	202	17	each	each	DET
eajbcs-667	202	18	time	time	NOUN
eajbcs-667	202	19	step	step	NOUN
eajbcs-667	202	20	.	.	PUNCT
eajbcs-667	203	1	lack	lack	VERB
eajbcs-667	203	2	common	common	ADJ
eajbcs-667	203	3	support	support	NOUN
eajbcs-667	203	4	for	for	ADP
eajbcs-667	203	5	|i−	|i−	NOUN
eajbcs-667	203	6	j|	j|	PROPN
eajbcs-667	203	7	>	>	X
eajbcs-667	203	8	1	1	NUM
eajbcs-667	203	9	only	only	ADV
eajbcs-667	203	10	mi	mi	PROPN
eajbcs-667	203	11	,	,	PUNCT
eajbcs-667	203	12	i	i	PROPN
eajbcs-667	203	13	,	,	PUNCT
eajbcs-667	203	14	mi	mi	PROPN
eajbcs-667	203	15	,	,	PUNCT
eajbcs-667	203	16	i+1	i+1	X
eajbcs-667	203	17	,	,	PUNCT
eajbcs-667	203	18	and	and	CCONJ
eajbcs-667	203	19	mi+1,i	mi+1,i	NOUN
eajbcs-667	203	20	need	need	VERB
eajbcs-667	203	21	to	to	PART
eajbcs-667	203	22	be	be	AUX
eajbcs-667	203	23	calculated	calculate	VERB
eajbcs-667	203	24	.	.	PUNCT
eajbcs-667	204	1	all	all	DET
eajbcs-667	204	2	other	other	ADJ
eajbcs-667	204	3	matrix	matrix	NOUN
eajbcs-667	204	4	entries	entry	NOUN
eajbcs-667	204	5	are	be	AUX
eajbcs-667	204	6	zero	zero	NUM
eajbcs-667	204	7	by	by	ADP
eajbcs-667	204	8	default	default	NOUN
eajbcs-667	204	9	.	.	PUNCT
eajbcs-667	205	1	this	this	PRON
eajbcs-667	205	2	is	be	AUX
eajbcs-667	205	3	clearly	clearly	ADV
eajbcs-667	205	4	seen	see	VERB
eajbcs-667	205	5	from	from	ADP
eajbcs-667	205	6	figure	figure	NOUN
eajbcs-667	205	7	2	2	NUM
eajbcs-667	205	8	,	,	PUNCT
eajbcs-667	205	9	(	(	PUNCT
eajbcs-667	205	10	larson	larson	PROPN
eajbcs-667	205	11	and	and	CCONJ
eajbcs-667	205	12	bengzon	bengzon	PROPN
eajbcs-667	205	13	,	,	PUNCT
eajbcs-667	205	14	2010	2010	NUM
eajbcs-667	205	15	)	)	PUNCT
eajbcs-667	205	16	showing	show	VERB
eajbcs-667	205	17	two	two	NUM
eajbcs-667	205	18	neighboring	neighboring	NOUN
eajbcs-667	205	19	hat	hat	NOUN
eajbcs-667	205	20	functions	function	NOUN
eajbcs-667	205	21	and	and	CCONJ
eajbcs-667	205	22	their	their	PRON
eajbcs-667	205	23	support	support	NOUN
eajbcs-667	205	24	.	.	PUNCT
eajbcs-667	206	1	as	as	ADP
eajbcs-667	206	2	a	a	DET
eajbcs-667	206	3	consequence	consequence	NOUN
eajbcs-667	206	4	,	,	PUNCT
eajbcs-667	206	5	the	the	DET
eajbcs-667	206	6	mass	mass	ADJ
eajbcs-667	206	7	matrix	matrix	NOUN
eajbcs-667	206	8	m	m	VERB
eajbcs-667	206	9	is	be	AUX
eajbcs-667	206	10	tridiagonal	tridiagonal	ADJ
eajbcs-667	206	11	.	.	PUNCT
eajbcs-667	207	1	let	let	VERB
eajbcs-667	207	2	us	we	PRON
eajbcs-667	207	3	now	now	ADV
eajbcs-667	207	4	go	go	VERB
eajbcs-667	207	5	through	through	ADP
eajbcs-667	207	6	the	the	DET
eajbcs-667	207	7	details	detail	NOUN
eajbcs-667	207	8	of	of	ADP
eajbcs-667	207	9	how	how	SCONJ
eajbcs-667	207	10	to	to	PART
eajbcs-667	207	11	assemble	assemble	VERB
eajbcs-667	207	12	the	the	DET
eajbcs-667	207	13	mass	mass	ADJ
eajbcs-667	207	14	matrix	matrix	NOUN
eajbcs-667	207	15	m	m	VERB
eajbcs-667	207	16	.	.	PUNCT
eajbcs-667	208	1	we	we	PRON
eajbcs-667	208	2	begin	begin	VERB
eajbcs-667	208	3	by	by	ADP
eajbcs-667	208	4	calculating	calculate	VERB
eajbcs-667	208	5	the	the	DET
eajbcs-667	208	6	entries	entry	NOUN
eajbcs-667	208	7	mi	mi	PROPN
eajbcs-667	208	8	,	,	PUNCT
eajbcs-667	208	9	j	j	PROPN
eajbcs-667	208	10	of	of	ADP
eajbcs-667	208	11	the	the	DET
eajbcs-667	208	12	mass	mass	ADJ
eajbcs-667	208	13	matrix	matrix	NOUN
eajbcs-667	208	14	,	,	PUNCT
eajbcs-667	208	15	which	which	PRON
eajbcs-667	208	16	involve	involve	VERB
eajbcs-667	208	17	products	product	NOUN
eajbcs-667	208	18	of	of	ADP
eajbcs-667	208	19	hat	hat	NOUN
eajbcs-667	208	20	functions	function	NOUN
eajbcs-667	208	21	given	give	VERB
eajbcs-667	208	22	in	in	ADP
eajbcs-667	208	23	eq	eq	ADJ
eajbcs-667	208	24	.	.	PROPN
eajbcs-667	208	25	7	7	X
eajbcs-667	208	26	.	.	X
eajbcs-667	208	27	since	since	SCONJ
eajbcs-667	208	28	each	each	DET
eajbcs-667	208	29	hat	hat	NOUN
eajbcs-667	208	30	is	be	AUX
eajbcs-667	208	31	a	a	DET
eajbcs-667	208	32	linear	linear	ADJ
eajbcs-667	208	33	polynomial	polynomial	NOUN
eajbcs-667	208	34	,	,	PUNCT
eajbcs-667	208	35	the	the	DET
eajbcs-667	208	36	product	product	NOUN
eajbcs-667	208	37	of	of	ADP
eajbcs-667	208	38	two	two	NUM
eajbcs-667	208	39	hats	hat	NOUN
eajbcs-667	208	40	is	be	AUX
eajbcs-667	208	41	a	a	DET
eajbcs-667	208	42	quadratic	quadratic	ADJ
eajbcs-667	208	43	polynomials	polynomial	NOUN
eajbcs-667	208	44	.	.	PUNCT
eajbcs-667	209	1	thus	thus	ADV
eajbcs-667	209	2	,	,	PUNCT
eajbcs-667	209	3	simpson	simpson	PROPN
eajbcs-667	209	4	’s	’s	PART
eajbcs-667	209	5	formula	formula	NOUN
eajbcs-667	209	6	(	(	PUNCT
eajbcs-667	209	7	eq	eq	NOUN
eajbcs-667	209	8	.	.	NOUN
eajbcs-667	209	9	18	18	NUM
eajbcs-667	209	10	)	)	PUNCT
eajbcs-667	209	11	can	can	AUX
eajbcs-667	209	12	be	be	AUX
eajbcs-667	209	13	used	use	VERB
eajbcs-667	209	14	to	to	PART
eajbcs-667	209	15	integrate	integrate	VERB
eajbcs-667	209	16	mi	mi	PROPN
eajbcs-667	209	17	,	,	PUNCT
eajbcs-667	209	18	j	j	PROPN
eajbcs-667	210	1	=	=	PUNCT
eajbcs-667	210	2	here	here	ADV
eajbcs-667	210	3	m	m	VERB
eajbcs-667	210	4	and	and	CCONJ
eajbcs-667	210	5	a	a	PRON
eajbcs-667	210	6	are	be	AUX
eajbcs-667	210	7	tridiagonal	tridiagonal	ADJ
eajbcs-667	210	8	matrices	matrix	NOUN
eajbcs-667	210	9	and	and	CCONJ
eajbcs-667	210	10	eq.12	eq.12	NOUN
eajbcs-667	210	11	is	be	AUX
eajbcs-667	210	12	a	a	DET
eajbcs-667	210	13	simple	simple	ADJ
eajbcs-667	210	14	system	system	NOUN
eajbcs-667	210	15	of	of	ADP
eajbcs-667	210	16	ode	ode	PROPN
eajbcs-667	210	17	.	.	PUNCT
eajbcs-667	211	1	a	a	DET
eajbcs-667	211	2	f(x)dx	f(x)dx	NUM
eajbcs-667	211	3	=	=	SYM
eajbcs-667	211	4	f(a	f(a	NOUN
eajbcs-667	211	5	)	)	PUNCT
eajbcs-667	212	1	+	+	CCONJ
eajbcs-667	212	2	4f(a+b	4f(a+b	NUM
eajbcs-667	212	3	2	2	NUM
eajbcs-667	212	4	)	)	PUNCT
eajbcs-667	213	1	+	+	CCONJ
eajbcs-667	213	2	f(b	f(b	X
eajbcs-667	213	3	)	)	PUNCT
eajbcs-667	213	4	6	6	NUM
eajbcs-667	213	5	(	(	PUNCT
eajbcs-667	213	6	b−a	b−a	NOUN
eajbcs-667	213	7	)	)	PUNCT
eajbcs-667	213	8	.	.	PUNCT
eajbcs-667	214	1	(	(	PUNCT
eajbcs-667	214	2	14	14	NUM
eajbcs-667	214	3	)	)	PUNCT
eajbcs-667	214	4	let	let	VERB
eajbcs-667	214	5	we	we	PRON
eajbcs-667	214	6	start	start	VERB
eajbcs-667	214	7	on	on	ADP
eajbcs-667	214	8	the	the	DET
eajbcs-667	214	9	diagonal	diagonal	ADJ
eajbcs-667	214	10	entries	entry	NOUN
eajbcs-667	214	11	of	of	ADP
eajbcs-667	214	12	m	m	PROPN
eajbcs-667	214	13	,	,	PUNCT
eajbcs-667	214	14	mi	mi	PROPN
eajbcs-667	214	15	,	,	PUNCT
eajbcs-667	214	16	i	i	PRON
eajbcs-667	214	17	using	use	VERB
eajbcs-667	214	18	simpson	simpson	PROPN
eajbcs-667	214	19	’s	’s	PART
eajbcs-667	214	20	formula	formula	NOUN
eajbcs-667	214	21	:	:	PUNCT
eajbcs-667	214	22	mi	mi	PROPN
eajbcs-667	214	23	,	,	PUNCT
eajbcs-667	215	1	i	i	PRON
eajbcs-667	215	2	=	=	NOUN
eajbcs-667	216	1	∫	∫	PROPN
eajbcs-667	216	2	ω	ω	NUM
eajbcs-667	217	1	ϕ2	ϕ2	ADV
eajbcs-667	217	2	i	i	PRON
eajbcs-667	217	3	dx	dx	PROPN
eajbcs-667	217	4	,	,	PUNCT
eajbcs-667	217	5	=	=	SYM
eajbcs-667	217	6	∫	∫	PROPN
eajbcs-667	217	7	xi	xi	INTJ
eajbcs-667	218	1	xi−1	xi−1	PROPN
eajbcs-667	219	1	ϕ2	ϕ2	ADV
eajbcs-667	219	2	i	i	PRON
eajbcs-667	219	3	dx+	dx+	ADV
eajbcs-667	219	4	∫	∫	PROPN
eajbcs-667	220	1	xi+1	xi+1	PROPN
eajbcs-667	220	2	xi	xi	PUNCT
eajbcs-667	221	1	ϕ2	ϕ2	ADV
eajbcs-667	221	2	i	i	PRON
eajbcs-667	221	3	dx	dx	PROPN
eajbcs-667	221	4	,	,	PUNCT
eajbcs-667	221	5	=	=	SYM
eajbcs-667	221	6	0	0	NUM
eajbcs-667	222	1	+	+	NUM
eajbcs-667	222	2	4(1	4(1	NUM
eajbcs-667	222	3	2	2	NUM
eajbcs-667	222	4	)	)	PUNCT
eajbcs-667	222	5	2	2	NUM
eajbcs-667	222	6	+	+	CCONJ
eajbcs-667	222	7	1	1	NUM
eajbcs-667	222	8	6	6	NUM
eajbcs-667	222	9	hi	hi	INTJ
eajbcs-667	222	10	+	+	CCONJ
eajbcs-667	222	11	1	1	NUM
eajbcs-667	222	12	+	+	NUM
eajbcs-667	222	13	4(1	4(1	NUM
eajbcs-667	222	14	2	2	NUM
eajbcs-667	222	15	)	)	PUNCT
eajbcs-667	222	16	2	2	NUM
eajbcs-667	222	17	+	+	CCONJ
eajbcs-667	222	18	0	0	NUM
eajbcs-667	222	19	6	6	NUM
eajbcs-667	222	20	hi+1	hi+1	NOUN
eajbcs-667	222	21	,	,	PUNCT
eajbcs-667	222	22	=	=	SYM
eajbcs-667	222	23	hi	hi	ADJ
eajbcs-667	222	24	3	3	NUM
eajbcs-667	222	25	+	+	CCONJ
eajbcs-667	222	26	hi+1	hi+1	SYM
eajbcs-667	222	27	3	3	NUM
eajbcs-667	222	28	,	,	PUNCT
eajbcs-667	222	29	for	for	ADP
eajbcs-667	222	30	i	i	PROPN
eajbcs-667	222	31	=	=	SYM
eajbcs-667	222	32	1	1	NUM
eajbcs-667	222	33	,	,	PUNCT
eajbcs-667	222	34	2	2	NUM
eajbcs-667	222	35	,	,	PUNCT
eajbcs-667	222	36	.	.	PUNCT
eajbcs-667	222	37	.	.	PUNCT
eajbcs-667	222	38	.	.	PUNCT
eajbcs-667	223	1	,	,	PUNCT
eajbcs-667	223	2	n−	n−	NOUN
eajbcs-667	223	3	1	1	NUM
eajbcs-667	223	4	,	,	PUNCT
eajbcs-667	223	5	where	where	SCONJ
eajbcs-667	223	6	hi+1	hi+1	NOUN
eajbcs-667	223	7	=	=	SYM
eajbcs-667	223	8	xi+1−xi	xi+1−xi	PROPN
eajbcs-667	223	9	and	and	CCONJ
eajbcs-667	223	10	hi	hi	INTJ
eajbcs-667	223	11	=	=	SYM
eajbcs-667	223	12	xi−xi−1	xi−xi−1	NOUN
eajbcs-667	223	13	,	,	PUNCT
eajbcs-667	223	14	but	but	CCONJ
eajbcs-667	223	15	in	in	ADP
eajbcs-667	223	16	our	our	PRON
eajbcs-667	223	17	case	case	NOUN
eajbcs-667	223	18	we	we	PRON
eajbcs-667	223	19	use	use	VERB
eajbcs-667	223	20	a	a	DET
eajbcs-667	223	21	uniform	uniform	ADJ
eajbcs-667	223	22	mesh	mesh	NOUN
eajbcs-667	223	23	length	length	NOUN
eajbcs-667	223	24	h	h	PROPN
eajbcs-667	224	1	=	=	SYM
eajbcs-667	224	2	xi+1	xi+1	PROPN
eajbcs-667	225	1	−	−	NOUN
eajbcs-667	225	2	xi	xi	X
eajbcs-667	226	1	=	=	PUNCT
eajbcs-667	226	2	xi	xi	PROPN
eajbcs-667	226	3	−	−	PROPN
eajbcs-667	226	4	xi−1	xi−1	PROPN
eajbcs-667	226	5	and	and	CCONJ
eajbcs-667	226	6	mii	mii	PROPN
eajbcs-667	226	7	=	=	SYM
eajbcs-667	226	8	2h	2h	NUM
eajbcs-667	226	9	3	3	NUM
eajbcs-667	226	10	.	.	PUNCT
eajbcs-667	227	1	the	the	DET
eajbcs-667	227	2	first	first	ADJ
eajbcs-667	227	3	and	and	CCONJ
eajbcs-667	227	4	last	last	ADJ
eajbcs-667	227	5	diagonal	diagonal	ADJ
eajbcs-667	227	6	entry	entry	NOUN
eajbcs-667	227	7	are	be	AUX
eajbcs-667	227	8	m00	m00	NOUN
eajbcs-667	227	9	=	=	NOUN
eajbcs-667	227	10	h1	h1	PROPN
eajbcs-667	227	11	3	3	NUM
eajbcs-667	227	12	and	and	CCONJ
eajbcs-667	227	13	mnn	mnn	NOUN
eajbcs-667	228	1	=	=	NOUN
eajbcs-667	228	2	hn	hn	PROPN
eajbcs-667	228	3	3	3	NUM
eajbcs-667	228	4	,	,	PUNCT
eajbcs-667	228	5	respectively	respectively	ADV
eajbcs-667	228	6	,	,	PUNCT
eajbcs-667	228	7	since	since	SCONJ
eajbcs-667	228	8	the	the	DET
eajbcs-667	228	9	hat	hat	NOUN
eajbcs-667	228	10	functions	function	NOUN
eajbcs-667	228	11	ϕ0	ϕ0	NOUN
eajbcs-667	228	12	and	and	CCONJ
eajbcs-667	228	13	ϕn	ϕn	NOUN
eajbcs-667	228	14	are	be	AUX
eajbcs-667	228	15	only	only	ADV
eajbcs-667	228	16	half	half	ADJ
eajbcs-667	228	17	.	.	PUNCT
eajbcs-667	229	1	formula	formula	NOUN
eajbcs-667	229	2	we	we	PRON
eajbcs-667	229	3	have	have	VERB
eajbcs-667	229	4	mi+1,i	mi+1,i	NOUN
eajbcs-667	229	5	=	=	SYM
eajbcs-667	229	6	∫	∫	PROPN
eajbcs-667	229	7	ω	ω	PROPN
eajbcs-667	229	8	ϕi+1ϕidx	ϕi+1ϕidx	PROPN
eajbcs-667	229	9	,	,	PUNCT
eajbcs-667	229	10	=	=	SYM
eajbcs-667	229	11	∫	∫	PROPN
eajbcs-667	229	12	xi+1	xi+1	PROPN
eajbcs-667	229	13	xi	xi	PROPN
eajbcs-667	229	14	ϕi+1ϕidx	ϕi+1ϕidx	PROPN
eajbcs-667	229	15	,	,	PUNCT
eajbcs-667	229	16	=	=	PUNCT
eajbcs-667	229	17	1.0	1.0	NUM
eajbcs-667	229	18	+	+	NUM
eajbcs-667	229	19	4(1	4(1	NUM
eajbcs-667	229	20	2	2	NUM
eajbcs-667	229	21	)	)	PUNCT
eajbcs-667	229	22	2	2	NUM
eajbcs-667	229	23	+	+	CCONJ
eajbcs-667	229	24	0.1	0.1	NUM
eajbcs-667	229	25	6	6	NUM
eajbcs-667	229	26	hi+1	hi+1	NOUN
eajbcs-667	229	27	=	=	SYM
eajbcs-667	229	28	hi+1	hi+1	SYM
eajbcs-667	229	29	6	6	NUM
eajbcs-667	229	30	,	,	PUNCT
eajbcs-667	229	31	for	for	ADP
eajbcs-667	229	32	i	i	PROPN
eajbcs-667	229	33	=	=	SYM
eajbcs-667	229	34	0	0	NUM
eajbcs-667	229	35	,	,	PUNCT
eajbcs-667	229	36	1	1	NUM
eajbcs-667	229	37	,	,	PUNCT
eajbcs-667	229	38	2	2	NUM
eajbcs-667	229	39	,	,	PUNCT
eajbcs-667	229	40	.	.	PUNCT
eajbcs-667	229	41	.	.	PUNCT
eajbcs-667	230	1	.	.	PUNCT
eajbcs-667	231	1	,	,	PUNCT
eajbcs-667	231	2	n−	n−	NOUN
eajbcs-667	231	3	1	1	NUM
eajbcs-667	231	4	.	.	PUNCT
eajbcs-667	232	1	by	by	ADP
eajbcs-667	232	2	using	use	VERB
eajbcs-667	232	3	a	a	DET
eajbcs-667	232	4	similar	similar	ADJ
eajbcs-667	232	5	calculation	calculation	NOUN
eajbcs-667	232	6	the	the	DET
eajbcs-667	232	7	super	super	ADV
eajbcs-667	232	8	diagonal	diagonal	ADJ
eajbcs-667	232	9	entries	entry	NOUN
eajbcs-667	232	10	mi	mi	PROPN
eajbcs-667	232	11	,	,	PUNCT
eajbcs-667	232	12	i+1	i+1	NUM
eajbcs-667	232	13	have	have	VERB
eajbcs-667	232	14	the	the	DET
eajbcs-667	232	15	values	value	NOUN
eajbcs-667	232	16	mi	mi	PROPN
eajbcs-667	232	17	,	,	PUNCT
eajbcs-667	232	18	i+1	i+1	ADP
eajbcs-667	232	19	=	=	SYM
eajbcs-667	232	20	hi+1	hi+1	SYM
eajbcs-667	232	21	6	6	NUM
eajbcs-667	232	22	,	,	PUNCT
eajbcs-667	232	23	for	for	ADP
eajbcs-667	232	24	i	i	PROPN
eajbcs-667	232	25	=	=	SYM
eajbcs-667	232	26	0	0	NUM
eajbcs-667	232	27	,	,	PUNCT
eajbcs-667	232	28	1	1	NUM
eajbcs-667	232	29	,	,	PUNCT
eajbcs-667	232	30	2	2	NUM
eajbcs-667	232	31	,	,	PUNCT
eajbcs-667	232	32	.	.	PUNCT
eajbcs-667	232	33	.	.	PUNCT
eajbcs-667	233	1	.	.	PUNCT
eajbcs-667	234	1	,	,	PUNCT
eajbcs-667	234	2	n−	n−	NOUN
eajbcs-667	234	3	1	1	NUM
eajbcs-667	234	4	.	.	PUNCT
eajbcs-667	235	1	hence	hence	ADV
eajbcs-667	235	2	,	,	PUNCT
eajbcs-667	235	3	the	the	DET
eajbcs-667	235	4	mass	mass	ADJ
eajbcs-667	235	5	matrix	matrix	NOUN
eajbcs-667	235	6	m	m	AUX
eajbcs-667	235	7	takes	take	VERB
eajbcs-667	235	8	the	the	DET
eajbcs-667	235	9	form	form	NOUN
eajbcs-667	235	10	m	m	NOUN
eajbcs-667	235	11	=	=	ADJ
eajbcs-667	235	12			NOUN
eajbcs-667	235	13	h1	h1	VERB
eajbcs-667	235	14	3	3	NUM
eajbcs-667	235	15	h1	h1	PROPN
eajbcs-667	235	16	6	6	NUM
eajbcs-667	235	17	h1	h1	NOUN
eajbcs-667	235	18	6	6	NUM
eajbcs-667	235	19	h1	h1	NOUN
eajbcs-667	235	20	3	3	NUM
eajbcs-667	235	21	+	+	NUM
eajbcs-667	235	22	h2	h2	PROPN
eajbcs-667	235	23	3	3	NUM
eajbcs-667	235	24	h2	h2	NOUN
eajbcs-667	235	25	6	6	NUM
eajbcs-667	235	26	h2	h2	NOUN
eajbcs-667	235	27	6	6	NUM
eajbcs-667	235	28	h2	h2	NOUN
eajbcs-667	235	29	3	3	NUM
eajbcs-667	235	30	+	+	NUM
eajbcs-667	235	31	h3	h3	NOUN
eajbcs-667	235	32	3	3	NUM
eajbcs-667	235	33	h3	h3	NOUN
eajbcs-667	235	34	6	6	NUM
eajbcs-667	235	35	.	.	PUNCT
eajbcs-667	235	36	.	.	PUNCT
eajbcs-667	235	37	.	.	PUNCT
eajbcs-667	235	38	.	.	PUNCT
eajbcs-667	235	39	.	.	PUNCT
eajbcs-667	235	40	.	.	PUNCT
eajbcs-667	235	41	.	.	PUNCT
eajbcs-667	235	42	.	.	PUNCT
eajbcs-667	235	43	.	.	PUNCT
eajbcs-667	236	1	hn−1	hn−1	ADJ
eajbcs-667	236	2	6	6	NUM
eajbcs-667	236	3	hn−1	hn−1	ADJ
eajbcs-667	236	4	3	3	NUM
eajbcs-667	236	5	+	+	CCONJ
eajbcs-667	236	6	hn	hn	PROPN
eajbcs-667	236	7	3	3	NUM
eajbcs-667	237	1	hn	hn	NOUN
eajbcs-667	237	2	6	6	NUM
eajbcs-667	237	3	hn	hn	NOUN
eajbcs-667	237	4	6	6	NUM
eajbcs-667	237	5	hn	hn	NOUN
eajbcs-667	237	6	3	3	NUM
eajbcs-667	237	7			NOUN
eajbcs-667	237	8	.	.	PUNCT
eajbcs-667	238	1	(	(	PUNCT
eajbcs-667	238	2	15	15	NUM
eajbcs-667	238	3	)	)	PUNCT
eajbcs-667	238	4	the	the	DET
eajbcs-667	238	5	global	global	ADJ
eajbcs-667	238	6	mass	mass	PROPN
eajbcs-667	238	7	matrix	matrix	NOUN
eajbcs-667	238	8	m	m	VERB
eajbcs-667	238	9	can	can	AUX
eajbcs-667	238	10	be	be	AUX
eajbcs-667	238	11	written	write	VERB
eajbcs-667	238	12	as	as	ADP
eajbcs-667	238	13	a	a	DET
eajbcs-667	238	14	sum	sum	NOUN
eajbcs-667	238	15	of	of	ADP
eajbcs-667	238	16	n	n	CCONJ
eajbcs-667	238	17	simpler	simple	ADJ
eajbcs-667	238	18	elemental	elemental	ADJ
eajbcs-667	238	19	matrices	matrix	NOUN
eajbcs-667	238	20	as	as	ADP
eajbcs-667	238	21	:	:	PUNCT
eajbcs-667	238	22	m	m	NOUN
eajbcs-667	238	23	=	=	ADJ
eajbcs-667	238	24			NOUN
eajbcs-667	238	25	h1	h1	VERB
eajbcs-667	238	26	3	3	NUM
eajbcs-667	238	27	h1	h1	PROPN
eajbcs-667	238	28	6	6	NUM
eajbcs-667	238	29	h1	h1	NOUN
eajbcs-667	238	30	6	6	NUM
eajbcs-667	238	31	h1	h1	NOUN
eajbcs-667	238	32	3	3	NUM
eajbcs-667	238	33	+	+	VERB
eajbcs-667	238	34			ADJ
eajbcs-667	238	35	h2	h2	NOUN
eajbcs-667	238	36	3	3	NUM
eajbcs-667	238	37	h2	h2	NOUN
eajbcs-667	238	38	6	6	NUM
eajbcs-667	238	39	h2	h2	NOUN
eajbcs-667	238	40	6	6	NUM
eajbcs-667	238	41	h2	h2	NOUN
eajbcs-667	238	42	3	3	NUM
eajbcs-667	238	43	+	+	VERB
eajbcs-667	238	44	.	.	PUNCT
eajbcs-667	238	45	.	.	PUNCT
eajbcs-667	239	1	.+	.+	NOUN
eajbcs-667	239	2			NOUN
eajbcs-667	240	1	hn	hn	NOUN
eajbcs-667	240	2	3	3	NUM
eajbcs-667	241	1	hn	hn	NOUN
eajbcs-667	241	2	6	6	NUM
eajbcs-667	241	3	hn	hn	NOUN
eajbcs-667	241	4	6	6	NUM
eajbcs-667	241	5	hn	hn	NOUN
eajbcs-667	241	6	3	3	NUM
eajbcs-667	241	7			NOUN
eajbcs-667	241	8	.	.	PUNCT
eajbcs-667	242	1	i.e.	i.e.	X
eajbcs-667	242	2	,	,	PUNCT
eajbcs-667	242	3	m	m	AUX
eajbcs-667	242	4	=	=	NOUN
eajbcs-667	242	5	mω1	mω1	NOUN
eajbcs-667	242	6	+	+	NOUN
eajbcs-667	242	7	mω2	mω2	NOUN
eajbcs-667	242	8	+	+	X
eajbcs-667	242	9	.	.	PUNCT
eajbcs-667	242	10	.	.	PUNCT
eajbcs-667	243	1	.+mωn	.+mωn	PUNCT
eajbcs-667	243	2	.	.	PUNCT
eajbcs-667	244	1	e	e	X
eajbcs-667	244	2	m	m	NOUN
eajbcs-667	244	3	e	e	NOUN
eajbcs-667	244	4	=	=	NOUN
eajbcs-667	244	5	h	h	PROPN
eajbcs-667	244	6	6	6	NUM
eajbcs-667	244	7	[	[	PUNCT
eajbcs-667	244	8	2	2	NUM
eajbcs-667	244	9	1	1	NUM
eajbcs-667	244	10	1	1	NUM
eajbcs-667	244	11	2	2	NUM
eajbcs-667	244	12	]	]	PUNCT
eajbcs-667	244	13	,	,	PUNCT
eajbcs-667	244	14	where	where	SCONJ
eajbcs-667	244	15	h	h	NOUN
eajbcs-667	244	16	is	be	AUX
eajbcs-667	244	17	the	the	DET
eajbcs-667	244	18	length	length	NOUN
eajbcs-667	244	19	of	of	ADP
eajbcs-667	244	20	e.	e.	PROPN
eajbcs-667	244	21	we	we	PRON
eajbcs-667	244	22	refer	refer	VERB
eajbcs-667	244	23	to	to	ADP
eajbcs-667	244	24	m	m	PROPN
eajbcs-667	244	25	e	e	NOUN
eajbcs-667	244	26	as	as	ADP
eajbcs-667	244	27	the	the	DET
eajbcs-667	244	28	local	local	ADJ
eajbcs-667	244	29	element	element	NOUN
eajbcs-667	244	30	mass	mass	NOUN
eajbcs-667	244	31	matrix	matrix	NOUN
eajbcs-667	244	32	.	.	PUNCT
eajbcs-667	245	1	the	the	DET
eajbcs-667	245	2	summation	summation	NOUN
eajbcs-667	245	3	of	of	ADP
eajbcs-667	245	4	the	the	DET
eajbcs-667	245	5	element	element	NOUN
eajbcs-667	245	6	mass	mass	NOUN
eajbcs-667	245	7	matrices	matrix	NOUN
eajbcs-667	245	8	into	into	ADP
eajbcs-667	245	9	the	the	DET
eajbcs-667	245	10	global	global	ADJ
eajbcs-667	245	11	mass	mass	NOUN
eajbcs-667	245	12	matrix	matrix	NOUN
eajbcs-667	245	13	is	be	AUX
eajbcs-667	245	14	called	call	VERB
eajbcs-667	245	15	assembling	assemble	VERB
eajbcs-667	245	16	.	.	PUNCT
eajbcs-667	246	1	east	east	PROPN
eajbcs-667	246	2	afr	afr	PROPN
eajbcs-667	246	3	.	.	PUNCT
eajbcs-667	247	1	j.	j.	PROPN
eajbcs-667	247	2	biophys	biophys	PROPN
eajbcs-667	247	3	.	.	PUNCT
eajbcs-667	248	1	comput	comput	NOUN
eajbcs-667	248	2	.	.	PUNCT
eajbcs-667	249	1	sci	sci	PROPN
eajbcs-667	249	2	.	.	PUNCT
eajbcs-667	249	3	(	(	PUNCT
eajbcs-667	249	4	2023	2023	NUM
eajbcs-667	249	5	)	)	PUNCT
eajbcs-667	249	6	,	,	PUNCT
eajbcs-667	249	7	vol	vol	NOUN
eajbcs-667	249	8	.	.	PROPN
eajbcs-667	249	9	4	4	NUM
eajbcs-667	249	10	,	,	PUNCT
eajbcs-667	249	11	no	no	INTJ
eajbcs-667	249	12	.	.	NOUN
eajbcs-667	249	13	1	1	NUM
eajbcs-667	249	14	,	,	PUNCT
eajbcs-667	249	15	52	52	NUM
eajbcs-667	249	16	-	-	SYM
eajbcs-667	249	17	74	74	NUM
eajbcs-667	249	18	60	60	NUM
eajbcs-667	249	19	on	on	ADP
eajbcs-667	249	20	the	the	DET
eajbcs-667	249	21	interval	interval	NOUN
eajbcs-667	250	1	i	i	PRON
eajbcs-667	250	2	=	=	PUNCT
eajbcs-667	250	3	(	(	PUNCT
eajbcs-667	250	4	a	a	DET
eajbcs-667	250	5	,	,	PUNCT
eajbcs-667	250	6	b	b	NOUN
eajbcs-667	250	7	)	)	PUNCT
eajbcs-667	250	8	,	,	PUNCT
eajbcs-667	250	9	simpson	simpson	PROPN
eajbcs-667	250	10	’s	’s	PART
eajbcs-667	250	11	formula	formula	NOUN
eajbcs-667	250	12	is	be	AUX
eajbcs-667	250	13	of	of	ADP
eajbcs-667	250	14	the	the	DET
eajbcs-667	250	15	form	form	NOUN
eajbcs-667	250	16	,	,	PUNCT
eajbcs-667	250	17	(	(	PUNCT
eajbcs-667	250	18	larson	larson	PROPN
eajbcs-667	250	19	and	and	CCONJ
eajbcs-667	250	20	bengzon	bengzon	PROPN
eajbcs-667	250	21	,	,	PUNCT
eajbcs-667	251	1	2010)∫	2010)∫	NUM
eajbcs-667	251	2	b	b	NOUN
eajbcs-667	251	3	we	we	PRON
eajbcs-667	251	4	know	know	VERB
eajbcs-667	251	5	continue	continue	VERB
eajbcs-667	251	6	with	with	ADP
eajbcs-667	251	7	the	the	DET
eajbcs-667	251	8	sub	sub	NOUN
eajbcs-667	251	9	diagonal	diagonal	ADJ
eajbcs-667	251	10	entries	entry	NOUN
eajbcs-667	251	11	mi+1,i	mi+1,i	NOUN
eajbcs-667	251	12	still	still	ADV
eajbcs-667	251	13	using	use	VERB
eajbcs-667	251	14	simpson	simpson	PROPN
eajbcs-667	251	15	’s	’s	PART
eajbcs-667	251	16	,	,	PUNCT
eajbcs-667	251	17	i	i	PRON
eajbcs-667	251	18	=	=	NOUN
eajbcs-667	251	19	1	1	NUM
eajbcs-667	251	20	,	,	PUNCT
eajbcs-667	251	21	2	2	NUM
eajbcs-667	251	22	,	,	PUNCT
eajbcs-667	251	23	.	.	PUNCT
eajbcs-667	251	24	.	.	PUNCT
eajbcs-667	252	1	.	.	PUNCT
eajbcs-667	253	1	,	,	PUNCT
eajbcs-667	253	2	n	n	CCONJ
eajbcs-667	253	3	,	,	PUNCT
eajbcs-667	253	4	is	be	AUX
eajbcs-667	253	5	obtained	obtain	VERB
eajbcs-667	253	6	by	by	ADP
eajbcs-667	253	7	restricting	restrict	VERB
eajbcs-667	253	8	the	the	DET
eajbcs-667	253	9	integration	integration	NOUN
eajbcs-667	253	10	to	to	ADP
eajbcs-667	253	11	one	one	NUM
eajbcs-667	253	12	sub	sub	NOUN
eajbcs-667	253	13	interval	interval	NOUN
eajbcs-667	253	14	or	or	CCONJ
eajbcs-667	253	15	element	element	NOUN
eajbcs-667	253	16	ωe	ωe	ADV
eajbcs-667	253	17	and	and	CCONJ
eajbcs-667	253	18	is	be	AUX
eajbcs-667	253	19	therefore	therefore	ADV
eajbcs-667	253	20	called	call	VERB
eajbcs-667	253	21	an	an	DET
eajbcs-667	253	22	element	element	ADJ
eajbcs-667	253	23	mass	mass	NOUN
eajbcs-667	253	24	matrix	matrix	NOUN
eajbcs-667	253	25	.	.	PUNCT
eajbcs-667	254	1	from	from	ADP
eajbcs-667	254	2	the	the	DET
eajbcs-667	254	3	sum	sum	NOUN
eajbcs-667	254	4	we	we	PRON
eajbcs-667	254	5	see	see	VERB
eajbcs-667	254	6	that	that	SCONJ
eajbcs-667	254	7	on	on	ADP
eajbcs-667	254	8	each	each	DET
eajbcs-667	254	9	element	element	NOUN
eajbcs-667	254	10	e	e	NOUN
eajbcs-667	254	11	this	this	DET
eajbcs-667	254	12	small	small	ADJ
eajbcs-667	254	13	block	block	NOUN
eajbcs-667	254	14	takes	take	VERB
eajbcs-667	254	15	the	the	DET
eajbcs-667	254	16	form	form	NOUN
eajbcs-667	254	17	:	:	PUNCT
eajbcs-667	254	18	each	each	DET
eajbcs-667	254	19	matrix	matrix	NOUN
eajbcs-667	254	20	mω	mω	VERB
eajbcs-667	254	21	implementation	implementation	NOUN
eajbcs-667	254	22	of	of	ADP
eajbcs-667	254	23	the	the	DET
eajbcs-667	254	24	1d	1d	NUM
eajbcs-667	254	25	diffusion	diffusion	NOUN
eajbcs-667	254	26	equation	equation	NOUN
eajbcs-667	254	27	let	let	VERB
eajbcs-667	254	28	we	we	PRON
eajbcs-667	254	29	now	now	ADV
eajbcs-667	254	30	consider	consider	VERB
eajbcs-667	254	31	the	the	DET
eajbcs-667	254	32	diffusion	diffusion	NOUN
eajbcs-667	254	33	equation	equation	NOUN
eajbcs-667	254	34	∂u	∂u	PROPN
eajbcs-667	254	35	∂t	∂t	PROPN
eajbcs-667	254	36	=	=	SYM
eajbcs-667	254	37	d	d	PROPN
eajbcs-667	254	38	∂2u	∂2u	PROPN
eajbcs-667	254	39	∂x2	∂x2	PROPN
eajbcs-667	254	40	.	.	PUNCT
eajbcs-667	255	1	we	we	PRON
eajbcs-667	255	2	multiply	multiply	VERB
eajbcs-667	255	3	the	the	DET
eajbcs-667	255	4	equation	equation	NOUN
eajbcs-667	255	5	by	by	ADP
eajbcs-667	255	6	a	a	DET
eajbcs-667	255	7	test	test	NOUN
eajbcs-667	255	8	function	function	NOUN
eajbcs-667	255	9	v(x	v(x	PROPN
eajbcs-667	255	10	)	)	PUNCT
eajbcs-667	255	11	,	,	PUNCT
eajbcs-667	255	12	which	which	PRON
eajbcs-667	255	13	satisfies	satisfy	VERB
eajbcs-667	255	14	the	the	DET
eajbcs-667	255	15	boundary	boundary	ADJ
eajbcs-667	255	16	conditions	condition	NOUN
eajbcs-667	255	17	v(0	v(0	NOUN
eajbcs-667	255	18	)	)	PUNCT
eajbcs-667	255	19	=	=	SYM
eajbcs-667	256	1	v(1	v(1	ADJ
eajbcs-667	256	2	)	)	PUNCT
eajbcs-667	256	3	=	=	SYM
eajbcs-667	256	4	0	0	PUNCT
eajbcs-667	256	5	and	and	CCONJ
eajbcs-667	256	6	then	then	ADV
eajbcs-667	256	7	integrating	integrate	VERB
eajbcs-667	256	8	by	by	ADP
eajbcs-667	256	9	parts	part	NOUN
eajbcs-667	256	10	we	we	PRON
eajbcs-667	256	11	have	have	VERB
eajbcs-667	256	12	that	that	PRON
eajbcs-667	256	13	:	:	PUNCT
eajbcs-667	257	1	∂u	∂u	PROPN
eajbcs-667	257	2	∂t	∂t	PROPN
eajbcs-667	257	3	.v	.v	PUNCT
eajbcs-667	258	1	=	=	PUNCT
eajbcs-667	258	2	d	d	PRON
eajbcs-667	258	3	∂2u	∂2u	PROPN
eajbcs-667	258	4	∂x2	∂x2	PROPN
eajbcs-667	258	5	.v	.v	NOUN
eajbcs-667	259	1	∫	∫	PROPN
eajbcs-667	259	2	1	1	NUM
eajbcs-667	259	3	0	0	NUM
eajbcs-667	260	1	(	(	PUNCT
eajbcs-667	260	2	∂u	∂u	PROPN
eajbcs-667	260	3	∂t	∂t	PROPN
eajbcs-667	260	4	.v	.v	PROPN
eajbcs-667	260	5	)	)	PUNCT
eajbcs-667	261	1	=	=	PUNCT
eajbcs-667	261	2	∫	∫	PROPN
eajbcs-667	262	1	1	1	NUM
eajbcs-667	262	2	0	0	NUM
eajbcs-667	263	1	(	(	PUNCT
eajbcs-667	263	2	d	d	PROPN
eajbcs-667	263	3	∂2u	∂2u	PROPN
eajbcs-667	263	4	∂x2	∂x2	PROPN
eajbcs-667	263	5	.v	.v	NOUN
eajbcs-667	263	6	)	)	PUNCT
eajbcs-667	263	7	hence	hence	ADV
eajbcs-667	263	8	by	by	ADP
eajbcs-667	263	9	using	use	VERB
eajbcs-667	263	10	integration	integration	NOUN
eajbcs-667	263	11	by	by	ADP
eajbcs-667	263	12	parts	part	NOUN
eajbcs-667	263	13	,	,	PUNCT
eajbcs-667	263	14	we	we	PRON
eajbcs-667	263	15	obtain∫	obtain∫	VERB
eajbcs-667	264	1	1	1	NUM
eajbcs-667	264	2	0	0	NUM
eajbcs-667	264	3	∂u	∂u	PROPN
eajbcs-667	264	4	∂t	∂t	PROPN
eajbcs-667	264	5	.v	.v	PROPN
eajbcs-667	265	1	=	=	PUNCT
eajbcs-667	265	2	−d	−d	PROPN
eajbcs-667	265	3	∫	∫	PROPN
eajbcs-667	265	4	1	1	NUM
eajbcs-667	265	5	0	0	NUM
eajbcs-667	265	6	∂u	∂u	PROPN
eajbcs-667	265	7	∂x	∂x	PROPN
eajbcs-667	265	8	.	.	PUNCT
eajbcs-667	266	1	∂v	∂v	PROPN
eajbcs-667	266	2	∂x	∂x	PROPN
eajbcs-667	266	3	,	,	PUNCT
eajbcs-667	266	4	(	(	PUNCT
eajbcs-667	266	5	16	16	NUM
eajbcs-667	266	6	)	)	PUNCT
eajbcs-667	266	7	which	which	PRON
eajbcs-667	266	8	is	be	AUX
eajbcs-667	266	9	the	the	DET
eajbcs-667	266	10	weak	weak	ADJ
eajbcs-667	266	11	formulation	formulation	NOUN
eajbcs-667	266	12	of	of	ADP
eajbcs-667	266	13	the	the	DET
eajbcs-667	266	14	one	one	NUM
eajbcs-667	266	15	dimensional	dimensional	ADJ
eajbcs-667	266	16	diffusion	diffusion	NOUN
eajbcs-667	266	17	equation	equation	NOUN
eajbcs-667	266	18	.	.	PUNCT
eajbcs-667	267	1	∫	∫	PROPN
eajbcs-667	267	2	1	1	NUM
eajbcs-667	267	3	0	0	NUM
eajbcs-667	267	4	∂	∂	NUM
eajbcs-667	268	1	∂t	∂t	PROPN
eajbcs-667	268	2	n−1∑	n−1∑	NUM
eajbcs-667	268	3	i=1	i=1	PROPN
eajbcs-667	268	4	uiϕi	uiϕi	ADJ
eajbcs-667	268	5	.	.	PUNCT
eajbcs-667	269	1	n−1∑	n−1∑	NUM
eajbcs-667	269	2	j=1	j=1	PROPN
eajbcs-667	269	3	vjϕj	vjϕj	NOUN
eajbcs-667	269	4	=	=	PUNCT
eajbcs-667	269	5	−d	−d	PROPN
eajbcs-667	269	6	∫	∫	PROPN
eajbcs-667	269	7	1	1	NUM
eajbcs-667	269	8	0	0	NUM
eajbcs-667	269	9	∂	∂	NUM
eajbcs-667	269	10	∂x	∂x	PROPN
eajbcs-667	269	11	n−1∑	n−1∑	NUM
eajbcs-667	269	12	i=1	i=1	PROPN
eajbcs-667	269	13	uiϕi	uiϕi	ADJ
eajbcs-667	269	14	.	.	PUNCT
eajbcs-667	270	1	∂	∂	NUM
eajbcs-667	271	1	∂x	∂x	PROPN
eajbcs-667	271	2	n−1∑	n−1∑	NUM
eajbcs-667	271	3	j=1	j=1	PROPN
eajbcs-667	271	4	vjϕj	vjϕj	NOUN
eajbcs-667	271	5	.	.	PUNCT
eajbcs-667	272	1	which	which	PRON
eajbcs-667	272	2	then	then	ADV
eajbcs-667	272	3	implies	imply	VERB
eajbcs-667	272	4	,	,	PUNCT
eajbcs-667	272	5	n−1∑	n−1∑	NUM
eajbcs-667	272	6	j=1	j=1	PROPN
eajbcs-667	272	7	vj	vj	X
eajbcs-667	272	8	(	(	PUNCT
eajbcs-667	272	9	∂	∂	NUM
eajbcs-667	272	10	∂t	∂t	PROPN
eajbcs-667	272	11	n−1∑	n−1∑	NUM
eajbcs-667	272	12	i=1	i=1	PROPN
eajbcs-667	272	13	ui	ui	PROPN
eajbcs-667	273	1	∫	∫	PROPN
eajbcs-667	273	2	1	1	NUM
eajbcs-667	273	3	0	0	NUM
eajbcs-667	273	4	ϕi.ϕj	ϕi.ϕj	NOUN
eajbcs-667	273	5	)	)	PUNCT
eajbcs-667	274	1	=	=	SYM
eajbcs-667	274	2	n−1∑	n−1∑	NUM
eajbcs-667	274	3	j=1	j=1	NOUN
eajbcs-667	274	4	vj	vj	PROPN
eajbcs-667	274	5	(	(	PUNCT
eajbcs-667	274	6	−d	−d	PROPN
eajbcs-667	274	7	n−1∑	n−1∑	PROPN
eajbcs-667	274	8	i=1	i=1	PROPN
eajbcs-667	274	9	ui	ui	PROPN
eajbcs-667	275	1	∫	∫	PROPN
eajbcs-667	275	2	1	1	NUM
eajbcs-667	275	3	0	0	NUM
eajbcs-667	275	4	ϕ′	ϕ′	X
eajbcs-667	276	1	i.ϕ	i.ϕ	INTJ
eajbcs-667	276	2	′	′	NUM
eajbcs-667	276	3	j	j	NOUN
eajbcs-667	276	4	)	)	PUNCT
eajbcs-667	276	5	.	.	PUNCT
eajbcs-667	277	1	that	that	PRON
eajbcs-667	277	2	is	be	AUX
eajbcs-667	277	3	∂	∂	NUM
eajbcs-667	277	4	∂t	∂t	PROPN
eajbcs-667	277	5	n−1∑	n−1∑	NUM
eajbcs-667	277	6	i=1	i=1	PROPN
eajbcs-667	278	1	ui	ui	PROPN
eajbcs-667	279	1	∫	∫	PROPN
eajbcs-667	280	1	1	1	NUM
eajbcs-667	280	2	0	0	NUM
eajbcs-667	280	3	ϕi.ϕj	ϕi.ϕj	NOUN
eajbcs-667	280	4	=	=	PRON
eajbcs-667	280	5	−d	−d	VERB
eajbcs-667	280	6	n−1∑	n−1∑	PROPN
eajbcs-667	280	7	i=1	i=1	PROPN
eajbcs-667	281	1	ui	ui	PROPN
eajbcs-667	282	1	∫	∫	PROPN
eajbcs-667	283	1	1	1	NUM
eajbcs-667	283	2	0	0	NUM
eajbcs-667	283	3	ϕ′	ϕ′	NOUN
eajbcs-667	284	1	i.ϕ	i.ϕ	INTJ
eajbcs-667	284	2	′	′	NOUN
eajbcs-667	284	3	j.	j.	NOUN
eajbcs-667	284	4	in	in	ADP
eajbcs-667	284	5	a	a	DET
eajbcs-667	284	6	matrix	matrix	NOUN
eajbcs-667	284	7	form	form	NOUN
eajbcs-667	284	8	it	it	PRON
eajbcs-667	284	9	can	can	AUX
eajbcs-667	284	10	be	be	AUX
eajbcs-667	284	11	written	write	VERB
eajbcs-667	284	12	as	as	ADP
eajbcs-667	284	13	:	:	PUNCT
eajbcs-667	284	14	mu̇	mu̇	X
eajbcs-667	285	1	+	+	NOUN
eajbcs-667	285	2	dau	dau	NOUN
eajbcs-667	285	3	=	=	SYM
eajbcs-667	285	4	0	0	NUM
eajbcs-667	285	5	,	,	PUNCT
eajbcs-667	285	6	(	(	PUNCT
eajbcs-667	285	7	17	17	NUM
eajbcs-667	285	8	)	)	PUNCT
eajbcs-667	285	9	assembly	assembly	NOUN
eajbcs-667	285	10	of	of	ADP
eajbcs-667	285	11	the	the	DET
eajbcs-667	285	12	stiffness	stiffness	ADJ
eajbcs-667	285	13	matrix	matrix	NOUN
eajbcs-667	285	14	in	in	ADP
eajbcs-667	285	15	1d	1d	NUM
eajbcs-667	285	16	the	the	DET
eajbcs-667	285	17	stiffness	stiffness	ADJ
eajbcs-667	285	18	matrix	matrix	NOUN
eajbcs-667	285	19	a	a	PRON
eajbcs-667	285	20	is	be	AUX
eajbcs-667	285	21	symmetric	symmetric	ADJ
eajbcs-667	285	22	for	for	ADP
eajbcs-667	285	23	this	this	DET
eajbcs-667	285	24	simple	simple	ADJ
eajbcs-667	285	25	problem	problem	NOUN
eajbcs-667	285	26	,	,	PUNCT
eajbcs-667	285	27	which	which	PRON
eajbcs-667	285	28	makes	make	VERB
eajbcs-667	285	29	the	the	DET
eajbcs-667	285	30	computation	computation	NOUN
eajbcs-667	285	31	of	of	ADP
eajbcs-667	285	32	the	the	DET
eajbcs-667	285	33	matrix	matrix	NOUN
eajbcs-667	285	34	faster	fast	ADV
eajbcs-667	285	35	since	since	SCONJ
eajbcs-667	285	36	we	we	PRON
eajbcs-667	285	37	do	do	AUX
eajbcs-667	285	38	n’t	not	PART
eajbcs-667	285	39	have	have	VERB
eajbcs-667	285	40	to	to	PART
eajbcs-667	285	41	compute	compute	VERB
eajbcs-667	285	42	all	all	PRON
eajbcs-667	285	43	of	of	ADP
eajbcs-667	285	44	the	the	DET
eajbcs-667	285	45	elements	element	NOUN
eajbcs-667	285	46	,	,	PUNCT
eajbcs-667	285	47	symmetric	symmetric	ADJ
eajbcs-667	285	48	matrices	matrix	NOUN
eajbcs-667	285	49	are	be	AUX
eajbcs-667	285	50	also	also	ADV
eajbcs-667	285	51	much	much	ADV
eajbcs-667	285	52	faster	fast	ADV
eajbcs-667	285	53	to	to	PART
eajbcs-667	285	54	invert	invert	VERB
eajbcs-667	285	55	.	.	PUNCT
eajbcs-667	286	1	here	here	ADV
eajbcs-667	286	2	ϕi	ϕi	ADP
eajbcs-667	286	3	’s	’	NOUN
eajbcs-667	286	4	are	be	AUX
eajbcs-667	286	5	the	the	DET
eajbcs-667	286	6	hat	hat	NOUN
eajbcs-667	286	7	functions	function	NOUN
eajbcs-667	286	8	given	give	VERB
eajbcs-667	286	9	in	in	ADP
eajbcs-667	286	10	eq	eq	ADJ
eajbcs-667	286	11	.	.	PROPN
eajbcs-667	286	12	7	7	NUM
eajbcs-667	286	13	,	,	PUNCT
eajbcs-667	286	14	the	the	DET
eajbcs-667	286	15	entries	entry	NOUN
eajbcs-667	286	16	of	of	ADP
eajbcs-667	286	17	each	each	DET
eajbcs-667	286	18	element	element	NOUN
eajbcs-667	286	19	of	of	ADP
eajbcs-667	286	20	the	the	DET
eajbcs-667	286	21	stiffness	stiffness	ADJ
eajbcs-667	286	22	matrix	matrix	NOUN
eajbcs-667	286	23	a	a	PRON
eajbcs-667	286	24	is	be	AUX
eajbcs-667	286	25	given	give	VERB
eajbcs-667	286	26	by	by	ADP
eajbcs-667	286	27	east	east	PROPN
eajbcs-667	286	28	afr	afr	PROPN
eajbcs-667	286	29	.	.	PUNCT
eajbcs-667	287	1	j.	j.	PROPN
eajbcs-667	287	2	biophys	biophys	PROPN
eajbcs-667	287	3	.	.	PUNCT
eajbcs-667	288	1	comput	comput	NOUN
eajbcs-667	288	2	.	.	PUNCT
eajbcs-667	289	1	sci	sci	PROPN
eajbcs-667	289	2	.	.	PUNCT
eajbcs-667	289	3	(	(	PUNCT
eajbcs-667	289	4	2023	2023	NUM
eajbcs-667	289	5	)	)	PUNCT
eajbcs-667	289	6	,	,	PUNCT
eajbcs-667	289	7	vol	vol	NOUN
eajbcs-667	289	8	.	.	PROPN
eajbcs-667	289	9	4	4	NUM
eajbcs-667	289	10	,	,	PUNCT
eajbcs-667	289	11	no	no	INTJ
eajbcs-667	289	12	.	.	NOUN
eajbcs-667	289	13	1	1	NUM
eajbcs-667	289	14	,	,	PUNCT
eajbcs-667	289	15	52	52	NUM
eajbcs-667	289	16	-	-	SYM
eajbcs-667	289	17	74	74	NUM
eajbcs-667	289	18	61	61	NUM
eajbcs-667	289	19	now	now	ADV
eajbcs-667	289	20	the	the	DET
eajbcs-667	289	21	system	system	NOUN
eajbcs-667	289	22	of	of	ADP
eajbcs-667	289	23	equation	equation	NOUN
eajbcs-667	289	24	,	,	PUNCT
eajbcs-667	289	25	that	that	PRON
eajbcs-667	289	26	is	be	AUX
eajbcs-667	289	27	eq.12	eq.12	PROPN
eajbcs-667	289	28	is	be	AUX
eajbcs-667	289	29	a	a	DET
eajbcs-667	289	30	simple	simple	ADJ
eajbcs-667	289	31	system	system	NOUN
eajbcs-667	289	32	of	of	ADP
eajbcs-667	289	33	ordinary	ordinary	ADJ
eajbcs-667	289	34	differential	differential	ADJ
eajbcs-667	289	35	equations	equation	NOUN
eajbcs-667	289	36	.	.	PUNCT
eajbcs-667	290	1	to	to	PART
eajbcs-667	290	2	solve	solve	VERB
eajbcs-667	290	3	this	this	DET
eajbcs-667	290	4	system	system	NOUN
eajbcs-667	290	5	of	of	ADP
eajbcs-667	290	6	ode	ode	PROPN
eajbcs-667	290	7	’s	’s	NOUN
eajbcs-667	290	8	,	,	PUNCT
eajbcs-667	290	9	we	we	PRON
eajbcs-667	290	10	have	have	VERB
eajbcs-667	290	11	to	to	PART
eajbcs-667	290	12	use	use	VERB
eajbcs-667	290	13	a	a	DET
eajbcs-667	290	14	back	back	ADJ
eajbcs-667	290	15	ward	ward	NOUN
eajbcs-667	290	16	euler	euler	NOUN
eajbcs-667	290	17	method	method	NOUN
eajbcs-667	290	18	and	and	CCONJ
eajbcs-667	290	19	a	a	DET
eajbcs-667	290	20	matlab	matlab	PROPN
eajbcs-667	290	21	soft	soft	ADJ
eajbcs-667	290	22	ware	ware	NOUN
eajbcs-667	290	23	to	to	PART
eajbcs-667	290	24	solve	solve	VERB
eajbcs-667	290	25	the	the	DET
eajbcs-667	290	26	system	system	NOUN
eajbcs-667	290	27	at	at	ADP
eajbcs-667	290	28	each	each	DET
eajbcs-667	290	29	time	time	NOUN
eajbcs-667	290	30	steps	step	NOUN
eajbcs-667	290	31	using	use	VERB
eajbcs-667	290	32	an	an	DET
eajbcs-667	290	33	initial	initial	ADJ
eajbcs-667	290	34	condition	condition	NOUN
eajbcs-667	290	35	.	.	PUNCT
eajbcs-667	291	1	where	where	SCONJ
eajbcs-667	291	2	m	m	NOUN
eajbcs-667	291	3	is	be	AUX
eajbcs-667	291	4	the	the	DET
eajbcs-667	291	5	mass	mass	ADJ
eajbcs-667	291	6	matrix	matrix	NOUN
eajbcs-667	291	7	with	with	ADP
eajbcs-667	291	8	entries	entry	NOUN
eajbcs-667	291	9	given	give	VERB
eajbcs-667	291	10	in	in	ADP
eajbcs-667	291	11	eq	eq	PROPN
eajbcs-667	291	12	.	.	PROPN
eajbcs-667	291	13	15	15	NUM
eajbcs-667	291	14	and	and	CCONJ
eajbcs-667	291	15	a	a	PRON
eajbcs-667	291	16	is	be	AUX
eajbcs-667	291	17	the	the	DET
eajbcs-667	291	18	stiffness	stiffness	ADJ
eajbcs-667	291	19	matrix	matrix	NOUN
eajbcs-667	291	20	with	with	ADP
eajbcs-667	291	21	entries	entry	NOUN
eajbcs-667	291	22	given	give	VERB
eajbcs-667	291	23	in	in	ADP
eajbcs-667	291	24	eq	eq	PROPN
eajbcs-667	291	25	.	.	PROPN
eajbcs-667	291	26	19	19	NUM
eajbcs-667	291	27	.	.	PUNCT
eajbcs-667	292	1	here	here	ADV
eajbcs-667	292	2	the	the	DET
eajbcs-667	292	3	matrix	matrix	NOUN
eajbcs-667	292	4	a	a	PRON
eajbcs-667	292	5	is	be	AUX
eajbcs-667	292	6	often	often	ADV
eajbcs-667	292	7	referred	refer	VERB
eajbcs-667	292	8	to	to	ADP
eajbcs-667	292	9	as	as	ADP
eajbcs-667	292	10	the	the	DET
eajbcs-667	292	11	stiffness	stiffness	ADJ
eajbcs-667	292	12	matrix	matrix	NOUN
eajbcs-667	292	13	,	,	PUNCT
eajbcs-667	292	14	a	a	DET
eajbcs-667	292	15	name	name	NOUN
eajbcs-667	292	16	coming	come	VERB
eajbcs-667	292	17	from	from	ADP
eajbcs-667	292	18	corresponding	correspond	VERB
eajbcs-667	292	19	matrices	matrix	NOUN
eajbcs-667	292	20	in	in	ADP
eajbcs-667	292	21	the	the	DET
eajbcs-667	292	22	context	context	NOUN
eajbcs-667	292	23	of	of	ADP
eajbcs-667	292	24	structural	structural	ADJ
eajbcs-667	292	25	problems	problem	NOUN
eajbcs-667	292	26	.	.	PUNCT
eajbcs-667	293	1	now	now	ADV
eajbcs-667	293	2	substituting	substitute	VERB
eajbcs-667	293	3	eq.10	eq.10	NOUN
eajbcs-667	293	4	and	and	CCONJ
eajbcs-667	293	5	eq.11	eq.11	NUM
eajbcs-667	293	6	in	in	ADP
eajbcs-667	293	7	the	the	DET
eajbcs-667	293	8	weak	weak	ADJ
eajbcs-667	293	9	formulation	formulation	NOUN
eajbcs-667	293	10	of	of	ADP
eajbcs-667	293	11	the	the	DET
eajbcs-667	293	12	equation	equation	NOUN
eajbcs-667	293	13	eq.16	eq.16	PROPN
eajbcs-667	293	14	,	,	PUNCT
eajbcs-667	293	15	we	we	PRON
eajbcs-667	293	16	have	have	AUX
eajbcs-667	293	17	:	:	PUNCT
eajbcs-667	293	18	ai	ai	VERB
eajbcs-667	293	19	,	,	PUNCT
eajbcs-667	293	20	j	j	PROPN
eajbcs-667	294	1	=	=	SYM
eajbcs-667	294	2	∫	∫	PROPN
eajbcs-667	294	3	1	1	NUM
eajbcs-667	295	1	0	0	NUM
eajbcs-667	295	2	ϕ′	ϕ′	NOUN
eajbcs-667	295	3	iϕ	iϕ	NOUN
eajbcs-667	295	4	′	′	NUM
eajbcs-667	295	5	jdx	jdx	PROPN
eajbcs-667	295	6	,	,	PUNCT
eajbcs-667	295	7	=	=	PUNCT
eajbcs-667	295	8	n−1∑	n−1∑	NUM
eajbcs-667	295	9	e=1	e=1	NOUN
eajbcs-667	295	10	∫	∫	PROPN
eajbcs-667	296	1	ωe	ωe	ADV
eajbcs-667	296	2	ϕ′	ϕ′	ADV
eajbcs-667	296	3	iϕ	iϕ	PROPN
eajbcs-667	296	4	′	′	NUM
eajbcs-667	296	5	jdx	jdx	PROPN
eajbcs-667	296	6	,	,	PUNCT
eajbcs-667	296	7	=	=	PUNCT
eajbcs-667	296	8	n−1∑	n−1∑	NUM
eajbcs-667	296	9	e=1	e=1	NOUN
eajbcs-667	297	1	ae	ae	PROPN
eajbcs-667	297	2	ij	ij	VERB
eajbcs-667	297	3	.	.	PUNCT
eajbcs-667	298	1	similarly	similarly	ADV
eajbcs-667	298	2	the	the	DET
eajbcs-667	298	3	load	load	NOUN
eajbcs-667	298	4	vector	vector	NOUN
eajbcs-667	298	5	fi	fi	NOUN
eajbcs-667	298	6	=	=	SYM
eajbcs-667	298	7	∫	∫	PROPN
eajbcs-667	298	8	1	1	NUM
eajbcs-667	298	9	0	0	NUM
eajbcs-667	298	10	fϕidx	fϕidx	NOUN
eajbcs-667	298	11	=	=	PUNCT
eajbcs-667	298	12	n−1∑	n−1∑	NUM
eajbcs-667	299	1	e=1	e=1	NOUN
eajbcs-667	299	2	∫	∫	PROPN
eajbcs-667	299	3	ωe	ωe	ADV
eajbcs-667	299	4	fϕidx	fϕidx	NOUN
eajbcs-667	299	5	=	=	PUNCT
eajbcs-667	299	6	n−1∑	n−1∑	NUM
eajbcs-667	299	7	e=1	e=1	PUNCT
eajbcs-667	299	8	f	f	PROPN
eajbcs-667	299	9	e	e	PROPN
eajbcs-667	299	10	i	i	PROPN
eajbcs-667	299	11	.	.	PUNCT
eajbcs-667	300	1	a	a	DET
eajbcs-667	300	2	f(x)dx	f(x)dx	NUM
eajbcs-667	300	3	=	=	SYM
eajbcs-667	300	4	f(a	f(a	NOUN
eajbcs-667	300	5	)	)	PUNCT
eajbcs-667	301	1	+	+	CCONJ
eajbcs-667	301	2	4f(a+b	4f(a+b	NUM
eajbcs-667	301	3	2	2	NUM
eajbcs-667	301	4	)	)	PUNCT
eajbcs-667	302	1	+	+	CCONJ
eajbcs-667	302	2	f(b	f(b	X
eajbcs-667	302	3	)	)	PUNCT
eajbcs-667	302	4	6	6	NUM
eajbcs-667	302	5	(	(	PUNCT
eajbcs-667	302	6	b−	b−	NOUN
eajbcs-667	302	7	a	a	PRON
eajbcs-667	302	8	)	)	PUNCT
eajbcs-667	302	9	.	.	PUNCT
eajbcs-667	303	1	(	(	PUNCT
eajbcs-667	303	2	18	18	NUM
eajbcs-667	303	3	)	)	PUNCT
eajbcs-667	303	4	then	then	ADV
eajbcs-667	303	5	we	we	PRON
eajbcs-667	303	6	can	can	AUX
eajbcs-667	303	7	be	be	AUX
eajbcs-667	303	8	illustrate	illustrate	ADJ
eajbcs-667	303	9	by	by	ADP
eajbcs-667	303	10	the	the	DET
eajbcs-667	303	11	hat	hat	NOUN
eajbcs-667	303	12	functions	function	NOUN
eajbcs-667	303	13	and	and	CCONJ
eajbcs-667	303	14	simpsons	simpson	NOUN
eajbcs-667	303	15	formula	formula	NOUN
eajbcs-667	303	16	as	as	SCONJ
eajbcs-667	303	17	follows	follow	VERB
eajbcs-667	303	18	:	:	PUNCT
eajbcs-667	303	19	ϕi(x	ϕi(x	NUM
eajbcs-667	303	20	)	)	PUNCT
eajbcs-667	304	1	=	=	PUNCT
eajbcs-667	305	1			PROPN
eajbcs-667	306	1	x−xi−1	x−xi−1	PROPN
eajbcs-667	307	1	hi	hi	INTJ
eajbcs-667	307	2	,	,	PUNCT
eajbcs-667	307	3	if	if	SCONJ
eajbcs-667	307	4	xi−1	xi−1	PROPN
eajbcs-667	307	5	≤	≤	ADV
eajbcs-667	307	6	x	x	PUNCT
eajbcs-667	307	7	≤	≤	NUM
eajbcs-667	307	8	xi	xi	X
eajbcs-667	307	9	xi+1−x	xi+1−x	PROPN
eajbcs-667	307	10	hi+1	hi+1	X
eajbcs-667	307	11	,	,	PUNCT
eajbcs-667	307	12	if	if	SCONJ
eajbcs-667	307	13	xi	xi	ADP
eajbcs-667	307	14	≤	≤	NUM
eajbcs-667	307	15	x	x	X
eajbcs-667	307	16	≤	≤	NUM
eajbcs-667	307	17	xi+1	xi+1	PROPN
eajbcs-667	307	18	0	0	NUM
eajbcs-667	307	19	,	,	PUNCT
eajbcs-667	307	20	other	other	ADJ
eajbcs-667	307	21	wise	wise	ADJ
eajbcs-667	307	22	,	,	PUNCT
eajbcs-667	307	23	ϕi−1(x	ϕi−1(x	NUM
eajbcs-667	307	24	)	)	PUNCT
eajbcs-667	307	25	=	=	PUNCT
eajbcs-667	308	1			PROPN
eajbcs-667	308	2	x−xi−2	x−xi−2	PROPN
eajbcs-667	308	3	hi−1	hi−1	PROPN
eajbcs-667	308	4	,	,	PUNCT
eajbcs-667	308	5	if	if	SCONJ
eajbcs-667	308	6	xi−2	xi−2	PROPN
eajbcs-667	308	7	≤	≤	NUM
eajbcs-667	308	8	x	x	X
eajbcs-667	308	9	≤	≤	NUM
eajbcs-667	308	10	xi−1	xi−1	ADP
eajbcs-667	308	11	xi−x	xi−x	PROPN
eajbcs-667	308	12	hi	hi	INTJ
eajbcs-667	308	13	,	,	PUNCT
eajbcs-667	308	14	if	if	SCONJ
eajbcs-667	308	15	xi−1	xi−1	PROPN
eajbcs-667	308	16	≤	≤	ADV
eajbcs-667	308	17	x	x	PUNCT
eajbcs-667	308	18	≤	≤	NUM
eajbcs-667	308	19	xi	xi	ADP
eajbcs-667	308	20	0	0	PROPN
eajbcs-667	308	21	,	,	PUNCT
eajbcs-667	308	22	other	other	ADJ
eajbcs-667	308	23	wise	wise	ADJ
eajbcs-667	308	24	,	,	PUNCT
eajbcs-667	308	25	and	and	CCONJ
eajbcs-667	308	26	ϕi+1(x	ϕi+1(x	NOUN
eajbcs-667	308	27	)	)	PUNCT
eajbcs-667	308	28	=	=	SYM
eajbcs-667	309	1			PRON
eajbcs-667	309	2	x−xi	x−xi	X
eajbcs-667	309	3	hi+1	hi+1	X
eajbcs-667	309	4	,	,	PUNCT
eajbcs-667	309	5	if	if	SCONJ
eajbcs-667	309	6	xi	xi	ADP
eajbcs-667	309	7	≤	≤	NUM
eajbcs-667	309	8	x	x	PUNCT
eajbcs-667	309	9	≤	≤	NUM
eajbcs-667	309	10	xi+1	xi+1	NUM
eajbcs-667	309	11	xi+2−x	xi+2−x	PROPN
eajbcs-667	309	12	hi+2	hi+2	PROPN
eajbcs-667	309	13	,	,	PUNCT
eajbcs-667	309	14	if	if	SCONJ
eajbcs-667	309	15	xi+1	xi+1	PROPN
eajbcs-667	309	16	≤	≤	NUM
eajbcs-667	309	17	x	x	SYM
eajbcs-667	309	18	≤	≤	NUM
eajbcs-667	309	19	xi+2	xi+2	NUM
eajbcs-667	309	20	0	0	NUM
eajbcs-667	309	21	,	,	PUNCT
eajbcs-667	309	22	other	other	ADJ
eajbcs-667	309	23	wise	wise	ADJ
eajbcs-667	309	24	.	.	PUNCT
eajbcs-667	310	1	now	now	ADV
eajbcs-667	310	2	ai	ai	VERB
eajbcs-667	310	3	,	,	PUNCT
eajbcs-667	310	4	i	i	PRON
eajbcs-667	310	5	=	=	PUNCT
eajbcs-667	310	6	a(ϕi	a(ϕi	X
eajbcs-667	310	7	,	,	PUNCT
eajbcs-667	310	8	ϕi	ϕi	ADJ
eajbcs-667	310	9	)	)	PUNCT
eajbcs-667	310	10	e	e	NOUN
eajbcs-667	310	11	=	=	SYM
eajbcs-667	310	12	∫	∫	PROPN
eajbcs-667	311	1	ωe	ωe	ADV
eajbcs-667	311	2	ϕ′	ϕ′	PUNCT
eajbcs-667	311	3	iϕ	iϕ	NOUN
eajbcs-667	311	4	′	′	NUM
eajbcs-667	311	5	idx	idx	NOUN
eajbcs-667	311	6	,	,	PUNCT
eajbcs-667	311	7	=	=	SYM
eajbcs-667	311	8	∫	∫	PROPN
eajbcs-667	311	9	xi	xi	PROPN
eajbcs-667	312	1	xi−1	xi−1	PROPN
eajbcs-667	312	2	ϕ′	ϕ′	PUNCT
eajbcs-667	312	3	iϕ	iϕ	VERB
eajbcs-667	312	4	′	′	NUM
eajbcs-667	312	5	idx+	idx+	NOUN
eajbcs-667	312	6	∫	∫	PROPN
eajbcs-667	312	7	xi+1	xi+1	PROPN
eajbcs-667	313	1	xi	xi	X
eajbcs-667	313	2	ϕ′	ϕ′	PROPN
eajbcs-667	313	3	iϕ	iϕ	NOUN
eajbcs-667	313	4	′	′	NUM
eajbcs-667	313	5	idx	idx	NOUN
eajbcs-667	313	6	,	,	PUNCT
eajbcs-667	313	7	=	=	SYM
eajbcs-667	313	8	∫	∫	PROPN
eajbcs-667	313	9	xi	xi	X
eajbcs-667	314	1	xi−1	xi−1	PROPN
eajbcs-667	314	2	1	1	NUM
eajbcs-667	314	3	hi	hi	INTJ
eajbcs-667	314	4	1	1	NUM
eajbcs-667	314	5	hi	hi	INTJ
eajbcs-667	314	6	dx+	dx+	PROPN
eajbcs-667	314	7	∫	∫	PROPN
eajbcs-667	314	8	xi+1	xi+1	PROPN
eajbcs-667	315	1	xi	xi	ADP
eajbcs-667	315	2	−1	−1	NOUN
eajbcs-667	315	3	hi+1	hi+1	ADP
eajbcs-667	315	4	−1	−1	NOUN
eajbcs-667	315	5	hi+1	hi+1	NOUN
eajbcs-667	315	6	dx	dx	PROPN
eajbcs-667	315	7	,	,	PUNCT
eajbcs-667	315	8	=	=	SYM
eajbcs-667	316	1	hi	hi	INTJ
eajbcs-667	316	2	6	6	NUM
eajbcs-667	316	3	(	(	PUNCT
eajbcs-667	316	4	1	1	NUM
eajbcs-667	316	5	h2	h2	NOUN
eajbcs-667	317	1	i	i	PRON
eajbcs-667	317	2	+	+	CCONJ
eajbcs-667	317	3	4	4	NUM
eajbcs-667	317	4	h2	h2	NOUN
eajbcs-667	317	5	i	i	PRON
eajbcs-667	317	6	+	+	CCONJ
eajbcs-667	317	7	1	1	NUM
eajbcs-667	317	8	h2	h2	NOUN
eajbcs-667	317	9	i	i	NOUN
eajbcs-667	317	10	)	)	PUNCT
eajbcs-667	318	1	+	+	CCONJ
eajbcs-667	318	2	hi+1	hi+1	NUM
eajbcs-667	318	3	6	6	NUM
eajbcs-667	318	4	(	(	PUNCT
eajbcs-667	318	5	1	1	NUM
eajbcs-667	318	6	h2	h2	NOUN
eajbcs-667	318	7	i+1	i+1	PRON
eajbcs-667	318	8	+	+	SYM
eajbcs-667	318	9	4	4	NUM
eajbcs-667	318	10	h2	h2	NOUN
eajbcs-667	318	11	i+1	i+1	PRON
eajbcs-667	318	12	+	+	SYM
eajbcs-667	318	13	1	1	NUM
eajbcs-667	318	14	h2	h2	NOUN
eajbcs-667	318	15	i+1	i+1	NUM
eajbcs-667	318	16	)	)	PUNCT
eajbcs-667	318	17	,	,	PUNCT
eajbcs-667	318	18	=	=	NOUN
eajbcs-667	318	19	1	1	NUM
eajbcs-667	318	20	hi	hi	INTJ
eajbcs-667	319	1	+	+	CCONJ
eajbcs-667	319	2	1	1	NUM
eajbcs-667	319	3	hi+1	hi+1	NOUN
eajbcs-667	319	4	.	.	PUNCT
eajbcs-667	320	1	,	,	PUNCT
eajbcs-667	320	2	ai−1,i	ai−1,i	ADP
eajbcs-667	320	3	=	=	SYM
eajbcs-667	320	4	a(ϕi−1	a(ϕi−1	PROPN
eajbcs-667	320	5	,	,	PUNCT
eajbcs-667	320	6	ϕi	ϕi	ADJ
eajbcs-667	320	7	)	)	PUNCT
eajbcs-667	320	8	e	e	NOUN
eajbcs-667	320	9	=	=	SYM
eajbcs-667	320	10	∫	∫	PROPN
eajbcs-667	321	1	ωe	ωe	ADV
eajbcs-667	321	2	ϕ′	ϕ′	PROPN
eajbcs-667	321	3	i−1ϕ	i−1ϕ	NOUN
eajbcs-667	321	4	′	′	NOUN
eajbcs-667	321	5	idx	idx	PROPN
eajbcs-667	321	6	,	,	PUNCT
eajbcs-667	321	7	=	=	SYM
eajbcs-667	321	8	∫	∫	PROPN
eajbcs-667	321	9	xi	xi	PROPN
eajbcs-667	321	10	xi−1	xi−1	PROPN
eajbcs-667	321	11	ϕ′	ϕ′	PART
eajbcs-667	321	12	i−1ϕ	i−1ϕ	NOUN
eajbcs-667	321	13	′	′	ADJ
eajbcs-667	321	14	idx+	idx+	NOUN
eajbcs-667	321	15	∫	∫	PROPN
eajbcs-667	321	16	xi+1	xi+1	PROPN
eajbcs-667	322	1	xi	xi	PROPN
eajbcs-667	322	2	ϕ′	ϕ′	PROPN
eajbcs-667	323	1	i−1ϕ	i−1ϕ	NOUN
eajbcs-667	323	2	′	′	NOUN
eajbcs-667	323	3	idx	idx	PROPN
eajbcs-667	323	4	,	,	PUNCT
eajbcs-667	323	5	=	=	SYM
eajbcs-667	323	6	∫	∫	PROPN
eajbcs-667	323	7	xi	xi	X
eajbcs-667	324	1	xi−1	xi−1	PROPN
eajbcs-667	324	2	−1	−1	ADV
eajbcs-667	324	3	hi	hi	INTJ
eajbcs-667	324	4	1	1	NUM
eajbcs-667	325	1	hi	hi	INTJ
eajbcs-667	325	2	dx	dx	PROPN
eajbcs-667	325	3	,	,	PUNCT
eajbcs-667	325	4	=	=	SYM
eajbcs-667	326	1	hi	hi	INTJ
eajbcs-667	326	2	6	6	NUM
eajbcs-667	326	3	(	(	PUNCT
eajbcs-667	326	4	−1	−1	NOUN
eajbcs-667	326	5	h2	h2	NOUN
eajbcs-667	327	1	i	i	PRON
eajbcs-667	327	2	+	+	CCONJ
eajbcs-667	327	3	−4	−4	X
eajbcs-667	327	4	h2	h2	NOUN
eajbcs-667	328	1	i	i	PRON
eajbcs-667	328	2	+	+	CCONJ
eajbcs-667	328	3	−1	−1	NOUN
eajbcs-667	328	4	h2	h2	NOUN
eajbcs-667	328	5	i	i	NOUN
eajbcs-667	328	6	)	)	PUNCT
eajbcs-667	329	1	=	=	PUNCT
eajbcs-667	329	2	−1	−1	NOUN
eajbcs-667	330	1	hi	hi	INTJ
eajbcs-667	330	2	.	.	PUNCT
eajbcs-667	331	1	east	east	PROPN
eajbcs-667	331	2	afr	afr	PROPN
eajbcs-667	331	3	.	.	PUNCT
eajbcs-667	332	1	j.	j.	PROPN
eajbcs-667	332	2	biophys	biophys	PROPN
eajbcs-667	332	3	.	.	PUNCT
eajbcs-667	333	1	comput	comput	NOUN
eajbcs-667	333	2	.	.	PUNCT
eajbcs-667	334	1	sci	sci	PROPN
eajbcs-667	334	2	.	.	PUNCT
eajbcs-667	334	3	(	(	PUNCT
eajbcs-667	334	4	2023	2023	NUM
eajbcs-667	334	5	)	)	PUNCT
eajbcs-667	334	6	,	,	PUNCT
eajbcs-667	334	7	vol	vol	NOUN
eajbcs-667	334	8	.	.	PROPN
eajbcs-667	334	9	4	4	NUM
eajbcs-667	334	10	,	,	PUNCT
eajbcs-667	334	11	no	no	INTJ
eajbcs-667	334	12	.	.	NOUN
eajbcs-667	334	13	1	1	NUM
eajbcs-667	334	14	,	,	PUNCT
eajbcs-667	334	15	52	52	NUM
eajbcs-667	334	16	-	-	SYM
eajbcs-667	334	17	74	74	NUM
eajbcs-667	334	18	62	62	NUM
eajbcs-667	334	19	on	on	ADP
eajbcs-667	334	20	the	the	DET
eajbcs-667	334	21	interval	interval	NOUN
eajbcs-667	335	1	i	i	PRON
eajbcs-667	335	2	=	=	PUNCT
eajbcs-667	335	3	(	(	PUNCT
eajbcs-667	335	4	a	a	PRON
eajbcs-667	335	5	,	,	PUNCT
eajbcs-667	335	6	b	b	NOUN
eajbcs-667	335	7	)	)	PUNCT
eajbcs-667	335	8	simpson	simpson	PROPN
eajbcs-667	335	9	’s	’s	PART
eajbcs-667	335	10	formula	formula	NOUN
eajbcs-667	335	11	is	be	AUX
eajbcs-667	335	12	of	of	ADP
eajbcs-667	335	13	the	the	DET
eajbcs-667	335	14	form	form	NOUN
eajbcs-667	335	15	,	,	PUNCT
eajbcs-667	335	16	(	(	PUNCT
eajbcs-667	335	17	larson	larson	PROPN
eajbcs-667	335	18	and	and	CCONJ
eajbcs-667	335	19	bengzon	bengzon	PROPN
eajbcs-667	335	20	,	,	PUNCT
eajbcs-667	335	21	2010)∫	2010)∫	NUM
eajbcs-667	335	22	b	b	NOUN
eajbcs-667	335	23	and	and	CCONJ
eajbcs-667	335	24	ai	ai	VERB
eajbcs-667	335	25	,	,	PUNCT
eajbcs-667	335	26	i+1	i+1	NOUN
eajbcs-667	335	27	=	=	ADJ
eajbcs-667	335	28	a(ϕi+1	a(ϕi+1	PROPN
eajbcs-667	335	29	,	,	PUNCT
eajbcs-667	335	30	ϕi	ϕi	ADJ
eajbcs-667	335	31	)	)	PUNCT
eajbcs-667	335	32	e	e	NOUN
eajbcs-667	336	1	=	=	SYM
eajbcs-667	336	2	∫	∫	PROPN
eajbcs-667	337	1	ωe	ωe	ADV
eajbcs-667	337	2	ϕ′	ϕ′	X
eajbcs-667	338	1	i+1ϕ	i+1ϕ	ADJ
eajbcs-667	338	2	′	′	ADJ
eajbcs-667	338	3	idx	idx	NOUN
eajbcs-667	338	4	,	,	PUNCT
eajbcs-667	338	5	=	=	SYM
eajbcs-667	338	6	∫	∫	PROPN
eajbcs-667	338	7	xi	xi	PROPN
eajbcs-667	339	1	xi−1	xi−1	PROPN
eajbcs-667	339	2	ϕ′	ϕ′	X
eajbcs-667	340	1	i+1ϕ	i+1ϕ	ADJ
eajbcs-667	340	2	′	′	NUM
eajbcs-667	340	3	idx+	idx+	NOUN
eajbcs-667	340	4	∫	∫	PROPN
eajbcs-667	340	5	xi+1	xi+1	PROPN
eajbcs-667	340	6	xi	xi	X
eajbcs-667	340	7	ϕ′	ϕ′	PROPN
eajbcs-667	341	1	i+1ϕ	i+1ϕ	ADJ
eajbcs-667	341	2	′	′	ADJ
eajbcs-667	341	3	idx	idx	NOUN
eajbcs-667	341	4	,	,	PUNCT
eajbcs-667	341	5	=	=	SYM
eajbcs-667	341	6	∫	∫	PROPN
eajbcs-667	341	7	xi+1	xi+1	PROPN
eajbcs-667	341	8	xi	xi	ADP
eajbcs-667	341	9	−1	−1	NOUN
eajbcs-667	341	10	hi+1	hi+1	ADV
eajbcs-667	341	11	1	1	NUM
eajbcs-667	341	12	hi+1	hi+1	NOUN
eajbcs-667	341	13	dx	dx	PROPN
eajbcs-667	341	14	,	,	PUNCT
eajbcs-667	341	15	=	=	SYM
eajbcs-667	341	16	hi+1	hi+1	SYM
eajbcs-667	341	17	6	6	NUM
eajbcs-667	341	18	(	(	PUNCT
eajbcs-667	341	19	−1	−1	NOUN
eajbcs-667	341	20	h2	h2	NOUN
eajbcs-667	341	21	i+1	i+1	PRON
eajbcs-667	341	22	+	+	CCONJ
eajbcs-667	341	23	−4	−4	X
eajbcs-667	341	24	h2	h2	NOUN
eajbcs-667	341	25	i+1	i+1	PRON
eajbcs-667	341	26	+	+	CCONJ
eajbcs-667	341	27	−1	−1	NOUN
eajbcs-667	341	28	h2	h2	NOUN
eajbcs-667	341	29	i+1	i+1	NUM
eajbcs-667	341	30	)	)	PUNCT
eajbcs-667	341	31	=	=	SYM
eajbcs-667	341	32	−1	−1	NOUN
eajbcs-667	341	33	hi+1	hi+1	NOUN
eajbcs-667	341	34	.	.	PUNCT
eajbcs-667	342	1	each	each	DET
eajbcs-667	342	2	generic	generic	ADJ
eajbcs-667	342	3	interior	interior	ADJ
eajbcs-667	342	4	element	element	NOUN
eajbcs-667	342	5	contributes	contribute	VERB
eajbcs-667	342	6	to	to	ADP
eajbcs-667	342	7	the	the	DET
eajbcs-667	342	8	stiffness	stiffness	ADJ
eajbcs-667	342	9	matrix	matrix	NOUN
eajbcs-667	342	10	of	of	ADP
eajbcs-667	342	11	a	a	DET
eajbcs-667	342	12	2×	2×	NUM
eajbcs-667	342	13	2	2	NUM
eajbcs-667	342	14	sub	sub	NOUN
eajbcs-667	342	15	matrix	matrix	NOUN
eajbcs-667	342	16	.	.	PUNCT
eajbcs-667	343	1	a	a	DET
eajbcs-667	343	2	=	=	PUNCT
eajbcs-667	343	3	∫	∫	PROPN
eajbcs-667	343	4	1	1	NUM
eajbcs-667	343	5	0	0	NUM
eajbcs-667	343	6	ϕ′	ϕ′	NOUN
eajbcs-667	343	7	iϕ	iϕ	NOUN
eajbcs-667	343	8	′	′	NUM
eajbcs-667	343	9	jdx	jdx	PROPN
eajbcs-667	343	10	=	=	PUNCT
eajbcs-667	344	1	n−1∑	n−1∑	NUM
eajbcs-667	344	2	e=1	e=1	NUM
eajbcs-667	344	3	ae	ae	PROPN
eajbcs-667	344	4	=	=	PUNCT
eajbcs-667	344	5			PROPN
eajbcs-667	344	6	1	1	NUM
eajbcs-667	344	7	h1	h1	NOUN
eajbcs-667	344	8	−1	−1	NOUN
eajbcs-667	344	9	h1−1	h1−1	PROPN
eajbcs-667	344	10	h1	h1	PROPN
eajbcs-667	344	11	1	1	NUM
eajbcs-667	344	12	h1	h1	NOUN
eajbcs-667	344	13	+	+	CCONJ
eajbcs-667	344	14	1	1	NUM
eajbcs-667	344	15	h2	h2	NOUN
eajbcs-667	344	16	−1	−1	NOUN
eajbcs-667	344	17	h2	h2	NOUN
eajbcs-667	344	18	−−1	−−1	PROPN
eajbcs-667	344	19	h2	h2	NOUN
eajbcs-667	344	20	1	1	NUM
eajbcs-667	344	21	h2	h2	NOUN
eajbcs-667	344	22	+	+	CCONJ
eajbcs-667	344	23	1	1	NUM
eajbcs-667	344	24	h3	h3	NOUN
eajbcs-667	344	25	−1	−1	NOUN
eajbcs-667	344	26	h3	h3	NOUN
eajbcs-667	344	27	.	.	PUNCT
eajbcs-667	344	28	.	.	PUNCT
eajbcs-667	344	29	.	.	PUNCT
eajbcs-667	344	30	.	.	PUNCT
eajbcs-667	344	31	.	.	PUNCT
eajbcs-667	344	32	.	.	PUNCT
eajbcs-667	344	33	.	.	PUNCT
eajbcs-667	344	34	.	.	PUNCT
eajbcs-667	344	35	.	.	PUNCT
eajbcs-667	345	1	−1	−1	NOUN
eajbcs-667	345	2	hn−1	hn−1	ADJ
eajbcs-667	345	3	1	1	NUM
eajbcs-667	345	4	hn−1	hn−1	ADJ
eajbcs-667	345	5	+	+	NUM
eajbcs-667	345	6	1	1	NUM
eajbcs-667	345	7	hn	hn	NOUN
eajbcs-667	345	8	−1	−1	NOUN
eajbcs-667	346	1	hn	hn	NOUN
eajbcs-667	346	2	−1	−1	NOUN
eajbcs-667	346	3	hn	hn	NOUN
eajbcs-667	346	4	1	1	NUM
eajbcs-667	346	5	hn	hn	NOUN
eajbcs-667	346	6			VERB
eajbcs-667	346	7	.	.	PUNCT
eajbcs-667	347	1	(	(	PUNCT
eajbcs-667	347	2	19	19	NUM
eajbcs-667	347	3	)	)	PUNCT
eajbcs-667	347	4	the	the	DET
eajbcs-667	347	5	global	global	ADJ
eajbcs-667	347	6	stiffness	stiffness	NOUN
eajbcs-667	347	7	matrix	matrix	NOUN
eajbcs-667	347	8	a	a	PRON
eajbcs-667	347	9	can	can	AUX
eajbcs-667	347	10	be	be	AUX
eajbcs-667	347	11	written	write	VERB
eajbcs-667	347	12	as	as	ADP
eajbcs-667	347	13	a	a	DET
eajbcs-667	347	14	sum	sum	NOUN
eajbcs-667	347	15	of	of	ADP
eajbcs-667	347	16	n	n	CCONJ
eajbcs-667	347	17	simpler	simple	ADJ
eajbcs-667	347	18	elemental	elemental	ADJ
eajbcs-667	347	19	matrices	matrix	NOUN
eajbcs-667	347	20	as	as	ADP
eajbcs-667	347	21	:	:	PUNCT
eajbcs-667	347	22	a	a	DET
eajbcs-667	347	23	=	=	SYM
eajbcs-667	347	24	1	1	NUM
eajbcs-667	347	25	h1	h1	VERB
eajbcs-667	347	26			NOUN
eajbcs-667	347	27	1	1	NUM
eajbcs-667	347	28	−1	−1	NOUN
eajbcs-667	347	29	−1	−1	NOUN
eajbcs-667	347	30	1	1	NUM
eajbcs-667	347	31	+	+	VERB
eajbcs-667	347	32	1	1	NUM
eajbcs-667	347	33	h2	h2	NOUN
eajbcs-667	347	34			NOUN
eajbcs-667	347	35	1	1	NUM
eajbcs-667	347	36	−1	−1	NOUN
eajbcs-667	347	37	−1	−1	NOUN
eajbcs-667	347	38	1	1	NUM
eajbcs-667	347	39	+	+	VERB
eajbcs-667	347	40	.	.	PUNCT
eajbcs-667	347	41	.	.	PUNCT
eajbcs-667	348	1	.+	.+	NOUN
eajbcs-667	348	2	1	1	NUM
eajbcs-667	348	3	hn	hn	NOUN
eajbcs-667	348	4			NOUN
eajbcs-667	348	5	1	1	NUM
eajbcs-667	348	6	−1	−1	NOUN
eajbcs-667	348	7	−1	−1	NOUN
eajbcs-667	348	8	1	1	NUM
eajbcs-667	348	9			NOUN
eajbcs-667	348	10	.	.	PUNCT
eajbcs-667	349	1	i.e.	i.e.	X
eajbcs-667	349	2	,	,	PUNCT
eajbcs-667	349	3	a	a	PRON
eajbcs-667	349	4	=	=	X
eajbcs-667	350	1	aω1	aω1	ADV
eajbcs-667	350	2	+	+	CCONJ
eajbcs-667	350	3	aω2	aω2	X
eajbcs-667	351	1	+	+	X
eajbcs-667	351	2	.	.	PUNCT
eajbcs-667	351	3	.	.	PUNCT
eajbcs-667	352	1	.+	.+	NOUN
eajbcs-667	352	2	aωn	aωn	INTJ
eajbcs-667	352	3	.	.	PUNCT
eajbcs-667	353	1	each	each	DET
eajbcs-667	353	2	matrix	matrix	NOUN
eajbcs-667	353	3	aωe	aωe	NOUN
eajbcs-667	353	4	,	,	PUNCT
eajbcs-667	353	5	e	e	X
eajbcs-667	353	6	=	=	SYM
eajbcs-667	353	7	1	1	NUM
eajbcs-667	353	8	,	,	PUNCT
eajbcs-667	353	9	2	2	NUM
eajbcs-667	353	10	,	,	PUNCT
eajbcs-667	353	11	.	.	PUNCT
eajbcs-667	353	12	.	.	PUNCT
eajbcs-667	353	13	.	.	PUNCT
eajbcs-667	353	14	,	,	PUNCT
eajbcs-667	353	15	n	n	CCONJ
eajbcs-667	353	16	,	,	PUNCT
eajbcs-667	353	17	is	be	AUX
eajbcs-667	353	18	obtained	obtain	VERB
eajbcs-667	353	19	by	by	ADP
eajbcs-667	353	20	restricting	restrict	VERB
eajbcs-667	353	21	the	the	DET
eajbcs-667	353	22	integration	integration	NOUN
eajbcs-667	353	23	to	to	ADP
eajbcs-667	353	24	one	one	NUM
eajbcs-667	353	25	sub	sub	NOUN
eajbcs-667	353	26	interval	interval	NOUN
eajbcs-667	353	27	or	or	CCONJ
eajbcs-667	353	28	element	element	NOUN
eajbcs-667	353	29	ωe	ωe	ADV
eajbcs-667	353	30	and	and	CCONJ
eajbcs-667	353	31	is	be	AUX
eajbcs-667	353	32	therefore	therefore	ADV
eajbcs-667	353	33	called	call	VERB
eajbcs-667	353	34	an	an	DET
eajbcs-667	353	35	element	element	ADJ
eajbcs-667	353	36	stiffness	stiffness	NOUN
eajbcs-667	353	37	matrix	matrix	NOUN
eajbcs-667	353	38	.	.	PUNCT
eajbcs-667	354	1	from	from	ADP
eajbcs-667	354	2	the	the	DET
eajbcs-667	354	3	sum	sum	NOUN
eajbcs-667	354	4	we	we	PRON
eajbcs-667	354	5	see	see	VERB
eajbcs-667	354	6	that	that	SCONJ
eajbcs-667	354	7	on	on	ADP
eajbcs-667	354	8	each	each	DET
eajbcs-667	354	9	element	element	NOUN
eajbcs-667	354	10	e	e	NOUN
eajbcs-667	354	11	this	this	DET
eajbcs-667	354	12	small	small	ADJ
eajbcs-667	354	13	block	block	NOUN
eajbcs-667	354	14	takes	take	VERB
eajbcs-667	354	15	the	the	DET
eajbcs-667	354	16	form	form	NOUN
eajbcs-667	354	17	:	:	PUNCT
eajbcs-667	354	18	ae	ae	PROPN
eajbcs-667	354	19	=	=	SYM
eajbcs-667	354	20	1	1	NUM
eajbcs-667	354	21	h	h	NOUN
eajbcs-667	354	22	[	[	PUNCT
eajbcs-667	354	23	1	1	NUM
eajbcs-667	354	24	−1	−1	NOUN
eajbcs-667	354	25	−1	−1	NOUN
eajbcs-667	354	26	1	1	NUM
eajbcs-667	354	27	]	]	PUNCT
eajbcs-667	354	28	,	,	PUNCT
eajbcs-667	354	29	where	where	SCONJ
eajbcs-667	354	30	h	h	NOUN
eajbcs-667	354	31	is	be	AUX
eajbcs-667	354	32	the	the	DET
eajbcs-667	354	33	length	length	NOUN
eajbcs-667	354	34	of	of	ADP
eajbcs-667	354	35	e.	e.	PROPN
eajbcs-667	354	36	we	we	PRON
eajbcs-667	354	37	refer	refer	VERB
eajbcs-667	354	38	to	to	ADP
eajbcs-667	354	39	ae	ae	PROPN
eajbcs-667	354	40	as	as	ADP
eajbcs-667	354	41	the	the	DET
eajbcs-667	354	42	local	local	ADJ
eajbcs-667	354	43	element	element	ADJ
eajbcs-667	354	44	stiffness	stiffness	NOUN
eajbcs-667	354	45	matrix	matrix	NOUN
eajbcs-667	354	46	.	.	PUNCT
eajbcs-667	355	1	implementation	implementation	NOUN
eajbcs-667	355	2	of	of	ADP
eajbcs-667	355	3	one	one	NUM
eajbcs-667	355	4	dimensional	dimensional	ADJ
eajbcs-667	355	5	advection	advection	NOUN
eajbcs-667	355	6	diffusion	diffusion	NOUN
eajbcs-667	355	7	equation	equation	NOUN
eajbcs-667	355	8	let	let	VERB
eajbcs-667	355	9	us	we	PRON
eajbcs-667	355	10	now	now	ADV
eajbcs-667	355	11	solve	solve	VERB
eajbcs-667	355	12	the	the	DET
eajbcs-667	355	13	1d	1d	NUM
eajbcs-667	355	14	governing	governing	NOUN
eajbcs-667	355	15	(	(	PUNCT
eajbcs-667	355	16	advection	advection	NOUN
eajbcs-667	355	17	diffusion	diffusion	NOUN
eajbcs-667	355	18	)	)	PUNCT
eajbcs-667	355	19	equation	equation	NOUN
eajbcs-667	355	20	∂u	∂u	PROPN
eajbcs-667	355	21	∂t	∂t	PROPN
eajbcs-667	356	1	+	+	CCONJ
eajbcs-667	356	2	a	a	DET
eajbcs-667	356	3	∂u	∂u	PROPN
eajbcs-667	356	4	∂x	∂x	PROPN
eajbcs-667	356	5	=	=	SYM
eajbcs-667	356	6	d	d	PROPN
eajbcs-667	356	7	∂2u	∂2u	X
eajbcs-667	356	8	∂x2	∂x2	PROPN
eajbcs-667	356	9	+	+	NOUN
eajbcs-667	356	10	f	f	PROPN
eajbcs-667	356	11	,	,	PUNCT
eajbcs-667	356	12	u(0	u(0	NOUN
eajbcs-667	356	13	)	)	PUNCT
eajbcs-667	356	14	=	=	SYM
eajbcs-667	356	15	u(1	u(1	PROPN
eajbcs-667	356	16	)	)	PUNCT
eajbcs-667	356	17	=	=	SYM
eajbcs-667	356	18	0	0	NUM
eajbcs-667	356	19	,	,	PUNCT
eajbcs-667	356	20	by	by	ADP
eajbcs-667	356	21	using	use	VERB
eajbcs-667	356	22	fem	fem	PROPN
eajbcs-667	356	23	.	.	PUNCT
eajbcs-667	357	1	let	let	VERB
eajbcs-667	357	2	we	we	PRON
eajbcs-667	357	3	find	find	VERB
eajbcs-667	357	4	the	the	DET
eajbcs-667	357	5	weak	weak	ADJ
eajbcs-667	357	6	formulation	formulation	NOUN
eajbcs-667	357	7	of	of	ADP
eajbcs-667	357	8	the	the	DET
eajbcs-667	357	9	equation	equation	NOUN
eajbcs-667	357	10	by	by	ADP
eajbcs-667	357	11	multiplying	multiply	VERB
eajbcs-667	357	12	the	the	DET
eajbcs-667	357	13	equation	equation	NOUN
eajbcs-667	357	14	with	with	ADP
eajbcs-667	357	15	a	a	DET
eajbcs-667	357	16	test	test	NOUN
eajbcs-667	357	17	function	function	NOUN
eajbcs-667	357	18	v(x	v(x	PROPN
eajbcs-667	357	19	)	)	PUNCT
eajbcs-667	357	20	,	,	PUNCT
eajbcs-667	357	21	which	which	PRON
eajbcs-667	357	22	satisfies	satisfy	VERB
eajbcs-667	357	23	the	the	DET
eajbcs-667	357	24	boundary	boundary	ADJ
eajbcs-667	357	25	conditions	condition	NOUN
eajbcs-667	357	26	v(0	v(0	NOUN
eajbcs-667	357	27	)	)	PUNCT
eajbcs-667	357	28	=	=	SYM
eajbcs-667	358	1	v(1	v(1	ADJ
eajbcs-667	358	2	)	)	PUNCT
eajbcs-667	358	3	=	=	SYM
eajbcs-667	358	4	0	0	PUNCT
eajbcs-667	358	5	and	and	CCONJ
eajbcs-667	358	6	then	then	ADV
eajbcs-667	358	7	integrating	integrate	VERB
eajbcs-667	358	8	by	by	ADP
eajbcs-667	358	9	parts	part	NOUN
eajbcs-667	358	10	as	as	ADP
eajbcs-667	358	11	the	the	DET
eajbcs-667	358	12	same	same	ADJ
eajbcs-667	358	13	procedure	procedure	NOUN
eajbcs-667	358	14	above	above	ADV
eajbcs-667	358	15	.	.	PUNCT
eajbcs-667	359	1	we	we	PRON
eajbcs-667	359	2	have	have	VERB
eajbcs-667	359	3	that	that	PRON
eajbcs-667	359	4	:	:	PUNCT
eajbcs-667	360	1	∂u	∂u	PROPN
eajbcs-667	360	2	∂t	∂t	PROPN
eajbcs-667	360	3	.v	.v	PROPN
eajbcs-667	361	1	+	+	CCONJ
eajbcs-667	361	2	a	a	DET
eajbcs-667	361	3	∂u	∂u	PROPN
eajbcs-667	361	4	∂x	∂x	PROPN
eajbcs-667	361	5	.v	.v	PUNCT
eajbcs-667	362	1	=	=	PUNCT
eajbcs-667	362	2	d	d	PRON
eajbcs-667	362	3	∂2u	∂2u	X
eajbcs-667	362	4	∂x2	∂x2	PROPN
eajbcs-667	362	5	.v	.v	NOUN
eajbcs-667	362	6	+	+	CCONJ
eajbcs-667	362	7	f.v,∫	f.v,∫	VERB
eajbcs-667	362	8	1	1	NUM
eajbcs-667	362	9	0	0	NUM
eajbcs-667	362	10	(	(	PUNCT
eajbcs-667	362	11	∂u	∂u	PROPN
eajbcs-667	362	12	∂t	∂t	PROPN
eajbcs-667	362	13	.v	.v	PROPN
eajbcs-667	363	1	+	+	CCONJ
eajbcs-667	363	2	a	a	DET
eajbcs-667	363	3	∂u	∂u	PROPN
eajbcs-667	363	4	∂x	∂x	PROPN
eajbcs-667	363	5	.v	.v	NOUN
eajbcs-667	363	6	)	)	PUNCT
eajbcs-667	364	1	=	=	PUNCT
eajbcs-667	364	2	∫	∫	PROPN
eajbcs-667	365	1	1	1	NUM
eajbcs-667	365	2	0	0	NUM
eajbcs-667	366	1	(	(	PUNCT
eajbcs-667	366	2	d	d	PROPN
eajbcs-667	366	3	∂2u	∂2u	PROPN
eajbcs-667	366	4	∂x2	∂x2	PROPN
eajbcs-667	366	5	.v	.v	PROPN
eajbcs-667	366	6	+	+	CCONJ
eajbcs-667	366	7	f.v	f.v	PROPN
eajbcs-667	366	8	)	)	PUNCT
eajbcs-667	366	9	.	.	PUNCT
eajbcs-667	367	1	using	use	VERB
eajbcs-667	367	2	integration	integration	NOUN
eajbcs-667	367	3	by	by	ADP
eajbcs-667	367	4	parts	part	NOUN
eajbcs-667	367	5	,	,	PUNCT
eajbcs-667	367	6	we	we	PRON
eajbcs-667	367	7	have	have	VERB
eajbcs-667	367	8	then	then	ADV
eajbcs-667	367	9	east	east	PROPN
eajbcs-667	367	10	afr	afr	PROPN
eajbcs-667	367	11	.	.	PUNCT
eajbcs-667	368	1	j.	j.	PROPN
eajbcs-667	368	2	biophys	biophys	PROPN
eajbcs-667	368	3	.	.	PUNCT
eajbcs-667	369	1	comput	comput	NOUN
eajbcs-667	369	2	.	.	PUNCT
eajbcs-667	370	1	sci	sci	PROPN
eajbcs-667	370	2	.	.	PUNCT
eajbcs-667	370	3	(	(	PUNCT
eajbcs-667	370	4	2023	2023	NUM
eajbcs-667	370	5	)	)	PUNCT
eajbcs-667	370	6	,	,	PUNCT
eajbcs-667	370	7	vol	vol	NOUN
eajbcs-667	370	8	.	.	PROPN
eajbcs-667	370	9	4	4	NUM
eajbcs-667	370	10	,	,	PUNCT
eajbcs-667	370	11	no	no	INTJ
eajbcs-667	370	12	.	.	NOUN
eajbcs-667	370	13	1	1	NUM
eajbcs-667	370	14	,	,	PUNCT
eajbcs-667	370	15	52	52	NUM
eajbcs-667	370	16	-	-	SYM
eajbcs-667	370	17	74	74	NUM
eajbcs-667	370	18	63	63	NUM
eajbcs-667	370	19	now	now	ADV
eajbcs-667	370	20	the	the	DET
eajbcs-667	370	21	system	system	NOUN
eajbcs-667	370	22	of	of	ADP
eajbcs-667	370	23	equation	equation	NOUN
eajbcs-667	370	24	,	,	PUNCT
eajbcs-667	370	25	that	that	PRON
eajbcs-667	370	26	is	be	AUX
eajbcs-667	370	27	eq.17	eq.17	PROPN
eajbcs-667	370	28	also	also	ADV
eajbcs-667	370	29	is	be	AUX
eajbcs-667	370	30	a	a	DET
eajbcs-667	370	31	simple	simple	ADJ
eajbcs-667	370	32	system	system	NOUN
eajbcs-667	370	33	of	of	ADP
eajbcs-667	370	34	ordinary	ordinary	ADJ
eajbcs-667	370	35	differential	differential	ADJ
eajbcs-667	370	36	equations	equation	NOUN
eajbcs-667	370	37	and	and	CCONJ
eajbcs-667	370	38	we	we	PRON
eajbcs-667	370	39	can	can	AUX
eajbcs-667	370	40	solve	solve	VERB
eajbcs-667	370	41	this	this	DET
eajbcs-667	370	42	system	system	NOUN
eajbcs-667	370	43	of	of	ADP
eajbcs-667	370	44	ode	ode	PROPN
eajbcs-667	370	45	’s	’s	NOUN
eajbcs-667	370	46	to	to	PART
eajbcs-667	370	47	get	get	VERB
eajbcs-667	370	48	the	the	DET
eajbcs-667	370	49	solution	solution	NOUN
eajbcs-667	370	50	of	of	ADP
eajbcs-667	370	51	the	the	DET
eajbcs-667	370	52	original	original	ADJ
eajbcs-667	370	53	pde	pde	NOUN
eajbcs-667	370	54	.	.	PUNCT
eajbcs-667	371	1	∫	∫	PROPN
eajbcs-667	372	1	1	1	NUM
eajbcs-667	372	2	0	0	NUM
eajbcs-667	372	3	∂u	∂u	PROPN
eajbcs-667	372	4	∂t	∂t	PROPN
eajbcs-667	372	5	.v	.v	PROPN
eajbcs-667	373	1	+	+	CCONJ
eajbcs-667	373	2	∫	∫	PROPN
eajbcs-667	373	3	1	1	NUM
eajbcs-667	373	4	0	0	NUM
eajbcs-667	373	5	a	a	DET
eajbcs-667	373	6	∂u	∂u	PROPN
eajbcs-667	373	7	∂x	∂x	PROPN
eajbcs-667	373	8	.v	.v	PUNCT
eajbcs-667	374	1	=	=	PUNCT
eajbcs-667	374	2	−d	−d	PROPN
eajbcs-667	374	3	∫	∫	PROPN
eajbcs-667	374	4	1	1	NUM
eajbcs-667	374	5	0	0	NUM
eajbcs-667	374	6	∂u	∂u	PROPN
eajbcs-667	374	7	∂x	∂x	PROPN
eajbcs-667	374	8	.	.	PUNCT
eajbcs-667	375	1	∂v	∂v	PROPN
eajbcs-667	375	2	∂x	∂x	PROPN
eajbcs-667	376	1	+	+	CCONJ
eajbcs-667	376	2	∫	∫	PROPN
eajbcs-667	376	3	1	1	NUM
eajbcs-667	376	4	0	0	NUM
eajbcs-667	376	5	f.v	f.v	PROPN
eajbcs-667	376	6	,	,	PUNCT
eajbcs-667	376	7	(	(	PUNCT
eajbcs-667	376	8	20	20	NUM
eajbcs-667	376	9	)	)	PUNCT
eajbcs-667	376	10	∫	∫	PROPN
eajbcs-667	376	11	1	1	NUM
eajbcs-667	376	12	0	0	NUM
eajbcs-667	376	13	∂	∂	NUM
eajbcs-667	376	14	∂t	∂t	PROPN
eajbcs-667	376	15	n−1∑	n−1∑	NUM
eajbcs-667	376	16	i=1	i=1	PROPN
eajbcs-667	376	17	uiϕi	uiϕi	ADJ
eajbcs-667	376	18	.	.	PUNCT
eajbcs-667	377	1	n−1∑	n−1∑	NUM
eajbcs-667	377	2	j=1	j=1	PROPN
eajbcs-667	377	3	vjϕj	vjϕj	NOUN
eajbcs-667	378	1	+	+	CCONJ
eajbcs-667	378	2	∫	∫	PROPN
eajbcs-667	378	3	1	1	NUM
eajbcs-667	378	4	0	0	NUM
eajbcs-667	378	5	a	a	DET
eajbcs-667	378	6	∂	∂	NOUN
eajbcs-667	378	7	∂x	∂x	PROPN
eajbcs-667	378	8	(	(	PUNCT
eajbcs-667	378	9	n−1∑	n−1∑	NUM
eajbcs-667	378	10	i=1	i=1	PROPN
eajbcs-667	378	11	uiϕi	uiϕi	ADJ
eajbcs-667	378	12	)	)	PUNCT
eajbcs-667	378	13	.	.	PUNCT
eajbcs-667	379	1	n−1∑	n−1∑	NUM
eajbcs-667	379	2	j=1	j=1	PROPN
eajbcs-667	379	3	vjϕj	vjϕj	NOUN
eajbcs-667	379	4	=	=	PUNCT
eajbcs-667	379	5	−d	−d	PROPN
eajbcs-667	379	6	∫	∫	PROPN
eajbcs-667	379	7	1	1	NUM
eajbcs-667	379	8	0	0	NUM
eajbcs-667	379	9	∂	∂	NUM
eajbcs-667	379	10	∂x	∂x	PROPN
eajbcs-667	379	11	n−1∑	n−1∑	NUM
eajbcs-667	379	12	i=1	i=1	PROPN
eajbcs-667	379	13	uiϕi	uiϕi	ADJ
eajbcs-667	379	14	.	.	PUNCT
eajbcs-667	380	1	∂	∂	NUM
eajbcs-667	381	1	∂x	∂x	PROPN
eajbcs-667	381	2	n−1∑	n−1∑	NUM
eajbcs-667	381	3	j=1	j=1	PROPN
eajbcs-667	381	4	vjϕj	vjϕj	NOUN
eajbcs-667	382	1	+	+	CCONJ
eajbcs-667	382	2	∫	∫	PROPN
eajbcs-667	382	3	1	1	NUM
eajbcs-667	382	4	0	0	NUM
eajbcs-667	382	5	f.	f.	PROPN
eajbcs-667	382	6	n−1∑	n−1∑	PROPN
eajbcs-667	382	7	j=1	j=1	PROPN
eajbcs-667	382	8	vjϕj	vjϕj	NOUN
eajbcs-667	382	9	.	.	PUNCT
eajbcs-667	383	1	which	which	PRON
eajbcs-667	383	2	then	then	ADV
eajbcs-667	383	3	implies	imply	VERB
eajbcs-667	383	4	,	,	PUNCT
eajbcs-667	383	5	n−1∑	n−1∑	NUM
eajbcs-667	383	6	j=1	j=1	PROPN
eajbcs-667	383	7	vj	vj	X
eajbcs-667	383	8	(	(	PUNCT
eajbcs-667	383	9	∂	∂	NUM
eajbcs-667	383	10	∂t	∂t	PROPN
eajbcs-667	383	11	n−1∑	n−1∑	NUM
eajbcs-667	383	12	i=1	i=1	PROPN
eajbcs-667	383	13	ui	ui	PROPN
eajbcs-667	384	1	∫	∫	PROPN
eajbcs-667	384	2	1	1	NUM
eajbcs-667	384	3	0	0	NUM
eajbcs-667	384	4	ϕi.ϕj	ϕi.ϕj	NOUN
eajbcs-667	384	5	+	+	CCONJ
eajbcs-667	384	6	a	a	DET
eajbcs-667	384	7	n−1∑	n−1∑	NUM
eajbcs-667	384	8	i=1	i=1	PROPN
eajbcs-667	384	9	ui	ui	PROPN
eajbcs-667	385	1	∫	∫	PROPN
eajbcs-667	385	2	1	1	NUM
eajbcs-667	385	3	0	0	X
eajbcs-667	385	4	ϕ′	ϕ′	PUNCT
eajbcs-667	386	1	i.ϕj	i.ϕj	INTJ
eajbcs-667	386	2	)	)	PUNCT
eajbcs-667	386	3	=	=	SYM
eajbcs-667	387	1	n−1∑	n−1∑	NUM
eajbcs-667	387	2	j=1	j=1	NOUN
eajbcs-667	387	3	vj	vj	PROPN
eajbcs-667	387	4	(	(	PUNCT
eajbcs-667	387	5	−d	−d	PROPN
eajbcs-667	387	6	n−1∑	n−1∑	PROPN
eajbcs-667	387	7	i=1	i=1	PROPN
eajbcs-667	388	1	ui	ui	PROPN
eajbcs-667	389	1	∫	∫	PROPN
eajbcs-667	390	1	1	1	NUM
eajbcs-667	390	2	0	0	NUM
eajbcs-667	390	3	ϕ′	ϕ′	NOUN
eajbcs-667	391	1	i.ϕ	i.ϕ	INTJ
eajbcs-667	391	2	′	′	NUM
eajbcs-667	391	3	j	j	PROPN
eajbcs-667	392	1	+	+	CCONJ
eajbcs-667	392	2	∫	∫	PROPN
eajbcs-667	392	3	1	1	NUM
eajbcs-667	392	4	0	0	X
eajbcs-667	392	5	f.ϕj	f.ϕj	ADV
eajbcs-667	392	6	)	)	PUNCT
eajbcs-667	392	7	.	.	PUNCT
eajbcs-667	393	1	(	(	PUNCT
eajbcs-667	393	2	21	21	NUM
eajbcs-667	393	3	)	)	PUNCT
eajbcs-667	393	4	that	that	PRON
eajbcs-667	393	5	is	be	AUX
eajbcs-667	393	6	∂	∂	NUM
eajbcs-667	393	7	∂t	∂t	PROPN
eajbcs-667	393	8	n−1∑	n−1∑	NUM
eajbcs-667	393	9	i=1	i=1	PROPN
eajbcs-667	394	1	ui	ui	PROPN
eajbcs-667	394	2	∫	∫	PROPN
eajbcs-667	394	3	1	1	NUM
eajbcs-667	394	4	0	0	NUM
eajbcs-667	394	5	ϕi.ϕj	ϕi.ϕj	NOUN
eajbcs-667	395	1	+	+	CCONJ
eajbcs-667	395	2	a	a	DET
eajbcs-667	395	3	n−1∑	n−1∑	NUM
eajbcs-667	395	4	i=1	i=1	PROPN
eajbcs-667	395	5	ui	ui	PROPN
eajbcs-667	395	6	∫	∫	PROPN
eajbcs-667	396	1	1	1	NUM
eajbcs-667	396	2	0	0	X
eajbcs-667	396	3	ϕ′	ϕ′	PUNCT
eajbcs-667	397	1	i.ϕj	i.ϕj	PROPN
eajbcs-667	397	2	=	=	PUNCT
eajbcs-667	397	3	−d	−d	PROPN
eajbcs-667	397	4	n−1∑	n−1∑	PROPN
eajbcs-667	397	5	i=1	i=1	PROPN
eajbcs-667	398	1	ui	ui	PROPN
eajbcs-667	399	1	∫	∫	PROPN
eajbcs-667	400	1	1	1	NUM
eajbcs-667	400	2	0	0	NUM
eajbcs-667	400	3	ϕ′	ϕ′	NOUN
eajbcs-667	401	1	i.ϕ	i.ϕ	INTJ
eajbcs-667	401	2	′	′	NUM
eajbcs-667	401	3	j	j	PROPN
eajbcs-667	402	1	+	+	CCONJ
eajbcs-667	402	2	∫	∫	PROPN
eajbcs-667	402	3	1	1	NUM
eajbcs-667	402	4	0	0	NUM
eajbcs-667	403	1	f.ϕj	f.ϕj	NOUN
eajbcs-667	403	2	.	.	PUNCT
eajbcs-667	404	1	in	in	ADP
eajbcs-667	404	2	a	a	DET
eajbcs-667	404	3	matrix	matrix	NOUN
eajbcs-667	404	4	form	form	NOUN
eajbcs-667	404	5	it	it	PRON
eajbcs-667	404	6	can	can	AUX
eajbcs-667	404	7	be	be	AUX
eajbcs-667	404	8	written	write	VERB
eajbcs-667	404	9	as	as	ADP
eajbcs-667	404	10	:	:	PUNCT
eajbcs-667	404	11	mu̇	mu̇	X
eajbcs-667	404	12	+	+	CCONJ
eajbcs-667	404	13	abu	abu	PROPN
eajbcs-667	404	14	+	+	NOUN
eajbcs-667	404	15	dau	dau	NOUN
eajbcs-667	404	16	=	=	SYM
eajbcs-667	404	17	f	f	X
eajbcs-667	404	18	,	,	PUNCT
eajbcs-667	404	19	(	(	PUNCT
eajbcs-667	404	20	22	22	NUM
eajbcs-667	404	21	)	)	PUNCT
eajbcs-667	404	22	assembly	assembly	NOUN
eajbcs-667	404	23	of	of	ADP
eajbcs-667	404	24	the	the	DET
eajbcs-667	404	25	load	load	NOUN
eajbcs-667	404	26	vector	vector	NOUN
eajbcs-667	404	27	in	in	ADP
eajbcs-667	404	28	1d	1d	NUM
eajbcs-667	404	29	the	the	DET
eajbcs-667	404	30	interval	interval	NOUN
eajbcs-667	404	31	[	[	X
eajbcs-667	404	32	a	a	X
eajbcs-667	404	33	,	,	PUNCT
eajbcs-667	404	34	b	b	NOUN
eajbcs-667	404	35	]	]	X
eajbcs-667	404	36	with	with	ADP
eajbcs-667	404	37	h	h	NOUN
eajbcs-667	404	38	=	=	NOUN
eajbcs-667	404	39	b−a	b−a	NOUN
eajbcs-667	404	40	n	n	NOUN
eajbcs-667	404	41	,	,	PUNCT
eajbcs-667	404	42	xi	xi	X
eajbcs-667	404	43	=	=	NOUN
eajbcs-667	404	44	a+ih	a+ih	NOUN
eajbcs-667	404	45	for	for	ADP
eajbcs-667	404	46	each	each	DET
eajbcs-667	404	47	i	i	NOUN
eajbcs-667	404	48	=	=	NOUN
eajbcs-667	404	49	0	0	NUM
eajbcs-667	404	50	,	,	PUNCT
eajbcs-667	404	51	1	1	NUM
eajbcs-667	404	52	,	,	PUNCT
eajbcs-667	404	53	.	.	PUNCT
eajbcs-667	404	54	.	.	PUNCT
eajbcs-667	405	1	.	.	PUNCT
eajbcs-667	406	1	,	,	PUNCT
eajbcs-667	406	2	n	n	PROPN
eajbcs-667	406	3	:	:	PUNCT
eajbcs-667	406	4	∫	∫	PROPN
eajbcs-667	406	5	b	b	PROPN
eajbcs-667	406	6	a	a	DET
eajbcs-667	406	7	f(x)dx	f(x)dx	PROPN
eajbcs-667	406	8	=	=	NOUN
eajbcs-667	406	9	h	h	NOUN
eajbcs-667	406	10	3	3	NUM
eajbcs-667	406	11	f(x0	f(x0	NOUN
eajbcs-667	406	12	)	)	PUNCT
eajbcs-667	407	1	+	+	CCONJ
eajbcs-667	407	2	2	2	NUM
eajbcs-667	407	3	n	n	SYM
eajbcs-667	407	4	2	2	NUM
eajbcs-667	407	5	−1∑	−1∑	PROPN
eajbcs-667	407	6	j=1	j=1	PROPN
eajbcs-667	407	7	f(x2j	f(x2j	PROPN
eajbcs-667	407	8	)	)	PUNCT
eajbcs-667	408	1	+	+	CCONJ
eajbcs-667	408	2	4	4	NUM
eajbcs-667	408	3	n	n	NUM
eajbcs-667	408	4	2∑	2∑	NUM
eajbcs-667	408	5	j=1	j=1	PROPN
eajbcs-667	408	6	f(x2j−1	f(x2j−1	PROPN
eajbcs-667	408	7	)	)	PUNCT
eajbcs-667	408	8	+	+	CCONJ
eajbcs-667	408	9	f(xn	f(xn	PROPN
eajbcs-667	408	10	)	)	PUNCT
eajbcs-667	408	11			PROPN
eajbcs-667	408	12	,	,	PUNCT
eajbcs-667	408	13	j	j	X
eajbcs-667	408	14	=	=	SYM
eajbcs-667	408	15	1	1	NUM
eajbcs-667	408	16	,	,	PUNCT
eajbcs-667	408	17	2	2	NUM
eajbcs-667	408	18	,	,	PUNCT
eajbcs-667	408	19	...	...	PUNCT
eajbcs-667	408	20	,	,	PUNCT
eajbcs-667	408	21	(	(	PUNCT
eajbcs-667	408	22	n	n	ADV
eajbcs-667	408	23	2	2	NUM
eajbcs-667	408	24	)	)	PUNCT
eajbcs-667	408	25	−	−	PROPN
eajbcs-667	409	1	1	1	X
eajbcs-667	409	2	.	.	PUNCT
eajbcs-667	409	3	(	(	PUNCT
eajbcs-667	409	4	23	23	NUM
eajbcs-667	409	5	)	)	PUNCT
eajbcs-667	409	6	east	east	PROPN
eajbcs-667	409	7	afr	afr	PROPN
eajbcs-667	409	8	.	.	PUNCT
eajbcs-667	410	1	j.	j.	PROPN
eajbcs-667	410	2	biophys	biophys	PROPN
eajbcs-667	410	3	.	.	PUNCT
eajbcs-667	411	1	comput	comput	NOUN
eajbcs-667	411	2	.	.	PUNCT
eajbcs-667	412	1	sci	sci	PROPN
eajbcs-667	412	2	.	.	PUNCT
eajbcs-667	412	3	(	(	PUNCT
eajbcs-667	412	4	2023	2023	NUM
eajbcs-667	412	5	)	)	PUNCT
eajbcs-667	412	6	,	,	PUNCT
eajbcs-667	412	7	vol	vol	NOUN
eajbcs-667	412	8	.	.	PROPN
eajbcs-667	412	9	4	4	NUM
eajbcs-667	412	10	,	,	PUNCT
eajbcs-667	412	11	no	no	INTJ
eajbcs-667	412	12	.	.	NOUN
eajbcs-667	412	13	1	1	NUM
eajbcs-667	412	14	,	,	PUNCT
eajbcs-667	412	15	52	52	NUM
eajbcs-667	412	16	-	-	SYM
eajbcs-667	412	17	74	74	NUM
eajbcs-667	412	18	64	64	NUM
eajbcs-667	412	19	which	which	PRON
eajbcs-667	412	20	is	be	AUX
eajbcs-667	412	21	the	the	DET
eajbcs-667	412	22	weak	weak	ADJ
eajbcs-667	412	23	formulation	formulation	NOUN
eajbcs-667	412	24	of	of	ADP
eajbcs-667	412	25	the	the	DET
eajbcs-667	412	26	one	one	NUM
eajbcs-667	412	27	dimensional	dimensional	ADJ
eajbcs-667	412	28	advection	advection	NOUN
eajbcs-667	412	29	-	-	PUNCT
eajbcs-667	412	30	diffusion	diffusion	NOUN
eajbcs-667	412	31	equation	equation	NOUN
eajbcs-667	412	32	.	.	PUNCT
eajbcs-667	413	1	substituting	substitute	VERB
eajbcs-667	413	2	eq.10	eq.10	NOUN
eajbcs-667	413	3	and	and	CCONJ
eajbcs-667	413	4	eq.11	eq.11	PRON
eajbcs-667	413	5	where	where	SCONJ
eajbcs-667	413	6	m	m	PROPN
eajbcs-667	413	7	is	be	AUX
eajbcs-667	413	8	the	the	DET
eajbcs-667	413	9	mass	mass	ADJ
eajbcs-667	413	10	matrix	matrix	NOUN
eajbcs-667	413	11	with	with	ADP
eajbcs-667	413	12	entries	entry	NOUN
eajbcs-667	413	13	given	give	VERB
eajbcs-667	413	14	in	in	ADP
eajbcs-667	413	15	eq	eq	PROPN
eajbcs-667	413	16	.	.	PROPN
eajbcs-667	413	17	15	15	NUM
eajbcs-667	413	18	,	,	PUNCT
eajbcs-667	413	19	b	b	NOUN
eajbcs-667	413	20	is	be	AUX
eajbcs-667	413	21	a	a	DET
eajbcs-667	413	22	matrix	matrix	NOUN
eajbcs-667	413	23	with	with	ADP
eajbcs-667	413	24	entries	entry	NOUN
eajbcs-667	413	25	given	give	VERB
eajbcs-667	413	26	in	in	ADP
eajbcs-667	413	27	eq	eq	PROPN
eajbcs-667	413	28	.	.	PROPN
eajbcs-667	413	29	13	13	NUM
eajbcs-667	413	30	,	,	PUNCT
eajbcs-667	413	31	a	a	PRON
eajbcs-667	413	32	is	be	AUX
eajbcs-667	413	33	the	the	DET
eajbcs-667	413	34	stiffness	stiffness	ADJ
eajbcs-667	413	35	matrix	matrix	NOUN
eajbcs-667	413	36	given	give	VERB
eajbcs-667	413	37	in	in	ADP
eajbcs-667	413	38	eq	eq	PROPN
eajbcs-667	413	39	.	.	PROPN
eajbcs-667	413	40	19	19	NUM
eajbcs-667	413	41	and	and	CCONJ
eajbcs-667	413	42	f	f	PROPN
eajbcs-667	413	43	is	be	AUX
eajbcs-667	413	44	a	a	DET
eajbcs-667	413	45	load	load	NOUN
eajbcs-667	413	46	vector	vector	NOUN
eajbcs-667	413	47	given	give	VERB
eajbcs-667	413	48	in	in	ADP
eajbcs-667	413	49	eq	eq	ADJ
eajbcs-667	413	50	.	.	PROPN
eajbcs-667	413	51	25	25	NUM
eajbcs-667	413	52	.	.	PUNCT
eajbcs-667	414	1	the	the	DET
eajbcs-667	414	2	right	right	ADJ
eajbcs-667	414	3	-	-	PUNCT
eajbcs-667	414	4	hand	hand	NOUN
eajbcs-667	414	5	-	-	PUNCT
eajbcs-667	414	6	side	side	NOUN
eajbcs-667	414	7	,	,	PUNCT
eajbcs-667	414	8	load	load	NOUN
eajbcs-667	414	9	vector	vector	NOUN
eajbcs-667	414	10	of	of	ADP
eajbcs-667	414	11	eq	eq	PROPN
eajbcs-667	414	12	.	.	PROPN
eajbcs-667	414	13	20	20	NUM
eajbcs-667	414	14	contains	contain	VERB
eajbcs-667	414	15	an	an	DET
eajbcs-667	414	16	integral	integral	ADJ
eajbcs-667	414	17	over	over	ADP
eajbcs-667	414	18	a	a	DET
eajbcs-667	414	19	function	function	NOUN
eajbcs-667	414	20	f(x	f(x	PROPN
eajbcs-667	414	21	)	)	PUNCT
eajbcs-667	414	22	.	.	PUNCT
eajbcs-667	415	1	in	in	ADP
eajbcs-667	415	2	general	general	ADJ
eajbcs-667	415	3	,	,	PUNCT
eajbcs-667	415	4	exactly	exactly	ADV
eajbcs-667	415	5	computing	compute	VERB
eajbcs-667	415	6	this	this	DET
eajbcs-667	415	7	integral	integral	NOUN
eajbcs-667	415	8	is	be	AUX
eajbcs-667	415	9	very	very	ADV
eajbcs-667	415	10	difficult	difficult	ADJ
eajbcs-667	415	11	,	,	PUNCT
eajbcs-667	415	12	so	so	ADV
eajbcs-667	415	13	another	another	DET
eajbcs-667	415	14	numerical	numerical	ADJ
eajbcs-667	415	15	approximation	approximation	NOUN
eajbcs-667	415	16	is	be	AUX
eajbcs-667	415	17	required	require	VERB
eajbcs-667	415	18	.	.	PUNCT
eajbcs-667	416	1	we	we	PRON
eajbcs-667	416	2	can	can	AUX
eajbcs-667	416	3	use	use	VERB
eajbcs-667	416	4	a	a	DET
eajbcs-667	416	5	well	well	ADV
eajbcs-667	416	6	known	know	VERB
eajbcs-667	416	7	integration	integration	NOUN
eajbcs-667	416	8	rule	rule	NOUN
eajbcs-667	416	9	composite	composite	ADJ
eajbcs-667	416	10	simpson	simpson	PROPN
eajbcs-667	416	11	rule	rule	NOUN
eajbcs-667	416	12	to	to	PART
eajbcs-667	416	13	approximate	approximate	VERB
eajbcs-667	416	14	these	these	DET
eajbcs-667	416	15	integration	integration	NOUN
eajbcs-667	416	16	whose	whose	DET
eajbcs-667	416	17	formula	formula	NOUN
eajbcs-667	416	18	(	(	PUNCT
eajbcs-667	416	19	for	for	ADP
eajbcs-667	416	20	more	more	ADJ
eajbcs-667	416	21	information	information	NOUN
eajbcs-667	416	22	you	you	PRON
eajbcs-667	416	23	can	can	AUX
eajbcs-667	416	24	see	see	VERB
eajbcs-667	416	25	,	,	PUNCT
eajbcs-667	416	26	(	(	PUNCT
eajbcs-667	416	27	?	?	PUNCT
eajbcs-667	416	28	)	)	PUNCT
eajbcs-667	416	29	)	)	PUNCT
eajbcs-667	416	30	is	be	AUX
eajbcs-667	416	31	given	give	VERB
eajbcs-667	416	32	in	in	ADP
eajbcs-667	416	33	eq	eq	ADP
eajbcs-667	416	34	.	.	PROPN
eajbcs-667	416	35	23	23	NUM
eajbcs-667	416	36	,	,	PUNCT
eajbcs-667	416	37	by	by	ADP
eajbcs-667	416	38	selecting	select	VERB
eajbcs-667	416	39	a	a	DET
eajbcs-667	416	40	set	set	NOUN
eajbcs-667	416	41	of	of	ADP
eajbcs-667	416	42	distinct	distinct	ADJ
eajbcs-667	416	43	n	n	PRON
eajbcs-667	416	44	nodes	node	NOUN
eajbcs-667	416	45	in	in	ADP
eajbcs-667	416	46	in	in	ADP
eajbcs-667	416	47	the	the	DET
eajbcs-667	416	48	weak	weak	ADJ
eajbcs-667	416	49	formulation	formulation	NOUN
eajbcs-667	416	50	of	of	ADP
eajbcs-667	416	51	the	the	DET
eajbcs-667	416	52	equation	equation	NOUN
eajbcs-667	416	53	eq.20	eq.20	NOUN
eajbcs-667	416	54	,	,	PUNCT
eajbcs-667	416	55	we	we	PRON
eajbcs-667	416	56	have	have	AUX
eajbcs-667	416	57	:	:	PUNCT
eajbcs-667	416	58	∫	∫	PROPN
eajbcs-667	416	59	b	b	PROPN
eajbcs-667	416	60	a	a	DET
eajbcs-667	416	61	f(x)dx	f(x)dx	PROPN
eajbcs-667	416	62	=	=	SYM
eajbcs-667	416	63	f(a	f(a	NOUN
eajbcs-667	416	64	)	)	PUNCT
eajbcs-667	416	65	+	+	CCONJ
eajbcs-667	416	66	f(b	f(b	X
eajbcs-667	416	67	)	)	PUNCT
eajbcs-667	416	68	2	2	NUM
eajbcs-667	416	69	(	(	PUNCT
eajbcs-667	416	70	b−	b−	NOUN
eajbcs-667	416	71	a	a	PRON
eajbcs-667	416	72	)	)	PUNCT
eajbcs-667	416	73	,	,	PUNCT
eajbcs-667	416	74	(	(	PUNCT
eajbcs-667	416	75	24	24	NUM
eajbcs-667	416	76	)	)	PUNCT
eajbcs-667	416	77	we	we	PRON
eajbcs-667	416	78	have	have	VERB
eajbcs-667	416	79	fi	fi	NOUN
eajbcs-667	416	80	=	=	SYM
eajbcs-667	416	81	∫	∫	PROPN
eajbcs-667	417	1	i	i	PRON
eajbcs-667	417	2	fϕidx	fϕidx	NOUN
eajbcs-667	417	3	,	,	PUNCT
eajbcs-667	417	4	=	=	SYM
eajbcs-667	417	5	∫	∫	PROPN
eajbcs-667	417	6	xi+1	xi+1	PROPN
eajbcs-667	418	1	xi−1	xi−1	PROPN
eajbcs-667	418	2	fϕidx	fϕidx	NOUN
eajbcs-667	418	3	,	,	PUNCT
eajbcs-667	418	4	=	=	SYM
eajbcs-667	418	5	∫	∫	PROPN
eajbcs-667	418	6	xi	xi	PROPN
eajbcs-667	419	1	xi−1	xi−1	PROPN
eajbcs-667	419	2	fϕidx+	fϕidx+	PROPN
eajbcs-667	419	3	∫	∫	PROPN
eajbcs-667	419	4	xi+1	xi+1	PROPN
eajbcs-667	419	5	xi	xi	PROPN
eajbcs-667	419	6	fϕidx	fϕidx	PROPN
eajbcs-667	419	7	,	,	PUNCT
eajbcs-667	419	8	≈	≈	PROPN
eajbcs-667	419	9	f(xi−1)ϕi(xi−1	f(xi−1)ϕi(xi−1	PROPN
eajbcs-667	419	10	)	)	PUNCT
eajbcs-667	420	1	+	+	SYM
eajbcs-667	420	2	f(xi)ϕi(xi	f(xi)ϕi(xi	X
eajbcs-667	420	3	)	)	PUNCT
eajbcs-667	420	4	2	2	NUM
eajbcs-667	420	5	hi	hi	INTJ
eajbcs-667	420	6	+	+	CCONJ
eajbcs-667	420	7	f(xi+1)ϕi(xi+1	f(xi+1)ϕi(xi+1	NOUN
eajbcs-667	420	8	)	)	PUNCT
eajbcs-667	420	9	+	+	SYM
eajbcs-667	421	1	f(xi)ϕi(xi	f(xi)ϕi(xi	X
eajbcs-667	421	2	)	)	PUNCT
eajbcs-667	421	3	2	2	NUM
eajbcs-667	421	4	hi+1	hi+1	NOUN
eajbcs-667	421	5	,	,	PUNCT
eajbcs-667	421	6	=	=	NOUN
eajbcs-667	421	7	0	0	PUNCT
eajbcs-667	421	8	+	+	CCONJ
eajbcs-667	421	9	f(xi	f(xi	NUM
eajbcs-667	421	10	)	)	PUNCT
eajbcs-667	421	11	2	2	NUM
eajbcs-667	421	12	hi	hi	INTJ
eajbcs-667	421	13	+	+	CCONJ
eajbcs-667	421	14	f(xi	f(xi	X
eajbcs-667	421	15	)	)	PUNCT
eajbcs-667	422	1	+	+	CCONJ
eajbcs-667	422	2	0	0	NUM
eajbcs-667	422	3	2	2	NUM
eajbcs-667	422	4	hi+1	hi+1	NOUN
eajbcs-667	422	5	,	,	PUNCT
eajbcs-667	422	6	=	=	SYM
eajbcs-667	422	7	f(xi	f(xi	X
eajbcs-667	422	8	)	)	PUNCT
eajbcs-667	422	9	(	(	PUNCT
eajbcs-667	422	10	hi	hi	INTJ
eajbcs-667	422	11	2	2	NUM
eajbcs-667	422	12	+	+	NUM
eajbcs-667	422	13	hi+1	hi+1	SYM
eajbcs-667	422	14	2	2	NUM
eajbcs-667	422	15	)	)	PUNCT
eajbcs-667	422	16	.	.	PUNCT
eajbcs-667	423	1	now	now	ADV
eajbcs-667	423	2	using	use	VERB
eajbcs-667	423	3	this	this	DET
eajbcs-667	423	4	trapezoidal	trapezoidal	ADJ
eajbcs-667	423	5	method	method	NOUN
eajbcs-667	423	6	,	,	PUNCT
eajbcs-667	423	7	the	the	DET
eajbcs-667	423	8	approximate	approximate	ADJ
eajbcs-667	423	9	load	load	NOUN
eajbcs-667	423	10	vector	vector	NOUN
eajbcs-667	423	11	takes	take	VERB
eajbcs-667	423	12	the	the	DET
eajbcs-667	423	13	form	form	NOUN
eajbcs-667	423	14	f	f	NOUN
eajbcs-667	423	15	=	=	SYM
eajbcs-667	423	16			ADJ
eajbcs-667	423	17	f(x0	f(x0	PROPN
eajbcs-667	423	18	)	)	PUNCT
eajbcs-667	423	19	h1	h1	VERB
eajbcs-667	423	20	2	2	NUM
eajbcs-667	423	21	f(x1	f(x1	NOUN
eajbcs-667	423	22	)	)	PUNCT
eajbcs-667	424	1	(	(	PUNCT
eajbcs-667	424	2	h1+h2	h1+h2	PROPN
eajbcs-667	424	3	2	2	NUM
eajbcs-667	424	4	)	)	PUNCT
eajbcs-667	424	5	f(x2	f(x2	NOUN
eajbcs-667	424	6	)	)	PUNCT
eajbcs-667	424	7	(	(	PUNCT
eajbcs-667	424	8	h2+h3	h2+h3	NOUN
eajbcs-667	424	9	2	2	NUM
eajbcs-667	424	10	)	)	PUNCT
eajbcs-667	424	11	...	...	PUNCT
eajbcs-667	425	1	f(xn−1	f(xn−1	X
eajbcs-667	425	2	)	)	PUNCT
eajbcs-667	425	3	(	(	PUNCT
eajbcs-667	425	4	hn−1+hn	hn−1+hn	PROPN
eajbcs-667	425	5	2	2	NUM
eajbcs-667	425	6	)	)	PUNCT
eajbcs-667	425	7	f(xn	f(xn	PROPN
eajbcs-667	425	8	)	)	PUNCT
eajbcs-667	425	9	hn	hn	PROPN
eajbcs-667	425	10	2	2	NUM
eajbcs-667	425	11			PROPN
eajbcs-667	425	12	.	.	PUNCT
eajbcs-667	426	1	(	(	PUNCT
eajbcs-667	426	2	25	25	NUM
eajbcs-667	426	3	)	)	PUNCT
eajbcs-667	426	4	splitting	split	VERB
eajbcs-667	426	5	f	f	PROPN
eajbcs-667	426	6	into	into	ADP
eajbcs-667	426	7	a	a	DET
eajbcs-667	426	8	sum	sum	NOUN
eajbcs-667	426	9	over	over	ADP
eajbcs-667	426	10	the	the	DET
eajbcs-667	426	11	elements	element	NOUN
eajbcs-667	426	12	yields	yield	VERB
eajbcs-667	426	13	the	the	DET
eajbcs-667	426	14	n	n	CCONJ
eajbcs-667	426	15	global	global	ADJ
eajbcs-667	426	16	element	element	NOUN
eajbcs-667	426	17	load	load	NOUN
eajbcs-667	426	18	vectors	vector	NOUN
eajbcs-667	426	19	fωe	fωe	VERB
eajbcs-667	426	20	:	:	PUNCT
eajbcs-667	426	21	f	f	X
eajbcs-667	426	22	=	=	PUNCT
eajbcs-667	426	23			PROPN
eajbcs-667	426	24	f(x0	f(x0	NOUN
eajbcs-667	426	25	)	)	PUNCT
eajbcs-667	426	26	f(x1	f(x1	ADJ
eajbcs-667	426	27	)	)	PUNCT
eajbcs-667	426	28			SCONJ
eajbcs-667	426	29	h1	h1	PROPN
eajbcs-667	426	30	2	2	NUM
eajbcs-667	426	31	+	+	NUM
eajbcs-667	426	32			NOUN
eajbcs-667	426	33	f(x1	f(x1	ADJ
eajbcs-667	426	34	)	)	PUNCT
eajbcs-667	426	35	f(x2	f(x2	NOUN
eajbcs-667	426	36	)	)	PUNCT
eajbcs-667	426	37			NOUN
eajbcs-667	426	38	h2	h2	NOUN
eajbcs-667	426	39	2	2	NUM
eajbcs-667	426	40	+	+	NOUN
eajbcs-667	426	41			ADJ
eajbcs-667	426	42	f(x2	f(x2	NOUN
eajbcs-667	426	43	)	)	PUNCT
eajbcs-667	426	44	f(x3	f(x3	PUNCT
eajbcs-667	426	45	)	)	PUNCT
eajbcs-667	426	46			NOUN
eajbcs-667	426	47	h3	h3	VERB
eajbcs-667	426	48	2	2	NUM
eajbcs-667	426	49	+	+	NUM
eajbcs-667	426	50	.	.	PUNCT
eajbcs-667	426	51	.	.	PUNCT
eajbcs-667	427	1	.+	.+	NOUN
eajbcs-667	427	2			VERB
eajbcs-667	427	3	f(xn−1	f(xn−1	NOUN
eajbcs-667	427	4	)	)	PUNCT
eajbcs-667	427	5	f(xn	f(xn	PROPN
eajbcs-667	427	6	)	)	PUNCT
eajbcs-667	427	7			ADP
eajbcs-667	428	1	hn	hn	PROPN
eajbcs-667	428	2	2	2	NUM
eajbcs-667	428	3	i.e.	i.e.	X
eajbcs-667	428	4	,	,	PUNCT
eajbcs-667	428	5	f	f	PROPN
eajbcs-667	428	6	=	=	PUNCT
eajbcs-667	428	7	fω1	fω1	PROPN
eajbcs-667	428	8	+	+	X
eajbcs-667	428	9	fω2	fω2	NOUN
eajbcs-667	428	10	+	+	X
eajbcs-667	428	11	.	.	PUNCT
eajbcs-667	428	12	.	.	PUNCT
eajbcs-667	428	13	.	.	PUNCT
eajbcs-667	429	1	+	+	CCONJ
eajbcs-667	429	2	fωn	fωn	NOUN
eajbcs-667	429	3	.	.	PUNCT
eajbcs-667	430	1	each	each	DET
eajbcs-667	430	2	vector	vector	NOUN
eajbcs-667	430	3	fωe	fωe	NOUN
eajbcs-667	430	4	,	,	PUNCT
eajbcs-667	430	5	e	e	X
eajbcs-667	430	6	=	=	SYM
eajbcs-667	430	7	1	1	NUM
eajbcs-667	430	8	,	,	PUNCT
eajbcs-667	430	9	2	2	NUM
eajbcs-667	430	10	,	,	PUNCT
eajbcs-667	430	11	.	.	PUNCT
eajbcs-667	430	12	.	.	PUNCT
eajbcs-667	430	13	.	.	PUNCT
eajbcs-667	431	1	,	,	PUNCT
eajbcs-667	431	2	n	n	CCONJ
eajbcs-667	431	3	,	,	PUNCT
eajbcs-667	431	4	is	be	AUX
eajbcs-667	431	5	formally	formally	ADV
eajbcs-667	431	6	derived	derive	VERB
eajbcs-667	431	7	by	by	ADP
eajbcs-667	431	8	restricting	restrict	VERB
eajbcs-667	431	9	the	the	DET
eajbcs-667	431	10	integration	integration	NOUN
eajbcs-667	431	11	to	to	PART
eajbcs-667	431	12	element	element	VERB
eajbcs-667	431	13	ωe	ωe	PRON
eajbcs-667	431	14	.	.	PUNCT
eajbcs-667	432	1	mu̇	mu̇	X
eajbcs-667	433	1	+	+	CCONJ
eajbcs-667	433	2	(	(	PUNCT
eajbcs-667	433	3	ab	ab	X
eajbcs-667	433	4	+	+	PROPN
eajbcs-667	433	5	da)u	da)u	PROPN
eajbcs-667	433	6	=	=	SYM
eajbcs-667	433	7	f	f	X
eajbcs-667	433	8	,	,	PUNCT
eajbcs-667	433	9	(	(	PUNCT
eajbcs-667	433	10	26	26	NUM
eajbcs-667	433	11	)	)	PUNCT
eajbcs-667	433	12	which	which	PRON
eajbcs-667	433	13	is	be	AUX
eajbcs-667	433	14	a	a	DET
eajbcs-667	433	15	simple	simple	ADJ
eajbcs-667	433	16	system	system	NOUN
eajbcs-667	433	17	of	of	ADP
eajbcs-667	433	18	ordinary	ordinary	ADJ
eajbcs-667	433	19	differential	differential	ADJ
eajbcs-667	433	20	equations	equation	NOUN
eajbcs-667	433	21	.	.	PUNCT
eajbcs-667	434	1	for	for	ADP
eajbcs-667	434	2	solving	solve	VERB
eajbcs-667	434	3	this	this	DET
eajbcs-667	434	4	system	system	NOUN
eajbcs-667	434	5	of	of	ADP
eajbcs-667	434	6	ode	ode	PROPN
eajbcs-667	434	7	’s	’s	NOUN
eajbcs-667	434	8	,	,	PUNCT
eajbcs-667	434	9	we	we	PRON
eajbcs-667	434	10	have	have	VERB
eajbcs-667	434	11	to	to	PART
eajbcs-667	434	12	use	use	VERB
eajbcs-667	434	13	a	a	DET
eajbcs-667	434	14	matlab	matlab	PROPN
eajbcs-667	434	15	soft	soft	ADJ
eajbcs-667	434	16	ware	ware	NOUN
eajbcs-667	434	17	in	in	ADP
eajbcs-667	434	18	which	which	PRON
eajbcs-667	434	19	it	it	PRON
eajbcs-667	434	20	has	have	VERB
eajbcs-667	434	21	a	a	DET
eajbcs-667	434	22	number	number	NOUN
eajbcs-667	434	23	of	of	ADP
eajbcs-667	434	24	tools	tool	NOUN
eajbcs-667	434	25	for	for	ADP
eajbcs-667	434	26	numerically	numerically	ADV
eajbcs-667	434	27	solving	solve	VERB
eajbcs-667	434	28	ordinary	ordinary	ADJ
eajbcs-667	434	29	differential	differential	ADJ
eajbcs-667	434	30	equations	equation	NOUN
eajbcs-667	434	31	.	.	PUNCT
eajbcs-667	435	1	we	we	PRON
eajbcs-667	435	2	would	would	AUX
eajbcs-667	435	3	focus	focus	VERB
eajbcs-667	435	4	on	on	ADP
eajbcs-667	435	5	the	the	DET
eajbcs-667	435	6	back	back	ADJ
eajbcs-667	435	7	ward	ward	NOUN
eajbcs-667	435	8	euler	euler	NOUN
eajbcs-667	435	9	method	method	NOUN
eajbcs-667	435	10	to	to	PART
eajbcs-667	435	11	descritize	descritize	VERB
eajbcs-667	435	12	time	time	NOUN
eajbcs-667	435	13	.	.	PUNCT
eajbcs-667	436	1	fem	fem	NOUN
eajbcs-667	436	2	implementation	implementation	NOUN
eajbcs-667	436	3	of	of	ADP
eajbcs-667	436	4	the	the	DET
eajbcs-667	436	5	two	two	NUM
eajbcs-667	436	6	dimensional	dimensional	ADJ
eajbcs-667	436	7	ad	ad	NOUN
eajbcs-667	436	8	equation	equation	NOUN
eajbcs-667	436	9	the	the	DET
eajbcs-667	436	10	2d	2d	NUM
eajbcs-667	436	11	advection	advection	NOUN
eajbcs-667	436	12	diffusion	diffusion	NOUN
eajbcs-667	436	13	equation	equation	NOUN
eajbcs-667	436	14	with	with	ADP
eajbcs-667	436	15	the	the	DET
eajbcs-667	436	16	same	same	ADJ
eajbcs-667	436	17	and	and	CCONJ
eajbcs-667	436	18	constant	constant	ADJ
eajbcs-667	436	19	velocity	velocity	NOUN
eajbcs-667	436	20	and	and	CCONJ
eajbcs-667	436	21	diffusion	diffusion	NOUN
eajbcs-667	436	22	term	term	NOUN
eajbcs-667	436	23	is	be	AUX
eajbcs-667	436	24	given	give	VERB
eajbcs-667	436	25	by	by	ADP
eajbcs-667	436	26	ut+a(ux+uy	ut+a(ux+uy	SYM
eajbcs-667	436	27	)	)	PUNCT
eajbcs-667	436	28	=	=	SYM
eajbcs-667	436	29	d(uxx+uyy)+f	d(uxx+uyy)+f	NOUN
eajbcs-667	436	30	,	,	PUNCT
eajbcs-667	436	31	(	(	PUNCT
eajbcs-667	436	32	x	x	NOUN
eajbcs-667	436	33	,	,	PUNCT
eajbcs-667	436	34	y	y	NOUN
eajbcs-667	436	35	)	)	PUNCT
eajbcs-667	436	36	∈	∈	PROPN
eajbcs-667	436	37	ω	ω	NOUN
eajbcs-667	436	38	=	=	PUNCT
eajbcs-667	437	1	[	[	X
eajbcs-667	437	2	0	0	NUM
eajbcs-667	437	3	,	,	PUNCT
eajbcs-667	437	4	1	1	NUM
eajbcs-667	437	5	]	]	PUNCT
eajbcs-667	437	6	,	,	PUNCT
eajbcs-667	437	7	east	east	PROPN
eajbcs-667	437	8	afr	afr	PROPN
eajbcs-667	437	9	.	.	PUNCT
eajbcs-667	438	1	j.	j.	PROPN
eajbcs-667	438	2	biophys	biophys	PROPN
eajbcs-667	438	3	.	.	PUNCT
eajbcs-667	439	1	comput	comput	NOUN
eajbcs-667	439	2	.	.	PUNCT
eajbcs-667	440	1	sci	sci	PROPN
eajbcs-667	440	2	.	.	PUNCT
eajbcs-667	440	3	(	(	PUNCT
eajbcs-667	440	4	2023	2023	NUM
eajbcs-667	440	5	)	)	PUNCT
eajbcs-667	440	6	,	,	PUNCT
eajbcs-667	440	7	vol	vol	NOUN
eajbcs-667	440	8	.	.	PROPN
eajbcs-667	440	9	4	4	NUM
eajbcs-667	440	10	,	,	PUNCT
eajbcs-667	440	11	no	no	INTJ
eajbcs-667	440	12	.	.	NOUN
eajbcs-667	440	13	1	1	NUM
eajbcs-667	440	14	,	,	PUNCT
eajbcs-667	440	15	52	52	NUM
eajbcs-667	440	16	-	-	SYM
eajbcs-667	440	17	74	74	NUM
eajbcs-667	440	18	65	65	NUM
eajbcs-667	440	19	using	use	VERB
eajbcs-667	440	20	the	the	DET
eajbcs-667	440	21	another	another	DET
eajbcs-667	440	22	quadrature	quadrature	NOUN
eajbcs-667	440	23	rule	rule	NOUN
eajbcs-667	440	24	for	for	ADP
eajbcs-667	440	25	simplicity	simplicity	NOUN
eajbcs-667	440	26	,	,	PUNCT
eajbcs-667	440	27	for	for	ADP
eajbcs-667	440	28	instance	instance	NOUN
eajbcs-667	440	29	,	,	PUNCT
eajbcs-667	440	30	using	use	VERB
eajbcs-667	440	31	the	the	DET
eajbcs-667	440	32	trapezoidal	trapezoidal	ADJ
eajbcs-667	440	33	rule	rule	NOUN
eajbcs-667	440	34	,	,	PUNCT
eajbcs-667	440	35	(	(	PUNCT
eajbcs-667	440	36	larson	larson	PROPN
eajbcs-667	440	37	and	and	CCONJ
eajbcs-667	440	38	bengzon	bengzon	PROPN
eajbcs-667	440	39	,	,	PUNCT
eajbcs-667	440	40	2010	2010	NUM
eajbcs-667	440	41	)	)	PUNCT
eajbcs-667	440	42	now	now	ADV
eajbcs-667	440	43	the	the	DET
eajbcs-667	440	44	system	system	NOUN
eajbcs-667	440	45	of	of	ADP
eajbcs-667	440	46	equation	equation	NOUN
eajbcs-667	440	47	,	,	PUNCT
eajbcs-667	440	48	that	that	PRON
eajbcs-667	440	49	is	is	ADV
eajbcs-667	440	50	eq.22	eq.22	NOUN
eajbcs-667	440	51	can	can	AUX
eajbcs-667	440	52	be	be	AUX
eajbcs-667	440	53	written	write	VERB
eajbcs-667	440	54	in	in	ADP
eajbcs-667	440	55	the	the	DET
eajbcs-667	440	56	form	form	NOUN
eajbcs-667	440	57	:	:	PUNCT
eajbcs-667	440	58	with	with	ADP
eajbcs-667	440	59	homogeneous	homogeneous	ADJ
eajbcs-667	440	60	boundary	boundary	ADJ
eajbcs-667	440	61	conditions	condition	NOUN
eajbcs-667	440	62	.	.	PUNCT
eajbcs-667	441	1	that	that	PRON
eajbcs-667	441	2	is	be	AUX
eajbcs-667	441	3	ut	ut	PROPN
eajbcs-667	441	4	+	+	CCONJ
eajbcs-667	441	5	a∇u	a∇u	X
eajbcs-667	441	6	=	=	SYM
eajbcs-667	441	7	d∇2u+	d∇2u+	PROPN
eajbcs-667	441	8	f.	f.	PROPN
eajbcs-667	441	9	now	now	ADV
eajbcs-667	441	10	to	to	PART
eajbcs-667	441	11	find	find	VERB
eajbcs-667	441	12	a	a	DET
eajbcs-667	441	13	weak	weak	ADJ
eajbcs-667	441	14	formulation	formulation	NOUN
eajbcs-667	441	15	for	for	ADP
eajbcs-667	441	16	this	this	DET
eajbcs-667	441	17	2d	2d	NUM
eajbcs-667	441	18	equation	equation	NOUN
eajbcs-667	441	19	,	,	PUNCT
eajbcs-667	441	20	we	we	PRON
eajbcs-667	441	21	multiply	multiply	VERB
eajbcs-667	441	22	both	both	DET
eajbcs-667	441	23	sides	side	NOUN
eajbcs-667	441	24	of	of	ADP
eajbcs-667	441	25	the	the	DET
eajbcs-667	441	26	equation	equation	NOUN
eajbcs-667	441	27	with	with	ADP
eajbcs-667	441	28	a	a	DET
eajbcs-667	441	29	test	test	NOUN
eajbcs-667	441	30	function	function	NOUN
eajbcs-667	441	31	v	v	ADP
eajbcs-667	441	32	=	=	PUNCT
eajbcs-667	441	33	v(x	v(x	PROPN
eajbcs-667	441	34	,	,	PUNCT
eajbcs-667	441	35	y	y	NOUN
eajbcs-667	441	36	)	)	PUNCT
eajbcs-667	441	37	∈	∈	PROPN
eajbcs-667	441	38	v	v	NOUN
eajbcs-667	441	39	which	which	PRON
eajbcs-667	441	40	satisfies	satisfy	VERB
eajbcs-667	441	41	the	the	DET
eajbcs-667	441	42	boundary	boundary	ADJ
eajbcs-667	441	43	conditions	condition	NOUN
eajbcs-667	441	44	.	.	PUNCT
eajbcs-667	442	1	ut.v	ut.v	ADJ
eajbcs-667	442	2	+	+	CCONJ
eajbcs-667	442	3	a∇u.v	a∇u.v	NOUN
eajbcs-667	442	4	=	=	PUNCT
eajbcs-667	442	5	d∇2u.v	d∇2u.v	PROPN
eajbcs-667	442	6	+	+	CCONJ
eajbcs-667	442	7	fv	fv	X
eajbcs-667	442	8	.	.	PUNCT
eajbcs-667	442	9	here	here	ADV
eajbcs-667	442	10	integrating	integrate	VERB
eajbcs-667	442	11	this	this	PRON
eajbcs-667	442	12	over	over	ADP
eajbcs-667	442	13	the	the	DET
eajbcs-667	442	14	domain	domain	NOUN
eajbcs-667	442	15	ω	ω	PROPN
eajbcs-667	442	16	yields	yield	VERB
eajbcs-667	442	17	the	the	DET
eajbcs-667	442	18	following	following	NOUN
eajbcs-667	442	19	:	:	PUNCT
eajbcs-667	442	20	∫	∫	PROPN
eajbcs-667	442	21	ω	ω	PROPN
eajbcs-667	442	22	(	(	PUNCT
eajbcs-667	442	23	ut.v	ut.v	ADJ
eajbcs-667	442	24	+	+	CCONJ
eajbcs-667	442	25	a∇u.v	a∇u.v	NOUN
eajbcs-667	442	26	)	)	PUNCT
eajbcs-667	443	1	=	=	SYM
eajbcs-667	443	2	∫	∫	PROPN
eajbcs-667	443	3	ω	ω	PROPN
eajbcs-667	443	4	(	(	PUNCT
eajbcs-667	443	5	d∇2u.v	d∇2u.v	PROPN
eajbcs-667	443	6	+	+	CCONJ
eajbcs-667	443	7	fv	fv	NOUN
eajbcs-667	443	8	)	)	PUNCT
eajbcs-667	443	9	.	.	PUNCT
eajbcs-667	444	1	we	we	PRON
eajbcs-667	444	2	see	see	VERB
eajbcs-667	444	3	from	from	ADP
eajbcs-667	444	4	the	the	DET
eajbcs-667	444	5	2d	2d	NUM
eajbcs-667	444	6	fem	fem	NOUN
eajbcs-667	444	7	of	of	ADP
eajbcs-667	444	8	poisson	poisson	PROPN
eajbcs-667	444	9	equation	equation	NOUN
eajbcs-667	444	10	(	(	PUNCT
eajbcs-667	444	11	using	use	VERB
eajbcs-667	444	12	gauss	gauss	NOUN
eajbcs-667	444	13	theorem	theorem	NOUN
eajbcs-667	444	14	and	and	CCONJ
eajbcs-667	444	15	the	the	DET
eajbcs-667	444	16	transformation	transformation	NOUN
eajbcs-667	444	17	of	of	ADP
eajbcs-667	444	18	a	a	DET
eajbcs-667	444	19	surface	surface	NOUN
eajbcs-667	444	20	integral	integral	ADJ
eajbcs-667	444	21	to	to	ADP
eajbcs-667	444	22	a	a	DET
eajbcs-667	444	23	line	line	NOUN
eajbcs-667	444	24	integral	integral	ADJ
eajbcs-667	444	25	)	)	PUNCT
eajbcs-667	444	26	that	that	SCONJ
eajbcs-667	444	27	∫	∫	PROPN
eajbcs-667	445	1	ω	ω	NUM
eajbcs-667	445	2	v∇2u	v∇2u	PROPN
eajbcs-667	445	3	=	=	SYM
eajbcs-667	446	1	−	−	PROPN
eajbcs-667	446	2	∫	∫	PROPN
eajbcs-667	446	3	ω	ω	PROPN
eajbcs-667	446	4	∇v∇u	∇v∇u	NUM
eajbcs-667	446	5	,	,	PUNCT
eajbcs-667	446	6	and	and	CCONJ
eajbcs-667	446	7	hence	hence	ADV
eajbcs-667	446	8	we	we	PRON
eajbcs-667	446	9	get	get	VERB
eajbcs-667	446	10	:	:	PUNCT
eajbcs-667	446	11	∫	∫	PROPN
eajbcs-667	446	12	ω	ω	PROPN
eajbcs-667	446	13	(	(	PUNCT
eajbcs-667	446	14	ut.v	ut.v	ADJ
eajbcs-667	446	15	+	+	CCONJ
eajbcs-667	446	16	a∇u.v	a∇u.v	NOUN
eajbcs-667	446	17	)	)	PUNCT
eajbcs-667	446	18	=	=	PUNCT
eajbcs-667	447	1	−d	−d	PROPN
eajbcs-667	447	2	∫	∫	PROPN
eajbcs-667	447	3	ω	ω	PROPN
eajbcs-667	447	4	(	(	PUNCT
eajbcs-667	447	5	∇v∇u	∇v∇u	NUM
eajbcs-667	447	6	)	)	PUNCT
eajbcs-667	448	1	+	+	NUM
eajbcs-667	448	2	∫	∫	PROPN
eajbcs-667	448	3	ω	ω	PROPN
eajbcs-667	448	4	fv.∫	fv.∫	PROPN
eajbcs-667	448	5	ω	ω	PROPN
eajbcs-667	448	6	ut.v	ut.v	ADJ
eajbcs-667	448	7	=	=	PUNCT
eajbcs-667	449	1	−	−	PROPN
eajbcs-667	449	2	∫	∫	PROPN
eajbcs-667	449	3	ω	ω	PROPN
eajbcs-667	449	4	(	(	PUNCT
eajbcs-667	449	5	d∇v∇u+	d∇v∇u+	X
eajbcs-667	449	6	a∇u.v	a∇u.v	PROPN
eajbcs-667	449	7	)	)	PUNCT
eajbcs-667	450	1	+	+	CCONJ
eajbcs-667	450	2	∫	∫	PROPN
eajbcs-667	450	3	ω	ω	NUM
eajbcs-667	450	4	fv	fv	PROPN
eajbcs-667	450	5	.	.	PROPN
eajbcs-667	451	1	(	(	PUNCT
eajbcs-667	451	2	27	27	NUM
eajbcs-667	451	3	)	)	PUNCT
eajbcs-667	451	4	(	(	PUNCT
eajbcs-667	451	5	ut	ut	PROPN
eajbcs-667	451	6	,	,	PUNCT
eajbcs-667	451	7	v	v	NOUN
eajbcs-667	451	8	)	)	PUNCT
eajbcs-667	451	9	=	=	SYM
eajbcs-667	451	10	l(u	l(u	PROPN
eajbcs-667	451	11	,	,	PUNCT
eajbcs-667	451	12	v	v	NOUN
eajbcs-667	451	13	)	)	PUNCT
eajbcs-667	451	14	+	+	CCONJ
eajbcs-667	451	15	(	(	PUNCT
eajbcs-667	451	16	f	f	X
eajbcs-667	451	17	,	,	PUNCT
eajbcs-667	451	18	v	v	NOUN
eajbcs-667	451	19	)	)	PUNCT
eajbcs-667	451	20	∀v	∀v	PROPN
eajbcs-667	451	21	∈	∈	PROPN
eajbcs-667	451	22	v	v	NOUN
eajbcs-667	451	23	,	,	PUNCT
eajbcs-667	451	24	(	(	PUNCT
eajbcs-667	451	25	28	28	NUM
eajbcs-667	451	26	)	)	PUNCT
eajbcs-667	451	27	where	where	SCONJ
eajbcs-667	451	28	(	(	PUNCT
eajbcs-667	451	29	ut	ut	PROPN
eajbcs-667	451	30	,	,	PUNCT
eajbcs-667	451	31	v	v	NOUN
eajbcs-667	451	32	)	)	PUNCT
eajbcs-667	451	33	=	=	SYM
eajbcs-667	452	1	∫	∫	PROPN
eajbcs-667	452	2	ω	ω	NUM
eajbcs-667	452	3	ut.v	ut.v	ADJ
eajbcs-667	452	4	and	and	CCONJ
eajbcs-667	452	5	l(u	l(u	PROPN
eajbcs-667	452	6	,	,	PUNCT
eajbcs-667	452	7	v	v	NOUN
eajbcs-667	452	8	)	)	PUNCT
eajbcs-667	452	9	=	=	PUNCT
eajbcs-667	453	1	−	−	PROPN
eajbcs-667	453	2	∫	∫	PROPN
eajbcs-667	453	3	ω	ω	PROPN
eajbcs-667	453	4	(	(	PUNCT
eajbcs-667	453	5	d∇v∇u+	d∇v∇u+	X
eajbcs-667	453	6	a∇u.v	a∇u.v	PROPN
eajbcs-667	453	7	)	)	PUNCT
eajbcs-667	453	8	.	.	PUNCT
eajbcs-667	454	1	given	give	VERB
eajbcs-667	454	2	a	a	DET
eajbcs-667	454	3	fe	fe	NOUN
eajbcs-667	454	4	space	space	NOUN
eajbcs-667	454	5	v	v	NOUN
eajbcs-667	454	6	,	,	PUNCT
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eajbcs-667	454	8	ϕi(x	ϕi(x	PROPN
eajbcs-667	454	9	,	,	PUNCT
eajbcs-667	454	10	y	y	PROPN
eajbcs-667	454	11	)	)	PUNCT
eajbcs-667	454	12	,	,	PUNCT
eajbcs-667	454	13	i	i	PRON
eajbcs-667	454	14	=	=	NOUN
eajbcs-667	454	15	1	1	NUM
eajbcs-667	454	16	,	,	PUNCT
eajbcs-667	454	17	2	2	NUM
eajbcs-667	454	18	,	,	PUNCT
eajbcs-667	454	19	.	.	PUNCT
eajbcs-667	454	20	.	.	PUNCT
eajbcs-667	454	21	.	.	PUNCT
eajbcs-667	455	1	,	,	PUNCT
eajbcs-667	456	1	n	n	CCONJ
eajbcs-667	456	2	denoting	denote	VERB
eajbcs-667	456	3	a	a	DET
eajbcs-667	456	4	set	set	NOUN
eajbcs-667	456	5	of	of	ADP
eajbcs-667	456	6	basis	basis	NOUN
eajbcs-667	456	7	functions	function	NOUN
eajbcs-667	456	8	for	for	ADP
eajbcs-667	456	9	v	v	NOUN
eajbcs-667	456	10	,	,	PUNCT
eajbcs-667	456	11	we	we	PRON
eajbcs-667	456	12	seek	seek	VERB
eajbcs-667	456	13	the	the	DET
eajbcs-667	456	14	fe	fe	NOUN
eajbcs-667	456	15	solution	solution	NOUN
eajbcs-667	456	16	of	of	ADP
eajbcs-667	456	17	form	form	NOUN
eajbcs-667	456	18	uh(x	uh(x	PROPN
eajbcs-667	456	19	,	,	PUNCT
eajbcs-667	456	20	y	y	PROPN
eajbcs-667	456	21	,	,	PUNCT
eajbcs-667	456	22	t	t	PROPN
eajbcs-667	456	23	)	)	PUNCT
eajbcs-667	456	24	=	=	PUNCT
eajbcs-667	457	1	n∑	n∑	NOUN
eajbcs-667	457	2	j=1	j=1	PROPN
eajbcs-667	457	3	uj(t)ϕj(x	uj(t)ϕj(x	PROPN
eajbcs-667	457	4	,	,	PUNCT
eajbcs-667	457	5	y	y	PROPN
eajbcs-667	457	6	)	)	PUNCT
eajbcs-667	457	7	.	.	PUNCT
eajbcs-667	458	1	(	(	PUNCT
eajbcs-667	458	2	29	29	NUM
eajbcs-667	458	3	)	)	PUNCT
eajbcs-667	458	4	and	and	CCONJ
eajbcs-667	458	5	taking	take	VERB
eajbcs-667	458	6	the	the	DET
eajbcs-667	458	7	test	test	NOUN
eajbcs-667	458	8	function	function	NOUN
eajbcs-667	458	9	v(x	v(x	PROPN
eajbcs-667	458	10	,	,	PUNCT
eajbcs-667	458	11	y	y	NOUN
eajbcs-667	458	12	)	)	PUNCT
eajbcs-667	458	13	as	as	ADP
eajbcs-667	458	14	a	a	DET
eajbcs-667	458	15	linear	linear	ADJ
eajbcs-667	458	16	combination	combination	NOUN
eajbcs-667	458	17	of	of	ADP
eajbcs-667	458	18	basis	basis	NOUN
eajbcs-667	458	19	functions	function	NOUN
eajbcs-667	458	20	v(x	v(x	PROPN
eajbcs-667	458	21	,	,	PUNCT
eajbcs-667	458	22	y	y	NOUN
eajbcs-667	458	23	)	)	PUNCT
eajbcs-667	458	24	=	=	SYM
eajbcs-667	459	1	n∑	n∑	PROPN
eajbcs-667	459	2	j=1	j=1	PROPN
eajbcs-667	459	3	vjϕj	vjϕj	NOUN
eajbcs-667	459	4	,	,	PUNCT
eajbcs-667	459	5	with	with	ADP
eajbcs-667	459	6	vj	vj	PROPN
eajbcs-667	459	7	are	be	AUX
eajbcs-667	459	8	constants	constant	NOUN
eajbcs-667	459	9	.	.	PUNCT
eajbcs-667	460	1	(	(	PUNCT
eajbcs-667	460	2	30	30	NUM
eajbcs-667	460	3	)	)	PUNCT
eajbcs-667	460	4	∑	∑	PUNCT
eajbcs-667	460	5	j=1	j=1	PROPN
eajbcs-667	460	6	u′	u′	PROPN
eajbcs-667	460	7	j(t)ϕj(x	j(t)ϕj(x	PROPN
eajbcs-667	460	8	,	,	PUNCT
eajbcs-667	460	9	y	y	PROPN
eajbcs-667	460	10	)	)	PUNCT
eajbcs-667	460	11	,	,	PUNCT
eajbcs-667	460	12	n∑	n∑	NOUN
eajbcs-667	460	13	j=1	j=1	NOUN
eajbcs-667	460	14	vjϕj	vjϕj	NOUN
eajbcs-667	460	15	)	)	PUNCT
eajbcs-667	461	1	=	=	PUNCT
eajbcs-667	461	2	l	l	NOUN
eajbcs-667	462	1	(	(	PUNCT
eajbcs-667	462	2	n∑	n∑	X
eajbcs-667	462	3	j=1	j=1	PROPN
eajbcs-667	462	4	uj(t)ϕj(x	uj(t)ϕj(x	PROPN
eajbcs-667	462	5	,	,	PUNCT
eajbcs-667	462	6	y	y	PROPN
eajbcs-667	462	7	)	)	PUNCT
eajbcs-667	462	8	,	,	PUNCT
eajbcs-667	462	9	n∑	n∑	NOUN
eajbcs-667	462	10	j=1	j=1	NOUN
eajbcs-667	462	11	vjϕj	vjϕj	NOUN
eajbcs-667	462	12	)	)	PUNCT
eajbcs-667	463	1	+	+	CCONJ
eajbcs-667	463	2	(	(	PUNCT
eajbcs-667	463	3	f	f	X
eajbcs-667	463	4	,	,	PUNCT
eajbcs-667	463	5	n∑	n∑	PROPN
eajbcs-667	463	6	j=1	j=1	PROPN
eajbcs-667	463	7	vjϕj	vjϕj	NOUN
eajbcs-667	463	8	)	)	PUNCT
eajbcs-667	463	9	.	.	PUNCT
eajbcs-667	464	1	(	(	PUNCT
eajbcs-667	464	2	31	31	NUM
eajbcs-667	464	3	)	)	PUNCT
eajbcs-667	464	4	then	then	ADV
eajbcs-667	464	5	we	we	PRON
eajbcs-667	464	6	get	get	VERB
eajbcs-667	464	7	the	the	DET
eajbcs-667	464	8	linear	linear	ADJ
eajbcs-667	464	9	system	system	NOUN
eajbcs-667	464	10	of	of	ADP
eajbcs-667	464	11	ordinary	ordinary	ADJ
eajbcs-667	464	12	differential	differential	ADJ
eajbcs-667	464	13	equations	equation	NOUN
eajbcs-667	464	14	in	in	ADP
eajbcs-667	464	15	the	the	DET
eajbcs-667	464	16	uj(t	uj(t	NOUN
eajbcs-667	464	17	)	)	PUNCT
eajbcs-667	464	18	as	as	ADP
eajbcs-667	464	19	:(	:(	PUNCT
eajbcs-667	464	20	u′	u′	PROPN
eajbcs-667	464	21	1(t)ϕ1(x	1(t)ϕ1(x	NUM
eajbcs-667	464	22	,	,	PUNCT
eajbcs-667	464	23	y	y	NOUN
eajbcs-667	464	24	)	)	PUNCT
eajbcs-667	465	1	+	+	CCONJ
eajbcs-667	465	2	n∑	n∑	PROPN
eajbcs-667	465	3	j=2	j=2	PROPN
eajbcs-667	465	4	u′	u′	PROPN
eajbcs-667	465	5	j(t)ϕj(x	j(t)ϕj(x	PROPN
eajbcs-667	465	6	,	,	PUNCT
eajbcs-667	465	7	y	y	PROPN
eajbcs-667	465	8	)	)	PUNCT
eajbcs-667	465	9	,	,	PUNCT
eajbcs-667	465	10	n∑	n∑	NOUN
eajbcs-667	465	11	j=1	j=1	NOUN
eajbcs-667	465	12	vjϕj	vjϕj	NOUN
eajbcs-667	465	13	)	)	PUNCT
eajbcs-667	466	1	=	=	SYM
eajbcs-667	466	2	l	l	NOUN
eajbcs-667	466	3	(	(	PUNCT
eajbcs-667	466	4	u1(t)ϕ1(x	u1(t)ϕ1(x	PROPN
eajbcs-667	466	5	,	,	PUNCT
eajbcs-667	466	6	y	y	PROPN
eajbcs-667	466	7	)	)	PUNCT
eajbcs-667	467	1	+	+	CCONJ
eajbcs-667	467	2	n∑	n∑	X
eajbcs-667	467	3	j=1	j=1	PROPN
eajbcs-667	467	4	uj(t)ϕj(x	uj(t)ϕj(x	PROPN
eajbcs-667	467	5	,	,	PUNCT
eajbcs-667	467	6	y	y	PROPN
eajbcs-667	467	7	)	)	PUNCT
eajbcs-667	467	8	,	,	PUNCT
eajbcs-667	467	9	n∑	n∑	NOUN
eajbcs-667	467	10	j=1	j=1	PROPN
eajbcs-667	467	11	vjϕj(x	vjϕj(x	PROPN
eajbcs-667	467	12	,	,	PUNCT
eajbcs-667	467	13	y	y	PROPN
eajbcs-667	467	14	)	)	PUNCT
eajbcs-667	467	15	)	)	PUNCT
eajbcs-667	468	1	+	+	CCONJ
eajbcs-667	468	2	(	(	PUNCT
eajbcs-667	468	3	f	f	X
eajbcs-667	468	4	,	,	PUNCT
eajbcs-667	468	5	n∑	n∑	NOUN
eajbcs-667	468	6	j=1	j=1	NOUN
eajbcs-667	468	7	vjϕj(x	vjϕj(x	PROPN
eajbcs-667	468	8	,	,	PUNCT
eajbcs-667	468	9	y	y	PROPN
eajbcs-667	468	10	)	)	PUNCT
eajbcs-667	468	11	)	)	PUNCT
eajbcs-667	468	12	,	,	PUNCT
eajbcs-667	468	13	east	east	PROPN
eajbcs-667	468	14	afr	afr	PROPN
eajbcs-667	468	15	.	.	PUNCT
eajbcs-667	469	1	j.	j.	PROPN
eajbcs-667	469	2	biophys	biophys	PROPN
eajbcs-667	469	3	.	.	PUNCT
eajbcs-667	470	1	comput	comput	NOUN
eajbcs-667	470	2	.	.	PUNCT
eajbcs-667	471	1	sci	sci	PROPN
eajbcs-667	471	2	.	.	PUNCT
eajbcs-667	471	3	(	(	PUNCT
eajbcs-667	471	4	2023	2023	NUM
eajbcs-667	471	5	)	)	PUNCT
eajbcs-667	471	6	,	,	PUNCT
eajbcs-667	471	7	vol	vol	NOUN
eajbcs-667	471	8	.	.	PROPN
eajbcs-667	471	9	4	4	NUM
eajbcs-667	471	10	,	,	PUNCT
eajbcs-667	471	11	no	no	INTJ
eajbcs-667	471	12	.	.	NOUN
eajbcs-667	471	13	1	1	NUM
eajbcs-667	471	14	,	,	PUNCT
eajbcs-667	471	15	52	52	NUM
eajbcs-667	471	16	-	-	SYM
eajbcs-667	471	17	74	74	NUM
eajbcs-667	471	18	66	66	NUM
eajbcs-667	471	19	substituting	substitute	VERB
eajbcs-667	471	20	this	this	DET
eajbcs-667	471	21	expression	expression	NOUN
eajbcs-667	471	22	(	(	PUNCT
eajbcs-667	471	23	29	29	NUM
eajbcs-667	471	24	and	and	CCONJ
eajbcs-667	471	25	30	30	NUM
eajbcs-667	471	26	)	)	PUNCT
eajbcs-667	471	27	into	into	ADP
eajbcs-667	471	28	eq.28	eq.28	NOUN
eajbcs-667	471	29	,	,	PUNCT
eajbcs-667	471	30	we	we	PRON
eajbcs-667	471	31	obtain	obtain	VERB
eajbcs-667	471	32	(	(	PUNCT
eajbcs-667	471	33	n	n	CCONJ
eajbcs-667	471	34	here	here	ADV
eajbcs-667	471	35	eq	eq	ADP
eajbcs-667	471	36	.	.	PROPN
eajbcs-667	471	37	27	27	NUM
eajbcs-667	471	38	is	be	AUX
eajbcs-667	471	39	the	the	DET
eajbcs-667	471	40	weak	weak	ADJ
eajbcs-667	471	41	formulation	formulation	NOUN
eajbcs-667	471	42	and	and	CCONJ
eajbcs-667	471	43	it	it	PRON
eajbcs-667	471	44	can	can	AUX
eajbcs-667	471	45	be	be	AUX
eajbcs-667	471	46	simplified	simplify	VERB
eajbcs-667	471	47	as	as	ADP
eajbcs-667	471	48	(	(	PUNCT
eajbcs-667	471	49	u′	u′	PROPN
eajbcs-667	471	50	1(t)ϕ1	1(t)ϕ1	PROPN
eajbcs-667	471	51	,	,	PUNCT
eajbcs-667	471	52	n∑	n∑	PRON
eajbcs-667	471	53	j=1	j=1	NOUN
eajbcs-667	471	54	vjϕj	vjϕj	NOUN
eajbcs-667	471	55	)	)	PUNCT
eajbcs-667	472	1	+	+	CCONJ
eajbcs-667	472	2	(	(	PUNCT
eajbcs-667	472	3	n∑	n∑	INTJ
eajbcs-667	472	4	j=2	j=2	PROPN
eajbcs-667	472	5	u′	u′	PROPN
eajbcs-667	472	6	j(t)ϕj	j(t)ϕj	PROPN
eajbcs-667	472	7	,	,	PUNCT
eajbcs-667	472	8	n∑	n∑	PRON
eajbcs-667	472	9	j=1	j=1	NOUN
eajbcs-667	472	10	vjϕj	vjϕj	NOUN
eajbcs-667	472	11	)	)	PUNCT
eajbcs-667	473	1	=	=	SYM
eajbcs-667	473	2	l	l	NOUN
eajbcs-667	473	3	(	(	PUNCT
eajbcs-667	473	4	u1(t)ϕ1	u1(t)ϕ1	ADJ
eajbcs-667	473	5	,	,	PUNCT
eajbcs-667	473	6	n∑	n∑	PROPN
eajbcs-667	473	7	j=1	j=1	NOUN
eajbcs-667	473	8	vjϕj	vjϕj	NOUN
eajbcs-667	473	9	)	)	PUNCT
eajbcs-667	474	1	+	+	CCONJ
eajbcs-667	474	2	l	l	NOUN
eajbcs-667	474	3	(	(	PUNCT
eajbcs-667	474	4	n∑	n∑	X
eajbcs-667	474	5	j=1	j=1	PROPN
eajbcs-667	474	6	uj(t)ϕj	uj(t)ϕj	PROPN
eajbcs-667	474	7	,	,	PUNCT
eajbcs-667	474	8	n∑	n∑	NOUN
eajbcs-667	474	9	j=1	j=1	NOUN
eajbcs-667	474	10	vjϕj	vjϕj	NOUN
eajbcs-667	474	11	)	)	PUNCT
eajbcs-667	475	1	+	+	CCONJ
eajbcs-667	475	2	(	(	PUNCT
eajbcs-667	475	3	f	f	X
eajbcs-667	475	4	,	,	PUNCT
eajbcs-667	475	5	n∑	n∑	PROPN
eajbcs-667	475	6	j=1	j=1	PROPN
eajbcs-667	475	7	vjϕj	vjϕj	NOUN
eajbcs-667	475	8	)	)	PUNCT
eajbcs-667	475	9	,	,	PUNCT
eajbcs-667	475	10	n∑	n∑	NOUN
eajbcs-667	475	11	j=1	j=1	NOUN
eajbcs-667	475	12	(	(	PUNCT
eajbcs-667	475	13	u′	u′	PROPN
eajbcs-667	475	14	j(t)ϕj	j(t)ϕj	PROPN
eajbcs-667	475	15	,	,	PUNCT
eajbcs-667	475	16	n∑	n∑	PRON
eajbcs-667	475	17	j=1	j=1	NOUN
eajbcs-667	475	18	vjϕj	vjϕj	NOUN
eajbcs-667	475	19	)	)	PUNCT
eajbcs-667	476	1	=	=	PUNCT
eajbcs-667	477	1	n∑	n∑	NOUN
eajbcs-667	477	2	j=1	j=1	ADJ
eajbcs-667	477	3	l	l	NOUN
eajbcs-667	477	4	(	(	PUNCT
eajbcs-667	477	5	uj(t)ϕj	uj(t)ϕj	ADJ
eajbcs-667	477	6	,	,	PUNCT
eajbcs-667	477	7	n∑	n∑	ADJ
eajbcs-667	477	8	j=1	j=1	NOUN
eajbcs-667	477	9	vjϕj	vjϕj	NOUN
eajbcs-667	477	10	)	)	PUNCT
eajbcs-667	478	1	+	+	CCONJ
eajbcs-667	478	2	(	(	PUNCT
eajbcs-667	478	3	f	f	X
eajbcs-667	478	4	,	,	PUNCT
eajbcs-667	478	5	n∑	n∑	PROPN
eajbcs-667	478	6	j=1	j=1	PROPN
eajbcs-667	478	7	vjϕj	vjϕj	NOUN
eajbcs-667	478	8	)	)	PUNCT
eajbcs-667	478	9	,	,	PUNCT
eajbcs-667	479	1	n∑	n∑	NOUN
eajbcs-667	479	2	j=1	j=1	PROPN
eajbcs-667	480	1	n∑	n∑	PROPN
eajbcs-667	480	2	j=1	j=1	NOUN
eajbcs-667	480	3	(	(	PUNCT
eajbcs-667	480	4	u′	u′	PROPN
eajbcs-667	480	5	j(t)ϕj	j(t)ϕj	PROPN
eajbcs-667	480	6	,	,	PUNCT
eajbcs-667	480	7	vjϕj	vjϕj	ADV
eajbcs-667	480	8	)	)	PUNCT
eajbcs-667	481	1	=	=	PUNCT
eajbcs-667	482	1	n∑	n∑	NOUN
eajbcs-667	482	2	j=1	j=1	PROPN
eajbcs-667	482	3	n∑	n∑	PROPN
eajbcs-667	482	4	j=1	j=1	PROPN
eajbcs-667	482	5	l	l	NOUN
eajbcs-667	482	6	(	(	PUNCT
eajbcs-667	482	7	uj(t)ϕj	uj(t)ϕj	ADJ
eajbcs-667	482	8	,	,	PUNCT
eajbcs-667	482	9	vjϕj	vjϕj	NOUN
eajbcs-667	482	10	)	)	PUNCT
eajbcs-667	483	1	+	+	CCONJ
eajbcs-667	483	2	n∑	n∑	ADJ
eajbcs-667	483	3	j=1	j=1	NOUN
eajbcs-667	483	4	(	(	PUNCT
eajbcs-667	483	5	f	f	X
eajbcs-667	483	6	,	,	PUNCT
eajbcs-667	483	7	vjϕj	vjϕj	NOUN
eajbcs-667	483	8	)	)	PUNCT
eajbcs-667	483	9	.	.	PUNCT
eajbcs-667	484	1	since	since	SCONJ
eajbcs-667	484	2	vj	vj	PROPN
eajbcs-667	484	3	’s	’s	PART
eajbcs-667	484	4	are	be	AUX
eajbcs-667	484	5	constants	constant	NOUN
eajbcs-667	484	6	we	we	PRON
eajbcs-667	484	7	have	have	VERB
eajbcs-667	484	8	also	also	ADV
eajbcs-667	484	9	that	that	PRON
eajbcs-667	484	10	:	:	PUNCT
eajbcs-667	484	11	n∑	n∑	PROPN
eajbcs-667	484	12	j=1	j=1	PROPN
eajbcs-667	484	13	vj	vj	INTJ
eajbcs-667	484	14	n∑	n∑	PROPN
eajbcs-667	484	15	j=1	j=1	PROPN
eajbcs-667	484	16	(	(	PUNCT
eajbcs-667	484	17	ϕj	ϕj	INTJ
eajbcs-667	484	18	,	,	PUNCT
eajbcs-667	484	19	ϕj)u	ϕj)u	PROPN
eajbcs-667	484	20	′	′	NUM
eajbcs-667	484	21	j(t	j(t	PROPN
eajbcs-667	484	22	)	)	PUNCT
eajbcs-667	485	1	=	=	PUNCT
eajbcs-667	486	1	n∑	n∑	NOUN
eajbcs-667	486	2	j=1	j=1	PROPN
eajbcs-667	486	3	vj	vj	INTJ
eajbcs-667	486	4	n∑	n∑	PROPN
eajbcs-667	486	5	j=1	j=1	PROPN
eajbcs-667	486	6	l	l	PROPN
eajbcs-667	486	7	(	(	PUNCT
eajbcs-667	486	8	ϕj	ϕj	PROPN
eajbcs-667	486	9	,	,	PUNCT
eajbcs-667	486	10	ϕj)uj(t	ϕj)uj(t	PROPN
eajbcs-667	486	11	)	)	PUNCT
eajbcs-667	487	1	+	+	PROPN
eajbcs-667	487	2	n∑	n∑	PROPN
eajbcs-667	487	3	j=1	j=1	NOUN
eajbcs-667	487	4	vj(f	vj(f	NUM
eajbcs-667	487	5	,	,	PUNCT
eajbcs-667	487	6	ϕj	ϕj	PROPN
eajbcs-667	487	7	)	)	PUNCT
eajbcs-667	487	8	.	.	PUNCT
eajbcs-667	488	1	the	the	DET
eajbcs-667	488	2	corresponding	corresponding	ADJ
eajbcs-667	488	3	problem	problem	NOUN
eajbcs-667	488	4	can	can	AUX
eajbcs-667	488	5	therefore	therefore	ADV
eajbcs-667	488	6	be	be	AUX
eajbcs-667	488	7	expressed	express	VERB
eajbcs-667	488	8	as	as	ADP
eajbcs-667	488	9	v	v	ADP
eajbcs-667	488	10	tmu̇	tmu̇	PROPN
eajbcs-667	488	11	=	=	SYM
eajbcs-667	488	12	v	v	ADP
eajbcs-667	488	13	tau	tau	PROPN
eajbcs-667	488	14	+	+	CCONJ
eajbcs-667	488	15	v	v	X
eajbcs-667	488	16	tf	tf	INTJ
eajbcs-667	488	17	.	.	PUNCT
eajbcs-667	489	1	that	that	PRON
eajbcs-667	489	2	is	be	AUX
eajbcs-667	489	3	mu̇	mu̇	NOUN
eajbcs-667	489	4	=	=	SYM
eajbcs-667	489	5	au	au	PROPN
eajbcs-667	490	1	+	+	CCONJ
eajbcs-667	490	2	f.	f.	PROPN
eajbcs-667	490	3	(	(	PUNCT
eajbcs-667	490	4	32	32	NUM
eajbcs-667	490	5	)	)	PUNCT
eajbcs-667	491	1	where	where	SCONJ
eajbcs-667	491	2	,	,	PUNCT
eajbcs-667	491	3	m	m	VERB
eajbcs-667	491	4	=	=	ADJ
eajbcs-667	491	5			PROPN
eajbcs-667	491	6	(	(	PUNCT
eajbcs-667	491	7	ϕ1	ϕ1	NOUN
eajbcs-667	491	8	,	,	PUNCT
eajbcs-667	491	9	ϕ1	ϕ1	NOUN
eajbcs-667	491	10	)	)	PUNCT
eajbcs-667	491	11	(	(	PUNCT
eajbcs-667	491	12	ϕ1	ϕ1	NOUN
eajbcs-667	491	13	,	,	PUNCT
eajbcs-667	491	14	ϕ2	ϕ2	ADV
eajbcs-667	491	15	)	)	PUNCT
eajbcs-667	491	16	.	.	PUNCT
eajbcs-667	491	17	.	.	PUNCT
eajbcs-667	491	18	.	.	PUNCT
eajbcs-667	492	1	(	(	PUNCT
eajbcs-667	492	2	ϕ1	ϕ1	NOUN
eajbcs-667	492	3	,	,	PUNCT
eajbcs-667	492	4	ϕn	ϕn	INTJ
eajbcs-667	492	5	)	)	PUNCT
eajbcs-667	492	6	(	(	PUNCT
eajbcs-667	492	7	ϕ2	ϕ2	ADV
eajbcs-667	492	8	,	,	PUNCT
eajbcs-667	492	9	ϕ1	ϕ1	NOUN
eajbcs-667	492	10	)	)	PUNCT
eajbcs-667	492	11	(	(	PUNCT
eajbcs-667	492	12	ϕ2	ϕ2	ADV
eajbcs-667	492	13	,	,	PUNCT
eajbcs-667	492	14	ϕ2	ϕ2	ADV
eajbcs-667	492	15	)	)	PUNCT
eajbcs-667	492	16	.	.	PUNCT
eajbcs-667	492	17	.	.	PUNCT
eajbcs-667	492	18	.	.	PUNCT
eajbcs-667	493	1	(	(	PUNCT
eajbcs-667	493	2	ϕ2	ϕ2	ADV
eajbcs-667	493	3	,	,	PUNCT
eajbcs-667	493	4	ϕn	ϕn	PROPN
eajbcs-667	493	5	)	)	PUNCT
eajbcs-667	493	6	...	...	PUNCT
eajbcs-667	493	7	...	...	PUNCT
eajbcs-667	493	8	...	...	PUNCT
eajbcs-667	494	1	(	(	PUNCT
eajbcs-667	494	2	ϕn	ϕn	INTJ
eajbcs-667	494	3	,	,	PUNCT
eajbcs-667	494	4	ϕ1	ϕ1	PROPN
eajbcs-667	494	5	)	)	PUNCT
eajbcs-667	494	6	(	(	PUNCT
eajbcs-667	494	7	ϕn	ϕn	INTJ
eajbcs-667	494	8	,	,	PUNCT
eajbcs-667	494	9	ϕ2	ϕ2	ADV
eajbcs-667	494	10	)	)	PUNCT
eajbcs-667	494	11	.	.	PUNCT
eajbcs-667	494	12	.	.	PUNCT
eajbcs-667	494	13	.	.	PUNCT
eajbcs-667	495	1	(	(	PUNCT
eajbcs-667	495	2	ϕn	ϕn	INTJ
eajbcs-667	495	3	,	,	PUNCT
eajbcs-667	495	4	ϕn	ϕn	INTJ
eajbcs-667	495	5	)	)	PUNCT
eajbcs-667	495	6			PROPN
eajbcs-667	495	7	,	,	PUNCT
eajbcs-667	495	8	a	a	DET
eajbcs-667	495	9	=	=	X
eajbcs-667	495	10			NOUN
eajbcs-667	495	11	l(ϕ1	l(ϕ1	ADJ
eajbcs-667	495	12	,	,	PUNCT
eajbcs-667	495	13	ϕ1	ϕ1	NOUN
eajbcs-667	495	14	)	)	PUNCT
eajbcs-667	495	15	l(ϕ1	l(ϕ1	ADJ
eajbcs-667	495	16	,	,	PUNCT
eajbcs-667	495	17	ϕ2	ϕ2	ADV
eajbcs-667	495	18	)	)	PUNCT
eajbcs-667	495	19	.	.	PUNCT
eajbcs-667	495	20	.	.	PUNCT
eajbcs-667	495	21	.	.	PUNCT
eajbcs-667	496	1	l(ϕ1	l(ϕ1	PROPN
eajbcs-667	496	2	,	,	PUNCT
eajbcs-667	496	3	ϕn	ϕn	NOUN
eajbcs-667	496	4	)	)	PUNCT
eajbcs-667	496	5	l(ϕ2	l(ϕ2	ADJ
eajbcs-667	496	6	,	,	PUNCT
eajbcs-667	496	7	ϕ1	ϕ1	NOUN
eajbcs-667	496	8	)	)	PUNCT
eajbcs-667	496	9	l(ϕ2	l(ϕ2	NOUN
eajbcs-667	496	10	,	,	PUNCT
eajbcs-667	496	11	ϕ2	ϕ2	ADV
eajbcs-667	496	12	)	)	PUNCT
eajbcs-667	496	13	.	.	PUNCT
eajbcs-667	496	14	.	.	PUNCT
eajbcs-667	497	1	.	.	PUNCT
eajbcs-667	498	1	l(ϕ2	l(ϕ2	ADJ
eajbcs-667	498	2	,	,	PUNCT
eajbcs-667	498	3	ϕn	ϕn	NOUN
eajbcs-667	498	4	)	)	PUNCT
eajbcs-667	498	5	...	...	PUNCT
eajbcs-667	498	6	...	...	PUNCT
eajbcs-667	498	7	...	...	PUNCT
eajbcs-667	499	1	l(ϕn	l(ϕn	NOUN
eajbcs-667	499	2	,	,	PUNCT
eajbcs-667	499	3	ϕ1	ϕ1	PROPN
eajbcs-667	499	4	)	)	PUNCT
eajbcs-667	499	5	l(ϕn	l(ϕn	NOUN
eajbcs-667	499	6	,	,	PUNCT
eajbcs-667	499	7	ϕ2	ϕ2	ADV
eajbcs-667	499	8	)	)	PUNCT
eajbcs-667	499	9	.	.	PUNCT
eajbcs-667	499	10	.	.	PUNCT
eajbcs-667	499	11	.	.	PUNCT
eajbcs-667	500	1	l(ϕn	l(ϕn	NOUN
eajbcs-667	500	2	,	,	PUNCT
eajbcs-667	500	3	ϕn	ϕn	INTJ
eajbcs-667	500	4	)	)	PUNCT
eajbcs-667	500	5			ADP
eajbcs-667	500	6	,	,	PUNCT
eajbcs-667	500	7	and	and	CCONJ
eajbcs-667	500	8	f	f	X
eajbcs-667	500	9	=	=	NOUN
eajbcs-667	500	10			PROPN
eajbcs-667	500	11	(	(	PUNCT
eajbcs-667	500	12	f	f	X
eajbcs-667	500	13	,	,	PUNCT
eajbcs-667	500	14	ϕ1	ϕ1	PROPN
eajbcs-667	500	15	)	)	PUNCT
eajbcs-667	500	16	(	(	PUNCT
eajbcs-667	500	17	f	f	X
eajbcs-667	500	18	,	,	PUNCT
eajbcs-667	500	19	ϕ2	ϕ2	ADV
eajbcs-667	500	20	)	)	PUNCT
eajbcs-667	500	21	...	...	PUNCT
eajbcs-667	501	1	(	(	PUNCT
eajbcs-667	501	2	f	f	X
eajbcs-667	501	3	,	,	PUNCT
eajbcs-667	501	4	ϕn	ϕn	INTJ
eajbcs-667	501	5	)	)	PUNCT
eajbcs-667	501	6			NOUN
eajbcs-667	501	7	.	.	PUNCT
eajbcs-667	502	1	in	in	ADP
eajbcs-667	502	2	this	this	DET
eajbcs-667	502	3	section	section	NOUN
eajbcs-667	502	4	,	,	PUNCT
eajbcs-667	502	5	we	we	PRON
eajbcs-667	502	6	are	be	AUX
eajbcs-667	502	7	compared	compare	VERB
eajbcs-667	502	8	for	for	ADP
eajbcs-667	502	9	the	the	DET
eajbcs-667	502	10	advection	advection	NOUN
eajbcs-667	502	11	-	-	PUNCT
eajbcs-667	502	12	diffusion	diffusion	NOUN
eajbcs-667	502	13	equations	equation	NOUN
eajbcs-667	502	14	with	with	ADP
eajbcs-667	502	15	an	an	DET
eajbcs-667	502	16	exact	exact	ADJ
eajbcs-667	502	17	solution	solution	NOUN
eajbcs-667	502	18	for	for	ADP
eajbcs-667	502	19	the	the	DET
eajbcs-667	502	20	given	give	VERB
eajbcs-667	502	21	finite	finite	ADJ
eajbcs-667	502	22	element	element	NOUN
eajbcs-667	502	23	methods	method	NOUN
eajbcs-667	502	24	and	and	CCONJ
eajbcs-667	502	25	then	then	ADV
eajbcs-667	502	26	we	we	PRON
eajbcs-667	502	27	solve	solve	VERB
eajbcs-667	502	28	the	the	DET
eajbcs-667	502	29	equation	equation	NOUN
eajbcs-667	502	30	with	with	ADP
eajbcs-667	502	31	out	out	ADP
eajbcs-667	502	32	knowing	know	VERB
eajbcs-667	502	33	the	the	DET
eajbcs-667	502	34	exact	exact	ADJ
eajbcs-667	502	35	solution	solution	NOUN
eajbcs-667	502	36	.	.	PUNCT
eajbcs-667	503	1	the	the	DET
eajbcs-667	503	2	comparison	comparison	NOUN
eajbcs-667	503	3	is	be	AUX
eajbcs-667	503	4	carried	carry	VERB
eajbcs-667	503	5	out	out	ADP
eajbcs-667	503	6	by	by	ADP
eajbcs-667	503	7	means	mean	NOUN
eajbcs-667	503	8	of	of	ADP
eajbcs-667	503	9	computed	computed	ADJ
eajbcs-667	503	10	solutions	solution	NOUN
eajbcs-667	503	11	for	for	ADP
eajbcs-667	503	12	a	a	DET
eajbcs-667	503	13	wide	wide	ADJ
eajbcs-667	503	14	range	range	NOUN
eajbcs-667	503	15	east	east	PROPN
eajbcs-667	503	16	afr	afr	PROPN
eajbcs-667	503	17	.	.	PUNCT
eajbcs-667	504	1	j.	j.	PROPN
eajbcs-667	504	2	biophys	biophys	PROPN
eajbcs-667	504	3	.	.	PUNCT
eajbcs-667	505	1	comput	comput	NOUN
eajbcs-667	505	2	.	.	PUNCT
eajbcs-667	506	1	sci	sci	PROPN
eajbcs-667	506	2	.	.	PUNCT
eajbcs-667	506	3	(	(	PUNCT
eajbcs-667	506	4	2023	2023	NUM
eajbcs-667	506	5	)	)	PUNCT
eajbcs-667	506	6	,	,	PUNCT
eajbcs-667	506	7	vol	vol	NOUN
eajbcs-667	506	8	.	.	PROPN
eajbcs-667	506	9	4	4	NUM
eajbcs-667	506	10	,	,	PUNCT
eajbcs-667	506	11	no	no	INTJ
eajbcs-667	506	12	.	.	NOUN
eajbcs-667	506	13	1	1	NUM
eajbcs-667	506	14	,	,	PUNCT
eajbcs-667	506	15	52	52	NUM
eajbcs-667	506	16	-	-	SYM
eajbcs-667	506	17	74	74	NUM
eajbcs-667	506	18	67	67	NUM
eajbcs-667	506	19	there	there	PRON
eajbcs-667	506	20	are	be	VERB
eajbcs-667	506	21	many	many	ADJ
eajbcs-667	506	22	methods	method	NOUN
eajbcs-667	506	23	to	to	PART
eajbcs-667	506	24	solve	solve	VERB
eajbcs-667	506	25	the	the	DET
eajbcs-667	506	26	above	above	ADJ
eajbcs-667	506	27	problem	problem	NOUN
eajbcs-667	506	28	involving	involve	VERB
eajbcs-667	506	29	the	the	DET
eajbcs-667	506	30	system	system	NOUN
eajbcs-667	506	31	of	of	ADP
eajbcs-667	506	32	first	first	ADJ
eajbcs-667	506	33	order	order	NOUN
eajbcs-667	506	34	ode	ode	ADJ
eajbcs-667	506	35	.	.	PUNCT
eajbcs-667	507	1	we	we	PRON
eajbcs-667	507	2	can	can	AUX
eajbcs-667	507	3	use	use	VERB
eajbcs-667	507	4	fd	fd	PROPN
eajbcs-667	507	5	methods	method	NOUN
eajbcs-667	507	6	that	that	PRON
eajbcs-667	507	7	will	will	AUX
eajbcs-667	507	8	descritize	descritize	VERB
eajbcs-667	507	9	in	in	ADP
eajbcs-667	507	10	time	time	NOUN
eajbcs-667	507	11	by	by	ADP
eajbcs-667	507	12	using	use	VERB
eajbcs-667	507	13	explicit	explicit	ADJ
eajbcs-667	507	14	euler	euler	NOUN
eajbcs-667	507	15	method	method	NOUN
eajbcs-667	507	16	,	,	PUNCT
eajbcs-667	507	17	implicit	implicit	ADJ
eajbcs-667	507	18	euler	euler	NOUN
eajbcs-667	507	19	method	method	NOUN
eajbcs-667	507	20	or	or	CCONJ
eajbcs-667	507	21	the	the	DET
eajbcs-667	507	22	crank	crank	NOUN
eajbcs-667	507	23	-	-	PUNCT
eajbcs-667	507	24	nicolson	nicolson	PROPN
eajbcs-667	507	25	method	method	NOUN
eajbcs-667	507	26	,	,	PUNCT
eajbcs-667	507	27	by	by	ADP
eajbcs-667	507	28	considering	consider	VERB
eajbcs-667	507	29	an	an	DET
eajbcs-667	507	30	appropriate	appropriate	ADJ
eajbcs-667	507	31	initial	initial	ADJ
eajbcs-667	507	32	condition	condition	NOUN
eajbcs-667	507	33	,	,	PUNCT
eajbcs-667	507	34	(	(	PUNCT
eajbcs-667	507	35	johnson	johnson	PROPN
eajbcs-667	507	36	,	,	PUNCT
eajbcs-667	507	37	2012	2012	NUM
eajbcs-667	507	38	)	)	PUNCT
eajbcs-667	507	39	.	.	PUNCT
eajbcs-667	508	1	but	but	CCONJ
eajbcs-667	508	2	for	for	ADP
eajbcs-667	508	3	this	this	DET
eajbcs-667	508	4	paper	paper	NOUN
eajbcs-667	508	5	in	in	ADP
eajbcs-667	508	6	the	the	DET
eajbcs-667	508	7	two	two	NUM
eajbcs-667	508	8	dimensional	dimensional	ADJ
eajbcs-667	508	9	case	case	NOUN
eajbcs-667	508	10	,	,	PUNCT
eajbcs-667	508	11	we	we	PRON
eajbcs-667	508	12	can	can	AUX
eajbcs-667	508	13	use	use	VERB
eajbcs-667	508	14	the	the	DET
eajbcs-667	508	15	ode	ode	ADJ
eajbcs-667	508	16	suite	suite	NOUN
eajbcs-667	508	17	in	in	ADP
eajbcs-667	508	18	matlab	matlab	PROPN
eajbcs-667	508	19	which	which	PRON
eajbcs-667	508	20	is	be	AUX
eajbcs-667	508	21	the	the	DET
eajbcs-667	508	22	matlab	matlab	PROPN
eajbcs-667	508	23	build	build	NOUN
eajbcs-667	508	24	in	in	ADP
eajbcs-667	508	25	system	system	NOUN
eajbcs-667	508	26	of	of	ADP
eajbcs-667	508	27	ode	ode	PROPN
eajbcs-667	508	28	solver	solver	NOUN
eajbcs-667	508	29	,	,	PUNCT
eajbcs-667	508	30	ode15i	ode15i	X
eajbcs-667	508	31	.	.	PUNCT
eajbcs-667	509	1	results	result	NOUN
eajbcs-667	509	2	and	and	CCONJ
eajbcs-667	509	3	discussion	discussion	NOUN
eajbcs-667	509	4	of	of	ADP
eajbcs-667	509	5	characteristic	characteristic	ADJ
eajbcs-667	509	6	parameters	parameter	NOUN
eajbcs-667	509	7	.	.	PUNCT
eajbcs-667	510	1	linear	linear	ADJ
eajbcs-667	510	2	elements	element	NOUN
eajbcs-667	510	3	are	be	AUX
eajbcs-667	510	4	employed	employ	VERB
eajbcs-667	510	5	at	at	ADP
eajbcs-667	510	6	the	the	DET
eajbcs-667	510	7	discretization	discretization	NOUN
eajbcs-667	510	8	in	in	ADP
eajbcs-667	510	9	case	case	NOUN
eajbcs-667	510	10	of	of	ADP
eajbcs-667	510	11	one	one	NUM
eajbcs-667	510	12	-	-	PUNCT
eajbcs-667	510	13	dimensional	dimensional	ADJ
eajbcs-667	510	14	problem	problem	NOUN
eajbcs-667	510	15	and	and	CCONJ
eajbcs-667	510	16	bilinear	bilinear	ADJ
eajbcs-667	510	17	elements	element	NOUN
eajbcs-667	510	18	in	in	ADP
eajbcs-667	510	19	case	case	NOUN
eajbcs-667	510	20	of	of	ADP
eajbcs-667	510	21	twodimensional	twodimensional	ADJ
eajbcs-667	510	22	problems	problem	NOUN
eajbcs-667	510	23	.	.	PUNCT
eajbcs-667	511	1	dirichlet	dirichlet	PROPN
eajbcs-667	511	2	and	and	CCONJ
eajbcs-667	511	3	general	general	ADJ
eajbcs-667	511	4	boundary	boundary	ADJ
eajbcs-667	511	5	conditions	condition	NOUN
eajbcs-667	511	6	are	be	AUX
eajbcs-667	511	7	considered	consider	VERB
eajbcs-667	511	8	with	with	ADP
eajbcs-667	511	9	different	different	ADJ
eajbcs-667	511	10	initial	initial	ADJ
eajbcs-667	511	11	conditions	condition	NOUN
eajbcs-667	511	12	and	and	CCONJ
eajbcs-667	511	13	different	different	ADJ
eajbcs-667	511	14	size	size	NOUN
eajbcs-667	511	15	of	of	ADP
eajbcs-667	511	16	computational	computational	ADJ
eajbcs-667	511	17	domain	domain	NOUN
eajbcs-667	511	18	.	.	PUNCT
eajbcs-667	512	1	x2	x2	NUM
eajbcs-667	512	2	)	)	PUNCT
eajbcs-667	513	1	+	+	NUM
eajbcs-667	513	2	t	t	PROPN
eajbcs-667	513	3	cos(πx)(1	cos(πx)(1	NOUN
eajbcs-667	513	4	−	−	NOUN
eajbcs-667	513	5	2x	2x	NUM
eajbcs-667	513	6	)	)	PUNCT
eajbcs-667	513	7	and	and	CCONJ
eajbcs-667	513	8	the	the	DET
eajbcs-667	513	9	velocity	velocity	NOUN
eajbcs-667	513	10	parameter	parameter	NOUN
eajbcs-667	513	11	a	a	DET
eajbcs-667	513	12	=	=	NOUN
eajbcs-667	513	13	3	3	NUM
eajbcs-667	513	14	,	,	PUNCT
eajbcs-667	513	15	with	with	ADP
eajbcs-667	513	16	homogeneous	homogeneous	ADJ
eajbcs-667	513	17	boundary	boundary	ADJ
eajbcs-667	513	18	conditions	condition	NOUN
eajbcs-667	513	19	.	.	PUNCT
eajbcs-667	514	1	the	the	DET
eajbcs-667	514	2	fm	fm	PROPN
eajbcs-667	514	3	solution	solution	NOUN
eajbcs-667	514	4	using	use	VERB
eajbcs-667	514	5	matlab	matlab	PROPN
eajbcs-667	514	6	with	with	ADP
eajbcs-667	514	7	a	a	DET
eajbcs-667	514	8	back	back	ADJ
eajbcs-667	514	9	ward	ward	NOUN
eajbcs-667	514	10	euler	euler	NOUN
eajbcs-667	514	11	descritization	descritization	NOUN
eajbcs-667	514	12	in	in	ADP
eajbcs-667	514	13	time	time	NOUN
eajbcs-667	514	14	is	be	AUX
eajbcs-667	514	15	given	give	VERB
eajbcs-667	514	16	in	in	ADP
eajbcs-667	514	17	fig	fig	NOUN
eajbcs-667	514	18	.	.	PUNCT
eajbcs-667	515	1	3(a	3(a	NUM
eajbcs-667	515	2	)	)	PUNCT
eajbcs-667	515	3	,	,	PUNCT
eajbcs-667	515	4	with	with	ADP
eajbcs-667	515	5	n	n	NOUN
eajbcs-667	515	6	=	=	SYM
eajbcs-667	515	7	100	100	NUM
eajbcs-667	515	8	nodes	node	NOUN
eajbcs-667	515	9	.	.	PUNCT
eajbcs-667	516	1	to	to	PART
eajbcs-667	516	2	saw	see	VERB
eajbcs-667	516	3	our	our	PRON
eajbcs-667	516	4	error	error	NOUN
eajbcs-667	516	5	the	the	DET
eajbcs-667	516	6	exact	exact	ADJ
eajbcs-667	516	7	solution	solution	NOUN
eajbcs-667	516	8	for	for	ADP
eajbcs-667	516	9	this	this	DET
eajbcs-667	516	10	equation	equation	NOUN
eajbcs-667	516	11	is	be	AUX
eajbcs-667	516	12	u(x	u(x	NOUN
eajbcs-667	516	13	,	,	PUNCT
eajbcs-667	516	14	t	t	NOUN
eajbcs-667	516	15	)	)	PUNCT
eajbcs-667	516	16	=	=	SYM
eajbcs-667	516	17	t	t	NOUN
eajbcs-667	516	18	cos(πx)(x−x2	cos(πx)(x−x2	PROPN
eajbcs-667	516	19	)	)	PUNCT
eajbcs-667	516	20	and	and	CCONJ
eajbcs-667	516	21	its	its	PRON
eajbcs-667	516	22	graph	graph	NOUN
eajbcs-667	516	23	is	be	AUX
eajbcs-667	516	24	of	of	ADP
eajbcs-667	516	25	nodes	node	NOUN
eajbcs-667	516	26	from	from	ADP
eajbcs-667	516	27	n	n	NOUN
eajbcs-667	516	28	=	=	NOUN
eajbcs-667	516	29	100	100	NUM
eajbcs-667	516	30	to	to	ADP
eajbcs-667	516	31	n	n	NOUN
eajbcs-667	516	32	=	=	SYM
eajbcs-667	516	33	1000	1000	NUM
eajbcs-667	516	34	,	,	PUNCT
eajbcs-667	516	35	then	then	ADV
eajbcs-667	516	36	our	our	PRON
eajbcs-667	516	37	numerical	numerical	ADJ
eajbcs-667	516	38	solution	solution	NOUN
eajbcs-667	516	39	becomes	become	VERB
eajbcs-667	516	40	more	more	ADV
eajbcs-667	516	41	accurate	accurate	ADJ
eajbcs-667	516	42	and	and	CCONJ
eajbcs-667	516	43	we	we	PRON
eajbcs-667	516	44	saw	see	VERB
eajbcs-667	516	45	that	that	SCONJ
eajbcs-667	516	46	the	the	DET
eajbcs-667	516	47	error	error	NOUN
eajbcs-667	516	48	is	be	AUX
eajbcs-667	516	49	an	an	DET
eajbcs-667	516	50	order	order	NOUN
eajbcs-667	516	51	of	of	ADP
eajbcs-667	516	52	10−7	10−7	NUM
eajbcs-667	516	53	by	by	ADP
eajbcs-667	516	54	modifying	modify	VERB
eajbcs-667	516	55	h	h	NOUN
eajbcs-667	516	56	=	=	NOUN
eajbcs-667	516	57	0.001	0.001	NUM
eajbcs-667	516	58	from	from	ADP
eajbcs-667	516	59	the	the	DET
eajbcs-667	516	60	algorithm	algorithm	NOUN
eajbcs-667	516	61	.	.	PUNCT
eajbcs-667	517	1	(	(	PUNCT
eajbcs-667	517	2	a	a	X
eajbcs-667	517	3	)	)	PUNCT
eajbcs-667	517	4	fem	fem	NOUN
eajbcs-667	517	5	solution	solution	NOUN
eajbcs-667	517	6	(	(	PUNCT
eajbcs-667	517	7	b	b	NOUN
eajbcs-667	517	8	)	)	PUNCT
eajbcs-667	517	9	exact	exact	ADJ
eajbcs-667	517	10	solution	solution	NOUN
eajbcs-667	517	11	figure	figure	NOUN
eajbcs-667	517	12	3	3	NUM
eajbcs-667	517	13	:	:	PUNCT
eajbcs-667	517	14	the	the	DET
eajbcs-667	517	15	matlab	matlab	PROPN
eajbcs-667	517	16	implementation	implementation	NOUN
eajbcs-667	517	17	of	of	ADP
eajbcs-667	517	18	the	the	DET
eajbcs-667	517	19	advection	advection	NOUN
eajbcs-667	517	20	dominated	dominate	VERB
eajbcs-667	517	21	equation	equation	NOUN
eajbcs-667	517	22	with	with	ADP
eajbcs-667	517	23	f(x	f(x	PROPN
eajbcs-667	517	24	,	,	PUNCT
eajbcs-667	517	25	t	t	PROPN
eajbcs-667	517	26	)	)	PUNCT
eajbcs-667	518	1	=	=	SYM
eajbcs-667	518	2	cos(πx)−	cos(πx)−	PROPN
eajbcs-667	518	3	atπ	atπ	VERB
eajbcs-667	518	4	sin(πx)(x−	sin(πx)(x−	NOUN
eajbcs-667	518	5	x2	x2	PROPN
eajbcs-667	518	6	)	)	PUNCT
eajbcs-667	519	1	+	+	NUM
eajbcs-667	519	2	t	t	PROPN
eajbcs-667	519	3	cos(πx)(1−	cos(πx)(1−	NOUN
eajbcs-667	519	4	2x	2x	NUM
eajbcs-667	519	5	)	)	PUNCT
eajbcs-667	519	6	.	.	PUNCT
eajbcs-667	520	1	tively	tively	ADV
eajbcs-667	520	2	large	large	ADJ
eajbcs-667	520	3	diffusion	diffusion	NOUN
eajbcs-667	520	4	coefficient	coefficient	NOUN
eajbcs-667	520	5	,	,	PUNCT
eajbcs-667	520	6	the	the	DET
eajbcs-667	520	7	diffusion	diffusion	NOUN
eajbcs-667	520	8	process	process	NOUN
eajbcs-667	520	9	is	be	AUX
eajbcs-667	520	10	faster	fast	ADJ
eajbcs-667	520	11	.	.	PUNCT
eajbcs-667	521	1	hence	hence	ADV
eajbcs-667	521	2	our	our	PRON
eajbcs-667	521	3	finite	finite	ADJ
eajbcs-667	521	4	element	element	NOUN
eajbcs-667	521	5	method	method	NOUN
eajbcs-667	521	6	is	be	AUX
eajbcs-667	521	7	reasonable	reasonable	ADJ
eajbcs-667	521	8	and	and	CCONJ
eajbcs-667	521	9	accurate	accurate	ADJ
eajbcs-667	521	10	.	.	PUNCT
eajbcs-667	522	1	east	east	PROPN
eajbcs-667	522	2	afr	afr	PROPN
eajbcs-667	522	3	.	.	PUNCT
eajbcs-667	523	1	j.	j.	PROPN
eajbcs-667	523	2	biophys	biophys	PROPN
eajbcs-667	523	3	.	.	PUNCT
eajbcs-667	524	1	comput	comput	NOUN
eajbcs-667	524	2	.	.	PUNCT
eajbcs-667	525	1	sci	sci	PROPN
eajbcs-667	525	2	.	.	PUNCT
eajbcs-667	525	3	(	(	PUNCT
eajbcs-667	525	4	2023	2023	NUM
eajbcs-667	525	5	)	)	PUNCT
eajbcs-667	525	6	,	,	PUNCT
eajbcs-667	525	7	vol	vol	NOUN
eajbcs-667	525	8	.	.	PROPN
eajbcs-667	525	9	4	4	NUM
eajbcs-667	525	10	,	,	PUNCT
eajbcs-667	525	11	no	no	INTJ
eajbcs-667	525	12	.	.	NOUN
eajbcs-667	525	13	1	1	NUM
eajbcs-667	525	14	,	,	PUNCT
eajbcs-667	525	15	52	52	NUM
eajbcs-667	525	16	-	-	SYM
eajbcs-667	525	17	74	74	NUM
eajbcs-667	525	18	68	68	NUM
eajbcs-667	525	19	fig	fig	NOUN
eajbcs-667	525	20	.	.	PUNCT
eajbcs-667	526	1	3(b	3(b	NUM
eajbcs-667	526	2	)	)	PUNCT
eajbcs-667	526	3	.	.	PUNCT
eajbcs-667	527	1	the	the	DET
eajbcs-667	527	2	error	error	NOUN
eajbcs-667	527	3	of	of	ADP
eajbcs-667	527	4	this	this	DET
eajbcs-667	527	5	equation	equation	NOUN
eajbcs-667	527	6	is	be	AUX
eajbcs-667	527	7	an	an	DET
eajbcs-667	527	8	order	order	NOUN
eajbcs-667	527	9	of	of	ADP
eajbcs-667	527	10	10−4	10−4	NUM
eajbcs-667	527	11	.	.	PUNCT
eajbcs-667	528	1	if	if	SCONJ
eajbcs-667	528	2	we	we	PRON
eajbcs-667	528	3	increase	increase	VERB
eajbcs-667	528	4	the	the	DET
eajbcs-667	528	5	number	number	NOUN
eajbcs-667	528	6	in	in	ADP
eajbcs-667	528	7	(	(	PUNCT
eajbcs-667	528	8	bergara	bergara	NOUN
eajbcs-667	528	9	,	,	PUNCT
eajbcs-667	528	10	2011	2011	NUM
eajbcs-667	528	11	)	)	PUNCT
eajbcs-667	528	12	,	,	PUNCT
eajbcs-667	528	13	there	there	PRON
eajbcs-667	528	14	is	be	VERB
eajbcs-667	528	15	a	a	DET
eajbcs-667	528	16	diffusion	diffusion	NOUN
eajbcs-667	528	17	(	(	PUNCT
eajbcs-667	528	18	ut	ut	PROPN
eajbcs-667	528	19	=	=	SYM
eajbcs-667	528	20	duxx	duxx	PROPN
eajbcs-667	528	21	)	)	PUNCT
eajbcs-667	528	22	example	example	NOUN
eajbcs-667	528	23	solved	solve	VERB
eajbcs-667	528	24	with	with	ADP
eajbcs-667	528	25	finite	finite	ADJ
eajbcs-667	528	26	difference	difference	NOUN
eajbcs-667	528	27	methods	method	NOUN
eajbcs-667	528	28	,	,	PUNCT
eajbcs-667	528	29	and	and	CCONJ
eajbcs-667	528	30	let	let	VERB
eajbcs-667	528	31	we	we	PRON
eajbcs-667	528	32	solve	solve	VERB
eajbcs-667	528	33	that	that	DET
eajbcs-667	528	34	equation	equation	NOUN
eajbcs-667	528	35	with	with	ADP
eajbcs-667	528	36	the	the	DET
eajbcs-667	528	37	finite	finite	ADJ
eajbcs-667	528	38	element	element	NOUN
eajbcs-667	528	39	method	method	NOUN
eajbcs-667	528	40	.	.	PUNCT
eajbcs-667	529	1	he	he	PRON
eajbcs-667	529	2	solves	solve	VERB
eajbcs-667	529	3	the	the	DET
eajbcs-667	529	4	diffusion	diffusion	NOUN
eajbcs-667	529	5	(	(	PUNCT
eajbcs-667	529	6	heat	heat	NOUN
eajbcs-667	529	7	)	)	PUNCT
eajbcs-667	529	8	equation	equation	NOUN
eajbcs-667	529	9	by	by	ADP
eajbcs-667	529	10	using	use	VERB
eajbcs-667	529	11	initial	initial	ADJ
eajbcs-667	529	12	condition	condition	NOUN
eajbcs-667	529	13	u(x	u(x	NOUN
eajbcs-667	529	14	,	,	PUNCT
eajbcs-667	529	15	0	0	NUM
eajbcs-667	529	16	)	)	PUNCT
eajbcs-667	529	17	=	=	SYM
eajbcs-667	529	18	sin(πx	sin(πx	NOUN
eajbcs-667	529	19	)	)	PUNCT
eajbcs-667	529	20	first	first	ADV
eajbcs-667	529	21	,	,	PUNCT
eajbcs-667	529	22	consider	consider	VERB
eajbcs-667	529	23	the	the	DET
eajbcs-667	529	24	advection	advection	NOUN
eajbcs-667	529	25	equation	equation	NOUN
eajbcs-667	529	26	ut	ut	PROPN
eajbcs-667	529	27	+	+	CCONJ
eajbcs-667	529	28	aux	aux	PROPN
eajbcs-667	529	29	=	=	SYM
eajbcs-667	529	30	f	f	PROPN
eajbcs-667	529	31	with	with	ADP
eajbcs-667	529	32	sources	source	NOUN
eajbcs-667	529	33	and	and	CCONJ
eajbcs-667	529	34	sinks	sink	NOUN
eajbcs-667	529	35	function	function	NOUN
eajbcs-667	529	36	,	,	PUNCT
eajbcs-667	529	37	f(x	f(x	PROPN
eajbcs-667	529	38	,	,	PUNCT
eajbcs-667	529	39	t	t	PROPN
eajbcs-667	529	40	)	)	PUNCT
eajbcs-667	530	1	=	=	SYM
eajbcs-667	530	2	cos(πx	cos(πx	NOUN
eajbcs-667	530	3	)	)	PUNCT
eajbcs-667	530	4	−	−	NOUN
eajbcs-667	530	5	atπ	atπ	VERB
eajbcs-667	530	6	sin(πx)(x	sin(πx)(x	PROPN
eajbcs-667	530	7	−	−	PROPN
eajbcs-667	530	8	when	when	SCONJ
eajbcs-667	530	9	we	we	PRON
eajbcs-667	530	10	saw	see	VERB
eajbcs-667	530	11	the	the	DET
eajbcs-667	530	12	results	result	NOUN
eajbcs-667	530	13	of	of	ADP
eajbcs-667	530	14	those	those	DET
eajbcs-667	530	15	figures	figure	NOUN
eajbcs-667	530	16	(	(	PUNCT
eajbcs-667	530	17	fig	fig	NOUN
eajbcs-667	530	18	.	.	NOUN
eajbcs-667	530	19	4	4	NUM
eajbcs-667	530	20	to	to	PART
eajbcs-667	530	21	fig	fig	VERB
eajbcs-667	530	22	.	.	PUNCT
eajbcs-667	531	1	6	6	X
eajbcs-667	531	2	)	)	PUNCT
eajbcs-667	531	3	we	we	PRON
eajbcs-667	531	4	can	can	AUX
eajbcs-667	531	5	observe	observe	VERB
eajbcs-667	531	6	the	the	DET
eajbcs-667	531	7	following	following	NOUN
eajbcs-667	531	8	.	.	PUNCT
eajbcs-667	532	1	if	if	SCONJ
eajbcs-667	532	2	we	we	PRON
eajbcs-667	532	3	use	use	VERB
eajbcs-667	532	4	a	a	DET
eajbcs-667	532	5	small	small	ADJ
eajbcs-667	532	6	amount	amount	NOUN
eajbcs-667	532	7	of	of	ADP
eajbcs-667	532	8	diffusion	diffusion	NOUN
eajbcs-667	532	9	coefficient	coefficient	NOUN
eajbcs-667	532	10	,	,	PUNCT
eajbcs-667	532	11	then	then	ADV
eajbcs-667	532	12	it	it	PRON
eajbcs-667	532	13	has	have	VERB
eajbcs-667	532	14	a	a	DET
eajbcs-667	532	15	mall	mall	NOUN
eajbcs-667	532	16	diffusion	diffusion	NOUN
eajbcs-667	532	17	process	process	NOUN
eajbcs-667	532	18	and	and	CCONJ
eajbcs-667	532	19	if	if	SCONJ
eajbcs-667	532	20	we	we	PRON
eajbcs-667	532	21	consider	consider	VERB
eajbcs-667	532	22	relaand	relaand	NOUN
eajbcs-667	532	23	with	with	ADP
eajbcs-667	532	24	homogeneous	homogeneous	ADJ
eajbcs-667	532	25	dirichlet	dirichlet	PROPN
eajbcs-667	532	26	boundary	boundary	ADJ
eajbcs-667	532	27	conditions	condition	NOUN
eajbcs-667	532	28	in	in	ADP
eajbcs-667	532	29	the	the	DET
eajbcs-667	532	30	interval	interval	NOUN
eajbcs-667	532	31	0	0	NUM
eajbcs-667	532	32	≤	≤	NUM
eajbcs-667	532	33	x	x	SYM
eajbcs-667	532	34	≤	≤	NUM
eajbcs-667	532	35	1	1	NUM
eajbcs-667	532	36	.	.	PUNCT
eajbcs-667	533	1	if	if	SCONJ
eajbcs-667	533	2	we	we	PRON
eajbcs-667	533	3	consider	consider	VERB
eajbcs-667	533	4	different	different	ADJ
eajbcs-667	533	5	values	value	NOUN
eajbcs-667	533	6	for	for	ADP
eajbcs-667	533	7	final	final	ADJ
eajbcs-667	533	8	time	time	NOUN
eajbcs-667	533	9	and	and	CCONJ
eajbcs-667	533	10	diffusion	diffusion	NOUN
eajbcs-667	533	11	coefficient	coefficient	NOUN
eajbcs-667	533	12	we	we	PRON
eajbcs-667	533	13	get	get	VERB
eajbcs-667	533	14	the	the	DET
eajbcs-667	533	15	following	follow	VERB
eajbcs-667	533	16	simulations	simulation	NOUN
eajbcs-667	533	17	(	(	PUNCT
eajbcs-667	533	18	fig	fig	NOUN
eajbcs-667	533	19	.	.	NOUN
eajbcs-667	533	20	4	4	NUM
eajbcs-667	533	21	to	to	PART
eajbcs-667	533	22	fig	fig	VERB
eajbcs-667	533	23	.	.	PUNCT
eajbcs-667	534	1	6	6	NUM
eajbcs-667	534	2	)	)	PUNCT
eajbcs-667	534	3	with	with	ADP
eajbcs-667	534	4	a	a	DET
eajbcs-667	534	5	similar	similar	ADJ
eajbcs-667	534	6	descritization	descritization	NOUN
eajbcs-667	534	7	of	of	ADP
eajbcs-667	534	8	space	space	NOUN
eajbcs-667	534	9	(	(	PUNCT
eajbcs-667	534	10	x	x	X
eajbcs-667	534	11	)	)	PUNCT
eajbcs-667	534	12	in	in	ADP
eajbcs-667	534	13	to	to	ADP
eajbcs-667	534	14	50	50	NUM
eajbcs-667	534	15	nodes	node	NOUN
eajbcs-667	534	16	.	.	PUNCT
eajbcs-667	535	1	(	(	PUNCT
eajbcs-667	535	2	a	a	X
eajbcs-667	535	3	)	)	PUNCT
eajbcs-667	535	4	fem	fem	NOUN
eajbcs-667	535	5	sol	sol	NOUN
eajbcs-667	535	6	.	.	PUNCT
eajbcs-667	536	1	with	with	ADP
eajbcs-667	536	2	d	d	PROPN
eajbcs-667	536	3	=	=	SYM
eajbcs-667	536	4	0.1	0.1	NUM
eajbcs-667	536	5	and	and	CCONJ
eajbcs-667	536	6	tf	tf	NOUN
eajbcs-667	536	7	=	=	SYM
eajbcs-667	536	8	0.5	0.5	NUM
eajbcs-667	536	9	(	(	PUNCT
eajbcs-667	536	10	b	b	NOUN
eajbcs-667	536	11	)	)	PUNCT
eajbcs-667	536	12	fem	fem	NOUN
eajbcs-667	536	13	sol	sol	NOUN
eajbcs-667	536	14	.	.	PUNCT
eajbcs-667	537	1	with	with	ADP
eajbcs-667	537	2	d	d	PROPN
eajbcs-667	537	3	=	=	SYM
eajbcs-667	537	4	1	1	NUM
eajbcs-667	537	5	and	and	CCONJ
eajbcs-667	537	6	tf	tf	NOUN
eajbcs-667	537	7	=	=	SYM
eajbcs-667	537	8	0.5	0.5	NUM
eajbcs-667	537	9	figure	figure	NOUN
eajbcs-667	537	10	4	4	NUM
eajbcs-667	537	11	:	:	PUNCT
eajbcs-667	537	12	solution	solution	NOUN
eajbcs-667	537	13	of	of	ADP
eajbcs-667	537	14	the	the	DET
eajbcs-667	537	15	diffusion	diffusion	NOUN
eajbcs-667	537	16	equation	equation	NOUN
eajbcs-667	537	17	using	use	VERB
eajbcs-667	537	18	d	d	PROPN
eajbcs-667	537	19	=	=	SYM
eajbcs-667	537	20	0.1	0.1	NUM
eajbcs-667	537	21	and	and	CCONJ
eajbcs-667	537	22	d	d	NOUN
eajbcs-667	537	23	=	=	SYM
eajbcs-667	537	24	1	1	NUM
eajbcs-667	537	25	for	for	ADP
eajbcs-667	537	26	a	a	DET
eajbcs-667	537	27	constant	constant	ADJ
eajbcs-667	537	28	final	final	ADJ
eajbcs-667	537	29	time	time	NOUN
eajbcs-667	537	30	of	of	ADP
eajbcs-667	537	31	tf	tf	PROPN
eajbcs-667	537	32	=	=	SYM
eajbcs-667	537	33	0.5	0.5	NUM
eajbcs-667	537	34	.	.	PUNCT
eajbcs-667	538	1	(	(	PUNCT
eajbcs-667	538	2	a	a	X
eajbcs-667	538	3	)	)	PUNCT
eajbcs-667	538	4	fem	fem	NOUN
eajbcs-667	538	5	sol	sol	NOUN
eajbcs-667	538	6	.	.	PUNCT
eajbcs-667	539	1	with	with	ADP
eajbcs-667	539	2	d	d	PROPN
eajbcs-667	539	3	=	=	SYM
eajbcs-667	539	4	10	10	NUM
eajbcs-667	539	5	and	and	CCONJ
eajbcs-667	539	6	tf	tf	X
eajbcs-667	539	7	=	=	SYM
eajbcs-667	539	8	0.5	0.5	NUM
eajbcs-667	539	9	(	(	PUNCT
eajbcs-667	539	10	b	b	NOUN
eajbcs-667	539	11	)	)	PUNCT
eajbcs-667	539	12	fem	fem	NOUN
eajbcs-667	539	13	sol	sol	NOUN
eajbcs-667	539	14	.	.	PUNCT
eajbcs-667	540	1	with	with	ADP
eajbcs-667	540	2	d	d	PROPN
eajbcs-667	540	3	=	=	SYM
eajbcs-667	540	4	0.1	0.1	NUM
eajbcs-667	540	5	and	and	CCONJ
eajbcs-667	540	6	tf	tf	NOUN
eajbcs-667	540	7	=	=	SYM
eajbcs-667	540	8	3	3	NUM
eajbcs-667	540	9	figure	figure	NOUN
eajbcs-667	540	10	5	5	NUM
eajbcs-667	540	11	:	:	PUNCT
eajbcs-667	540	12	solution	solution	NOUN
eajbcs-667	540	13	of	of	ADP
eajbcs-667	540	14	the	the	DET
eajbcs-667	540	15	diffusion	diffusion	NOUN
eajbcs-667	540	16	equation	equation	NOUN
eajbcs-667	540	17	using	use	VERB
eajbcs-667	540	18	d	d	PROPN
eajbcs-667	540	19	=	=	SYM
eajbcs-667	540	20	10	10	NUM
eajbcs-667	540	21	with	with	ADP
eajbcs-667	540	22	tf	tf	PROPN
eajbcs-667	540	23	=	=	SYM
eajbcs-667	540	24	0.5	0.5	NUM
eajbcs-667	540	25	and	and	CCONJ
eajbcs-667	540	26	d	d	NOUN
eajbcs-667	540	27	=	=	SYM
eajbcs-667	540	28	0.1	0.1	NUM
eajbcs-667	540	29	for	for	ADP
eajbcs-667	540	30	final	final	ADJ
eajbcs-667	540	31	time	time	NOUN
eajbcs-667	540	32	of	of	ADP
eajbcs-667	540	33	tf	tf	PROPN
eajbcs-667	540	34	=	=	SYM
eajbcs-667	540	35	3	3	NUM
eajbcs-667	540	36	.	.	NOUN
eajbcs-667	540	37	sin(πx	sin(πx	NOUN
eajbcs-667	540	38	)	)	PUNCT
eajbcs-667	540	39	(	(	PUNCT
eajbcs-667	540	40	1	1	NUM
eajbcs-667	540	41	+	+	NOUN
eajbcs-667	540	42	dπ2	dπ2	NOUN
eajbcs-667	540	43	t	t	PROPN
eajbcs-667	540	44	)	)	PUNCT
eajbcs-667	541	1	+	+	NUM
eajbcs-667	541	2	aπt	aπt	NOUN
eajbcs-667	541	3	cos(πx	cos(πx	NOUN
eajbcs-667	541	4	)	)	PUNCT
eajbcs-667	541	5	,	,	PUNCT
eajbcs-667	541	6	u(x	u(x	NOUN
eajbcs-667	541	7	,	,	PUNCT
eajbcs-667	541	8	0	0	NUM
eajbcs-667	541	9	)	)	PUNCT
eajbcs-667	541	10	=	=	SYM
eajbcs-667	541	11	0	0	NUM
eajbcs-667	541	12	,	,	PUNCT
eajbcs-667	541	13	u(0	u(0	PROPN
eajbcs-667	541	14	,	,	PUNCT
eajbcs-667	541	15	t	t	PROPN
eajbcs-667	541	16	)	)	PUNCT
eajbcs-667	541	17	=	=	SYM
eajbcs-667	542	1	u(1	u(1	PROPN
eajbcs-667	542	2	,	,	PUNCT
eajbcs-667	542	3	t	t	PROPN
eajbcs-667	542	4	)	)	PUNCT
eajbcs-667	542	5	=	=	SYM
eajbcs-667	542	6	0	0	X
eajbcs-667	542	7	.	.	PUNCT
eajbcs-667	543	1	by	by	ADP
eajbcs-667	543	2	using	use	VERB
eajbcs-667	543	3	the	the	DET
eajbcs-667	543	4	forward	forward	ADJ
eajbcs-667	543	5	euler	euler	NOUN
eajbcs-667	543	6	descritization	descritization	NOUN
eajbcs-667	543	7	in	in	ADP
eajbcs-667	543	8	pe	pe	PROPN
eajbcs-667	543	9	=	=	PUNCT
eajbcs-667	543	10	al	al	PROPN
eajbcs-667	543	11	d	d	PROPN
eajbcs-667	543	12	=	=	NOUN
eajbcs-667	543	13	0.003	0.003	NUM
eajbcs-667	543	14	≪	≪	X
eajbcs-667	543	15	1	1	NUM
eajbcs-667	543	16	which	which	PRON
eajbcs-667	543	17	is	be	AUX
eajbcs-667	543	18	diffusion	diffusion	NOUN
eajbcs-667	543	19	dominated	dominate	VERB
eajbcs-667	543	20	for	for	ADP
eajbcs-667	543	21	this	this	DET
eajbcs-667	543	22	choice	choice	NOUN
eajbcs-667	543	23	of	of	ADP
eajbcs-667	543	24	a	a	PRON
eajbcs-667	543	25	and	and	CCONJ
eajbcs-667	543	26	d.	d.	PROPN
eajbcs-667	543	27	east	east	PROPN
eajbcs-667	543	28	afr	afr	PROPN
eajbcs-667	543	29	.	.	PUNCT
eajbcs-667	544	1	j.	j.	PROPN
eajbcs-667	544	2	biophys	biophys	PROPN
eajbcs-667	544	3	.	.	PUNCT
eajbcs-667	545	1	comput	comput	NOUN
eajbcs-667	545	2	.	.	PUNCT
eajbcs-667	546	1	sci	sci	PROPN
eajbcs-667	546	2	.	.	PUNCT
eajbcs-667	546	3	(	(	PUNCT
eajbcs-667	546	4	2023	2023	NUM
eajbcs-667	546	5	)	)	PUNCT
eajbcs-667	546	6	,	,	PUNCT
eajbcs-667	546	7	vol	vol	NOUN
eajbcs-667	546	8	.	.	PROPN
eajbcs-667	546	9	4	4	NUM
eajbcs-667	546	10	,	,	PUNCT
eajbcs-667	546	11	no	no	INTJ
eajbcs-667	546	12	.	.	NOUN
eajbcs-667	546	13	1	1	NUM
eajbcs-667	546	14	,	,	PUNCT
eajbcs-667	546	15	52	52	NUM
eajbcs-667	546	16	-	-	SYM
eajbcs-667	546	17	74	74	NUM
eajbcs-667	546	18	69	69	NUM
eajbcs-667	546	19	time	time	NOUN
eajbcs-667	546	20	the	the	DET
eajbcs-667	546	21	matlab	matlab	PROPN
eajbcs-667	546	22	implementation	implementation	NOUN
eajbcs-667	546	23	with	with	ADP
eajbcs-667	546	24	a	a	DET
eajbcs-667	546	25	=	=	SYM
eajbcs-667	546	26	0.03	0.03	NUM
eajbcs-667	546	27	and	and	CCONJ
eajbcs-667	546	28	d	d	NOUN
eajbcs-667	546	29	=	=	SYM
eajbcs-667	546	30	10	10	NUM
eajbcs-667	546	31	for	for	ADP
eajbcs-667	546	32	this	this	DET
eajbcs-667	546	33	problem	problem	NOUN
eajbcs-667	546	34	is	be	AUX
eajbcs-667	546	35	given	give	VERB
eajbcs-667	546	36	in	in	ADP
eajbcs-667	546	37	fig	fig	NOUN
eajbcs-667	546	38	.	.	PUNCT
eajbcs-667	547	1	7	7	X
eajbcs-667	547	2	.	.	X
eajbcs-667	547	3	the	the	DET
eajbcs-667	547	4	peclet	peclet	NOUN
eajbcs-667	547	5	’s	’s	PART
eajbcs-667	547	6	number	number	NOUN
eajbcs-667	547	7	let	let	VERB
eajbcs-667	547	8	we	we	PRON
eajbcs-667	547	9	know	know	VERB
eajbcs-667	547	10	consider	consider	VERB
eajbcs-667	547	11	the	the	DET
eajbcs-667	547	12	ade	ade	X
eajbcs-667	547	13	ut	ut	PROPN
eajbcs-667	548	1	+	+	PROPN
eajbcs-667	549	1	aux	aux	PROPN
eajbcs-667	550	1	−	−	PROPN
eajbcs-667	551	1	duxx	duxx	PROPN
eajbcs-667	552	1	=	=	SYM
eajbcs-667	552	2	f	f	PROPN
eajbcs-667	552	3	with	with	ADP
eajbcs-667	552	4	f(x	f(x	PROPN
eajbcs-667	552	5	,	,	PUNCT
eajbcs-667	552	6	t	t	PROPN
eajbcs-667	552	7	)	)	PUNCT
eajbcs-667	553	1	=	=	PUNCT
eajbcs-667	553	2	since	since	SCONJ
eajbcs-667	553	3	we	we	PRON
eajbcs-667	553	4	implements	implement	VERB
eajbcs-667	553	5	zero	zero	NUM
eajbcs-667	553	6	flux	flux	NOUN
eajbcs-667	553	7	concentrations	concentration	NOUN
eajbcs-667	553	8	of	of	ADP
eajbcs-667	553	9	the	the	DET
eajbcs-667	553	10	pollutants	pollutant	NOUN
eajbcs-667	553	11	in	in	ADP
eajbcs-667	553	12	the	the	DET
eajbcs-667	553	13	boundaries	boundary	NOUN
eajbcs-667	553	14	,	,	PUNCT
eajbcs-667	553	15	the	the	DET
eajbcs-667	553	16	total	total	ADJ
eajbcs-667	553	17	mass	mass	NOUN
eajbcs-667	553	18	of	of	ADP
eajbcs-667	553	19	the	the	DET
eajbcs-667	553	20	pollution	pollution	NOUN
eajbcs-667	553	21	should	should	AUX
eajbcs-667	553	22	remain	remain	VERB
eajbcs-667	553	23	constant	constant	ADJ
eajbcs-667	553	24	.	.	PUNCT
eajbcs-667	554	1	these	these	DET
eajbcs-667	554	2	boundaries	boundary	NOUN
eajbcs-667	554	3	physically	physically	ADV
eajbcs-667	554	4	correspond	correspond	VERB
eajbcs-667	554	5	to	to	ADP
eajbcs-667	554	6	a	a	DET
eajbcs-667	554	7	system	system	NOUN
eajbcs-667	554	8	where	where	SCONJ
eajbcs-667	554	9	the	the	DET
eajbcs-667	554	10	species	specie	NOUN
eajbcs-667	554	11	is	be	AUX
eajbcs-667	554	12	enclosed	enclose	VERB
eajbcs-667	554	13	inside	inside	ADP
eajbcs-667	554	14	a	a	DET
eajbcs-667	554	15	mesh	mesh	NOUN
eajbcs-667	554	16	that	that	PRON
eajbcs-667	554	17	it	it	PRON
eajbcs-667	554	18	can	can	AUX
eajbcs-667	554	19	not	not	PART
eajbcs-667	554	20	penetrate	penetrate	VERB
eajbcs-667	554	21	,	,	PUNCT
eajbcs-667	554	22	however	however	ADV
eajbcs-667	554	23	,	,	PUNCT
eajbcs-667	554	24	the	the	DET
eajbcs-667	554	25	mesh	mesh	NOUN
eajbcs-667	554	26	allows	allow	VERB
eajbcs-667	554	27	the	the	DET
eajbcs-667	554	28	fluids	fluid	NOUN
eajbcs-667	554	29	to	to	PART
eajbcs-667	554	30	diffuse	diffuse	VERB
eajbcs-667	554	31	through	through	ADP
eajbcs-667	554	32	the	the	DET
eajbcs-667	554	33	flow	flow	NOUN
eajbcs-667	554	34	field	field	NOUN
eajbcs-667	554	35	.	.	PUNCT
eajbcs-667	555	1	we	we	PRON
eajbcs-667	555	2	see	see	VERB
eajbcs-667	555	3	that	that	SCONJ
eajbcs-667	555	4	all	all	DET
eajbcs-667	555	5	masses	masse	NOUN
eajbcs-667	555	6	eventually	eventually	ADV
eajbcs-667	555	7	concentrate	concentrate	VERB
eajbcs-667	555	8	along	along	ADP
eajbcs-667	555	9	the	the	DET
eajbcs-667	555	10	domain	domain	NOUN
eajbcs-667	555	11	as	as	SCONJ
eajbcs-667	555	12	we	we	PRON
eajbcs-667	555	13	have	have	AUX
eajbcs-667	555	14	seen	see	VERB
eajbcs-667	555	15	in	in	ADP
eajbcs-667	555	16	fig	fig	NOUN
eajbcs-667	555	17	.	.	PUNCT
eajbcs-667	556	1	7	7	X
eajbcs-667	556	2	.	.	X
eajbcs-667	556	3	this	this	DET
eajbcs-667	556	4	steady	steady	ADJ
eajbcs-667	556	5	state	state	NOUN
eajbcs-667	556	6	corresponds	correspond	VERB
eajbcs-667	556	7	to	to	ADP
eajbcs-667	556	8	diffusive	diffusive	ADJ
eajbcs-667	556	9	and	and	CCONJ
eajbcs-667	556	10	advective	advective	ADJ
eajbcs-667	556	11	fluxes	flux	NOUN
eajbcs-667	556	12	balancing	balance	VERB
eajbcs-667	556	13	each	each	DET
eajbcs-667	556	14	other	other	ADJ
eajbcs-667	556	15	.	.	PUNCT
eajbcs-667	557	1	the	the	DET
eajbcs-667	557	2	flow	flow	NOUN
eajbcs-667	557	3	carries	carry	VERB
eajbcs-667	557	4	additional	additional	ADJ
eajbcs-667	557	5	mass	mass	NOUN
eajbcs-667	557	6	towards	towards	ADP
eajbcs-667	557	7	the	the	DET
eajbcs-667	557	8	domain	domain	NOUN
eajbcs-667	557	9	,	,	PUNCT
eajbcs-667	557	10	but	but	CCONJ
eajbcs-667	557	11	the	the	DET
eajbcs-667	557	12	density	density	NOUN
eajbcs-667	557	13	gradient	gradient	NOUN
eajbcs-667	557	14	limits	limit	VERB
eajbcs-667	557	15	how	how	SCONJ
eajbcs-667	557	16	much	much	ADV
eajbcs-667	557	17	more	more	ADJ
eajbcs-667	557	18	mass	mass	NOUN
eajbcs-667	557	19	can	can	AUX
eajbcs-667	557	20	be	be	AUX
eajbcs-667	557	21	deposited	deposit	VERB
eajbcs-667	557	22	.	.	PUNCT
eajbcs-667	558	1	(	(	PUNCT
eajbcs-667	558	2	a	a	X
eajbcs-667	558	3	)	)	PUNCT
eajbcs-667	558	4	fem	fem	NOUN
eajbcs-667	558	5	sol	sol	NOUN
eajbcs-667	558	6	.	.	PUNCT
eajbcs-667	559	1	with	with	ADP
eajbcs-667	559	2	d	d	PROPN
eajbcs-667	559	3	=	=	SYM
eajbcs-667	559	4	1	1	NUM
eajbcs-667	559	5	and	and	CCONJ
eajbcs-667	559	6	tf	tf	NOUN
eajbcs-667	559	7	=	=	SYM
eajbcs-667	559	8	3	3	NUM
eajbcs-667	559	9	(	(	PUNCT
eajbcs-667	559	10	b	b	NOUN
eajbcs-667	559	11	)	)	PUNCT
eajbcs-667	559	12	fem	fem	NOUN
eajbcs-667	559	13	sol	sol	NOUN
eajbcs-667	559	14	.	.	PUNCT
eajbcs-667	560	1	with	with	ADP
eajbcs-667	560	2	d	d	PROPN
eajbcs-667	560	3	=	=	SYM
eajbcs-667	560	4	10	10	NUM
eajbcs-667	560	5	and	and	CCONJ
eajbcs-667	560	6	tf	tf	NOUN
eajbcs-667	560	7	=	=	SYM
eajbcs-667	560	8	3	3	NUM
eajbcs-667	560	9	figure	figure	NOUN
eajbcs-667	560	10	6	6	NUM
eajbcs-667	560	11	:	:	PUNCT
eajbcs-667	560	12	solution	solution	NOUN
eajbcs-667	560	13	of	of	ADP
eajbcs-667	560	14	the	the	DET
eajbcs-667	560	15	diffusion	diffusion	NOUN
eajbcs-667	560	16	equation	equation	NOUN
eajbcs-667	560	17	using	use	VERB
eajbcs-667	560	18	d	d	PROPN
eajbcs-667	560	19	=	=	SYM
eajbcs-667	560	20	1	1	NUM
eajbcs-667	560	21	and	and	CCONJ
eajbcs-667	560	22	d	d	NOUN
eajbcs-667	560	23	=	=	SYM
eajbcs-667	560	24	10	10	NUM
eajbcs-667	560	25	for	for	ADP
eajbcs-667	560	26	a	a	DET
eajbcs-667	560	27	constant	constant	ADJ
eajbcs-667	560	28	final	final	ADJ
eajbcs-667	560	29	time	time	NOUN
eajbcs-667	560	30	of	of	ADP
eajbcs-667	560	31	tf	tf	PROPN
eajbcs-667	560	32	=	=	SYM
eajbcs-667	560	33	0.5	0.5	NUM
eajbcs-667	560	34	.	.	PUNCT
eajbcs-667	561	1	(	(	PUNCT
eajbcs-667	561	2	a	a	X
eajbcs-667	561	3	)	)	PUNCT
eajbcs-667	561	4	fem	fem	NOUN
eajbcs-667	561	5	solution	solution	NOUN
eajbcs-667	561	6	(	(	PUNCT
eajbcs-667	561	7	b	b	NOUN
eajbcs-667	561	8	)	)	PUNCT
eajbcs-667	561	9	bar	bar	NOUN
eajbcs-667	561	10	plot	plot	NOUN
eajbcs-667	561	11	of	of	ADP
eajbcs-667	561	12	the	the	DET
eajbcs-667	561	13	solution	solution	NOUN
eajbcs-667	561	14	figure	figure	NOUN
eajbcs-667	561	15	7	7	NUM
eajbcs-667	561	16	:	:	PUNCT
eajbcs-667	561	17	implementation	implementation	NOUN
eajbcs-667	561	18	of	of	ADP
eajbcs-667	561	19	1d	1d	NUM
eajbcs-667	561	20	ad	ad	NOUN
eajbcs-667	561	21	equation	equation	NOUN
eajbcs-667	561	22	with	with	ADP
eajbcs-667	561	23	the	the	DET
eajbcs-667	561	24	source	source	NOUN
eajbcs-667	561	25	functionf(x	functionf(x	PROPN
eajbcs-667	561	26	,	,	PUNCT
eajbcs-667	561	27	t	t	PROPN
eajbcs-667	561	28	)	)	PUNCT
eajbcs-667	561	29	=	=	SYM
eajbcs-667	561	30	sin(πx	sin(πx	NOUN
eajbcs-667	561	31	)	)	PUNCT
eajbcs-667	561	32	(	(	PUNCT
eajbcs-667	561	33	1	1	NUM
eajbcs-667	561	34	+	+	NOUN
eajbcs-667	561	35	dπ2	dπ2	NOUN
eajbcs-667	561	36	t	t	PROPN
eajbcs-667	561	37	)	)	PUNCT
eajbcs-667	561	38	+	+	NUM
eajbcs-667	561	39	aπt	aπt	NOUN
eajbcs-667	561	40	cos(πx	cos(πx	NOUN
eajbcs-667	561	41	)	)	PUNCT
eajbcs-667	561	42	and	and	CCONJ
eajbcs-667	561	43	homogeneous	homogeneous	ADJ
eajbcs-667	561	44	dirichlet	dirichlet	PROPN
eajbcs-667	561	45	boundary	boundary	ADJ
eajbcs-667	561	46	conditions	condition	NOUN
eajbcs-667	561	47	and	and	CCONJ
eajbcs-667	561	48	a	a	DET
eajbcs-667	561	49	=	=	NOUN
eajbcs-667	561	50	0.03	0.03	NUM
eajbcs-667	561	51	,	,	PUNCT
eajbcs-667	561	52	d	d	NOUN
eajbcs-667	561	53	=	=	SYM
eajbcs-667	561	54	10	10	NUM
eajbcs-667	561	55	,	,	PUNCT
eajbcs-667	561	56	in	in	ADP
eajbcs-667	561	57	which	which	PRON
eajbcs-667	561	58	the	the	DET
eajbcs-667	561	59	peclet	peclet	NOUN
eajbcs-667	561	60	’s	’s	PART
eajbcs-667	561	61	number	number	NOUN
eajbcs-667	561	62	pe	pe	PROPN
eajbcs-667	561	63	=	=	PUNCT
eajbcs-667	561	64	al	al	PROPN
eajbcs-667	561	65	d	d	PROPN
eajbcs-667	561	66	=	=	NOUN
eajbcs-667	561	67	0.003	0.003	NUM
eajbcs-667	561	68	≪	≪	X
eajbcs-667	561	69	1	1	NUM
eajbcs-667	561	70	which	which	PRON
eajbcs-667	561	71	is	be	AUX
eajbcs-667	561	72	diffusion	diffusion	NOUN
eajbcs-667	561	73	dominated	dominate	VERB
eajbcs-667	561	74	.	.	PUNCT
eajbcs-667	562	1	ary	ary	PROPN
eajbcs-667	562	2	condition	condition	NOUN
eajbcs-667	562	3	dux(0	dux(0	PROPN
eajbcs-667	562	4	,	,	PUNCT
eajbcs-667	562	5	t	t	PROPN
eajbcs-667	562	6	)	)	PUNCT
eajbcs-667	562	7	=	=	SYM
eajbcs-667	562	8	k0(u(0	k0(u(0	PROPN
eajbcs-667	562	9	,	,	PUNCT
eajbcs-667	562	10	t	t	PROPN
eajbcs-667	562	11	)	)	PUNCT
eajbcs-667	562	12	+	+	CCONJ
eajbcs-667	562	13	g0	g0	ADJ
eajbcs-667	562	14	and	and	CCONJ
eajbcs-667	562	15	−dux(1	−dux(1	NOUN
eajbcs-667	562	16	,	,	PUNCT
eajbcs-667	562	17	t	t	PROPN
eajbcs-667	562	18	)	)	PUNCT
eajbcs-667	562	19	=	=	SYM
eajbcs-667	562	20	k1(u(1	k1(u(1	PROPN
eajbcs-667	562	21	,	,	PUNCT
eajbcs-667	562	22	t	t	PROPN
eajbcs-667	562	23	)	)	PUNCT
eajbcs-667	562	24	+	+	CCONJ
eajbcs-667	562	25	g1	g1	NOUN
eajbcs-667	562	26	and	and	CCONJ
eajbcs-667	562	27	by	by	ADP
eajbcs-667	562	28	considering	consider	VERB
eajbcs-667	562	29	different	different	ADJ
eajbcs-667	562	30	robin	robin	PROPN
eajbcs-667	562	31	boundary	boundary	PROPN
eajbcs-667	562	32	paeast	paeast	PROPN
eajbcs-667	562	33	afr	afr	PROPN
eajbcs-667	562	34	.	.	PUNCT
eajbcs-667	563	1	j.	j.	PROPN
eajbcs-667	563	2	biophys	biophys	PROPN
eajbcs-667	563	3	.	.	PUNCT
eajbcs-667	564	1	comput	comput	NOUN
eajbcs-667	564	2	.	.	PUNCT
eajbcs-667	565	1	sci	sci	PROPN
eajbcs-667	565	2	.	.	PUNCT
eajbcs-667	565	3	(	(	PUNCT
eajbcs-667	565	4	2023	2023	NUM
eajbcs-667	565	5	)	)	PUNCT
eajbcs-667	565	6	,	,	PUNCT
eajbcs-667	565	7	vol	vol	NOUN
eajbcs-667	565	8	.	.	PROPN
eajbcs-667	565	9	4	4	NUM
eajbcs-667	565	10	,	,	PUNCT
eajbcs-667	565	11	no	no	INTJ
eajbcs-667	565	12	.	.	NOUN
eajbcs-667	565	13	1	1	NUM
eajbcs-667	565	14	,	,	PUNCT
eajbcs-667	565	15	52	52	NUM
eajbcs-667	565	16	-	-	SYM
eajbcs-667	565	17	74	74	NUM
eajbcs-667	565	18	70	70	NUM
eajbcs-667	565	19	we	we	PRON
eajbcs-667	565	20	see	see	VERB
eajbcs-667	565	21	know	know	VERB
eajbcs-667	565	22	on	on	ADP
eajbcs-667	565	23	an	an	DET
eajbcs-667	565	24	arbitrary	arbitrary	ADJ
eajbcs-667	565	25	input	input	NOUN
eajbcs-667	565	26	concentration	concentration	NOUN
eajbcs-667	565	27	by	by	ADP
eajbcs-667	565	28	using	use	VERB
eajbcs-667	565	29	neumann	neumann	PROPN
eajbcs-667	565	30	and	and	CCONJ
eajbcs-667	565	31	robin	robin	PROPN
eajbcs-667	565	32	boundary	boundary	PROPN
eajbcs-667	565	33	conditions	condition	NOUN
eajbcs-667	565	34	.	.	PUNCT
eajbcs-667	566	1	here	here	ADV
eajbcs-667	566	2	,	,	PUNCT
eajbcs-667	566	3	the	the	DET
eajbcs-667	566	4	problem	problem	NOUN
eajbcs-667	566	5	is	be	AUX
eajbcs-667	566	6	undetermined	undetermined	ADJ
eajbcs-667	566	7	because	because	SCONJ
eajbcs-667	566	8	of	of	ADP
eajbcs-667	566	9	an	an	DET
eajbcs-667	566	10	unknown	unknown	ADJ
eajbcs-667	566	11	input	input	NOUN
eajbcs-667	566	12	concentration	concentration	NOUN
eajbcs-667	566	13	and	and	CCONJ
eajbcs-667	566	14	hence	hence	ADV
eajbcs-667	566	15	unknown	unknown	ADJ
eajbcs-667	566	16	exit	exit	NOUN
eajbcs-667	566	17	concentration	concentration	NOUN
eajbcs-667	566	18	and	and	CCONJ
eajbcs-667	566	19	depends	depend	VERB
eajbcs-667	566	20	on	on	ADP
eajbcs-667	566	21	the	the	DET
eajbcs-667	566	22	parameters	parameter	NOUN
eajbcs-667	566	23	.	.	PUNCT
eajbcs-667	567	1	authors	author	NOUN
eajbcs-667	567	2	of	of	ADP
eajbcs-667	567	3	previous	previous	ADJ
eajbcs-667	567	4	works	work	NOUN
eajbcs-667	567	5	on	on	ADP
eajbcs-667	567	6	problems	problem	NOUN
eajbcs-667	567	7	of	of	ADP
eajbcs-667	567	8	this	this	DET
eajbcs-667	567	9	type	type	NOUN
eajbcs-667	567	10	have	have	AUX
eajbcs-667	567	11	done	do	VERB
eajbcs-667	567	12	on	on	ADP
eajbcs-667	567	13	a	a	DET
eajbcs-667	567	14	known	know	VERB
eajbcs-667	567	15	exit	exit	NOUN
eajbcs-667	567	16	concentration	concentration	NOUN
eajbcs-667	567	17	by	by	ADP
eajbcs-667	567	18	assuming	assume	VERB
eajbcs-667	567	19	a	a	DET
eajbcs-667	567	20	continuous	continuous	ADJ
eajbcs-667	567	21	and	and	CCONJ
eajbcs-667	567	22	constant	constant	ADJ
eajbcs-667	567	23	concentration	concentration	NOUN
eajbcs-667	567	24	at	at	ADP
eajbcs-667	567	25	the	the	DET
eajbcs-667	567	26	flow	flow	NOUN
eajbcs-667	567	27	boundary	boundary	NOUN
eajbcs-667	567	28	of	of	ADP
eajbcs-667	567	29	dirichlet	dirichlet	PROPN
eajbcs-667	567	30	type	type	NOUN
eajbcs-667	567	31	.	.	PUNCT
eajbcs-667	568	1	this	this	DET
eajbcs-667	568	2	yields	yield	VERB
eajbcs-667	568	3	we	we	PRON
eajbcs-667	568	4	to	to	PART
eajbcs-667	568	5	consider	consider	VERB
eajbcs-667	568	6	a	a	DET
eajbcs-667	568	7	problem	problem	NOUN
eajbcs-667	568	8	by	by	ADP
eajbcs-667	568	9	forcing	force	VERB
eajbcs-667	568	10	the	the	DET
eajbcs-667	568	11	flow	flow	NOUN
eajbcs-667	568	12	boundaries	boundary	NOUN
eajbcs-667	568	13	using	use	VERB
eajbcs-667	568	14	neumann	neumann	PROPN
eajbcs-667	568	15	and	and	CCONJ
eajbcs-667	568	16	robin	robin	PROPN
eajbcs-667	568	17	boundary	boundary	PROPN
eajbcs-667	568	18	conditions	condition	NOUN
eajbcs-667	568	19	.	.	PUNCT
eajbcs-667	569	1	this	this	DET
eajbcs-667	569	2	conditions	condition	NOUN
eajbcs-667	569	3	yields	yield	VERB
eajbcs-667	569	4	the	the	DET
eajbcs-667	569	5	flow	flow	NOUN
eajbcs-667	569	6	of	of	ADP
eajbcs-667	569	7	the	the	DET
eajbcs-667	569	8	concentration	concentration	NOUN
eajbcs-667	569	9	to	to	PART
eajbcs-667	569	10	move	move	VERB
eajbcs-667	569	11	freely	freely	ADV
eajbcs-667	569	12	.	.	PUNCT
eajbcs-667	570	1	here	here	ADV
eajbcs-667	570	2	the	the	DET
eajbcs-667	570	3	velocity	velocity	NOUN
eajbcs-667	570	4	term	term	NOUN
eajbcs-667	570	5	and	and	CCONJ
eajbcs-667	570	6	the	the	DET
eajbcs-667	570	7	diffusion	diffusion	NOUN
eajbcs-667	570	8	coefficient	coefficient	NOUN
eajbcs-667	570	9	highly	highly	ADV
eajbcs-667	570	10	affects	affect	VERB
eajbcs-667	570	11	the	the	DET
eajbcs-667	570	12	flow	flow	NOUN
eajbcs-667	570	13	of	of	ADP
eajbcs-667	570	14	the	the	DET
eajbcs-667	570	15	concentration	concentration	NOUN
eajbcs-667	570	16	.	.	PUNCT
eajbcs-667	571	1	we	we	PRON
eajbcs-667	571	2	saw	see	VERB
eajbcs-667	571	3	it	it	PRON
eajbcs-667	571	4	by	by	ADP
eajbcs-667	571	5	giving	give	VERB
eajbcs-667	571	6	a	a	DET
eajbcs-667	571	7	source	source	NOUN
eajbcs-667	571	8	function	function	NOUN
eajbcs-667	571	9	f(x	f(x	PROPN
eajbcs-667	571	10	,	,	PUNCT
eajbcs-667	571	11	t	t	PROPN
eajbcs-667	571	12	)	)	PUNCT
eajbcs-667	571	13	=	=	VERB
eajbcs-667	572	1	a(2x	a(2x	VERB
eajbcs-667	572	2	+	+	NOUN
eajbcs-667	572	3	1	1	X
eajbcs-667	572	4	)	)	PUNCT
eajbcs-667	572	5	−	−	PROPN
eajbcs-667	572	6	2d	2d	NOUN
eajbcs-667	572	7	−	−	NOUN
eajbcs-667	572	8	1	1	NUM
eajbcs-667	572	9	,	,	PUNCT
eajbcs-667	572	10	0	0	NUM
eajbcs-667	572	11	≤	≤	NUM
eajbcs-667	572	12	x	x	SYM
eajbcs-667	572	13	≤	≤	NUM
eajbcs-667	572	14	1	1	NUM
eajbcs-667	572	15	,	,	PUNCT
eajbcs-667	572	16	0	0	NUM
eajbcs-667	572	17	≤	≤	NUM
eajbcs-667	572	18	t	t	NOUN
eajbcs-667	572	19	≤	≤	NOUN
eajbcs-667	572	20	tf	tf	NUM
eajbcs-667	572	21	with	with	ADP
eajbcs-667	572	22	a	a	DET
eajbcs-667	572	23	robin	robin	PROPN
eajbcs-667	572	24	boundrameters	boundrameter	NOUN
eajbcs-667	572	25	.	.	PUNCT
eajbcs-667	573	1	2	2	NUM
eajbcs-667	573	2	(	(	PUNCT
eajbcs-667	573	3	a	a	X
eajbcs-667	573	4	)	)	PUNCT
eajbcs-667	573	5	fem	fem	NOUN
eajbcs-667	573	6	solution	solution	NOUN
eajbcs-667	573	7	with	with	ADP
eajbcs-667	573	8	tf	tf	PROPN
eajbcs-667	573	9	=	=	SYM
eajbcs-667	573	10	1	1	NUM
eajbcs-667	573	11	(	(	PUNCT
eajbcs-667	573	12	b	b	NOUN
eajbcs-667	573	13	)	)	PUNCT
eajbcs-667	573	14	fem	fem	NOUN
eajbcs-667	573	15	solution	solution	NOUN
eajbcs-667	573	16	with	with	ADP
eajbcs-667	573	17	tf	tf	PROPN
eajbcs-667	573	18	=	=	SYM
eajbcs-667	573	19	3	3	NUM
eajbcs-667	573	20	figure	figure	NOUN
eajbcs-667	573	21	8	8	NUM
eajbcs-667	573	22	:	:	PUNCT
eajbcs-667	573	23	numerical	numerical	ADJ
eajbcs-667	573	24	solution	solution	NOUN
eajbcs-667	573	25	using	use	VERB
eajbcs-667	573	26	fem	fem	NOUN
eajbcs-667	573	27	for	for	ADP
eajbcs-667	573	28	the	the	DET
eajbcs-667	573	29	advection	advection	NOUN
eajbcs-667	573	30	diffusion	diffusion	NOUN
eajbcs-667	573	31	equation	equation	NOUN
eajbcs-667	573	32	with	with	ADP
eajbcs-667	573	33	f(x	f(x	PROPN
eajbcs-667	573	34	,	,	PUNCT
eajbcs-667	573	35	t	t	PROPN
eajbcs-667	573	36	)	)	PUNCT
eajbcs-667	573	37	=	=	VERB
eajbcs-667	573	38	a(2x	a(2x	VERB
eajbcs-667	573	39	+	+	NOUN
eajbcs-667	573	40	1	1	X
eajbcs-667	573	41	)	)	PUNCT
eajbcs-667	573	42	−	−	PROPN
eajbcs-667	573	43	2d	2d	NOUN
eajbcs-667	573	44	−	−	NOUN
eajbcs-667	573	45	1	1	NUM
eajbcs-667	573	46	and	and	CCONJ
eajbcs-667	573	47	robin	robin	PROPN
eajbcs-667	573	48	boundary	boundary	PROPN
eajbcs-667	573	49	conditions	condition	NOUN
eajbcs-667	573	50	,	,	PUNCT
eajbcs-667	573	51	with	with	ADP
eajbcs-667	573	52	a	a	DET
eajbcs-667	573	53	diffusion	diffusion	NOUN
eajbcs-667	573	54	coefficient	coefficient	NOUN
eajbcs-667	574	1	d	d	NOUN
eajbcs-667	574	2	=	=	NOUN
eajbcs-667	574	3	0.02	0.02	NUM
eajbcs-667	574	4	and	and	CCONJ
eajbcs-667	574	5	velocity	velocity	NOUN
eajbcs-667	574	6	term	term	NOUN
eajbcs-667	574	7	a	a	DET
eajbcs-667	574	8	=	=	NOUN
eajbcs-667	574	9	1	1	NUM
eajbcs-667	574	10	with	with	ADP
eajbcs-667	574	11	time	time	NOUN
eajbcs-667	574	12	tf	tf	X
eajbcs-667	574	13	=	=	SYM
eajbcs-667	574	14	1	1	NUM
eajbcs-667	574	15	for	for	ADP
eajbcs-667	574	16	(	(	PUNCT
eajbcs-667	574	17	a	a	NOUN
eajbcs-667	574	18	)	)	PUNCT
eajbcs-667	574	19	and	and	CCONJ
eajbcs-667	574	20	tf	tf	X
eajbcs-667	574	21	=	=	SYM
eajbcs-667	574	22	3	3	NUM
eajbcs-667	574	23	for	for	ADP
eajbcs-667	574	24	(	(	PUNCT
eajbcs-667	574	25	b	b	NOUN
eajbcs-667	574	26	)	)	PUNCT
eajbcs-667	574	27	.	.	PUNCT
eajbcs-667	575	1	u(t	u(t	PROPN
eajbcs-667	575	2	,	,	PUNCT
eajbcs-667	575	3	x	x	NOUN
eajbcs-667	575	4	,	,	PUNCT
eajbcs-667	575	5	y	y	NOUN
eajbcs-667	575	6	)	)	PUNCT
eajbcs-667	575	7	=	=	SYM
eajbcs-667	575	8	e−t	e−t	NOUN
eajbcs-667	575	9	sin(πx	sin(πx	NOUN
eajbcs-667	575	10	)	)	PUNCT
eajbcs-667	575	11	sin(πy	sin(πy	NOUN
eajbcs-667	575	12	)	)	PUNCT
eajbcs-667	575	13	be	be	VERB
eajbcs-667	575	14	the	the	DET
eajbcs-667	575	15	given	give	VERB
eajbcs-667	575	16	exact	exact	ADJ
eajbcs-667	575	17	solution	solution	NOUN
eajbcs-667	575	18	for	for	ADP
eajbcs-667	575	19	the	the	DET
eajbcs-667	575	20	diffusion	diffusion	NOUN
eajbcs-667	575	21	equation	equation	NOUN
eajbcs-667	576	1	ut	ut	PROPN
eajbcs-667	576	2	−	−	PROPN
eajbcs-667	576	3	d∆u	d∆u	PROPN
eajbcs-667	576	4	=	=	SYM
eajbcs-667	576	5	f	f	PROPN
eajbcs-667	576	6	,	,	PUNCT
eajbcs-667	576	7	then	then	ADV
eajbcs-667	576	8	our	our	PRON
eajbcs-667	576	9	source	source	NOUN
eajbcs-667	576	10	function	function	NOUN
eajbcs-667	576	11	f	f	PROPN
eajbcs-667	576	12	becomes	become	VERB
eajbcs-667	576	13	f(t	f(t	PROPN
eajbcs-667	576	14	,	,	PUNCT
eajbcs-667	576	15	x	x	NOUN
eajbcs-667	576	16	,	,	PUNCT
eajbcs-667	576	17	y	y	NOUN
eajbcs-667	576	18	)	)	PUNCT
eajbcs-667	576	19	=	=	PUNCT
eajbcs-667	577	1	(	(	PUNCT
eajbcs-667	577	2	2dπ2	2dπ2	NUM
eajbcs-667	577	3	−	−	PROPN
eajbcs-667	577	4	1)e−t	1)e−t	PROPN
eajbcs-667	577	5	sin(πx	sin(πx	NOUN
eajbcs-667	577	6	)	)	PUNCT
eajbcs-667	577	7	sin(πy	sin(πy	NOUN
eajbcs-667	577	8	)	)	PUNCT
eajbcs-667	577	9	.	.	PUNCT
eajbcs-667	578	1	now	now	ADV
eajbcs-667	578	2	by	by	ADP
eajbcs-667	578	3	using	use	VERB
eajbcs-667	578	4	the	the	DET
eajbcs-667	578	5	east	east	PROPN
eajbcs-667	578	6	afr	afr	PROPN
eajbcs-667	578	7	.	.	PUNCT
eajbcs-667	579	1	j.	j.	PROPN
eajbcs-667	579	2	biophys	biophys	PROPN
eajbcs-667	579	3	.	.	PUNCT
eajbcs-667	580	1	comput	comput	NOUN
eajbcs-667	580	2	.	.	PUNCT
eajbcs-667	581	1	sci	sci	PROPN
eajbcs-667	581	2	.	.	PUNCT
eajbcs-667	581	3	(	(	PUNCT
eajbcs-667	581	4	2023	2023	NUM
eajbcs-667	581	5	)	)	PUNCT
eajbcs-667	581	6	,	,	PUNCT
eajbcs-667	581	7	vol	vol	NOUN
eajbcs-667	581	8	.	.	PROPN
eajbcs-667	581	9	4	4	NUM
eajbcs-667	581	10	,	,	PUNCT
eajbcs-667	581	11	no	no	INTJ
eajbcs-667	581	12	.	.	NOUN
eajbcs-667	581	13	1	1	NUM
eajbcs-667	581	14	,	,	PUNCT
eajbcs-667	581	15	52	52	NUM
eajbcs-667	581	16	-	-	SYM
eajbcs-667	581	17	74	74	NUM
eajbcs-667	581	18	71	71	NUM
eajbcs-667	581	19	let	let	VERB
eajbcs-667	581	20	we	we	PRON
eajbcs-667	581	21	change	change	VERB
eajbcs-667	581	22	the	the	DET
eajbcs-667	581	23	velocity	velocity	NOUN
eajbcs-667	581	24	term	term	NOUN
eajbcs-667	581	25	and	and	CCONJ
eajbcs-667	581	26	the	the	DET
eajbcs-667	581	27	diffusion	diffusion	NOUN
eajbcs-667	581	28	coefficient	coefficient	NOUN
eajbcs-667	581	29	for	for	ADP
eajbcs-667	581	30	the	the	DET
eajbcs-667	581	31	given	give	VERB
eajbcs-667	581	32	source	source	NOUN
eajbcs-667	581	33	term	term	NOUN
eajbcs-667	581	34	using	use	VERB
eajbcs-667	581	35	d	d	PROPN
eajbcs-667	581	36	=	=	PROPN
eajbcs-667	581	37	0.02	0.02	NUM
eajbcs-667	581	38	and	and	CCONJ
eajbcs-667	581	39	a	a	DET
eajbcs-667	581	40	=	=	ADJ
eajbcs-667	581	41	1	1	NUM
eajbcs-667	581	42	,	,	PUNCT
eajbcs-667	581	43	(	(	PUNCT
eajbcs-667	581	44	g0	g0	ADJ
eajbcs-667	581	45	,	,	PUNCT
eajbcs-667	581	46	g1	g1	PROPN
eajbcs-667	581	47	)	)	PUNCT
eajbcs-667	582	1	=	=	SYM
eajbcs-667	582	2	(	(	PUNCT
eajbcs-667	582	3	0.02	0.02	NUM
eajbcs-667	582	4	,	,	PUNCT
eajbcs-667	582	5	0.06	0.06	NUM
eajbcs-667	582	6	)	)	PUNCT
eajbcs-667	582	7	.	.	PUNCT
eajbcs-667	583	1	the	the	DET
eajbcs-667	583	2	solution	solution	NOUN
eajbcs-667	583	3	for	for	ADP
eajbcs-667	583	4	sion	sion	NOUN
eajbcs-667	583	5	equation	equation	NOUN
eajbcs-667	583	6	with	with	ADP
eajbcs-667	583	7	a	a	DET
eajbcs-667	583	8	source	source	NOUN
eajbcs-667	583	9	function	function	NOUN
eajbcs-667	583	10	given	give	VERB
eajbcs-667	583	11	as	as	ADP
eajbcs-667	583	12	f(x	f(x	PROPN
eajbcs-667	583	13	,	,	PUNCT
eajbcs-667	583	14	y	y	NOUN
eajbcs-667	583	15	)	)	PUNCT
eajbcs-667	583	16	=	=	SYM
eajbcs-667	583	17	dπ2(sin(πx)+sin(πy	dπ2(sin(πx)+sin(πy	ADJ
eajbcs-667	583	18	)	)	PUNCT
eajbcs-667	583	19	)	)	PUNCT
eajbcs-667	583	20	to	to	PART
eajbcs-667	583	21	imlet	imlet	VERB
eajbcs-667	583	22	we	we	PRON
eajbcs-667	583	23	consider	consider	VERB
eajbcs-667	583	24	the	the	DET
eajbcs-667	583	25	2d	2d	NUM
eajbcs-667	583	26	steady	steady	ADJ
eajbcs-667	583	27	diffuin	diffuin	NOUN
eajbcs-667	583	28	order	order	NOUN
eajbcs-667	583	29	to	to	PART
eajbcs-667	583	30	consider	consider	VERB
eajbcs-667	583	31	the	the	DET
eajbcs-667	583	32	finite	finite	ADJ
eajbcs-667	583	33	element	element	NOUN
eajbcs-667	583	34	method	method	NOUN
eajbcs-667	583	35	of	of	ADP
eajbcs-667	583	36	the	the	DET
eajbcs-667	583	37	time	time	NOUN
eajbcs-667	583	38	dependent	dependent	ADJ
eajbcs-667	583	39	two	two	NUM
eajbcs-667	583	40	dimensional	dimensional	ADJ
eajbcs-667	583	41	equations	equation	NOUN
eajbcs-667	583	42	,	,	PUNCT
eajbcs-667	583	43	we	we	PRON
eajbcs-667	583	44	use	use	VERB
eajbcs-667	583	45	the	the	DET
eajbcs-667	583	46	unsteady	unsteady	ADJ
eajbcs-667	583	47	2d	2d	NUM
eajbcs-667	583	48	diffusion	diffusion	NOUN
eajbcs-667	583	49	equation	equation	NOUN
eajbcs-667	583	50	.	.	PUNCT
eajbcs-667	584	1	let	let	VERB
eajbcs-667	584	2	we	we	PRON
eajbcs-667	584	3	test	test	VERB
eajbcs-667	584	4	it	it	PRON
eajbcs-667	584	5	with	with	ADP
eajbcs-667	584	6	a	a	DET
eajbcs-667	584	7	known	know	VERB
eajbcs-667	584	8	exact	exact	ADJ
eajbcs-667	584	9	function	function	NOUN
eajbcs-667	584	10	and	and	CCONJ
eajbcs-667	584	11	use	use	VERB
eajbcs-667	584	12	the	the	DET
eajbcs-667	584	13	source	source	NOUN
eajbcs-667	584	14	terms	term	NOUN
eajbcs-667	584	15	and	and	CCONJ
eajbcs-667	584	16	boundary	boundary	ADJ
eajbcs-667	584	17	conditions	condition	NOUN
eajbcs-667	584	18	from	from	ADP
eajbcs-667	584	19	that	that	DET
eajbcs-667	584	20	function	function	NOUN
eajbcs-667	584	21	.	.	PUNCT
eajbcs-667	585	1	let	let	VERB
eajbcs-667	585	2	plement	plement	VERB
eajbcs-667	585	3	the	the	DET
eajbcs-667	585	4	2d	2d	PROPN
eajbcs-667	585	5	problem	problem	NOUN
eajbcs-667	585	6	.	.	PUNCT
eajbcs-667	586	1	the	the	DET
eajbcs-667	586	2	fem	fem	PROPN
eajbcs-667	586	3	numerical	numerical	PROPN
eajbcs-667	586	4	simulation	simulation	PROPN
eajbcs-667	586	5	of	of	ADP
eajbcs-667	586	6	the	the	DET
eajbcs-667	586	7	pde	pde	NOUN
eajbcs-667	586	8	is	be	AUX
eajbcs-667	586	9	given	give	VERB
eajbcs-667	586	10	in	in	ADP
eajbcs-667	586	11	fig	fig	NOUN
eajbcs-667	586	12	.	.	PUNCT
eajbcs-667	587	1	9	9	NUM
eajbcs-667	587	2	.	.	NUM
eajbcs-667	587	3	ode15i	ode15i	PROPN
eajbcs-667	587	4	,	,	PUNCT
eajbcs-667	587	5	ode	ode	PROPN
eajbcs-667	587	6	solver	solver	ADV
eajbcs-667	587	7	to	to	PART
eajbcs-667	587	8	integrate	integrate	VERB
eajbcs-667	587	9	for	for	ADP
eajbcs-667	587	10	time	time	NOUN
eajbcs-667	587	11	in	in	ADP
eajbcs-667	587	12	the	the	DET
eajbcs-667	587	13	final	final	ADJ
eajbcs-667	587	14	finite	finite	NOUN
eajbcs-667	587	15	element	element	NOUN
eajbcs-667	587	16	method	method	NOUN
eajbcs-667	587	17	descritization	descritization	NOUN
eajbcs-667	587	18	the	the	DET
eajbcs-667	587	19	solutions	solution	NOUN
eajbcs-667	587	20	for	for	ADP
eajbcs-667	587	21	the	the	DET
eajbcs-667	587	22	equation	equation	NOUN
eajbcs-667	587	23	is	be	AUX
eajbcs-667	587	24	given	give	VERB
eajbcs-667	587	25	in	in	ADP
eajbcs-667	587	26	fig	fig	NOUN
eajbcs-667	587	27	.	.	PUNCT
eajbcs-667	588	1	10	10	NUM
eajbcs-667	588	2	.	.	PUNCT
eajbcs-667	589	1	the	the	DET
eajbcs-667	589	2	surface	surface	NOUN
eajbcs-667	589	3	plots	plot	NOUN
eajbcs-667	589	4	of	of	ADP
eajbcs-667	589	5	those	those	DET
eajbcs-667	589	6	figures	figure	NOUN
eajbcs-667	589	7	are	be	AUX
eajbcs-667	589	8	the	the	DET
eajbcs-667	589	9	exact	exact	ADJ
eajbcs-667	589	10	solutions	solution	NOUN
eajbcs-667	589	11	,	,	PUNCT
eajbcs-667	589	12	the	the	DET
eajbcs-667	589	13	finite	finite	PROPN
eajbcs-667	589	14	element	element	NOUN
eajbcs-667	589	15	method	method	NOUN
eajbcs-667	589	16	solutions	solution	NOUN
eajbcs-667	589	17	and	and	CCONJ
eajbcs-667	589	18	the	the	DET
eajbcs-667	589	19	color	color	NOUN
eajbcs-667	589	20	bar	bar	NOUN
eajbcs-667	589	21	of	of	ADP
eajbcs-667	589	22	the	the	DET
eajbcs-667	589	23	finite	finite	ADJ
eajbcs-667	589	24	element	element	NOUN
eajbcs-667	589	25	solution	solution	NOUN
eajbcs-667	589	26	to	to	PART
eajbcs-667	589	27	view	view	VERB
eajbcs-667	589	28	its	its	PRON
eajbcs-667	589	29	properties	property	NOUN
eajbcs-667	589	30	.	.	PUNCT
eajbcs-667	590	1	this	this	DET
eajbcs-667	590	2	conditions	condition	NOUN
eajbcs-667	590	3	is	be	AUX
eajbcs-667	590	4	given	give	VERB
eajbcs-667	590	5	in	in	ADP
eajbcs-667	590	6	fig	fig	NOUN
eajbcs-667	590	7	.	.	PUNCT
eajbcs-667	591	1	8(a	8(a	NUM
eajbcs-667	591	2	)	)	PUNCT
eajbcs-667	591	3	using	use	VERB
eajbcs-667	591	4	a	a	DET
eajbcs-667	591	5	total	total	ADJ
eajbcs-667	591	6	time	time	NOUN
eajbcs-667	591	7	of	of	ADP
eajbcs-667	591	8	tf	tf	PROPN
eajbcs-667	591	9	=	=	SYM
eajbcs-667	591	10	1	1	NUM
eajbcs-667	591	11	and	and	CCONJ
eajbcs-667	591	12	8(b	8(b	NUM
eajbcs-667	591	13	)	)	PUNCT
eajbcs-667	591	14	using	use	VERB
eajbcs-667	591	15	a	a	DET
eajbcs-667	591	16	total	total	ADJ
eajbcs-667	591	17	time	time	NOUN
eajbcs-667	591	18	of	of	ADP
eajbcs-667	591	19	tf	tf	PROPN
eajbcs-667	591	20	=	=	SYM
eajbcs-667	591	21	3	3	X
eajbcs-667	591	22	.	.	PUNCT
eajbcs-667	592	1	this	this	PRON
eajbcs-667	592	2	is	be	AUX
eajbcs-667	592	3	an	an	DET
eajbcs-667	592	4	advection	advection	NOUN
eajbcs-667	592	5	dominated	dominate	VERB
eajbcs-667	592	6	example	example	NOUN
eajbcs-667	592	7	in	in	ADP
eajbcs-667	592	8	which	which	PRON
eajbcs-667	592	9	the	the	DET
eajbcs-667	592	10	movement	movement	NOUN
eajbcs-667	592	11	of	of	ADP
eajbcs-667	592	12	the	the	DET
eajbcs-667	592	13	concentration	concentration	NOUN
eajbcs-667	592	14	is	be	AUX
eajbcs-667	592	15	faster	fast	ADJ
eajbcs-667	592	16	in	in	ADP
eajbcs-667	592	17	advection	advection	NOUN
eajbcs-667	592	18	terms	term	NOUN
eajbcs-667	592	19	relative	relative	ADJ
eajbcs-667	592	20	to	to	ADP
eajbcs-667	592	21	diffusion	diffusion	NOUN
eajbcs-667	592	22	.	.	PUNCT
eajbcs-667	593	1	(	(	PUNCT
eajbcs-667	593	2	a	a	X
eajbcs-667	593	3	)	)	PUNCT
eajbcs-667	593	4	fem	fem	NOUN
eajbcs-667	593	5	solution	solution	NOUN
eajbcs-667	593	6	(	(	PUNCT
eajbcs-667	593	7	b	b	NOUN
eajbcs-667	593	8	)	)	PUNCT
eajbcs-667	593	9	bar	bar	NOUN
eajbcs-667	593	10	plot	plot	NOUN
eajbcs-667	593	11	of	of	ADP
eajbcs-667	593	12	the	the	DET
eajbcs-667	593	13	solution	solution	NOUN
eajbcs-667	593	14	figure	figure	NOUN
eajbcs-667	593	15	9	9	NUM
eajbcs-667	593	16	:	:	PUNCT
eajbcs-667	593	17	the	the	DET
eajbcs-667	593	18	numerical	numerical	PROPN
eajbcs-667	593	19	simulation	simulation	PROPN
eajbcs-667	593	20	using	use	VERB
eajbcs-667	593	21	fem	fem	NOUN
eajbcs-667	593	22	for	for	ADP
eajbcs-667	593	23	the	the	DET
eajbcs-667	593	24	steady	steady	ADJ
eajbcs-667	593	25	2d	2d	NOUN
eajbcs-667	593	26	diffusion	diffusion	NOUN
eajbcs-667	593	27	equation	equation	NOUN
eajbcs-667	593	28	with	with	ADP
eajbcs-667	593	29	f(x	f(x	PROPN
eajbcs-667	593	30	,	,	PUNCT
eajbcs-667	593	31	y	y	NOUN
eajbcs-667	593	32	)	)	PUNCT
eajbcs-667	593	33	=	=	SYM
eajbcs-667	593	34	dπ2(sin(πx	dπ2(sin(πx	X
eajbcs-667	593	35	)	)	PUNCT
eajbcs-667	593	36	+	+	NUM
eajbcs-667	593	37	sin(πy	sin(πy	NOUN
eajbcs-667	593	38	)	)	PUNCT
eajbcs-667	593	39	)	)	PUNCT
eajbcs-667	593	40	,	,	PUNCT
eajbcs-667	593	41	with	with	ADP
eajbcs-667	593	42	a	a	DET
eajbcs-667	593	43	diffusion	diffusion	NOUN
eajbcs-667	593	44	coefficient	coefficient	NOUN
eajbcs-667	593	45	d	d	NOUN
eajbcs-667	593	46	=	=	SYM
eajbcs-667	593	47	0.5	0.5	NUM
eajbcs-667	593	48	.	.	PUNCT
eajbcs-667	594	1	(	(	PUNCT
eajbcs-667	594	2	a	a	X
eajbcs-667	594	3	)	)	PUNCT
eajbcs-667	594	4	fem	fem	NOUN
eajbcs-667	594	5	&	&	CCONJ
eajbcs-667	594	6	exact	exact	PROPN
eajbcs-667	594	7	sol	sol	NOUN
eajbcs-667	594	8	(	(	PUNCT
eajbcs-667	594	9	b	b	NOUN
eajbcs-667	594	10	)	)	PUNCT
eajbcs-667	594	11	colour	colour	NOUN
eajbcs-667	594	12	bar	bar	NOUN
eajbcs-667	594	13	of	of	ADP
eajbcs-667	594	14	the	the	DET
eajbcs-667	594	15	fem	fem	NOUN
eajbcs-667	594	16	solution	solution	NOUN
eajbcs-667	594	17	figure	figure	NOUN
eajbcs-667	594	18	10	10	NUM
eajbcs-667	594	19	:	:	PUNCT
eajbcs-667	594	20	the	the	DET
eajbcs-667	594	21	surface	surface	NOUN
eajbcs-667	594	22	of	of	ADP
eajbcs-667	594	23	time	time	NOUN
eajbcs-667	594	24	dependent	dependent	ADJ
eajbcs-667	594	25	two	two	NUM
eajbcs-667	594	26	dimensional	dimensional	ADJ
eajbcs-667	594	27	diffusion	diffusion	NOUN
eajbcs-667	594	28	equation	equation	NOUN
eajbcs-667	594	29	with	with	ADP
eajbcs-667	594	30	d	d	PROPN
eajbcs-667	594	31	=	=	NOUN
eajbcs-667	594	32	100	100	NUM
eajbcs-667	594	33	with	with	ADP
eajbcs-667	594	34	its	its	PRON
eajbcs-667	594	35	color	color	NOUN
eajbcs-667	594	36	bar	bar	NOUN
eajbcs-667	594	37	plot	plot	NOUN
eajbcs-667	594	38	for	for	ADP
eajbcs-667	594	39	its	its	PRON
eajbcs-667	594	40	source	source	NOUN
eajbcs-667	594	41	term	term	NOUN
eajbcs-667	594	42	f	f	PROPN
eajbcs-667	594	43	,	,	PUNCT
eajbcs-667	594	44	f(t	f(t	PROPN
eajbcs-667	594	45	,	,	PUNCT
eajbcs-667	594	46	x	x	NOUN
eajbcs-667	594	47	,	,	PUNCT
eajbcs-667	594	48	y	y	NOUN
eajbcs-667	594	49	)	)	PUNCT
eajbcs-667	594	50	=	=	SYM
eajbcs-667	594	51	(	(	PUNCT
eajbcs-667	594	52	2dπ2−1)e−t	2dπ2−1)e−t	NUM
eajbcs-667	594	53	sin(πx	sin(πx	NOUN
eajbcs-667	594	54	)	)	PUNCT
eajbcs-667	594	55	sin(πy	sin(πy	NOUN
eajbcs-667	594	56	)	)	PUNCT
eajbcs-667	594	57	.	.	PUNCT
eajbcs-667	595	1	method	method	NOUN
eajbcs-667	595	2	to	to	PART
eajbcs-667	595	3	solve	solve	VERB
eajbcs-667	595	4	any	any	DET
eajbcs-667	595	5	differential	differential	ADJ
eajbcs-667	595	6	equations	equation	NOUN
eajbcs-667	595	7	numerically	numerically	ADV
eajbcs-667	595	8	in	in	ADP
eajbcs-667	595	9	any	any	DET
eajbcs-667	595	10	type	type	NOUN
eajbcs-667	595	11	of	of	ADP
eajbcs-667	595	12	geometries	geometry	NOUN
eajbcs-667	595	13	.	.	PUNCT
eajbcs-667	596	1	east	east	PROPN
eajbcs-667	596	2	afr	afr	PROPN
eajbcs-667	596	3	.	.	PUNCT
eajbcs-667	597	1	j.	j.	PROPN
eajbcs-667	597	2	biophys	biophys	PROPN
eajbcs-667	597	3	.	.	PUNCT
eajbcs-667	598	1	comput	comput	NOUN
eajbcs-667	598	2	.	.	PUNCT
eajbcs-667	599	1	sci	sci	PROPN
eajbcs-667	599	2	.	.	PUNCT
eajbcs-667	599	3	(	(	PUNCT
eajbcs-667	599	4	2023	2023	NUM
eajbcs-667	599	5	)	)	PUNCT
eajbcs-667	599	6	,	,	PUNCT
eajbcs-667	599	7	vol	vol	NOUN
eajbcs-667	599	8	.	.	PROPN
eajbcs-667	599	9	4	4	NUM
eajbcs-667	599	10	,	,	PUNCT
eajbcs-667	599	11	no	no	INTJ
eajbcs-667	599	12	.	.	NOUN
eajbcs-667	599	13	1	1	NUM
eajbcs-667	599	14	,	,	PUNCT
eajbcs-667	599	15	52	52	NUM
eajbcs-667	599	16	-	-	SYM
eajbcs-667	599	17	74	74	NUM
eajbcs-667	599	18	72	72	NUM
eajbcs-667	599	19	if	if	SCONJ
eajbcs-667	599	20	we	we	PRON
eajbcs-667	599	21	consider	consider	VERB
eajbcs-667	599	22	another	another	DET
eajbcs-667	599	23	linear	linear	ADJ
eajbcs-667	599	24	test	test	NOUN
eajbcs-667	599	25	example	example	NOUN
eajbcs-667	599	26	with	with	ADP
eajbcs-667	599	27	exact	exact	ADJ
eajbcs-667	599	28	solution	solution	NOUN
eajbcs-667	599	29	u(x	u(x	NOUN
eajbcs-667	599	30	,	,	PUNCT
eajbcs-667	599	31	y	y	PROPN
eajbcs-667	599	32	,	,	PUNCT
eajbcs-667	599	33	t	t	PROPN
eajbcs-667	599	34	)	)	PUNCT
eajbcs-667	599	35	=	=	SYM
eajbcs-667	599	36	x+	x+	PUNCT
eajbcs-667	599	37	y	y	PROPN
eajbcs-667	599	38	+	+	PROPN
eajbcs-667	599	39	t	t	PROPN
eajbcs-667	599	40	,	,	PUNCT
eajbcs-667	599	41	then	then	ADV
eajbcs-667	599	42	its	its	PRON
eajbcs-667	599	43	source	source	NOUN
eajbcs-667	599	44	function	function	NOUN
eajbcs-667	599	45	becomes	become	VERB
eajbcs-667	599	46	f(x	f(x	PROPN
eajbcs-667	599	47	,	,	PUNCT
eajbcs-667	599	48	y	y	PROPN
eajbcs-667	599	49	,	,	PUNCT
eajbcs-667	599	50	t	t	PROPN
eajbcs-667	599	51	)	)	PUNCT
eajbcs-667	599	52	=	=	SYM
eajbcs-667	600	1	1	1	X
eajbcs-667	600	2	.	.	PUNCT
eajbcs-667	601	1	the	the	DET
eajbcs-667	601	2	finite	finite	PROPN
eajbcs-667	601	3	element	element	NOUN
eajbcs-667	601	4	solution	solution	NOUN
eajbcs-667	601	5	the	the	DET
eajbcs-667	601	6	numerical	numerical	ADJ
eajbcs-667	601	7	results	result	NOUN
eajbcs-667	601	8	in	in	ADP
eajbcs-667	601	9	all	all	DET
eajbcs-667	601	10	the	the	DET
eajbcs-667	601	11	above	above	ADJ
eajbcs-667	601	12	mentioned	mention	VERB
eajbcs-667	601	13	discussions	discussion	NOUN
eajbcs-667	601	14	using	use	VERB
eajbcs-667	601	15	the	the	DET
eajbcs-667	601	16	finite	finite	ADJ
eajbcs-667	601	17	element	element	NOUN
eajbcs-667	601	18	method	method	NOUN
eajbcs-667	601	19	are	be	AUX
eajbcs-667	601	20	almost	almost	ADV
eajbcs-667	601	21	closed	close	VERB
eajbcs-667	601	22	to	to	ADP
eajbcs-667	601	23	the	the	DET
eajbcs-667	601	24	analytical	analytical	ADJ
eajbcs-667	601	25	solution	solution	NOUN
eajbcs-667	601	26	in	in	ADP
eajbcs-667	601	27	the	the	DET
eajbcs-667	601	28	case	case	NOUN
eajbcs-667	601	29	of	of	ADP
eajbcs-667	601	30	the	the	DET
eajbcs-667	601	31	test	test	NOUN
eajbcs-667	601	32	examples	example	NOUN
eajbcs-667	601	33	.	.	PUNCT
eajbcs-667	602	1	this	this	PRON
eajbcs-667	602	2	shows	show	VERB
eajbcs-667	602	3	us	we	PRON
eajbcs-667	602	4	that	that	SCONJ
eajbcs-667	602	5	the	the	DET
eajbcs-667	602	6	finite	finite	PROPN
eajbcs-667	602	7	element	element	NOUN
eajbcs-667	602	8	method	method	NOUN
eajbcs-667	602	9	is	be	AUX
eajbcs-667	602	10	one	one	NUM
eajbcs-667	602	11	of	of	ADP
eajbcs-667	602	12	the	the	DET
eajbcs-667	602	13	best	good	ADJ
eajbcs-667	602	14	numerical	numerical	ADJ
eajbcs-667	602	15	for	for	ADP
eajbcs-667	602	16	this	this	DET
eajbcs-667	602	17	2d	2d	NUM
eajbcs-667	602	18	unsteady	unsteady	ADJ
eajbcs-667	602	19	diffusion	diffusion	NOUN
eajbcs-667	602	20	equation	equation	NOUN
eajbcs-667	602	21	with	with	ADP
eajbcs-667	602	22	its	its	PRON
eajbcs-667	602	23	exact	exact	ADJ
eajbcs-667	602	24	solution	solution	NOUN
eajbcs-667	602	25	and	and	CCONJ
eajbcs-667	602	26	color	color	NOUN
eajbcs-667	602	27	bar	bar	NOUN
eajbcs-667	602	28	is	be	AUX
eajbcs-667	602	29	given	give	VERB
eajbcs-667	602	30	in	in	ADP
eajbcs-667	602	31	fig	fig	NOUN
eajbcs-667	602	32	.	.	PUNCT
eajbcs-667	603	1	11	11	NUM
eajbcs-667	603	2	by	by	ADP
eajbcs-667	603	3	ode	ode	PROPN
eajbcs-667	603	4	solver	solver	PROPN
eajbcs-667	603	5	ode15i	ode15i	NOUN
eajbcs-667	603	6	.	.	PUNCT
eajbcs-667	604	1	(	(	PUNCT
eajbcs-667	604	2	a	a	X
eajbcs-667	604	3	)	)	PUNCT
eajbcs-667	604	4	fem	fem	NOUN
eajbcs-667	604	5	&	&	CCONJ
eajbcs-667	604	6	exact	exact	PROPN
eajbcs-667	604	7	sol	sol	NOUN
eajbcs-667	604	8	(	(	PUNCT
eajbcs-667	604	9	b	b	NOUN
eajbcs-667	604	10	)	)	PUNCT
eajbcs-667	604	11	colour	colour	NOUN
eajbcs-667	604	12	bar	bar	NOUN
eajbcs-667	604	13	of	of	ADP
eajbcs-667	604	14	the	the	DET
eajbcs-667	604	15	fem	fem	NOUN
eajbcs-667	604	16	solution	solution	NOUN
eajbcs-667	604	17	figure	figure	NOUN
eajbcs-667	604	18	11	11	NUM
eajbcs-667	604	19	:	:	PUNCT
eajbcs-667	604	20	the	the	DET
eajbcs-667	604	21	surface	surface	NOUN
eajbcs-667	604	22	of	of	ADP
eajbcs-667	604	23	time	time	NOUN
eajbcs-667	604	24	dependent	dependent	ADJ
eajbcs-667	604	25	two	two	NUM
eajbcs-667	604	26	dimensional	dimensional	ADJ
eajbcs-667	604	27	diffusion	diffusion	NOUN
eajbcs-667	604	28	equation	equation	NOUN
eajbcs-667	604	29	with	with	ADP
eajbcs-667	604	30	d	d	PROPN
eajbcs-667	604	31	=	=	SYM
eajbcs-667	604	32	1	1	NUM
eajbcs-667	604	33	with	with	ADP
eajbcs-667	604	34	its	its	PRON
eajbcs-667	604	35	color	color	NOUN
eajbcs-667	604	36	bar	bar	NOUN
eajbcs-667	604	37	plot	plot	NOUN
eajbcs-667	604	38	for	for	ADP
eajbcs-667	604	39	its	its	PRON
eajbcs-667	604	40	source	source	NOUN
eajbcs-667	604	41	term	term	NOUN
eajbcs-667	604	42	f	f	PROPN
eajbcs-667	604	43	,	,	PUNCT
eajbcs-667	604	44	f(t	f(t	PROPN
eajbcs-667	604	45	,	,	PUNCT
eajbcs-667	604	46	x	x	NOUN
eajbcs-667	604	47	,	,	PUNCT
eajbcs-667	604	48	y	y	NOUN
eajbcs-667	604	49	)	)	PUNCT
eajbcs-667	604	50	=	=	SYM
eajbcs-667	605	1	1	1	X
eajbcs-667	605	2	.	.	NUM
eajbcs-667	605	3	sional	sional	ADJ
eajbcs-667	605	4	and	and	CCONJ
eajbcs-667	605	5	two	two	NUM
eajbcs-667	605	6	dimensional	dimensional	ADJ
eajbcs-667	605	7	cases	case	NOUN
eajbcs-667	605	8	with	with	ADP
eajbcs-667	605	9	the	the	DET
eajbcs-667	605	10	weighted	weight	VERB
eajbcs-667	605	11	residual	residual	ADJ
eajbcs-667	605	12	(	(	PUNCT
eajbcs-667	605	13	galerkin	galerkin	ADJ
eajbcs-667	605	14	)	)	PUNCT
eajbcs-667	605	15	method	method	NOUN
eajbcs-667	605	16	of	of	ADP
eajbcs-667	605	17	finite	finite	ADJ
eajbcs-667	605	18	elements	element	NOUN
eajbcs-667	605	19	with	with	ADP
eajbcs-667	605	20	constant	constant	ADJ
eajbcs-667	605	21	velocity	velocity	NOUN
eajbcs-667	605	22	term	term	NOUN
eajbcs-667	605	23	and	and	CCONJ
eajbcs-667	605	24	diffusion	diffusion	NOUN
eajbcs-667	605	25	coefficient	coefficient	NOUN
eajbcs-667	605	26	.	.	PUNCT
eajbcs-667	606	1	these	these	DET
eajbcs-667	606	2	statements	statement	NOUN
eajbcs-667	606	3	are	be	AUX
eajbcs-667	606	4	supported	support	VERB
eajbcs-667	606	5	by	by	ADP
eajbcs-667	606	6	our	our	PRON
eajbcs-667	606	7	test	test	NOUN
eajbcs-667	606	8	numerical	numerical	ADJ
eajbcs-667	606	9	investigations	investigation	NOUN
eajbcs-667	606	10	for	for	ADP
eajbcs-667	606	11	the	the	DET
eajbcs-667	606	12	one	one	NUM
eajbcs-667	606	13	and	and	CCONJ
eajbcs-667	606	14	two	two	NUM
eajbcs-667	606	15	dimensional	dimensional	ADJ
eajbcs-667	606	16	poisson	poisson	NOUN
eajbcs-667	606	17	equation	equation	NOUN
eajbcs-667	606	18	,	,	PUNCT
eajbcs-667	606	19	the	the	DET
eajbcs-667	606	20	advection	advection	NOUN
eajbcs-667	606	21	equation	equation	NOUN
eajbcs-667	606	22	,	,	PUNCT
eajbcs-667	606	23	diffusion	diffusion	NOUN
eajbcs-667	606	24	equation	equation	NOUN
eajbcs-667	606	25	and	and	CCONJ
eajbcs-667	606	26	the	the	DET
eajbcs-667	606	27	advection	advection	NOUN
eajbcs-667	606	28	-	-	PUNCT
eajbcs-667	606	29	diffusion	diffusion	NOUN
eajbcs-667	606	30	equation	equation	NOUN
eajbcs-667	606	31	.	.	PUNCT
eajbcs-667	607	1	we	we	PRON
eajbcs-667	607	2	have	have	AUX
eajbcs-667	607	3	also	also	ADV
eajbcs-667	607	4	seen	see	VERB
eajbcs-667	607	5	the	the	DET
eajbcs-667	607	6	simulations	simulation	NOUN
eajbcs-667	607	7	of	of	ADP
eajbcs-667	607	8	the	the	DET
eajbcs-667	607	9	equations	equation	NOUN
eajbcs-667	607	10	by	by	ADP
eajbcs-667	607	11	using	use	VERB
eajbcs-667	607	12	the	the	DET
eajbcs-667	607	13	language	language	NOUN
eajbcs-667	607	14	of	of	ADP
eajbcs-667	607	15	technical	technical	ADJ
eajbcs-667	607	16	computing	computing	NOUN
eajbcs-667	607	17	called	call	VERB
eajbcs-667	607	18	matlab	matlab	PROPN
eajbcs-667	607	19	and	and	CCONJ
eajbcs-667	607	20	for	for	ADP
eajbcs-667	607	21	time	time	NOUN
eajbcs-667	607	22	dependent	dependent	ADJ
eajbcs-667	607	23	equation	equation	NOUN
eajbcs-667	607	24	using	use	VERB
eajbcs-667	607	25	backward	backward	ADV
eajbcs-667	607	26	(	(	PUNCT
eajbcs-667	607	27	implicit	implicit	ADJ
eajbcs-667	607	28	)	)	PUNCT
eajbcs-667	607	29	euler	euler	NOUN
eajbcs-667	607	30	finite	finite	ADJ
eajbcs-667	607	31	difference	difference	NOUN
eajbcs-667	607	32	method	method	NOUN
eajbcs-667	607	33	for	for	ADP
eajbcs-667	607	34	one	one	NUM
eajbcs-667	607	35	dimensional	dimensional	ADJ
eajbcs-667	607	36	problem	problem	NOUN
eajbcs-667	607	37	and	and	CCONJ
eajbcs-667	607	38	the	the	DET
eajbcs-667	607	39	builtin	builtin	NOUN
eajbcs-667	607	40	function	function	NOUN
eajbcs-667	607	41	ode15i	ode15i	X
eajbcs-667	607	42	to	to	PART
eajbcs-667	607	43	solve	solve	VERB
eajbcs-667	607	44	the	the	DET
eajbcs-667	607	45	system	system	NOUN
eajbcs-667	607	46	of	of	ADP
eajbcs-667	607	47	ode	ode	PROPN
eajbcs-667	607	48	’s	’s	NOUN
eajbcs-667	607	49	for	for	ADP
eajbcs-667	607	50	two	two	NUM
eajbcs-667	607	51	dimensional	dimensional	ADJ
eajbcs-667	607	52	equations	equation	NOUN
eajbcs-667	607	53	.	.	PUNCT
eajbcs-667	608	1	east	east	PROPN
eajbcs-667	608	2	afr	afr	PROPN
eajbcs-667	608	3	.	.	PUNCT
eajbcs-667	609	1	j.	j.	PROPN
eajbcs-667	609	2	biophys	biophys	PROPN
eajbcs-667	609	3	.	.	PUNCT
eajbcs-667	610	1	comput	comput	NOUN
eajbcs-667	610	2	.	.	PUNCT
eajbcs-667	611	1	sci	sci	PROPN
eajbcs-667	611	2	.	.	PUNCT
eajbcs-667	611	3	(	(	PUNCT
eajbcs-667	611	4	2023	2023	NUM
eajbcs-667	611	5	)	)	PUNCT
eajbcs-667	611	6	,	,	PUNCT
eajbcs-667	611	7	vol	vol	NOUN
eajbcs-667	611	8	.	.	PROPN
eajbcs-667	611	9	4	4	NUM
eajbcs-667	611	10	,	,	PUNCT
eajbcs-667	611	11	no	no	INTJ
eajbcs-667	611	12	.	.	NOUN
eajbcs-667	611	13	1	1	NUM
eajbcs-667	611	14	,	,	PUNCT
eajbcs-667	611	15	52	52	NUM
eajbcs-667	611	16	-	-	SYM
eajbcs-667	611	17	74	74	NUM
eajbcs-667	611	18	73	73	NUM
eajbcs-667	611	19	the	the	DET
eajbcs-667	611	20	generalization	generalization	NOUN
eajbcs-667	611	21	of	of	ADP
eajbcs-667	611	22	the	the	DET
eajbcs-667	611	23	proposed	propose	VERB
eajbcs-667	611	24	galerkin	galerkin	ADJ
eajbcs-667	611	25	method	method	NOUN
eajbcs-667	611	26	to	to	ADP
eajbcs-667	611	27	the	the	DET
eajbcs-667	611	28	three	three	NUM
eajbcs-667	611	29	-	-	PUNCT
eajbcs-667	611	30	dimensional	dimensional	ADJ
eajbcs-667	611	31	advection	advection	NOUN
eajbcs-667	611	32	diffusion	diffusion	NOUN
eajbcs-667	611	33	is	be	AUX
eajbcs-667	611	34	obtained	obtain	VERB
eajbcs-667	611	35	within	within	ADP
eajbcs-667	611	36	the	the	DET
eajbcs-667	611	37	current	current	ADJ
eajbcs-667	611	38	framework	framework	NOUN
eajbcs-667	611	39	and	and	CCONJ
eajbcs-667	611	40	an	an	DET
eajbcs-667	611	41	interested	interested	ADJ
eajbcs-667	611	42	body	body	NOUN
eajbcs-667	611	43	will	will	AUX
eajbcs-667	611	44	be	be	AUX
eajbcs-667	611	45	done	do	VERB
eajbcs-667	611	46	on	on	ADP
eajbcs-667	611	47	this	this	DET
eajbcs-667	611	48	approach	approach	NOUN
eajbcs-667	611	49	by	by	ADP
eajbcs-667	611	50	including	include	VERB
eajbcs-667	611	51	the	the	DET
eajbcs-667	611	52	reaction	reaction	NOUN
eajbcs-667	611	53	term	term	NOUN
eajbcs-667	611	54	.	.	PUNCT
eajbcs-667	612	1	to	to	PART
eajbcs-667	612	2	obtain	obtain	VERB
eajbcs-667	612	3	the	the	DET
eajbcs-667	612	4	solutions	solution	NOUN
eajbcs-667	612	5	to	to	ADP
eajbcs-667	612	6	this	this	DET
eajbcs-667	612	7	problems	problem	NOUN
eajbcs-667	612	8	,	,	PUNCT
eajbcs-667	612	9	the	the	DET
eajbcs-667	612	10	method	method	NOUN
eajbcs-667	612	11	can	can	AUX
eajbcs-667	612	12	be	be	AUX
eajbcs-667	612	13	extended	extend	VERB
eajbcs-667	612	14	to	to	ADP
eajbcs-667	612	15	the	the	DET
eajbcs-667	612	16	least	least	ADJ
eajbcs-667	612	17	square	square	ADJ
eajbcs-667	612	18	finite	finite	ADJ
eajbcs-667	612	19	element	element	NOUN
eajbcs-667	612	20	method	method	NOUN
eajbcs-667	612	21	,	,	PUNCT
eajbcs-667	612	22	and	and	CCONJ
eajbcs-667	612	23	another	another	DET
eajbcs-667	612	24	interested	interested	ADJ
eajbcs-667	612	25	body	body	NOUN
eajbcs-667	612	26	will	will	AUX
eajbcs-667	612	27	also	also	ADV
eajbcs-667	612	28	be	be	AUX
eajbcs-667	612	29	done	do	VERB
eajbcs-667	612	30	on	on	ADP
eajbcs-667	612	31	this	this	DET
eajbcs-667	612	32	approach	approach	NOUN
eajbcs-667	612	33	.	.	PUNCT
eajbcs-667	613	1	one	one	PRON
eajbcs-667	613	2	can	can	AUX
eajbcs-667	613	3	also	also	ADV
eajbcs-667	613	4	extend	extend	VERB
eajbcs-667	613	5	this	this	DET
eajbcs-667	613	6	method	method	NOUN
eajbcs-667	613	7	using	use	VERB
eajbcs-667	613	8	variable	variable	ADJ
eajbcs-667	613	9	velocities	velocity	NOUN
eajbcs-667	613	10	and	and	CCONJ
eajbcs-667	613	11	diffusion	diffusion	NOUN
eajbcs-667	613	12	coefficients	coefficient	NOUN
eajbcs-667	613	13	whichwe	whichwe	PROPN
eajbcs-667	613	14	have	have	AUX
eajbcs-667	613	15	solved	solve	VERB
eajbcs-667	613	16	the	the	DET
eajbcs-667	613	17	advectiondiffusion	advectiondiffusion	NOUN
eajbcs-667	613	18	e	e	NOUN
eajbcs-667	613	19	quation	quation	NOUN
eajbcs-667	613	20	f	f	PROPN
eajbcs-667	613	21	or	or	CCONJ
eajbcs-667	613	22	b	b	PROPN
eajbcs-667	613	23	oth	oth	PROPN
eajbcs-667	613	24	o	o	PROPN
eajbcs-667	613	25	ne	ne	PROPN
eajbcs-667	613	26	dimenin	dimenin	VERB
eajbcs-667	613	27	this	this	DET
eajbcs-667	613	28	paper	paper	NOUN
eajbcs-667	613	29	,	,	PUNCT
eajbcs-667	613	30	we	we	PRON
eajbcs-667	613	31	study	study	VERB
eajbcs-667	613	32	the	the	DET
eajbcs-667	613	33	finite	finite	PROPN
eajbcs-667	613	34	element	element	NOUN
eajbcs-667	613	35	solution	solution	NOUN
eajbcs-667	613	36	of	of	ADP
eajbcs-667	613	37	the	the	DET
eajbcs-667	613	38	advection	advection	NOUN
eajbcs-667	613	39	-	-	PUNCT
eajbcs-667	613	40	diffusion	diffusion	NOUN
eajbcs-667	613	41	equation	equation	NOUN
eajbcs-667	613	42	.	.	PUNCT
eajbcs-667	614	1	in	in	ADP
eajbcs-667	614	2	the	the	DET
eajbcs-667	614	3	finite	finite	PROPN
eajbcs-667	614	4	e	e	PROPN
eajbcs-667	614	5	lement	lement	PROPN
eajbcs-667	614	6	analysis	analysis	NOUN
eajbcs-667	614	7	,	,	PUNCT
eajbcs-667	614	8	we	we	PRON
eajbcs-667	614	9	approximate	approximate	VERB
eajbcs-667	614	10	a	a	DET
eajbcs-667	614	11	function	function	NOUN
eajbcs-667	614	12	defined	define	VERB
eajbcs-667	614	13	in	in	ADP
eajbcs-667	614	14	a	a	DET
eajbcs-667	614	15	domain	domain	NOUN
eajbcs-667	614	16	,	,	PUNCT
eajbcs-667	614	17	ω	ω	NOUN
eajbcs-667	614	18	,	,	PUNCT
eajbcs-667	614	19	with	with	ADP
eajbcs-667	614	20	a	a	DET
eajbcs-667	614	21	set	set	NOUN
eajbcs-667	614	22	of	of	ADP
eajbcs-667	614	23	orthogonal	orthogonal	ADJ
eajbcs-667	614	24	basis	basis	NOUN
eajbcs-667	614	25	functions	function	NOUN
eajbcs-667	614	26	with	with	ADP
eajbcs-667	614	27	coefficients	coefficient	NOUN
eajbcs-667	614	28	corresponding	correspond	VERB
eajbcs-667	614	29	to	to	ADP
eajbcs-667	614	30	the	the	DET
eajbcs-667	614	31	functional	functional	ADJ
eajbcs-667	614	32	values	value	NOUN
eajbcs-667	614	33	at	at	ADP
eajbcs-667	614	34	some	some	DET
eajbcs-667	614	35	node	node	ADJ
eajbcs-667	614	36	points	point	NOUN
eajbcs-667	614	37	.	.	PUNCT
eajbcs-667	615	1	we	we	PRON
eajbcs-667	615	2	deal	deal	VERB
eajbcs-667	615	3	with	with	ADP
eajbcs-667	615	4	the	the	DET
eajbcs-667	615	5	numerical	numerical	PROPN
eajbcs-667	615	6	simulation	simulation	NOUN
eajbcs-667	615	7	of	of	ADP
eajbcs-667	615	8	the	the	DET
eajbcs-667	615	9	advection	advection	NOUN
eajbcs-667	615	10	-	-	PUNCT
eajbcs-667	615	11	diffusion	diffusion	NOUN
eajbcs-667	615	12	equation	equation	NOUN
eajbcs-667	615	13	using	use	VERB
eajbcs-667	615	14	the	the	DET
eajbcs-667	615	15	finite	finite	ADJ
eajbcs-667	615	16	e	e	PROPN
eajbcs-667	615	17	lement	lement	PROPN
eajbcs-667	615	18	method	method	NOUN
eajbcs-667	615	19	scheme	scheme	NOUN
eajbcs-667	615	20	in	in	ADP
eajbcs-667	615	21	space	space	NOUN
eajbcs-667	615	22	and	and	CCONJ
eajbcs-667	615	23	the	the	DET
eajbcs-667	615	24	backward	backward	PROPN
eajbcs-667	615	25	euler	euler	NOUN
eajbcs-667	615	26	method	method	NOUN
eajbcs-667	615	27	in	in	ADP
eajbcs-667	615	28	time	time	NOUN
eajbcs-667	615	29	.	.	PUNCT
eajbcs-667	616	1	the	the	DET
eajbcs-667	616	2	main	main	ADJ
eajbcs-667	616	3	focus	focus	NOUN
eajbcs-667	616	4	of	of	ADP
eajbcs-667	616	5	the	the	DET
eajbcs-667	616	6	paper	paper	NOUN
eajbcs-667	616	7	has	have	AUX
eajbcs-667	616	8	been	be	AUX
eajbcs-667	616	9	on	on	ADP
eajbcs-667	616	10	the	the	DET
eajbcs-667	616	11	variational	variational	ADJ
eajbcs-667	616	12	formulation	formulation	NOUN
eajbcs-667	616	13	techniques	technique	NOUN
eajbcs-667	616	14	for	for	ADP
eajbcs-667	616	15	the	the	DET
eajbcs-667	616	16	solution	solution	NOUN
eajbcs-667	616	17	of	of	ADP
eajbcs-667	616	18	the	the	DET
eajbcs-667	616	19	discrete	discrete	ADJ
eajbcs-667	616	20	galerkin	galerkin	ADJ
eajbcs-667	616	21	method	method	NOUN
eajbcs-667	616	22	and	and	CCONJ
eajbcs-667	616	23	the	the	DET
eajbcs-667	616	24	other	other	ADJ
eajbcs-667	616	25	hand	hand	NOUN
eajbcs-667	616	26	on	on	ADP
eajbcs-667	616	27	computational	computational	ADJ
eajbcs-667	616	28	analysis	analysis	NOUN
eajbcs-667	616	29	of	of	ADP
eajbcs-667	616	30	different	different	ADJ
eajbcs-667	616	31	one	one	NUM
eajbcs-667	616	32	and	and	CCONJ
eajbcs-667	616	33	two	two	NUM
eajbcs-667	616	34	dimensional	dimensional	ADJ
eajbcs-667	616	35	pde	pde	NOUN
eajbcs-667	616	36	’s	’s	NOUN
eajbcs-667	616	37	.	.	PUNCT
eajbcs-667	617	1	we	we	PRON
eajbcs-667	617	2	have	have	AUX
eajbcs-667	617	3	done	do	VERB
eajbcs-667	617	4	numerical	numerical	ADJ
eajbcs-667	617	5	simulations	simulation	NOUN
eajbcs-667	617	6	for	for	ADP
eajbcs-667	617	7	the	the	DET
eajbcs-667	617	8	above	above	ADJ
eajbcs-667	617	9	mentioned	mention	VERB
eajbcs-667	617	10	equations	equation	NOUN
eajbcs-667	617	11	by	by	ADP
eajbcs-667	617	12	considering	consider	VERB
eajbcs-667	617	13	the	the	DET
eajbcs-667	617	14	technique	technique	NOUN
eajbcs-667	617	15	and	and	CCONJ
eajbcs-667	617	16	using	use	VERB
eajbcs-667	617	17	different	different	ADJ
eajbcs-667	617	18	test	test	NOUN
eajbcs-667	617	19	examples	example	NOUN
eajbcs-667	617	20	.	.	PUNCT
eajbcs-667	618	1	the	the	DET
eajbcs-667	618	2	solution	solution	NOUN
eajbcs-667	618	3	for	for	ADP
eajbcs-667	618	4	the	the	DET
eajbcs-667	618	5	values	value	NOUN
eajbcs-667	618	6	at	at	ADP
eajbcs-667	618	7	the	the	DET
eajbcs-667	618	8	nodes	node	NOUN
eajbcs-667	618	9	for	for	ADP
eajbcs-667	618	10	the	the	DET
eajbcs-667	618	11	partial	partial	ADJ
eajbcs-667	618	12	differential	differential	NOUN
eajbcs-667	618	13	equations	equation	NOUN
eajbcs-667	618	14	can	can	AUX
eajbcs-667	618	15	be	be	AUX
eajbcs-667	618	16	obtained	obtain	VERB
eajbcs-667	618	17	by	by	ADP
eajbcs-667	618	18	solving	solve	VERB
eajbcs-667	618	19	a	a	DET
eajbcs-667	618	20	linear	linear	ADJ
eajbcs-667	618	21	system	system	NOUN
eajbcs-667	618	22	of	of	ADP
eajbcs-667	618	23	equations	equation	NOUN
eajbcs-667	618	24	involving	involve	VERB
eajbcs-667	618	25	the	the	DET
eajbcs-667	618	26	inversion	inversion	NOUN
eajbcs-667	618	27	of	of	ADP
eajbcs-667	618	28	the	the	DET
eajbcs-667	618	29	sparse	sparse	ADJ
eajbcs-667	618	30	matrices	matrix	NOUN
eajbcs-667	618	31	.	.	PUNCT
eajbcs-667	619	1	conclusions	conclusion	NOUN
eajbcs-667	619	2	east	east	PROPN
eajbcs-667	619	3	afr	afr	PROPN
eajbcs-667	619	4	.	.	PUNCT
eajbcs-667	620	1	j.	j.	PROPN
eajbcs-667	620	2	biophys	biophys	PROPN
eajbcs-667	620	3	.	.	PUNCT
eajbcs-667	621	1	comput	comput	NOUN
eajbcs-667	621	2	.	.	PUNCT
eajbcs-667	622	1	sci	sci	PROPN
eajbcs-667	622	2	.	.	PUNCT
eajbcs-667	622	3	(	(	PUNCT
eajbcs-667	622	4	2023	2023	NUM
eajbcs-667	622	5	)	)	PUNCT
eajbcs-667	622	6	,	,	PUNCT
eajbcs-667	622	7	vol	vol	NOUN
eajbcs-667	622	8	.	.	PROPN
eajbcs-667	622	9	4	4	NUM
eajbcs-667	622	10	,	,	PUNCT
eajbcs-667	622	11	no	no	INTJ
eajbcs-667	622	12	.	.	NOUN
eajbcs-667	622	13	1	1	NUM
eajbcs-667	622	14	,	,	PUNCT
eajbcs-667	622	15	52	52	NUM
eajbcs-667	622	16	-	-	SYM
eajbcs-667	622	17	74	74	NUM
eajbcs-667	622	18	hundsdorfer	hundsdorfer	NOUN
eajbcs-667	622	19	w.1996	w.1996	PROPN
eajbcs-667	622	20	.	.	PUNCT
eajbcs-667	623	1	numerical	numerical	ADJ
eajbcs-667	623	2	solution	solution	NOUN
eajbcs-667	623	3	of	of	ADP
eajbcs-667	623	4	advectiondiffusion	advectiondiffusion	NOUN
eajbcs-667	623	5	-	-	PUNCT
eajbcs-667	623	6	reaction	reaction	NOUN
eajbcs-667	623	7	equations	equation	NOUN
eajbcs-667	623	8	:	:	PUNCT
eajbcs-667	623	9	lecture	lecture	NOUN
eajbcs-667	623	10	notes	note	NOUN
eajbcs-667	623	11	for	for	ADP
eajbcs-667	623	12	ph	ph	PROPN
eajbcs-667	623	13	.	.	PROPN
eajbcs-667	623	14	d.	d.	PROPN
eajbcs-667	623	15	course	course	PROPN
eajbcs-667	623	16	,	,	PUNCT
eajbcs-667	623	17	1996	1996	NUM
eajbcs-667	623	18	,	,	PUNCT
eajbcs-667	623	19	thomas	thomas	PROPN
eajbcs-667	623	20	stieltjes	stieltjes	PROPN
eajbcs-667	623	21	institute	institute	PROPN
eajbcs-667	623	22	.	.	PUNCT
eajbcs-667	624	1	department	department	PROPN
eajbcs-667	624	2	of	of	ADP
eajbcs-667	624	3	numerical	numerical	PROPN
eajbcs-667	624	4	mathematics	mathematics	PROPN
eajbcs-667	625	1	[	[	X
eajbcs-667	625	2	nm	nm	X
eajbcs-667	625	3	]	]	X
eajbcs-667	625	4	,	,	PUNCT
eajbcs-667	625	5	(	(	PUNCT
eajbcs-667	625	6	9603	9603	NUM
eajbcs-667	625	7	)	)	PUNCT
eajbcs-667	625	8	.	.	PUNCT
eajbcs-667	626	1	johnson	johnson	PROPN
eajbcs-667	626	2	c.	c.	PROPN
eajbcs-667	626	3	2012	2012	NUM
eajbcs-667	626	4	.	.	PUNCT
eajbcs-667	627	1	numerical	numerical	ADJ
eajbcs-667	627	2	solution	solution	NOUN
eajbcs-667	627	3	of	of	ADP
eajbcs-667	627	4	partial	partial	ADJ
eajbcs-667	627	5	differential	differential	ADJ
eajbcs-667	627	6	equations	equation	NOUN
eajbcs-667	627	7	by	by	ADP
eajbcs-667	627	8	the	the	DET
eajbcs-667	627	9	finite	finite	PROPN
eajbcs-667	627	10	element	element	NOUN
eajbcs-667	627	11	method	method	NOUN
eajbcs-667	627	12	.	.	PUNCT
eajbcs-667	628	1	courier	courier	NOUN
eajbcs-667	628	2	corporation	corporation	NOUN
eajbcs-667	628	3	.	.	PUNCT
eajbcs-667	629	1	langtangen	langtangen	PROPN
eajbcs-667	629	2	h.p	h.p	PROPN
eajbcs-667	629	3	.	.	PROPN
eajbcs-667	629	4	2003	2003	NUM
eajbcs-667	629	5	.	.	PUNCT
eajbcs-667	630	1	computational	computational	ADJ
eajbcs-667	630	2	partial	partial	ADJ
eajbcs-667	630	3	differential	differential	NOUN
eajbcs-667	630	4	equations	equation	NOUN
eajbcs-667	630	5	:	:	PUNCT
eajbcs-667	630	6	numerical	numerical	ADJ
eajbcs-667	630	7	methods	method	NOUN
eajbcs-667	630	8	and	and	CCONJ
eajbcs-667	630	9	diffpack	diffpack	NOUN
eajbcs-667	630	10	programming	programming	NOUN
eajbcs-667	630	11	(	(	PUNCT
eajbcs-667	630	12	vol	vol	NOUN
eajbcs-667	630	13	.	.	NOUN
eajbcs-667	630	14	2	2	NUM
eajbcs-667	630	15	)	)	PUNCT
eajbcs-667	630	16	.	.	PUNCT
eajbcs-667	631	1	berlin	berlin	PROPN
eajbcs-667	631	2	:	:	PUNCT
eajbcs-667	631	3	springer	springer	NOUN
eajbcs-667	631	4	.	.	PUNCT
eajbcs-667	632	1	larson	larson	PROPN
eajbcs-667	632	2	m.g	m.g	PROPN
eajbcs-667	632	3	.	.	PROPN
eajbcs-667	632	4	and	and	CCONJ
eajbcs-667	632	5	bengzon	bengzon	PROPN
eajbcs-667	632	6	f.	f.	PROPN
eajbcs-667	632	7	2010	2010	NUM
eajbcs-667	632	8	.	.	PUNCT
eajbcs-667	633	1	the	the	DET
eajbcs-667	633	2	finite	finite	PROPN
eajbcs-667	633	3	element	element	NOUN
eajbcs-667	633	4	method	method	NOUN
eajbcs-667	633	5	:	:	PUNCT
eajbcs-667	633	6	theory	theory	NOUN
eajbcs-667	633	7	,	,	PUNCT
eajbcs-667	633	8	implementation	implementation	NOUN
eajbcs-667	633	9	,	,	PUNCT
eajbcs-667	633	10	and	and	CCONJ
eajbcs-667	633	11	practice	practice	NOUN
eajbcs-667	633	12	.	.	PUNCT
eajbcs-667	634	1	texts	text	NOUN
eajbcs-667	634	2	in	in	ADP
eajbcs-667	634	3	computational	computational	ADJ
eajbcs-667	634	4	science	science	NOUN
eajbcs-667	634	5	and	and	CCONJ
eajbcs-667	634	6	engineering	engineering	NOUN
eajbcs-667	634	7	,	,	PUNCT
eajbcs-667	634	8	10	10	NUM
eajbcs-667	634	9	,	,	PUNCT
eajbcs-667	634	10	23	23	NUM
eajbcs-667	634	11	-	-	SYM
eajbcs-667	634	12	44	44	NUM
eajbcs-667	634	13	.	.	PUNCT
eajbcs-667	635	1	lian	lian	PROPN
eajbcs-667	635	2	y.	y.	PROPN
eajbcs-667	635	3	,	,	PUNCT
eajbcs-667	635	4	ying	ying	PROPN
eajbcs-667	635	5	y.	y.	PROPN
eajbcs-667	635	6	,	,	PUNCT
eajbcs-667	635	7	tang	tang	PROPN
eajbcs-667	635	8	s.	s.	PROPN
eajbcs-667	635	9	,	,	PUNCT
eajbcs-667	635	10	lin	lin	PROPN
eajbcs-667	635	11	s.	s.	PROPN
eajbcs-667	635	12	,	,	PUNCT
eajbcs-667	635	13	wagner	wagner	PROPN
eajbcs-667	635	14	g.j	g.j	PROPN
eajbcs-667	635	15	.	.	PROPN
eajbcs-667	635	16	and	and	CCONJ
eajbcs-667	635	17	liu	liu	PROPN
eajbcs-667	635	18	w.k	w.k	PROPN
eajbcs-667	635	19	.	.	PROPN
eajbcs-667	635	20	2016	2016	NUM
eajbcs-667	635	21	.	.	PUNCT
eajbcs-667	636	1	a	a	DET
eajbcs-667	636	2	petrov	petrov	PROPN
eajbcs-667	636	3	–	–	PUNCT
eajbcs-667	636	4	galerkin	galerkin	ADJ
eajbcs-667	636	5	finite	finite	PROPN
eajbcs-667	636	6	element	element	NOUN
eajbcs-667	636	7	method	method	NOUN
eajbcs-667	636	8	for	for	ADP
eajbcs-667	636	9	the	the	DET
eajbcs-667	636	10	fractional	fractional	ADJ
eajbcs-667	636	11	advection	advection	NOUN
eajbcs-667	636	12	–	–	PUNCT
eajbcs-667	636	13	diffusion	diffusion	NOUN
eajbcs-667	636	14	equation	equation	NOUN
eajbcs-667	636	15	.	.	PUNCT
eajbcs-667	637	1	comput	comput	NOUN
eajbcs-667	637	2	.	.	PUNCT
eajbcs-667	638	1	methods	method	NOUN
eajbcs-667	638	2	appl	appl	PROPN
eajbcs-667	638	3	.	.	PROPN
eajbcs-667	638	4	mech	mech	PROPN
eajbcs-667	638	5	.	.	PUNCT
eajbcs-667	639	1	eng	eng	PROPN
eajbcs-667	639	2	.	.	PROPN
eajbcs-667	639	3	,	,	PUNCT
eajbcs-667	639	4	309	309	NUM
eajbcs-667	639	5	:	:	PUNCT
eajbcs-667	639	6	388	388	NUM
eajbcs-667	639	7	-	-	SYM
eajbcs-667	639	8	410	410	NUM
eajbcs-667	639	9	.	.	PUNCT
eajbcs-667	640	1	lima	lima	PROPN
eajbcs-667	640	2	s.a	s.a	PROPN
eajbcs-667	640	3	.	.	PROPN
eajbcs-667	640	4	,	,	PUNCT
eajbcs-667	640	5	kamrujjaman	kamrujjaman	NOUN
eajbcs-667	640	6	m.	m.	NOUN
eajbcs-667	640	7	and	and	CCONJ
eajbcs-667	640	8	islam	islam	PROPN
eajbcs-667	640	9	m.s	m.s	PROPN
eajbcs-667	640	10	.	.	PROPN
eajbcs-667	640	11	2021	2021	NUM
eajbcs-667	640	12	.	.	PUNCT
eajbcs-667	641	1	numerical	numerical	ADJ
eajbcs-667	641	2	solution	solution	NOUN
eajbcs-667	641	3	of	of	ADP
eajbcs-667	641	4	convection	convection	NOUN
eajbcs-667	641	5	–	–	PUNCT
eajbcs-667	641	6	diffusion	diffusion	NOUN
eajbcs-667	641	7	–	–	PUNCT
eajbcs-667	641	8	reaction	reaction	NOUN
eajbcs-667	641	9	equations	equation	NOUN
eajbcs-667	641	10	by	by	ADP
eajbcs-667	641	11	a	a	DET
eajbcs-667	641	12	finite	finite	ADJ
eajbcs-667	641	13	element	element	NOUN
eajbcs-667	641	14	method	method	NOUN
eajbcs-667	641	15	with	with	ADP
eajbcs-667	641	16	error	error	NOUN
eajbcs-667	641	17	correlation	correlation	NOUN
eajbcs-667	641	18	.	.	PUNCT
eajbcs-667	642	1	aip	aip	PROPN
eajbcs-667	642	2	advances	advance	VERB
eajbcs-667	642	3	11(8	11(8	NUM
eajbcs-667	642	4	):	):	PUNCT
eajbcs-667	642	5	085225	085225	NUM
eajbcs-667	642	6	.	.	PUNCT
eajbcs-667	643	1	pochai	pochai	PROPN
eajbcs-667	643	2	n.	n.	PROPN
eajbcs-667	643	3	and	and	CCONJ
eajbcs-667	643	4	deepana	deepana	PROPN
eajbcs-667	643	5	r.	r.	PROPN
eajbcs-667	643	6	2011	2011	NUM
eajbcs-667	643	7	.	.	PUNCT
eajbcs-667	644	1	a	a	DET
eajbcs-667	644	2	numerical	numerical	ADJ
eajbcs-667	644	3	computation	computation	NOUN
eajbcs-667	644	4	of	of	ADP
eajbcs-667	644	5	water	water	NOUN
eajbcs-667	644	6	quality	quality	NOUN
eajbcs-667	644	7	measurement	measurement	NOUN
eajbcs-667	644	8	in	in	ADP
eajbcs-667	644	9	a	a	DET
eajbcs-667	644	10	uniform	uniform	ADJ
eajbcs-667	644	11	channel	channel	NOUN
eajbcs-667	644	12	using	use	VERB
eajbcs-667	644	13	a	a	DET
eajbcs-667	644	14	finite	finite	ADJ
eajbcs-667	644	15	difference	difference	NOUN
eajbcs-667	644	16	method	method	NOUN
eajbcs-667	644	17	.	.	PUNCT
eajbcs-667	645	1	procedia	procedia	PROPN
eajbcs-667	645	2	eng	eng	PROPN
eajbcs-667	645	3	.	.	PROPN
eajbcs-667	645	4	8	8	NUM
eajbcs-667	645	5	:	:	SYM
eajbcs-667	645	6	85	85	NUM
eajbcs-667	645	7	-	-	SYM
eajbcs-667	645	8	88	88	NUM
eajbcs-667	645	9	.	.	PUNCT
eajbcs-667	646	1	quarteroni	quarteroni	PROPN
eajbcs-667	646	2	a.	a.	PROPN
eajbcs-667	646	3	and	and	CCONJ
eajbcs-667	646	4	quarteroni	quarteroni	PROPN
eajbcs-667	646	5	s.	s.	PROPN
eajbcs-667	646	6	2009	2009	NUM
eajbcs-667	646	7	.	.	PUNCT
eajbcs-667	647	1	numerical	numerical	ADJ
eajbcs-667	647	2	models	model	NOUN
eajbcs-667	647	3	for	for	ADP
eajbcs-667	647	4	differential	differential	ADJ
eajbcs-667	647	5	problems	problem	NOUN
eajbcs-667	647	6	(	(	PUNCT
eajbcs-667	647	7	vol	vol	NOUN
eajbcs-667	647	8	.	.	NOUN
eajbcs-667	647	9	2	2	NUM
eajbcs-667	647	10	)	)	PUNCT
eajbcs-667	647	11	.	.	PUNCT
eajbcs-667	648	1	milan	milan	PROPN
eajbcs-667	648	2	:	:	PUNCT
eajbcs-667	648	3	springer	springer	NOUN
eajbcs-667	648	4	.	.	PUNCT
eajbcs-667	649	1	szymkiewicz	szymkiewicz	PROPN
eajbcs-667	649	2	r.	r.	PROPN
eajbcs-667	649	3	and	and	CCONJ
eajbcs-667	649	4	gąsiorowski	gąsiorowski	PROPN
eajbcs-667	649	5	d.	d.	PROPN
eajbcs-667	649	6	2021	2021	NUM
eajbcs-667	649	7	.	.	PUNCT
eajbcs-667	650	1	adaptive	adaptive	ADJ
eajbcs-667	650	2	method	method	NOUN
eajbcs-667	650	3	for	for	ADP
eajbcs-667	650	4	the	the	DET
eajbcs-667	650	5	solution	solution	NOUN
eajbcs-667	650	6	of	of	ADP
eajbcs-667	650	7	1d	1d	NUM
eajbcs-667	650	8	and	and	CCONJ
eajbcs-667	650	9	2d	2d	NUM
eajbcs-667	650	10	advection	advection	NOUN
eajbcs-667	650	11	–	–	PUNCT
eajbcs-667	650	12	diffusion	diffusion	NOUN
eajbcs-667	650	13	equations	equation	NOUN
eajbcs-667	650	14	used	use	VERB
eajbcs-667	650	15	in	in	ADP
eajbcs-667	650	16	environmental	environmental	ADJ
eajbcs-667	650	17	engineering	engineering	NOUN
eajbcs-667	650	18	.	.	PUNCT
eajbcs-667	651	1	j.	j.	PROPN
eajbcs-667	651	2	hydroinformatics	hydroinformatics	PROPN
eajbcs-667	651	3	23(6	23(6	NUM
eajbcs-667	651	4	):	):	PUNCT
eajbcs-667	651	5	12901311	12901311	NUM
eajbcs-667	651	6	.	.	PUNCT
eajbcs-667	652	1	yang	yang	PROPN
eajbcs-667	652	2	w.y	w.y	PROPN
eajbcs-667	652	3	.	.	PROPN
eajbcs-667	652	4	,	,	PUNCT
eajbcs-667	652	5	cao	cao	PROPN
eajbcs-667	652	6	w.	w.	PROPN
eajbcs-667	652	7	,	,	PUNCT
eajbcs-667	652	8	chung	chung	PROPN
eajbcs-667	652	9	t.s	t.s	PROPN
eajbcs-667	652	10	.	.	PROPN
eajbcs-667	652	11	and	and	CCONJ
eajbcs-667	652	12	morris	morris	PROPN
eajbcs-667	652	13	j.	j.	PROPN
eajbcs-667	652	14	2005	2005	NUM
eajbcs-667	652	15	.	.	PUNCT
eajbcs-667	653	1	applied	apply	VERB
eajbcs-667	653	2	numerical	numerical	ADJ
eajbcs-667	653	3	methods	method	NOUN
eajbcs-667	653	4	using	use	VERB
eajbcs-667	653	5	matlab	matlab	PROPN
eajbcs-667	653	6	.	.	PUNCT
eajbcs-667	654	1	a	a	DET
eajbcs-667	654	2	john	john	PROPN
eajbcs-667	654	3	wiley	wiley	PROPN
eajbcs-667	654	4	&	&	CCONJ
eajbcs-667	654	5	sons	son	NOUN
eajbcs-667	654	6	.	.	PUNCT
eajbcs-667	655	1	hoboken	hoboken	PROPN
eajbcs-667	655	2	,	,	PUNCT
eajbcs-667	655	3	nj	nj	PROPN
eajbcs-667	655	4	.	.	PUNCT
eajbcs-667	655	5	vary	vary	VERB
eajbcs-667	655	6	with	with	ADP
eajbcs-667	655	7	the	the	DET
eajbcs-667	655	8	time	time	NOUN
eajbcs-667	655	9	and	and	CCONJ
eajbcs-667	655	10	space	space	NOUN
eajbcs-667	655	11	of	of	ADP
eajbcs-667	655	12	the	the	DET
eajbcs-667	655	13	given	give	VERB
eajbcs-667	655	14	dimension	dimension	NOUN
eajbcs-667	655	15	.	.	PUNCT
eajbcs-667	656	1	the	the	DET
eajbcs-667	656	2	finite	finite	PROPN
eajbcs-667	656	3	element	element	NOUN
eajbcs-667	656	4	method	method	NOUN
eajbcs-667	656	5	descretizes	descretize	VERB
eajbcs-667	656	6	only	only	ADV
eajbcs-667	656	7	in	in	ADP
eajbcs-667	656	8	space	space	NOUN
eajbcs-667	656	9	and	and	CCONJ
eajbcs-667	656	10	the	the	DET
eajbcs-667	656	11	time	time	NOUN
eajbcs-667	656	12	descritization	descritization	NOUN
eajbcs-667	656	13	is	be	AUX
eajbcs-667	656	14	based	base	VERB
eajbcs-667	656	15	on	on	ADP
eajbcs-667	656	16	the	the	DET
eajbcs-667	656	17	finite	finite	ADJ
eajbcs-667	656	18	difference	difference	NOUN
eajbcs-667	656	19	methods	method	NOUN
eajbcs-667	656	20	.	.	PUNCT
eajbcs-667	657	1	hence	hence	ADV
eajbcs-667	657	2	the	the	DET
eajbcs-667	657	3	finite	finite	PROPN
eajbcs-667	657	4	element	element	NOUN
eajbcs-667	657	5	method	method	NOUN
eajbcs-667	657	6	needs	need	VERB
eajbcs-667	657	7	human	human	ADJ
eajbcs-667	657	8	power	power	NOUN
eajbcs-667	657	9	still	still	ADV
eajbcs-667	657	10	now	now	ADV
eajbcs-667	657	11	for	for	ADP
eajbcs-667	657	12	the	the	DET
eajbcs-667	657	13	descritization	descritization	NOUN
eajbcs-667	657	14	process	process	NOUN
eajbcs-667	657	15	of	of	ADP
eajbcs-667	657	16	time	time	NOUN
eajbcs-667	657	17	.	.	PUNCT
eajbcs-667	658	1	so	so	ADV
eajbcs-667	658	2	this	this	PRON
eajbcs-667	658	3	is	be	AUX
eajbcs-667	658	4	a	a	DET
eajbcs-667	658	5	very	very	ADV
eajbcs-667	658	6	interesting	interesting	ADJ
eajbcs-667	658	7	area	area	NOUN
eajbcs-667	658	8	to	to	PART
eajbcs-667	658	9	do	do	VERB
eajbcs-667	658	10	researches	research	NOUN
eajbcs-667	658	11	in	in	ADP
eajbcs-667	658	12	the	the	DET
eajbcs-667	658	13	future	future	NOUN
eajbcs-667	658	14	to	to	PART
eajbcs-667	658	15	include	include	VERB
eajbcs-667	658	16	the	the	DET
eajbcs-667	658	17	time	time	NOUN
eajbcs-667	658	18	descritization	descritization	NOUN
eajbcs-667	658	19	in	in	ADP
eajbcs-667	658	20	the	the	DET
eajbcs-667	658	21	method	method	NOUN
eajbcs-667	658	22	.	.	PUNCT
eajbcs-667	659	1	ahsan	ahsan	PROPN
eajbcs-667	659	2	m.	m.	PROPN
eajbcs-667	659	3	2012	2012	NUM
eajbcs-667	659	4	.	.	PUNCT
eajbcs-667	660	1	numerical	numerical	ADJ
eajbcs-667	660	2	solution	solution	NOUN
eajbcs-667	660	3	of	of	ADP
eajbcs-667	660	4	the	the	DET
eajbcs-667	660	5	advectiondiffusion	advectiondiffusion	NOUN
eajbcs-667	660	6	equation	equation	NOUN
eajbcs-667	660	7	using	use	VERB
eajbcs-667	660	8	laplace	laplace	NOUN
eajbcs-667	660	9	transform	transform	VERB
eajbcs-667	660	10	finite	finite	ADJ
eajbcs-667	660	11	analytical	analytical	ADJ
eajbcs-667	660	12	method	method	NOUN
eajbcs-667	660	13	.	.	PUNCT
eajbcs-667	661	1	int	int	NOUN
eajbcs-667	661	2	.	.	PUNCT
eajbcs-667	662	1	j.	j.	PROPN
eajbcs-667	662	2	river	river	PROPN
eajbcs-667	662	3	basin	basin	PROPN
eajbcs-667	662	4	manag	manag	NOUN
eajbcs-667	662	5	.	.	PUNCT
eajbcs-667	663	1	10(2	10(2	NUM
eajbcs-667	663	2	):	):	PUNCT
eajbcs-667	663	3	177188	177188	NUM
eajbcs-667	663	4	.	.	PUNCT
eajbcs-667	664	1	aragonés	aragonés	PROPN
eajbcs-667	664	2	l.	l.	PROPN
eajbcs-667	664	3	,	,	PUNCT
eajbcs-667	664	4	pagán	pagán	NOUN
eajbcs-667	664	5	j.i	j.i	PROPN
eajbcs-667	664	6	.	.	PROPN
eajbcs-667	664	7	,	,	PUNCT
eajbcs-667	664	8	lópez	lópez	PROPN
eajbcs-667	664	9	i.	i.	PROPN
eajbcs-667	664	10	,	,	PUNCT
eajbcs-667	664	11	navarrogonzález	navarrogonzález	PROPN
eajbcs-667	665	1	f.j	f.j	PROPN
eajbcs-667	665	2	.	.	PROPN
eajbcs-667	666	1	and	and	CCONJ
eajbcs-667	666	2	villacampa	villacampa	VERB
eajbcs-667	666	3	y.	y.	PROPN
eajbcs-667	666	4	2019	2019	NUM
eajbcs-667	666	5	.	.	PUNCT
eajbcs-667	667	1	galerkin	galerkin	PROPN
eajbcs-667	667	2	's	's	PART
eajbcs-667	667	3	formulation	formulation	NOUN
eajbcs-667	667	4	of	of	ADP
eajbcs-667	667	5	the	the	DET
eajbcs-667	667	6	finite	finite	ADJ
eajbcs-667	667	7	elements	element	NOUN
eajbcs-667	667	8	method	method	VERB
eajbcs-667	667	9	to	to	PART
eajbcs-667	667	10	obtain	obtain	VERB
eajbcs-667	667	11	the	the	DET
eajbcs-667	667	12	depth	depth	NOUN
eajbcs-667	667	13	of	of	ADP
eajbcs-667	667	14	closure	closure	NOUN
eajbcs-667	667	15	.	.	PUNCT
eajbcs-667	668	1	sci	sci	PROPN
eajbcs-667	668	2	.	.	PUNCT
eajbcs-667	668	3	total	total	PROPN
eajbcs-667	668	4	environ	environ	PROPN
eajbcs-667	668	5	.	.	PUNCT
eajbcs-667	669	1	660	660	NUM
eajbcs-667	669	2	:	:	PUNCT
eajbcs-667	669	3	1256	1256	NUM
eajbcs-667	669	4	-	-	SYM
eajbcs-667	669	5	1263	1263	NUM
eajbcs-667	669	6	.	.	PUNCT
eajbcs-667	669	7	bajellan	bajellan	PROPN
eajbcs-667	669	8	a.a.f	a.a.f	PROPN
eajbcs-667	669	9	.	.	PUNCT
eajbcs-667	670	1	2015	2015	NUM
eajbcs-667	670	2	.	.	PUNCT
eajbcs-667	671	1	computation	computation	NOUN
eajbcs-667	671	2	of	of	ADP
eajbcs-667	671	3	the	the	DET
eajbcs-667	671	4	convection	convection	NOUN
eajbcs-667	671	5	diffusion	diffusion	NOUN
eajbcs-667	671	6	equation	equation	NOUN
eajbcs-667	671	7	by	by	ADP
eajbcs-667	671	8	the	the	DET
eajbcs-667	671	9	fourth	fourth	ADJ
eajbcs-667	671	10	order	order	NOUN
eajbcs-667	671	11	compact	compact	ADJ
eajbcs-667	671	12	finite	finite	ADJ
eajbcs-667	671	13	difference	difference	NOUN
eajbcs-667	671	14	method	method	NOUN
eajbcs-667	671	15	.	.	PUNCT
eajbcs-667	672	1	doctoral	doctoral	ADJ
eajbcs-667	672	2	dissertation	dissertation	NOUN
eajbcs-667	672	3	,	,	PUNCT
eajbcs-667	672	4	izmir	izmir	PROPN
eajbcs-667	672	5	institute	institute	PROPN
eajbcs-667	672	6	of	of	ADP
eajbcs-667	672	7	technology	technology	PROPN
eajbcs-667	672	8	,	,	PUNCT
eajbcs-667	672	9	turkey	turkey	PROPN
eajbcs-667	672	10	.	.	PUNCT
eajbcs-667	673	1	bergara	bergara	PROPN
eajbcs-667	673	2	a.	a.	PROPN
eajbcs-667	673	3	2011	2011	NUM
eajbcs-667	673	4	.	.	PUNCT
eajbcs-667	674	1	finite	finite	PROPN
eajbcs-667	674	2	difference	difference	NOUN
eajbcs-667	674	3	numerical	numerical	ADJ
eajbcs-667	674	4	methods	method	NOUN
eajbcs-667	674	5	of	of	ADP
eajbcs-667	674	6	partial	partial	ADJ
eajbcs-667	674	7	differential	differential	ADJ
eajbcs-667	674	8	equations	equation	NOUN
eajbcs-667	674	9	in	in	ADP
eajbcs-667	674	10	finance	finance	NOUN
eajbcs-667	674	11	with	with	ADP
eajbcs-667	674	12	matlab	matlab	PROPN
eajbcs-667	674	13	.	.	PUNCT
eajbcs-667	675	1	master	master	PROPN
eajbcs-667	675	2	and	and	CCONJ
eajbcs-667	675	3	banca	banca	PROPN
eajbcs-667	675	4	.	.	PUNCT
eajbcs-667	676	1	new	new	PROPN
eajbcs-667	676	2	york	york	PROPN
eajbcs-667	676	3	,	,	PUNCT
eajbcs-667	676	4	springer	springer	NOUN
eajbcs-667	676	5	.	.	PUNCT
eajbcs-667	677	1	brenner	brenner	PROPN
eajbcs-667	677	2	s.c	s.c	PROPN
eajbcs-667	677	3	.	.	PROPN
eajbcs-667	677	4	,	,	PUNCT
eajbcs-667	677	5	scott	scott	PROPN
eajbcs-667	677	6	l.r	l.r	PROPN
eajbcs-667	677	7	.	.	PROPN
eajbcs-667	677	8	,	,	PUNCT
eajbcs-667	677	9	&	&	CCONJ
eajbcs-667	677	10	scott	scott	PROPN
eajbcs-667	677	11	l.r	l.r	PROPN
eajbcs-667	677	12	.	.	PROPN
eajbcs-667	677	13	2008	2008	NUM
eajbcs-667	677	14	.	.	PUNCT
eajbcs-667	678	1	the	the	DET
eajbcs-667	678	2	mathematical	mathematical	ADJ
eajbcs-667	678	3	theory	theory	NOUN
eajbcs-667	678	4	of	of	ADP
eajbcs-667	678	5	finite	finite	ADJ
eajbcs-667	678	6	element	element	NOUN
eajbcs-667	678	7	methods	method	NOUN
eajbcs-667	678	8	.	.	PUNCT
eajbcs-667	679	1	vol	vol	NOUN
eajbcs-667	679	2	,	,	PUNCT
eajbcs-667	679	3	pp	pp	ADJ
eajbcs-667	679	4	263	263	NUM
eajbcs-667	679	5	-	-	SYM
eajbcs-667	679	6	291	291	NUM
eajbcs-667	679	7	.	.	PUNCT
eajbcs-667	679	8	springer	springer	NOUN
eajbcs-667	679	9	,	,	PUNCT
eajbcs-667	679	10	new	new	PROPN
eajbcs-667	679	11	york	york	PROPN
eajbcs-667	679	12	.	.	PUNCT
eajbcs-667	680	1	donea	donea	PROPN
eajbcs-667	680	2	j.	j.	PROPN
eajbcs-667	680	3	and	and	CCONJ
eajbcs-667	680	4	huerta	huerta	PROPN
eajbcs-667	680	5	a.	a.	PROPN
eajbcs-667	680	6	2003	2003	NUM
eajbcs-667	680	7	.	.	PUNCT
eajbcs-667	681	1	finite	finite	PROPN
eajbcs-667	681	2	element	element	NOUN
eajbcs-667	681	3	methods	method	NOUN
eajbcs-667	681	4	for	for	ADP
eajbcs-667	681	5	flow	flow	NOUN
eajbcs-667	681	6	problems	problem	NOUN
eajbcs-667	681	7	.	.	PUNCT
eajbcs-667	682	1	john	john	PROPN
eajbcs-667	682	2	wiley	wiley	PROPN
eajbcs-667	682	3	&	&	CCONJ
eajbcs-667	682	4	sons	son	NOUN
eajbcs-667	682	5	.	.	PUNCT
eajbcs-667	683	1	74	74	NUM
eajbcs-667	683	2	the	the	DET
eajbcs-667	683	3	core	core	NOUN
eajbcs-667	683	4	of	of	ADP
eajbcs-667	683	5	this	this	DET
eajbcs-667	683	6	work	work	NOUN
eajbcs-667	683	7	was	be	AUX
eajbcs-667	683	8	completed	complete	VERB
eajbcs-667	683	9	during	during	ADP
eajbcs-667	683	10	the	the	DET
eajbcs-667	683	11	master	master	NOUN
eajbcs-667	683	12	study	study	NOUN
eajbcs-667	683	13	at	at	ADP
eajbcs-667	683	14	hawassa	hawassa	PROPN
eajbcs-667	683	15	university	university	PROPN
eajbcs-667	683	16	and	and	CCONJ
eajbcs-667	683	17	presented	present	VERB
eajbcs-667	683	18	to	to	ADP
eajbcs-667	683	19	former	former	ADJ
eajbcs-667	683	20	school	school	NOUN
eajbcs-667	683	21	of	of	ADP
eajbcs-667	683	22	mathematics	mathematic	NOUN
eajbcs-667	683	23	and	and	CCONJ
eajbcs-667	683	24	statistical	statistical	ADJ
eajbcs-667	683	25	science	science	NOUN
eajbcs-667	683	26	(	(	PUNCT
eajbcs-667	683	27	currently	currently	ADV
eajbcs-667	683	28	department	department	NOUN
eajbcs-667	683	29	of	of	ADP
eajbcs-667	683	30	mathematics	mathematic	NOUN
eajbcs-667	683	31	)	)	PUNCT
eajbcs-667	683	32	in	in	ADP
eajbcs-667	683	33	partial	partial	ADJ
eajbcs-667	683	34	fulfillment	fulfillment	NOUN
eajbcs-667	683	35	of	of	ADP
eajbcs-667	683	36	the	the	DET
eajbcs-667	683	37	requirement	requirement	NOUN
eajbcs-667	683	38	for	for	ADP
eajbcs-667	683	39	the	the	DET
eajbcs-667	683	40	degree	degree	NOUN
eajbcs-667	683	41	of	of	ADP
eajbcs-667	683	42	master	master	NOUN
eajbcs-667	683	43	.	.	PUNCT
eajbcs-667	684	1	the	the	DET
eajbcs-667	684	2	authers	auther	NOUN
eajbcs-667	684	3	would	would	AUX
eajbcs-667	684	4	like	like	VERB
eajbcs-667	684	5	to	to	PART
eajbcs-667	684	6	thank	thank	VERB
eajbcs-667	684	7	the	the	DET
eajbcs-667	684	8	department	department	NOUN
eajbcs-667	684	9	of	of	ADP
eajbcs-667	684	10	mathematics	mathematics	PROPN
eajbcs-667	684	11	at	at	ADP
eajbcs-667	684	12	the	the	DET
eajbcs-667	684	13	university	university	PROPN
eajbcs-667	684	14	of	of	ADP
eajbcs-667	684	15	hawassa	hawassa	PROPN
eajbcs-667	684	16	university	university	PROPN
eajbcs-667	684	17	.	.	PUNCT
eajbcs-667	685	1	this	this	DET
eajbcs-667	685	2	work	work	NOUN
eajbcs-667	685	3	of	of	ADP
eajbcs-667	685	4	has	have	AUX
eajbcs-667	685	5	been	be	AUX
eajbcs-667	685	6	partially	partially	ADV
eajbcs-667	685	7	supported	support	VERB
eajbcs-667	685	8	by	by	ADP
eajbcs-667	685	9	hawassa	hawassa	PROPN
eajbcs-667	685	10	university	university	NOUN
eajbcs-667	685	11	.	.	PUNCT
eajbcs-667	686	1	references	reference	NOUN
eajbcs-667	686	2	acknowledgments	acknowledgment	NOUN
eajbcs-667	686	3	juliet	juliet	ADJ
eajbcs-667	686	4	and	and	CCONJ
eajbcs-667	686	5	florence	florence	NOUN
eajbcs-667	686	6	_	_	PUNCT
eajbcs-667	686	7	beef	beef	NOUN
eajbcs-667	686	8	handling	handling	NOUN
eajbcs-667	686	9	practices	practice	NOUN
eajbcs-667	686	10	at	at	ADP
eajbcs-667	686	11	abattoir	abattoir	NOUN
eajbcs-667	686	12	and	and	CCONJ
eajbcs-667	686	13	bucher	bucher	PROPN
eajbcs-667	686	14	shops_doi.pdf	shops_doi.pdf	NOUN
eajbcs-667	686	15	(	(	PUNCT
eajbcs-667	686	16	p.1	p.1	NOUN
eajbcs-667	686	17	-	-	PUNCT
eajbcs-667	686	18	17	17	NUM
eajbcs-667	686	19	)	)	PUNCT
eajbcs-667	686	20	amanuel	amanuel	NOUN
eajbcs-667	686	21	_	_	PUNCT
eajbcs-667	687	1	study	study	NOUN
eajbcs-667	687	2	of	of	ADP
eajbcs-667	687	3	reaction	reaction	NOUN
eajbcs-667	687	4	mechanisms	mechanism	NOUN
eajbcs-667	687	5	in	in	ADP
eajbcs-667	687	6	α	α	PROPN
eajbcs-667	687	7	+	+	CCONJ
eajbcs-667	687	8	69ga	69ga	ADJ
eajbcs-667	687	9	reaction	reaction	NOUN
eajbcs-667	687	10	at	at	ADP
eajbcs-667	687	11	≈10	≈10	PROPN
eajbcs-667	687	12	–	–	PUNCT
eajbcs-667	687	13	50	50	NUM
eajbcs-667	687	14	mev_doi.pdf	mev_doi.pdf	NOUN
eajbcs-667	687	15	(	(	PUNCT
eajbcs-667	687	16	p.18	p.18	NOUN
eajbcs-667	687	17	-	-	ADJ
eajbcs-667	687	18	27	27	NUM
eajbcs-667	687	19	)	)	PUNCT
eajbcs-667	687	20	buchale	buchale	NOUN
eajbcs-667	687	21	and	and	CCONJ
eajbcs-667	687	22	atnafu_population	atnafu_population	NOUN
eajbcs-667	687	23	dynamics	dynamic	NOUN
eajbcs-667	687	24	and	and	CCONJ
eajbcs-667	687	25	yield	yield	NOUN
eajbcs-667	687	26	estimation	estimation	NOUN
eajbcs-667	687	27	of	of	ADP
eajbcs-667	687	28	common	common	ADJ
eajbcs-667	687	29	carp.pdf	carp.pdf	X
eajbcs-667	687	30	(	(	PUNCT
eajbcs-667	687	31	p.28	p.28	NOUN
eajbcs-667	687	32	-	-	PUNCT
eajbcs-667	687	33	41	41	NUM
eajbcs-667	687	34	)	)	PUNCT
eajbcs-667	687	35	haftom	haftom	NOUN
eajbcs-667	687	36	et	et	NOUN
eajbcs-667	687	37	al_synthesis	al_synthesis	VERB
eajbcs-667	687	38	characterization	characterization	NOUN
eajbcs-667	687	39	and	and	CCONJ
eajbcs-667	687	40	antibacterial	antibacterial	ADJ
eajbcs-667	687	41	activity	activity	NOUN
eajbcs-667	687	42	of	of	ADP
eajbcs-667	687	43	benzimidazole.pdf	benzimidazole.pdf	X
eajbcs-667	687	44	(	(	PUNCT
eajbcs-667	687	45	p.42	p.42	NOUN
eajbcs-667	687	46	-	-	PUNCT
eajbcs-667	687	47	51	51	NUM
eajbcs-667	687	48	)	)	PUNCT
eajbcs-667	687	49	kassahun	kassahun	PROPN
eajbcs-667	687	50	and	and	CCONJ
eajbcs-667	687	51	zerihun	zerihun	PROPN
eajbcs-667	687	52	_	_	PRON
eajbcs-667	687	53	numerical	numerical	ADJ
eajbcs-667	687	54	solutions	solution	NOUN
eajbcs-667	687	55	of	of	ADP
eajbcs-667	687	56	advection	advection	NOUN
eajbcs-667	687	57	diffusion	diffusion	NOUN
eajbcs-667	687	58	equations.pdf	equations.pdf	NOUN
eajbcs-667	687	59	(	(	PUNCT
eajbcs-667	687	60	p.52	p.52	NOUN
eajbcs-667	687	61	-	-	ADJ
eajbcs-667	687	62	74	74	NUM
eajbcs-667	687	63	)	)	PUNCT