id	sid	tid	token	lemma	pos
iajs-4097	1	1	376	376	NUM
iajs-4097	1	2	©	©	ADP
iajs-4097	1	3	2025	2025	NUM
iajs-4097	1	4	the	the	DET
iajs-4097	1	5	author(s	author(s	NOUN
iajs-4097	1	6	)	)	PUNCT
iajs-4097	1	7	.	.	PUNCT
iajs-4097	2	1	published	publish	VERB
iajs-4097	2	2	by	by	ADP
iajs-4097	2	3	college	college	NOUN
iajs-4097	2	4	of	of	ADP
iajs-4097	2	5	education	education	NOUN
iajs-4097	2	6	for	for	ADP
iajs-4097	2	7	pure	pure	ADJ
iajs-4097	2	8	science	science	NOUN
iajs-4097	2	9	(	(	PUNCT
iajs-4097	2	10	ibn	ibn	PROPN
iajs-4097	2	11	al	al	PROPN
iajs-4097	2	12	-	-	PUNCT
iajs-4097	2	13	haitham	haitham	PROPN
iajs-4097	2	14	)	)	PUNCT
iajs-4097	2	15	,	,	PUNCT
iajs-4097	2	16	university	university	NOUN
iajs-4097	2	17	of	of	ADP
iajs-4097	2	18	baghdad	baghdad	PROPN
iajs-4097	2	19	.	.	PUNCT
iajs-4097	3	1	this	this	PRON
iajs-4097	3	2	is	be	AUX
iajs-4097	3	3	an	an	DET
iajs-4097	3	4	open	open	ADJ
iajs-4097	3	5	-	-	PUNCT
iajs-4097	3	6	access	access	NOUN
iajs-4097	3	7	article	article	NOUN
iajs-4097	3	8	distributed	distribute	VERB
iajs-4097	3	9	under	under	ADP
iajs-4097	3	10	the	the	DET
iajs-4097	3	11	terms	term	NOUN
iajs-4097	3	12	of	of	ADP
iajs-4097	3	13	the	the	DET
iajs-4097	3	14	creative	creative	ADJ
iajs-4097	3	15	commons	common	NOUN
iajs-4097	3	16	attribution	attribution	NOUN
iajs-4097	3	17	4.0	4.0	NUM
iajs-4097	3	18	international	international	ADJ
iajs-4097	3	19	license	license	NOUN
iajs-4097	3	20	a	a	DET
iajs-4097	3	21	numerical	numerical	ADJ
iajs-4097	3	22	study	study	NOUN
iajs-4097	3	23	of	of	ADP
iajs-4097	3	24	initial	initial	ADJ
iajs-4097	3	25	and	and	CCONJ
iajs-4097	3	26	boundary	boundary	ADJ
iajs-4097	3	27	conditions	condition	NOUN
iajs-4097	3	28	problem	problem	NOUN
iajs-4097	3	29	for	for	ADP
iajs-4097	3	30	wave	wave	NOUN
iajs-4097	3	31	equation	equation	NOUN
iajs-4097	3	32	using	use	VERB
iajs-4097	3	33	the	the	DET
iajs-4097	3	34	operational	operational	ADJ
iajs-4097	3	35	matrices	matrix	NOUN
iajs-4097	3	36	myasar	myasar	VERB
iajs-4097	3	37	obaid	obaid	PROPN
iajs-4097	3	38	enadi	enadi	PROPN
iajs-4097	3	39	1	1	NUM
iajs-4097	3	40	*	*	PUNCT
iajs-4097	3	41	and	and	CCONJ
iajs-4097	3	42	majeed	majeed	PROPN
iajs-4097	3	43	a.	a.	PROPN
iajs-4097	3	44	al	al	PROPN
iajs-4097	3	45	-	-	PUNCT
iajs-4097	3	46	jawary	jawary	ADJ
iajs-4097	3	47	2	2	NUM
iajs-4097	3	48	1,2	1,2	NUM
iajs-4097	3	49	department	department	NOUN
iajs-4097	3	50	of	of	ADP
iajs-4097	3	51	mathematics	mathematic	NOUN
iajs-4097	3	52	,	,	PUNCT
iajs-4097	3	53	college	college	NOUN
iajs-4097	3	54	of	of	ADP
iajs-4097	3	55	education	education	NOUN
iajs-4097	3	56	for	for	ADP
iajs-4097	3	57	pure	pure	ADJ
iajs-4097	3	58	sciences	science	NOUN
iajs-4097	3	59	(	(	PUNCT
iajs-4097	3	60	ibn	ibn	PROPN
iajs-4097	3	61	al	al	PROPN
iajs-4097	3	62	-	-	PUNCT
iajs-4097	3	63	haitham	haitham	PROPN
iajs-4097	3	64	)	)	PUNCT
iajs-4097	3	65	,	,	PUNCT
iajs-4097	3	66	university	university	NOUN
iajs-4097	3	67	of	of	ADP
iajs-4097	3	68	baghdad	baghdad	PROPN
iajs-4097	3	69	,	,	PUNCT
iajs-4097	3	70	baghdad	baghdad	PROPN
iajs-4097	3	71	,	,	PUNCT
iajs-4097	3	72	iraq	iraq	PROPN
iajs-4097	3	73	.	.	PUNCT
iajs-4097	4	1	*	*	PUNCT
iajs-4097	4	2	corresponding	correspond	VERB
iajs-4097	4	3	author	author	NOUN
iajs-4097	4	4	.	.	PUNCT
iajs-4097	5	1	received	receive	VERB
iajs-4097	5	2	:	:	PUNCT
iajs-4097	5	3	6	6	NUM
iajs-4097	5	4	december	december	PROPN
iajs-4097	5	5	2024	2024	NUM
iajs-4097	5	6	accepted	accept	VERB
iajs-4097	5	7	:	:	PUNCT
iajs-4097	5	8	12	12	NUM
iajs-4097	5	9	may	may	AUX
iajs-4097	5	10	2025	2025	NUM
iajs-4097	5	11	published	publish	VERB
iajs-4097	5	12	:	:	PUNCT
iajs-4097	5	13	20	20	NUM
iajs-4097	5	14	october	october	NOUN
iajs-4097	5	15	2025	2025	NUM
iajs-4097	5	16	doi.org/10.30526/38.4.4097	doi.org/10.30526/38.4.4097	NOUN
iajs-4097	5	17	abstract	abstract	ADJ
iajs-4097	5	18	in	in	ADP
iajs-4097	5	19	this	this	DET
iajs-4097	5	20	paper	paper	NOUN
iajs-4097	5	21	,	,	PUNCT
iajs-4097	5	22	orthogonal	orthogonal	ADJ
iajs-4097	5	23	polynomials	polynomial	NOUN
iajs-4097	5	24	and	and	CCONJ
iajs-4097	5	25	their	their	PRON
iajs-4097	5	26	operational	operational	ADJ
iajs-4097	5	27	matrices	matrix	NOUN
iajs-4097	5	28	will	will	AUX
iajs-4097	5	29	be	be	AUX
iajs-4097	5	30	utilized	utilize	VERB
iajs-4097	5	31	to	to	PART
iajs-4097	5	32	address	address	VERB
iajs-4097	5	33	the	the	DET
iajs-4097	5	34	initial	initial	ADJ
iajs-4097	5	35	and	and	CCONJ
iajs-4097	5	36	boundary	boundary	ADJ
iajs-4097	5	37	value	value	NOUN
iajs-4097	5	38	problems	problem	NOUN
iajs-4097	5	39	of	of	ADP
iajs-4097	5	40	the	the	DET
iajs-4097	5	41	one	one	NUM
iajs-4097	5	42	-	-	PUNCT
iajs-4097	5	43	dimensional	dimensional	ADJ
iajs-4097	5	44	wave	wave	NOUN
iajs-4097	5	45	problem	problem	NOUN
iajs-4097	5	46	,	,	PUNCT
iajs-4097	5	47	where	where	SCONJ
iajs-4097	5	48	the	the	DET
iajs-4097	5	49	domain	domain	NOUN
iajs-4097	5	50	of	of	ADP
iajs-4097	5	51	the	the	DET
iajs-4097	5	52	space	space	NOUN
iajs-4097	5	53	variable	variable	NOUN
iajs-4097	5	54	is	be	AUX
iajs-4097	5	55	bounded	bound	VERB
iajs-4097	5	56	,	,	PUNCT
iajs-4097	5	57	which	which	PRON
iajs-4097	5	58	covers	cover	VERB
iajs-4097	5	59	a	a	DET
iajs-4097	5	60	variety	variety	NOUN
iajs-4097	5	61	of	of	ADP
iajs-4097	5	62	scientific	scientific	ADJ
iajs-4097	5	63	and	and	CCONJ
iajs-4097	5	64	engineering	engineering	NOUN
iajs-4097	5	65	operations	operation	NOUN
iajs-4097	5	66	.	.	PUNCT
iajs-4097	6	1	six	six	NUM
iajs-4097	6	2	types	type	NOUN
iajs-4097	6	3	of	of	ADP
iajs-4097	6	4	orthogonal	orthogonal	ADJ
iajs-4097	6	5	polynomials	polynomial	NOUN
iajs-4097	6	6	,	,	PUNCT
iajs-4097	6	7	include	include	VERB
iajs-4097	6	8	instead	instead	ADV
iajs-4097	6	9	of	of	ADP
iajs-4097	6	10	such	such	ADJ
iajs-4097	6	11	as	as	ADP
iajs-4097	6	12	the	the	DET
iajs-4097	6	13	genocchi	genocchi	NOUN
iajs-4097	6	14	,	,	PUNCT
iajs-4097	6	15	bernoulli	bernoulli	PROPN
iajs-4097	6	16	,	,	PUNCT
iajs-4097	6	17	legendre	legendre	PROPN
iajs-4097	6	18	,	,	PUNCT
iajs-4097	6	19	boubaker	boubaker	PROPN
iajs-4097	6	20	,	,	PUNCT
iajs-4097	6	21	chebyshev	chebyshev	NOUN
iajs-4097	6	22	and	and	CCONJ
iajs-4097	6	23	standard	standard	ADJ
iajs-4097	6	24	polynomials	polynomial	NOUN
iajs-4097	6	25	.	.	PUNCT
iajs-4097	7	1	the	the	DET
iajs-4097	7	2	linear	linear	ADJ
iajs-4097	7	3	problem	problem	NOUN
iajs-4097	7	4	with	with	ADP
iajs-4097	7	5	its	its	PRON
iajs-4097	7	6	initial	initial	ADJ
iajs-4097	7	7	and	and	CCONJ
iajs-4097	7	8	boundary	boundary	ADJ
iajs-4097	7	9	conditions	condition	NOUN
iajs-4097	7	10	are	be	AUX
iajs-4097	7	11	transformed	transform	VERB
iajs-4097	7	12	to	to	ADP
iajs-4097	7	13	a	a	DET
iajs-4097	7	14	linear	linear	ADJ
iajs-4097	7	15	algebraic	algebraic	ADJ
iajs-4097	7	16	equations	equation	NOUN
iajs-4097	7	17	,	,	PUNCT
iajs-4097	7	18	which	which	PRON
iajs-4097	7	19	can	can	AUX
iajs-4097	7	20	then	then	ADV
iajs-4097	7	21	be	be	AUX
iajs-4097	7	22	solved	solve	VERB
iajs-4097	7	23	by	by	ADP
iajs-4097	7	24	utilizing	utilize	VERB
iajs-4097	7	25	to	to	PART
iajs-4097	7	26	get	get	VERB
iajs-4097	7	27	an	an	DET
iajs-4097	7	28	approximate	approximate	ADJ
iajs-4097	7	29	solution	solution	NOUN
iajs-4097	7	30	for	for	ADP
iajs-4097	7	31	this	this	DET
iajs-4097	7	32	problem	problem	NOUN
iajs-4097	7	33	.	.	PUNCT
iajs-4097	8	1	some	some	DET
iajs-4097	8	2	test	test	NOUN
iajs-4097	8	3	problems	problem	NOUN
iajs-4097	8	4	related	relate	VERB
iajs-4097	8	5	to	to	ADP
iajs-4097	8	6	the	the	DET
iajs-4097	8	7	one	one	NUM
iajs-4097	8	8	-	-	PUNCT
iajs-4097	8	9	dimensional	dimensional	ADJ
iajs-4097	8	10	wave	wave	NOUN
iajs-4097	8	11	equation	equation	NOUN
iajs-4097	8	12	with	with	ADP
iajs-4097	8	13	different	different	ADJ
iajs-4097	8	14	conditions	condition	NOUN
iajs-4097	8	15	are	be	AUX
iajs-4097	8	16	discussed	discuss	VERB
iajs-4097	8	17	and	and	CCONJ
iajs-4097	8	18	solved	solve	VERB
iajs-4097	8	19	to	to	PART
iajs-4097	8	20	show	show	VERB
iajs-4097	8	21	how	how	SCONJ
iajs-4097	8	22	reliable	reliable	ADJ
iajs-4097	8	23	and	and	CCONJ
iajs-4097	8	24	efficient	efficient	ADJ
iajs-4097	8	25	the	the	DET
iajs-4097	8	26	proposed	propose	VERB
iajs-4097	8	27	methods	method	NOUN
iajs-4097	8	28	.	.	PUNCT
iajs-4097	9	1	the	the	DET
iajs-4097	9	2	error	error	NOUN
iajs-4097	9	3	norm	norm	NOUN
iajs-4097	9	4	and	and	CCONJ
iajs-4097	9	5	the	the	DET
iajs-4097	9	6	mean	mean	ADJ
iajs-4097	9	7	square	square	ADJ
iajs-4097	9	8	error	error	NOUN
iajs-4097	9	9	,	,	PUNCT
iajs-4097	9	10	were	be	AUX
iajs-4097	9	11	computed	compute	VERB
iajs-4097	9	12	;	;	PUNCT
iajs-4097	9	13	these	these	PRON
iajs-4097	9	14	are	be	AUX
iajs-4097	9	15	presented	present	VERB
iajs-4097	9	16	through	through	ADP
iajs-4097	9	17	analytical	analytical	ADJ
iajs-4097	9	18	tables	table	NOUN
iajs-4097	9	19	and	and	CCONJ
iajs-4097	9	20	graphics	graphic	NOUN
iajs-4097	9	21	showing	show	VERB
iajs-4097	9	22	the	the	DET
iajs-4097	9	23	rapid	rapid	ADJ
iajs-4097	9	24	convergence	convergence	NOUN
iajs-4097	9	25	for	for	ADP
iajs-4097	9	26	these	these	DET
iajs-4097	9	27	methods	method	NOUN
iajs-4097	9	28	.	.	PUNCT
iajs-4097	10	1	keywords	keyword	NOUN
iajs-4097	10	2	:	:	PUNCT
iajs-4097	10	3	wave	wave	NOUN
iajs-4097	10	4	problem	problem	NOUN
iajs-4097	10	5	;	;	PUNCT
iajs-4097	10	6	operational	operational	ADJ
iajs-4097	10	7	matrices	matrix	NOUN
iajs-4097	10	8	,	,	PUNCT
iajs-4097	10	9	orthogonal	orthogonal	ADJ
iajs-4097	10	10	polynomials	polynomial	NOUN
iajs-4097	10	11	,	,	PUNCT
iajs-4097	10	12	approximate	approximate	ADJ
iajs-4097	10	13	solutions	solution	NOUN
iajs-4097	10	14	.	.	PUNCT
iajs-4097	11	1	1	1	X
iajs-4097	11	2	.	.	X
iajs-4097	11	3	introduction	introduction	NOUN
iajs-4097	11	4	the	the	DET
iajs-4097	11	5	mathematical	mathematical	ADJ
iajs-4097	11	6	foundation	foundation	NOUN
iajs-4097	11	7	for	for	ADP
iajs-4097	11	8	many	many	ADJ
iajs-4097	11	9	problems	problem	NOUN
iajs-4097	11	10	in	in	ADP
iajs-4097	11	11	mathematics	mathematic	NOUN
iajs-4097	11	12	,	,	PUNCT
iajs-4097	11	13	physics	physics	NOUN
iajs-4097	11	14	,	,	PUNCT
iajs-4097	11	15	engineering	engineering	NOUN
iajs-4097	11	16	,	,	PUNCT
iajs-4097	11	17	and	and	CCONJ
iajs-4097	11	18	chemistry	chemistry	NOUN
iajs-4097	11	19	can	can	AUX
iajs-4097	11	20	be	be	AUX
iajs-4097	11	21	explained	explain	VERB
iajs-4097	11	22	mathematically	mathematically	ADV
iajs-4097	11	23	using	use	VERB
iajs-4097	11	24	partial	partial	ADJ
iajs-4097	11	25	differential	differential	ADJ
iajs-4097	11	26	equations	equation	NOUN
iajs-4097	11	27	(	(	PUNCT
iajs-4097	11	28	pdes	pde	NOUN
iajs-4097	11	29	)	)	PUNCT
iajs-4097	11	30	.	.	PUNCT
iajs-4097	12	1	in	in	ADP
iajs-4097	12	2	physics	physics	PROPN
iajs-4097	12	3	,	,	PUNCT
iajs-4097	12	4	for	for	ADP
iajs-4097	12	5	example	example	NOUN
iajs-4097	12	6	,	,	PUNCT
iajs-4097	12	7	partial	partial	ADJ
iajs-4097	12	8	differential	differential	ADJ
iajs-4097	12	9	equations	equation	NOUN
iajs-4097	12	10	clear	clear	ADJ
iajs-4097	12	11	describe	describe	VERB
iajs-4097	12	12	wave	wave	NOUN
iajs-4097	12	13	propagation	propagation	NOUN
iajs-4097	12	14	and	and	CCONJ
iajs-4097	12	15	heat	heat	NOUN
iajs-4097	12	16	transfer	transfer	NOUN
iajs-4097	12	17	.	.	PUNCT
iajs-4097	13	1	furthermore	furthermore	ADV
iajs-4097	13	2	,	,	PUNCT
iajs-4097	13	3	partial	partial	ADJ
iajs-4097	13	4	differential	differential	NOUN
iajs-4097	13	5	equations	equation	NOUN
iajs-4097	13	6	were	be	AUX
iajs-4097	13	7	utilized	utilize	VERB
iajs-4097	13	8	to	to	PART
iajs-4097	13	9	describe	describe	VERB
iajs-4097	13	10	the	the	DET
iajs-4097	13	11	majority	majority	NOUN
iajs-4097	13	12	of	of	ADP
iajs-4097	13	13	physical	physical	ADJ
iajs-4097	13	14	processes	process	NOUN
iajs-4097	13	15	in	in	ADP
iajs-4097	13	16	domains	domain	NOUN
iajs-4097	13	17	such	such	ADJ
iajs-4097	13	18	as	as	ADP
iajs-4097	13	19	electricity	electricity	NOUN
iajs-4097	13	20	,	,	PUNCT
iajs-4097	13	21	quantum	quantum	NOUN
iajs-4097	13	22	mechanics	mechanic	NOUN
iajs-4097	13	23	,	,	PUNCT
iajs-4097	13	24	wave	wave	NOUN
iajs-4097	13	25	propagation	propagation	NOUN
iajs-4097	13	26	in	in	ADP
iajs-4097	13	27	shallow	shallow	ADJ
iajs-4097	13	28	water	water	NOUN
iajs-4097	13	29	,	,	PUNCT
iajs-4097	13	30	fluid	fluid	ADJ
iajs-4097	13	31	dynamics	dynamic	NOUN
iajs-4097	13	32	,	,	PUNCT
iajs-4097	13	33	plasma	plasma	NOUN
iajs-4097	13	34	physics	physics	NOUN
iajs-4097	13	35	,	,	PUNCT
iajs-4097	13	36	and	and	CCONJ
iajs-4097	13	37	others	other	NOUN
iajs-4097	13	38	(	(	PUNCT
iajs-4097	13	39	1	1	NUM
iajs-4097	13	40	)	)	PUNCT
iajs-4097	13	41	.	.	PUNCT
iajs-4097	14	1	there	there	PRON
iajs-4097	14	2	is	be	VERB
iajs-4097	14	3	a	a	DET
iajs-4097	14	4	need	need	NOUN
iajs-4097	14	5	to	to	PART
iajs-4097	14	6	find	find	VERB
iajs-4097	14	7	effective	effective	ADJ
iajs-4097	14	8	and	and	CCONJ
iajs-4097	14	9	reliable	reliable	ADJ
iajs-4097	14	10	approximate	approximate	ADJ
iajs-4097	14	11	or	or	CCONJ
iajs-4097	14	12	analytical	analytical	ADJ
iajs-4097	14	13	techniques	technique	NOUN
iajs-4097	14	14	that	that	PRON
iajs-4097	14	15	can	can	AUX
iajs-4097	14	16	handle	handle	VERB
iajs-4097	14	17	pdes	pde	NOUN
iajs-4097	14	18	because	because	SCONJ
iajs-4097	14	19	of	of	ADP
iajs-4097	14	20	these	these	DET
iajs-4097	14	21	massive	massive	ADJ
iajs-4097	14	22	applications	application	NOUN
iajs-4097	14	23	,	,	PUNCT
iajs-4097	14	24	there	there	PRON
iajs-4097	14	25	is	be	VERB
iajs-4097	14	26	a	a	DET
iajs-4097	14	27	need	need	NOUN
iajs-4097	14	28	to	to	PART
iajs-4097	14	29	develop	develop	VERB
iajs-4097	14	30	effective	effective	ADJ
iajs-4097	14	31	and	and	CCONJ
iajs-4097	14	32	trustworthy	trustworthy	ADJ
iajs-4097	14	33	approximate	approximate	ADJ
iajs-4097	14	34	or	or	CCONJ
iajs-4097	14	35	analytical	analytical	ADJ
iajs-4097	14	36	procedures	procedure	NOUN
iajs-4097	14	37	for	for	ADP
iajs-4097	14	38	dealing	deal	VERB
iajs-4097	14	39	with	with	ADP
iajs-4097	14	40	pdes	pde	NOUN
iajs-4097	14	41	.	.	PUNCT
iajs-4097	15	1	many	many	ADJ
iajs-4097	15	2	mathematicians	mathematician	NOUN
iajs-4097	15	3	and	and	CCONJ
iajs-4097	15	4	engineers	engineer	NOUN
iajs-4097	15	5	have	have	AUX
iajs-4097	15	6	solved	solve	VERB
iajs-4097	15	7	a	a	DET
iajs-4097	15	8	wide	wide	ADJ
iajs-4097	15	9	range	range	NOUN
iajs-4097	15	10	of	of	ADP
iajs-4097	15	11	functional	functional	ADJ
iajs-4097	15	12	equations	equation	NOUN
iajs-4097	15	13	using	use	VERB
iajs-4097	15	14	the	the	DET
iajs-4097	15	15	vim	vim	NOUN
iajs-4097	15	16	by	by	ADP
iajs-4097	15	17	he	he	PRON
iajs-4097	15	18	(	(	PUNCT
iajs-4097	15	19	2–4	2–4	NUM
iajs-4097	15	20	)	)	PUNCT
iajs-4097	15	21	.	.	PUNCT
iajs-4097	16	1	many	many	ADJ
iajs-4097	16	2	tapes	tape	NOUN
iajs-4097	16	3	of	of	ADP
iajs-4097	16	4	scientific	scientific	ADJ
iajs-4097	16	5	,	,	PUNCT
iajs-4097	16	6	physical	physical	ADJ
iajs-4097	16	7	,	,	PUNCT
iajs-4097	16	8	and	and	CCONJ
iajs-4097	16	9	engineering	engineering	NOUN
iajs-4097	16	10	problems	problem	NOUN
iajs-4097	16	11	can	can	AUX
iajs-4097	16	12	be	be	AUX
iajs-4097	16	13	classified	classify	VERB
iajs-4097	16	14	as	as	ADP
iajs-4097	16	15	initial	initial	ADJ
iajs-4097	16	16	boundary	boundary	ADJ
iajs-4097	16	17	value	value	NOUN
iajs-4097	16	18	problems	problem	NOUN
iajs-4097	16	19	(	(	PUNCT
iajs-4097	16	20	ibvp	ibvp	PROPN
iajs-4097	16	21	)	)	PUNCT
iajs-4097	16	22	.	.	PUNCT
iajs-4097	17	1	with	with	ADP
iajs-4097	17	2	a	a	DET
iajs-4097	17	3	few	few	ADJ
iajs-4097	17	4	exceptions	exception	NOUN
iajs-4097	17	5	,	,	PUNCT
iajs-4097	17	6	we	we	PRON
iajs-4097	17	7	are	be	AUX
iajs-4097	17	8	unable	unable	ADJ
iajs-4097	17	9	to	to	PART
iajs-4097	17	10	find	find	VERB
iajs-4097	17	11	accurate	accurate	ADJ
iajs-4097	17	12	analytical	analytical	ADJ
iajs-4097	17	13	answers	answer	NOUN
iajs-4097	17	14	to	to	ADP
iajs-4097	17	15	the	the	DET
iajs-4097	17	16	majority	majority	NOUN
iajs-4097	17	17	of	of	ADP
iajs-4097	17	18	these	these	DET
iajs-4097	17	19	difficulties	difficulty	NOUN
iajs-4097	17	20	.	.	PUNCT
iajs-4097	18	1	there	there	PRON
iajs-4097	18	2	have	have	AUX
iajs-4097	18	3	been	be	AUX
iajs-4097	18	4	several	several	ADJ
iajs-4097	18	5	attempts	attempt	NOUN
iajs-4097	18	6	to	to	PART
iajs-4097	18	7	https://orcid.org/0009-0005-2426-4558	https://orcid.org/0009-0005-2426-4558	VERB
iajs-4097	18	8	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-4097	18	9	https://orcid.org/0000-0003-3967-0012	https://orcid.org/0000-0003-3967-0012	PROPN
iajs-4097	18	10	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4097	18	11	https://orcid.org/0009-0005-2426-4558	https://orcid.org/0009-0005-2426-4558	VERB
iajs-4097	18	12	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-4097	18	13	https://orcid.org/0000-0003-3967-0012	https://orcid.org/0000-0003-3967-0012	PROPN
iajs-4097	18	14	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4097	18	15	https://orcid.org/0009-0005-2426-4558	https://orcid.org/0009-0005-2426-4558	VERB
iajs-4097	18	16	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-4097	18	17	https://orcid.org/0000-0003-3967-0012	https://orcid.org/0000-0003-3967-0012	PROPN
iajs-4097	18	18	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4097	18	19	https://orcid.org/0009-0005-2426-4558	https://orcid.org/0009-0005-2426-4558	VERB
iajs-4097	18	20	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-4097	18	21	https://orcid.org/0000-0003-3967-0012	https://orcid.org/0000-0003-3967-0012	PROPN
iajs-4097	18	22	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4097	18	23	https://orcid.org/0009-0005-2426-4558	https://orcid.org/0009-0005-2426-4558	VERB
iajs-4097	18	24	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-4097	18	25	https://orcid.org/0000-0003-3967-0012	https://orcid.org/0000-0003-3967-0012	PROPN
iajs-4097	18	26	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-4097	18	27	https://orcid.org/0009-0005-2426-4558	https://orcid.org/0009-0005-2426-4558	VERB
iajs-4097	18	28	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	mailto:moysar.obaid1103a@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-4097	18	29	https://orcid.org/0000-0003-3967-0012	https://orcid.org/0000-0003-3967-0012	PROPN
iajs-4097	18	30	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	mailto:majeed.a.w@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-4097	18	31	ihjpas	ihjpa	NOUN
iajs-4097	18	32	.	.	PUNCT
iajs-4097	19	1	2025,38(4	2025,38(4	NOUN
iajs-4097	19	2	)	)	PUNCT
iajs-4097	19	3	377	377	NUM
iajs-4097	19	4	develop	develop	VERB
iajs-4097	19	5	approximate	approximate	ADJ
iajs-4097	19	6	and	and	CCONJ
iajs-4097	19	7	analytical	analytical	ADJ
iajs-4097	19	8	methods	method	NOUN
iajs-4097	19	9	for	for	ADP
iajs-4097	19	10	solving	solve	VERB
iajs-4097	19	11	the	the	DET
iajs-4097	19	12	non	non	ADJ
iajs-4097	19	13	-	-	ADJ
iajs-4097	19	14	linear	linear	ADJ
iajs-4097	19	15	diffusion	diffusion	NOUN
iajs-4097	19	16	and	and	CCONJ
iajs-4097	19	17	wave	wave	NOUN
iajs-4097	19	18	problems	problem	NOUN
iajs-4097	19	19	,	,	PUNCT
iajs-4097	19	20	see	see	VERB
iajs-4097	19	21	(	(	PUNCT
iajs-4097	19	22	5–8	5–8	NOUN
iajs-4097	19	23	)	)	PUNCT
iajs-4097	19	24	.	.	PUNCT
iajs-4097	20	1	wave	wave	NOUN
iajs-4097	20	2	equations	equation	NOUN
iajs-4097	20	3	can	can	AUX
iajs-4097	20	4	only	only	ADV
iajs-4097	20	5	be	be	AUX
iajs-4097	20	6	solved	solve	VERB
iajs-4097	20	7	analytically	analytically	ADV
iajs-4097	20	8	in	in	ADP
iajs-4097	20	9	highly	highly	ADV
iajs-4097	20	10	particular	particular	ADJ
iajs-4097	20	11	situations	situation	NOUN
iajs-4097	20	12	,	,	PUNCT
iajs-4097	20	13	hence	hence	ADV
iajs-4097	20	14	many	many	ADJ
iajs-4097	20	15	realistic	realistic	ADJ
iajs-4097	20	16	cases	case	NOUN
iajs-4097	20	17	can	can	AUX
iajs-4097	20	18	not	not	PART
iajs-4097	20	19	be	be	AUX
iajs-4097	20	20	solved	solve	VERB
iajs-4097	20	21	using	use	VERB
iajs-4097	20	22	commonly	commonly	ADV
iajs-4097	20	23	used	use	VERB
iajs-4097	20	24	analytical	analytical	ADJ
iajs-4097	20	25	methods	method	NOUN
iajs-4097	20	26	.	.	PUNCT
iajs-4097	21	1	in	in	ADP
iajs-4097	21	2	addition	addition	NOUN
iajs-4097	21	3	to	to	ADP
iajs-4097	21	4	discretization	discretization	VERB
iajs-4097	21	5	methods	method	NOUN
iajs-4097	21	6	such	such	ADJ
iajs-4097	21	7	as	as	ADP
iajs-4097	21	8	finite	finite	ADJ
iajs-4097	21	9	difference	difference	NOUN
iajs-4097	21	10	,	,	PUNCT
iajs-4097	21	11	finite	finite	ADJ
iajs-4097	21	12	volume	volume	NOUN
iajs-4097	21	13	,	,	PUNCT
iajs-4097	21	14	and	and	CCONJ
iajs-4097	21	15	finite	finite	ADJ
iajs-4097	21	16	element	element	NOUN
iajs-4097	21	17	approaches	approach	NOUN
iajs-4097	21	18	,	,	PUNCT
iajs-4097	21	19	various	various	ADJ
iajs-4097	21	20	more	more	ADJ
iajs-4097	21	21	ways	way	NOUN
iajs-4097	21	22	have	have	AUX
iajs-4097	21	23	been	be	AUX
iajs-4097	21	24	proposed	propose	VERB
iajs-4097	21	25	to	to	PART
iajs-4097	21	26	solve	solve	VERB
iajs-4097	21	27	the	the	DET
iajs-4097	21	28	wave	wave	NOUN
iajs-4097	21	29	problem	problem	NOUN
iajs-4097	21	30	.	.	PUNCT
iajs-4097	22	1	for	for	ADP
iajs-4097	22	2	example	example	NOUN
iajs-4097	22	3	,	,	PUNCT
iajs-4097	22	4	the	the	DET
iajs-4097	22	5	analytical	analytical	ADJ
iajs-4097	22	6	approach	approach	NOUN
iajs-4097	22	7	to	to	ADP
iajs-4097	22	8	solving	solve	VERB
iajs-4097	22	9	the	the	DET
iajs-4097	22	10	wave	wave	NOUN
iajs-4097	22	11	equation	equation	NOUN
iajs-4097	22	12	(	(	PUNCT
iajs-4097	22	13	9	9	NUM
iajs-4097	22	14	)	)	PUNCT
iajs-4097	22	15	,	,	PUNCT
iajs-4097	22	16	the	the	DET
iajs-4097	22	17	numerical	numerical	ADJ
iajs-4097	22	18	solution	solution	NOUN
iajs-4097	22	19	of	of	ADP
iajs-4097	22	20	the	the	DET
iajs-4097	22	21	wave	wave	NOUN
iajs-4097	22	22	issue	issue	NOUN
iajs-4097	22	23	(	(	PUNCT
iajs-4097	22	24	10,11	10,11	NOUN
iajs-4097	22	25	)	)	PUNCT
iajs-4097	22	26	,	,	PUNCT
iajs-4097	22	27	and	and	CCONJ
iajs-4097	22	28	the	the	DET
iajs-4097	22	29	optimal	optimal	ADJ
iajs-4097	22	30	homotopy	homotopy	NOUN
iajs-4097	22	31	strategy	strategy	NOUN
iajs-4097	22	32	for	for	ADP
iajs-4097	22	33	solving	solve	VERB
iajs-4097	22	34	the	the	DET
iajs-4097	22	35	nonlinear	nonlinear	ADJ
iajs-4097	22	36	wave	wave	NOUN
iajs-4097	22	37	equation	equation	NOUN
iajs-4097	22	38	(	(	PUNCT
iajs-4097	22	39	12	12	NUM
iajs-4097	22	40	)	)	PUNCT
iajs-4097	22	41	.	.	PUNCT
iajs-4097	23	1	furthermore	furthermore	ADV
iajs-4097	23	2	,	,	PUNCT
iajs-4097	23	3	there	there	PRON
iajs-4097	23	4	are	be	VERB
iajs-4097	23	5	numerous	numerous	ADJ
iajs-4097	23	6	techniques	technique	NOUN
iajs-4097	23	7	that	that	PRON
iajs-4097	23	8	offer	offer	VERB
iajs-4097	23	9	an	an	DET
iajs-4097	23	10	approximate	approximate	ADJ
iajs-4097	23	11	resolve	resolve	NOUN
iajs-4097	23	12	for	for	ADP
iajs-4097	23	13	the	the	DET
iajs-4097	23	14	differing	differ	VERB
iajs-4097	23	15	kinds	kind	NOUN
iajs-4097	23	16	of	of	ADP
iajs-4097	23	17	the	the	DET
iajs-4097	23	18	differential	differential	ADJ
iajs-4097	23	19	equations	equation	NOUN
iajs-4097	23	20	,	,	PUNCT
iajs-4097	23	21	see	see	VERB
iajs-4097	23	22	,	,	PUNCT
iajs-4097	23	23	(	(	PUNCT
iajs-4097	23	24	13–15	13–15	NUM
iajs-4097	23	25	)	)	PUNCT
iajs-4097	23	26	.	.	PUNCT
iajs-4097	24	1	corrington	corrington	PROPN
iajs-4097	24	2	suggested	suggest	VERB
iajs-4097	24	3	in	in	ADP
iajs-4097	24	4	1973	1973	NUM
iajs-4097	24	5	that	that	SCONJ
iajs-4097	24	6	a	a	DET
iajs-4097	24	7	system	system	NOUN
iajs-4097	24	8	of	of	ADP
iajs-4097	24	9	algebraic	algebraic	ADJ
iajs-4097	24	10	linear	linear	ADJ
iajs-4097	24	11	equations	equation	NOUN
iajs-4097	24	12	could	could	AUX
iajs-4097	24	13	be	be	AUX
iajs-4097	24	14	created	create	VERB
iajs-4097	24	15	from	from	ADP
iajs-4097	24	16	linear	linear	PROPN
iajs-4097	24	17	differential	differential	NOUN
iajs-4097	24	18	and	and	CCONJ
iajs-4097	24	19	integral	integral	ADJ
iajs-4097	24	20	equations	equation	NOUN
iajs-4097	24	21	by	by	ADP
iajs-4097	24	22	using	use	VERB
iajs-4097	24	23	a	a	DET
iajs-4097	24	24	least	least	ADJ
iajs-4097	24	25	squares	square	NOUN
iajs-4097	24	26	approximation	approximation	NOUN
iajs-4097	24	27	and	and	CCONJ
iajs-4097	24	28	repeatedly	repeatedly	ADV
iajs-4097	24	29	integrating	integrate	VERB
iajs-4097	24	30	walsh	walsh	NOUN
iajs-4097	24	31	functions	function	NOUN
iajs-4097	24	32	(	(	PUNCT
iajs-4097	24	33	16	16	NUM
iajs-4097	24	34	)	)	PUNCT
iajs-4097	24	35	.	.	PUNCT
iajs-4097	25	1	in	in	ADP
iajs-4097	25	2	addition	addition	NOUN
iajs-4097	25	3	,	,	PUNCT
iajs-4097	25	4	sparis	sparis	NOUN
iajs-4097	25	5	and	and	CCONJ
iajs-4097	25	6	mouroutsos	mouroutsos	NOUN
iajs-4097	25	7	in	in	ADP
iajs-4097	25	8	1986	1986	NUM
iajs-4097	25	9	using	use	VERB
iajs-4097	25	10	the	the	DET
iajs-4097	25	11	orthogonal	orthogonal	ADJ
iajs-4097	25	12	polynomial	polynomial	ADJ
iajs-4097	25	13	series	series	NOUN
iajs-4097	25	14	operational	operational	ADJ
iajs-4097	25	15	matrices	matrix	NOUN
iajs-4097	25	16	to	to	PART
iajs-4097	25	17	solve	solve	VERB
iajs-4097	25	18	differential	differential	ADJ
iajs-4097	25	19	equations	equation	NOUN
iajs-4097	25	20	(	(	PUNCT
iajs-4097	25	21	17	17	NUM
iajs-4097	25	22	)	)	PUNCT
iajs-4097	25	23	,	,	PUNCT
iajs-4097	25	24	many	many	ADJ
iajs-4097	25	25	researchers	researcher	NOUN
iajs-4097	25	26	have	have	AUX
iajs-4097	25	27	used	use	VERB
iajs-4097	25	28	orthogonal	orthogonal	ADJ
iajs-4097	25	29	polynomials	polynomial	NOUN
iajs-4097	25	30	to	to	PART
iajs-4097	25	31	find	find	VERB
iajs-4097	25	32	approximate	approximate	ADJ
iajs-4097	25	33	solutions	solution	NOUN
iajs-4097	25	34	for	for	ADP
iajs-4097	25	35	many	many	ADJ
iajs-4097	25	36	applications	application	NOUN
iajs-4097	25	37	.	.	PUNCT
iajs-4097	26	1	see	see	VERB
iajs-4097	26	2	;	;	PUNCT
iajs-4097	26	3	(	(	PUNCT
iajs-4097	26	4	18–28	18–28	NUM
iajs-4097	26	5	)	)	PUNCT
iajs-4097	26	6	.	.	PUNCT
iajs-4097	27	1	the	the	DET
iajs-4097	27	2	authors	author	NOUN
iajs-4097	27	3	were	be	AUX
iajs-4097	27	4	highly	highly	ADV
iajs-4097	27	5	interested	interested	ADJ
iajs-4097	27	6	in	in	ADP
iajs-4097	27	7	since	since	SCONJ
iajs-4097	27	8	they	they	PRON
iajs-4097	27	9	were	be	AUX
iajs-4097	27	10	practical	practical	ADJ
iajs-4097	27	11	methods	method	NOUN
iajs-4097	27	12	for	for	ADP
iajs-4097	27	13	resolving	resolve	VERB
iajs-4097	27	14	an	an	DET
iajs-4097	27	15	extensive	extensive	ADJ
iajs-4097	27	16	number	number	NOUN
iajs-4097	27	17	of	of	ADP
iajs-4097	27	18	approximation	approximation	NOUN
iajs-4097	27	19	theory	theory	NOUN
iajs-4097	27	20	and	and	CCONJ
iajs-4097	27	21	numerical	numerical	ADJ
iajs-4097	27	22	analysis	analysis	NOUN
iajs-4097	27	23	problems	problem	NOUN
iajs-4097	27	24	.	.	PUNCT
iajs-4097	28	1	on	on	ADP
iajs-4097	28	2	the	the	DET
iajs-4097	28	3	other	other	ADJ
iajs-4097	28	4	hand	hand	NOUN
iajs-4097	28	5	,	,	PUNCT
iajs-4097	28	6	the	the	DET
iajs-4097	28	7	orthogonal	orthogonal	ADJ
iajs-4097	28	8	polynomials	polynomial	NOUN
iajs-4097	28	9	and	and	CCONJ
iajs-4097	28	10	operational	operational	ADJ
iajs-4097	28	11	matrices	matrix	NOUN
iajs-4097	28	12	stand	stand	VERB
iajs-4097	28	13	out	out	ADP
iajs-4097	28	14	above	above	ADP
iajs-4097	28	15	other	other	ADJ
iajs-4097	28	16	types	type	NOUN
iajs-4097	28	17	due	due	ADP
iajs-4097	28	18	to	to	ADP
iajs-4097	28	19	their	their	PRON
iajs-4097	28	20	effective	effective	ADJ
iajs-4097	28	21	reduction	reduction	NOUN
iajs-4097	28	22	of	of	ADP
iajs-4097	28	23	the	the	DET
iajs-4097	28	24	required	require	VERB
iajs-4097	28	25	solution	solution	NOUN
iajs-4097	28	26	,	,	PUNCT
iajs-4097	28	27	which	which	PRON
iajs-4097	28	28	is	be	AUX
iajs-4097	28	29	achieved	achieve	VERB
iajs-4097	28	30	by	by	ADP
iajs-4097	28	31	applying	apply	VERB
iajs-4097	28	32	the	the	DET
iajs-4097	28	33	operational	operational	ADJ
iajs-4097	28	34	matrices	matrix	NOUN
iajs-4097	28	35	technique	technique	NOUN
iajs-4097	28	36	to	to	PART
iajs-4097	28	37	transform	transform	VERB
iajs-4097	28	38	the	the	DET
iajs-4097	28	39	linear	linear	ADJ
iajs-4097	28	40	differential	differential	ADJ
iajs-4097	28	41	equations	equation	NOUN
iajs-4097	28	42	into	into	ADP
iajs-4097	28	43	linear	linear	ADJ
iajs-4097	28	44	algebraic	algebraic	ADJ
iajs-4097	28	45	systems	system	NOUN
iajs-4097	28	46	of	of	ADP
iajs-4097	28	47	equations	equation	NOUN
iajs-4097	28	48	,	,	PUNCT
iajs-4097	28	49	where	where	SCONJ
iajs-4097	28	50	any	any	DET
iajs-4097	28	51	computer	computer	NOUN
iajs-4097	28	52	program	program	NOUN
iajs-4097	28	53	can	can	AUX
iajs-4097	28	54	be	be	AUX
iajs-4097	28	55	used	use	VERB
iajs-4097	28	56	to	to	PART
iajs-4097	28	57	solve	solve	VERB
iajs-4097	28	58	them	they	PRON
iajs-4097	28	59	.	.	PUNCT
iajs-4097	29	1	farther	far	ADV
iajs-4097	29	2	more	more	ADV
iajs-4097	29	3	,	,	PUNCT
iajs-4097	29	4	turkyilmazoglu	turkyilmazoglu	NOUN
iajs-4097	29	5	in	in	ADP
iajs-4097	29	6	2013	2013	NUM
iajs-4097	29	7	proposed	propose	VERB
iajs-4097	29	8	an	an	DET
iajs-4097	29	9	approximate	approximate	ADJ
iajs-4097	29	10	analytical	analytical	ADJ
iajs-4097	29	11	technique	technique	NOUN
iajs-4097	29	12	for	for	ADP
iajs-4097	29	13	resolving	resolve	VERB
iajs-4097	29	14	differential	differential	ADJ
iajs-4097	29	15	equations	equation	NOUN
iajs-4097	29	16	based	base	VERB
iajs-4097	29	17	on	on	ADP
iajs-4097	29	18	standard	standard	ADJ
iajs-4097	29	19	polynomials	polynomial	NOUN
iajs-4097	29	20	,	,	PUNCT
iajs-4097	29	21	which	which	PRON
iajs-4097	29	22	is	be	AUX
iajs-4097	29	23	used	use	VERB
iajs-4097	29	24	on	on	ADP
iajs-4097	29	25	various	various	ADJ
iajs-4097	29	26	types	type	NOUN
iajs-4097	29	27	of	of	ADP
iajs-4097	29	28	problems	problem	NOUN
iajs-4097	29	29	(	(	PUNCT
iajs-4097	29	30	29–33	29–33	NOUN
iajs-4097	29	31	)	)	PUNCT
iajs-4097	29	32	.	.	PUNCT
iajs-4097	30	1	in	in	ADP
iajs-4097	30	2	2023	2023	NUM
iajs-4097	30	3	,	,	PUNCT
iajs-4097	30	4	othman	othman	PROPN
iajs-4097	30	5	,	,	PUNCT
iajs-4097	30	6	et	et	PROPN
iajs-4097	30	7	al	al	PROPN
iajs-4097	30	8	.	.	PROPN
iajs-4097	30	9	introduced	introduce	VERB
iajs-4097	30	10	several	several	ADJ
iajs-4097	30	11	orthogonal	orthogonal	ADJ
iajs-4097	30	12	polynomials	polynomial	NOUN
iajs-4097	30	13	,	,	PUNCT
iajs-4097	30	14	for	for	ADP
iajs-4097	30	15	instance	instance	NOUN
iajs-4097	30	16	,	,	PUNCT
iajs-4097	30	17	hermite	hermite	PROPN
iajs-4097	30	18	,	,	PUNCT
iajs-4097	30	19	laguerre	laguerre	NOUN
iajs-4097	30	20	,	,	PUNCT
iajs-4097	30	21	chebyshev	chebyshev	NOUN
iajs-4097	30	22	,	,	PUNCT
iajs-4097	30	23	and	and	CCONJ
iajs-4097	30	24	others	other	NOUN
iajs-4097	30	25	polynomials	polynomial	VERB
iajs-4097	30	26	with	with	ADP
iajs-4097	30	27	inner	inner	ADJ
iajs-4097	30	28	product	product	NOUN
iajs-4097	30	29	,	,	PUNCT
iajs-4097	30	30	to	to	PART
iajs-4097	30	31	develop	develop	VERB
iajs-4097	30	32	the	the	DET
iajs-4097	30	33	computational	computational	ADJ
iajs-4097	30	34	method	method	NOUN
iajs-4097	30	35	(	(	PUNCT
iajs-4097	30	36	cm	cm	NOUN
iajs-4097	30	37	)	)	PUNCT
iajs-4097	30	38	(	(	PUNCT
iajs-4097	30	39	34	34	NUM
iajs-4097	30	40	)	)	PUNCT
iajs-4097	30	41	.	.	PUNCT
iajs-4097	31	1	myasar	myasar	VERB
iajs-4097	31	2	,	,	PUNCT
iajs-4097	31	3	et	et	PROPN
iajs-4097	31	4	al	al	PROPN
iajs-4097	31	5	.	.	PUNCT
iajs-4097	32	1	(	(	PUNCT
iajs-4097	32	2	35,36	35,36	NUM
iajs-4097	32	3	)	)	PUNCT
iajs-4097	32	4	,	,	PUNCT
iajs-4097	32	5	further	far	ADV
iajs-4097	32	6	introduced	introduce	VERB
iajs-4097	32	7	the	the	DET
iajs-4097	32	8	genocchi	genocchi	NOUN
iajs-4097	32	9	,	,	PUNCT
iajs-4097	32	10	bernoulli	bernoulli	PROPN
iajs-4097	32	11	,	,	PUNCT
iajs-4097	32	12	and	and	CCONJ
iajs-4097	32	13	boubaker	boubaker	NOUN
iajs-4097	32	14	polynomials	polynomial	NOUN
iajs-4097	32	15	,	,	PUNCT
iajs-4097	32	16	which	which	PRON
iajs-4097	32	17	contribute	contribute	VERB
iajs-4097	32	18	to	to	ADP
iajs-4097	32	19	the	the	DET
iajs-4097	32	20	effective	effective	ADJ
iajs-4097	32	21	computational	computational	ADJ
iajs-4097	32	22	method	method	NOUN
iajs-4097	32	23	(	(	PUNCT
iajs-4097	32	24	ecm	ecm	PROPN
iajs-4097	32	25	)	)	PUNCT
iajs-4097	32	26	.	.	PUNCT
iajs-4097	33	1	the	the	DET
iajs-4097	33	2	outline	outline	NOUN
iajs-4097	33	3	for	for	ADP
iajs-4097	33	4	this	this	DET
iajs-4097	33	5	paper	paper	NOUN
iajs-4097	33	6	is	be	AUX
iajs-4097	33	7	as	as	SCONJ
iajs-4097	33	8	follows	follow	VERB
iajs-4097	33	9	:	:	PUNCT
iajs-4097	33	10	section	section	NOUN
iajs-4097	33	11	two	two	NUM
iajs-4097	33	12	shows	show	VERB
iajs-4097	33	13	the	the	DET
iajs-4097	33	14	second	second	ADJ
iajs-4097	33	15	-	-	PUNCT
iajs-4097	33	16	order	order	NOUN
iajs-4097	33	17	linear	linear	ADJ
iajs-4097	33	18	wave	wave	NOUN
iajs-4097	33	19	equation	equation	NOUN
iajs-4097	33	20	's	's	PART
iajs-4097	33	21	mathematical	mathematical	ADJ
iajs-4097	33	22	formulation	formulation	NOUN
iajs-4097	33	23	.	.	PUNCT
iajs-4097	34	1	section	section	NOUN
iajs-4097	34	2	three	three	NUM
iajs-4097	34	3	:	:	PUNCT
iajs-4097	34	4	preliminary	preliminary	ADJ
iajs-4097	34	5	of	of	ADP
iajs-4097	34	6	the	the	DET
iajs-4097	34	7	orthogonal	orthogonal	ADJ
iajs-4097	34	8	polynomials	polynomial	NOUN
iajs-4097	34	9	.	.	PUNCT
iajs-4097	35	1	section	section	NOUN
iajs-4097	35	2	four	four	NUM
iajs-4097	35	3	:	:	PUNCT
iajs-4097	35	4	main	main	ADJ
iajs-4097	35	5	results	result	NOUN
iajs-4097	35	6	and	and	CCONJ
iajs-4097	35	7	applications	application	NOUN
iajs-4097	35	8	of	of	ADP
iajs-4097	35	9	orthogonal	orthogonal	ADJ
iajs-4097	35	10	polynomials	polynomial	NOUN
iajs-4097	35	11	to	to	PART
iajs-4097	35	12	resolve	resolve	VERB
iajs-4097	35	13	some	some	DET
iajs-4097	35	14	examples	example	NOUN
iajs-4097	35	15	for	for	ADP
iajs-4097	35	16	the	the	DET
iajs-4097	35	17	wave	wave	NOUN
iajs-4097	35	18	problem	problem	NOUN
iajs-4097	35	19	.	.	PUNCT
iajs-4097	36	1	section	section	NOUN
iajs-4097	36	2	five	five	NUM
iajs-4097	36	3	:	:	PUNCT
iajs-4097	36	4	gives	give	VERB
iajs-4097	36	5	the	the	DET
iajs-4097	36	6	conclusions	conclusion	NOUN
iajs-4097	36	7	.	.	PUNCT
iajs-4097	37	1	2	2	X
iajs-4097	37	2	.	.	X
iajs-4097	37	3	the	the	DET
iajs-4097	37	4	mathematical	mathematical	ADJ
iajs-4097	37	5	model	model	NOUN
iajs-4097	37	6	of	of	ADP
iajs-4097	37	7	the	the	DET
iajs-4097	37	8	wave	wave	NOUN
iajs-4097	37	9	equation	equation	NOUN
iajs-4097	37	10	let	let	VERB
iajs-4097	37	11	us	we	PRON
iajs-4097	37	12	consider	consider	VERB
iajs-4097	37	13	the	the	DET
iajs-4097	37	14	one	one	NUM
iajs-4097	37	15	-	-	PUNCT
iajs-4097	37	16	dimensional	dimensional	ADJ
iajs-4097	37	17	wave	wave	NOUN
iajs-4097	37	18	problem	problem	NOUN
iajs-4097	37	19	(	(	PUNCT
iajs-4097	37	20	7	7	NUM
iajs-4097	37	21	)	)	PUNCT
iajs-4097	37	22	:	:	PUNCT
iajs-4097	37	23	(	(	PUNCT
iajs-4097	37	24	)	)	PUNCT
iajs-4097	37	25	(	(	PUNCT
iajs-4097	37	26	1	1	X
iajs-4097	37	27	)	)	PUNCT
iajs-4097	37	28	with	with	ADP
iajs-4097	37	29	initial	initial	ADJ
iajs-4097	37	30	conditions	condition	NOUN
iajs-4097	37	31	:	:	PUNCT
iajs-4097	37	32	(	(	PUNCT
iajs-4097	37	33	)	)	PUNCT
iajs-4097	37	34	(	(	PUNCT
iajs-4097	37	35	)	)	PUNCT
iajs-4097	37	36	(	(	PUNCT
iajs-4097	37	37	)	)	PUNCT
iajs-4097	37	38	(	(	PUNCT
iajs-4097	37	39	)	)	PUNCT
iajs-4097	37	40	(	(	PUNCT
iajs-4097	37	41	2	2	NUM
iajs-4097	37	42	)	)	PUNCT
iajs-4097	37	43	and	and	CCONJ
iajs-4097	37	44	boundary	boundary	ADJ
iajs-4097	37	45	conditions	condition	NOUN
iajs-4097	37	46	:	:	PUNCT
iajs-4097	37	47	(	(	PUNCT
iajs-4097	37	48	)	)	PUNCT
iajs-4097	37	49	̂	̂	X
iajs-4097	37	50	(	(	PUNCT
iajs-4097	37	51	)	)	PUNCT
iajs-4097	37	52	(	(	PUNCT
iajs-4097	37	53	3	3	X
iajs-4097	37	54	)	)	PUNCT
iajs-4097	37	55	(	(	PUNCT
iajs-4097	37	56	)	)	PUNCT
iajs-4097	37	57	̂	̂	X
iajs-4097	37	58	(	(	PUNCT
iajs-4097	37	59	)	)	PUNCT
iajs-4097	37	60	.	.	PUNCT
iajs-4097	38	1	(	(	PUNCT
iajs-4097	38	2	4	4	X
iajs-4097	38	3	)	)	PUNCT
iajs-4097	38	4	where	where	SCONJ
iajs-4097	38	5	(	(	PUNCT
iajs-4097	38	6	)	)	PUNCT
iajs-4097	38	7	is	be	AUX
iajs-4097	38	8	a	a	DET
iajs-4097	38	9	particular	particular	ADJ
iajs-4097	38	10	function	function	NOUN
iajs-4097	38	11	,	,	PUNCT
iajs-4097	38	12	ω	ω	PROPN
iajs-4097	38	13	is	be	AUX
iajs-4097	38	14	the	the	DET
iajs-4097	38	15	bounded	bounded	ADJ
iajs-4097	38	16	domain	domain	NOUN
iajs-4097	38	17	,	,	PUNCT
iajs-4097	38	18	is	be	AUX
iajs-4097	38	19	a	a	DET
iajs-4097	38	20	constant	constant	ADJ
iajs-4097	38	21	,	,	PUNCT
iajs-4097	38	22	̂	̂	X
iajs-4097	38	23	(	(	PUNCT
iajs-4097	38	24	)	)	PUNCT
iajs-4097	38	25	and	and	CCONJ
iajs-4097	38	26	̂	̂	X
iajs-4097	38	27	(	(	PUNCT
iajs-4097	38	28	)	)	PUNCT
iajs-4097	38	29	are	be	AUX
iajs-4097	38	30	known	know	VERB
iajs-4097	38	31	functions	function	NOUN
iajs-4097	38	32	,	,	PUNCT
iajs-4097	38	33	,	,	PUNCT
iajs-4097	38	34	-	-	PUNCT
iajs-4097	38	35	(	(	PUNCT
iajs-4097	38	36	)	)	PUNCT
iajs-4097	38	37	(	(	PUNCT
iajs-4097	38	38	)	)	PUNCT
iajs-4097	38	39	,	,	PUNCT
iajs-4097	38	40	and	and	CCONJ
iajs-4097	38	41	(	(	PUNCT
iajs-4097	38	42	)	)	PUNCT
iajs-4097	38	43	the	the	DET
iajs-4097	38	44	external	external	ADJ
iajs-4097	38	45	normal	normal	ADJ
iajs-4097	38	46	vector	vector	NOUN
iajs-4097	38	47	to	to	ADP
iajs-4097	38	48	the	the	DET
iajs-4097	38	49	boundary	boundary	ADJ
iajs-4097	38	50	,	,	PUNCT
iajs-4097	38	51	when	when	SCONJ
iajs-4097	38	52	equation	equation	NOUN
iajs-4097	38	53	1	1	NUM
iajs-4097	38	54	will	will	AUX
iajs-4097	38	55	be	be	AUX
iajs-4097	38	56	homogeneous	homogeneous	ADJ
iajs-4097	38	57	wave	wave	NOUN
iajs-4097	38	58	equation	equation	NOUN
iajs-4097	38	59	,	,	PUNCT
iajs-4097	38	60	and	and	CCONJ
iajs-4097	38	61	if	if	SCONJ
iajs-4097	38	62	equation	equation	NOUN
iajs-4097	38	63	1	1	NUM
iajs-4097	38	64	will	will	AUX
iajs-4097	38	65	be	be	AUX
iajs-4097	38	66	non	non	ADJ
iajs-4097	38	67	-	-	ADJ
iajs-4097	38	68	homogeneous	homogeneous	ADJ
iajs-4097	38	69	wave	wave	NOUN
iajs-4097	38	70	problem	problem	NOUN
iajs-4097	38	71	.	.	PUNCT
iajs-4097	39	1	ihjpas	ihjpas	PROPN
iajs-4097	39	2	.	.	PUNCT
iajs-4097	40	1	2025,38(4	2025,38(4	NOUN
iajs-4097	40	2	)	)	PUNCT
iajs-4097	41	1	378	378	NUM
iajs-4097	41	2	3	3	NUM
iajs-4097	41	3	.	.	PUNCT
iajs-4097	41	4	preliminary	preliminary	ADJ
iajs-4097	41	5	of	of	ADP
iajs-4097	41	6	the	the	DET
iajs-4097	41	7	orthogonal	orthogonal	ADJ
iajs-4097	41	8	polynomials	polynomial	NOUN
iajs-4097	41	9	and	and	CCONJ
iajs-4097	41	10	their	their	PRON
iajs-4097	41	11	operational	operational	ADJ
iajs-4097	41	12	matrices	matrix	NOUN
iajs-4097	41	13	the	the	DET
iajs-4097	41	14	orthogonal	orthogonal	ADJ
iajs-4097	41	15	polynomials	polynomial	NOUN
iajs-4097	41	16	and	and	CCONJ
iajs-4097	41	17	their	their	PRON
iajs-4097	41	18	operational	operational	ADJ
iajs-4097	41	19	matrices	matrix	NOUN
iajs-4097	41	20	play	play	VERB
iajs-4097	41	21	an	an	DET
iajs-4097	41	22	important	important	ADJ
iajs-4097	41	23	part	part	NOUN
iajs-4097	41	24	applied	apply	VERB
iajs-4097	41	25	and	and	CCONJ
iajs-4097	41	26	pure	pure	ADJ
iajs-4097	41	27	mathematics	mathematic	NOUN
iajs-4097	41	28	as	as	ADV
iajs-4097	41	29	well	well	ADV
iajs-4097	41	30	as	as	ADP
iajs-4097	41	31	numerical	numerical	ADJ
iajs-4097	41	32	calculation	calculation	NOUN
iajs-4097	41	33	.	.	PUNCT
iajs-4097	42	1	six	six	NUM
iajs-4097	42	2	types	type	NOUN
iajs-4097	42	3	of	of	ADP
iajs-4097	42	4	these	these	DET
iajs-4097	42	5	orthogonal	orthogonal	ADJ
iajs-4097	42	6	polynomials	polynomial	NOUN
iajs-4097	42	7	will	will	AUX
iajs-4097	42	8	be	be	AUX
iajs-4097	42	9	used	use	VERB
iajs-4097	42	10	:	:	PUNCT
iajs-4097	42	11	standard	standard	ADJ
iajs-4097	42	12	polynomials	polynomial	NOUN
iajs-4097	42	13	(	(	PUNCT
iajs-4097	42	14	)	)	PUNCT
iajs-4097	42	15	,	,	PUNCT
iajs-4097	42	16	legendre	legendre	PROPN
iajs-4097	42	17	,	,	PUNCT
iajs-4097	42	18	chebyshev	chebyshev	PROPN
iajs-4097	42	19	,	,	PUNCT
iajs-4097	42	20	boubaker	boubaker	NOUN
iajs-4097	42	21	,	,	PUNCT
iajs-4097	42	22	genocchi	genocchi	NOUN
iajs-4097	42	23	,	,	PUNCT
iajs-4097	42	24	and	and	CCONJ
iajs-4097	42	25	bernoulli	bernoulli	NOUN
iajs-4097	42	26	polynomials	polynomial	NOUN
iajs-4097	42	27	to	to	PART
iajs-4097	42	28	resolve	resolve	VERB
iajs-4097	42	29	the	the	DET
iajs-4097	42	30	wave	wave	NOUN
iajs-4097	42	31	problem	problem	NOUN
iajs-4097	42	32	with	with	ADP
iajs-4097	42	33	initial	initial	ADJ
iajs-4097	42	34	and	and	CCONJ
iajs-4097	42	35	boundary	boundary	ADJ
iajs-4097	42	36	conditions	condition	NOUN
iajs-4097	42	37	.	.	PUNCT
iajs-4097	43	1	3.1	3.1	NUM
iajs-4097	43	2	.	.	PUNCT
iajs-4097	44	1	the	the	DET
iajs-4097	44	2	operational	operational	ADJ
iajs-4097	44	3	matrices	matrix	NOUN
iajs-4097	44	4	for	for	ADP
iajs-4097	44	5	standard	standard	ADJ
iajs-4097	44	6	polynomials	polynomial	NOUN
iajs-4097	44	7	assume	assume	VERB
iajs-4097	44	8	that	that	SCONJ
iajs-4097	44	9	the	the	DET
iajs-4097	44	10	wave	wave	NOUN
iajs-4097	44	11	problem	problem	NOUN
iajs-4097	44	12	(	(	PUNCT
iajs-4097	44	13	1	1	NUM
iajs-4097	44	14	)	)	PUNCT
iajs-4097	44	15	,	,	PUNCT
iajs-4097	44	16	with	with	ADP
iajs-4097	44	17	the	the	DET
iajs-4097	44	18	initial	initial	ADJ
iajs-4097	44	19	and	and	CCONJ
iajs-4097	44	20	boundary	boundary	ADJ
iajs-4097	44	21	conditions	condition	NOUN
iajs-4097	44	22	(	(	PUNCT
iajs-4097	44	23	2	2	NUM
iajs-4097	44	24	-	-	SYM
iajs-4097	44	25	4	4	NUM
iajs-4097	44	26	)	)	PUNCT
iajs-4097	44	27	,	,	PUNCT
iajs-4097	44	28	has	have	VERB
iajs-4097	44	29	a	a	DET
iajs-4097	44	30	unique	unique	ADJ
iajs-4097	44	31	solution	solution	NOUN
iajs-4097	44	32	,	,	PUNCT
iajs-4097	44	33	this	this	PRON
iajs-4097	44	34	can	can	AUX
iajs-4097	44	35	be	be	AUX
iajs-4097	44	36	implemented	implement	VERB
iajs-4097	44	37	using	use	VERB
iajs-4097	44	38	a	a	DET
iajs-4097	44	39	suitable	suitable	ADJ
iajs-4097	44	40	linear	linear	ADJ
iajs-4097	44	41	transformation	transformation	NOUN
iajs-4097	44	42	.	.	PUNCT
iajs-4097	45	1	taking	take	VERB
iajs-4097	45	2	the	the	DET
iajs-4097	45	3	basic	basic	ADJ
iajs-4097	45	4	functions	function	NOUN
iajs-4097	45	5	*	*	PUNCT
iajs-4097	45	6	(	(	PUNCT
iajs-4097	45	7	)	)	PUNCT
iajs-4097	45	8	(	(	PUNCT
iajs-4097	45	9	)	)	PUNCT
iajs-4097	45	10	(	(	PUNCT
iajs-4097	45	11	)	)	PUNCT
iajs-4097	46	1	+	+	ADV
iajs-4097	46	2	,	,	PUNCT
iajs-4097	46	3	and	and	CCONJ
iajs-4097	46	4	*	*	PUNCT
iajs-4097	46	5	(	(	PUNCT
iajs-4097	46	6	)	)	PUNCT
iajs-4097	46	7	(	(	PUNCT
iajs-4097	46	8	)	)	PUNCT
iajs-4097	46	9	(	(	PUNCT
iajs-4097	46	10	)	)	PUNCT
iajs-4097	46	11	+	+	ADV
iajs-4097	46	12	,	,	PUNCT
iajs-4097	46	13	let	let	VERB
iajs-4097	46	14	the	the	DET
iajs-4097	46	15	solution	solution	NOUN
iajs-4097	46	16	of	of	ADP
iajs-4097	46	17	equation	equation	NOUN
iajs-4097	46	18	1	1	NUM
iajs-4097	46	19	,	,	PUNCT
iajs-4097	46	20	may	may	AUX
iajs-4097	46	21	be	be	AUX
iajs-4097	46	22	presented	present	VERB
iajs-4097	46	23	by	by	ADP
iajs-4097	46	24	using	use	VERB
iajs-4097	46	25	the	the	DET
iajs-4097	46	26	double	double	ADJ
iajs-4097	46	27	series	series	NOUN
iajs-4097	46	28	extension	extension	NOUN
iajs-4097	46	29	in	in	ADP
iajs-4097	46	30	terms	term	NOUN
iajs-4097	46	31	of	of	ADP
iajs-4097	46	32	a	a	DET
iajs-4097	46	33	base	base	NOUN
iajs-4097	46	34	functions	function	NOUN
iajs-4097	46	35	(	(	PUNCT
iajs-4097	46	36	)	)	PUNCT
iajs-4097	46	37	∑	∑	ADP
iajs-4097	46	38	∑	∑	PUNCT
iajs-4097	46	39	(	(	PUNCT
iajs-4097	46	40	)	)	PUNCT
iajs-4097	46	41	(	(	PUNCT
iajs-4097	46	42	)	)	PUNCT
iajs-4097	46	43	,	,	PUNCT
iajs-4097	46	44	(	(	PUNCT
iajs-4097	46	45	5	5	X
iajs-4097	46	46	)	)	PUNCT
iajs-4097	46	47	the	the	DET
iajs-4097	46	48	equation	equation	NOUN
iajs-4097	46	49	5	5	NUM
iajs-4097	46	50	may	may	AUX
iajs-4097	46	51	be	be	AUX
iajs-4097	46	52	written	write	VERB
iajs-4097	46	53	using	use	VERB
iajs-4097	46	54	the	the	DET
iajs-4097	46	55	dot	dot	NOUN
iajs-4097	46	56	product	product	NOUN
iajs-4097	46	57	,	,	PUNCT
iajs-4097	46	58	as	as	SCONJ
iajs-4097	46	59	follows	follow	VERB
iajs-4097	46	60	:	:	PUNCT
iajs-4097	46	61	(	(	PUNCT
iajs-4097	46	62	)	)	PUNCT
iajs-4097	46	63	∑	∑	ADP
iajs-4097	46	64	∑	∑	PUNCT
iajs-4097	46	65	(	(	PUNCT
iajs-4097	46	66	)	)	PUNCT
iajs-4097	46	67	(	(	PUNCT
iajs-4097	46	68	)	)	PUNCT
iajs-4097	46	69	(	(	PUNCT
iajs-4097	46	70	)	)	PUNCT
iajs-4097	46	71	(	(	PUNCT
iajs-4097	46	72	)	)	PUNCT
iajs-4097	46	73	,	,	PUNCT
iajs-4097	46	74	(	(	PUNCT
iajs-4097	46	75	6	6	NUM
iajs-4097	46	76	)	)	PUNCT
iajs-4097	46	77	where	where	SCONJ
iajs-4097	46	78	the	the	DET
iajs-4097	46	79	coefficients	coefficient	NOUN
iajs-4097	46	80	will	will	AUX
iajs-4097	46	81	be	be	AUX
iajs-4097	46	82	evaluated	evaluate	VERB
iajs-4097	46	83	later	later	ADV
iajs-4097	46	84	,	,	PUNCT
iajs-4097	46	85	by	by	ADP
iajs-4097	46	86	use	use	VERB
iajs-4097	46	87	the	the	DET
iajs-4097	46	88	definition	definition	NOUN
iajs-4097	46	89	:	:	PUNCT
iajs-4097	46	90	(	(	PUNCT
iajs-4097	46	91	)	)	PUNCT
iajs-4097	46	92	,	,	PUNCT
iajs-4097	46	93	-	-	PUNCT
iajs-4097	46	94	,	,	PUNCT
iajs-4097	46	95	(	(	PUNCT
iajs-4097	46	96	)	)	PUNCT
iajs-4097	46	97	,	,	PUNCT
iajs-4097	46	98	,	,	PUNCT
iajs-4097	46	99	and	and	CCONJ
iajs-4097	46	100	[	[	PUNCT
iajs-4097	46	101	]	]	X
iajs-4097	46	102	,	,	PUNCT
iajs-4097	46	103	furthermore	furthermore	ADV
iajs-4097	46	104	,	,	PUNCT
iajs-4097	46	105	the	the	DET
iajs-4097	46	106	-order	-order	ADJ
iajs-4097	46	107	partial	partial	ADJ
iajs-4097	46	108	derivatives	derivative	NOUN
iajs-4097	46	109	for	for	ADP
iajs-4097	46	110	and	and	CCONJ
iajs-4097	46	111	with	with	ADP
iajs-4097	46	112	respect	respect	NOUN
iajs-4097	46	113	and	and	CCONJ
iajs-4097	46	114	are	be	AUX
iajs-4097	46	115	obtained	obtain	VERB
iajs-4097	46	116	by	by	ADP
iajs-4097	46	117	:	:	PUNCT
iajs-4097	46	118	(	(	PUNCT
iajs-4097	46	119	)	)	PUNCT
iajs-4097	46	120	(	(	PUNCT
iajs-4097	46	121	)	)	PUNCT
iajs-4097	46	122	,	,	PUNCT
iajs-4097	46	123	(	(	PUNCT
iajs-4097	46	124	)	)	PUNCT
iajs-4097	46	125	(	(	PUNCT
iajs-4097	46	126	)	)	PUNCT
iajs-4097	46	127	(	(	PUNCT
iajs-4097	46	128	)	)	PUNCT
iajs-4097	46	129	,	,	PUNCT
iajs-4097	46	130	where	where	SCONJ
iajs-4097	46	131	is	be	AUX
iajs-4097	46	132	the	the	DET
iajs-4097	46	133	derivative	derivative	ADJ
iajs-4097	46	134	matrix	matrix	NOUN
iajs-4097	46	135	that	that	PRON
iajs-4097	46	136	is	be	AUX
iajs-4097	46	137	dimensional	dimensional	ADJ
iajs-4097	46	138	(	(	PUNCT
iajs-4097	46	139	)	)	PUNCT
iajs-4097	46	140	(	(	PUNCT
iajs-4097	46	141	)	)	PUNCT
iajs-4097	46	142	is	be	AUX
iajs-4097	46	143	given	give	VERB
iajs-4097	46	144	by	by	ADP
iajs-4097	46	145	:	:	PUNCT
iajs-4097	46	146	[	[	PUNCT
iajs-4097	46	147	]	]	X
iajs-4097	46	148	,	,	PUNCT
iajs-4097	46	149	thus	thus	ADV
iajs-4097	46	150	,	,	PUNCT
iajs-4097	46	151	the	the	DET
iajs-4097	46	152	derivatives	derivative	NOUN
iajs-4097	46	153	of	of	ADP
iajs-4097	46	154	(	(	PUNCT
iajs-4097	46	155	)	)	PUNCT
iajs-4097	46	156	will	will	AUX
iajs-4097	46	157	be	be	AUX
iajs-4097	46	158	obtained	obtain	VERB
iajs-4097	46	159	using	use	VERB
iajs-4097	46	160	equation	equation	NOUN
iajs-4097	46	161	6	6	NUM
iajs-4097	46	162	have	have	VERB
iajs-4097	46	163	the	the	DET
iajs-4097	46	164	following	follow	VERB
iajs-4097	46	165	forms	form	NOUN
iajs-4097	46	166	:	:	PUNCT
iajs-4097	46	167	(	(	PUNCT
iajs-4097	46	168	)	)	PUNCT
iajs-4097	46	169	(	(	PUNCT
iajs-4097	46	170	)	)	PUNCT
iajs-4097	46	171	(	(	PUNCT
iajs-4097	46	172	)	)	PUNCT
iajs-4097	46	173	(	(	PUNCT
iajs-4097	46	174	)	)	PUNCT
iajs-4097	46	175	(	(	PUNCT
iajs-4097	46	176	)	)	PUNCT
iajs-4097	46	177	(	(	PUNCT
iajs-4097	46	178	)	)	PUNCT
iajs-4097	46	179	(	(	PUNCT
iajs-4097	46	180	)	)	PUNCT
iajs-4097	46	181	(	(	PUNCT
iajs-4097	46	182	7	7	NUM
iajs-4097	46	183	)	)	PUNCT
iajs-4097	46	184	3.2	3.2	NUM
iajs-4097	46	185	.	.	PUNCT
iajs-4097	47	1	the	the	DET
iajs-4097	47	2	operational	operational	ADJ
iajs-4097	47	3	matrices	matrix	NOUN
iajs-4097	47	4	for	for	ADP
iajs-4097	47	5	boubaker	boubaker	NOUN
iajs-4097	47	6	polynomials	polynomial	NOUN
iajs-4097	47	7	the	the	DET
iajs-4097	47	8	-degree	-degree	NOUN
iajs-4097	47	9	of	of	ADP
iajs-4097	47	10	boubaker	boubaker	NOUN
iajs-4097	47	11	polynomials	polynomial	NOUN
iajs-4097	47	12	(	(	PUNCT
iajs-4097	47	13	)	)	PUNCT
iajs-4097	47	14	are	be	AUX
iajs-4097	47	15	defined	define	VERB
iajs-4097	47	16	as	as	ADP
iajs-4097	47	17	follows	follow	VERB
iajs-4097	47	18	(	(	PUNCT
iajs-4097	47	19	28,37	28,37	NOUN
iajs-4097	47	20	)	)	PUNCT
iajs-4097	47	21	:	:	PUNCT
iajs-4097	47	22	(	(	PUNCT
iajs-4097	47	23	)	)	PUNCT
iajs-4097	47	24	∑	∑	PUNCT
iajs-4097	47	25	(	(	PUNCT
iajs-4097	47	26	)	)	PUNCT
iajs-4097	47	27	(	(	PUNCT
iajs-4097	47	28	)	)	PUNCT
iajs-4097	47	29	(	(	PUNCT
iajs-4097	47	30	⁄	⁄	PROPN
iajs-4097	47	31	)	)	PUNCT
iajs-4097	47	32	also	also	ADV
iajs-4097	47	33	boubaker	boubaker	NOUN
iajs-4097	47	34	polynomials	polynomial	NOUN
iajs-4097	47	35	can	can	AUX
iajs-4097	47	36	be	be	AUX
iajs-4097	47	37	computed	compute	VERB
iajs-4097	47	38	by	by	ADP
iajs-4097	47	39	used	use	VERB
iajs-4097	47	40	the	the	DET
iajs-4097	47	41	iterative	iterative	NOUN
iajs-4097	47	42	procedure	procedure	NOUN
iajs-4097	47	43	as	as	SCONJ
iajs-4097	47	44	follows	follow	VERB
iajs-4097	47	45	:	:	PUNCT
iajs-4097	47	46	(	(	PUNCT
iajs-4097	47	47	)	)	PUNCT
iajs-4097	47	48	(	(	PUNCT
iajs-4097	47	49	)	)	PUNCT
iajs-4097	47	50	(	(	PUNCT
iajs-4097	47	51	)	)	PUNCT
iajs-4097	47	52	,	,	PUNCT
iajs-4097	47	53	.	.	PUNCT
iajs-4097	48	1	furthermore	furthermore	ADV
iajs-4097	48	2	,	,	PUNCT
iajs-4097	48	3	suppose	suppose	VERB
iajs-4097	48	4	that	that	SCONJ
iajs-4097	48	5	the	the	DET
iajs-4097	48	6	unknown	unknown	ADJ
iajs-4097	48	7	function	function	NOUN
iajs-4097	48	8	(	(	PUNCT
iajs-4097	48	9	)	)	PUNCT
iajs-4097	48	10	,	,	PUNCT
iajs-4097	48	11	may	may	AUX
iajs-4097	48	12	be	be	AUX
iajs-4097	48	13	approximated	approximate	VERB
iajs-4097	48	14	by	by	ADP
iajs-4097	48	15	applying	apply	VERB
iajs-4097	48	16	the	the	DET
iajs-4097	48	17	double	double	ADJ
iajs-4097	48	18	series	series	NOUN
iajs-4097	48	19	based	base	VERB
iajs-4097	48	20	on	on	ADP
iajs-4097	48	21	the	the	DET
iajs-4097	48	22	boubaker	boubaker	NOUN
iajs-4097	48	23	polynomial	polynomial	NOUN
iajs-4097	48	24	:	:	PUNCT
iajs-4097	48	25	(	(	PUNCT
iajs-4097	48	26	)	)	PUNCT
iajs-4097	48	27	∑	∑	ADP
iajs-4097	48	28	∑	∑	PUNCT
iajs-4097	48	29	(	(	PUNCT
iajs-4097	48	30	)	)	PUNCT
iajs-4097	48	31	(	(	PUNCT
iajs-4097	48	32	)	)	PUNCT
iajs-4097	48	33	(	(	PUNCT
iajs-4097	48	34	)	)	PUNCT
iajs-4097	48	35	(	(	PUNCT
iajs-4097	48	36	)	)	PUNCT
iajs-4097	48	37	,	,	PUNCT
iajs-4097	48	38	(	(	PUNCT
iajs-4097	48	39	8)	8)	NUM
iajs-4097	48	40	where	where	SCONJ
iajs-4097	48	41	(	(	PUNCT
iajs-4097	48	42	)	)	PUNCT
iajs-4097	48	43	,	,	PUNCT
iajs-4097	48	44	(	(	PUNCT
iajs-4097	48	45	)	)	PUNCT
iajs-4097	48	46	(	(	PUNCT
iajs-4097	48	47	)	)	PUNCT
iajs-4097	48	48	-	-	PUNCT
iajs-4097	48	49	,	,	PUNCT
iajs-4097	48	50	(	(	PUNCT
iajs-4097	48	51	)	)	PUNCT
iajs-4097	48	52	,	,	PUNCT
iajs-4097	48	53	(	(	PUNCT
iajs-4097	48	54	)	)	PUNCT
iajs-4097	48	55	(	(	PUNCT
iajs-4097	48	56	)	)	PUNCT
iajs-4097	48	57	,	,	PUNCT
iajs-4097	48	58	and	and	CCONJ
iajs-4097	48	59	[	[	PUNCT
iajs-4097	48	60	]	]	X
iajs-4097	48	61	,	,	PUNCT
iajs-4097	48	62	where	where	SCONJ
iajs-4097	48	63	are	be	AUX
iajs-4097	48	64	the	the	DET
iajs-4097	48	65	boubaker	boubaker	NOUN
iajs-4097	48	66	polynomials	polynomial	NOUN
iajs-4097	48	67	coefficients	coefficient	NOUN
iajs-4097	48	68	will	will	AUX
iajs-4097	48	69	be	be	AUX
iajs-4097	48	70	evaluated	evaluate	VERB
iajs-4097	48	71	latter	latter	ADV
iajs-4097	48	72	.	.	PUNCT
iajs-4097	49	1	ihjpas	ihjpas	PROPN
iajs-4097	49	2	.	.	PUNCT
iajs-4097	50	1	2025,38(4	2025,38(4	X
iajs-4097	50	2	)	)	PUNCT
iajs-4097	51	1	379	379	NUM
iajs-4097	51	2	the	the	DET
iajs-4097	51	3	derivatives	derivative	NOUN
iajs-4097	51	4	of	of	ADP
iajs-4097	51	5	(	(	PUNCT
iajs-4097	51	6	)	)	PUNCT
iajs-4097	51	7	may	may	AUX
iajs-4097	51	8	be	be	AUX
iajs-4097	51	9	transformed	transform	VERB
iajs-4097	51	10	into	into	ADP
iajs-4097	51	11	matrices	matrix	NOUN
iajs-4097	51	12	by	by	ADP
iajs-4097	51	13	utilizing	utilize	VERB
iajs-4097	51	14	the	the	DET
iajs-4097	51	15	following	follow	VERB
iajs-4097	51	16	formulation	formulation	NOUN
iajs-4097	51	17	:	:	PUNCT
iajs-4097	51	18	(	(	PUNCT
iajs-4097	51	19	)	)	PUNCT
iajs-4097	51	20	(	(	PUNCT
iajs-4097	51	21	)	)	PUNCT
iajs-4097	51	22	(	(	PUNCT
iajs-4097	51	23	)	)	PUNCT
iajs-4097	51	24	(	(	PUNCT
iajs-4097	51	25	)	)	PUNCT
iajs-4097	51	26	(	(	PUNCT
iajs-4097	51	27	)	)	PUNCT
iajs-4097	51	28	(	(	PUNCT
iajs-4097	51	29	)	)	PUNCT
iajs-4097	51	30	(	(	PUNCT
iajs-4097	51	31	)	)	PUNCT
iajs-4097	51	32	(	(	PUNCT
iajs-4097	51	33	9	9	X
iajs-4097	51	34	)	)	PUNCT
iajs-4097	51	35	the	the	DET
iajs-4097	51	36	derivatives	derivative	NOUN
iajs-4097	51	37	matrix	matrix	NOUN
iajs-4097	51	38	of	of	ADP
iajs-4097	51	39	boubaker	boubaker	NOUN
iajs-4097	51	40	polynomials	polynomial	NOUN
iajs-4097	51	41	denoted	denote	VERB
iajs-4097	51	42	by	by	ADP
iajs-4097	51	43	,	,	PUNCT
iajs-4097	51	44	and	and	CCONJ
iajs-4097	51	45	if	if	SCONJ
iajs-4097	51	46	is	be	AUX
iajs-4097	51	47	even	even	ADV
iajs-4097	51	48	or	or	CCONJ
iajs-4097	51	49	is	be	AUX
iajs-4097	51	50	odd	odd	ADJ
iajs-4097	51	51	,	,	PUNCT
iajs-4097	51	52	defined	define	VERB
iajs-4097	51	53	as	as	ADP
iajs-4097	51	54	follows	follow	VERB
iajs-4097	51	55	,	,	PUNCT
iajs-4097	51	56	respectively	respectively	ADV
iajs-4097	51	57	:	:	PUNCT
iajs-4097	51	58	[	[	PUNCT
iajs-4097	51	59	]	]	X
iajs-4097	51	60	,	,	PUNCT
iajs-4097	51	61	[	[	X
iajs-4097	51	62	]	]	X
iajs-4097	51	63	.	.	PUNCT
iajs-4097	52	1	the	the	DET
iajs-4097	52	2	following	follow	VERB
iajs-4097	52	3	relations	relation	NOUN
iajs-4097	52	4	can	can	AUX
iajs-4097	52	5	be	be	AUX
iajs-4097	52	6	used	use	VERB
iajs-4097	52	7	to	to	PART
iajs-4097	52	8	calculate	calculate	VERB
iajs-4097	52	9	the	the	DET
iajs-4097	52	10	components	component	NOUN
iajs-4097	52	11	{	{	PUNCT
iajs-4097	52	12	}	}	PUNCT
iajs-4097	52	13	0	0	NUM
iajs-4097	52	14	1	1	NUM
iajs-4097	52	15	:	:	PUNCT
iajs-4097	52	16	(	(	PUNCT
iajs-4097	52	17	)	)	PUNCT
iajs-4097	52	18	(	(	PUNCT
iajs-4097	52	19	)	)	PUNCT
iajs-4097	52	20	∏	∏	PROPN
iajs-4097	52	21	(	(	PUNCT
iajs-4097	52	22	)	)	PUNCT
iajs-4097	52	23	0	0	NUM
iajs-4097	52	24	1	1	NUM
iajs-4097	52	25	.	.	ADP
iajs-4097	52	26	3.3	3.3	NUM
iajs-4097	52	27	.	.	PUNCT
iajs-4097	53	1	the	the	DET
iajs-4097	53	2	operational	operational	ADJ
iajs-4097	53	3	matrices	matrix	NOUN
iajs-4097	53	4	for	for	ADP
iajs-4097	53	5	bernoulli	bernoulli	NOUN
iajs-4097	53	6	polynomials	polynomial	VERB
iajs-4097	53	7	bernoulli	bernoulli	NOUN
iajs-4097	53	8	polynomials	polynomial	NOUN
iajs-4097	53	9	(	(	PUNCT
iajs-4097	53	10	)	)	PUNCT
iajs-4097	53	11	of	of	ADP
iajs-4097	53	12	degree	degree	NOUN
iajs-4097	53	13	is	be	AUX
iajs-4097	53	14	given	give	VERB
iajs-4097	53	15	by	by	ADP
iajs-4097	53	16	(	(	PUNCT
iajs-4097	53	17	27,38,39	27,38,39	NUM
iajs-4097	53	18	)	)	PUNCT
iajs-4097	53	19	:	:	PUNCT
iajs-4097	53	20	(	(	PUNCT
iajs-4097	53	21	)	)	PUNCT
iajs-4097	53	22	∑	∑	INTJ
iajs-4097	53	23	.	.	PUNCT
iajs-4097	54	1	/	/	PUNCT
iajs-4097	54	2	,	,	PUNCT
iajs-4097	54	3	where	where	SCONJ
iajs-4097	54	4	is	be	AUX
iajs-4097	54	5	the	the	DET
iajs-4097	54	6	bernoulli	bernoulli	NOUN
iajs-4097	54	7	number	number	NOUN
iajs-4097	54	8	,	,	PUNCT
iajs-4097	54	9	such	such	ADJ
iajs-4097	54	10	that	that	SCONJ
iajs-4097	54	11	∑	∑	PUNCT
iajs-4097	54	12	(	(	PUNCT
iajs-4097	54	13	)	)	PUNCT
iajs-4097	54	14	.	.	PUNCT
iajs-4097	55	1	/∑	/∑	PRON
iajs-4097	55	2	,	,	PUNCT
iajs-4097	55	3	for	for	ADP
iajs-4097	55	4	,	,	PUNCT
iajs-4097	55	5	,	,	PUNCT
iajs-4097	55	6	the	the	DET
iajs-4097	55	7	first	first	ADJ
iajs-4097	55	8	bernoulli	bernoulli	NOUN
iajs-4097	55	9	numbers	number	NOUN
iajs-4097	55	10	define	define	VERB
iajs-4097	55	11	as	as	SCONJ
iajs-4097	55	12	follows	follow	VERB
iajs-4097	55	13	:	:	PUNCT
iajs-4097	55	14	.	.	PUNCT
iajs-4097	56	1	additionally	additionally	ADV
iajs-4097	56	2	,	,	PUNCT
iajs-4097	56	3	the	the	DET
iajs-4097	56	4	unknown	unknown	ADJ
iajs-4097	56	5	function	function	NOUN
iajs-4097	56	6	(	(	PUNCT
iajs-4097	56	7	)	)	PUNCT
iajs-4097	56	8	can	can	AUX
iajs-4097	56	9	be	be	AUX
iajs-4097	56	10	approximated	approximate	VERB
iajs-4097	56	11	by	by	ADP
iajs-4097	56	12	using	use	VERB
iajs-4097	56	13	the	the	DET
iajs-4097	56	14	following	follow	VERB
iajs-4097	56	15	dot	dot	NOUN
iajs-4097	56	16	product	product	NOUN
iajs-4097	56	17	based	base	VERB
iajs-4097	56	18	on	on	ADP
iajs-4097	56	19	the	the	DET
iajs-4097	56	20	bernoulli	bernoulli	NOUN
iajs-4097	56	21	polynomials	polynomial	NOUN
iajs-4097	56	22	:	:	PUNCT
iajs-4097	56	23	(	(	PUNCT
iajs-4097	56	24	)	)	PUNCT
iajs-4097	56	25	∑	∑	ADP
iajs-4097	56	26	∑	∑	PUNCT
iajs-4097	56	27	(	(	PUNCT
iajs-4097	56	28	)	)	PUNCT
iajs-4097	56	29	(	(	PUNCT
iajs-4097	56	30	)	)	PUNCT
iajs-4097	56	31	(	(	PUNCT
iajs-4097	56	32	)	)	PUNCT
iajs-4097	56	33	(	(	PUNCT
iajs-4097	56	34	)	)	PUNCT
iajs-4097	56	35	,	,	PUNCT
iajs-4097	56	36	(	(	PUNCT
iajs-4097	56	37	10	10	NUM
iajs-4097	56	38	)	)	PUNCT
iajs-4097	56	39	where	where	SCONJ
iajs-4097	56	40	(	(	PUNCT
iajs-4097	56	41	)	)	PUNCT
iajs-4097	56	42	,	,	PUNCT
iajs-4097	56	43	(	(	PUNCT
iajs-4097	56	44	)	)	PUNCT
iajs-4097	56	45	(	(	PUNCT
iajs-4097	56	46	)	)	PUNCT
iajs-4097	56	47	-	-	PUNCT
iajs-4097	56	48	,	,	PUNCT
iajs-4097	56	49	(	(	PUNCT
iajs-4097	56	50	)	)	PUNCT
iajs-4097	56	51	,	,	PUNCT
iajs-4097	56	52	(	(	PUNCT
iajs-4097	56	53	)	)	PUNCT
iajs-4097	56	54	(	(	PUNCT
iajs-4097	56	55	)	)	PUNCT
iajs-4097	56	56	,	,	PUNCT
iajs-4097	56	57	and	and	CCONJ
iajs-4097	56	58	[	[	PUNCT
iajs-4097	56	59	]	]	X
iajs-4097	56	60	,	,	PUNCT
iajs-4097	56	61	where	where	SCONJ
iajs-4097	56	62	is	be	AUX
iajs-4097	56	63	the	the	DET
iajs-4097	56	64	unknown	unknown	ADJ
iajs-4097	56	65	coefficients	coefficient	NOUN
iajs-4097	56	66	of	of	ADP
iajs-4097	56	67	bernoulli	bernoulli	NOUN
iajs-4097	56	68	polynomials	polynomial	NOUN
iajs-4097	56	69	determined	determine	VERB
iajs-4097	56	70	latter	latter	ADV
iajs-4097	56	71	.	.	PUNCT
iajs-4097	57	1	moreover	moreover	ADV
iajs-4097	57	2	,	,	PUNCT
iajs-4097	57	3	the	the	DET
iajs-4097	57	4	derivatives	derivative	NOUN
iajs-4097	57	5	of	of	ADP
iajs-4097	57	6	(	(	PUNCT
iajs-4097	57	7	)	)	PUNCT
iajs-4097	57	8	can	can	AUX
iajs-4097	57	9	be	be	AUX
iajs-4097	57	10	written	write	VERB
iajs-4097	57	11	in	in	ADP
iajs-4097	57	12	matrices	matrix	NOUN
iajs-4097	57	13	form	form	NOUN
iajs-4097	57	14	by	by	ADP
iajs-4097	57	15	the	the	DET
iajs-4097	57	16	following	follow	VERB
iajs-4097	57	17	formula	formula	NOUN
iajs-4097	57	18	:	:	PUNCT
iajs-4097	57	19	(	(	PUNCT
iajs-4097	57	20	)	)	PUNCT
iajs-4097	57	21	(	(	PUNCT
iajs-4097	57	22	)	)	PUNCT
iajs-4097	57	23	(	(	PUNCT
iajs-4097	57	24	)	)	PUNCT
iajs-4097	57	25	(	(	PUNCT
iajs-4097	57	26	)	)	PUNCT
iajs-4097	57	27	(	(	PUNCT
iajs-4097	57	28	)	)	PUNCT
iajs-4097	57	29	(	(	PUNCT
iajs-4097	57	30	)	)	PUNCT
iajs-4097	57	31	(	(	PUNCT
iajs-4097	57	32	)	)	PUNCT
iajs-4097	57	33	(	(	PUNCT
iajs-4097	57	34	11	11	NUM
iajs-4097	57	35	)	)	PUNCT
iajs-4097	57	36	where	where	SCONJ
iajs-4097	57	37	is	be	AUX
iajs-4097	57	38	the	the	DET
iajs-4097	57	39	derivative	derivative	ADJ
iajs-4097	57	40	matrix	matrix	NOUN
iajs-4097	57	41	of	of	ADP
iajs-4097	57	42	bernoulli	bernoulli	NOUN
iajs-4097	57	43	polynomials	polynomial	NOUN
iajs-4097	57	44	,	,	PUNCT
iajs-4097	57	45	is	be	AUX
iajs-4097	57	46	given	give	VERB
iajs-4097	57	47	as	as	ADP
iajs-4097	57	48	below	below	ADV
iajs-4097	57	49	:	:	PUNCT
iajs-4097	57	50	ihjpas	ihjpas	PROPN
iajs-4097	57	51	.	.	PUNCT
iajs-4097	58	1	2025,38(4	2025,38(4	NOUN
iajs-4097	58	2	)	)	PUNCT
iajs-4097	58	3	380	380	NUM
iajs-4097	58	4	[	[	PUNCT
iajs-4097	58	5	]	]	X
iajs-4097	58	6	,	,	PUNCT
iajs-4097	58	7	3.4	3.4	NUM
iajs-4097	58	8	.	.	PUNCT
iajs-4097	59	1	the	the	DET
iajs-4097	59	2	operational	operational	ADJ
iajs-4097	59	3	matrices	matrix	NOUN
iajs-4097	59	4	for	for	ADP
iajs-4097	59	5	legendre	legendre	PROPN
iajs-4097	59	6	polynomials	polynomials	PROPN
iajs-4097	59	7	legendre	legendre	PROPN
iajs-4097	59	8	polynomials	polynomial	NOUN
iajs-4097	59	9	(	(	PUNCT
iajs-4097	59	10	)	)	PUNCT
iajs-4097	59	11	of	of	ADP
iajs-4097	59	12	degree	degree	NOUN
iajs-4097	59	13	is	be	AUX
iajs-4097	59	14	given	give	VERB
iajs-4097	59	15	as	as	ADP
iajs-4097	59	16	follows	follow	NOUN
iajs-4097	59	17	(	(	PUNCT
iajs-4097	59	18	24,40	24,40	NUM
iajs-4097	59	19	)	)	PUNCT
iajs-4097	59	20	:	:	PUNCT
iajs-4097	59	21	(	(	PUNCT
iajs-4097	59	22	)	)	PUNCT
iajs-4097	59	23	,	,	PUNCT
iajs-4097	59	24	(	(	PUNCT
iajs-4097	59	25	)	)	PUNCT
iajs-4097	59	26	,	,	PUNCT
iajs-4097	59	27	…	…	PUNCT
iajs-4097	59	28	.	.	PUNCT
iajs-4097	59	29	,	,	PUNCT
iajs-4097	59	30	(	(	PUNCT
iajs-4097	59	31	)	)	PUNCT
iajs-4097	59	32	(	(	PUNCT
iajs-4097	59	33	)	)	PUNCT
iajs-4097	59	34	(	(	PUNCT
iajs-4097	59	35	)	)	PUNCT
iajs-4097	59	36	(	(	PUNCT
iajs-4097	59	37	)	)	PUNCT
iajs-4097	59	38	,	,	PUNCT
iajs-4097	59	39	.	.	PUNCT
iajs-4097	60	1	furthermore	furthermore	ADV
iajs-4097	60	2	,	,	PUNCT
iajs-4097	60	3	the	the	DET
iajs-4097	60	4	legendre	legendre	PROPN
iajs-4097	60	5	polynomials	polynomial	NOUN
iajs-4097	60	6	will	will	AUX
iajs-4097	60	7	be	be	AUX
iajs-4097	60	8	get	get	VERB
iajs-4097	60	9	by	by	ADP
iajs-4097	60	10	using	use	VERB
iajs-4097	60	11	the	the	DET
iajs-4097	60	12	formula	formula	NOUN
iajs-4097	60	13	as	as	SCONJ
iajs-4097	60	14	follows	follow	VERB
iajs-4097	60	15	:	:	PUNCT
iajs-4097	60	16	(	(	PUNCT
iajs-4097	60	17	)	)	PUNCT
iajs-4097	60	18	∑	∑	PUNCT
iajs-4097	60	19	(	(	PUNCT
iajs-4097	60	20	)	)	PUNCT
iajs-4097	60	21	(	(	PUNCT
iajs-4097	60	22	)	)	PUNCT
iajs-4097	60	23	(	(	PUNCT
iajs-4097	60	24	)	)	PUNCT
iajs-4097	60	25	(	(	PUNCT
iajs-4097	60	26	)	)	PUNCT
iajs-4097	60	27	(	(	PUNCT
iajs-4097	60	28	)	)	PUNCT
iajs-4097	60	29	by	by	ADP
iajs-4097	60	30	using	use	VERB
iajs-4097	60	31	,	,	PUNCT
iajs-4097	60	32	the	the	DET
iajs-4097	60	33	linear	linear	ADJ
iajs-4097	60	34	combination	combination	NOUN
iajs-4097	60	35	as	as	ADP
iajs-4097	60	36	follow	follow	NOUN
iajs-4097	60	37	can	can	AUX
iajs-4097	60	38	be	be	AUX
iajs-4097	60	39	approximated	approximate	VERB
iajs-4097	60	40	the	the	DET
iajs-4097	60	41	unknown	unknown	ADJ
iajs-4097	60	42	function	function	NOUN
iajs-4097	60	43	(	(	PUNCT
iajs-4097	60	44	)	)	PUNCT
iajs-4097	60	45	.	.	PUNCT
iajs-4097	61	1	(	(	PUNCT
iajs-4097	61	2	)	)	PUNCT
iajs-4097	61	3	∑	∑	ADP
iajs-4097	61	4	∑	∑	PUNCT
iajs-4097	61	5	(	(	PUNCT
iajs-4097	61	6	)	)	PUNCT
iajs-4097	61	7	(	(	PUNCT
iajs-4097	61	8	)	)	PUNCT
iajs-4097	61	9	(	(	PUNCT
iajs-4097	61	10	)	)	PUNCT
iajs-4097	61	11	(	(	PUNCT
iajs-4097	61	12	)	)	PUNCT
iajs-4097	61	13	,	,	PUNCT
iajs-4097	61	14	(	(	PUNCT
iajs-4097	61	15	12	12	NUM
iajs-4097	61	16	)	)	PUNCT
iajs-4097	61	17	where	where	SCONJ
iajs-4097	61	18	(	(	PUNCT
iajs-4097	61	19	)	)	PUNCT
iajs-4097	61	20	,	,	PUNCT
iajs-4097	61	21	(	(	PUNCT
iajs-4097	61	22	)	)	PUNCT
iajs-4097	61	23	(	(	PUNCT
iajs-4097	61	24	)	)	PUNCT
iajs-4097	61	25	-	-	PUNCT
iajs-4097	61	26	,	,	PUNCT
iajs-4097	61	27	(	(	PUNCT
iajs-4097	61	28	)	)	PUNCT
iajs-4097	61	29	,	,	PUNCT
iajs-4097	61	30	(	(	PUNCT
iajs-4097	61	31	)	)	PUNCT
iajs-4097	61	32	(	(	PUNCT
iajs-4097	61	33	)	)	PUNCT
iajs-4097	61	34	,	,	PUNCT
iajs-4097	61	35	and	and	CCONJ
iajs-4097	61	36	[	[	PUNCT
iajs-4097	61	37	]	]	X
iajs-4097	61	38	,	,	PUNCT
iajs-4097	61	39	moreover	moreover	ADV
iajs-4097	61	40	,	,	PUNCT
iajs-4097	61	41	is	be	AUX
iajs-4097	61	42	the	the	DET
iajs-4097	61	43	coefficients	coefficient	NOUN
iajs-4097	61	44	of	of	ADP
iajs-4097	61	45	legendre	legendre	PROPN
iajs-4097	61	46	polynomials	polynomial	NOUN
iajs-4097	61	47	evaluated	evaluate	VERB
iajs-4097	61	48	latter	latter	ADV
iajs-4097	61	49	.	.	PUNCT
iajs-4097	62	1	the	the	DET
iajs-4097	62	2	derivative	derivative	NOUN
iajs-4097	62	3	of	of	ADP
iajs-4097	62	4	(	(	PUNCT
iajs-4097	62	5	)	)	PUNCT
iajs-4097	62	6	with	with	ADP
iajs-4097	62	7	a	a	DET
iajs-4097	62	8	respect	respect	NOUN
iajs-4097	62	9	to	to	ADP
iajs-4097	62	10	and	and	CCONJ
iajs-4097	62	11	may	may	AUX
iajs-4097	62	12	be	be	AUX
iajs-4097	62	13	written	write	VERB
iajs-4097	62	14	in	in	ADP
iajs-4097	62	15	matrices	matrix	NOUN
iajs-4097	62	16	as	as	SCONJ
iajs-4097	62	17	follows	follow	VERB
iajs-4097	62	18	:	:	PUNCT
iajs-4097	62	19	(	(	PUNCT
iajs-4097	62	20	)	)	PUNCT
iajs-4097	62	21	(	(	PUNCT
iajs-4097	62	22	)	)	PUNCT
iajs-4097	62	23	(	(	PUNCT
iajs-4097	62	24	)	)	PUNCT
iajs-4097	62	25	(	(	PUNCT
iajs-4097	62	26	)	)	PUNCT
iajs-4097	62	27	(	(	PUNCT
iajs-4097	62	28	)	)	PUNCT
iajs-4097	62	29	(	(	PUNCT
iajs-4097	62	30	)	)	PUNCT
iajs-4097	62	31	(	(	PUNCT
iajs-4097	62	32	)	)	PUNCT
iajs-4097	62	33	(	(	PUNCT
iajs-4097	62	34	13	13	NUM
iajs-4097	62	35	)	)	PUNCT
iajs-4097	62	36	the	the	DET
iajs-4097	62	37	derivatives	derivative	NOUN
iajs-4097	62	38	matrix	matrix	NOUN
iajs-4097	62	39	of	of	ADP
iajs-4097	62	40	the	the	DET
iajs-4097	62	41	legendre	legendre	PROPN
iajs-4097	62	42	polynomials	polynomial	NOUN
iajs-4097	62	43	denoted	denote	VERB
iajs-4097	62	44	by	by	ADP
iajs-4097	62	45	,	,	PUNCT
iajs-4097	62	46	and	and	CCONJ
iajs-4097	62	47	we	we	PRON
iajs-4097	62	48	obtained	obtain	VERB
iajs-4097	62	49	by	by	ADP
iajs-4097	62	50	(	(	PUNCT
iajs-4097	62	51	34	34	NUM
iajs-4097	62	52	):	):	SYM
iajs-4097	62	53	2	2	NUM
iajs-4097	62	54	(	(	PUNCT
iajs-4097	62	55	)	)	PUNCT
iajs-4097	62	56	if	if	SCONJ
iajs-4097	62	57	is	be	AUX
iajs-4097	62	58	even	even	ADV
iajs-4097	62	59	then	then	ADV
iajs-4097	62	60	,	,	PUNCT
iajs-4097	62	61	and	and	CCONJ
iajs-4097	62	62	if	if	SCONJ
iajs-4097	62	63	is	be	AUX
iajs-4097	62	64	odd	odd	ADJ
iajs-4097	62	65	then	then	ADV
iajs-4097	62	66	.	.	PUNCT
iajs-4097	63	1	3.5	3.5	NUM
iajs-4097	63	2	.	.	PUNCT
iajs-4097	64	1	the	the	DET
iajs-4097	64	2	operational	operational	ADJ
iajs-4097	64	3	matrices	matrix	NOUN
iajs-4097	64	4	for	for	ADP
iajs-4097	64	5	genocchi	genocchi	PROPN
iajs-4097	64	6	polynomials	polynomial	VERB
iajs-4097	64	7	the	the	DET
iajs-4097	64	8	genocchi	genocchi	PROPN
iajs-4097	64	9	polynomials	polynomial	VERB
iajs-4097	64	10	(	(	PUNCT
iajs-4097	64	11	)	)	PUNCT
iajs-4097	64	12	of	of	ADP
iajs-4097	64	13	degree	degree	NOUN
iajs-4097	64	14	is	be	AUX
iajs-4097	64	15	given	give	VERB
iajs-4097	64	16	by	by	ADP
iajs-4097	64	17	(	(	PUNCT
iajs-4097	64	18	20	20	NUM
iajs-4097	64	19	)	)	PUNCT
iajs-4097	64	20	:	:	PUNCT
iajs-4097	64	21	(	(	PUNCT
iajs-4097	64	22	)	)	PUNCT
iajs-4097	64	23	∑	∑	PUNCT
iajs-4097	64	24	.	.	PUNCT
iajs-4097	64	25	/	/	PUNCT
iajs-4097	64	26	,	,	PUNCT
iajs-4097	64	27	by	by	ADP
iajs-4097	64	28	using	use	VERB
iajs-4097	64	29	,	,	PUNCT
iajs-4097	64	30	the	the	DET
iajs-4097	64	31	linear	linear	ADJ
iajs-4097	64	32	combination	combination	NOUN
iajs-4097	64	33	as	as	ADP
iajs-4097	64	34	follow	follow	NOUN
iajs-4097	64	35	can	can	AUX
iajs-4097	64	36	be	be	AUX
iajs-4097	64	37	approximated	approximate	VERB
iajs-4097	64	38	the	the	DET
iajs-4097	64	39	function	function	NOUN
iajs-4097	64	40	(	(	PUNCT
iajs-4097	64	41	)	)	PUNCT
iajs-4097	64	42	.	.	PUNCT
iajs-4097	65	1	(	(	PUNCT
iajs-4097	65	2	)	)	PUNCT
iajs-4097	65	3	∑	∑	ADP
iajs-4097	65	4	∑	∑	PUNCT
iajs-4097	65	5	(	(	PUNCT
iajs-4097	65	6	)	)	PUNCT
iajs-4097	65	7	(	(	PUNCT
iajs-4097	65	8	)	)	PUNCT
iajs-4097	65	9	(	(	PUNCT
iajs-4097	65	10	)	)	PUNCT
iajs-4097	65	11	(	(	PUNCT
iajs-4097	65	12	)	)	PUNCT
iajs-4097	65	13	,	,	PUNCT
iajs-4097	65	14	(	(	PUNCT
iajs-4097	65	15	14	14	NUM
iajs-4097	65	16	)	)	PUNCT
iajs-4097	65	17	where	where	SCONJ
iajs-4097	65	18	(	(	PUNCT
iajs-4097	65	19	)	)	PUNCT
iajs-4097	65	20	,	,	PUNCT
iajs-4097	65	21	(	(	PUNCT
iajs-4097	65	22	)	)	PUNCT
iajs-4097	65	23	(	(	PUNCT
iajs-4097	65	24	)	)	PUNCT
iajs-4097	65	25	,	,	PUNCT
iajs-4097	65	26	(	(	PUNCT
iajs-4097	65	27	)	)	PUNCT
iajs-4097	65	28	(	(	PUNCT
iajs-4097	65	29	(	(	PUNCT
iajs-4097	65	30	)	)	PUNCT
iajs-4097	65	31	(	(	PUNCT
iajs-4097	65	32	)	)	PUNCT
iajs-4097	65	33	)	)	PUNCT
iajs-4097	66	1	and	and	CCONJ
iajs-4097	66	2	[	[	PUNCT
iajs-4097	66	3	]	]	X
iajs-4097	66	4	moreover	moreover	ADV
iajs-4097	66	5	,	,	PUNCT
iajs-4097	66	6	is	be	AUX
iajs-4097	66	7	the	the	DET
iajs-4097	66	8	unknown	unknown	ADJ
iajs-4097	66	9	coefficient	coefficient	NOUN
iajs-4097	66	10	of	of	ADP
iajs-4097	66	11	genocchi	genocchi	PROPN
iajs-4097	66	12	polynomials	polynomial	NOUN
iajs-4097	66	13	evaluated	evaluate	VERB
iajs-4097	66	14	latter	latter	ADV
iajs-4097	66	15	.	.	PUNCT
iajs-4097	67	1	the	the	DET
iajs-4097	67	2	derivatives	derivative	NOUN
iajs-4097	67	3	of	of	ADP
iajs-4097	67	4	(	(	PUNCT
iajs-4097	67	5	)	)	PUNCT
iajs-4097	67	6	can	can	AUX
iajs-4097	67	7	be	be	AUX
iajs-4097	67	8	written	write	VERB
iajs-4097	67	9	in	in	ADP
iajs-4097	67	10	matrices	matrix	NOUN
iajs-4097	67	11	form	form	NOUN
iajs-4097	67	12	as	as	SCONJ
iajs-4097	67	13	follows	follow	VERB
iajs-4097	67	14	:	:	PUNCT
iajs-4097	67	15	(	(	PUNCT
iajs-4097	67	16	)	)	PUNCT
iajs-4097	67	17	(	(	PUNCT
iajs-4097	67	18	)	)	PUNCT
iajs-4097	67	19	(	(	PUNCT
iajs-4097	67	20	)	)	PUNCT
iajs-4097	67	21	(	(	PUNCT
iajs-4097	67	22	)	)	PUNCT
iajs-4097	67	23	(	(	PUNCT
iajs-4097	67	24	)	)	PUNCT
iajs-4097	67	25	(	(	PUNCT
iajs-4097	67	26	)	)	PUNCT
iajs-4097	67	27	(	(	PUNCT
iajs-4097	67	28	)	)	PUNCT
iajs-4097	67	29	(	(	PUNCT
iajs-4097	67	30	15	15	X
iajs-4097	67	31	)	)	PUNCT
iajs-4097	67	32	the	the	DET
iajs-4097	67	33	derivative	derivative	ADJ
iajs-4097	67	34	matrix	matrix	NOUN
iajs-4097	67	35	of	of	ADP
iajs-4097	67	36	genocchi	genocchi	PROPN
iajs-4097	67	37	polynomials	polynomial	NOUN
iajs-4097	67	38	is	be	AUX
iajs-4097	67	39	denoted	denote	VERB
iajs-4097	67	40	by	by	ADP
iajs-4097	67	41	and	and	CCONJ
iajs-4097	67	42	given	give	VERB
iajs-4097	67	43	by	by	ADP
iajs-4097	67	44	:	:	PUNCT
iajs-4097	67	45	[	[	PUNCT
iajs-4097	67	46	]	]	X
iajs-4097	67	47	,	,	PUNCT
iajs-4097	67	48	ihjpas	ihjpas	PROPN
iajs-4097	67	49	.	.	PUNCT
iajs-4097	68	1	2025,38(4	2025,38(4	NOUN
iajs-4097	68	2	)	)	PUNCT
iajs-4097	69	1	381	381	NUM
iajs-4097	69	2	3.6	3.6	NUM
iajs-4097	69	3	.	.	PUNCT
iajs-4097	70	1	the	the	DET
iajs-4097	70	2	operational	operational	ADJ
iajs-4097	70	3	matrices	matrix	NOUN
iajs-4097	70	4	for	for	ADP
iajs-4097	70	5	chebyshev	chebyshev	NOUN
iajs-4097	70	6	polynomials	polynomial	NOUN
iajs-4097	70	7	chebyshev	chebyshev	NOUN
iajs-4097	70	8	polynomials	polynomial	NOUN
iajs-4097	70	9	(	(	PUNCT
iajs-4097	70	10	)	)	PUNCT
iajs-4097	70	11	of	of	ADP
iajs-4097	70	12	degree	degree	NOUN
iajs-4097	70	13	are	be	AUX
iajs-4097	70	14	defined	define	VERB
iajs-4097	70	15	as	as	ADP
iajs-4097	70	16	follows	follow	VERB
iajs-4097	70	17	(	(	PUNCT
iajs-4097	70	18	25,41	25,41	NUM
iajs-4097	70	19	)	)	PUNCT
iajs-4097	70	20	:	:	PUNCT
iajs-4097	70	21	(	(	PUNCT
iajs-4097	70	22	)	)	PUNCT
iajs-4097	70	23	∑	∑	PUNCT
iajs-4097	70	24	(	(	PUNCT
iajs-4097	70	25	)	)	PUNCT
iajs-4097	70	26	(	(	PUNCT
iajs-4097	70	27	)	)	PUNCT
iajs-4097	70	28	(	(	PUNCT
iajs-4097	70	29	)	)	PUNCT
iajs-4097	70	30	(	(	PUNCT
iajs-4097	70	31	)	)	PUNCT
iajs-4097	70	32	(	(	PUNCT
iajs-4097	70	33	)	)	PUNCT
iajs-4097	70	34	,	,	PUNCT
iajs-4097	70	35	in	in	ADP
iajs-4097	70	36	addition	addition	NOUN
iajs-4097	70	37	,	,	PUNCT
iajs-4097	70	38	the	the	DET
iajs-4097	70	39	function	function	NOUN
iajs-4097	70	40	(	(	PUNCT
iajs-4097	70	41	)	)	PUNCT
iajs-4097	70	42	may	may	AUX
iajs-4097	70	43	be	be	AUX
iajs-4097	70	44	approximated	approximate	VERB
iajs-4097	70	45	by	by	ADP
iajs-4097	70	46	using	use	VERB
iajs-4097	70	47	the	the	DET
iajs-4097	70	48	linear	linear	ADJ
iajs-4097	70	49	combination	combination	NOUN
iajs-4097	70	50	as	as	ADP
iajs-4097	70	51	following	follow	VERB
iajs-4097	70	52	:	:	PUNCT
iajs-4097	70	53	(	(	PUNCT
iajs-4097	70	54	)	)	PUNCT
iajs-4097	70	55	∑	∑	ADP
iajs-4097	70	56	∑	∑	PUNCT
iajs-4097	70	57	(	(	PUNCT
iajs-4097	70	58	)	)	PUNCT
iajs-4097	70	59	(	(	PUNCT
iajs-4097	70	60	)	)	PUNCT
iajs-4097	70	61	(	(	PUNCT
iajs-4097	70	62	)	)	PUNCT
iajs-4097	70	63	(	(	PUNCT
iajs-4097	70	64	)	)	PUNCT
iajs-4097	70	65	,	,	PUNCT
iajs-4097	70	66	(	(	PUNCT
iajs-4097	70	67	16	16	NUM
iajs-4097	70	68	)	)	PUNCT
iajs-4097	70	69	where	where	SCONJ
iajs-4097	70	70	(	(	PUNCT
iajs-4097	70	71	)	)	PUNCT
iajs-4097	70	72	,	,	PUNCT
iajs-4097	70	73	(	(	PUNCT
iajs-4097	70	74	)	)	PUNCT
iajs-4097	70	75	(	(	PUNCT
iajs-4097	70	76	)	)	PUNCT
iajs-4097	70	77	,	,	PUNCT
iajs-4097	70	78	(	(	PUNCT
iajs-4097	70	79	)	)	PUNCT
iajs-4097	70	80	,	,	PUNCT
iajs-4097	70	81	(	(	PUNCT
iajs-4097	70	82	)	)	PUNCT
iajs-4097	70	83	(	(	PUNCT
iajs-4097	70	84	)	)	PUNCT
iajs-4097	70	85	-	-	PUNCT
iajs-4097	70	86	,	,	PUNCT
iajs-4097	70	87	and	and	CCONJ
iajs-4097	70	88	[	[	PUNCT
iajs-4097	70	89	]	]	X
iajs-4097	70	90	,	,	PUNCT
iajs-4097	70	91	furthermore	furthermore	ADV
iajs-4097	70	92	,	,	PUNCT
iajs-4097	70	93	represents	represent	VERB
iajs-4097	70	94	the	the	DET
iajs-4097	70	95	unknown	unknown	ADJ
iajs-4097	70	96	coefficients	coefficient	NOUN
iajs-4097	70	97	of	of	ADP
iajs-4097	70	98	chebyshev	chebyshev	NOUN
iajs-4097	70	99	polynomials	polynomial	NOUN
iajs-4097	70	100	calculated	calculate	VERB
iajs-4097	70	101	latter	latter	ADV
iajs-4097	70	102	.	.	PUNCT
iajs-4097	71	1	the	the	DET
iajs-4097	71	2	derivatives	derivative	NOUN
iajs-4097	71	3	of	of	ADP
iajs-4097	71	4	(	(	PUNCT
iajs-4097	71	5	)	)	PUNCT
iajs-4097	71	6	may	may	AUX
iajs-4097	71	7	be	be	AUX
iajs-4097	71	8	rewritten	rewrite	VERB
iajs-4097	71	9	in	in	ADP
iajs-4097	71	10	matrices	matrix	NOUN
iajs-4097	71	11	using	use	VERB
iajs-4097	71	12	the	the	DET
iajs-4097	71	13	following	follow	VERB
iajs-4097	71	14	formula	formula	NOUN
iajs-4097	71	15	:	:	PUNCT
iajs-4097	71	16	(	(	PUNCT
iajs-4097	71	17	)	)	PUNCT
iajs-4097	71	18	(	(	PUNCT
iajs-4097	71	19	)	)	PUNCT
iajs-4097	71	20	(	(	PUNCT
iajs-4097	71	21	)	)	PUNCT
iajs-4097	71	22	(	(	PUNCT
iajs-4097	71	23	)	)	PUNCT
iajs-4097	71	24	(	(	PUNCT
iajs-4097	71	25	)	)	PUNCT
iajs-4097	71	26	(	(	PUNCT
iajs-4097	71	27	)	)	PUNCT
iajs-4097	71	28	(	(	PUNCT
iajs-4097	71	29	)	)	PUNCT
iajs-4097	71	30	(	(	PUNCT
iajs-4097	71	31	17	17	NUM
iajs-4097	71	32	)	)	PUNCT
iajs-4097	71	33	the	the	DET
iajs-4097	71	34	derivative	derivative	ADJ
iajs-4097	71	35	matrix	matrix	NOUN
iajs-4097	71	36	of	of	ADP
iajs-4097	71	37	the	the	DET
iajs-4097	71	38	chebyshev	chebyshev	NOUN
iajs-4097	71	39	polynomials	polynomial	NOUN
iajs-4097	71	40	is	be	AUX
iajs-4097	71	41	denoted	denote	VERB
iajs-4097	71	42	by	by	ADP
iajs-4097	71	43	,	,	PUNCT
iajs-4097	71	44	and	and	CCONJ
iajs-4097	71	45	obtained	obtain	VERB
iajs-4097	71	46	by	by	ADP
iajs-4097	71	47	:	:	PUNCT
iajs-4097	71	48	{	{	PUNCT
iajs-4097	71	49	if	if	SCONJ
iajs-4097	71	50	is	be	AUX
iajs-4097	71	51	odd	odd	ADJ
iajs-4097	71	52	then	then	ADV
iajs-4097	71	53	and	and	CCONJ
iajs-4097	71	54	if	if	SCONJ
iajs-4097	71	55	is	be	AUX
iajs-4097	71	56	even	even	ADV
iajs-4097	71	57	then	then	ADV
iajs-4097	71	58	,	,	PUNCT
iajs-4097	71	59	and	and	CCONJ
iajs-4097	71	60	,	,	PUNCT
iajs-4097	71	61	and	and	CCONJ
iajs-4097	71	62	for	for	ADP
iajs-4097	71	63	.	.	PROPN
iajs-4097	72	1	3.7	3.7	NUM
iajs-4097	72	2	.	.	PUNCT
iajs-4097	73	1	algorithm	algorithm	PROPN
iajs-4097	73	2	1	1	NUM
iajs-4097	73	3	.	.	PUNCT
iajs-4097	74	1	input	input	NOUN
iajs-4097	74	2	(	(	PUNCT
iajs-4097	74	3	integer	integer	NOUN
iajs-4097	74	4	)	)	PUNCT
iajs-4097	74	5	n.	n.	NOUN
iajs-4097	74	6	input	input	NOUN
iajs-4097	74	7	(	(	PUNCT
iajs-4097	74	8	double	double	ADJ
iajs-4097	74	9	series	series	NOUN
iajs-4097	74	10	)	)	PUNCT
iajs-4097	74	11	tools	tool	NOUN
iajs-4097	74	12	.	.	PUNCT
iajs-4097	75	1	input	input	NOUN
iajs-4097	75	2	(	(	PUNCT
iajs-4097	75	3	array	array	NOUN
iajs-4097	75	4	)	)	PUNCT
iajs-4097	75	5	=	=	SYM
iajs-4097	75	6	,	,	PUNCT
iajs-4097	75	7	(	(	PUNCT
iajs-4097	75	8	initial	initial	ADJ
iajs-4097	75	9	approximation	approximation	NOUN
iajs-4097	75	10	,	,	PUNCT
iajs-4097	75	11	with	with	ADP
iajs-4097	75	12	dimension	dimension	NOUN
iajs-4097	75	13	,	,	PUNCT
iajs-4097	75	14	are	be	AUX
iajs-4097	75	15	chosen	choose	VERB
iajs-4097	75	16	so	so	SCONJ
iajs-4097	75	17	that	that	SCONJ
iajs-4097	75	18	the	the	DET
iajs-4097	75	19	boundary	boundary	ADJ
iajs-4097	75	20	conditions	condition	NOUN
iajs-4097	75	21	are	be	AUX
iajs-4097	75	22	satisfied	satisfied	ADJ
iajs-4097	75	23	)	)	PUNCT
iajs-4097	75	24	.	.	PUNCT
iajs-4097	76	1	2	2	X
iajs-4097	76	2	.	.	PUNCT
iajs-4097	76	3	̂	̂	PUNCT
iajs-4097	76	4	(	(	PUNCT
iajs-4097	76	5	)	)	PUNCT
iajs-4097	76	6	̂	̂	VERB
iajs-4097	76	7	is	be	AUX
iajs-4097	76	8	a	a	DET
iajs-4097	76	9	system	system	NOUN
iajs-4097	76	10	of	of	ADP
iajs-4097	76	11	linear	linear	PROPN
iajs-4097	76	12	algebraic	algebraic	ADJ
iajs-4097	76	13	equations	equation	NOUN
iajs-4097	76	14	that	that	PRON
iajs-4097	76	15	has	have	AUX
iajs-4097	76	16	been	be	AUX
iajs-4097	76	17	solved	solve	VERB
iajs-4097	76	18	and	and	CCONJ
iajs-4097	76	19	is	be	AUX
iajs-4097	76	20	obtained	obtain	VERB
iajs-4097	76	21	.	.	PUNCT
iajs-4097	77	1	go	go	VERB
iajs-4097	77	2	to	to	ADP
iajs-4097	77	3	(	(	PUNCT
iajs-4097	77	4	2	2	NUM
iajs-4097	77	5	)	)	PUNCT
iajs-4097	77	6	.	.	PUNCT
iajs-4097	78	1	2.1	2.1	NUM
iajs-4097	78	2	if	if	SCONJ
iajs-4097	78	3	|	|	ADV
iajs-4097	78	4	|	|	ADV
iajs-4097	78	5	tol	tol	VERB
iajs-4097	78	6	then	then	ADV
iajs-4097	78	7	,	,	PUNCT
iajs-4097	78	8	break	break	NOUN
iajs-4097	78	9	(	(	PUNCT
iajs-4097	78	10	the	the	DET
iajs-4097	78	11	program	program	NOUN
iajs-4097	78	12	is	be	AUX
iajs-4097	78	13	finished	finish	VERB
iajs-4097	78	14	)	)	PUNCT
iajs-4097	78	15	.	.	PUNCT
iajs-4097	79	1	2.2	2.2	NUM
iajs-4097	79	2	else	else	ADV
iajs-4097	79	3	then	then	ADV
iajs-4097	79	4	.	.	PUNCT
iajs-4097	80	1	3	3	X
iajs-4097	80	2	.	.	X
iajs-4097	80	3	go	go	VERB
iajs-4097	80	4	to	to	ADP
iajs-4097	80	5	(	(	PUNCT
iajs-4097	80	6	2	2	NUM
iajs-4097	80	7	)	)	PUNCT
iajs-4097	80	8	.	.	PUNCT
iajs-4097	81	1	4	4	X
iajs-4097	81	2	.	.	X
iajs-4097	81	3	main	main	ADJ
iajs-4097	81	4	results	result	NOUN
iajs-4097	81	5	and	and	CCONJ
iajs-4097	81	6	applications	application	NOUN
iajs-4097	81	7	of	of	ADP
iajs-4097	81	8	orthogonal	orthogonal	ADJ
iajs-4097	81	9	polynomials	polynomial	NOUN
iajs-4097	81	10	in	in	ADP
iajs-4097	81	11	applied	apply	VERB
iajs-4097	81	12	science	science	NOUN
iajs-4097	81	13	in	in	ADP
iajs-4097	81	14	this	this	DET
iajs-4097	81	15	section	section	NOUN
iajs-4097	81	16	,	,	PUNCT
iajs-4097	81	17	will	will	AUX
iajs-4097	81	18	be	be	AUX
iajs-4097	81	19	applied	apply	VERB
iajs-4097	81	20	the	the	DET
iajs-4097	81	21	proposed	propose	VERB
iajs-4097	81	22	methods	method	NOUN
iajs-4097	81	23	to	to	PART
iajs-4097	81	24	resolve	resolve	VERB
iajs-4097	81	25	some	some	DET
iajs-4097	81	26	test	test	NOUN
iajs-4097	81	27	problems	problem	NOUN
iajs-4097	81	28	of	of	ADP
iajs-4097	81	29	the	the	DET
iajs-4097	81	30	non	non	ADJ
iajs-4097	81	31	-	-	ADJ
iajs-4097	81	32	homogeneous	homogeneous	ADJ
iajs-4097	81	33	wave	wave	NOUN
iajs-4097	81	34	equation	equation	NOUN
iajs-4097	81	35	to	to	PART
iajs-4097	81	36	get	get	VERB
iajs-4097	81	37	the	the	DET
iajs-4097	81	38	approximate	approximate	ADJ
iajs-4097	81	39	solution	solution	NOUN
iajs-4097	81	40	.	.	PUNCT
iajs-4097	82	1	we	we	PRON
iajs-4097	82	2	use	use	VERB
iajs-4097	82	3	two	two	NUM
iajs-4097	82	4	distinct	distinct	ADJ
iajs-4097	82	5	error	error	NOUN
iajs-4097	82	6	criteria	criterion	NOUN
iajs-4097	82	7	to	to	PART
iajs-4097	82	8	assess	assess	VERB
iajs-4097	82	9	the	the	DET
iajs-4097	82	10	accuracy	accuracy	NOUN
iajs-4097	82	11	of	of	ADP
iajs-4097	82	12	suggested	suggest	VERB
iajs-4097	82	13	methods	method	NOUN
iajs-4097	82	14	,	,	PUNCT
iajs-4097	82	15	which	which	PRON
iajs-4097	82	16	were	be	AUX
iajs-4097	82	17	calculated	calculate	VERB
iajs-4097	82	18	by	by	ADP
iajs-4097	82	19	using	use	VERB
iajs-4097	82	20	.	.	PUNCT
iajs-4097	83	1	the	the	DET
iajs-4097	83	2	norm	norm	NOUN
iajs-4097	83	3	is	be	AUX
iajs-4097	83	4	defined	define	VERB
iajs-4097	83	5	by	by	ADP
iajs-4097	83	6	:	:	PUNCT
iajs-4097	83	7	(	(	PUNCT
iajs-4097	83	8	∑	∑	PUNCT
iajs-4097	83	9	.	.	PUNCT
iajs-4097	84	1	(	(	PUNCT
iajs-4097	84	2	)	)	PUNCT
iajs-4097	84	3	(	(	PUNCT
iajs-4097	84	4	)	)	PUNCT
iajs-4097	84	5	/	/	SYM
iajs-4097	84	6	∑	∑	PROPN
iajs-4097	84	7	.	.	PUNCT
iajs-4097	85	1	(	(	PUNCT
iajs-4097	85	2	)	)	PUNCT
iajs-4097	85	3	/	/	SYM
iajs-4097	85	4	)	)	PUNCT
iajs-4097	85	5	⁄	⁄	PROPN
iajs-4097	85	6	,	,	PUNCT
iajs-4097	85	7	and	and	CCONJ
iajs-4097	85	8	the	the	DET
iajs-4097	85	9	norm	norm	NOUN
iajs-4097	85	10	which	which	PRON
iajs-4097	85	11	is	be	AUX
iajs-4097	85	12	defined	define	VERB
iajs-4097	85	13	by	by	ADP
iajs-4097	85	14	:	:	PUNCT
iajs-4097	85	15	|	|	INTJ
iajs-4097	85	16	(	(	PUNCT
iajs-4097	85	17	)	)	PUNCT
iajs-4097	85	18	(	(	PUNCT
iajs-4097	85	19	)	)	PUNCT
iajs-4097	85	20	|	|	INTJ
iajs-4097	85	21	.	.	PUNCT
iajs-4097	86	1	where	where	SCONJ
iajs-4097	86	2	and	and	CCONJ
iajs-4097	86	3	represent	represent	VERB
iajs-4097	86	4	the	the	DET
iajs-4097	86	5	numerical	numerical	ADJ
iajs-4097	86	6	and	and	CCONJ
iajs-4097	86	7	analytical	analytical	ADJ
iajs-4097	86	8	solutions	solution	NOUN
iajs-4097	86	9	,	,	PUNCT
iajs-4097	86	10	respectively	respectively	ADV
iajs-4097	86	11	.	.	PUNCT
iajs-4097	86	12	example	example	NOUN
iajs-4097	86	13	1	1	NUM
iajs-4097	86	14	.	.	X
iajs-4097	87	1	consider	consider	VERB
iajs-4097	87	2	the	the	DET
iajs-4097	87	3	following	follow	VERB
iajs-4097	87	4	inhomogeneous	inhomogeneous	ADJ
iajs-4097	87	5	wave	wave	NOUN
iajs-4097	87	6	problem	problem	NOUN
iajs-4097	87	7	:	:	PUNCT
iajs-4097	87	8	(	(	PUNCT
iajs-4097	87	9	18	18	NUM
iajs-4097	87	10	)	)	PUNCT
iajs-4097	87	11	with	with	ADP
iajs-4097	87	12	initial	initial	ADJ
iajs-4097	87	13	conditions	condition	NOUN
iajs-4097	87	14	:	:	PUNCT
iajs-4097	87	15	ihjpas	ihjpas	PROPN
iajs-4097	87	16	.	.	PUNCT
iajs-4097	88	1	2025,38(4	2025,38(4	NOUN
iajs-4097	88	2	)	)	PUNCT
iajs-4097	89	1	382	382	NUM
iajs-4097	89	2	(	(	PUNCT
iajs-4097	89	3	)	)	PUNCT
iajs-4097	89	4	(	(	PUNCT
iajs-4097	89	5	)	)	PUNCT
iajs-4097	89	6	(	(	PUNCT
iajs-4097	89	7	19	19	NUM
iajs-4097	89	8	)	)	PUNCT
iajs-4097	89	9	and	and	CCONJ
iajs-4097	89	10	dirichlet	dirichlet	PROPN
iajs-4097	89	11	boundary	boundary	ADJ
iajs-4097	89	12	conditions	condition	NOUN
iajs-4097	89	13	:	:	PUNCT
iajs-4097	89	14	(	(	PUNCT
iajs-4097	89	15	)	)	PUNCT
iajs-4097	89	16	(	(	PUNCT
iajs-4097	89	17	)	)	PUNCT
iajs-4097	89	18	.	.	PUNCT
iajs-4097	90	1	(	(	PUNCT
iajs-4097	90	2	20	20	NUM
iajs-4097	90	3	)	)	PUNCT
iajs-4097	90	4	and	and	CCONJ
iajs-4097	90	5	the	the	DET
iajs-4097	90	6	analytical	analytical	ADJ
iajs-4097	90	7	solution	solution	NOUN
iajs-4097	90	8	is	be	AUX
iajs-4097	90	9	:	:	PUNCT
iajs-4097	90	10	(	(	PUNCT
iajs-4097	90	11	)	)	PUNCT
iajs-4097	90	12	.	.	PUNCT
iajs-4097	91	1	by	by	ADP
iajs-4097	91	2	applying	apply	VERB
iajs-4097	91	3	the	the	DET
iajs-4097	91	4	proposed	propose	VERB
iajs-4097	91	5	method	method	NOUN
iajs-4097	91	6	to	to	PART
iajs-4097	91	7	resolve	resolve	VERB
iajs-4097	91	8	this	this	DET
iajs-4097	91	9	problem	problem	NOUN
iajs-4097	91	10	with	with	ADP
iajs-4097	91	11	initial	initial	ADJ
iajs-4097	91	12	and	and	CCONJ
iajs-4097	91	13	boundary	boundary	ADJ
iajs-4097	91	14	condition	condition	NOUN
iajs-4097	91	15	gives	give	VERB
iajs-4097	91	16	in	in	ADP
iajs-4097	91	17	equations	equation	NOUN
iajs-4097	91	18	19	19	NUM
iajs-4097	91	19	,	,	PUNCT
iajs-4097	91	20	and	and	CCONJ
iajs-4097	91	21	20	20	NUM
iajs-4097	91	22	,	,	PUNCT
iajs-4097	91	23	respectively	respectively	ADV
iajs-4097	91	24	.	.	PUNCT
iajs-4097	92	1	more	more	ADV
iajs-4097	92	2	specifically	specifically	ADV
iajs-4097	92	3	,	,	PUNCT
iajs-4097	92	4	by	by	ADP
iajs-4097	92	5	converting	convert	VERB
iajs-4097	92	6	the	the	DET
iajs-4097	92	7	unknown	unknown	ADJ
iajs-4097	92	8	function	function	NOUN
iajs-4097	92	9	(	(	PUNCT
iajs-4097	92	10	)	)	PUNCT
iajs-4097	92	11	with	with	ADP
iajs-4097	92	12	partial	partial	ADJ
iajs-4097	92	13	derivatives	derivative	NOUN
iajs-4097	92	14	in	in	ADP
iajs-4097	92	15	to	to	ADP
iajs-4097	92	16	linear	linear	PROPN
iajs-4097	92	17	equations	equation	NOUN
iajs-4097	92	18	system	system	NOUN
iajs-4097	92	19	.	.	PUNCT
iajs-4097	93	1	furthermore	furthermore	ADV
iajs-4097	93	2	,	,	PUNCT
iajs-4097	93	3	assume	assume	VERB
iajs-4097	93	4	that	that	PRON
iajs-4097	93	5	.	.	PUNCT
iajs-4097	94	1	first	first	ADV
iajs-4097	94	2	:	:	PUNCT
iajs-4097	94	3	applying	apply	VERB
iajs-4097	94	4	the	the	DET
iajs-4097	94	5	standard	standard	ADJ
iajs-4097	94	6	polynomials	polynomial	NOUN
iajs-4097	94	7	.	.	PUNCT
iajs-4097	95	1	by	by	ADP
iajs-4097	95	2	inserting	insert	VERB
iajs-4097	95	3	the	the	DET
iajs-4097	95	4	equations	equation	NOUN
iajs-4097	95	5	6	6	NUM
iajs-4097	95	6	,	,	PUNCT
iajs-4097	95	7	and	and	CCONJ
iajs-4097	95	8	7	7	NUM
iajs-4097	95	9	into	into	ADP
iajs-4097	95	10	equations	equation	NOUN
iajs-4097	95	11	18	18	NUM
iajs-4097	95	12	,	,	PUNCT
iajs-4097	95	13	19	19	NUM
iajs-4097	95	14	and	and	CCONJ
iajs-4097	95	15	20	20	NUM
iajs-4097	95	16	the	the	DET
iajs-4097	95	17	-order	-order	ADJ
iajs-4097	95	18	partial	partial	ADJ
iajs-4097	95	19	derivatives	derivative	NOUN
iajs-4097	95	20	and	and	CCONJ
iajs-4097	95	21	conditions	condition	NOUN
iajs-4097	95	22	are	be	AUX
iajs-4097	95	23	converted	convert	VERB
iajs-4097	95	24	into	into	ADP
iajs-4097	95	25	matrices	matrix	NOUN
iajs-4097	95	26	,	,	PUNCT
iajs-4097	95	27	such	such	ADJ
iajs-4097	95	28	that	that	SCONJ
iajs-4097	95	29	:	:	PUNCT
iajs-4097	95	30	(	(	PUNCT
iajs-4097	95	31	)	)	PUNCT
iajs-4097	95	32	(	(	PUNCT
iajs-4097	95	33	)	)	PUNCT
iajs-4097	95	34	(	(	PUNCT
iajs-4097	95	35	)	)	PUNCT
iajs-4097	95	36	(	(	PUNCT
iajs-4097	95	37	)	)	PUNCT
iajs-4097	95	38	(	(	PUNCT
iajs-4097	95	39	)	)	PUNCT
iajs-4097	95	40	,	,	PUNCT
iajs-4097	95	41	[	[	PUNCT
iajs-4097	95	42	]	]	X
iajs-4097	95	43	[	[	PUNCT
iajs-4097	95	44	]	]	X
iajs-4097	95	45	,	,	PUNCT
iajs-4097	95	46	,	,	PUNCT
iajs-4097	95	47	[	[	PUNCT
iajs-4097	95	48	]	]	X
iajs-4097	95	49	[	[	PUNCT
iajs-4097	95	50	]	]	X
iajs-4097	95	51	,	,	PUNCT
iajs-4097	95	52	,	,	PUNCT
iajs-4097	95	53	,	,	PUNCT
iajs-4097	95	54	(	(	PUNCT
iajs-4097	95	55	21	21	NUM
iajs-4097	95	56	)	)	PUNCT
iajs-4097	95	57	for	for	ADP
iajs-4097	95	58	the	the	DET
iajs-4097	95	59	equation	equation	NOUN
iajs-4097	95	60	6	6	NUM
iajs-4097	95	61	,	,	PUNCT
iajs-4097	95	62	becomes	become	VERB
iajs-4097	95	63	as	as	ADP
iajs-4097	95	64	following	follow	VERB
iajs-4097	95	65	:	:	PUNCT
iajs-4097	95	66	(	(	PUNCT
iajs-4097	95	67	)	)	PUNCT
iajs-4097	95	68	,	,	PUNCT
iajs-4097	95	69	[	[	PUNCT
iajs-4097	95	70	]	]	X
iajs-4097	95	71	,	,	PUNCT
iajs-4097	95	72	,	,	PUNCT
iajs-4097	95	73	(	(	PUNCT
iajs-4097	95	74	)	)	PUNCT
iajs-4097	95	75	(	(	PUNCT
iajs-4097	95	76	22	22	NUM
iajs-4097	95	77	)	)	PUNCT
iajs-4097	95	78	from	from	ADP
iajs-4097	95	79	equation	equation	NOUN
iajs-4097	95	80	7	7	NUM
iajs-4097	95	81	,	,	PUNCT
iajs-4097	95	82	the	the	DET
iajs-4097	95	83	first	first	ADJ
iajs-4097	95	84	derivatives	derivative	NOUN
iajs-4097	95	85	of	of	ADP
iajs-4097	95	86	(	(	PUNCT
iajs-4097	95	87	)	)	PUNCT
iajs-4097	95	88	with	with	ADP
iajs-4097	95	89	a	a	DET
iajs-4097	95	90	respective	respective	ADJ
iajs-4097	95	91	its	its	PRON
iajs-4097	95	92	:	:	PUNCT
iajs-4097	95	93	(	(	PUNCT
iajs-4097	95	94	)	)	PUNCT
iajs-4097	95	95	(	(	PUNCT
iajs-4097	95	96	)	)	PUNCT
iajs-4097	95	97	(	(	PUNCT
iajs-4097	95	98	)	)	PUNCT
iajs-4097	95	99	(	(	PUNCT
iajs-4097	95	100	)	)	PUNCT
iajs-4097	95	101	,	,	PUNCT
iajs-4097	95	102	[	[	PUNCT
iajs-4097	95	103	]	]	X
iajs-4097	95	104	[	[	PUNCT
iajs-4097	95	105	]	]	X
iajs-4097	95	106	,	,	PUNCT
iajs-4097	95	107	,	,	PUNCT
iajs-4097	95	108	,	,	PUNCT
iajs-4097	95	109	(	(	PUNCT
iajs-4097	95	110	23	23	NUM
iajs-4097	95	111	)	)	PUNCT
iajs-4097	95	112	by	by	ADP
iajs-4097	95	113	substituting	substitute	VERB
iajs-4097	95	114	the	the	DET
iajs-4097	95	115	ic	ic	PROPN
iajs-4097	95	116	from	from	ADP
iajs-4097	95	117	equation	equation	NOUN
iajs-4097	95	118	19	19	NUM
iajs-4097	95	119	,	,	PUNCT
iajs-4097	95	120	when	when	SCONJ
iajs-4097	95	121	in	in	ADP
iajs-4097	95	122	to	to	ADP
iajs-4097	95	123	equations	equation	NOUN
iajs-4097	95	124	22	22	NUM
iajs-4097	95	125	,	,	PUNCT
iajs-4097	95	126	and	and	CCONJ
iajs-4097	95	127	23	23	NUM
iajs-4097	95	128	,	,	PUNCT
iajs-4097	95	129	will	will	AUX
iajs-4097	95	130	be	be	AUX
iajs-4097	95	131	obtained	obtain	VERB
iajs-4097	95	132	:	:	PUNCT
iajs-4097	95	133	(	(	PUNCT
iajs-4097	95	134	)	)	PUNCT
iajs-4097	95	135	(	(	PUNCT
iajs-4097	95	136	)	)	PUNCT
iajs-4097	95	137	(	(	PUNCT
iajs-4097	95	138	)	)	PUNCT
iajs-4097	95	139	,	,	PUNCT
iajs-4097	95	140	(	(	PUNCT
iajs-4097	95	141	24	24	NUM
iajs-4097	95	142	)	)	PUNCT
iajs-4097	95	143	(	(	PUNCT
iajs-4097	95	144	)	)	PUNCT
iajs-4097	95	145	(	(	PUNCT
iajs-4097	95	146	)	)	PUNCT
iajs-4097	95	147	(	(	PUNCT
iajs-4097	95	148	)	)	PUNCT
iajs-4097	95	149	(	(	PUNCT
iajs-4097	95	150	)	)	PUNCT
iajs-4097	95	151	,	,	PUNCT
iajs-4097	95	152	(	(	PUNCT
iajs-4097	95	153	25	25	NUM
iajs-4097	95	154	)	)	PUNCT
iajs-4097	95	155	also	also	ADV
iajs-4097	95	156	,	,	PUNCT
iajs-4097	95	157	by	by	ADP
iajs-4097	95	158	inserting	insert	VERB
iajs-4097	95	159	the	the	DET
iajs-4097	95	160	bc	bc	PROPN
iajs-4097	95	161	from	from	ADP
iajs-4097	95	162	equation	equation	NOUN
iajs-4097	95	163	20	20	NUM
iajs-4097	95	164	when	when	SCONJ
iajs-4097	95	165	in	in	ADP
iajs-4097	95	166	to	to	ADP
iajs-4097	95	167	equation	equation	NOUN
iajs-4097	95	168	22	22	NUM
iajs-4097	95	169	,	,	PUNCT
iajs-4097	95	170	will	will	AUX
iajs-4097	95	171	be	be	AUX
iajs-4097	95	172	get	get	VERB
iajs-4097	95	173	:	:	PUNCT
iajs-4097	95	174	(	(	PUNCT
iajs-4097	95	175	)	)	PUNCT
iajs-4097	95	176	(	(	PUNCT
iajs-4097	95	177	)	)	PUNCT
iajs-4097	95	178	(	(	PUNCT
iajs-4097	95	179	)	)	PUNCT
iajs-4097	95	180	,	,	PUNCT
iajs-4097	95	181	(	(	PUNCT
iajs-4097	95	182	26	26	NUM
iajs-4097	95	183	)	)	PUNCT
iajs-4097	95	184	(	(	PUNCT
iajs-4097	95	185	)	)	PUNCT
iajs-4097	95	186	(	(	PUNCT
iajs-4097	95	187	)	)	PUNCT
iajs-4097	95	188	(	(	PUNCT
iajs-4097	95	189	)	)	PUNCT
iajs-4097	95	190	,	,	PUNCT
iajs-4097	95	191	(	(	PUNCT
iajs-4097	95	192	)	)	PUNCT
iajs-4097	95	193	(	(	PUNCT
iajs-4097	95	194	)	)	PUNCT
iajs-4097	95	195	(	(	PUNCT
iajs-4097	95	196	)	)	PUNCT
iajs-4097	95	197	(	(	PUNCT
iajs-4097	95	198	)	)	PUNCT
iajs-4097	95	199	,	,	PUNCT
iajs-4097	95	200	(	(	PUNCT
iajs-4097	95	201	27	27	NUM
iajs-4097	95	202	)	)	PUNCT
iajs-4097	95	203	by	by	ADP
iajs-4097	95	204	resolve	resolve	VERB
iajs-4097	95	205	the	the	DET
iajs-4097	95	206	linear	linear	ADJ
iajs-4097	95	207	algebraic	algebraic	ADJ
iajs-4097	95	208	system	system	NOUN
iajs-4097	95	209	consisting	consist	VERB
iajs-4097	95	210	of	of	ADP
iajs-4097	95	211	the	the	DET
iajs-4097	95	212	equations	equation	NOUN
iajs-4097	95	213	21	21	NUM
iajs-4097	95	214	,	,	PUNCT
iajs-4097	95	215	24	24	NUM
iajs-4097	95	216	,	,	PUNCT
iajs-4097	95	217	25	25	NUM
iajs-4097	95	218	,	,	PUNCT
iajs-4097	95	219	26	26	NUM
iajs-4097	95	220	,	,	PUNCT
iajs-4097	95	221	and	and	CCONJ
iajs-4097	95	222	27	27	NUM
iajs-4097	95	223	,	,	PUNCT
iajs-4097	95	224	to	to	PART
iajs-4097	95	225	find	find	VERB
iajs-4097	95	226	the	the	DET
iajs-4097	95	227	values	value	NOUN
iajs-4097	95	228	of	of	ADP
iajs-4097	95	229	,	,	PUNCT
iajs-4097	95	230	,	,	PUNCT
iajs-4097	95	231	,	,	PUNCT
iajs-4097	95	232	,	,	PUNCT
iajs-4097	95	233	,	,	PUNCT
iajs-4097	95	234	,	,	PUNCT
iajs-4097	95	235	,	,	PUNCT
iajs-4097	95	236	and	and	CCONJ
iajs-4097	95	237	,	,	PUNCT
iajs-4097	95	238	using	use	VERB
iajs-4097	95	239	the	the	PRON
iajs-4097	95	240	,	,	PUNCT
iajs-4097	95	241	we	we	PRON
iajs-4097	95	242	get	get	VERB
iajs-4097	95	243	:	:	PUNCT
iajs-4097	95	244	[	[	PUNCT
iajs-4097	95	245	]	]	X
iajs-4097	95	246	[	[	PUNCT
iajs-4097	95	247	]	]	X
iajs-4097	95	248	.	.	PUNCT
iajs-4097	96	1	by	by	ADP
iajs-4097	96	2	replacing	replace	VERB
iajs-4097	96	3	these	these	DET
iajs-4097	96	4	values	value	NOUN
iajs-4097	96	5	in	in	ADP
iajs-4097	96	6	equations	equation	NOUN
iajs-4097	96	7	6	6	NUM
iajs-4097	96	8	or	or	CCONJ
iajs-4097	96	9	22	22	NUM
iajs-4097	96	10	,	,	PUNCT
iajs-4097	96	11	therefore	therefore	ADV
iajs-4097	96	12	,	,	PUNCT
iajs-4097	96	13	the	the	DET
iajs-4097	96	14	approximate	approximate	ADJ
iajs-4097	96	15	solution	solution	NOUN
iajs-4097	96	16	using	use	VERB
iajs-4097	96	17	ihjpas	ihjpas	PROPN
iajs-4097	96	18	.	.	PUNCT
iajs-4097	97	1	2025,38(4	2025,38(4	NOUN
iajs-4097	97	2	)	)	PUNCT
iajs-4097	97	3	383	383	NUM
iajs-4097	97	4	standard	standard	ADJ
iajs-4097	97	5	polynomials	polynomial	NOUN
iajs-4097	97	6	(	(	PUNCT
iajs-4097	97	7	)	)	PUNCT
iajs-4097	97	8	as	as	SCONJ
iajs-4097	97	9	follows	follow	VERB
iajs-4097	97	10	:	:	PUNCT
iajs-4097	97	11	(	(	PUNCT
iajs-4097	97	12	)	)	PUNCT
iajs-4097	97	13	(	(	PUNCT
iajs-4097	97	14	)	)	PUNCT
iajs-4097	97	15	(	(	PUNCT
iajs-4097	97	16	)	)	PUNCT
iajs-4097	97	17	finally	finally	ADV
iajs-4097	97	18	,	,	PUNCT
iajs-4097	97	19	by	by	ADP
iajs-4097	97	20	using	use	VERB
iajs-4097	97	21	the	the	DET
iajs-4097	97	22	other	other	ADJ
iajs-4097	97	23	proposed	propose	VERB
iajs-4097	97	24	methods	method	NOUN
iajs-4097	97	25	such	such	ADJ
iajs-4097	97	26	as	as	ADP
iajs-4097	97	27	:	:	PUNCT
iajs-4097	97	28	boubaker	boubaker	NOUN
iajs-4097	97	29	,	,	PUNCT
iajs-4097	97	30	bernoulli	bernoulli	PROPN
iajs-4097	97	31	,	,	PUNCT
iajs-4097	97	32	chebyshev	chebyshev	PROPN
iajs-4097	97	33	,	,	PUNCT
iajs-4097	97	34	genocchi	genocchi	NOUN
iajs-4097	97	35	,	,	PUNCT
iajs-4097	97	36	and	and	CCONJ
iajs-4097	97	37	legendre	legendre	PROPN
iajs-4097	97	38	,	,	PUNCT
iajs-4097	97	39	the	the	DET
iajs-4097	97	40	approximate	approximate	ADJ
iajs-4097	97	41	solution	solution	NOUN
iajs-4097	97	42	for	for	ADP
iajs-4097	97	43	test	test	NOUN
iajs-4097	97	44	example	example	NOUN
iajs-4097	97	45	1	1	NUM
iajs-4097	97	46	as	as	SCONJ
iajs-4097	97	47	follows	follow	VERB
iajs-4097	97	48	:	:	PUNCT
iajs-4097	97	49	(	(	PUNCT
iajs-4097	97	50	)	)	PUNCT
iajs-4097	97	51	(	(	PUNCT
iajs-4097	97	52	)	)	PUNCT
iajs-4097	97	53	(	(	PUNCT
iajs-4097	97	54	)	)	PUNCT
iajs-4097	97	55	(	(	PUNCT
iajs-4097	97	56	)	)	PUNCT
iajs-4097	97	57	(	(	PUNCT
iajs-4097	97	58	)	)	PUNCT
iajs-4097	97	59	(	(	PUNCT
iajs-4097	97	60	)	)	PUNCT
iajs-4097	97	61	(	(	PUNCT
iajs-4097	97	62	)	)	PUNCT
iajs-4097	97	63	(	(	PUNCT
iajs-4097	97	64	)	)	PUNCT
iajs-4097	97	65	(	(	PUNCT
iajs-4097	97	66	)	)	PUNCT
iajs-4097	97	67	(	(	PUNCT
iajs-4097	97	68	)	)	PUNCT
iajs-4097	97	69	(	(	PUNCT
iajs-4097	97	70	)	)	PUNCT
iajs-4097	97	71	(	(	PUNCT
iajs-4097	97	72	)	)	PUNCT
iajs-4097	97	73	(	(	PUNCT
iajs-4097	97	74	)	)	PUNCT
iajs-4097	97	75	(	(	PUNCT
iajs-4097	97	76	)	)	PUNCT
iajs-4097	97	77	(	(	PUNCT
iajs-4097	97	78	)	)	PUNCT
iajs-4097	97	79	table	table	NOUN
iajs-4097	97	80	1	1	NUM
iajs-4097	97	81	.	.	PUNCT
iajs-4097	98	1	the	the	DET
iajs-4097	98	2	values	value	NOUN
iajs-4097	98	3	of	of	ADP
iajs-4097	98	4	and	and	CCONJ
iajs-4097	98	5	for	for	ADP
iajs-4097	98	6	example	example	NOUN
iajs-4097	98	7	1	1	NUM
iajs-4097	98	8	,	,	PUNCT
iajs-4097	98	9	when	when	SCONJ
iajs-4097	98	10	.	.	PUNCT
iajs-4097	98	11	ihjpas	ihjpas	PROPN
iajs-4097	98	12	.	.	PUNCT
iajs-4097	99	1	2025,38(4	2025,38(4	X
iajs-4097	99	2	)	)	PUNCT
iajs-4097	99	3	384	384	NUM
iajs-4097	99	4	norm	norm	NOUN
iajs-4097	99	5	.	.	PUNCT
iajs-4097	100	1	norm	norm	NOUN
iajs-4097	100	2	.	.	PUNCT
iajs-4097	101	1	figure	figure	NOUN
iajs-4097	101	2	1	1	NUM
iajs-4097	101	3	.	.	PUNCT
iajs-4097	102	1	the	the	DET
iajs-4097	102	2	plots	plot	NOUN
iajs-4097	102	3	logarithmic	logarithmic	ADJ
iajs-4097	102	4	of	of	ADP
iajs-4097	102	5	and	and	CCONJ
iajs-4097	102	6	,	,	PUNCT
iajs-4097	102	7	for	for	ADP
iajs-4097	102	8	example	example	NOUN
iajs-4097	102	9	1	1	NUM
iajs-4097	102	10	,	,	PUNCT
iajs-4097	102	11	when	when	SCONJ
iajs-4097	102	12	.	.	PUNCT
iajs-4097	103	1	in	in	ADP
iajs-4097	103	2	table	table	NOUN
iajs-4097	103	3	1	1	NUM
iajs-4097	103	4	and	and	CCONJ
iajs-4097	103	5	figure	figure	NOUN
iajs-4097	103	6	1	1	NUM
iajs-4097	103	7	,	,	PUNCT
iajs-4097	103	8	it	it	PRON
iajs-4097	103	9	can	can	AUX
iajs-4097	103	10	be	be	AUX
iajs-4097	103	11	seen	see	VERB
iajs-4097	103	12	the	the	DET
iajs-4097	103	13	values	value	NOUN
iajs-4097	103	14	of	of	ADP
iajs-4097	103	15	the	the	DET
iajs-4097	103	16	norm	norm	NOUN
iajs-4097	103	17	and	and	CCONJ
iajs-4097	103	18	.	.	PUNCT
iajs-4097	104	1	they	they	PRON
iajs-4097	104	2	depend	depend	VERB
iajs-4097	104	3	on	on	ADP
iajs-4097	104	4	the	the	DET
iajs-4097	104	5	value	value	NOUN
iajs-4097	104	6	of	of	ADP
iajs-4097	104	7	,	,	PUNCT
iajs-4097	104	8	which	which	PRON
iajs-4097	104	9	represents	represent	VERB
iajs-4097	104	10	the	the	DET
iajs-4097	104	11	degree	degree	NOUN
iajs-4097	104	12	of	of	ADP
iajs-4097	104	13	the	the	DET
iajs-4097	104	14	polynomial	polynomial	NOUN
iajs-4097	104	15	.	.	PUNCT
iajs-4097	105	1	the	the	PRON
iajs-4097	105	2	higher	high	ADJ
iajs-4097	105	3	the	the	DET
iajs-4097	105	4	value	value	NOUN
iajs-4097	105	5	of	of	ADP
iajs-4097	105	6	gives	give	VERB
iajs-4097	105	7	the	the	DET
iajs-4097	105	8	lower	low	ADJ
iajs-4097	105	9	error	error	NOUN
iajs-4097	105	10	.	.	PUNCT
iajs-4097	106	1	the	the	DET
iajs-4097	106	2	best	good	ADJ
iajs-4097	106	3	results	result	NOUN
iajs-4097	106	4	were	be	AUX
iajs-4097	106	5	achieved	achieve	VERB
iajs-4097	106	6	using	use	VERB
iajs-4097	106	7	the	the	DET
iajs-4097	106	8	standard	standard	ADJ
iajs-4097	106	9	polynomials	polynomial	NOUN
iajs-4097	106	10	when	when	SCONJ
iajs-4097	106	11	,	,	PUNCT
iajs-4097	106	12	,	,	PUNCT
iajs-4097	106	13	the	the	DET
iajs-4097	106	14	values	value	NOUN
iajs-4097	106	15	of	of	ADP
iajs-4097	106	16	and	and	CCONJ
iajs-4097	106	17	equals	equal	VERB
iajs-4097	106	18	,	,	PUNCT
iajs-4097	106	19	,	,	PUNCT
iajs-4097	106	20	,	,	PUNCT
iajs-4097	106	21	respectively	respectively	ADV
iajs-4097	106	22	,	,	PUNCT
iajs-4097	106	23	followed	follow	VERB
iajs-4097	106	24	by	by	ADP
iajs-4097	106	25	bernoulli	bernoulli	NOUN
iajs-4097	106	26	polynomials	polynomial	NOUN
iajs-4097	106	27	.	.	PUNCT
iajs-4097	107	1	figure	figure	NOUN
iajs-4097	107	2	2	2	NUM
iajs-4097	107	3	.	.	PUNCT
iajs-4097	107	4	plots	plot	VERB
iajs-4097	107	5	the	the	DET
iajs-4097	107	6	norm	norm	NOUN
iajs-4097	107	7	for	for	ADP
iajs-4097	107	8	example	example	NOUN
iajs-4097	107	9	1	1	NUM
iajs-4097	107	10	,	,	PUNCT
iajs-4097	107	11	.	.	PUNCT
iajs-4097	108	1	exact	exact	ADJ
iajs-4097	108	2	solution	solution	NOUN
iajs-4097	108	3	approximate	approximate	ADJ
iajs-4097	108	4	solution	solution	NOUN
iajs-4097	108	5	figure	figure	NOUN
iajs-4097	108	6	3	3	NUM
iajs-4097	108	7	.	.	NOUN
iajs-4097	108	8	comparison	comparison	NOUN
iajs-4097	108	9	between	between	ADP
iajs-4097	108	10	analytical	analytical	ADJ
iajs-4097	108	11	and	and	CCONJ
iajs-4097	108	12	numerical	numerical	ADJ
iajs-4097	108	13	solution	solution	NOUN
iajs-4097	108	14	for	for	ADP
iajs-4097	108	15	example	example	NOUN
iajs-4097	108	16	1	1	NUM
iajs-4097	108	17	by	by	ADP
iajs-4097	108	18	the	the	DET
iajs-4097	108	19	standard	standard	ADJ
iajs-4097	108	20	poly	poly	NOUN
iajs-4097	108	21	.	.	PUNCT
iajs-4097	109	1	ihjpas	ihjpas	PROPN
iajs-4097	109	2	.	.	PUNCT
iajs-4097	110	1	2025,38(4	2025,38(4	NOUN
iajs-4097	110	2	)	)	PUNCT
iajs-4097	110	3	385	385	NUM
iajs-4097	110	4	example	example	NOUN
iajs-4097	110	5	2	2	NUM
iajs-4097	110	6	.	.	X
iajs-4097	110	7	consider	consider	VERB
iajs-4097	110	8	the	the	DET
iajs-4097	110	9	inhomogeneous	inhomogeneous	ADJ
iajs-4097	110	10	wave	wave	NOUN
iajs-4097	110	11	problem	problem	NOUN
iajs-4097	110	12	as	as	SCONJ
iajs-4097	110	13	follows	follow	VERB
iajs-4097	110	14	:	:	PUNCT
iajs-4097	110	15	(	(	PUNCT
iajs-4097	110	16	)	)	PUNCT
iajs-4097	110	17	(	(	PUNCT
iajs-4097	110	18	)	)	PUNCT
iajs-4097	110	19	(	(	PUNCT
iajs-4097	110	20	28	28	NUM
iajs-4097	110	21	)	)	PUNCT
iajs-4097	110	22	with	with	ADP
iajs-4097	110	23	initial	initial	ADJ
iajs-4097	110	24	conditions	condition	NOUN
iajs-4097	110	25	:	:	PUNCT
iajs-4097	110	26	(	(	PUNCT
iajs-4097	110	27	)	)	PUNCT
iajs-4097	110	28	(	(	PUNCT
iajs-4097	110	29	)	)	PUNCT
iajs-4097	110	30	(	(	PUNCT
iajs-4097	110	31	29	29	NUM
iajs-4097	110	32	)	)	PUNCT
iajs-4097	110	33	and	and	CCONJ
iajs-4097	110	34	dirichlet	dirichlet	PROPN
iajs-4097	110	35	boundary	boundary	ADJ
iajs-4097	110	36	conditions	condition	NOUN
iajs-4097	110	37	:	:	PUNCT
iajs-4097	110	38	(	(	PUNCT
iajs-4097	110	39	)	)	PUNCT
iajs-4097	110	40	(	(	PUNCT
iajs-4097	110	41	)	)	PUNCT
iajs-4097	110	42	(	(	PUNCT
iajs-4097	110	43	)	)	PUNCT
iajs-4097	110	44	.	.	PUNCT
iajs-4097	111	1	(	(	PUNCT
iajs-4097	111	2	30	30	NUM
iajs-4097	111	3	)	)	PUNCT
iajs-4097	111	4	and	and	CCONJ
iajs-4097	111	5	the	the	DET
iajs-4097	111	6	analytical	analytical	ADJ
iajs-4097	111	7	solution	solution	NOUN
iajs-4097	111	8	is	be	AUX
iajs-4097	111	9	:	:	PUNCT
iajs-4097	111	10	(	(	PUNCT
iajs-4097	111	11	)	)	PUNCT
iajs-4097	111	12	(	(	PUNCT
iajs-4097	111	13	)	)	PUNCT
iajs-4097	111	14	.	.	PUNCT
iajs-4097	112	1	(	(	PUNCT
iajs-4097	112	2	)	)	PUNCT
iajs-4097	112	3	(	(	PUNCT
iajs-4097	112	4	)	)	PUNCT
iajs-4097	112	5	(	(	PUNCT
iajs-4097	112	6	)	)	PUNCT
iajs-4097	112	7	(	(	PUNCT
iajs-4097	112	8	)	)	PUNCT
iajs-4097	112	9	(	(	PUNCT
iajs-4097	112	10	)	)	PUNCT
iajs-4097	112	11	(	(	PUNCT
iajs-4097	112	12	)	)	PUNCT
iajs-4097	112	13	(	(	PUNCT
iajs-4097	112	14	)	)	PUNCT
iajs-4097	112	15	(	(	PUNCT
iajs-4097	112	16	)	)	PUNCT
iajs-4097	112	17	(	(	PUNCT
iajs-4097	112	18	)	)	PUNCT
iajs-4097	112	19	(	(	PUNCT
iajs-4097	112	20	)	)	PUNCT
iajs-4097	112	21	(	(	PUNCT
iajs-4097	112	22	)	)	PUNCT
iajs-4097	112	23	(	(	PUNCT
iajs-4097	112	24	)	)	PUNCT
iajs-4097	112	25	(	(	PUNCT
iajs-4097	112	26	)	)	PUNCT
iajs-4097	112	27	(	(	PUNCT
iajs-4097	112	28	)	)	PUNCT
iajs-4097	112	29	(	(	PUNCT
iajs-4097	112	30	)	)	PUNCT
iajs-4097	112	31	(	(	PUNCT
iajs-4097	112	32	)	)	PUNCT
iajs-4097	112	33	(	(	PUNCT
iajs-4097	112	34	)	)	PUNCT
iajs-4097	112	35	(	(	PUNCT
iajs-4097	112	36	)	)	PUNCT
iajs-4097	112	37	(	(	PUNCT
iajs-4097	112	38	)	)	PUNCT
iajs-4097	112	39	(	(	PUNCT
iajs-4097	112	40	)	)	PUNCT
iajs-4097	112	41	(	(	PUNCT
iajs-4097	112	42	)	)	PUNCT
iajs-4097	112	43	such	such	ADJ
iajs-4097	112	44	that	that	PRON
iajs-4097	112	45	represented	represent	VERB
iajs-4097	112	46	the	the	DET
iajs-4097	112	47	operational	operational	ADJ
iajs-4097	112	48	matrices	matrix	NOUN
iajs-4097	112	49	of	of	ADP
iajs-4097	112	50	all	all	DET
iajs-4097	112	51	proposed	propose	VERB
iajs-4097	112	52	methods	method	NOUN
iajs-4097	112	53	.	.	PUNCT
iajs-4097	113	1	let	let	VERB
iajs-4097	113	2	us	we	PRON
iajs-4097	113	3	use	use	VERB
iajs-4097	113	4	the	the	PRON
iajs-4097	113	5	to	to	PART
iajs-4097	113	6	solve	solve	VERB
iajs-4097	113	7	the	the	DET
iajs-4097	113	8	given	give	VERB
iajs-4097	113	9	test	test	NOUN
iajs-4097	113	10	problem	problem	NOUN
iajs-4097	113	11	and	and	CCONJ
iajs-4097	113	12	determine	determine	VERB
iajs-4097	113	13	the	the	PRON
iajs-4097	113	14	and	and	CCONJ
iajs-4097	113	15	:	:	PUNCT
iajs-4097	113	16	table	table	NOUN
iajs-4097	113	17	2	2	NUM
iajs-4097	113	18	.	.	PUNCT
iajs-4097	114	1	the	the	DET
iajs-4097	114	2	values	value	NOUN
iajs-4097	114	3	of	of	ADP
iajs-4097	114	4	and	and	CCONJ
iajs-4097	114	5	for	for	ADP
iajs-4097	114	6	example	example	NOUN
iajs-4097	114	7	2	2	NUM
iajs-4097	114	8	,	,	PUNCT
iajs-4097	114	9	.	.	PUNCT
iajs-4097	115	1	the	the	DET
iajs-4097	115	2	error	error	NOUN
iajs-4097	115	3	norm	norm	NOUN
iajs-4097	115	4	and	and	CCONJ
iajs-4097	115	5	mean	mean	ADJ
iajs-4097	115	6	square	square	ADJ
iajs-4097	115	7	error	error	NOUN
iajs-4097	115	8	values	value	NOUN
iajs-4097	115	9	are	be	AUX
iajs-4097	115	10	readily	readily	ADV
iajs-4097	115	11	seen	see	VERB
iajs-4097	115	12	in	in	ADP
iajs-4097	115	13	table	table	NOUN
iajs-4097	115	14	2	2	NUM
iajs-4097	115	15	,	,	PUNCT
iajs-4097	115	16	and	and	CCONJ
iajs-4097	115	17	figure	figure	VERB
iajs-4097	115	18	4	4	NUM
iajs-4097	115	19	,	,	PUNCT
iajs-4097	115	20	which	which	PRON
iajs-4097	115	21	can	can	AUX
iajs-4097	115	22	be	be	AUX
iajs-4097	115	23	obtained	obtain	VERB
iajs-4097	115	24	by	by	ADP
iajs-4097	115	25	solving	solve	VERB
iajs-4097	115	26	example	example	NOUN
iajs-4097	115	27	2	2	NUM
iajs-4097	115	28	using	use	VERB
iajs-4097	115	29	mathematica	mathematica	PROPN
iajs-4097	115	30	.	.	PUNCT
iajs-4097	116	1	some	some	DET
iajs-4097	116	2	methods	method	NOUN
iajs-4097	116	3	gave	give	VERB
iajs-4097	116	4	close	close	ADJ
iajs-4097	116	5	ihjpas	ihjpa	NOUN
iajs-4097	116	6	.	.	PUNCT
iajs-4097	117	1	2025,38(4	2025,38(4	X
iajs-4097	117	2	)	)	PUNCT
iajs-4097	117	3	386	386	NUM
iajs-4097	117	4	results	result	NOUN
iajs-4097	117	5	,	,	PUNCT
iajs-4097	117	6	but	but	CCONJ
iajs-4097	117	7	the	the	DET
iajs-4097	117	8	bernoulli	bernoulli	NOUN
iajs-4097	117	9	polynomials	polynomial	NOUN
iajs-4097	117	10	gave	give	VERB
iajs-4097	117	11	better	well	ADJ
iajs-4097	117	12	results	result	NOUN
iajs-4097	117	13	when	when	SCONJ
iajs-4097	117	14	,	,	PUNCT
iajs-4097	117	15	such	such	ADJ
iajs-4097	117	16	that	that	PRON
iajs-4097	117	17	equals	equal	VERB
iajs-4097	117	18	,	,	PUNCT
iajs-4097	117	19	respectively	respectively	ADV
iajs-4097	117	20	,	,	PUNCT
iajs-4097	117	21	but	but	CCONJ
iajs-4097	117	22	when	when	SCONJ
iajs-4097	117	23	the	the	DET
iajs-4097	117	24	bernoulli	bernoulli	PROPN
iajs-4097	117	25	and	and	CCONJ
iajs-4097	117	26	standard	standard	ADJ
iajs-4097	117	27	polynomials	polynomial	NOUN
iajs-4097	117	28	give	give	VERB
iajs-4097	117	29	the	the	DET
iajs-4097	117	30	same	same	ADJ
iajs-4097	117	31	results	result	NOUN
iajs-4097	117	32	.	.	PUNCT
iajs-4097	118	1	norm	norm	NOUN
iajs-4097	118	2	.	.	PUNCT
iajs-4097	119	1	norm	norm	NOUN
iajs-4097	119	2	.	.	PUNCT
iajs-4097	120	1	figure	figure	VERB
iajs-4097	120	2	4	4	NUM
iajs-4097	120	3	.	.	PUNCT
iajs-4097	121	1	the	the	DET
iajs-4097	121	2	plots	plot	NOUN
iajs-4097	121	3	logarithmic	logarithmic	ADJ
iajs-4097	121	4	of	of	ADP
iajs-4097	121	5	and	and	CCONJ
iajs-4097	121	6	,	,	PUNCT
iajs-4097	121	7	for	for	ADP
iajs-4097	121	8	example	example	NOUN
iajs-4097	121	9	2	2	NUM
iajs-4097	121	10	,	,	PUNCT
iajs-4097	121	11	when	when	SCONJ
iajs-4097	121	12	.	.	PUNCT
iajs-4097	122	1	figure	figure	NOUN
iajs-4097	122	2	5	5	NUM
iajs-4097	122	3	.	.	PUNCT
iajs-4097	123	1	plots	plot	VERB
iajs-4097	123	2	the	the	DET
iajs-4097	123	3	norm	norm	NOUN
iajs-4097	123	4	for	for	ADP
iajs-4097	123	5	example	example	NOUN
iajs-4097	123	6	2	2	NUM
iajs-4097	123	7	,	,	PUNCT
iajs-4097	123	8	.	.	PUNCT
iajs-4097	124	1	exact	exact	ADJ
iajs-4097	124	2	solution	solution	NOUN
iajs-4097	124	3	approximate	approximate	ADJ
iajs-4097	124	4	solution	solution	NOUN
iajs-4097	124	5	figure	figure	NOUN
iajs-4097	124	6	6	6	NUM
iajs-4097	124	7	.	.	PUNCT
iajs-4097	124	8	comparison	comparison	NOUN
iajs-4097	124	9	between	between	ADP
iajs-4097	124	10	analytical	analytical	ADJ
iajs-4097	124	11	and	and	CCONJ
iajs-4097	124	12	numerical	numerical	ADJ
iajs-4097	124	13	solution	solution	NOUN
iajs-4097	124	14	for	for	ADP
iajs-4097	124	15	example	example	NOUN
iajs-4097	124	16	2	2	NUM
iajs-4097	124	17	by	by	ADP
iajs-4097	124	18	the	the	DET
iajs-4097	124	19	bernoulli	bernoulli	PROPN
iajs-4097	124	20	poly	poly	NOUN
iajs-4097	124	21	.	.	PUNCT
iajs-4097	125	1	example	example	NOUN
iajs-4097	126	1	3	3	X
iajs-4097	126	2	.	.	X
iajs-4097	126	3	consider	consider	VERB
iajs-4097	126	4	the	the	DET
iajs-4097	126	5	homogeneous	homogeneous	ADJ
iajs-4097	126	6	wave	wave	NOUN
iajs-4097	126	7	problem	problem	NOUN
iajs-4097	126	8	as	as	SCONJ
iajs-4097	126	9	follows	follow	VERB
iajs-4097	126	10	(	(	PUNCT
iajs-4097	126	11	7	7	NUM
iajs-4097	126	12	):	):	PUNCT
iajs-4097	126	13	(	(	PUNCT
iajs-4097	126	14	31	31	NUM
iajs-4097	126	15	)	)	PUNCT
iajs-4097	126	16	ihjpas	ihjpa	NOUN
iajs-4097	126	17	.	.	PUNCT
iajs-4097	127	1	2025,38(4	2025,38(4	NOUN
iajs-4097	127	2	)	)	PUNCT
iajs-4097	128	1	387	387	NUM
iajs-4097	128	2	with	with	ADP
iajs-4097	128	3	initial	initial	ADJ
iajs-4097	128	4	conditions	condition	NOUN
iajs-4097	128	5	:	:	PUNCT
iajs-4097	128	6	(	(	PUNCT
iajs-4097	128	7	)	)	PUNCT
iajs-4097	128	8	(	(	PUNCT
iajs-4097	128	9	)	)	PUNCT
iajs-4097	128	10	(	(	PUNCT
iajs-4097	128	11	)	)	PUNCT
iajs-4097	128	12	(	(	PUNCT
iajs-4097	128	13	32	32	NUM
iajs-4097	128	14	)	)	PUNCT
iajs-4097	128	15	and	and	CCONJ
iajs-4097	128	16	neumann	neumann	PROPN
iajs-4097	128	17	boundary	boundary	ADJ
iajs-4097	128	18	conditions	condition	NOUN
iajs-4097	128	19	:	:	PUNCT
iajs-4097	128	20	(	(	PUNCT
iajs-4097	128	21	)	)	PUNCT
iajs-4097	128	22	(	(	PUNCT
iajs-4097	128	23	)	)	PUNCT
iajs-4097	128	24	(	(	PUNCT
iajs-4097	128	25	)	)	PUNCT
iajs-4097	128	26	(	(	PUNCT
iajs-4097	128	27	)	)	PUNCT
iajs-4097	128	28	.	.	PUNCT
iajs-4097	129	1	(	(	PUNCT
iajs-4097	129	2	33	33	NUM
iajs-4097	129	3	)	)	PUNCT
iajs-4097	129	4	and	and	CCONJ
iajs-4097	129	5	the	the	DET
iajs-4097	129	6	analytical	analytical	ADJ
iajs-4097	129	7	solution	solution	NOUN
iajs-4097	129	8	is	be	AUX
iajs-4097	129	9	:	:	PUNCT
iajs-4097	129	10	(	(	PUNCT
iajs-4097	129	11	)	)	PUNCT
iajs-4097	129	12	(	(	PUNCT
iajs-4097	129	13	)	)	PUNCT
iajs-4097	129	14	(	(	PUNCT
iajs-4097	129	15	)	)	PUNCT
iajs-4097	129	16	.	.	PUNCT
iajs-4097	130	1	by	by	ADP
iajs-4097	130	2	implementing	implement	VERB
iajs-4097	130	3	the	the	DET
iajs-4097	130	4	proposed	propose	VERB
iajs-4097	130	5	methods	method	NOUN
iajs-4097	130	6	will	will	AUX
iajs-4097	130	7	be	be	AUX
iajs-4097	130	8	get	get	VERB
iajs-4097	130	9	the	the	DET
iajs-4097	130	10	following	follow	VERB
iajs-4097	130	11	linear	linear	PROPN
iajs-4097	130	12	equations	equation	NOUN
iajs-4097	130	13	system	system	NOUN
iajs-4097	130	14	:	:	PUNCT
iajs-4097	130	15	(	(	PUNCT
iajs-4097	130	16	)	)	PUNCT
iajs-4097	130	17	(	(	PUNCT
iajs-4097	130	18	)	)	PUNCT
iajs-4097	130	19	(	(	PUNCT
iajs-4097	130	20	)	)	PUNCT
iajs-4097	130	21	(	(	PUNCT
iajs-4097	130	22	)	)	PUNCT
iajs-4097	130	23	(	(	PUNCT
iajs-4097	130	24	)	)	PUNCT
iajs-4097	130	25	(	(	PUNCT
iajs-4097	130	26	)	)	PUNCT
iajs-4097	130	27	(	(	PUNCT
iajs-4097	130	28	)	)	PUNCT
iajs-4097	130	29	(	(	PUNCT
iajs-4097	130	30	)	)	PUNCT
iajs-4097	130	31	(	(	PUNCT
iajs-4097	130	32	)	)	PUNCT
iajs-4097	130	33	(	(	PUNCT
iajs-4097	130	34	)	)	PUNCT
iajs-4097	130	35	(	(	PUNCT
iajs-4097	130	36	)	)	PUNCT
iajs-4097	130	37	(	(	PUNCT
iajs-4097	130	38	)	)	PUNCT
iajs-4097	130	39	(	(	PUNCT
iajs-4097	130	40	)	)	PUNCT
iajs-4097	130	41	(	(	PUNCT
iajs-4097	130	42	)	)	PUNCT
iajs-4097	130	43	(	(	PUNCT
iajs-4097	130	44	)	)	PUNCT
iajs-4097	130	45	(	(	PUNCT
iajs-4097	130	46	)	)	PUNCT
iajs-4097	130	47	(	(	PUNCT
iajs-4097	130	48	)	)	PUNCT
iajs-4097	130	49	(	(	PUNCT
iajs-4097	130	50	)	)	PUNCT
iajs-4097	130	51	(	(	PUNCT
iajs-4097	130	52	)	)	PUNCT
iajs-4097	130	53	(	(	PUNCT
iajs-4097	130	54	)	)	PUNCT
iajs-4097	130	55	(	(	PUNCT
iajs-4097	130	56	)	)	PUNCT
iajs-4097	130	57	.	.	PUNCT
iajs-4097	131	1	table	table	NOUN
iajs-4097	131	2	3	3	NUM
iajs-4097	131	3	.	.	PUNCT
iajs-4097	132	1	the	the	DET
iajs-4097	132	2	values	value	NOUN
iajs-4097	132	3	of	of	ADP
iajs-4097	132	4	and	and	CCONJ
iajs-4097	132	5	for	for	ADP
iajs-4097	132	6	example	example	NOUN
iajs-4097	132	7	3	3	NUM
iajs-4097	132	8	,	,	PUNCT
iajs-4097	132	9	when	when	SCONJ
iajs-4097	132	10	.	.	PUNCT
iajs-4097	133	1	from	from	ADP
iajs-4097	133	2	table	table	NOUN
iajs-4097	133	3	3	3	NUM
iajs-4097	133	4	.	.	PUNCT
iajs-4097	134	1	and	and	CCONJ
iajs-4097	134	2	figure	figure	VERB
iajs-4097	134	3	7	7	NUM
iajs-4097	134	4	.	.	NUM
iajs-4097	134	5	,	,	PUNCT
iajs-4097	134	6	the	the	DET
iajs-4097	134	7	error	error	NOUN
iajs-4097	134	8	norm	norm	NOUN
iajs-4097	134	9	and	and	CCONJ
iajs-4097	134	10	mean	mean	ADJ
iajs-4097	134	11	square	square	ADJ
iajs-4097	134	12	error	error	NOUN
iajs-4097	134	13	values	value	NOUN
iajs-4097	134	14	are	be	AUX
iajs-4097	134	15	visible	visible	ADJ
iajs-4097	134	16	,	,	PUNCT
iajs-4097	134	17	as	as	SCONJ
iajs-4097	134	18	these	these	DET
iajs-4097	134	19	values	value	NOUN
iajs-4097	134	20	decrease	decrease	VERB
iajs-4097	134	21	as	as	ADP
iajs-4097	134	22	the	the	DET
iajs-4097	134	23	value	value	NOUN
iajs-4097	134	24	of	of	ADP
iajs-4097	134	25	increases	increase	NOUN
iajs-4097	134	26	,	,	PUNCT
iajs-4097	134	27	which	which	PRON
iajs-4097	134	28	represents	represent	VERB
iajs-4097	134	29	the	the	DET
iajs-4097	134	30	degree	degree	NOUN
iajs-4097	134	31	of	of	ADP
iajs-4097	134	32	the	the	DET
iajs-4097	134	33	polynomial	polynomial	NOUN
iajs-4097	134	34	.	.	PUNCT
iajs-4097	135	1	the	the	DET
iajs-4097	135	2	best	good	ADJ
iajs-4097	135	3	methods	method	NOUN
iajs-4097	135	4	for	for	ADP
iajs-4097	135	5	approximation	approximation	NOUN
iajs-4097	135	6	are	be	AUX
iajs-4097	135	7	the	the	DET
iajs-4097	135	8	bernoulli	bernoulli	NOUN
iajs-4097	135	9	polynomials	polynomial	NOUN
iajs-4097	135	10	,	,	PUNCT
iajs-4097	135	11	such	such	ADJ
iajs-4097	135	12	that	that	SCONJ
iajs-4097	135	13	when	when	SCONJ
iajs-4097	135	14	,	,	PUNCT
iajs-4097	135	15	the	the	DET
iajs-4097	135	16	values	value	NOUN
iajs-4097	135	17	of	of	ADP
iajs-4097	135	18	and	and	CCONJ
iajs-4097	135	19	equals	equal	VERB
iajs-4097	135	20	,	,	PUNCT
iajs-4097	135	21	,	,	PUNCT
iajs-4097	135	22	ihjpas	ihjpas	PROPN
iajs-4097	135	23	.	.	PUNCT
iajs-4097	136	1	2025,38(4	2025,38(4	X
iajs-4097	136	2	)	)	PUNCT
iajs-4097	136	3	388	388	NUM
iajs-4097	136	4	respectively	respectively	ADV
iajs-4097	136	5	,	,	PUNCT
iajs-4097	136	6	but	but	CCONJ
iajs-4097	136	7	when	when	SCONJ
iajs-4097	136	8	,	,	PUNCT
iajs-4097	136	9	it	it	PRON
iajs-4097	136	10	can	can	AUX
iajs-4097	136	11	be	be	AUX
iajs-4097	136	12	seen	see	VERB
iajs-4097	136	13	the	the	DET
iajs-4097	136	14	values	value	NOUN
iajs-4097	136	15	of	of	ADP
iajs-4097	136	16	and	and	CCONJ
iajs-4097	136	17	gives	give	VERB
iajs-4097	136	18	,	,	PUNCT
iajs-4097	136	19	.	.	PUNCT
iajs-4097	137	1	norm	norm	NOUN
iajs-4097	137	2	.	.	PUNCT
iajs-4097	138	1	norm	norm	NOUN
iajs-4097	138	2	.	.	PUNCT
iajs-4097	139	1	figure	figure	VERB
iajs-4097	139	2	7	7	NUM
iajs-4097	139	3	.	.	PUNCT
iajs-4097	140	1	the	the	DET
iajs-4097	140	2	plots	plot	NOUN
iajs-4097	140	3	logarithmic	logarithmic	ADJ
iajs-4097	140	4	of	of	ADP
iajs-4097	140	5	and	and	CCONJ
iajs-4097	140	6	,	,	PUNCT
iajs-4097	140	7	for	for	ADP
iajs-4097	140	8	example	example	NOUN
iajs-4097	140	9	3	3	NUM
iajs-4097	140	10	,	,	PUNCT
iajs-4097	140	11	when	when	SCONJ
iajs-4097	140	12	.	.	PUNCT
iajs-4097	141	1	figure	figure	NOUN
iajs-4097	141	2	8	8	NUM
iajs-4097	141	3	.	.	PUNCT
iajs-4097	142	1	plots	plot	VERB
iajs-4097	142	2	the	the	DET
iajs-4097	142	3	norm	norm	NOUN
iajs-4097	142	4	for	for	ADP
iajs-4097	142	5	example	example	NOUN
iajs-4097	142	6	3	3	NUM
iajs-4097	142	7	,	,	PUNCT
iajs-4097	142	8	.	.	PUNCT
iajs-4097	143	1	exact	exact	ADJ
iajs-4097	143	2	solution	solution	NOUN
iajs-4097	143	3	approximate	approximate	ADJ
iajs-4097	143	4	solution	solution	NOUN
iajs-4097	143	5	figure	figure	NOUN
iajs-4097	143	6	9	9	NUM
iajs-4097	143	7	.	.	PUNCT
iajs-4097	143	8	comparison	comparison	NOUN
iajs-4097	143	9	between	between	ADP
iajs-4097	143	10	analytical	analytical	ADJ
iajs-4097	143	11	and	and	CCONJ
iajs-4097	143	12	numerical	numerical	ADJ
iajs-4097	143	13	solution	solution	NOUN
iajs-4097	143	14	for	for	ADP
iajs-4097	143	15	example	example	NOUN
iajs-4097	143	16	3	3	NUM
iajs-4097	143	17	by	by	ADP
iajs-4097	143	18	the	the	DET
iajs-4097	143	19	bernoulli	bernoulli	PROPN
iajs-4097	143	20	poly	poly	NOUN
iajs-4097	143	21	.	.	PUNCT
iajs-4097	144	1	example	example	NOUN
iajs-4097	144	2	4	4	NUM
iajs-4097	144	3	:	:	PUNCT
iajs-4097	144	4	consider	consider	VERB
iajs-4097	144	5	the	the	DET
iajs-4097	144	6	following	follow	VERB
iajs-4097	144	7	inhomogeneous	inhomogeneous	ADJ
iajs-4097	144	8	wave	wave	NOUN
iajs-4097	144	9	problem	problem	NOUN
iajs-4097	144	10	(	(	PUNCT
iajs-4097	144	11	7	7	NUM
iajs-4097	144	12	):	):	PUNCT
iajs-4097	144	13	(	(	PUNCT
iajs-4097	144	14	34	34	NUM
iajs-4097	144	15	)	)	PUNCT
iajs-4097	144	16	with	with	ADP
iajs-4097	144	17	initial	initial	ADJ
iajs-4097	144	18	conditions	condition	NOUN
iajs-4097	144	19	:	:	PUNCT
iajs-4097	144	20	ihjpas	ihjpas	PROPN
iajs-4097	144	21	.	.	PUNCT
iajs-4097	145	1	2025,38(4	2025,38(4	X
iajs-4097	145	2	)	)	PUNCT
iajs-4097	146	1	389	389	NUM
iajs-4097	146	2	(	(	PUNCT
iajs-4097	146	3	)	)	PUNCT
iajs-4097	146	4	(	(	PUNCT
iajs-4097	146	5	)	)	PUNCT
iajs-4097	146	6	(	(	PUNCT
iajs-4097	146	7	)	)	PUNCT
iajs-4097	146	8	(	(	PUNCT
iajs-4097	146	9	35	35	NUM
iajs-4097	146	10	)	)	PUNCT
iajs-4097	146	11	and	and	CCONJ
iajs-4097	146	12	neumann	neumann	PROPN
iajs-4097	146	13	boundary	boundary	ADJ
iajs-4097	146	14	conditions	condition	NOUN
iajs-4097	146	15	:	:	PUNCT
iajs-4097	146	16	(	(	PUNCT
iajs-4097	146	17	)	)	PUNCT
iajs-4097	146	18	(	(	PUNCT
iajs-4097	146	19	)	)	PUNCT
iajs-4097	146	20	(	(	PUNCT
iajs-4097	146	21	)	)	PUNCT
iajs-4097	146	22	(	(	PUNCT
iajs-4097	146	23	)	)	PUNCT
iajs-4097	146	24	.	.	PUNCT
iajs-4097	147	1	(	(	PUNCT
iajs-4097	147	2	36	36	NUM
iajs-4097	147	3	)	)	PUNCT
iajs-4097	147	4	and	and	CCONJ
iajs-4097	147	5	the	the	DET
iajs-4097	147	6	analytical	analytical	ADJ
iajs-4097	147	7	solution	solution	NOUN
iajs-4097	147	8	is	be	AUX
iajs-4097	147	9	:	:	PUNCT
iajs-4097	147	10	(	(	PUNCT
iajs-4097	147	11	)	)	PUNCT
iajs-4097	147	12	(	(	PUNCT
iajs-4097	147	13	)	)	PUNCT
iajs-4097	147	14	(	(	PUNCT
iajs-4097	147	15	)	)	PUNCT
iajs-4097	147	16	.	.	PUNCT
iajs-4097	148	1	(	(	PUNCT
iajs-4097	148	2	)	)	PUNCT
iajs-4097	148	3	(	(	PUNCT
iajs-4097	148	4	)	)	PUNCT
iajs-4097	148	5	(	(	PUNCT
iajs-4097	148	6	)	)	PUNCT
iajs-4097	148	7	(	(	PUNCT
iajs-4097	148	8	)	)	PUNCT
iajs-4097	148	9	(	(	PUNCT
iajs-4097	148	10	)	)	PUNCT
iajs-4097	148	11	(	(	PUNCT
iajs-4097	148	12	)	)	PUNCT
iajs-4097	148	13	(	(	PUNCT
iajs-4097	148	14	)	)	PUNCT
iajs-4097	148	15	(	(	PUNCT
iajs-4097	148	16	)	)	PUNCT
iajs-4097	148	17	(	(	PUNCT
iajs-4097	148	18	)	)	PUNCT
iajs-4097	148	19	(	(	PUNCT
iajs-4097	148	20	)	)	PUNCT
iajs-4097	148	21	(	(	PUNCT
iajs-4097	148	22	)	)	PUNCT
iajs-4097	148	23	(	(	PUNCT
iajs-4097	148	24	)	)	PUNCT
iajs-4097	148	25	(	(	PUNCT
iajs-4097	148	26	)	)	PUNCT
iajs-4097	148	27	(	(	PUNCT
iajs-4097	148	28	)	)	PUNCT
iajs-4097	148	29	(	(	PUNCT
iajs-4097	148	30	)	)	PUNCT
iajs-4097	148	31	(	(	PUNCT
iajs-4097	148	32	)	)	PUNCT
iajs-4097	148	33	(	(	PUNCT
iajs-4097	148	34	)	)	PUNCT
iajs-4097	148	35	(	(	PUNCT
iajs-4097	148	36	)	)	PUNCT
iajs-4097	148	37	(	(	PUNCT
iajs-4097	148	38	)	)	PUNCT
iajs-4097	148	39	(	(	PUNCT
iajs-4097	148	40	)	)	PUNCT
iajs-4097	148	41	(	(	PUNCT
iajs-4097	148	42	)	)	PUNCT
iajs-4097	148	43	.	.	PUNCT
iajs-4097	149	1	table	table	NOUN
iajs-4097	149	2	4	4	NUM
iajs-4097	149	3	.	.	PUNCT
iajs-4097	150	1	the	the	DET
iajs-4097	150	2	values	value	NOUN
iajs-4097	150	3	error	error	NOUN
iajs-4097	150	4	norm	norm	NOUN
iajs-4097	150	5	and	and	CCONJ
iajs-4097	150	6	for	for	ADP
iajs-4097	150	7	example	example	NOUN
iajs-4097	150	8	4	4	NUM
iajs-4097	150	9	,	,	PUNCT
iajs-4097	150	10	when	when	SCONJ
iajs-4097	150	11	.	.	PUNCT
iajs-4097	151	1	from	from	ADP
iajs-4097	151	2	table	table	NOUN
iajs-4097	151	3	4	4	NUM
iajs-4097	151	4	,	,	PUNCT
iajs-4097	151	5	and	and	CCONJ
iajs-4097	151	6	figure	figure	VERB
iajs-4097	151	7	10	10	NUM
iajs-4097	151	8	,	,	PUNCT
iajs-4097	151	9	it	it	PRON
iajs-4097	151	10	can	can	AUX
iajs-4097	151	11	be	be	AUX
iajs-4097	151	12	seen	see	VERB
iajs-4097	151	13	the	the	DET
iajs-4097	151	14	values	value	NOUN
iajs-4097	151	15	of	of	ADP
iajs-4097	151	16	the	the	PRON
iajs-4097	151	17	and	and	CCONJ
iajs-4097	151	18	.	.	PUNCT
iajs-4097	152	1	the	the	DET
iajs-4097	152	2	best	good	ADJ
iajs-4097	152	3	approximation	approximation	NOUN
iajs-4097	152	4	method	method	NOUN
iajs-4097	152	5	was	be	AUX
iajs-4097	152	6	the	the	DET
iajs-4097	152	7	bernoulli	bernoulli	NOUN
iajs-4097	152	8	polynomial	polynomial	NOUN
iajs-4097	152	9	,	,	PUNCT
iajs-4097	152	10	where	where	SCONJ
iajs-4097	152	11	the	the	DET
iajs-4097	152	12	value	value	NOUN
iajs-4097	152	13	of	of	ADP
iajs-4097	152	14	the	the	DET
iajs-4097	152	15	error	error	NOUN
iajs-4097	152	16	norm	norm	NOUN
iajs-4097	152	17	was	be	AUX
iajs-4097	152	18	equal	equal	ADJ
iajs-4097	152	19	to	to	ADP
iajs-4097	152	20	and	and	CCONJ
iajs-4097	152	21	the	the	DET
iajs-4097	152	22	value	value	NOUN
iajs-4097	152	23	of	of	ADP
iajs-4097	152	24	the	the	DET
iajs-4097	152	25	mean	mean	ADJ
iajs-4097	152	26	square	square	ADJ
iajs-4097	152	27	error	error	NOUN
iajs-4097	152	28	was	be	AUX
iajs-4097	152	29	equal	equal	ADJ
iajs-4097	152	30	to	to	ADP
iajs-4097	152	31	when	when	SCONJ
iajs-4097	152	32	,	,	PUNCT
iajs-4097	152	33	also	also	ADV
iajs-4097	152	34	when	when	SCONJ
iajs-4097	152	35	,	,	PUNCT
iajs-4097	152	36	the	the	DET
iajs-4097	152	37	values	value	NOUN
iajs-4097	152	38	of	of	ADP
iajs-4097	152	39	,	,	PUNCT
iajs-4097	152	40	and	and	CCONJ
iajs-4097	152	41	are	be	AUX
iajs-4097	152	42	,	,	PUNCT
iajs-4097	152	43	,	,	PUNCT
iajs-4097	152	44	respectively	respectively	ADV
iajs-4097	152	45	.	.	PUNCT
iajs-4097	153	1	ihjpas	ihjpas	PROPN
iajs-4097	153	2	.	.	PUNCT
iajs-4097	154	1	2025,38(4	2025,38(4	X
iajs-4097	154	2	)	)	PUNCT
iajs-4097	154	3	390	390	NUM
iajs-4097	154	4	norm	norm	NOUN
iajs-4097	154	5	.	.	PUNCT
iajs-4097	155	1	norm	norm	NOUN
iajs-4097	155	2	.	.	PUNCT
iajs-4097	156	1	figure	figure	VERB
iajs-4097	156	2	10	10	NUM
iajs-4097	156	3	.	.	PUNCT
iajs-4097	157	1	the	the	DET
iajs-4097	157	2	plots	plot	NOUN
iajs-4097	157	3	logarithmic	logarithmic	ADJ
iajs-4097	157	4	of	of	ADP
iajs-4097	157	5	and	and	CCONJ
iajs-4097	157	6	,	,	PUNCT
iajs-4097	157	7	for	for	ADP
iajs-4097	157	8	example	example	NOUN
iajs-4097	157	9	4	4	NUM
iajs-4097	157	10	,	,	PUNCT
iajs-4097	157	11	when	when	SCONJ
iajs-4097	157	12	.	.	PUNCT
iajs-4097	158	1	figure	figure	NOUN
iajs-4097	158	2	11	11	NUM
iajs-4097	158	3	.	.	PUNCT
iajs-4097	159	1	plots	plot	VERB
iajs-4097	159	2	the	the	DET
iajs-4097	159	3	norm	norm	NOUN
iajs-4097	159	4	for	for	ADP
iajs-4097	159	5	example	example	NOUN
iajs-4097	159	6	4	4	NUM
iajs-4097	159	7	,	,	PUNCT
iajs-4097	159	8	.	.	PUNCT
iajs-4097	160	1	exact	exact	ADJ
iajs-4097	160	2	solution	solution	NOUN
iajs-4097	160	3	approximate	approximate	ADJ
iajs-4097	160	4	solution	solution	NOUN
iajs-4097	160	5	figure	figure	NOUN
iajs-4097	160	6	12	12	NUM
iajs-4097	160	7	.	.	PUNCT
iajs-4097	161	1	comparison	comparison	NOUN
iajs-4097	161	2	between	between	ADP
iajs-4097	161	3	analytical	analytical	ADJ
iajs-4097	161	4	and	and	CCONJ
iajs-4097	161	5	numerical	numerical	ADJ
iajs-4097	161	6	solution	solution	NOUN
iajs-4097	161	7	for	for	ADP
iajs-4097	161	8	example	example	NOUN
iajs-4097	161	9	4	4	NUM
iajs-4097	161	10	by	by	ADP
iajs-4097	161	11	the	the	DET
iajs-4097	161	12	bernoulli	bernoulli	PROPN
iajs-4097	161	13	poly	poly	NOUN
iajs-4097	161	14	.	.	PUNCT
iajs-4097	162	1	example	example	NOUN
iajs-4097	162	2	5	5	NUM
iajs-4097	162	3	.	.	PUNCT
iajs-4097	162	4	consider	consider	VERB
iajs-4097	162	5	the	the	DET
iajs-4097	162	6	following	follow	VERB
iajs-4097	162	7	homogeneous	homogeneous	ADJ
iajs-4097	162	8	wave	wave	NOUN
iajs-4097	162	9	problem	problem	NOUN
iajs-4097	162	10	:	:	PUNCT
iajs-4097	162	11	,	,	PUNCT
iajs-4097	162	12	(	(	PUNCT
iajs-4097	162	13	37	37	NUM
iajs-4097	162	14	)	)	PUNCT
iajs-4097	162	15	with	with	ADP
iajs-4097	162	16	initial	initial	ADJ
iajs-4097	162	17	conditions	condition	NOUN
iajs-4097	162	18	:	:	PUNCT
iajs-4097	162	19	(	(	PUNCT
iajs-4097	162	20	)	)	PUNCT
iajs-4097	162	21	(	(	PUNCT
iajs-4097	162	22	)	)	PUNCT
iajs-4097	162	23	(	(	PUNCT
iajs-4097	162	24	38	38	NUM
iajs-4097	162	25	)	)	PUNCT
iajs-4097	162	26	ihjpas	ihjpa	NOUN
iajs-4097	162	27	.	.	PUNCT
iajs-4097	163	1	2025,38(4	2025,38(4	NOUN
iajs-4097	163	2	)	)	PUNCT
iajs-4097	163	3	391	391	NUM
iajs-4097	163	4	and	and	CCONJ
iajs-4097	163	5	mixed	mixed	ADJ
iajs-4097	163	6	boundary	boundary	ADJ
iajs-4097	163	7	conditions	condition	NOUN
iajs-4097	163	8	:	:	PUNCT
iajs-4097	163	9	(	(	PUNCT
iajs-4097	163	10	)	)	PUNCT
iajs-4097	163	11	(	(	PUNCT
iajs-4097	163	12	)	)	PUNCT
iajs-4097	163	13	.	.	PUNCT
iajs-4097	164	1	(	(	PUNCT
iajs-4097	164	2	39	39	NUM
iajs-4097	164	3	)	)	PUNCT
iajs-4097	164	4	and	and	CCONJ
iajs-4097	164	5	the	the	DET
iajs-4097	164	6	analytical	analytical	ADJ
iajs-4097	164	7	solution	solution	NOUN
iajs-4097	164	8	is	be	AUX
iajs-4097	164	9	:	:	PUNCT
iajs-4097	164	10	(	(	PUNCT
iajs-4097	164	11	)	)	PUNCT
iajs-4097	164	12	.	.	PUNCT
iajs-4097	165	1	(	(	PUNCT
iajs-4097	165	2	)	)	PUNCT
iajs-4097	165	3	(	(	PUNCT
iajs-4097	165	4	)	)	PUNCT
iajs-4097	165	5	(	(	PUNCT
iajs-4097	165	6	)	)	PUNCT
iajs-4097	165	7	(	(	PUNCT
iajs-4097	165	8	)	)	PUNCT
iajs-4097	165	9	(	(	PUNCT
iajs-4097	165	10	)	)	PUNCT
iajs-4097	165	11	(	(	PUNCT
iajs-4097	165	12	)	)	PUNCT
iajs-4097	165	13	(	(	PUNCT
iajs-4097	165	14	)	)	PUNCT
iajs-4097	165	15	(	(	PUNCT
iajs-4097	165	16	)	)	PUNCT
iajs-4097	165	17	(	(	PUNCT
iajs-4097	165	18	)	)	PUNCT
iajs-4097	165	19	(	(	PUNCT
iajs-4097	165	20	)	)	PUNCT
iajs-4097	165	21	(	(	PUNCT
iajs-4097	165	22	)	)	PUNCT
iajs-4097	165	23	(	(	PUNCT
iajs-4097	165	24	)	)	PUNCT
iajs-4097	165	25	(	(	PUNCT
iajs-4097	165	26	)	)	PUNCT
iajs-4097	165	27	(	(	PUNCT
iajs-4097	165	28	)	)	PUNCT
iajs-4097	165	29	(	(	PUNCT
iajs-4097	165	30	)	)	PUNCT
iajs-4097	165	31	(	(	PUNCT
iajs-4097	165	32	)	)	PUNCT
iajs-4097	165	33	(	(	PUNCT
iajs-4097	165	34	)	)	PUNCT
iajs-4097	165	35	(	(	PUNCT
iajs-4097	165	36	)	)	PUNCT
iajs-4097	165	37	.	.	PUNCT
iajs-4097	166	1	table	table	NOUN
iajs-4097	166	2	5	5	NUM
iajs-4097	166	3	.	.	PUNCT
iajs-4097	167	1	the	the	DET
iajs-4097	167	2	values	value	NOUN
iajs-4097	167	3	and	and	CCONJ
iajs-4097	167	4	for	for	ADP
iajs-4097	167	5	example	example	NOUN
iajs-4097	167	6	5	5	NUM
iajs-4097	167	7	,	,	PUNCT
iajs-4097	167	8	when	when	SCONJ
iajs-4097	167	9	.	.	PUNCT
iajs-4097	168	1	the	the	DET
iajs-4097	168	2	mean	mean	ADJ
iajs-4097	168	3	square	square	ADJ
iajs-4097	168	4	error	error	NOUN
iajs-4097	168	5	and	and	CCONJ
iajs-4097	168	6	the	the	DET
iajs-4097	168	7	error	error	NOUN
iajs-4097	168	8	norm	norm	NOUN
iajs-4097	168	9	values	value	NOUN
iajs-4097	168	10	can	can	AUX
iajs-4097	168	11	be	be	AUX
iajs-4097	168	12	easily	easily	ADV
iajs-4097	168	13	observed	observe	VERB
iajs-4097	168	14	in	in	ADP
iajs-4097	168	15	table	table	NOUN
iajs-4097	168	16	5	5	NUM
iajs-4097	168	17	,	,	PUNCT
iajs-4097	168	18	and	and	CCONJ
iajs-4097	168	19	figure	figure	VERB
iajs-4097	168	20	13	13	NUM
iajs-4097	168	21	,	,	PUNCT
iajs-4097	168	22	which	which	PRON
iajs-4097	168	23	we	we	PRON
iajs-4097	168	24	obtained	obtain	VERB
iajs-4097	168	25	by	by	ADP
iajs-4097	168	26	solving	solve	VERB
iajs-4097	168	27	example	example	NOUN
iajs-4097	168	28	5	5	NUM
iajs-4097	168	29	using	use	VERB
iajs-4097	168	30	mathematica	mathematica	PROPN
iajs-4097	168	31	.	.	PUNCT
iajs-4097	169	1	some	some	DET
iajs-4097	169	2	methods	method	NOUN
iajs-4097	169	3	gave	give	VERB
iajs-4097	169	4	close	close	ADJ
iajs-4097	169	5	results	result	NOUN
iajs-4097	169	6	;	;	PUNCT
iajs-4097	169	7	except	except	SCONJ
iajs-4097	169	8	for	for	ADP
iajs-4097	169	9	the	the	DET
iajs-4097	169	10	bernoulli	bernoulli	NOUN
iajs-4097	169	11	polynomial	polynomial	NOUN
iajs-4097	169	12	,	,	PUNCT
iajs-4097	169	13	which	which	PRON
iajs-4097	169	14	gave	give	VERB
iajs-4097	169	15	better	well	ADJ
iajs-4097	169	16	results	result	NOUN
iajs-4097	169	17	in	in	ADP
iajs-4097	169	18	general	general	ADJ
iajs-4097	169	19	.	.	PUNCT
iajs-4097	170	1	the	the	DET
iajs-4097	170	2	values	value	NOUN
iajs-4097	170	3	of	of	ADP
iajs-4097	170	4	,	,	PUNCT
iajs-4097	170	5	and	and	CCONJ
iajs-4097	170	6	when	when	SCONJ
iajs-4097	170	7	can	can	AUX
iajs-4097	170	8	be	be	AUX
iajs-4097	170	9	observed	observe	VERB
iajs-4097	170	10	to	to	PART
iajs-4097	170	11	be	be	AUX
iajs-4097	170	12	equals	equal	VERB
iajs-4097	170	13	to	to	PART
iajs-4097	170	14	,	,	PUNCT
iajs-4097	170	15	and	and	CCONJ
iajs-4097	170	16	equals	equal	VERB
iajs-4097	170	17	,	,	PUNCT
iajs-4097	170	18	,	,	PUNCT
iajs-4097	171	1	respectively	respectively	ADV
iajs-4097	171	2	,	,	PUNCT
iajs-4097	171	3	when	when	SCONJ
iajs-4097	171	4	.	.	PUNCT
iajs-4097	171	5	ihjpas	ihjpas	PROPN
iajs-4097	171	6	.	.	PUNCT
iajs-4097	172	1	2025,38(4	2025,38(4	X
iajs-4097	172	2	)	)	PUNCT
iajs-4097	172	3	392	392	NUM
iajs-4097	172	4	norm	norm	NOUN
iajs-4097	172	5	.	.	PUNCT
iajs-4097	173	1	norm	norm	NOUN
iajs-4097	173	2	.	.	PUNCT
iajs-4097	174	1	figure	figure	VERB
iajs-4097	174	2	13	13	NUM
iajs-4097	174	3	.	.	PUNCT
iajs-4097	175	1	the	the	DET
iajs-4097	175	2	plots	plot	NOUN
iajs-4097	175	3	logarithmic	logarithmic	ADJ
iajs-4097	175	4	of	of	ADP
iajs-4097	175	5	and	and	CCONJ
iajs-4097	175	6	,	,	PUNCT
iajs-4097	175	7	for	for	ADP
iajs-4097	175	8	example	example	NOUN
iajs-4097	175	9	5	5	NUM
iajs-4097	175	10	,	,	PUNCT
iajs-4097	175	11	when	when	SCONJ
iajs-4097	175	12	.	.	PUNCT
iajs-4097	176	1	figure	figure	NOUN
iajs-4097	176	2	14	14	NUM
iajs-4097	176	3	.	.	PUNCT
iajs-4097	177	1	plots	plot	VERB
iajs-4097	177	2	the	the	DET
iajs-4097	177	3	norm	norm	NOUN
iajs-4097	177	4	for	for	ADP
iajs-4097	177	5	example	example	NOUN
iajs-4097	177	6	5	5	NUM
iajs-4097	177	7	,	,	PUNCT
iajs-4097	177	8	when	when	SCONJ
iajs-4097	177	9	.	.	PUNCT
iajs-4097	178	1	exact	exact	ADJ
iajs-4097	178	2	solution	solution	NOUN
iajs-4097	178	3	approximate	approximate	ADJ
iajs-4097	178	4	solution	solution	NOUN
iajs-4097	178	5	figure	figure	NOUN
iajs-4097	178	6	15	15	NUM
iajs-4097	178	7	.	.	PUNCT
iajs-4097	179	1	comparison	comparison	NOUN
iajs-4097	179	2	between	between	ADP
iajs-4097	179	3	analytical	analytical	ADJ
iajs-4097	179	4	and	and	CCONJ
iajs-4097	179	5	numerical	numerical	ADJ
iajs-4097	179	6	solution	solution	NOUN
iajs-4097	179	7	for	for	ADP
iajs-4097	179	8	example	example	NOUN
iajs-4097	179	9	5	5	NUM
iajs-4097	179	10	by	by	ADP
iajs-4097	179	11	the	the	DET
iajs-4097	179	12	bernoulli	bernoulli	PROPN
iajs-4097	179	13	poly	poly	NOUN
iajs-4097	179	14	.	.	PUNCT
iajs-4097	180	1	example	example	NOUN
iajs-4097	180	2	6	6	NUM
iajs-4097	180	3	.	.	PUNCT
iajs-4097	180	4	consider	consider	VERB
iajs-4097	180	5	the	the	DET
iajs-4097	180	6	following	follow	VERB
iajs-4097	180	7	homogeneous	homogeneous	ADJ
iajs-4097	180	8	wave	wave	NOUN
iajs-4097	180	9	problem	problem	NOUN
iajs-4097	180	10	:	:	PUNCT
iajs-4097	180	11	(	(	PUNCT
iajs-4097	180	12	40	40	NUM
iajs-4097	180	13	)	)	PUNCT
iajs-4097	180	14	with	with	ADP
iajs-4097	180	15	initial	initial	ADJ
iajs-4097	180	16	conditions	condition	NOUN
iajs-4097	180	17	:	:	PUNCT
iajs-4097	180	18	(	(	PUNCT
iajs-4097	180	19	)	)	PUNCT
iajs-4097	180	20	(	(	PUNCT
iajs-4097	180	21	)	)	PUNCT
iajs-4097	180	22	(	(	PUNCT
iajs-4097	180	23	)	)	PUNCT
iajs-4097	180	24	(	(	PUNCT
iajs-4097	180	25	41	41	NUM
iajs-4097	180	26	)	)	PUNCT
iajs-4097	180	27	and	and	CCONJ
iajs-4097	180	28	mixed	mixed	ADJ
iajs-4097	180	29	boundary	boundary	ADJ
iajs-4097	180	30	conditions	condition	NOUN
iajs-4097	180	31	:	:	PUNCT
iajs-4097	180	32	(	(	PUNCT
iajs-4097	180	33	)	)	PUNCT
iajs-4097	180	34	(	(	PUNCT
iajs-4097	180	35	)	)	PUNCT
iajs-4097	180	36	(	(	PUNCT
iajs-4097	180	37	)	)	PUNCT
iajs-4097	180	38	(	(	PUNCT
iajs-4097	180	39	)	)	PUNCT
iajs-4097	180	40	(	(	PUNCT
iajs-4097	180	41	)	)	PUNCT
iajs-4097	180	42	(	(	PUNCT
iajs-4097	180	43	42	42	X
iajs-4097	180	44	)	)	PUNCT
iajs-4097	180	45	ihjpas	ihjpa	NOUN
iajs-4097	180	46	.	.	PUNCT
iajs-4097	181	1	2025,38(4	2025,38(4	NOUN
iajs-4097	181	2	)	)	PUNCT
iajs-4097	181	3	393	393	NUM
iajs-4097	182	1	and	and	CCONJ
iajs-4097	182	2	the	the	DET
iajs-4097	182	3	analytical	analytical	ADJ
iajs-4097	182	4	solution	solution	NOUN
iajs-4097	182	5	is	be	AUX
iajs-4097	182	6	:	:	PUNCT
iajs-4097	182	7	(	(	PUNCT
iajs-4097	182	8	)	)	PUNCT
iajs-4097	182	9	(	(	PUNCT
iajs-4097	182	10	)	)	PUNCT
iajs-4097	182	11	(	(	PUNCT
iajs-4097	182	12	)	)	PUNCT
iajs-4097	182	13	.	.	PUNCT
iajs-4097	183	1	(	(	PUNCT
iajs-4097	183	2	)	)	PUNCT
iajs-4097	183	3	(	(	PUNCT
iajs-4097	183	4	)	)	PUNCT
iajs-4097	183	5	(	(	PUNCT
iajs-4097	183	6	)	)	PUNCT
iajs-4097	183	7	(	(	PUNCT
iajs-4097	183	8	)	)	PUNCT
iajs-4097	183	9	(	(	PUNCT
iajs-4097	183	10	)	)	PUNCT
iajs-4097	183	11	(	(	PUNCT
iajs-4097	183	12	)	)	PUNCT
iajs-4097	183	13	(	(	PUNCT
iajs-4097	183	14	)	)	PUNCT
iajs-4097	183	15	(	(	PUNCT
iajs-4097	183	16	)	)	PUNCT
iajs-4097	183	17	(	(	PUNCT
iajs-4097	183	18	)	)	PUNCT
iajs-4097	183	19	(	(	PUNCT
iajs-4097	183	20	)	)	PUNCT
iajs-4097	183	21	(	(	PUNCT
iajs-4097	183	22	)	)	PUNCT
iajs-4097	183	23	(	(	PUNCT
iajs-4097	183	24	)	)	PUNCT
iajs-4097	183	25	(	(	PUNCT
iajs-4097	183	26	)	)	PUNCT
iajs-4097	183	27	(	(	PUNCT
iajs-4097	183	28	)	)	PUNCT
iajs-4097	183	29	(	(	PUNCT
iajs-4097	183	30	)	)	PUNCT
iajs-4097	183	31	(	(	PUNCT
iajs-4097	183	32	)	)	PUNCT
iajs-4097	183	33	(	(	PUNCT
iajs-4097	183	34	)	)	PUNCT
iajs-4097	183	35	(	(	PUNCT
iajs-4097	183	36	)	)	PUNCT
iajs-4097	183	37	(	(	PUNCT
iajs-4097	183	38	)	)	PUNCT
iajs-4097	183	39	(	(	PUNCT
iajs-4097	183	40	)	)	PUNCT
iajs-4097	183	41	(	(	PUNCT
iajs-4097	183	42	)	)	PUNCT
iajs-4097	183	43	(	(	PUNCT
iajs-4097	183	44	)	)	PUNCT
iajs-4097	183	45	.	.	PUNCT
iajs-4097	184	1	table	table	NOUN
iajs-4097	184	2	6	6	NUM
iajs-4097	184	3	.	.	PUNCT
iajs-4097	185	1	the	the	DET
iajs-4097	185	2	values	value	NOUN
iajs-4097	185	3	of	of	ADP
iajs-4097	185	4	and	and	CCONJ
iajs-4097	185	5	for	for	ADP
iajs-4097	185	6	example	example	NOUN
iajs-4097	185	7	6	6	NUM
iajs-4097	185	8	,	,	PUNCT
iajs-4097	185	9	when	when	SCONJ
iajs-4097	185	10	.	.	PUNCT
iajs-4097	186	1	from	from	ADP
iajs-4097	186	2	table	table	NOUN
iajs-4097	186	3	6	6	NUM
iajs-4097	186	4	,	,	PUNCT
iajs-4097	186	5	and	and	CCONJ
iajs-4097	186	6	figure	figure	VERB
iajs-4097	186	7	16	16	NUM
iajs-4097	186	8	,	,	PUNCT
iajs-4097	186	9	the	the	DET
iajs-4097	186	10	error	error	NOUN
iajs-4097	186	11	norm	norm	NOUN
iajs-4097	186	12	and	and	CCONJ
iajs-4097	186	13	mean	mean	ADJ
iajs-4097	186	14	square	square	ADJ
iajs-4097	186	15	error	error	NOUN
iajs-4097	186	16	values	value	NOUN
iajs-4097	186	17	are	be	AUX
iajs-4097	186	18	presented	present	VERB
iajs-4097	186	19	.	.	PUNCT
iajs-4097	187	1	the	the	DET
iajs-4097	187	2	best	good	ADJ
iajs-4097	187	3	approximation	approximation	NOUN
iajs-4097	187	4	method	method	NOUN
iajs-4097	187	5	was	be	AUX
iajs-4097	187	6	the	the	DET
iajs-4097	187	7	bernoulli	bernoulli	NOUN
iajs-4097	187	8	polynomial	polynomial	NOUN
iajs-4097	187	9	,	,	PUNCT
iajs-4097	187	10	where	where	SCONJ
iajs-4097	187	11	the	the	DET
iajs-4097	187	12	value	value	NOUN
iajs-4097	187	13	of	of	ADP
iajs-4097	187	14	the	the	DET
iajs-4097	187	15	error	error	NOUN
iajs-4097	187	16	norm	norm	NOUN
iajs-4097	187	17	was	be	AUX
iajs-4097	187	18	equal	equal	ADJ
iajs-4097	187	19	to	to	ADP
iajs-4097	187	20	and	and	CCONJ
iajs-4097	187	21	the	the	DET
iajs-4097	187	22	value	value	NOUN
iajs-4097	187	23	of	of	ADP
iajs-4097	187	24	the	the	DET
iajs-4097	187	25	mean	mean	ADJ
iajs-4097	187	26	square	square	ADJ
iajs-4097	187	27	error	error	NOUN
iajs-4097	187	28	was	be	AUX
iajs-4097	187	29	equal	equal	ADJ
iajs-4097	187	30	to	to	ADP
iajs-4097	187	31	when	when	SCONJ
iajs-4097	187	32	.	.	PUNCT
iajs-4097	188	1	ihjpas	ihjpas	PROPN
iajs-4097	188	2	.	.	PUNCT
iajs-4097	189	1	2025,38(4	2025,38(4	NOUN
iajs-4097	189	2	)	)	PUNCT
iajs-4097	189	3	394	394	NUM
iajs-4097	189	4	norm	norm	NOUN
iajs-4097	189	5	.	.	PUNCT
iajs-4097	190	1	norm	norm	NOUN
iajs-4097	190	2	.	.	PUNCT
iajs-4097	191	1	figure	figure	VERB
iajs-4097	191	2	16	16	NUM
iajs-4097	191	3	.	.	PUNCT
iajs-4097	192	1	the	the	DET
iajs-4097	192	2	plots	plot	NOUN
iajs-4097	192	3	logarithmic	logarithmic	ADJ
iajs-4097	192	4	of	of	ADP
iajs-4097	192	5	and	and	CCONJ
iajs-4097	192	6	,	,	PUNCT
iajs-4097	192	7	for	for	ADP
iajs-4097	192	8	example	example	NOUN
iajs-4097	192	9	6	6	NUM
iajs-4097	192	10	,	,	PUNCT
iajs-4097	192	11	when	when	SCONJ
iajs-4097	192	12	.	.	PUNCT
iajs-4097	193	1	figure	figure	NOUN
iajs-4097	193	2	17	17	NUM
iajs-4097	193	3	.	.	PUNCT
iajs-4097	194	1	plots	plot	VERB
iajs-4097	194	2	the	the	DET
iajs-4097	194	3	norm	norm	NOUN
iajs-4097	194	4	for	for	ADP
iajs-4097	194	5	example	example	NOUN
iajs-4097	194	6	6	6	NUM
iajs-4097	194	7	,	,	PUNCT
iajs-4097	194	8	when	when	SCONJ
iajs-4097	194	9	.	.	PUNCT
iajs-4097	195	1	exact	exact	ADJ
iajs-4097	195	2	solution	solution	NOUN
iajs-4097	195	3	approximate	approximate	ADJ
iajs-4097	195	4	solution	solution	NOUN
iajs-4097	195	5	figure	figure	NOUN
iajs-4097	195	6	18	18	NUM
iajs-4097	195	7	.	.	PUNCT
iajs-4097	196	1	comparison	comparison	NOUN
iajs-4097	196	2	between	between	ADP
iajs-4097	196	3	analytical	analytical	ADJ
iajs-4097	196	4	and	and	CCONJ
iajs-4097	196	5	numerical	numerical	ADJ
iajs-4097	196	6	solution	solution	NOUN
iajs-4097	196	7	for	for	ADP
iajs-4097	196	8	example	example	NOUN
iajs-4097	196	9	6	6	NUM
iajs-4097	196	10	by	by	ADP
iajs-4097	196	11	the	the	DET
iajs-4097	196	12	bernoulli	bernoulli	PROPN
iajs-4097	196	13	poly	poly	NOUN
iajs-4097	196	14	.	.	PUNCT
iajs-4097	197	1	5	5	X
iajs-4097	197	2	.	.	X
iajs-4097	197	3	conclusion	conclusion	NOUN
iajs-4097	197	4	in	in	ADP
iajs-4097	197	5	this	this	DET
iajs-4097	197	6	work	work	NOUN
iajs-4097	197	7	,	,	PUNCT
iajs-4097	197	8	the	the	DET
iajs-4097	197	9	orthogonal	orthogonal	ADJ
iajs-4097	197	10	polynomials	polynomial	NOUN
iajs-4097	197	11	and	and	CCONJ
iajs-4097	197	12	operational	operational	ADJ
iajs-4097	197	13	matrices	matrix	NOUN
iajs-4097	197	14	based	base	VERB
iajs-4097	197	15	on	on	ADP
iajs-4097	197	16	standard	standard	ADJ
iajs-4097	197	17	polynomials	polynomial	NOUN
iajs-4097	197	18	were	be	AUX
iajs-4097	197	19	proposed	propose	VERB
iajs-4097	197	20	and	and	CCONJ
iajs-4097	197	21	applied	apply	VERB
iajs-4097	197	22	to	to	PART
iajs-4097	197	23	resolve	resolve	VERB
iajs-4097	197	24	non	non	ADJ
iajs-4097	197	25	-	-	ADJ
iajs-4097	197	26	homogeneous	homogeneous	ADJ
iajs-4097	197	27	wave	wave	NOUN
iajs-4097	197	28	problems	problem	NOUN
iajs-4097	197	29	.	.	PUNCT
iajs-4097	198	1	the	the	DET
iajs-4097	198	2	approximate	approximate	ADJ
iajs-4097	198	3	solutions	solution	NOUN
iajs-4097	198	4	were	be	AUX
iajs-4097	198	5	achieved	achieve	VERB
iajs-4097	198	6	and	and	CCONJ
iajs-4097	198	7	showed	show	VERB
iajs-4097	198	8	to	to	PART
iajs-4097	198	9	be	be	AUX
iajs-4097	198	10	reliable	reliable	ADJ
iajs-4097	198	11	and	and	CCONJ
iajs-4097	198	12	effective	effective	ADJ
iajs-4097	198	13	,	,	PUNCT
iajs-4097	198	14	even	even	ADV
iajs-4097	198	15	for	for	ADP
iajs-4097	198	16	loworder	loword	ADJ
iajs-4097	198	17	polynomials	polynomial	NOUN
iajs-4097	198	18	.	.	PUNCT
iajs-4097	199	1	additionally	additionally	ADV
iajs-4097	199	2	,	,	PUNCT
iajs-4097	199	3	the	the	DET
iajs-4097	199	4	mean	mean	ADJ
iajs-4097	199	5	square	square	ADJ
iajs-4097	199	6	error	error	NOUN
iajs-4097	199	7	and	and	CCONJ
iajs-4097	199	8	the	the	DET
iajs-4097	199	9	norm	norm	NOUN
iajs-4097	199	10	were	be	AUX
iajs-4097	199	11	calculated	calculate	VERB
iajs-4097	199	12	to	to	PART
iajs-4097	199	13	assess	assess	VERB
iajs-4097	199	14	the	the	DET
iajs-4097	199	15	accuracy	accuracy	NOUN
iajs-4097	199	16	,	,	PUNCT
iajs-4097	199	17	dependability	dependability	NOUN
iajs-4097	199	18	,	,	PUNCT
iajs-4097	199	19	and	and	CCONJ
iajs-4097	199	20	validity	validity	NOUN
iajs-4097	199	21	of	of	ADP
iajs-4097	199	22	the	the	DET
iajs-4097	199	23	approaches	approach	NOUN
iajs-4097	199	24	.	.	PUNCT
iajs-4097	200	1	the	the	DET
iajs-4097	200	2	results	result	NOUN
iajs-4097	200	3	ihjpas	ihjpa	VERB
iajs-4097	200	4	.	.	PUNCT
iajs-4097	201	1	2025,38(4	2025,38(4	NOUN
iajs-4097	201	2	)	)	PUNCT
iajs-4097	201	3	395	395	NUM
iajs-4097	201	4	show	show	VERB
iajs-4097	201	5	that	that	SCONJ
iajs-4097	201	6	the	the	DET
iajs-4097	201	7	recommended	recommend	VERB
iajs-4097	201	8	solutions	solution	NOUN
iajs-4097	201	9	have	have	VERB
iajs-4097	201	10	high	high	ADJ
iajs-4097	201	11	accuracy	accuracy	NOUN
iajs-4097	201	12	and	and	CCONJ
iajs-4097	201	13	lower	low	ADJ
iajs-4097	201	14	error	error	NOUN
iajs-4097	201	15	rates	rate	NOUN
iajs-4097	201	16	.	.	PUNCT
iajs-4097	202	1	in	in	ADP
iajs-4097	202	2	all	all	DET
iajs-4097	202	3	situations	situation	NOUN
iajs-4097	202	4	,	,	PUNCT
iajs-4097	202	5	the	the	DET
iajs-4097	202	6	mean	mean	ADJ
iajs-4097	202	7	square	square	ADJ
iajs-4097	202	8	error	error	NOUN
iajs-4097	202	9	and	and	CCONJ
iajs-4097	202	10	error	error	NOUN
iajs-4097	202	11	norm	norm	NOUN
iajs-4097	202	12	values	value	NOUN
iajs-4097	202	13	drop	drop	VERB
iajs-4097	202	14	as	as	ADP
iajs-4097	202	15	grows	grow	NOUN
iajs-4097	202	16	.	.	PUNCT
iajs-4097	203	1	furthermore	furthermore	ADV
iajs-4097	203	2	,	,	PUNCT
iajs-4097	203	3	it	it	PRON
iajs-4097	203	4	is	be	AUX
iajs-4097	203	5	showed	show	VERB
iajs-4097	203	6	that	that	SCONJ
iajs-4097	203	7	the	the	DET
iajs-4097	203	8	results	result	NOUN
iajs-4097	203	9	of	of	ADP
iajs-4097	203	10	the	the	PRON
iajs-4097	203	11	,	,	PUNCT
iajs-4097	203	12	and	and	CCONJ
iajs-4097	203	13	by	by	ADP
iajs-4097	203	14	the	the	DET
iajs-4097	203	15	recommended	recommend	VERB
iajs-4097	203	16	methods	method	NOUN
iajs-4097	203	17	bernoulli	bernoulli	PROPN
iajs-4097	203	18	and	and	CCONJ
iajs-4097	203	19	standard	standard	ADJ
iajs-4097	203	20	polynomials	polynomial	NOUN
iajs-4097	203	21	fell	fall	VERB
iajs-4097	203	22	greatly	greatly	ADV
iajs-4097	203	23	in	in	ADP
iajs-4097	203	24	comparison	comparison	NOUN
iajs-4097	203	25	to	to	ADP
iajs-4097	203	26	the	the	DET
iajs-4097	203	27	other	other	ADJ
iajs-4097	203	28	orthogonal	orthogonal	ADJ
iajs-4097	203	29	polynomials	polynomial	NOUN
iajs-4097	203	30	.	.	PUNCT
iajs-4097	204	1	the	the	DET
iajs-4097	204	2	standard	standard	ADJ
iajs-4097	204	3	polynomial	polynomial	NOUN
iajs-4097	204	4	was	be	AUX
iajs-4097	204	5	the	the	DET
iajs-4097	204	6	best	good	ADJ
iajs-4097	204	7	in	in	ADP
iajs-4097	204	8	the	the	DET
iajs-4097	204	9	first	first	ADJ
iajs-4097	204	10	example	example	NOUN
iajs-4097	204	11	when	when	SCONJ
iajs-4097	204	12	to	to	PART
iajs-4097	204	13	,	,	PUNCT
iajs-4097	204	14	and	and	CCONJ
iajs-4097	204	15	the	the	DET
iajs-4097	204	16	values	value	NOUN
iajs-4097	204	17	of	of	ADP
iajs-4097	204	18	the	the	PRON
iajs-4097	204	19	,	,	PUNCT
iajs-4097	204	20	and	and	CCONJ
iajs-4097	204	21	were	be	AUX
iajs-4097	204	22	equal	equal	ADJ
iajs-4097	204	23	to	to	ADP
iajs-4097	204	24	,	,	PUNCT
iajs-4097	204	25	,	,	PUNCT
iajs-4097	204	26	respectively	respectively	ADV
iajs-4097	204	27	,	,	PUNCT
iajs-4097	204	28	when	when	SCONJ
iajs-4097	204	29	,	,	PUNCT
iajs-4097	204	30	while	while	SCONJ
iajs-4097	204	31	the	the	DET
iajs-4097	204	32	bernoulli	bernoulli	PROPN
iajs-4097	204	33	polynomial	polynomial	NOUN
iajs-4097	204	34	was	be	AUX
iajs-4097	204	35	the	the	DET
iajs-4097	204	36	best	good	ADJ
iajs-4097	204	37	in	in	ADP
iajs-4097	204	38	the	the	DET
iajs-4097	204	39	remaining	remain	VERB
iajs-4097	204	40	examples	example	NOUN
iajs-4097	204	41	as	as	SCONJ
iajs-4097	204	42	it	it	PRON
iajs-4097	204	43	gave	give	VERB
iajs-4097	204	44	the	the	DET
iajs-4097	204	45	lowest	low	ADJ
iajs-4097	204	46	error	error	NOUN
iajs-4097	204	47	when	when	SCONJ
iajs-4097	204	48	to	to	PART
iajs-4097	204	49	.	.	PUNCT
iajs-4097	205	1	acknowledgment	acknowledgment	NOUN
iajs-4097	205	2	the	the	DET
iajs-4097	205	3	authors	author	NOUN
iajs-4097	205	4	are	be	AUX
iajs-4097	205	5	very	very	ADV
iajs-4097	205	6	grateful	grateful	ADJ
iajs-4097	205	7	to	to	ADP
iajs-4097	205	8	the	the	DET
iajs-4097	205	9	referees	referee	NOUN
iajs-4097	205	10	for	for	ADP
iajs-4097	205	11	giving	give	VERB
iajs-4097	205	12	them	they	PRON
iajs-4097	205	13	insightful	insightful	ADJ
iajs-4097	205	14	comments	comment	NOUN
iajs-4097	205	15	and	and	CCONJ
iajs-4097	205	16	ideas	idea	NOUN
iajs-4097	205	17	for	for	ADP
iajs-4097	205	18	enhancing	enhance	VERB
iajs-4097	205	19	the	the	DET
iajs-4097	205	20	paper	paper	NOUN
iajs-4097	205	21	.	.	PUNCT
iajs-4097	206	1	conflict	conflict	NOUN
iajs-4097	206	2	of	of	ADP
iajs-4097	206	3	interest	interest	NOUN
iajs-4097	206	4	the	the	DET
iajs-4097	206	5	authors	author	NOUN
iajs-4097	206	6	declare	declare	VERB
iajs-4097	206	7	that	that	SCONJ
iajs-4097	206	8	they	they	PRON
iajs-4097	206	9	have	have	VERB
iajs-4097	206	10	no	no	DET
iajs-4097	206	11	conflicts	conflict	NOUN
iajs-4097	206	12	of	of	ADP
iajs-4097	206	13	interest	interest	NOUN
iajs-4097	206	14	.	.	PUNCT
iajs-4097	207	1	funding	funding	NOUN
iajs-4097	207	2	there	there	PRON
iajs-4097	207	3	is	be	VERB
iajs-4097	207	4	no	no	DET
iajs-4097	207	5	monetary	monetary	ADJ
iajs-4097	207	6	support	support	NOUN
iajs-4097	207	7	in	in	ADP
iajs-4097	207	8	preparing	prepare	VERB
iajs-4097	207	9	for	for	ADP
iajs-4097	207	10	the	the	DET
iajs-4097	207	11	publication	publication	NOUN
iajs-4097	207	12	.	.	PUNCT
iajs-4097	208	1	references	reference	NOUN
iajs-4097	208	2	1	1	NUM
iajs-4097	208	3	.	.	PUNCT
iajs-4097	208	4	wazwaz	wazwaz	PROPN
iajs-4097	208	5	am	am	PROPN
iajs-4097	208	6	.	.	PUNCT
iajs-4097	209	1	linear	linear	ADJ
iajs-4097	209	2	and	and	CCONJ
iajs-4097	209	3	nonlinear	nonlinear	ADJ
iajs-4097	209	4	integral	integral	ADJ
iajs-4097	209	5	equations	equation	NOUN
iajs-4097	209	6	.	.	PUNCT
iajs-4097	210	1	berlin	berlin	ADJ
iajs-4097	210	2	:	:	PUNCT
iajs-4097	210	3	springer	springer	NOUN
iajs-4097	210	4	;	;	PUNCT
iajs-4097	210	5	2011	2011	NUM
iajs-4097	210	6	.	.	PUNCT
iajs-4097	211	1	2	2	X
iajs-4097	211	2	.	.	X
iajs-4097	212	1	he	he	PRON
iajs-4097	212	2	jh	jh	PROPN
iajs-4097	212	3	.	.	PUNCT
iajs-4097	213	1	variational	variational	ADJ
iajs-4097	213	2	iteration	iteration	NOUN
iajs-4097	213	3	method	method	NOUN
iajs-4097	213	4	for	for	ADP
iajs-4097	213	5	delay	delay	NOUN
iajs-4097	213	6	differential	differential	ADJ
iajs-4097	213	7	equations	equation	NOUN
iajs-4097	213	8	.	.	PUNCT
iajs-4097	214	1	commun	commun	PROPN
iajs-4097	214	2	nonlinear	nonlinear	PROPN
iajs-4097	214	3	sci	sci	PROPN
iajs-4097	214	4	numer	numer	PROPN
iajs-4097	214	5	simul	simul	PROPN
iajs-4097	214	6	.	.	PUNCT
iajs-4097	215	1	1997;2(4):235–236	1997;2(4):235–236	NUM
iajs-4097	215	2	.	.	X
iajs-4097	215	3	https://doi.org/10.1016/s1007-5704(97)90008-3	https://doi.org/10.1016/s1007-5704(97)90008-3	PROPN
iajs-4097	215	4	3	3	NUM
iajs-4097	215	5	.	.	PUNCT
iajs-4097	216	1	he	he	PRON
iajs-4097	216	2	jh	jh	PROPN
iajs-4097	216	3	.	.	PUNCT
iajs-4097	217	1	variational	variational	ADJ
iajs-4097	217	2	iteration	iteration	NOUN
iajs-4097	217	3	method	method	NOUN
iajs-4097	217	4	for	for	ADP
iajs-4097	217	5	autonomous	autonomous	ADJ
iajs-4097	217	6	ordinary	ordinary	ADJ
iajs-4097	217	7	differential	differential	NOUN
iajs-4097	217	8	systems	system	NOUN
iajs-4097	217	9	.	.	PUNCT
iajs-4097	218	1	appl	appl	PROPN
iajs-4097	218	2	math	math	PROPN
iajs-4097	218	3	comput	comput	NOUN
iajs-4097	218	4	.	.	PUNCT
iajs-4097	219	1	2000;114(2–3):115–123	2000;114(2–3):115–123	NUM
iajs-4097	219	2	.	.	PUNCT
iajs-4097	220	1	https://doi.org/10.1016/s0096-3003(99)00104-6	https://doi.org/10.1016/s0096-3003(99)00104-6	NOUN
iajs-4097	220	2	4	4	NUM
iajs-4097	220	3	.	.	PUNCT
iajs-4097	221	1	he	he	PRON
iajs-4097	221	2	jh	jh	PROPN
iajs-4097	221	3	.	.	PUNCT
iajs-4097	222	1	variational	variational	ADJ
iajs-4097	222	2	iteration	iteration	NOUN
iajs-4097	222	3	method	method	NOUN
iajs-4097	222	4	-	-	PUNCT
iajs-4097	222	5	some	some	DET
iajs-4097	222	6	recent	recent	ADJ
iajs-4097	222	7	results	result	NOUN
iajs-4097	222	8	and	and	CCONJ
iajs-4097	222	9	new	new	ADJ
iajs-4097	222	10	interpretations	interpretation	NOUN
iajs-4097	222	11	.	.	PUNCT
iajs-4097	223	1	j	j	PROPN
iajs-4097	223	2	comput	comput	PROPN
iajs-4097	223	3	appl	appl	PROPN
iajs-4097	223	4	math	math	PROPN
iajs-4097	223	5	.	.	PUNCT
iajs-4097	224	1	2007;207(1):3–17	2007;207(1):3–17	NOUN
iajs-4097	224	2	.	.	PUNCT
iajs-4097	225	1	https://doi.org/10.1016/j.cam.2006.07.009	https://doi.org/10.1016/j.cam.2006.07.009	PROPN
iajs-4097	225	2	5	5	NUM
iajs-4097	225	3	.	.	PUNCT
iajs-4097	225	4	wazwaz	wazwaz	PROPN
iajs-4097	225	5	am	am	PROPN
iajs-4097	225	6	.	.	PUNCT
iajs-4097	226	1	the	the	DET
iajs-4097	226	2	variational	variational	ADJ
iajs-4097	226	3	iteration	iteration	NOUN
iajs-4097	226	4	method	method	NOUN
iajs-4097	226	5	:	:	PUNCT
iajs-4097	226	6	a	a	DET
iajs-4097	226	7	powerful	powerful	ADJ
iajs-4097	226	8	scheme	scheme	NOUN
iajs-4097	226	9	for	for	ADP
iajs-4097	226	10	handling	handle	VERB
iajs-4097	226	11	linear	linear	ADJ
iajs-4097	226	12	and	and	CCONJ
iajs-4097	226	13	nonlinear	nonlinear	ADJ
iajs-4097	226	14	diffusion	diffusion	NOUN
iajs-4097	226	15	equations	equation	NOUN
iajs-4097	226	16	.	.	PUNCT
iajs-4097	227	1	comput	comput	PROPN
iajs-4097	227	2	math	math	NOUN
iajs-4097	227	3	with	with	ADP
iajs-4097	227	4	appl	appl	NOUN
iajs-4097	227	5	.	.	PUNCT
iajs-4097	228	1	2007;54(7–8):933–939	2007;54(7–8):933–939	NUM
iajs-4097	228	2	.	.	PUNCT
iajs-4097	229	1	https://doi.org/10.1016/j.camwa.2006.12.038	https://doi.org/10.1016/j.camwa.2006.12.038	NOUN
iajs-4097	229	2	6	6	NUM
iajs-4097	229	3	.	.	PUNCT
iajs-4097	230	1	wazwaz	wazwaz	PROPN
iajs-4097	230	2	am	am	PROPN
iajs-4097	230	3	.	.	PUNCT
iajs-4097	231	1	the	the	DET
iajs-4097	231	2	variational	variational	ADJ
iajs-4097	231	3	iteration	iteration	NOUN
iajs-4097	231	4	method	method	NOUN
iajs-4097	231	5	:	:	PUNCT
iajs-4097	231	6	a	a	DET
iajs-4097	231	7	reliable	reliable	ADJ
iajs-4097	231	8	analytic	analytic	ADJ
iajs-4097	231	9	tool	tool	NOUN
iajs-4097	231	10	for	for	ADP
iajs-4097	231	11	solving	solve	VERB
iajs-4097	231	12	linear	linear	NOUN
iajs-4097	231	13	and	and	CCONJ
iajs-4097	231	14	nonlinear	nonlinear	ADJ
iajs-4097	231	15	wave	wave	NOUN
iajs-4097	231	16	equations	equation	NOUN
iajs-4097	231	17	.	.	PUNCT
iajs-4097	232	1	comput	comput	PROPN
iajs-4097	232	2	math	math	NOUN
iajs-4097	232	3	with	with	ADP
iajs-4097	232	4	appl	appl	NOUN
iajs-4097	232	5	.	.	PUNCT
iajs-4097	233	1	2007;54(7–8):926–932	2007;54(7–8):926–932	NUM
iajs-4097	233	2	.	.	PUNCT
iajs-4097	233	3	https://doi.org/10.1016/j.camwa.2006.12.038	https://doi.org/10.1016/j.camwa.2006.12.038	PROPN
iajs-4097	233	4	7	7	NUM
iajs-4097	233	5	.	.	PUNCT
iajs-4097	234	1	wazwaz	wazwaz	PROPN
iajs-4097	234	2	am	am	PROPN
iajs-4097	234	3	.	.	PUNCT
iajs-4097	235	1	partial	partial	ADJ
iajs-4097	235	2	differential	differential	ADJ
iajs-4097	235	3	equations	equation	NOUN
iajs-4097	235	4	and	and	CCONJ
iajs-4097	235	5	solitary	solitary	ADJ
iajs-4097	235	6	waves	wave	NOUN
iajs-4097	235	7	theory	theory	NOUN
iajs-4097	235	8	.	.	PUNCT
iajs-4097	236	1	springer	springer	NOUN
iajs-4097	236	2	science	science	PROPN
iajs-4097	236	3	&	&	CCONJ
iajs-4097	236	4	business	business	NOUN
iajs-4097	236	5	media	medium	NOUN
iajs-4097	236	6	;	;	PUNCT
iajs-4097	236	7	2010	2010	NUM
iajs-4097	236	8	.	.	PUNCT
iajs-4097	237	1	8	8	X
iajs-4097	237	2	.	.	PUNCT
iajs-4097	238	1	mansour	mansour	PROPN
iajs-4097	238	2	mb	mb	PROPN
iajs-4097	238	3	.	.	PROPN
iajs-4097	238	4	accurate	accurate	ADJ
iajs-4097	238	5	computation	computation	NOUN
iajs-4097	238	6	of	of	ADP
iajs-4097	238	7	traveling	travel	VERB
iajs-4097	238	8	wave	wave	NOUN
iajs-4097	238	9	solutions	solution	NOUN
iajs-4097	238	10	of	of	ADP
iajs-4097	238	11	some	some	DET
iajs-4097	238	12	nonlinear	nonlinear	ADJ
iajs-4097	238	13	diffusion	diffusion	NOUN
iajs-4097	238	14	equations	equation	NOUN
iajs-4097	238	15	.	.	PUNCT
iajs-4097	239	1	wave	wave	NOUN
iajs-4097	239	2	motion	motion	NOUN
iajs-4097	239	3	.	.	PUNCT
iajs-4097	240	1	2007;44(3	2007;44(3	NUM
iajs-4097	240	2	):	):	PUNCT
iajs-4097	240	3	222	222	NUM
iajs-4097	240	4	-	-	SYM
iajs-4097	240	5	230	230	NUM
iajs-4097	240	6	.	.	PUNCT
iajs-4097	241	1	doi:10.1016	doi:10.1016	PROPN
iajs-4097	241	2	/	/	SYM
iajs-4097	241	3	j.wavemoti.2006.10.001	j.wavemoti.2006.10.001	PROPN
iajs-4097	241	4	9	9	NUM
iajs-4097	241	5	.	.	X
iajs-4097	241	6	biazar	biazar	PROPN
iajs-4097	241	7	j	j	PROPN
iajs-4097	241	8	,	,	PUNCT
iajs-4097	241	9	ghazvini	ghazvini	PROPN
iajs-4097	241	10	h.	h.	PROPN
iajs-4097	242	1	an	an	DET
iajs-4097	242	2	analytical	analytical	ADJ
iajs-4097	242	3	approximation	approximation	NOUN
iajs-4097	242	4	to	to	ADP
iajs-4097	242	5	the	the	DET
iajs-4097	242	6	solution	solution	NOUN
iajs-4097	242	7	of	of	ADP
iajs-4097	242	8	a	a	DET
iajs-4097	242	9	wave	wave	NOUN
iajs-4097	242	10	equation	equation	NOUN
iajs-4097	242	11	by	by	ADP
iajs-4097	242	12	a	a	DET
iajs-4097	242	13	variational	variational	ADJ
iajs-4097	242	14	iteration	iteration	NOUN
iajs-4097	242	15	method	method	NOUN
iajs-4097	242	16	.	.	PUNCT
iajs-4097	243	1	appl	appl	PROPN
iajs-4097	243	2	math	math	PROPN
iajs-4097	243	3	lett	lett	PROPN
iajs-4097	243	4	.	.	PUNCT
iajs-4097	244	1	2008;21(8):780–5	2008;21(8):780–5	NUM
iajs-4097	244	2	.	.	PUNCT
iajs-4097	245	1	https://doi.org/10.1016/j.aml.2007.08.004	https://doi.org/10.1016/j.aml.2007.08.004	PROPN
iajs-4097	245	2	10	10	NUM
iajs-4097	245	3	.	.	PUNCT
iajs-4097	246	1	britt	britt	PROPN
iajs-4097	246	2	s	s	PROPN
iajs-4097	246	3	,	,	PUNCT
iajs-4097	246	4	tsynkov	tsynkov	PROPN
iajs-4097	246	5	s	s	PART
iajs-4097	246	6	,	,	PUNCT
iajs-4097	246	7	turkel	turkel	PROPN
iajs-4097	246	8	e.	e.	PROPN
iajs-4097	246	9	numerical	numerical	PROPN
iajs-4097	246	10	solution	solution	NOUN
iajs-4097	246	11	of	of	ADP
iajs-4097	246	12	the	the	DET
iajs-4097	246	13	wave	wave	NOUN
iajs-4097	246	14	equation	equation	NOUN
iajs-4097	246	15	with	with	ADP
iajs-4097	246	16	variable	variable	ADJ
iajs-4097	246	17	wave	wave	NOUN
iajs-4097	246	18	speed	speed	NOUN
iajs-4097	246	19	on	on	ADP
iajs-4097	246	20	nonconforming	nonconforming	NOUN
iajs-4097	246	21	domains	domain	NOUN
iajs-4097	246	22	by	by	ADP
iajs-4097	246	23	high	high	ADJ
iajs-4097	246	24	-	-	PUNCT
iajs-4097	246	25	order	order	NOUN
iajs-4097	246	26	difference	difference	NOUN
iajs-4097	246	27	potentials	potential	NOUN
iajs-4097	246	28	.	.	PUNCT
iajs-4097	247	1	j	j	PROPN
iajs-4097	247	2	comput	comput	PROPN
iajs-4097	247	3	phys	phy	NOUN
iajs-4097	247	4	.	.	PUNCT
iajs-4097	248	1	2018;354:26–42	2018;354:26–42	NUM
iajs-4097	248	2	.	.	PUNCT
iajs-4097	248	3	https://doi.org/10.1016/j.jcp.2017.10.049	https://doi.org/10.1016/j.jcp.2017.10.049	NOUN
iajs-4097	248	4	11	11	NUM
iajs-4097	248	5	.	.	PUNCT
iajs-4097	249	1	cheniguel	cheniguel	PROPN
iajs-4097	249	2	a.	a.	PROPN
iajs-4097	249	3	numerical	numerical	PROPN
iajs-4097	249	4	method	method	PROPN
iajs-4097	249	5	for	for	ADP
iajs-4097	249	6	solving	solve	VERB
iajs-4097	249	7	wave	wave	NOUN
iajs-4097	249	8	equation	equation	NOUN
iajs-4097	249	9	with	with	ADP
iajs-4097	249	10	non	non	PRON
iajs-4097	249	11	local	local	ADJ
iajs-4097	249	12	boundary	boundary	ADJ
iajs-4097	249	13	conditions	condition	NOUN
iajs-4097	249	14	.	.	PUNCT
iajs-4097	250	1	proc	proc	NOUN
iajs-4097	250	2	int	int	PROPN
iajs-4097	250	3	multiconference	multiconference	ADV
iajs-4097	250	4	eng	eng	PROPN
iajs-4097	250	5	comput	comput	PROPN
iajs-4097	250	6	sci	sci	PROPN
iajs-4097	250	7	.	.	PUNCT
iajs-4097	250	8	2013;2:13–5	2013;2:13–5	PROPN
iajs-4097	250	9	.	.	PUNCT
iajs-4097	251	1	12	12	NUM
iajs-4097	251	2	.	.	PUNCT
iajs-4097	252	1	ullah	ullah	PROPN
iajs-4097	252	2	,	,	PUNCT
iajs-4097	252	3	h.	h.	PROPN
iajs-4097	252	4	,	,	PUNCT
iajs-4097	252	5	islam	islam	PROPN
iajs-4097	252	6	,	,	PUNCT
iajs-4097	252	7	s.	s.	PROPN
iajs-4097	252	8	,	,	PUNCT
iajs-4097	252	9	dennis	dennis	PROPN
iajs-4097	252	10	,	,	PUNCT
iajs-4097	252	11	l.c.c	l.c.c	PROPN
iajs-4097	252	12	.	.	PUNCT
iajs-4097	252	13	,	,	PUNCT
iajs-4097	252	14	abdelhameed	abdelhameed	PROPN
iajs-4097	252	15	,	,	PUNCT
iajs-4097	252	16	t.n	t.n	PROPN
iajs-4097	252	17	.	.	PROPN
iajs-4097	252	18	,	,	PUNCT
iajs-4097	252	19	khan	khan	PROPN
iajs-4097	252	20	,	,	PUNCT
iajs-4097	252	21	i.	i.	PROPN
iajs-4097	252	22	and	and	CCONJ
iajs-4097	252	23	fiza	fiza	ADJ
iajs-4097	252	24	m.	m.	NOUN
iajs-4097	252	25	approximate	approximate	ADJ
iajs-4097	252	26	solution	solution	NOUN
iajs-4097	252	27	of	of	ADP
iajs-4097	252	28	two	two	NUM
iajs-4097	252	29	-	-	PUNCT
iajs-4097	252	30	dimensional	dimensional	ADJ
iajs-4097	252	31	nonlinear	nonlinear	ADJ
iajs-4097	252	32	wave	wave	NOUN
iajs-4097	252	33	equation	equation	NOUN
iajs-4097	252	34	by	by	ADP
iajs-4097	252	35	optimal	optimal	ADJ
iajs-4097	252	36	homotopy	homotopy	NOUN
iajs-4097	252	37	asymptotic	asymptotic	ADJ
iajs-4097	252	38	method	method	NOUN
iajs-4097	252	39	.	.	PUNCT
iajs-4097	253	1	math	math	PROPN
iajs-4097	253	2	probl	probl	PROPN
iajs-4097	253	3	eng	eng	PROPN
iajs-4097	253	4	.	.	PROPN
iajs-4097	253	5	2015;2015(1):380104	2015;2015(1):380104	PROPN
iajs-4097	253	6	.	.	PUNCT
iajs-4097	254	1	https://doi.org/10.1155/2015/380104	https://doi.org/10.1155/2015/380104	NOUN
iajs-4097	254	2	13	13	NUM
iajs-4097	254	3	.	.	PUNCT
iajs-4097	255	1	abolarin	abolarin	PROPN
iajs-4097	255	2	oe	oe	PROPN
iajs-4097	255	3	.	.	PUNCT
iajs-4097	256	1	new	new	ADJ
iajs-4097	256	2	improved	improve	VERB
iajs-4097	256	3	variational	variational	ADJ
iajs-4097	256	4	homotopy	homotopy	NOUN
iajs-4097	256	5	perturbation	perturbation	NOUN
iajs-4097	256	6	method	method	NOUN
iajs-4097	256	7	for	for	ADP
iajs-4097	256	8	bratu	bratu	NOUN
iajs-4097	256	9	-	-	PUNCT
iajs-4097	256	10	type	type	NOUN
iajs-4097	256	11	problems	problem	NOUN
iajs-4097	256	12	.	.	PUNCT
iajs-4097	257	1	am	be	AUX
iajs-4097	257	2	j	j	PROPN
iajs-4097	257	3	comput	comput	PROPN
iajs-4097	257	4	math	math	NOUN
iajs-4097	257	5	.	.	PUNCT
iajs-4097	258	1	2013;03(02):110–113	2013;03(02):110–113	NUM
iajs-4097	258	2	.	.	PUNCT
iajs-4097	259	1	https://doi.org/10.4236/ajcm.2013.32018	https://doi.org/10.4236/ajcm.2013.32018	NOUN
iajs-4097	259	2	14	14	NUM
iajs-4097	259	3	.	.	PUNCT
iajs-4097	260	1	kumar	kumar	PROPN
iajs-4097	260	2	s	s	PROPN
iajs-4097	260	3	,	,	PUNCT
iajs-4097	260	4	kumar	kumar	PROPN
iajs-4097	260	5	a	a	PROPN
iajs-4097	260	6	,	,	PUNCT
iajs-4097	260	7	momani	momani	X
iajs-4097	260	8	s	s	PROPN
iajs-4097	260	9	,	,	PUNCT
iajs-4097	260	10	aldhaifallah	aldhaifallah	PROPN
iajs-4097	260	11	m	m	PROPN
iajs-4097	260	12	,	,	PUNCT
iajs-4097	260	13	nisar	nisar	PROPN
iajs-4097	260	14	ks	ks	PROPN
iajs-4097	260	15	.	.	PROPN
iajs-4097	260	16	numerical	numerical	PROPN
iajs-4097	260	17	solutions	solution	NOUN
iajs-4097	260	18	of	of	ADP
iajs-4097	260	19	nonlinear	nonlinear	ADJ
iajs-4097	260	20	https://doi.org/10.1016/s1007-5704(97)90008-3	https://doi.org/10.1016/s1007-5704(97)90008-3	PROPN
iajs-4097	260	21	https://doi.org/10.1016/s0096-3003(99)00104-6	https://doi.org/10.1016/s0096-3003(99)00104-6	NOUN
iajs-4097	260	22	https://doi.org/10.1016/j.cam.2006.07.009	https://doi.org/10.1016/j.cam.2006.07.009	PROPN
iajs-4097	260	23	https://doi.org/10.1016/j.camwa.2006.12.038	https://doi.org/10.1016/j.camwa.2006.12.038	PROPN
iajs-4097	260	24	https://doi.org/10.1016/j.camwa.2006.12.038	https://doi.org/10.1016/j.camwa.2006.12.038	PROPN
iajs-4097	260	25	mailto	mailto	PROPN
iajs-4097	260	26	:	:	PUNCT
iajs-4097	260	27	doi:10.1016	doi:10.1016	PROPN
iajs-4097	260	28	/	/	SYM
iajs-4097	260	29	j.wavemoti.2006.10.001	j.wavemoti.2006.10.001	PROPN
iajs-4097	260	30	mailto	mailto	PROPN
iajs-4097	260	31	:	:	PUNCT
iajs-4097	260	32	doi:10.1016	doi:10.1016	PROPN
iajs-4097	260	33	/	/	SYM
iajs-4097	260	34	j.wavemoti.2006.10.001	j.wavemoti.2006.10.001	PROPN
iajs-4097	260	35	https://doi.org/10.1016/j.aml.2007.08.004	https://doi.org/10.1016/j.aml.2007.08.004	PROPN
iajs-4097	260	36	https://doi.org/10.1016/j.jcp.2017.10.049	https://doi.org/10.1016/j.jcp.2017.10.049	NOUN
iajs-4097	260	37	https://doi.org/10.1155/2015/380104	https://doi.org/10.1155/2015/380104	PROPN
iajs-4097	260	38	https://doi.org/10.4236/ajcm.2013.32018	https://doi.org/10.4236/ajcm.2013.32018	PROPN
iajs-4097	260	39	ihjpas	ihjpa	VERB
iajs-4097	260	40	.	.	PUNCT
iajs-4097	261	1	2025,38(4	2025,38(4	NUM
iajs-4097	261	2	)	)	PUNCT
iajs-4097	261	3	396	396	NUM
iajs-4097	261	4	fractional	fractional	ADJ
iajs-4097	261	5	model	model	NOUN
iajs-4097	261	6	arising	arise	VERB
iajs-4097	261	7	in	in	ADP
iajs-4097	261	8	the	the	DET
iajs-4097	261	9	appearance	appearance	NOUN
iajs-4097	261	10	of	of	ADP
iajs-4097	261	11	the	the	DET
iajs-4097	261	12	stripe	stripe	NOUN
iajs-4097	261	13	patterns	pattern	NOUN
iajs-4097	261	14	in	in	ADP
iajs-4097	261	15	two	two	NUM
iajs-4097	261	16	-	-	PUNCT
iajs-4097	261	17	dimensional	dimensional	ADJ
iajs-4097	261	18	systems	system	NOUN
iajs-4097	261	19	.	.	PUNCT
iajs-4097	262	1	adv	adv	PROPN
iajs-4097	262	2	differ	differ	VERB
iajs-4097	262	3	equations	equation	NOUN
iajs-4097	262	4	.	.	PUNCT
iajs-4097	263	1	2019;2019(1):1	2019;2019(1):1	NUM
iajs-4097	263	2	-	-	SYM
iajs-4097	263	3	19	19	NUM
iajs-4097	263	4	.	.	PUNCT
iajs-4097	263	5	https://doi.org/10.1186/s13662-019-2334-7	https://doi.org/10.1186/s13662-019-2334-7	NUM
iajs-4097	263	6	15	15	NUM
iajs-4097	263	7	.	.	PUNCT
iajs-4097	264	1	kumar	kumar	PROPN
iajs-4097	264	2	s	s	PROPN
iajs-4097	264	3	,	,	PUNCT
iajs-4097	264	4	kumar	kumar	PROPN
iajs-4097	264	5	a	a	PROPN
iajs-4097	264	6	,	,	PUNCT
iajs-4097	264	7	abbas	abbas	PROPN
iajs-4097	264	8	s	s	PART
iajs-4097	264	9	,	,	PUNCT
iajs-4097	264	10	al	al	PROPN
iajs-4097	264	11	qurashi	qurashi	PROPN
iajs-4097	264	12	m	m	PROPN
iajs-4097	264	13	,	,	PUNCT
iajs-4097	264	14	baleanu	baleanu	PROPN
iajs-4097	264	15	d.	d.	PROPN
iajs-4097	264	16	a	a	DET
iajs-4097	264	17	modified	modify	VERB
iajs-4097	264	18	analytical	analytical	ADJ
iajs-4097	264	19	approach	approach	NOUN
iajs-4097	264	20	with	with	ADP
iajs-4097	264	21	existence	existence	NOUN
iajs-4097	264	22	and	and	CCONJ
iajs-4097	264	23	uniqueness	uniqueness	NOUN
iajs-4097	264	24	for	for	ADP
iajs-4097	264	25	fractional	fractional	ADJ
iajs-4097	264	26	cauchy	cauchy	ADJ
iajs-4097	264	27	reaction	reaction	NOUN
iajs-4097	264	28	–	–	PUNCT
iajs-4097	264	29	diffusion	diffusion	NOUN
iajs-4097	264	30	equations	equation	NOUN
iajs-4097	264	31	.	.	PUNCT
iajs-4097	265	1	adv	adv	PROPN
iajs-4097	265	2	differ	differ	VERB
iajs-4097	265	3	equations	equation	NOUN
iajs-4097	265	4	.	.	PUNCT
iajs-4097	266	1	2020;2020(1):28	2020;2020(1):28	NUM
iajs-4097	266	2	.	.	PUNCT
iajs-4097	267	1	https://doi.org/10.1186/s13662-019-2488-3	https://doi.org/10.1186/s13662-019-2488-3	NUM
iajs-4097	267	2	16	16	NUM
iajs-4097	267	3	.	.	PUNCT
iajs-4097	268	1	corrington	corrington	PROPN
iajs-4097	268	2	ms	ms	PROPN
iajs-4097	268	3	.	.	PROPN
iajs-4097	268	4	solution	solution	NOUN
iajs-4097	268	5	of	of	ADP
iajs-4097	268	6	differential	differential	ADJ
iajs-4097	268	7	and	and	CCONJ
iajs-4097	268	8	integral	integral	ADJ
iajs-4097	268	9	equations	equation	NOUN
iajs-4097	268	10	with	with	ADP
iajs-4097	268	11	walsh	walsh	NOUN
iajs-4097	268	12	functions	function	NOUN
iajs-4097	268	13	.	.	PUNCT
iajs-4097	269	1	ieee	ieee	PROPN
iajs-4097	269	2	trans	trans	PROPN
iajs-4097	269	3	circuit	circuit	PROPN
iajs-4097	269	4	theory	theory	NOUN
iajs-4097	269	5	.	.	PUNCT
iajs-4097	270	1	1973;20(5):470–476	1973;20(5):470–476	NUM
iajs-4097	270	2	.	.	NOUN
iajs-4097	270	3	17	17	NUM
iajs-4097	270	4	.	.	PUNCT
iajs-4097	270	5	sparis	sparis	PROPN
iajs-4097	270	6	pd	pd	PROPN
iajs-4097	270	7	,	,	PUNCT
iajs-4097	270	8	mouroutsos	mouroutsos	NOUN
iajs-4097	270	9	sg	sg	PROPN
iajs-4097	270	10	.	.	PUNCT
iajs-4097	271	1	the	the	DET
iajs-4097	271	2	operational	operational	ADJ
iajs-4097	271	3	matrix	matrix	NOUN
iajs-4097	271	4	of	of	ADP
iajs-4097	271	5	differentiation	differentiation	NOUN
iajs-4097	271	6	for	for	ADP
iajs-4097	271	7	orthogonal	orthogonal	ADJ
iajs-4097	271	8	polynomial	polynomial	ADJ
iajs-4097	271	9	series	series	NOUN
iajs-4097	271	10	.	.	PUNCT
iajs-4097	272	1	int	int	PROPN
iajs-4097	272	2	j	j	PROPN
iajs-4097	272	3	control	control	PROPN
iajs-4097	272	4	.	.	PUNCT
iajs-4097	273	1	1986;44(1):1–15	1986;44(1):1–15	X
iajs-4097	273	2	.	.	PUNCT
iajs-4097	273	3	https://doi.org/10.1080/00207178608933579	https://doi.org/10.1080/00207178608933579	NUM
iajs-4097	274	1	18	18	NUM
iajs-4097	274	2	.	.	PUNCT
iajs-4097	274	3	kosunalp	kosunalp	PROPN
iajs-4097	274	4	hy	hy	PROPN
iajs-4097	274	5	,	,	PUNCT
iajs-4097	274	6	g¨ulsu	g¨ulsu	NOUN
iajs-4097	274	7	m.	m.	NOUN
iajs-4097	274	8	operational	operational	ADJ
iajs-4097	274	9	matrix	matrix	NOUN
iajs-4097	274	10	by	by	ADP
iajs-4097	274	11	hermite	hermite	ADJ
iajs-4097	274	12	polynomials	polynomial	NOUN
iajs-4097	274	13	for	for	ADP
iajs-4097	274	14	solving	solve	VERB
iajs-4097	274	15	nonlinear	nonlinear	ADJ
iajs-4097	274	16	riccati	riccati	PROPN
iajs-4097	274	17	differential	differential	PROPN
iajs-4097	274	18	equations	equation	NOUN
iajs-4097	274	19	.	.	PUNCT
iajs-4097	275	1	int	int	PROPN
iajs-4097	275	2	j	j	PROPN
iajs-4097	275	3	math	math	PROPN
iajs-4097	275	4	comput	comput	PROPN
iajs-4097	275	5	sci	sci	PROPN
iajs-4097	275	6	.	.	PUNCT
iajs-4097	276	1	2020;16(2):525–536	2020;16(2):525–536	NUM
iajs-4097	276	2	.	.	PUNCT
iajs-4097	277	1	19	19	NUM
iajs-4097	277	2	.	.	PUNCT
iajs-4097	277	3	bagherpoorfard	bagherpoorfard	PROPN
iajs-4097	277	4	m	m	PROPN
iajs-4097	277	5	,	,	PUNCT
iajs-4097	277	6	ghassabzade	ghassabzade	ADJ
iajs-4097	277	7	fa	fa	PROPN
iajs-4097	277	8	.	.	PUNCT
iajs-4097	278	1	hermite	hermite	ADJ
iajs-4097	278	2	matrix	matrix	NOUN
iajs-4097	278	3	polynomial	polynomial	ADJ
iajs-4097	278	4	collocation	collocation	NOUN
iajs-4097	278	5	method	method	NOUN
iajs-4097	278	6	for	for	ADP
iajs-4097	278	7	linear	linear	ADJ
iajs-4097	278	8	complex	complex	ADJ
iajs-4097	278	9	differential	differential	ADJ
iajs-4097	278	10	equations	equation	NOUN
iajs-4097	278	11	and	and	CCONJ
iajs-4097	278	12	some	some	DET
iajs-4097	278	13	comparisons	comparison	NOUN
iajs-4097	278	14	.	.	PUNCT
iajs-4097	279	1	j	j	PROPN
iajs-4097	279	2	appl	appl	PROPN
iajs-4097	279	3	math	math	PROPN
iajs-4097	279	4	phys	phys	PROPN
iajs-4097	279	5	.	.	PUNCT
iajs-4097	280	1	2013;01(05):58–64	2013;01(05):58–64	PROPN
iajs-4097	280	2	.	.	PROPN
iajs-4097	280	3	http://dx.doi.org/10.4236/jamp.2013.15009	http://dx.doi.org/10.4236/jamp.2013.15009	NOUN
iajs-4097	280	4	20	20	NUM
iajs-4097	280	5	.	.	PUNCT
iajs-4097	281	1	isah	isah	PROPN
iajs-4097	281	2	a	a	PRON
iajs-4097	281	3	,	,	PUNCT
iajs-4097	281	4	phang	phang	PROPN
iajs-4097	281	5	c.	c.	PROPN
iajs-4097	281	6	operational	operational	PROPN
iajs-4097	281	7	matrix	matrix	NOUN
iajs-4097	281	8	based	base	VERB
iajs-4097	281	9	on	on	ADP
iajs-4097	281	10	genocchi	genocchi	PROPN
iajs-4097	281	11	polynomials	polynomial	NOUN
iajs-4097	281	12	for	for	ADP
iajs-4097	281	13	solution	solution	NOUN
iajs-4097	281	14	of	of	ADP
iajs-4097	281	15	delay	delay	NOUN
iajs-4097	281	16	differential	differential	ADJ
iajs-4097	281	17	equations	equation	NOUN
iajs-4097	281	18	.	.	PUNCT
iajs-4097	282	1	ain	ain	PROPN
iajs-4097	282	2	shams	shams	PROPN
iajs-4097	282	3	eng	eng	PROPN
iajs-4097	282	4	j.	j.	PROPN
iajs-4097	282	5	2018;9(4):2123–2128	2018;9(4):2123–2128	PROPN
iajs-4097	282	6	.	.	PUNCT
iajs-4097	283	1	https://doi.org/10.1016/j.asej.2016.09.015	https://doi.org/10.1016/j.asej.2016.09.015	PROPN
iajs-4097	283	2	21	21	NUM
iajs-4097	283	3	.	.	PUNCT
iajs-4097	284	1	abdelkawy	abdelkawy	PROPN
iajs-4097	284	2	ma	ma	PROPN
iajs-4097	284	3	,	,	PUNCT
iajs-4097	284	4	taha	taha	PROPN
iajs-4097	284	5	tm	tm	PROPN
iajs-4097	284	6	.	.	PROPN
iajs-4097	285	1	an	an	DET
iajs-4097	285	2	operational	operational	ADJ
iajs-4097	285	3	matrix	matrix	NOUN
iajs-4097	285	4	of	of	ADP
iajs-4097	285	5	fractional	fractional	ADJ
iajs-4097	285	6	derivatives	derivative	NOUN
iajs-4097	285	7	of	of	ADP
iajs-4097	285	8	laguerre	laguerre	NOUN
iajs-4097	285	9	polynomials	polynomial	NOUN
iajs-4097	285	10	.	.	PUNCT
iajs-4097	286	1	walailak	walailak	ADJ
iajs-4097	286	2	j	j	PROPN
iajs-4097	286	3	sci	sci	PROPN
iajs-4097	286	4	technol	technol	NOUN
iajs-4097	286	5	.	.	PUNCT
iajs-4097	286	6	2014;11(12):1041–1055	2014;11(12):1041–1055	NUM
iajs-4097	286	7	.	.	PUNCT
iajs-4097	287	1	http://dx.doi.org/10.14456/wjst.2014.59	http://dx.doi.org/10.14456/wjst.2014.59	VERB
iajs-4097	287	2	22	22	NUM
iajs-4097	287	3	.	.	PUNCT
iajs-4097	288	1	yousefi	yousefi	PROPN
iajs-4097	288	2	sa	sa	PROPN
iajs-4097	288	3	,	,	PUNCT
iajs-4097	288	4	behroozifar	behroozifar	ADV
iajs-4097	288	5	m.	m.	NOUN
iajs-4097	288	6	operational	operational	ADJ
iajs-4097	288	7	matrices	matrix	NOUN
iajs-4097	288	8	of	of	ADP
iajs-4097	288	9	bernstein	bernstein	PROPN
iajs-4097	288	10	polynomials	polynomial	NOUN
iajs-4097	288	11	and	and	CCONJ
iajs-4097	288	12	their	their	PRON
iajs-4097	288	13	applications	application	NOUN
iajs-4097	288	14	.	.	PUNCT
iajs-4097	289	1	int	int	PROPN
iajs-4097	289	2	j	j	PROPN
iajs-4097	289	3	syst	syst	PROPN
iajs-4097	289	4	sci	sci	PROPN
iajs-4097	289	5	.	.	PUNCT
iajs-4097	290	1	2010;41(6):709–716	2010;41(6):709–716	X
iajs-4097	290	2	.	.	PUNCT
iajs-4097	291	1	https://doi.org/10.1080/00207720903154783	https://doi.org/10.1080/00207720903154783	NUM
iajs-4097	291	2	23	23	NUM
iajs-4097	291	3	.	.	PUNCT
iajs-4097	292	1	yousefi	yousefi	PROPN
iajs-4097	292	2	sa	sa	PROPN
iajs-4097	292	3	,	,	PUNCT
iajs-4097	292	4	behroozifar	behroozifar	ADV
iajs-4097	292	5	m	m	PRON
iajs-4097	292	6	,	,	PUNCT
iajs-4097	292	7	dehghan	dehghan	PROPN
iajs-4097	292	8	m.	m.	NOUN
iajs-4097	292	9	the	the	DET
iajs-4097	292	10	operational	operational	ADJ
iajs-4097	292	11	matrices	matrix	NOUN
iajs-4097	292	12	of	of	ADP
iajs-4097	292	13	bernstein	bernstein	PROPN
iajs-4097	292	14	polynomials	polynomial	NOUN
iajs-4097	292	15	for	for	ADP
iajs-4097	292	16	solving	solve	VERB
iajs-4097	292	17	the	the	DET
iajs-4097	292	18	parabolic	parabolic	ADJ
iajs-4097	292	19	equation	equation	NOUN
iajs-4097	292	20	subject	subject	ADJ
iajs-4097	292	21	to	to	ADP
iajs-4097	292	22	specification	specification	NOUN
iajs-4097	292	23	of	of	ADP
iajs-4097	292	24	the	the	DET
iajs-4097	292	25	mass	mass	PROPN
iajs-4097	292	26	.	.	PROPN
iajs-4097	293	1	journal	journal	PROPN
iajs-4097	293	2	of	of	ADP
iajs-4097	293	3	computational	computational	ADJ
iajs-4097	293	4	and	and	CCONJ
iajs-4097	293	5	applied	applied	ADJ
iajs-4097	293	6	mathematics	mathematic	NOUN
iajs-4097	293	7	.	.	PUNCT
iajs-4097	294	1	2011;235(17):5272	2011;235(17):5272	X
iajs-4097	294	2	-	-	PUNCT
iajs-4097	294	3	83	83	NUM
iajs-4097	294	4	.	.	PUNCT
iajs-4097	294	5	doi:10.1016	doi:10.1016	PROPN
iajs-4097	294	6	/	/	SYM
iajs-4097	294	7	j.cam.2011.05.038	j.cam.2011.05.038	PROPN
iajs-4097	294	8	24	24	NUM
iajs-4097	294	9	.	.	PUNCT
iajs-4097	294	10	ibraheem	ibraheem	PROPN
iajs-4097	294	11	gh	gh	PROPN
iajs-4097	294	12	,	,	PUNCT
iajs-4097	294	13	al	al	PROPN
iajs-4097	294	14	-	-	PUNCT
iajs-4097	294	15	jawary	jawary	PROPN
iajs-4097	294	16	ma	ma	PROPN
iajs-4097	294	17	.	.	PUNCT
iajs-4097	295	1	the	the	DET
iajs-4097	295	2	operational	operational	ADJ
iajs-4097	295	3	matrix	matrix	NOUN
iajs-4097	295	4	of	of	ADP
iajs-4097	295	5	legendre	legendre	PROPN
iajs-4097	295	6	polynomials	polynomial	NOUN
iajs-4097	295	7	for	for	ADP
iajs-4097	295	8	solving	solve	VERB
iajs-4097	295	9	nonlinear	nonlinear	ADJ
iajs-4097	295	10	thin	thin	ADJ
iajs-4097	295	11	film	film	NOUN
iajs-4097	295	12	flow	flow	NOUN
iajs-4097	295	13	problems	problem	NOUN
iajs-4097	295	14	.	.	PUNCT
iajs-4097	296	1	alexandria	alexandria	PROPN
iajs-4097	296	2	eng	eng	PROPN
iajs-4097	296	3	j.	j.	PROPN
iajs-4097	296	4	2020;59(5):4027–4033	2020;59(5):4027–4033	PROPN
iajs-4097	296	5	.	.	PUNCT
iajs-4097	297	1	https://doi.org/10.1016/j.aej.2020.07.008	https://doi.org/10.1016/j.aej.2020.07.008	NOUN
iajs-4097	297	2	25	25	NUM
iajs-4097	297	3	.	.	PUNCT
iajs-4097	298	1	sharma	sharma	PROPN
iajs-4097	298	2	b	b	PROPN
iajs-4097	298	3	,	,	PUNCT
iajs-4097	298	4	kumar	kumar	PROPN
iajs-4097	298	5	s	s	PROPN
iajs-4097	298	6	,	,	PUNCT
iajs-4097	298	7	paswan	paswan	PROPN
iajs-4097	298	8	mk	mk	NOUN
iajs-4097	298	9	,	,	PUNCT
iajs-4097	298	10	and	and	CCONJ
iajs-4097	298	11	mahato	mahato	PROPN
iajs-4097	298	12	d.	d.	PROPN
iajs-4097	298	13	chebyshev	chebyshev	PROPN
iajs-4097	298	14	operational	operational	ADJ
iajs-4097	298	15	matrix	matrix	NOUN
iajs-4097	298	16	method	method	NOUN
iajs-4097	298	17	for	for	ADP
iajs-4097	298	18	lane	lane	NOUN
iajs-4097	298	19	-	-	PUNCT
iajs-4097	298	20	emden	emden	NOUN
iajs-4097	298	21	problem	problem	NOUN
iajs-4097	298	22	.	.	PUNCT
iajs-4097	299	1	nonlinear	nonlinear	PROPN
iajs-4097	299	2	eng	eng	PROPN
iajs-4097	299	3	.	.	PUNCT
iajs-4097	300	1	2019;8(1):1–9	2019;8(1):1–9	NUM
iajs-4097	300	2	.	.	PUNCT
iajs-4097	301	1	https://doi.org/10.1515/nleng-2017-0157	https://doi.org/10.1515/nleng-2017-0157	NOUN
iajs-4097	301	2	26	26	NUM
iajs-4097	301	3	.	.	PUNCT
iajs-4097	302	1	el	el	PROPN
iajs-4097	302	2	-	-	PUNCT
iajs-4097	302	3	gamel	gamel	PROPN
iajs-4097	302	4	m	m	PROPN
iajs-4097	302	5	e	e	PROPN
iajs-4097	302	6	,	,	PUNCT
iajs-4097	302	7	adel	adel	PROPN
iajs-4097	302	8	w	w	PROPN
iajs-4097	302	9	,	,	PUNCT
iajs-4097	302	10	el	el	PROPN
iajs-4097	302	11	-	-	PUNCT
iajs-4097	302	12	azab	azab	PROPN
iajs-4097	302	13	ms	ms	PROPN
iajs-4097	302	14	.	.	PROPN
iajs-4097	302	15	collocation	collocation	NOUN
iajs-4097	302	16	method	method	NOUN
iajs-4097	302	17	based	base	VERB
iajs-4097	302	18	on	on	ADP
iajs-4097	302	19	bernoulli	bernoulli	PROPN
iajs-4097	302	20	polynomial	polynomial	ADJ
iajs-4097	302	21	and	and	CCONJ
iajs-4097	302	22	shifted	shift	VERB
iajs-4097	302	23	chebychev	chebychev	NOUN
iajs-4097	302	24	for	for	ADP
iajs-4097	302	25	solving	solve	VERB
iajs-4097	302	26	the	the	DET
iajs-4097	302	27	bratu	bratu	NOUN
iajs-4097	302	28	equation	equation	NOUN
iajs-4097	302	29	.	.	PUNCT
iajs-4097	303	1	j	j	PROPN
iajs-4097	303	2	appl	appl	PROPN
iajs-4097	303	3	comput	comput	PROPN
iajs-4097	303	4	math	math	PROPN
iajs-4097	303	5	.	.	PUNCT
iajs-4097	304	1	2018;07(03	2018;07(03	NUM
iajs-4097	304	2	)	)	PUNCT
iajs-4097	304	3	.	.	PUNCT
iajs-4097	305	1	27	27	NUM
iajs-4097	305	2	.	.	PUNCT
iajs-4097	306	1	singh	singh	PROPN
iajs-4097	306	2	s	s	PROPN
iajs-4097	306	3	,	,	PUNCT
iajs-4097	306	4	patel	patel	PROPN
iajs-4097	306	5	vk	vk	PROPN
iajs-4097	306	6	,	,	PUNCT
iajs-4097	306	7	singh	singh	PROPN
iajs-4097	306	8	vk	vk	PROPN
iajs-4097	306	9	,	,	PUNCT
iajs-4097	306	10	tohidi	tohidi	PROPN
iajs-4097	306	11	e.	e.	PROPN
iajs-4097	306	12	application	application	PROPN
iajs-4097	306	13	of	of	ADP
iajs-4097	306	14	bernoulli	bernoulli	PROPN
iajs-4097	306	15	matrix	matrix	NOUN
iajs-4097	306	16	method	method	NOUN
iajs-4097	306	17	for	for	ADP
iajs-4097	306	18	solving	solve	VERB
iajs-4097	306	19	twodimensional	twodimensional	ADJ
iajs-4097	306	20	hyperbolic	hyperbolic	ADJ
iajs-4097	306	21	telegraph	telegraph	NOUN
iajs-4097	306	22	equations	equation	NOUN
iajs-4097	306	23	with	with	ADP
iajs-4097	306	24	dirichlet	dirichlet	PROPN
iajs-4097	306	25	boundary	boundary	PROPN
iajs-4097	306	26	conditions	condition	NOUN
iajs-4097	306	27	.	.	PUNCT
iajs-4097	307	1	comput	comput	ADJ
iajs-4097	307	2	math	math	NOUN
iajs-4097	307	3	with	with	ADP
iajs-4097	307	4	appl	appl	NOUN
iajs-4097	307	5	.	.	PUNCT
iajs-4097	308	1	2018;75(7):2280–2294	2018;75(7):2280–2294	NOUN
iajs-4097	308	2	.	.	PUNCT
iajs-4097	309	1	https://doi.org/10.1016/j.camwa.2017.12.003	https://doi.org/10.1016/j.camwa.2017.12.003	PROPN
iajs-4097	309	2	28	28	NUM
iajs-4097	309	3	.	.	PUNCT
iajs-4097	310	1	bencheikh	bencheikh	PROPN
iajs-4097	310	2	a	a	DET
iajs-4097	310	3	,	,	PUNCT
iajs-4097	310	4	chiter	chiter	PROPN
iajs-4097	310	5	l.	l.	PROPN
iajs-4097	310	6	boubaker	boubaker	PROPN
iajs-4097	310	7	operational	operational	ADJ
iajs-4097	310	8	matrix	matrix	NOUN
iajs-4097	310	9	method	method	NOUN
iajs-4097	310	10	for	for	ADP
iajs-4097	310	11	fractional	fractional	ADJ
iajs-4097	310	12	emden	emden	ADJ
iajs-4097	310	13	-	-	PUNCT
iajs-4097	310	14	fowler	fowler	PROPN
iajs-4097	310	15	problem	problem	NOUN
iajs-4097	310	16	.	.	PUNCT
iajs-4097	311	1	j	j	PROPN
iajs-4097	311	2	sib	sib	PROPN
iajs-4097	311	3	fed	feed	VERB
iajs-4097	311	4	univ	univ	PROPN
iajs-4097	311	5	math	math	PROPN
iajs-4097	311	6	phys	phys	PROPN
iajs-4097	311	7	.	.	PUNCT
iajs-4097	312	1	2023;16(3):357–369	2023;16(3):357–369	NUM
iajs-4097	312	2	.	.	PUNCT
iajs-4097	313	1	https://www.mathnet.ru/eng/jsfu1085	https://www.mathnet.ru/eng/jsfu1085	NOUN
iajs-4097	313	2	29	29	NUM
iajs-4097	313	3	.	.	PUNCT
iajs-4097	313	4	turkyilmazoglu	turkyilmazoglu	NOUN
iajs-4097	313	5	m.	m.	NOUN
iajs-4097	313	6	effective	effective	ADJ
iajs-4097	313	7	computation	computation	NOUN
iajs-4097	313	8	of	of	ADP
iajs-4097	313	9	exact	exact	ADJ
iajs-4097	313	10	and	and	CCONJ
iajs-4097	313	11	analytic	analytic	ADJ
iajs-4097	313	12	approximate	approximate	ADJ
iajs-4097	313	13	solutions	solution	NOUN
iajs-4097	313	14	to	to	PART
iajs-4097	313	15	singular	singular	VERB
iajs-4097	313	16	nonlinear	nonlinear	ADJ
iajs-4097	313	17	equations	equation	NOUN
iajs-4097	313	18	of	of	ADP
iajs-4097	313	19	lane	lane	NOUN
iajs-4097	313	20	-	-	PUNCT
iajs-4097	313	21	emden	emden	VERB
iajs-4097	313	22	-	-	PUNCT
iajs-4097	313	23	fowler	fowler	NOUN
iajs-4097	313	24	type	type	NOUN
iajs-4097	313	25	.	.	PUNCT
iajs-4097	314	1	appl	appl	PROPN
iajs-4097	314	2	math	math	PROPN
iajs-4097	314	3	model	model	PROPN
iajs-4097	314	4	.	.	PUNCT
iajs-4097	315	1	2013;37(14–15):7539–7548	2013;37(14–15):7539–7548	PROPN
iajs-4097	315	2	.	.	PUNCT
iajs-4097	316	1	https://doi.org/10.1016/j.apm.2013.02.014	https://doi.org/10.1016/j.apm.2013.02.014	PROPN
iajs-4097	316	2	30	30	NUM
iajs-4097	316	3	.	.	PUNCT
iajs-4097	317	1	turkyilmazoglu	turkyilmazoglu	NOUN
iajs-4097	317	2	m.	m.	NOUN
iajs-4097	317	3	an	an	DET
iajs-4097	317	4	effective	effective	ADJ
iajs-4097	317	5	approach	approach	NOUN
iajs-4097	317	6	for	for	ADP
iajs-4097	317	7	numerical	numerical	ADJ
iajs-4097	317	8	solutions	solution	NOUN
iajs-4097	317	9	of	of	ADP
iajs-4097	317	10	high	high	ADJ
iajs-4097	317	11	-	-	PUNCT
iajs-4097	317	12	order	order	NOUN
iajs-4097	317	13	fredholm	fredholm	NOUN
iajs-4097	317	14	integrodifferential	integrodifferential	ADJ
iajs-4097	317	15	equations	equation	NOUN
iajs-4097	317	16	.	.	PUNCT
iajs-4097	318	1	appl	appl	PROPN
iajs-4097	318	2	math	math	PROPN
iajs-4097	318	3	comput	comput	NOUN
iajs-4097	318	4	.	.	PUNCT
iajs-4097	319	1	2014;227:384–398	2014;227:384–398	NUM
iajs-4097	319	2	.	.	PUNCT
iajs-4097	320	1	https://doi.org/10.1016/j.amc.2013.10.079	https://doi.org/10.1016/j.amc.2013.10.079	PROPN
iajs-4097	320	2	31	31	NUM
iajs-4097	320	3	.	.	PUNCT
iajs-4097	321	1	turkyilmazoglu	turkyilmazoglu	NOUN
iajs-4097	321	2	m.	m.	NOUN
iajs-4097	321	3	effective	effective	ADJ
iajs-4097	321	4	computation	computation	NOUN
iajs-4097	321	5	of	of	ADP
iajs-4097	321	6	solutions	solution	NOUN
iajs-4097	321	7	for	for	ADP
iajs-4097	321	8	nonlinear	nonlinear	ADJ
iajs-4097	321	9	heat	heat	NOUN
iajs-4097	321	10	transfer	transfer	NOUN
iajs-4097	321	11	problems	problem	NOUN
iajs-4097	321	12	in	in	ADP
iajs-4097	321	13	fins	fin	NOUN
iajs-4097	321	14	.	.	PUNCT
iajs-4097	322	1	j	j	PROPN
iajs-4097	322	2	heat	heat	NOUN
iajs-4097	322	3	transfer	transfer	NOUN
iajs-4097	322	4	.	.	PUNCT
iajs-4097	323	1	2014;136(9):1–7	2014;136(9):1–7	AUX
iajs-4097	323	2	.	.	PUNCT
iajs-4097	323	3	https://doi.org/10.1115/1.4027772	https://doi.org/10.1115/1.4027772	PROPN
iajs-4097	323	4	32	32	NUM
iajs-4097	323	5	.	.	PUNCT
iajs-4097	323	6	turkyilmazoglu	turkyilmazoglu	NOUN
iajs-4097	323	7	m.	m.	NOUN
iajs-4097	323	8	solution	solution	NOUN
iajs-4097	323	9	of	of	ADP
iajs-4097	323	10	initial	initial	ADJ
iajs-4097	323	11	and	and	CCONJ
iajs-4097	323	12	boundary	boundary	ADJ
iajs-4097	323	13	value	value	NOUN
iajs-4097	323	14	problems	problem	NOUN
iajs-4097	323	15	by	by	ADP
iajs-4097	323	16	an	an	DET
iajs-4097	323	17	effective	effective	ADJ
iajs-4097	323	18	accurate	accurate	ADJ
iajs-4097	323	19	method	method	NOUN
iajs-4097	323	20	.	.	PUNCT
iajs-4097	324	1	int	int	NOUN
iajs-4097	324	2	j	j	PROPN
iajs-4097	324	3	comput	comput	ADJ
iajs-4097	324	4	methods	method	NOUN
iajs-4097	324	5	.	.	PUNCT
iajs-4097	325	1	2017;14(6	2017;14(6	NUM
iajs-4097	325	2	)	)	PUNCT
iajs-4097	325	3	.	.	PUNCT
iajs-4097	326	1	https://doi.org/10.1142/s0219876217500694	https://doi.org/10.1142/s0219876217500694	NUM
iajs-4097	326	2	33	33	NUM
iajs-4097	326	3	.	.	PUNCT
iajs-4097	326	4	turkyilmazoglu	turkyilmazoglu	PROPN
iajs-4097	326	5	m.	m.	PROPN
iajs-4097	326	6	parabolic	parabolic	PROPN
iajs-4097	327	1	partial	partial	ADJ
iajs-4097	327	2	differential	differential	NOUN
iajs-4097	327	3	equations	equation	NOUN
iajs-4097	327	4	with	with	ADP
iajs-4097	327	5	nonlocal	nonlocal	ADJ
iajs-4097	327	6	initial	initial	ADJ
iajs-4097	327	7	and	and	CCONJ
iajs-4097	327	8	boundary	boundary	ADJ
iajs-4097	327	9	values	value	NOUN
iajs-4097	327	10	.	.	PUNCT
iajs-4097	328	1	int	int	NOUN
iajs-4097	328	2	j	j	PROPN
iajs-4097	328	3	comput	comput	PROPN
iajs-4097	328	4	methods	method	NOUN
iajs-4097	328	5	.	.	PUNCT
iajs-4097	329	1	2015;12(5):1–10	2015;12(5):1–10	X
iajs-4097	329	2	.	.	PUNCT
iajs-4097	330	1	https://doi.org/10.1142/s0219876215500243	https://doi.org/10.1142/s0219876215500243	NUM
iajs-4097	330	2	34	34	NUM
iajs-4097	330	3	.	.	PUNCT
iajs-4097	331	1	salih	salih	PROPN
iajs-4097	331	2	om	om	PROPN
iajs-4097	331	3	,	,	PUNCT
iajs-4097	331	4	al	al	PROPN
iajs-4097	331	5	-	-	PUNCT
iajs-4097	331	6	jawary	jawary	PROPN
iajs-4097	331	7	ma	ma	PROPN
iajs-4097	331	8	.	.	PROPN
iajs-4097	331	9	computational	computational	ADJ
iajs-4097	331	10	methods	method	NOUN
iajs-4097	331	11	for	for	ADP
iajs-4097	331	12	solving	solve	VERB
iajs-4097	331	13	nonlinear	nonlinear	ADJ
iajs-4097	331	14	ordinary	ordinary	ADJ
iajs-4097	331	15	differential	differential	ADJ
iajs-4097	331	16	equations	equation	NOUN
iajs-4097	331	17	arising	arise	VERB
iajs-4097	331	18	in	in	ADP
iajs-4097	331	19	engineering	engineering	NOUN
iajs-4097	331	20	and	and	CCONJ
iajs-4097	331	21	applied	apply	VERB
iajs-4097	331	22	sciences	science	NOUN
iajs-4097	331	23	.	.	PUNCT
iajs-4097	332	1	iraqi	iraqi	PROPN
iajs-4097	332	2	j	j	PROPN
iajs-4097	332	3	sci	sci	PROPN
iajs-4097	332	4	.	.	PUNCT
iajs-4097	333	1	2023;64(8):4070–4091	2023;64(8):4070–4091	NUM
iajs-4097	333	2	.	.	PUNCT
iajs-4097	334	1	doi	doi	NOUN
iajs-4097	334	2	:	:	PUNCT
iajs-4097	334	3	10.24996	10.24996	NUM
iajs-4097	334	4	/	/	SYM
iajs-4097	334	5	ijs.2023.64.8.30	ijs.2023.64.8.30	PROPN
iajs-4097	334	6	https://doi.org/10.1186/s13662-019-2334-7	https://doi.org/10.1186/s13662-019-2334-7	NUM
iajs-4097	334	7	https://doi.org/10.1186/s13662-019-2488-3	https://doi.org/10.1186/s13662-019-2488-3	NUM
iajs-4097	334	8	https://doi.org/10.1080/00207178608933579	https://doi.org/10.1080/00207178608933579	NUM
iajs-4097	334	9	http://dx.doi.org/10.4236/jamp.2013.15009	http://dx.doi.org/10.4236/jamp.2013.15009	NOUN
iajs-4097	334	10	https://doi.org/10.1016/j.asej.2016.09.015	https://doi.org/10.1016/j.asej.2016.09.015	PROPN
iajs-4097	334	11	http://dx.doi.org/10.14456/wjst.2014.59	http://dx.doi.org/10.14456/wjst.2014.59	PUNCT
iajs-4097	334	12	https://doi.org/10.1080/00207720903154783	https://doi.org/10.1080/00207720903154783	PROPN
iajs-4097	334	13	mailto	mailto	PROPN
iajs-4097	334	14	:	:	PUNCT
iajs-4097	334	15	doi:10.1016	doi:10.1016	PROPN
iajs-4097	334	16	/	/	SYM
iajs-4097	334	17	j.cam.2011.05.038	j.cam.2011.05.038	PROPN
iajs-4097	334	18	https://doi.org/10.1016/j.aej.2020.07.008	https://doi.org/10.1016/j.aej.2020.07.008	NOUN
iajs-4097	335	1	https://doi.org/10.1515/nleng-2017-0157	https://doi.org/10.1515/nleng-2017-0157	PROPN
iajs-4097	335	2	https://doi.org/10.1016/j.camwa.2017.12.003	https://doi.org/10.1016/j.camwa.2017.12.003	PROPN
iajs-4097	335	3	mailto	mailto	PROPN
iajs-4097	335	4	:	:	PUNCT
iajs-4097	335	5	https://www.mathnet.ru	https://www.mathnet.ru	PROPN
iajs-4097	335	6	/	/	SYM
iajs-4097	335	7	eng	eng	PROPN
iajs-4097	335	8	/	/	SYM
iajs-4097	335	9	jsfu1085	jsfu1085	PROPN
iajs-4097	335	10	https://doi.org/10.1016/j.apm.2013.02.014	https://doi.org/10.1016/j.apm.2013.02.014	PROPN
iajs-4097	335	11	https://doi.org/10.1016/j.amc.2013.10.079	https://doi.org/10.1016/j.amc.2013.10.079	PROPN
iajs-4097	335	12	https://doi.org/10.1115/1.4027772	https://doi.org/10.1115/1.4027772	PROPN
iajs-4097	336	1	https://doi.org/10.1142/s0219876217500694	https://doi.org/10.1142/s0219876217500694	PROPN
iajs-4097	336	2	https://doi.org/10.1142/s0219876215500243	https://doi.org/10.1142/s0219876215500243	NUM
iajs-4097	336	3	doi:%2010.24996	doi:%2010.24996	PROPN
iajs-4097	336	4	/	/	SYM
iajs-4097	336	5	ijs.2023.64.8.30	ijs.2023.64.8.30	PROPN
iajs-4097	336	6	doi:%2010.24996	doi:%2010.24996	PROPN
iajs-4097	336	7	/	/	SYM
iajs-4097	336	8	ijs.2023.64.8.30	ijs.2023.64.8.30	PROPN
iajs-4097	336	9	ihjpas	ihjpa	VERB
iajs-4097	336	10	.	.	PUNCT
iajs-4097	337	1	2025,38(4	2025,38(4	NOUN
iajs-4097	337	2	)	)	PUNCT
iajs-4097	337	3	397	397	NUM
iajs-4097	337	4	35	35	NUM
iajs-4097	337	5	.	.	PUNCT
iajs-4097	338	1	enadi	enadi	PROPN
iajs-4097	338	2	mo	mo	PROPN
iajs-4097	338	3	,	,	PUNCT
iajs-4097	338	4	al	al	PROPN
iajs-4097	338	5	-	-	PUNCT
iajs-4097	338	6	jawary	jawary	PROPN
iajs-4097	338	7	ma	ma	PROPN
iajs-4097	338	8	.	.	PUNCT
iajs-4097	339	1	the	the	DET
iajs-4097	339	2	operational	operational	ADJ
iajs-4097	339	3	matrices	matrix	NOUN
iajs-4097	339	4	for	for	ADP
iajs-4097	339	5	elliptic	elliptic	ADJ
iajs-4097	339	6	partial	partial	ADJ
iajs-4097	339	7	differential	differential	ADJ
iajs-4097	339	8	equations	equation	NOUN
iajs-4097	339	9	with	with	ADP
iajs-4097	339	10	mixed	mixed	ADJ
iajs-4097	339	11	boundary	boundary	ADJ
iajs-4097	339	12	conditions	condition	NOUN
iajs-4097	339	13	.	.	PUNCT
iajs-4097	340	1	j	j	PROPN
iajs-4097	340	2	phys	phy	NOUN
iajs-4097	340	3	conf	conf	NOUN
iajs-4097	340	4	ser	ser	NOUN
iajs-4097	340	5	.	.	PROPN
iajs-4097	340	6	2024;2857(1):1–18	2024;2857(1):1–18	NUM
iajs-4097	340	7	.	.	PUNCT
iajs-4097	341	1	doi	doi	X
iajs-4097	341	2	10.1088/17426596/2857/1/012027	10.1088/17426596/2857/1/012027	NOUN
iajs-4097	341	3	36	36	NUM
iajs-4097	341	4	.	.	PUNCT
iajs-4097	342	1	enadi	enadi	PROPN
iajs-4097	342	2	mo	mo	PROPN
iajs-4097	342	3	,	,	PUNCT
iajs-4097	342	4	al	al	PROPN
iajs-4097	342	5	-	-	PUNCT
iajs-4097	342	6	jawary	jawary	PROPN
iajs-4097	342	7	ma	ma	PROPN
iajs-4097	342	8	,	,	PUNCT
iajs-4097	342	9	turkyilmazoglu	turkyilmazoglu	NOUN
iajs-4097	342	10	m.	m.	NOUN
iajs-4097	342	11	the	the	DET
iajs-4097	342	12	operational	operational	ADJ
iajs-4097	342	13	matrices	matrix	NOUN
iajs-4097	342	14	methods	method	NOUN
iajs-4097	342	15	for	for	ADP
iajs-4097	342	16	solving	solve	VERB
iajs-4097	342	17	the	the	DET
iajs-4097	342	18	poisson	poisson	NOUN
iajs-4097	342	19	equation	equation	NOUN
iajs-4097	342	20	with	with	ADP
iajs-4097	342	21	nonlocal	nonlocal	ADJ
iajs-4097	342	22	boundary	boundary	ADJ
iajs-4097	342	23	conditions	condition	NOUN
iajs-4097	342	24	.	.	PUNCT
iajs-4097	343	1	j	j	PROPN
iajs-4097	343	2	appl	appl	PROPN
iajs-4097	343	3	math	math	PROPN
iajs-4097	343	4	comput	comput	NOUN
iajs-4097	343	5	.	.	PUNCT
iajs-4097	344	1	2025;71(2):2385	2025;71(2):2385	NUM
iajs-4097	344	2	–	–	PUNCT
iajs-4097	344	3	2432	2432	NUM
iajs-4097	344	4	.	.	PUNCT
iajs-4097	345	1	https://doi.org/10.1007/s12190-024-02313-y	https://doi.org/10.1007/s12190-024-02313-y	PROPN
iajs-4097	345	2	37	37	NUM
iajs-4097	345	3	.	.	PUNCT
iajs-4097	346	1	davaeifar	davaeifar	PROPN
iajs-4097	346	2	s	s	PROPN
iajs-4097	346	3	,	,	PUNCT
iajs-4097	346	4	rashidinia	rashidinia	PROPN
iajs-4097	346	5	j.	j.	PROPN
iajs-4097	346	6	boubaker	boubaker	PROPN
iajs-4097	346	7	polynomials	polynomial	NOUN
iajs-4097	346	8	collocation	collocation	NOUN
iajs-4097	346	9	approach	approach	NOUN
iajs-4097	346	10	for	for	ADP
iajs-4097	346	11	solving	solve	VERB
iajs-4097	346	12	systems	system	NOUN
iajs-4097	346	13	of	of	ADP
iajs-4097	346	14	nonlinear	nonlinear	PROPN
iajs-4097	346	15	volterra	volterra	PROPN
iajs-4097	346	16	–	–	PUNCT
iajs-4097	346	17	fredholm	fredholm	ADJ
iajs-4097	346	18	integral	integral	ADJ
iajs-4097	346	19	equations	equation	NOUN
iajs-4097	346	20	.	.	PUNCT
iajs-4097	347	1	j	j	PROPN
iajs-4097	347	2	taibah	taibah	PROPN
iajs-4097	347	3	univ	univ	PROPN
iajs-4097	347	4	sci	sci	PROPN
iajs-4097	347	5	.	.	PUNCT
iajs-4097	347	6	2017;11(6):1182–1199	2017;11(6):1182–1199	PROPN
iajs-4097	347	7	.	.	PUNCT
iajs-4097	348	1	https://doi.org/10.1016/j.jtusci.2017.05.002	https://doi.org/10.1016/j.jtusci.2017.05.002	NOUN
iajs-4097	348	2	38	38	NUM
iajs-4097	348	3	.	.	PUNCT
iajs-4097	349	1	bhrawy	bhrawy	PROPN
iajs-4097	349	2	ah	ah	INTJ
iajs-4097	349	3	,	,	PUNCT
iajs-4097	349	4	tohidi	tohidi	NOUN
iajs-4097	349	5	e	e	NOUN
iajs-4097	349	6	,	,	PUNCT
iajs-4097	349	7	soleymani	soleymani	PROPN
iajs-4097	349	8	f.	f.	PROPN
iajs-4097	350	1	a	a	DET
iajs-4097	350	2	new	new	ADJ
iajs-4097	350	3	bernoulli	bernoulli	NOUN
iajs-4097	350	4	matrix	matrix	NOUN
iajs-4097	350	5	method	method	NOUN
iajs-4097	350	6	for	for	ADP
iajs-4097	350	7	solving	solve	VERB
iajs-4097	350	8	high	high	ADJ
iajs-4097	350	9	-	-	PUNCT
iajs-4097	350	10	order	order	NOUN
iajs-4097	350	11	linear	linear	NOUN
iajs-4097	350	12	and	and	CCONJ
iajs-4097	350	13	nonlinear	nonlinear	ADJ
iajs-4097	350	14	fredholm	fredholm	NOUN
iajs-4097	350	15	integro	integro	ADJ
iajs-4097	350	16	-	-	PUNCT
iajs-4097	350	17	differential	differential	NOUN
iajs-4097	350	18	equations	equation	NOUN
iajs-4097	350	19	with	with	ADP
iajs-4097	350	20	piecewise	piecewise	NOUN
iajs-4097	350	21	intervals	interval	NOUN
iajs-4097	350	22	.	.	PUNCT
iajs-4097	351	1	appl	appl	PROPN
iajs-4097	351	2	math	math	PROPN
iajs-4097	351	3	comput	comput	NOUN
iajs-4097	351	4	.	.	PUNCT
iajs-4097	352	1	2012;219(2):482–497	2012;219(2):482–497	X
iajs-4097	352	2	.	.	NUM
iajs-4097	352	3	https://doi.org/10.1016/j.amc.2012.06.020	https://doi.org/10.1016/j.amc.2012.06.020	NOUN
iajs-4097	352	4	39	39	NUM
iajs-4097	352	5	.	.	PUNCT
iajs-4097	353	1	toutounian	toutounian	PROPN
iajs-4097	353	2	f	f	NOUN
iajs-4097	353	3	,	,	PUNCT
iajs-4097	353	4	tohidi	tohidi	PROPN
iajs-4097	353	5	e.	e.	PROPN
iajs-4097	354	1	a	a	DET
iajs-4097	354	2	new	new	ADJ
iajs-4097	354	3	bernoulli	bernoulli	NOUN
iajs-4097	354	4	matrix	matrix	NOUN
iajs-4097	354	5	method	method	NOUN
iajs-4097	354	6	for	for	ADP
iajs-4097	354	7	solving	solve	VERB
iajs-4097	354	8	second	second	ADJ
iajs-4097	354	9	order	order	NOUN
iajs-4097	354	10	linear	linear	VERB
iajs-4097	354	11	partial	partial	ADJ
iajs-4097	354	12	differential	differential	NOUN
iajs-4097	354	13	equations	equation	NOUN
iajs-4097	354	14	with	with	ADP
iajs-4097	354	15	the	the	DET
iajs-4097	354	16	convergence	convergence	NOUN
iajs-4097	354	17	analysis	analysis	NOUN
iajs-4097	354	18	.	.	PUNCT
iajs-4097	355	1	appl	appl	PROPN
iajs-4097	355	2	math	math	PROPN
iajs-4097	355	3	comput	comput	PROPN
iajs-4097	355	4	.	.	PUNCT
iajs-4097	356	1	2013;223:298–310	2013;223:298–310	X
iajs-4097	356	2	.	.	PUNCT
iajs-4097	357	1	https://doi.org/10.1016/j.amc.2013.07.094	https://doi.org/10.1016/j.amc.2013.07.094	VERB
iajs-4097	357	2	40	40	NUM
iajs-4097	357	3	.	.	PUNCT
iajs-4097	358	1	jung	jung	PROPN
iajs-4097	358	2	,	,	PUNCT
iajs-4097	358	3	c.y	c.y	PROPN
iajs-4097	358	4	.	.	PROPN
iajs-4097	358	5	,	,	PUNCT
iajs-4097	358	6	liu	liu	PROPN
iajs-4097	358	7	,	,	PUNCT
iajs-4097	358	8	z.	z.	PROPN
iajs-4097	358	9	,	,	PUNCT
iajs-4097	358	10	rafiq	rafiq	PROPN
iajs-4097	358	11	,	,	PUNCT
iajs-4097	358	12	a.	a.	PROPN
iajs-4097	358	13	,	,	PUNCT
iajs-4097	358	14	ali	ali	PROPN
iajs-4097	358	15	,	,	PUNCT
iajs-4097	358	16	f.	f.	PROPN
iajs-4097	358	17	and	and	CCONJ
iajs-4097	358	18	kang	kang	PROPN
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iajs-4097	358	21	solution	solution	NOUN
iajs-4097	358	22	of	of	ADP
iajs-4097	358	23	second	second	ADJ
iajs-4097	358	24	order	order	NOUN
iajs-4097	358	25	linear	linear	ADJ
iajs-4097	358	26	and	and	CCONJ
iajs-4097	358	27	nonlinear	nonlinear	ADJ
iajs-4097	358	28	ordinary	ordinary	ADJ
iajs-4097	358	29	differential	differential	ADJ
iajs-4097	358	30	equations	equation	NOUN
iajs-4097	358	31	using	use	VERB
iajs-4097	358	32	legendre	legendre	PROPN
iajs-4097	358	33	operational	operational	ADJ
iajs-4097	358	34	matrix	matrix	NOUN
iajs-4097	358	35	of	of	ADP
iajs-4097	358	36	differentiation	differentiation	NOUN
iajs-4097	358	37	.	.	PUNCT
iajs-4097	359	1	int	int	NOUN
iajs-4097	359	2	j	j	PROPN
iajs-4097	359	3	pure	pure	ADJ
iajs-4097	359	4	appl	appl	PROPN
iajs-4097	359	5	math	math	NOUN
iajs-4097	359	6	.	.	PUNCT
iajs-4097	360	1	2014;93(2):285–295	2014;93(2):285–295	X
iajs-4097	360	2	.	.	X
iajs-4097	360	3	doi	doi	NOUN
iajs-4097	360	4	:	:	PUNCT
iajs-4097	360	5	http://dx.doi.org/10.12732/ijpam.v93i2.12	http://dx.doi.org/10.12732/ijpam.v93i2.12	PROPN
iajs-4097	360	6	41	41	NUM
iajs-4097	360	7	.	.	PUNCT
iajs-4097	361	1	doha	doha	PROPN
iajs-4097	361	2	eh	eh	INTJ
iajs-4097	361	3	,	,	PUNCT
iajs-4097	361	4	bhrawy	bhrawy	NOUN
iajs-4097	361	5	ah	ah	INTJ
iajs-4097	361	6	,	,	PUNCT
iajs-4097	361	7	ezz	ezz	ADJ
iajs-4097	361	8	-	-	PUNCT
iajs-4097	361	9	eldien	eldien	NOUN
iajs-4097	361	10	ss	ss	PROPN
iajs-4097	361	11	.	.	PUNCT
iajs-4097	362	1	a	a	DET
iajs-4097	362	2	chebyshev	chebyshev	PROPN
iajs-4097	362	3	spectral	spectral	ADJ
iajs-4097	362	4	method	method	NOUN
iajs-4097	362	5	based	base	VERB
iajs-4097	362	6	on	on	ADP
iajs-4097	362	7	operational	operational	ADJ
iajs-4097	362	8	matrix	matrix	NOUN
iajs-4097	362	9	for	for	ADP
iajs-4097	362	10	initial	initial	ADJ
iajs-4097	362	11	and	and	CCONJ
iajs-4097	362	12	boundary	boundary	ADJ
iajs-4097	362	13	value	value	NOUN
iajs-4097	362	14	problems	problem	NOUN
iajs-4097	362	15	of	of	ADP
iajs-4097	362	16	fractional	fractional	ADJ
iajs-4097	362	17	order	order	NOUN
iajs-4097	362	18	.	.	PUNCT
iajs-4097	363	1	comput	comput	ADJ
iajs-4097	363	2	math	math	NOUN
iajs-4097	363	3	with	with	ADP
iajs-4097	363	4	appl	appl	NOUN
iajs-4097	363	5	.	.	PUNCT
iajs-4097	364	1	2011;62(5):2364–2373	2011;62(5):2364–2373	NOUN
iajs-4097	364	2	.	.	PUNCT
iajs-4097	365	1	https://doi.org/10.1016/j.camwa.2011.07.024	https://doi.org/10.1016/j.camwa.2011.07.024	VERB
iajs-4097	365	2	.	.	PUNCT
iajs-4097	366	1	file:///c:/users	file:///c:/user	NOUN
iajs-4097	366	2	/	/	SYM
iajs-4097	366	3	hp	hp	NOUN
iajs-4097	366	4	/	/	SYM
iajs-4097	366	5	desktop	desktop	NOUN
iajs-4097	366	6	/	/	SYM
iajs-4097	366	7	البحوث%20المرسلة	البحوث%20المرسلة	NOUN
iajs-4097	366	8	/	/	SYM
iajs-4097	366	9	wave%20بحث%20ابن%20الهيثم%20بعد%20التعديلات	wave%20بحث%20ابن%20الهيثم%20بعد%20التعديلات	PROPN
iajs-4097	366	10	/	/	SYM
iajs-4097	366	11	doi%2010.1088/1742	doi%2010.1088/1742	NOUN
iajs-4097	366	12	-	-	PUNCT
iajs-4097	366	13	6596/2857/1/012027	6596/2857/1/012027	NUM
iajs-4097	366	14	file:///c:/users	file:///c:/user	NOUN
iajs-4097	366	15	/	/	SYM
iajs-4097	366	16	hp	hp	NOUN
iajs-4097	366	17	/	/	SYM
iajs-4097	366	18	desktop	desktop	NOUN
iajs-4097	366	19	/	/	SYM
iajs-4097	366	20	البحوث%20المرسلة	البحوث%20المرسلة	NOUN
iajs-4097	366	21	/	/	SYM
iajs-4097	366	22	wave%20بحث%20ابن%20الهيثم%20بعد%20التعديلات	wave%20بحث%20ابن%20الهيثم%20بعد%20التعديلات	PROPN
iajs-4097	366	23	/	/	SYM
iajs-4097	366	24	doi%2010.1088/1742	doi%2010.1088/1742	NOUN
iajs-4097	366	25	-	-	PUNCT
iajs-4097	366	26	6596/2857/1/012027	6596/2857/1/012027	NUM
iajs-4097	366	27	https://doi.org/10.1007/s12190-024-02313-y	https://doi.org/10.1007/s12190-024-02313-y	X
iajs-4097	366	28	https://doi.org/10.1016/j.jtusci.2017.05.002	https://doi.org/10.1016/j.jtusci.2017.05.002	PROPN
iajs-4097	366	29	https://doi.org/10.1016/j.amc.2012.06.020	https://doi.org/10.1016/j.amc.2012.06.020	NOUN
iajs-4097	366	30	https://doi.org/10.1016/j.amc.2013.07.094	https://doi.org/10.1016/j.amc.2013.07.094	VERB
iajs-4097	366	31	doi:%20http://dx.doi.org/10.12732	doi:%20http://dx.doi.org/10.12732	ADJ
iajs-4097	366	32	/	/	SYM
iajs-4097	366	33	ijpam.v93i2.12	ijpam.v93i2.12	NOUN
iajs-4097	366	34	https://doi.org/10.1016/j.camwa.2011.07.024	https://doi.org/10.1016/j.camwa.2011.07.024	NOUN