id	sid	tid	token	lemma	pos
ejpam-3423	1	1	european	european	PROPN
ejpam-3423	1	2	journal	journal	PROPN
ejpam-3423	1	3	of	of	ADP
ejpam-3423	1	4	pure	pure	ADJ
ejpam-3423	1	5	and	and	CCONJ
ejpam-3423	1	6	applied	apply	VERB
ejpam-3423	1	7	mathematics	mathematic	NOUN
ejpam-3423	1	8	vol	vol	NOUN
ejpam-3423	1	9	.	.	PROPN
ejpam-3423	2	1	12	12	NUM
ejpam-3423	2	2	,	,	PUNCT
ejpam-3423	2	3	no	no	INTJ
ejpam-3423	2	4	.	.	NOUN
ejpam-3423	2	5	3	3	NUM
ejpam-3423	2	6	,	,	PUNCT
ejpam-3423	2	7	2019	2019	NUM
ejpam-3423	2	8	,	,	PUNCT
ejpam-3423	2	9	1215	1215	NUM
ejpam-3423	2	10	-	-	SYM
ejpam-3423	2	11	1230	1230	NUM
ejpam-3423	2	12	issn	issn	PROPN
ejpam-3423	2	13	1307	1307	NUM
ejpam-3423	2	14	-	-	SYM
ejpam-3423	2	15	5543	5543	NUM
ejpam-3423	2	16	–	–	PUNCT
ejpam-3423	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3423	2	18	published	publish	VERB
ejpam-3423	2	19	by	by	ADP
ejpam-3423	2	20	new	new	PROPN
ejpam-3423	2	21	york	york	PROPN
ejpam-3423	2	22	business	business	PROPN
ejpam-3423	2	23	global	global	ADJ
ejpam-3423	2	24	convergence	convergence	NOUN
ejpam-3423	2	25	of	of	ADP
ejpam-3423	2	26	an	an	DET
ejpam-3423	2	27	exponential	exponential	ADJ
ejpam-3423	2	28	runge	runge	NOUN
ejpam-3423	2	29	–	–	PUNCT
ejpam-3423	2	30	kutta	kutta	NOUN
ejpam-3423	2	31	method	method	NOUN
ejpam-3423	2	32	for	for	ADP
ejpam-3423	2	33	non	non	ADJ
ejpam-3423	2	34	-	-	ADJ
ejpam-3423	2	35	smooth	smooth	ADJ
ejpam-3423	2	36	initial	initial	ADJ
ejpam-3423	2	37	data	datum	NOUN
ejpam-3423	2	38	muhammad	muhammad	PROPN
ejpam-3423	2	39	asif	asif	PROPN
ejpam-3423	2	40	gondal1,∗	gondal1,∗	PROPN
ejpam-3423	2	41	,	,	PUNCT
ejpam-3423	2	42	inayatur	inayatur	PROPN
ejpam-3423	2	43	rehman1	rehman1	PROPN
ejpam-3423	2	44	,	,	PUNCT
ejpam-3423	2	45	asima	asima	PROPN
ejpam-3423	2	46	razzaque2	razzaque2	NOUN
ejpam-3423	2	47	1	1	NUM
ejpam-3423	2	48	department	department	NOUN
ejpam-3423	2	49	of	of	ADP
ejpam-3423	2	50	mathematics	mathematic	NOUN
ejpam-3423	2	51	and	and	CCONJ
ejpam-3423	2	52	sciences	science	NOUN
ejpam-3423	2	53	,	,	PUNCT
ejpam-3423	2	54	dhofar	dhofar	ADJ
ejpam-3423	2	55	university	university	PROPN
ejpam-3423	2	56	,	,	PUNCT
ejpam-3423	2	57	salalah	salalah	PROPN
ejpam-3423	2	58	,	,	PUNCT
ejpam-3423	2	59	oman	oman	PROPN
ejpam-3423	2	60	2	2	NUM
ejpam-3423	2	61	department	department	NOUN
ejpam-3423	2	62	of	of	ADP
ejpam-3423	2	63	mathematics	mathematic	NOUN
ejpam-3423	2	64	,	,	PUNCT
ejpam-3423	2	65	university	university	NOUN
ejpam-3423	2	66	of	of	ADP
ejpam-3423	2	67	education	education	NOUN
ejpam-3423	2	68	,	,	PUNCT
ejpam-3423	2	69	lahore	lahore	NOUN
ejpam-3423	2	70	,	,	PUNCT
ejpam-3423	2	71	pakistan	pakistan	PROPN
ejpam-3423	2	72	abstract	abstract	NOUN
ejpam-3423	2	73	.	.	PUNCT
ejpam-3423	3	1	the	the	DET
ejpam-3423	3	2	paper	paper	NOUN
ejpam-3423	3	3	presents	present	VERB
ejpam-3423	3	4	error	error	NOUN
ejpam-3423	3	5	bounds	bound	NOUN
ejpam-3423	3	6	for	for	ADP
ejpam-3423	3	7	the	the	DET
ejpam-3423	3	8	second	second	ADJ
ejpam-3423	3	9	order	order	NOUN
ejpam-3423	3	10	exponential	exponential	ADJ
ejpam-3423	3	11	runge	runge	NOUN
ejpam-3423	3	12	-	-	PUNCT
ejpam-3423	3	13	kutta	kutta	NOUN
ejpam-3423	3	14	method	method	NOUN
ejpam-3423	3	15	for	for	ADP
ejpam-3423	3	16	parabolic	parabolic	ADJ
ejpam-3423	3	17	abstract	abstract	ADJ
ejpam-3423	3	18	linear	linear	ADJ
ejpam-3423	3	19	time	time	NOUN
ejpam-3423	3	20	-	-	PUNCT
ejpam-3423	3	21	dependent	dependent	ADJ
ejpam-3423	3	22	differential	differential	ADJ
ejpam-3423	3	23	equations	equation	NOUN
ejpam-3423	3	24	incorporating	incorporate	VERB
ejpam-3423	3	25	non	non	ADJ
ejpam-3423	3	26	-	-	ADJ
ejpam-3423	3	27	smooth	smooth	ADJ
ejpam-3423	3	28	initial	initial	ADJ
ejpam-3423	3	29	data	datum	NOUN
ejpam-3423	3	30	.	.	PUNCT
ejpam-3423	4	1	as	as	ADP
ejpam-3423	4	2	an	an	DET
ejpam-3423	4	3	example	example	NOUN
ejpam-3423	4	4	for	for	ADP
ejpam-3423	4	5	this	this	DET
ejpam-3423	4	6	particular	particular	ADJ
ejpam-3423	4	7	type	type	NOUN
ejpam-3423	4	8	of	of	ADP
ejpam-3423	4	9	problems	problem	NOUN
ejpam-3423	4	10	,	,	PUNCT
ejpam-3423	4	11	the	the	DET
ejpam-3423	4	12	paper	paper	NOUN
ejpam-3423	4	13	presents	present	VERB
ejpam-3423	4	14	a	a	DET
ejpam-3423	4	15	spatial	spatial	ADJ
ejpam-3423	4	16	discretization	discretization	NOUN
ejpam-3423	4	17	of	of	ADP
ejpam-3423	4	18	a	a	DET
ejpam-3423	4	19	partial	partial	ADJ
ejpam-3423	4	20	integro	integro	ADJ
ejpam-3423	4	21	-	-	PUNCT
ejpam-3423	4	22	differential	differential	NOUN
ejpam-3423	4	23	equation	equation	NOUN
ejpam-3423	4	24	arising	arise	VERB
ejpam-3423	4	25	in	in	ADP
ejpam-3423	4	26	financial	financial	ADJ
ejpam-3423	4	27	mathematics	mathematic	NOUN
ejpam-3423	4	28	,	,	PUNCT
ejpam-3423	4	29	where	where	SCONJ
ejpam-3423	4	30	non	non	ADJ
ejpam-3423	4	31	-	-	ADJ
ejpam-3423	4	32	smooth	smooth	ADJ
ejpam-3423	4	33	initial	initial	ADJ
ejpam-3423	4	34	conditions	condition	NOUN
ejpam-3423	4	35	occur	occur	VERB
ejpam-3423	4	36	in	in	ADP
ejpam-3423	4	37	option	option	NOUN
ejpam-3423	4	38	pricing	pricing	NOUN
ejpam-3423	4	39	models	model	NOUN
ejpam-3423	4	40	.	.	PUNCT
ejpam-3423	5	1	for	for	ADP
ejpam-3423	5	2	this	this	DET
ejpam-3423	5	3	example	example	NOUN
ejpam-3423	5	4	,	,	PUNCT
ejpam-3423	5	5	numerical	numerical	ADJ
ejpam-3423	5	6	studies	study	NOUN
ejpam-3423	5	7	of	of	ADP
ejpam-3423	5	8	the	the	DET
ejpam-3423	5	9	convergence	convergence	NOUN
ejpam-3423	5	10	rate	rate	NOUN
ejpam-3423	5	11	are	be	AUX
ejpam-3423	5	12	given	give	VERB
ejpam-3423	5	13	.	.	PUNCT
ejpam-3423	6	1	2010	2010	NUM
ejpam-3423	6	2	mathematics	mathematic	NOUN
ejpam-3423	6	3	subject	subject	NOUN
ejpam-3423	6	4	classifications	classification	NOUN
ejpam-3423	6	5	:	:	PUNCT
ejpam-3423	6	6	35k90	35k90	NUM
ejpam-3423	6	7	key	key	ADJ
ejpam-3423	6	8	words	word	NOUN
ejpam-3423	6	9	and	and	CCONJ
ejpam-3423	6	10	phrases	phrase	NOUN
ejpam-3423	6	11	:	:	PUNCT
ejpam-3423	6	12	exponential	exponential	ADJ
ejpam-3423	6	13	integrators	integrator	NOUN
ejpam-3423	6	14	,	,	PUNCT
ejpam-3423	6	15	runge	runge	NOUN
ejpam-3423	6	16	–	–	PUNCT
ejpam-3423	6	17	kutta	kutta	NOUN
ejpam-3423	6	18	methods	method	NOUN
ejpam-3423	6	19	,	,	PUNCT
ejpam-3423	6	20	integro	integro	ADJ
ejpam-3423	6	21	-	-	PUNCT
ejpam-3423	6	22	differential	differential	NOUN
ejpam-3423	6	23	equations	equation	NOUN
ejpam-3423	6	24	1	1	NUM
ejpam-3423	6	25	.	.	PUNCT
ejpam-3423	7	1	introduction	introduction	NOUN
ejpam-3423	7	2	to	to	PART
ejpam-3423	7	3	give	give	VERB
ejpam-3423	7	4	numerical	numerical	ADJ
ejpam-3423	7	5	solution	solution	NOUN
ejpam-3423	7	6	of	of	ADP
ejpam-3423	7	7	stiff	stiff	ADJ
ejpam-3423	7	8	differential	differential	ADJ
ejpam-3423	7	9	equations	equation	NOUN
ejpam-3423	7	10	,	,	PUNCT
ejpam-3423	7	11	exponential	exponential	ADJ
ejpam-3423	7	12	integrators	integrator	NOUN
ejpam-3423	7	13	have	have	AUX
ejpam-3423	7	14	been	be	AUX
ejpam-3423	7	15	constructed	construct	VERB
ejpam-3423	7	16	.	.	PUNCT
ejpam-3423	8	1	through	through	ADP
ejpam-3423	8	2	exponential	exponential	ADJ
ejpam-3423	8	3	integrators	integrator	NOUN
ejpam-3423	8	4	,	,	PUNCT
ejpam-3423	8	5	unlike	unlike	ADP
ejpam-3423	8	6	standard	standard	ADJ
ejpam-3423	8	7	numerical	numerical	ADJ
ejpam-3423	8	8	integrators	integrator	NOUN
ejpam-3423	8	9	,	,	PUNCT
ejpam-3423	8	10	the	the	DET
ejpam-3423	8	11	exponential	exponential	ADJ
ejpam-3423	8	12	and	and	CCONJ
ejpam-3423	8	13	related	related	ADJ
ejpam-3423	8	14	functions	function	NOUN
ejpam-3423	8	15	(	(	PUNCT
ejpam-3423	8	16	often	often	ADV
ejpam-3423	8	17	called	call	VERB
ejpam-3423	8	18	ϕ-functions	ϕ-function	NOUN
ejpam-3423	8	19	)	)	PUNCT
ejpam-3423	8	20	of	of	ADP
ejpam-3423	8	21	large	large	ADJ
ejpam-3423	8	22	matrices	matrix	NOUN
ejpam-3423	8	23	can	can	AUX
ejpam-3423	8	24	be	be	AUX
ejpam-3423	8	25	used	use	VERB
ejpam-3423	8	26	explicitly	explicitly	ADV
ejpam-3423	8	27	.	.	PUNCT
ejpam-3423	9	1	the	the	DET
ejpam-3423	9	2	exponential	exponential	ADJ
ejpam-3423	9	3	runge	runge	NOUN
ejpam-3423	9	4	-	-	PUNCT
ejpam-3423	9	5	kutta	kutta	NOUN
ejpam-3423	9	6	methods	method	NOUN
ejpam-3423	9	7	of	of	ADP
ejpam-3423	9	8	collocation	collocation	NOUN
ejpam-3423	9	9	type	type	NOUN
ejpam-3423	9	10	have	have	AUX
ejpam-3423	9	11	been	be	AUX
ejpam-3423	9	12	constructed	construct	VERB
ejpam-3423	9	13	by	by	ADP
ejpam-3423	9	14	hochbruck	hochbruck	NOUN
ejpam-3423	9	15	&	&	CCONJ
ejpam-3423	9	16	ostermann	ostermann	PROPN
ejpam-3423	10	1	[	[	X
ejpam-3423	10	2	9	9	NUM
ejpam-3423	10	3	]	]	PUNCT
ejpam-3423	10	4	and	and	CCONJ
ejpam-3423	10	5	their	their	PRON
ejpam-3423	10	6	convergence	convergence	NOUN
ejpam-3423	10	7	properties	property	NOUN
ejpam-3423	10	8	were	be	AUX
ejpam-3423	10	9	analyzed	analyze	VERB
ejpam-3423	10	10	for	for	ADP
ejpam-3423	10	11	linear	linear	ADJ
ejpam-3423	10	12	and	and	CCONJ
ejpam-3423	10	13	semi	semi	ADJ
ejpam-3423	10	14	-	-	ADJ
ejpam-3423	10	15	linear	linear	ADJ
ejpam-3423	10	16	parabolic	parabolic	NOUN
ejpam-3423	10	17	problems	problem	NOUN
ejpam-3423	10	18	.	.	PUNCT
ejpam-3423	11	1	hochbruck	hochbruck	NOUN
ejpam-3423	11	2	&	&	CCONJ
ejpam-3423	11	3	ostermann	ostermann	PROPN
ejpam-3423	12	1	[	[	X
ejpam-3423	12	2	8	8	NUM
ejpam-3423	12	3	]	]	PUNCT
ejpam-3423	12	4	also	also	ADV
ejpam-3423	12	5	studied	study	VERB
ejpam-3423	12	6	explicit	explicit	ADJ
ejpam-3423	12	7	exponential	exponential	ADJ
ejpam-3423	12	8	rung	rung	NOUN
ejpam-3423	12	9	-	-	PUNCT
ejpam-3423	12	10	kutta	kutta	NOUN
ejpam-3423	12	11	methods	method	NOUN
ejpam-3423	12	12	for	for	ADP
ejpam-3423	12	13	the	the	DET
ejpam-3423	12	14	time	time	NOUN
ejpam-3423	12	15	integration	integration	NOUN
ejpam-3423	12	16	of	of	ADP
ejpam-3423	12	17	semi	semi	ADJ
ejpam-3423	12	18	-	-	ADJ
ejpam-3423	12	19	linear	linear	ADJ
ejpam-3423	12	20	parabolic	parabolic	NOUN
ejpam-3423	12	21	problems	problem	NOUN
ejpam-3423	12	22	.	.	PUNCT
ejpam-3423	13	1	gondal	gondal	NOUN
ejpam-3423	13	2	[	[	X
ejpam-3423	13	3	4	4	NUM
ejpam-3423	13	4	]	]	PUNCT
ejpam-3423	13	5	considered	consider	VERB
ejpam-3423	13	6	exponential	exponential	ADJ
ejpam-3423	13	7	rosenbrock	rosenbrock	NOUN
ejpam-3423	13	8	integrators	integrator	NOUN
ejpam-3423	13	9	for	for	ADP
ejpam-3423	13	10	option	option	NOUN
ejpam-3423	13	11	pricing	pricing	NOUN
ejpam-3423	13	12	.	.	PUNCT
ejpam-3423	14	1	different	different	ADJ
ejpam-3423	14	2	types	type	NOUN
ejpam-3423	14	3	of	of	ADP
ejpam-3423	14	4	exponential	exponential	ADJ
ejpam-3423	14	5	integrators	integrator	NOUN
ejpam-3423	14	6	and	and	CCONJ
ejpam-3423	14	7	their	their	PRON
ejpam-3423	14	8	applications	application	NOUN
ejpam-3423	14	9	are	be	AUX
ejpam-3423	14	10	discussed	discuss	VERB
ejpam-3423	14	11	in	in	ADP
ejpam-3423	14	12	details	detail	NOUN
ejpam-3423	14	13	in	in	ADP
ejpam-3423	14	14	[	[	X
ejpam-3423	14	15	6	6	NUM
ejpam-3423	14	16	,	,	PUNCT
ejpam-3423	14	17	10	10	NUM
ejpam-3423	14	18	,	,	PUNCT
ejpam-3423	14	19	11	11	NUM
ejpam-3423	14	20	,	,	PUNCT
ejpam-3423	14	21	18	18	NUM
ejpam-3423	14	22	,	,	PUNCT
ejpam-3423	14	23	19	19	NUM
ejpam-3423	14	24	]	]	PUNCT
ejpam-3423	14	25	.	.	PUNCT
ejpam-3423	15	1	henry	henry	PROPN
ejpam-3423	16	1	[	[	X
ejpam-3423	16	2	7	7	NUM
ejpam-3423	16	3	]	]	PUNCT
ejpam-3423	16	4	and	and	CCONJ
ejpam-3423	16	5	pazy	pazy	NOUN
ejpam-3423	16	6	[	[	X
ejpam-3423	16	7	16	16	NUM
ejpam-3423	16	8	]	]	PUNCT
ejpam-3423	16	9	studied	study	VERB
ejpam-3423	16	10	semi	semi	ADJ
ejpam-3423	16	11	-	-	ADJ
ejpam-3423	16	12	linear	linear	ADJ
ejpam-3423	16	13	problems	problem	NOUN
ejpam-3423	16	14	and	and	CCONJ
ejpam-3423	16	15	contributed	contribute	VERB
ejpam-3423	16	16	significantly	significantly	ADV
ejpam-3423	16	17	.	.	PUNCT
ejpam-3423	17	1	le	le	AUX
ejpam-3423	17	2	roux	roux	VERB
ejpam-3423	17	3	[	[	X
ejpam-3423	17	4	17	17	NUM
ejpam-3423	17	5	]	]	PUNCT
ejpam-3423	17	6	introduced	introduce	VERB
ejpam-3423	17	7	for	for	ADP
ejpam-3423	17	8	the	the	DET
ejpam-3423	17	9	first	first	ADJ
ejpam-3423	17	10	time	time	NOUN
ejpam-3423	17	11	non	non	ADJ
ejpam-3423	17	12	-	-	ADJ
ejpam-3423	17	13	smooth	smooth	ADJ
ejpam-3423	17	14	data	datum	NOUN
ejpam-3423	17	15	error	error	NOUN
ejpam-3423	17	16	estimates	estimate	NOUN
ejpam-3423	17	17	for	for	ADP
ejpam-3423	17	18	time	time	NOUN
ejpam-3423	17	19	discretizations	discretization	NOUN
ejpam-3423	17	20	of	of	ADP
ejpam-3423	17	21	linear	linear	ADJ
ejpam-3423	17	22	parabolic	parabolic	NOUN
ejpam-3423	17	23	problems	problem	NOUN
ejpam-3423	17	24	.	.	PUNCT
ejpam-3423	18	1	the	the	DET
ejpam-3423	18	2	error	error	NOUN
ejpam-3423	18	3	bounds	bound	VERB
ejpam-3423	18	4	for	for	ADP
ejpam-3423	18	5	time	time	NOUN
ejpam-3423	18	6	discretizations	discretization	NOUN
ejpam-3423	18	7	of	of	ADP
ejpam-3423	18	8	∗corresponding	∗corresponde	VERB
ejpam-3423	18	9	author	author	NOUN
ejpam-3423	18	10	.	.	PUNCT
ejpam-3423	19	1	doi	doi	NOUN
ejpam-3423	19	2	:	:	PUNCT
ejpam-3423	19	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3423	https://doi.org/10.29020/nybg.ejpam.v12i3.3423	NOUN
ejpam-3423	19	4	email	email	NOUN
ejpam-3423	19	5	addresses	address	NOUN
ejpam-3423	19	6	:	:	PUNCT
ejpam-3423	19	7	mgondal@du.edu.om	mgondal@du.edu.om	NOUN
ejpam-3423	19	8	(	(	PUNCT
ejpam-3423	19	9	m.	m.	NOUN
ejpam-3423	19	10	a.	a.	NOUN
ejpam-3423	19	11	gondal	gondal	PROPN
ejpam-3423	19	12	)	)	PUNCT
ejpam-3423	19	13	,	,	PUNCT
ejpam-3423	19	14	irehman@du.edu.om	irehman@du.edu.om	PROPN
ejpam-3423	19	15	(	(	PUNCT
ejpam-3423	19	16	i.	i.	PROPN
ejpam-3423	19	17	rehman	rehman	PROPN
ejpam-3423	19	18	)	)	PUNCT
ejpam-3423	19	19	,	,	PUNCT
ejpam-3423	19	20	asima.razzaque@yahoo.com	asima.razzaque@yahoo.com	X
ejpam-3423	19	21	(	(	PUNCT
ejpam-3423	19	22	a.	a.	NOUN
ejpam-3423	19	23	razzaque	razzaque	NOUN
ejpam-3423	19	24	)	)	PUNCT
ejpam-3423	19	25	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3423	20	1	1215	1215	NUM
ejpam-3423	20	2	c	c	X
ejpam-3423	20	3	©	©	PROPN
ejpam-3423	20	4	2019	2019	NUM
ejpam-3423	20	5	ejpam	ejpam	NOUN
ejpam-3423	20	6	all	all	DET
ejpam-3423	20	7	rights	right	NOUN
ejpam-3423	20	8	reserved	reserve	VERB
ejpam-3423	20	9	.	.	PUNCT
ejpam-3423	21	1	m.	m.	NOUN
ejpam-3423	21	2	a.	a.	PROPN
ejpam-3423	21	3	gondal	gondal	PROPN
ejpam-3423	21	4	,	,	PUNCT
ejpam-3423	21	5	i.	i.	PROPN
ejpam-3423	21	6	rehman	rehman	PROPN
ejpam-3423	21	7	,	,	PUNCT
ejpam-3423	21	8	a.	a.	NOUN
ejpam-3423	21	9	razzaque	razzaque	NOUN
ejpam-3423	21	10	/	/	SYM
ejpam-3423	21	11	eur	eur	NOUN
ejpam-3423	21	12	.	.	PUNCT
ejpam-3423	22	1	j.	j.	PROPN
ejpam-3423	22	2	pure	pure	PROPN
ejpam-3423	22	3	appl	appl	PROPN
ejpam-3423	22	4	.	.	PROPN
ejpam-3423	22	5	math	math	PROPN
ejpam-3423	22	6	,	,	PUNCT
ejpam-3423	22	7	12	12	NUM
ejpam-3423	22	8	(	(	PUNCT
ejpam-3423	22	9	3	3	NUM
ejpam-3423	22	10	)	)	PUNCT
ejpam-3423	22	11	(	(	PUNCT
ejpam-3423	22	12	2019	2019	NUM
ejpam-3423	22	13	)	)	PUNCT
ejpam-3423	22	14	,	,	PUNCT
ejpam-3423	22	15	1215	1215	NUM
ejpam-3423	22	16	-	-	SYM
ejpam-3423	22	17	1230	1230	NUM
ejpam-3423	22	18	1216	1216	NUM
ejpam-3423	22	19	semi	semi	ADJ
ejpam-3423	22	20	-	-	ADJ
ejpam-3423	22	21	linear	linear	ADJ
ejpam-3423	22	22	parabolic	parabolic	ADJ
ejpam-3423	22	23	equations	equation	NOUN
ejpam-3423	22	24	with	with	ADP
ejpam-3423	22	25	non	non	ADJ
ejpam-3423	22	26	-	-	ADJ
ejpam-3423	22	27	smooth	smooth	ADJ
ejpam-3423	22	28	initial	initial	ADJ
ejpam-3423	22	29	data	datum	NOUN
ejpam-3423	22	30	have	have	AUX
ejpam-3423	22	31	been	be	AUX
ejpam-3423	22	32	inferred	infer	VERB
ejpam-3423	22	33	in	in	ADP
ejpam-3423	22	34	[	[	X
ejpam-3423	22	35	12	12	NUM
ejpam-3423	22	36	]	]	PUNCT
ejpam-3423	22	37	.	.	PUNCT
ejpam-3423	23	1	linearly	linearly	ADV
ejpam-3423	23	2	implicit	implicit	ADJ
ejpam-3423	23	3	time	time	NOUN
ejpam-3423	23	4	discretization	discretization	NOUN
ejpam-3423	23	5	of	of	ADP
ejpam-3423	23	6	semi	semi	ADJ
ejpam-3423	23	7	-	-	ADJ
ejpam-3423	23	8	linear	linear	ADJ
ejpam-3423	23	9	parabolic	parabolic	ADJ
ejpam-3423	23	10	equations	equation	NOUN
ejpam-3423	23	11	with	with	ADP
ejpam-3423	23	12	non	non	ADJ
ejpam-3423	23	13	-	-	ADJ
ejpam-3423	23	14	smooth	smooth	ADJ
ejpam-3423	23	15	initial	initial	ADJ
ejpam-3423	23	16	data	datum	NOUN
ejpam-3423	23	17	was	be	AUX
ejpam-3423	23	18	studied	study	VERB
ejpam-3423	23	19	by	by	ADP
ejpam-3423	23	20	the	the	DET
ejpam-3423	23	21	authors	author	NOUN
ejpam-3423	23	22	in	in	ADP
ejpam-3423	23	23	[	[	X
ejpam-3423	23	24	15	15	NUM
ejpam-3423	23	25	]	]	PUNCT
ejpam-3423	23	26	.	.	PUNCT
ejpam-3423	24	1	in	in	ADP
ejpam-3423	24	2	[	[	X
ejpam-3423	24	3	5	5	NUM
ejpam-3423	24	4	]	]	PUNCT
ejpam-3423	24	5	,	,	PUNCT
ejpam-3423	24	6	author	author	NOUN
ejpam-3423	24	7	proved	prove	VERB
ejpam-3423	24	8	convergence	convergence	NOUN
ejpam-3423	24	9	result	result	NOUN
ejpam-3423	24	10	of	of	ADP
ejpam-3423	24	11	an	an	DET
ejpam-3423	24	12	exponential	exponential	NOUN
ejpam-3423	24	13	euler	euler	NOUN
ejpam-3423	24	14	method	method	NOUN
ejpam-3423	24	15	using	use	VERB
ejpam-3423	24	16	non	non	ADJ
ejpam-3423	24	17	-	-	ADJ
ejpam-3423	24	18	smooth	smooth	ADJ
ejpam-3423	24	19	initial	initial	ADJ
ejpam-3423	24	20	data	datum	NOUN
ejpam-3423	24	21	for	for	ADP
ejpam-3423	24	22	option	option	NOUN
ejpam-3423	24	23	pricing	pricing	NOUN
ejpam-3423	24	24	.	.	PUNCT
ejpam-3423	25	1	we	we	PRON
ejpam-3423	25	2	mean	mean	VERB
ejpam-3423	25	3	to	to	PART
ejpam-3423	25	4	examine	examine	VERB
ejpam-3423	25	5	convergence	convergence	NOUN
ejpam-3423	25	6	properties	property	NOUN
ejpam-3423	25	7	of	of	ADP
ejpam-3423	25	8	exponential	exponential	ADJ
ejpam-3423	25	9	runge	runge	NOUN
ejpam-3423	25	10	-	-	PUNCT
ejpam-3423	25	11	kutta	kutta	NOUN
ejpam-3423	25	12	method	method	NOUN
ejpam-3423	25	13	for	for	ADP
ejpam-3423	25	14	linear	linear	PROPN
ejpam-3423	25	15	parabolic	parabolic	ADJ
ejpam-3423	25	16	problems	problem	NOUN
ejpam-3423	25	17	that	that	PRON
ejpam-3423	25	18	spring	spring	VERB
ejpam-3423	25	19	up	up	ADP
ejpam-3423	25	20	in	in	ADP
ejpam-3423	25	21	financial	financial	ADJ
ejpam-3423	25	22	problems	problem	NOUN
ejpam-3423	25	23	.	.	PUNCT
ejpam-3423	26	1	to	to	PART
ejpam-3423	26	2	evaluate	evaluate	VERB
ejpam-3423	26	3	this	this	PRON
ejpam-3423	26	4	,	,	PUNCT
ejpam-3423	26	5	we	we	PRON
ejpam-3423	26	6	cultivate	cultivate	VERB
ejpam-3423	26	7	in	in	ADP
ejpam-3423	26	8	an	an	DET
ejpam-3423	26	9	abstract	abstract	ADJ
ejpam-3423	26	10	banach	banach	NOUN
ejpam-3423	26	11	space	space	NOUN
ejpam-3423	26	12	framework	framework	NOUN
ejpam-3423	26	13	of	of	ADP
ejpam-3423	26	14	sectorial	sectorial	ADJ
ejpam-3423	26	15	operators	operator	NOUN
ejpam-3423	26	16	and	and	CCONJ
ejpam-3423	26	17	analytic	analytic	ADJ
ejpam-3423	26	18	semi	semi	NOUN
ejpam-3423	26	19	-	-	NOUN
ejpam-3423	26	20	groups	group	NOUN
ejpam-3423	26	21	and	and	CCONJ
ejpam-3423	26	22	prove	prove	VERB
ejpam-3423	26	23	convergence	convergence	NOUN
ejpam-3423	26	24	for	for	ADP
ejpam-3423	26	25	exponential	exponential	ADJ
ejpam-3423	26	26	runge	runge	NOUN
ejpam-3423	26	27	-	-	PUNCT
ejpam-3423	26	28	kutta	kutta	NOUN
ejpam-3423	26	29	method	method	NOUN
ejpam-3423	26	30	for	for	ADP
ejpam-3423	26	31	non	non	ADJ
ejpam-3423	26	32	-	-	ADJ
ejpam-3423	26	33	smooth	smooth	ADJ
ejpam-3423	26	34	initial	initial	ADJ
ejpam-3423	26	35	data	datum	NOUN
ejpam-3423	26	36	.	.	PUNCT
ejpam-3423	27	1	the	the	DET
ejpam-3423	27	2	jump	jump	NOUN
ejpam-3423	27	3	diffusion	diffusion	NOUN
ejpam-3423	27	4	model	model	NOUN
ejpam-3423	27	5	,	,	PUNCT
ejpam-3423	27	6	proposed	propose	VERB
ejpam-3423	27	7	in	in	ADP
ejpam-3423	27	8	[	[	X
ejpam-3423	27	9	14	14	NUM
ejpam-3423	27	10	]	]	PUNCT
ejpam-3423	27	11	,	,	PUNCT
ejpam-3423	27	12	is	be	AUX
ejpam-3423	27	13	chosen	choose	VERB
ejpam-3423	27	14	as	as	ADP
ejpam-3423	27	15	an	an	DET
ejpam-3423	27	16	application	application	NOUN
ejpam-3423	27	17	of	of	ADP
ejpam-3423	27	18	our	our	PRON
ejpam-3423	27	19	analysis	analysis	NOUN
ejpam-3423	27	20	.	.	PUNCT
ejpam-3423	28	1	in	in	ADP
ejpam-3423	28	2	particular	particular	ADJ
ejpam-3423	28	3	,	,	PUNCT
ejpam-3423	28	4	we	we	PRON
ejpam-3423	28	5	discuss	discuss	VERB
ejpam-3423	28	6	the	the	DET
ejpam-3423	28	7	partial	partial	ADJ
ejpam-3423	28	8	integro	integro	ADJ
ejpam-3423	28	9	-	-	PUNCT
ejpam-3423	28	10	differential	differential	NOUN
ejpam-3423	28	11	equations(pide	equations(pide	NOUN
ejpam-3423	28	12	)	)	PUNCT
ejpam-3423	28	13	for	for	ADP
ejpam-3423	28	14	mertons	merton	NOUN
ejpam-3423	28	15	model	model	NOUN
ejpam-3423	28	16	.	.	PUNCT
ejpam-3423	29	1	briani	briani	PROPN
ejpam-3423	29	2	,	,	PUNCT
ejpam-3423	29	3	la	la	PROPN
ejpam-3423	29	4	chioma	chioma	PROPN
ejpam-3423	29	5	&	&	CCONJ
ejpam-3423	29	6	natalini	natalini	PROPN
ejpam-3423	30	1	[	[	X
ejpam-3423	30	2	13	13	NUM
ejpam-3423	30	3	]	]	PUNCT
ejpam-3423	30	4	used	use	VERB
ejpam-3423	30	5	an	an	DET
ejpam-3423	30	6	explicit	explicit	ADJ
ejpam-3423	30	7	method	method	NOUN
ejpam-3423	30	8	to	to	PART
ejpam-3423	30	9	solve	solve	VERB
ejpam-3423	30	10	mertons	merton	NOUN
ejpam-3423	30	11	model	model	NOUN
ejpam-3423	30	12	and	and	CCONJ
ejpam-3423	30	13	constituted	constitute	VERB
ejpam-3423	30	14	a	a	DET
ejpam-3423	30	15	convergence	convergence	NOUN
ejpam-3423	30	16	theory	theory	NOUN
ejpam-3423	30	17	for	for	ADP
ejpam-3423	30	18	explicit	explicit	ADJ
ejpam-3423	30	19	schemes	scheme	NOUN
ejpam-3423	30	20	for	for	ADP
ejpam-3423	30	21	varied	varied	ADJ
ejpam-3423	30	22	integro	integro	ADJ
ejpam-3423	30	23	-	-	PUNCT
ejpam-3423	30	24	differential	differential	NOUN
ejpam-3423	30	25	cauchy	cauchy	NOUN
ejpam-3423	30	26	problems	problem	NOUN
ejpam-3423	30	27	.	.	PUNCT
ejpam-3423	31	1	cont	cont	PROPN
ejpam-3423	31	2	&	&	CCONJ
ejpam-3423	31	3	voltchkova	voltchkova	VERB
ejpam-3423	32	1	[	[	X
ejpam-3423	32	2	2	2	X
ejpam-3423	32	3	]	]	PUNCT
ejpam-3423	32	4	used	use	VERB
ejpam-3423	32	5	implicit	implicit	ADJ
ejpam-3423	32	6	-	-	PUNCT
ejpam-3423	32	7	explicit	explicit	ADJ
ejpam-3423	32	8	finite	finite	ADJ
ejpam-3423	32	9	difference	difference	NOUN
ejpam-3423	32	10	methods	method	NOUN
ejpam-3423	32	11	successfully	successfully	ADV
ejpam-3423	32	12	for	for	ADP
ejpam-3423	32	13	european	european	ADJ
ejpam-3423	32	14	and	and	CCONJ
ejpam-3423	32	15	barrier	barrier	NOUN
ejpam-3423	32	16	options	option	NOUN
ejpam-3423	32	17	in	in	ADP
ejpam-3423	32	18	jump	jump	NOUN
ejpam-3423	32	19	diffusion	diffusion	NOUN
ejpam-3423	32	20	and	and	CCONJ
ejpam-3423	32	21	exponential	exponential	ADJ
ejpam-3423	32	22	levy	levy	NOUN
ejpam-3423	32	23	models	model	NOUN
ejpam-3423	32	24	.	.	PUNCT
ejpam-3423	33	1	in	in	ADP
ejpam-3423	33	2	a	a	DET
ejpam-3423	33	3	paper	paper	NOUN
ejpam-3423	33	4	[	[	X
ejpam-3423	33	5	3	3	NUM
ejpam-3423	33	6	]	]	PUNCT
ejpam-3423	33	7	,	,	PUNCT
ejpam-3423	33	8	one	one	PRON
ejpam-3423	33	9	can	can	AUX
ejpam-3423	33	10	find	find	VERB
ejpam-3423	33	11	different	different	ADJ
ejpam-3423	33	12	option	option	NOUN
ejpam-3423	33	13	pricing	pricing	NOUN
ejpam-3423	33	14	problems	problem	NOUN
ejpam-3423	33	15	solved	solve	VERB
ejpam-3423	33	16	numerically	numerically	ADV
ejpam-3423	33	17	through	through	ADP
ejpam-3423	33	18	chebychev	chebychev	NOUN
ejpam-3423	33	19	discretisation	discretisation	NOUN
ejpam-3423	33	20	schemes	scheme	NOUN
ejpam-3423	33	21	and	and	CCONJ
ejpam-3423	33	22	exponential	exponential	ADJ
ejpam-3423	33	23	integrators	integrator	NOUN
ejpam-3423	33	24	.	.	PUNCT
ejpam-3423	34	1	this	this	DET
ejpam-3423	34	2	article	article	NOUN
ejpam-3423	34	3	is	be	AUX
ejpam-3423	34	4	attributed	attribute	VERB
ejpam-3423	34	5	to	to	ADP
ejpam-3423	34	6	a	a	DET
ejpam-3423	34	7	theoretical	theoretical	ADJ
ejpam-3423	34	8	convergence	convergence	NOUN
ejpam-3423	34	9	analysis	analysis	NOUN
ejpam-3423	34	10	of	of	ADP
ejpam-3423	34	11	exponential	exponential	ADJ
ejpam-3423	34	12	integrators	integrator	NOUN
ejpam-3423	34	13	which	which	PRON
ejpam-3423	34	14	is	be	AUX
ejpam-3423	34	15	transported	transport	VERB
ejpam-3423	34	16	out	out	ADP
ejpam-3423	34	17	within	within	ADP
ejpam-3423	34	18	the	the	DET
ejpam-3423	34	19	framework	framework	NOUN
ejpam-3423	34	20	of	of	ADP
ejpam-3423	34	21	evolution	evolution	NOUN
ejpam-3423	34	22	equations	equation	NOUN
ejpam-3423	34	23	in	in	ADP
ejpam-3423	34	24	banach	banach	NOUN
ejpam-3423	34	25	spaces.in	spaces.in	X
ejpam-3423	34	26	financial	financial	ADJ
ejpam-3423	34	27	applications	application	NOUN
ejpam-3423	34	28	,	,	PUNCT
ejpam-3423	34	29	the	the	DET
ejpam-3423	34	30	initial	initial	ADJ
ejpam-3423	34	31	information	information	NOUN
ejpam-3423	34	32	is	be	AUX
ejpam-3423	34	33	generally	generally	ADV
ejpam-3423	34	34	non	non	ADJ
ejpam-3423	34	35	-	-	ADJ
ejpam-3423	34	36	smooth	smooth	ADJ
ejpam-3423	34	37	and	and	CCONJ
ejpam-3423	34	38	lies	lie	NOUN
ejpam-3423	34	39	of	of	ADP
ejpam-3423	34	40	the	the	DET
ejpam-3423	34	41	payoff	payoff	NOUN
ejpam-3423	34	42	function	function	NOUN
ejpam-3423	34	43	of	of	ADP
ejpam-3423	34	44	the	the	DET
ejpam-3423	34	45	option	option	NOUN
ejpam-3423	34	46	.	.	PUNCT
ejpam-3423	35	1	therefore	therefore	ADV
ejpam-3423	35	2	,	,	PUNCT
ejpam-3423	35	3	in	in	ADP
ejpam-3423	35	4	case	case	NOUN
ejpam-3423	35	5	of	of	ADP
ejpam-3423	35	6	non	non	ADJ
ejpam-3423	35	7	-	-	ADJ
ejpam-3423	35	8	smooth	smooth	ADJ
ejpam-3423	35	9	initial	initial	ADJ
ejpam-3423	35	10	data	datum	NOUN
ejpam-3423	35	11	,	,	PUNCT
ejpam-3423	35	12	the	the	DET
ejpam-3423	35	13	matter	matter	NOUN
ejpam-3423	35	14	of	of	ADP
ejpam-3423	35	15	concern	concern	NOUN
ejpam-3423	35	16	is	be	AUX
ejpam-3423	35	17	to	to	PART
ejpam-3423	35	18	have	have	VERB
ejpam-3423	35	19	practical	practical	ADJ
ejpam-3423	35	20	error	error	NOUN
ejpam-3423	35	21	bounds	bound	NOUN
ejpam-3423	35	22	.	.	PUNCT
ejpam-3423	36	1	a	a	DET
ejpam-3423	36	2	bound	bind	VERB
ejpam-3423	36	3	of	of	ADP
ejpam-3423	36	4	this	this	DET
ejpam-3423	36	5	nature	nature	NOUN
ejpam-3423	36	6	is	be	AUX
ejpam-3423	36	7	developed	develop	VERB
ejpam-3423	36	8	in	in	ADP
ejpam-3423	36	9	[	[	X
ejpam-3423	36	10	5	5	NUM
ejpam-3423	36	11	]	]	PUNCT
ejpam-3423	36	12	of	of	ADP
ejpam-3423	36	13	order	order	NOUN
ejpam-3423	36	14	one	one	NUM
ejpam-3423	36	15	for	for	ADP
ejpam-3423	36	16	the	the	DET
ejpam-3423	36	17	exponential	exponential	NOUN
ejpam-3423	36	18	euler	euler	NOUN
ejpam-3423	36	19	method	method	NOUN
ejpam-3423	36	20	.	.	PUNCT
ejpam-3423	37	1	following	follow	VERB
ejpam-3423	37	2	,	,	PUNCT
ejpam-3423	37	3	in	in	ADP
ejpam-3423	37	4	section	section	NOUN
ejpam-3423	37	5	3	3	NUM
ejpam-3423	37	6	,	,	PUNCT
ejpam-3423	37	7	a	a	DET
ejpam-3423	37	8	error	error	NOUN
ejpam-3423	37	9	bound	bind	VERB
ejpam-3423	37	10	is	be	AUX
ejpam-3423	37	11	established	establish	VERB
ejpam-3423	37	12	for	for	ADP
ejpam-3423	37	13	the	the	DET
ejpam-3423	37	14	method	method	NOUN
ejpam-3423	37	15	called	call	VERB
ejpam-3423	37	16	an	an	DET
ejpam-3423	37	17	exponential	exponential	ADJ
ejpam-3423	37	18	rungekutta	rungekutta	NOUN
ejpam-3423	37	19	method	method	NOUN
ejpam-3423	37	20	of	of	ADP
ejpam-3423	37	21	order	order	NOUN
ejpam-3423	37	22	two	two	NUM
ejpam-3423	37	23	,	,	PUNCT
ejpam-3423	37	24	and	and	CCONJ
ejpam-3423	37	25	the	the	DET
ejpam-3423	37	26	result	result	NOUN
ejpam-3423	37	27	is	be	AUX
ejpam-3423	37	28	given	give	VERB
ejpam-3423	37	29	in	in	ADP
ejpam-3423	37	30	theorem	theorem	NOUN
ejpam-3423	37	31	1	1	NUM
ejpam-3423	37	32	.	.	PUNCT
ejpam-3423	38	1	besides	besides	SCONJ
ejpam-3423	38	2	this	this	DET
ejpam-3423	38	3	preamble	preamble	NOUN
ejpam-3423	38	4	,	,	PUNCT
ejpam-3423	38	5	the	the	DET
ejpam-3423	38	6	paper	paper	NOUN
ejpam-3423	38	7	comprises	comprise	NOUN
ejpam-3423	38	8	of	of	ADP
ejpam-3423	38	9	four	four	NUM
ejpam-3423	38	10	sections	section	NOUN
ejpam-3423	38	11	.	.	PUNCT
ejpam-3423	39	1	section	section	NOUN
ejpam-3423	39	2	2	2	NUM
ejpam-3423	39	3	describes	describe	VERB
ejpam-3423	39	4	the	the	DET
ejpam-3423	39	5	exponential	exponential	ADJ
ejpam-3423	39	6	rung	rung	NOUN
ejpam-3423	39	7	-	-	PUNCT
ejpam-3423	39	8	kutta	kutta	PROPN
ejpam-3423	39	9	and	and	CCONJ
ejpam-3423	39	10	exponential	exponential	NOUN
ejpam-3423	39	11	euler	euler	NOUN
ejpam-3423	39	12	time	time	NOUN
ejpam-3423	39	13	integrators	integrator	NOUN
ejpam-3423	39	14	.	.	PUNCT
ejpam-3423	40	1	section	section	NOUN
ejpam-3423	40	2	3	3	NUM
ejpam-3423	40	3	present	present	VERB
ejpam-3423	40	4	the	the	DET
ejpam-3423	40	5	main	main	ADJ
ejpam-3423	40	6	results	result	NOUN
ejpam-3423	40	7	and	and	CCONJ
ejpam-3423	40	8	originate	originate	VERB
ejpam-3423	40	9	new	new	ADJ
ejpam-3423	40	10	error	error	NOUN
ejpam-3423	40	11	bounds	bound	NOUN
ejpam-3423	40	12	.	.	PUNCT
ejpam-3423	41	1	although	although	SCONJ
ejpam-3423	41	2	in	in	ADP
ejpam-3423	41	3	case	case	NOUN
ejpam-3423	41	4	of	of	ADP
ejpam-3423	41	5	non	non	ADJ
ejpam-3423	41	6	-	-	ADJ
ejpam-3423	41	7	smooth	smooth	ADJ
ejpam-3423	41	8	initial	initial	ADJ
ejpam-3423	41	9	data	datum	NOUN
ejpam-3423	41	10	,	,	PUNCT
ejpam-3423	41	11	error	error	NOUN
ejpam-3423	41	12	bounds	bound	NOUN
ejpam-3423	41	13	derived	derive	VERB
ejpam-3423	41	14	in	in	ADP
ejpam-3423	41	15	[	[	X
ejpam-3423	41	16	12	12	NUM
ejpam-3423	41	17	]	]	PUNCT
ejpam-3423	41	18	and	and	CCONJ
ejpam-3423	41	19	[	[	X
ejpam-3423	41	20	15	15	NUM
ejpam-3423	41	21	]	]	PUNCT
ejpam-3423	41	22	.	.	PUNCT
ejpam-3423	42	1	but	but	CCONJ
ejpam-3423	42	2	the	the	DET
ejpam-3423	42	3	results	result	NOUN
ejpam-3423	42	4	for	for	ADP
ejpam-3423	42	5	exponential	exponential	ADJ
ejpam-3423	42	6	integrators	integrator	NOUN
ejpam-3423	42	7	,	,	PUNCT
ejpam-3423	42	8	however	however	ADV
ejpam-3423	42	9	,	,	PUNCT
ejpam-3423	42	10	have	have	AUX
ejpam-3423	42	11	not	not	PART
ejpam-3423	42	12	been	be	AUX
ejpam-3423	42	13	experienced	experience	VERB
ejpam-3423	42	14	.	.	PUNCT
ejpam-3423	43	1	for	for	ADP
ejpam-3423	43	2	the	the	DET
ejpam-3423	43	3	application	application	NOUN
ejpam-3423	43	4	of	of	ADP
ejpam-3423	43	5	analysis	analysis	NOUN
ejpam-3423	43	6	,	,	PUNCT
ejpam-3423	43	7	section	section	NOUN
ejpam-3423	43	8	4	4	NUM
ejpam-3423	43	9	offers	offer	VERB
ejpam-3423	43	10	an	an	DET
ejpam-3423	43	11	example	example	NOUN
ejpam-3423	43	12	from	from	ADP
ejpam-3423	43	13	the	the	DET
ejpam-3423	43	14	mertons	merton	NOUN
ejpam-3423	43	15	models	model	NOUN
ejpam-3423	43	16	.	.	PUNCT
ejpam-3423	44	1	the	the	DET
ejpam-3423	44	2	conclusion	conclusion	NOUN
ejpam-3423	44	3	includes	include	VERB
ejpam-3423	44	4	few	few	ADJ
ejpam-3423	44	5	final	final	ADJ
ejpam-3423	44	6	remarks	remark	NOUN
ejpam-3423	44	7	.	.	PUNCT
ejpam-3423	45	1	2	2	X
ejpam-3423	45	2	.	.	X
ejpam-3423	45	3	numerical	numerical	ADJ
ejpam-3423	45	4	method	method	NOUN
ejpam-3423	45	5	in	in	ADP
ejpam-3423	45	6	this	this	DET
ejpam-3423	45	7	section	section	NOUN
ejpam-3423	45	8	,	,	PUNCT
ejpam-3423	45	9	the	the	DET
ejpam-3423	45	10	abstract	abstract	ADJ
ejpam-3423	45	11	form	form	NOUN
ejpam-3423	45	12	of	of	ADP
ejpam-3423	45	13	evaluation	evaluation	NOUN
ejpam-3423	45	14	equation	equation	NOUN
ejpam-3423	45	15	that	that	PRON
ejpam-3423	45	16	results	result	VERB
ejpam-3423	45	17	from	from	ADP
ejpam-3423	45	18	partial	partial	ADJ
ejpam-3423	45	19	integrodifferential	integrodifferential	ADJ
ejpam-3423	45	20	equations	equation	NOUN
ejpam-3423	45	21	,	,	PUNCT
ejpam-3423	45	22	that	that	PRON
ejpam-3423	45	23	arise	arise	VERB
ejpam-3423	45	24	in	in	ADP
ejpam-3423	45	25	financial	financial	ADJ
ejpam-3423	45	26	mathematics	mathematic	NOUN
ejpam-3423	45	27	,	,	PUNCT
ejpam-3423	45	28	is	be	AUX
ejpam-3423	45	29	considered	consider	VERB
ejpam-3423	45	30	as	as	SCONJ
ejpam-3423	45	31	follows	follow	VERB
ejpam-3423	45	32	:	:	PUNCT
ejpam-3423	45	33	u′(t	u′(t	X
ejpam-3423	45	34	)	)	PUNCT
ejpam-3423	46	1	=	=	SYM
ejpam-3423	46	2	au(t	au(t	X
ejpam-3423	46	3	)	)	PUNCT
ejpam-3423	47	1	+	+	NOUN
ejpam-3423	47	2	bu(t	bu(t	X
ejpam-3423	47	3	)	)	PUNCT
ejpam-3423	47	4	+	+	CCONJ
ejpam-3423	47	5	g(t	g(t	PROPN
ejpam-3423	47	6	)	)	PUNCT
ejpam-3423	47	7	,	,	PUNCT
ejpam-3423	47	8	u(t0	u(t0	NOUN
ejpam-3423	47	9	)	)	PUNCT
ejpam-3423	47	10	=	=	SYM
ejpam-3423	47	11	u0	u0	ADJ
ejpam-3423	47	12	,	,	PUNCT
ejpam-3423	47	13	0	0	PUNCT
ejpam-3423	47	14	<	<	X
ejpam-3423	47	15	t	t	X
ejpam-3423	47	16	≤	≤	PROPN
ejpam-3423	47	17	t	t	PROPN
ejpam-3423	47	18	,	,	PUNCT
ejpam-3423	47	19	(	(	PUNCT
ejpam-3423	47	20	1	1	X
ejpam-3423	47	21	)	)	PUNCT
ejpam-3423	47	22	the	the	DET
ejpam-3423	47	23	variation	variation	NOUN
ejpam-3423	47	24	-	-	PUNCT
ejpam-3423	47	25	of	of	ADP
ejpam-3423	47	26	-	-	PUNCT
ejpam-3423	47	27	constants	constant	NOUN
ejpam-3423	47	28	formula	formula	NOUN
ejpam-3423	47	29	with	with	ADP
ejpam-3423	47	30	the	the	DET
ejpam-3423	47	31	exact	exact	ADJ
ejpam-3423	47	32	solution	solution	NOUN
ejpam-3423	47	33	representation	representation	NOUN
ejpam-3423	47	34	of	of	ADP
ejpam-3423	47	35	(	(	PUNCT
ejpam-3423	47	36	1	1	NUM
ejpam-3423	47	37	)	)	PUNCT
ejpam-3423	47	38	is	be	AUX
ejpam-3423	47	39	u(tn+1	u(tn+1	ADJ
ejpam-3423	47	40	)	)	PUNCT
ejpam-3423	47	41	=	=	SYM
ejpam-3423	47	42	ehau(tn	ehau(tn	PROPN
ejpam-3423	47	43	)	)	PUNCT
ejpam-3423	47	44	+	+	NUM
ejpam-3423	48	1	∫	∫	PROPN
ejpam-3423	48	2	h	h	NOUN
ejpam-3423	48	3	0	0	PROPN
ejpam-3423	49	1	e(h−τ)ab	e(h−τ)ab	PROPN
ejpam-3423	49	2	·	·	PUNCT
ejpam-3423	49	3	u(tn	u(tn	PROPN
ejpam-3423	49	4	+	+	PUNCT
ejpam-3423	50	1	τ)dτ	τ)dτ	PROPN
ejpam-3423	50	2	+	+	NUM
ejpam-3423	50	3	∫	∫	PROPN
ejpam-3423	50	4	h	h	NOUN
ejpam-3423	50	5	0	0	NUM
ejpam-3423	50	6	e(h−τ)ag	e(h−τ)ag	PROPN
ejpam-3423	50	7	(	(	PUNCT
ejpam-3423	50	8	tn	tn	PROPN
ejpam-3423	50	9	+	+	CCONJ
ejpam-3423	50	10	τ	τ	PROPN
ejpam-3423	50	11	)	)	PUNCT
ejpam-3423	50	12	dτ	dτ	PROPN
ejpam-3423	50	13	.	.	PROPN
ejpam-3423	50	14	m.	m.	PROPN
ejpam-3423	50	15	a.	a.	PROPN
ejpam-3423	50	16	gondal	gondal	PROPN
ejpam-3423	50	17	,	,	PUNCT
ejpam-3423	50	18	i.	i.	PROPN
ejpam-3423	50	19	rehman	rehman	PROPN
ejpam-3423	50	20	,	,	PUNCT
ejpam-3423	50	21	a.	a.	NOUN
ejpam-3423	50	22	razzaque	razzaque	NOUN
ejpam-3423	50	23	/	/	SYM
ejpam-3423	50	24	eur	eur	NOUN
ejpam-3423	50	25	.	.	PUNCT
ejpam-3423	51	1	j.	j.	PROPN
ejpam-3423	51	2	pure	pure	PROPN
ejpam-3423	51	3	appl	appl	PROPN
ejpam-3423	51	4	.	.	PROPN
ejpam-3423	51	5	math	math	PROPN
ejpam-3423	51	6	,	,	PUNCT
ejpam-3423	51	7	12	12	NUM
ejpam-3423	51	8	(	(	PUNCT
ejpam-3423	51	9	3	3	NUM
ejpam-3423	51	10	)	)	PUNCT
ejpam-3423	51	11	(	(	PUNCT
ejpam-3423	51	12	2019	2019	NUM
ejpam-3423	51	13	)	)	PUNCT
ejpam-3423	51	14	,	,	PUNCT
ejpam-3423	51	15	1215	1215	NUM
ejpam-3423	51	16	-	-	SYM
ejpam-3423	51	17	1230	1230	NUM
ejpam-3423	51	18	1217	1217	NUM
ejpam-3423	51	19	the	the	DET
ejpam-3423	51	20	approximation	approximation	NOUN
ejpam-3423	51	21	obtained	obtain	VERB
ejpam-3423	51	22	through	through	ADP
ejpam-3423	51	23	the	the	DET
ejpam-3423	51	24	left	left	ADJ
ejpam-3423	51	25	rectangular	rectangular	ADJ
ejpam-3423	51	26	rule	rule	NOUN
ejpam-3423	51	27	is	be	AUX
ejpam-3423	51	28	u(tn+1	u(tn+1	ADJ
ejpam-3423	51	29	)	)	PUNCT
ejpam-3423	52	1	≈	≈	PROPN
ejpam-3423	52	2	ehau(tn	ehau(tn	PROPN
ejpam-3423	52	3	)	)	PUNCT
ejpam-3423	52	4	+	+	NUM
ejpam-3423	52	5	∫	∫	PROPN
ejpam-3423	52	6	h	h	NOUN
ejpam-3423	52	7	0	0	PROPN
ejpam-3423	52	8	e(h−τ)ab	e(h−τ)ab	PROPN
ejpam-3423	52	9	·	·	PUNCT
ejpam-3423	52	10	u(tn)dτ	u(tn)dτ	NOUN
ejpam-3423	53	1	+	+	CCONJ
ejpam-3423	53	2	∫	∫	PROPN
ejpam-3423	53	3	h	h	NOUN
ejpam-3423	53	4	0	0	NUM
ejpam-3423	53	5	e(h−τ)ag	e(h−τ)ag	PROPN
ejpam-3423	53	6	(	(	PUNCT
ejpam-3423	53	7	tn	tn	PROPN
ejpam-3423	53	8	)	)	PUNCT
ejpam-3423	53	9	dτ	dτ	NOUN
ejpam-3423	53	10	and	and	CCONJ
ejpam-3423	53	11	un+1	un+1	NOUN
ejpam-3423	53	12	=	=	SYM
ejpam-3423	53	13	ehaun	ehaun	NOUN
ejpam-3423	53	14	+	+	NUM
ejpam-3423	53	15	hϕ1(ha)(bun	hϕ1(ha)(bun	NOUN
ejpam-3423	53	16	+	+	CCONJ
ejpam-3423	53	17	g(tn	g(tn	NOUN
ejpam-3423	53	18	)	)	PUNCT
ejpam-3423	53	19	)	)	PUNCT
ejpam-3423	53	20	,	,	PUNCT
ejpam-3423	53	21	ϕ1(ha	ϕ1(ha	X
ejpam-3423	53	22	)	)	PUNCT
ejpam-3423	53	23	=	=	SYM
ejpam-3423	54	1	1	1	NUM
ejpam-3423	54	2	h	h	NOUN
ejpam-3423	54	3	∫	∫	PROPN
ejpam-3423	54	4	h	h	NOUN
ejpam-3423	54	5	0	0	NUM
ejpam-3423	54	6	e(h−τ)adτ	e(h−τ)adτ	NOUN
ejpam-3423	54	7	.	.	PUNCT
ejpam-3423	55	1	(	(	PUNCT
ejpam-3423	55	2	2	2	X
ejpam-3423	55	3	)	)	PUNCT
ejpam-3423	55	4	which	which	PRON
ejpam-3423	55	5	is	be	AUX
ejpam-3423	55	6	known	know	VERB
ejpam-3423	55	7	as	as	ADP
ejpam-3423	55	8	the	the	DET
ejpam-3423	55	9	exponential	exponential	PROPN
ejpam-3423	55	10	euler	euler	NOUN
ejpam-3423	55	11	method	method	NOUN
ejpam-3423	55	12	of	of	ADP
ejpam-3423	55	13	order	order	NOUN
ejpam-3423	55	14	one	one	NUM
ejpam-3423	55	15	for	for	ADP
ejpam-3423	55	16	problem	problem	NOUN
ejpam-3423	55	17	given	give	VERB
ejpam-3423	55	18	in	in	ADP
ejpam-3423	55	19	(	(	PUNCT
ejpam-3423	55	20	1	1	NUM
ejpam-3423	55	21	)	)	PUNCT
ejpam-3423	55	22	.	.	PUNCT
ejpam-3423	56	1	now	now	ADV
ejpam-3423	56	2	for	for	ADP
ejpam-3423	56	3	(	(	PUNCT
ejpam-3423	56	4	1	1	NUM
ejpam-3423	56	5	)	)	PUNCT
ejpam-3423	56	6	,	,	PUNCT
ejpam-3423	56	7	we	we	PRON
ejpam-3423	56	8	assume	assume	VERB
ejpam-3423	56	9	the	the	DET
ejpam-3423	56	10	following	follow	VERB
ejpam-3423	56	11	exponential	exponential	ADJ
ejpam-3423	56	12	runge	runge	NOUN
ejpam-3423	56	13	–	–	PUNCT
ejpam-3423	56	14	kutta	kutta	NOUN
ejpam-3423	56	15	methods	method	NOUN
ejpam-3423	56	16	un+1	un+1	NOUN
ejpam-3423	57	1	=	=	SYM
ejpam-3423	57	2	ehaun	ehaun	NOUN
ejpam-3423	58	1	+	+	CCONJ
ejpam-3423	58	2	h	h	NOUN
ejpam-3423	58	3	s∑	s∑	PROPN
ejpam-3423	58	4	i=1	i=1	PROPN
ejpam-3423	58	5	bi(ha	bi(ha	PROPN
ejpam-3423	58	6	)	)	PUNCT
ejpam-3423	58	7	(	(	PUNCT
ejpam-3423	58	8	buni	buni	X
ejpam-3423	58	9	+	+	PUNCT
ejpam-3423	58	10	g(tn	g(tn	PROPN
ejpam-3423	58	11	+	+	CCONJ
ejpam-3423	58	12	cih	cih	NOUN
ejpam-3423	58	13	)	)	PUNCT
ejpam-3423	58	14	)	)	PUNCT
ejpam-3423	58	15	,	,	PUNCT
ejpam-3423	58	16	(	(	PUNCT
ejpam-3423	58	17	3	3	X
ejpam-3423	58	18	)	)	PUNCT
ejpam-3423	58	19	uni	uni	NOUN
ejpam-3423	58	20	=	=	NOUN
ejpam-3423	58	21	ecihaun	ecihaun	PROPN
ejpam-3423	58	22	+	+	CCONJ
ejpam-3423	58	23	h	h	NOUN
ejpam-3423	58	24	i−1∑	i−1∑	NOUN
ejpam-3423	58	25	j=1	j=1	PROPN
ejpam-3423	58	26	aij(ha	aij(ha	X
ejpam-3423	58	27	)	)	PUNCT
ejpam-3423	58	28	(	(	PUNCT
ejpam-3423	58	29	bunj	bunj	VERB
ejpam-3423	58	30	+	+	NOUN
ejpam-3423	58	31	g(tn	g(tn	NOUN
ejpam-3423	58	32	+	+	CCONJ
ejpam-3423	58	33	cjh	cjh	NOUN
ejpam-3423	58	34	)	)	PUNCT
ejpam-3423	58	35	)	)	PUNCT
ejpam-3423	58	36	,	,	PUNCT
ejpam-3423	58	37	1	1	NUM
ejpam-3423	58	38	≤	≤	NUM
ejpam-3423	58	39	i	i	PRON
ejpam-3423	58	40	≤	≤	VERB
ejpam-3423	58	41	s.	s.	PROPN
ejpam-3423	59	1	the	the	DET
ejpam-3423	59	2	exponential	exponential	ADJ
ejpam-3423	59	3	runge	runge	NOUN
ejpam-3423	59	4	–	–	PUNCT
ejpam-3423	59	5	kutta	kutta	NOUN
ejpam-3423	59	6	method	method	NOUN
ejpam-3423	59	7	(	(	PUNCT
ejpam-3423	59	8	3	3	NUM
ejpam-3423	59	9	)	)	PUNCT
ejpam-3423	59	10	for	for	ADP
ejpam-3423	59	11	a	a	DET
ejpam-3423	59	12	second	second	ADJ
ejpam-3423	59	13	-	-	PUNCT
ejpam-3423	59	14	order	order	NOUN
ejpam-3423	59	15	method	method	NOUN
ejpam-3423	59	16	with	with	ADP
ejpam-3423	59	17	two	two	NUM
ejpam-3423	59	18	stages	stage	NOUN
ejpam-3423	59	19	can	can	AUX
ejpam-3423	59	20	be	be	AUX
ejpam-3423	59	21	written	write	VERB
ejpam-3423	59	22	as	as	ADP
ejpam-3423	59	23	un+1	un+1	NOUN
ejpam-3423	59	24	=	=	SYM
ejpam-3423	59	25	ehaun	ehaun	NOUN
ejpam-3423	59	26	+	+	NUM
ejpam-3423	59	27	h	h	NOUN
ejpam-3423	59	28	(	(	PUNCT
ejpam-3423	59	29	b1(ha	b1(ha	PROPN
ejpam-3423	59	30	)	)	PUNCT
ejpam-3423	59	31	(	(	PUNCT
ejpam-3423	59	32	bun1	bun1	X
ejpam-3423	59	33	+	+	CCONJ
ejpam-3423	59	34	g(tn	g(tn	NOUN
ejpam-3423	59	35	+	+	CCONJ
ejpam-3423	59	36	c1h	c1h	NOUN
ejpam-3423	59	37	)	)	PUNCT
ejpam-3423	59	38	)	)	PUNCT
ejpam-3423	60	1	+	+	CCONJ
ejpam-3423	60	2	b2(ha	b2(ha	ADP
ejpam-3423	60	3	)	)	PUNCT
ejpam-3423	60	4	(	(	PUNCT
ejpam-3423	60	5	bun2	bun2	PROPN
ejpam-3423	60	6	+	+	PUNCT
ejpam-3423	60	7	g(tn	g(tn	NOUN
ejpam-3423	60	8	+	+	CCONJ
ejpam-3423	60	9	c2h	c2h	NOUN
ejpam-3423	60	10	)	)	PUNCT
ejpam-3423	60	11	)	)	PUNCT
ejpam-3423	60	12	)	)	PUNCT
ejpam-3423	60	13	,	,	PUNCT
ejpam-3423	60	14	(	(	PUNCT
ejpam-3423	60	15	4	4	X
ejpam-3423	60	16	)	)	PUNCT
ejpam-3423	60	17	un1	un1	NOUN
ejpam-3423	60	18	=	=	SYM
ejpam-3423	60	19	ec1haun	ec1haun	PROPN
ejpam-3423	60	20	,	,	PUNCT
ejpam-3423	60	21	un2	un2	ADJ
ejpam-3423	60	22	=	=	PUNCT
ejpam-3423	60	23	ec2haun	ec2haun	PROPN
ejpam-3423	61	1	+	+	NUM
ejpam-3423	61	2	ha21(ha	ha21(ha	NOUN
ejpam-3423	61	3	)	)	PUNCT
ejpam-3423	61	4	(	(	PUNCT
ejpam-3423	61	5	bun1	bun1	NOUN
ejpam-3423	61	6	+	+	CCONJ
ejpam-3423	61	7	g(tn	g(tn	NOUN
ejpam-3423	61	8	+	+	CCONJ
ejpam-3423	61	9	c1h	c1h	NOUN
ejpam-3423	61	10	)	)	PUNCT
ejpam-3423	61	11	)	)	PUNCT
ejpam-3423	61	12	.	.	PUNCT
ejpam-3423	62	1	further	far	ADV
ejpam-3423	62	2	we	we	PRON
ejpam-3423	62	3	know	know	VERB
ejpam-3423	62	4	that	that	SCONJ
ejpam-3423	62	5	for	for	ADP
ejpam-3423	62	6	a	a	DET
ejpam-3423	62	7	second	second	ADJ
ejpam-3423	62	8	-	-	PUNCT
ejpam-3423	62	9	order	order	NOUN
ejpam-3423	62	10	method	method	NOUN
ejpam-3423	62	11	with	with	ADP
ejpam-3423	62	12	two	two	NUM
ejpam-3423	62	13	stages	stage	NOUN
ejpam-3423	62	14	it	it	PRON
ejpam-3423	62	15	must	must	AUX
ejpam-3423	62	16	satisfy	satisfy	VERB
ejpam-3423	62	17	the	the	DET
ejpam-3423	62	18	following	follow	VERB
ejpam-3423	62	19	three	three	NUM
ejpam-3423	62	20	order	order	NOUN
ejpam-3423	62	21	conditions	condition	NOUN
ejpam-3423	62	22	given	give	VERB
ejpam-3423	62	23	in	in	ADP
ejpam-3423	62	24	hochbruck	hochbruck	NOUN
ejpam-3423	62	25	&	&	CCONJ
ejpam-3423	62	26	ostermann	ostermann	PROPN
ejpam-3423	63	1	[	[	X
ejpam-3423	63	2	9	9	NUM
ejpam-3423	63	3	]	]	SYM
ejpam-3423	63	4	b1(ha	b1(ha	NOUN
ejpam-3423	63	5	)	)	PUNCT
ejpam-3423	63	6	+	+	PUNCT
ejpam-3423	63	7	b2(ha	b2(ha	PROPN
ejpam-3423	63	8	)	)	PUNCT
ejpam-3423	63	9	=	=	SYM
ejpam-3423	64	1	ϕ1(ha	ϕ1(ha	PROPN
ejpam-3423	64	2	)	)	PUNCT
ejpam-3423	64	3	,	,	PUNCT
ejpam-3423	64	4	c1b1(ha	c1b1(ha	PROPN
ejpam-3423	64	5	)	)	PUNCT
ejpam-3423	65	1	+	+	NUM
ejpam-3423	65	2	c2b2(ha	c2b2(ha	NOUN
ejpam-3423	65	3	)	)	PUNCT
ejpam-3423	65	4	=	=	SYM
ejpam-3423	65	5	ϕ2(ha	ϕ2(ha	PROPN
ejpam-3423	65	6	)	)	PUNCT
ejpam-3423	65	7	,	,	PUNCT
ejpam-3423	65	8	(	(	PUNCT
ejpam-3423	65	9	5	5	X
ejpam-3423	65	10	)	)	PUNCT
ejpam-3423	65	11	a21(ha	a21(ha	NUM
ejpam-3423	65	12	)	)	PUNCT
ejpam-3423	65	13	=	=	PUNCT
ejpam-3423	65	14	c2ϕ1(c2	c2ϕ1(c2	NOUN
ejpam-3423	65	15	ha	ha	X
ejpam-3423	65	16	)	)	PUNCT
ejpam-3423	65	17	,	,	PUNCT
ejpam-3423	65	18	where	where	SCONJ
ejpam-3423	65	19	ϕ1(z	ϕ1(z	VERB
ejpam-3423	65	20	)	)	PUNCT
ejpam-3423	65	21	=	=	SYM
ejpam-3423	65	22	ez	ez	PROPN
ejpam-3423	66	1	−	−	PROPN
ejpam-3423	66	2	1	1	NUM
ejpam-3423	66	3	z	z	NOUN
ejpam-3423	66	4	,	,	PUNCT
ejpam-3423	66	5	ϕ2(z	ϕ2(z	X
ejpam-3423	66	6	)	)	PUNCT
ejpam-3423	66	7	=	=	PUNCT
ejpam-3423	67	1	ϕ1(z)−	ϕ1(z)−	PUNCT
ejpam-3423	67	2	1	1	NUM
ejpam-3423	67	3	z	z	NOUN
ejpam-3423	67	4	.	.	PUNCT
ejpam-3423	68	1	by	by	ADP
ejpam-3423	68	2	taking	take	VERB
ejpam-3423	68	3	c1	c1	PROPN
ejpam-3423	68	4	=	=	PROPN
ejpam-3423	68	5	0	0	PUNCT
ejpam-3423	68	6	and	and	CCONJ
ejpam-3423	68	7	c2	c2	PROPN
ejpam-3423	68	8	=	=	SYM
ejpam-3423	68	9	1	1	NUM
ejpam-3423	68	10	,	,	PUNCT
ejpam-3423	68	11	we	we	PRON
ejpam-3423	68	12	find	find	VERB
ejpam-3423	68	13	the	the	DET
ejpam-3423	68	14	values	value	NOUN
ejpam-3423	68	15	of	of	ADP
ejpam-3423	68	16	b1	b1	NOUN
ejpam-3423	68	17	=	=	SYM
ejpam-3423	68	18	ϕ1(ha	ϕ1(ha	PROPN
ejpam-3423	68	19	)	)	PUNCT
ejpam-3423	68	20	−	−	PROPN
ejpam-3423	68	21	ϕ2(ha	ϕ2(ha	PROPN
ejpam-3423	68	22	)	)	PUNCT
ejpam-3423	68	23	,	,	PUNCT
ejpam-3423	68	24	b2	b2	NOUN
ejpam-3423	68	25	=	=	SYM
ejpam-3423	68	26	ϕ2(ha	ϕ2(ha	PROPN
ejpam-3423	68	27	)	)	PUNCT
ejpam-3423	68	28	and	and	CCONJ
ejpam-3423	68	29	a21	a21	NOUN
ejpam-3423	68	30	=	=	SYM
ejpam-3423	68	31	ϕ1(ha	ϕ1(ha	PROPN
ejpam-3423	68	32	)	)	PUNCT
ejpam-3423	68	33	.	.	PUNCT
ejpam-3423	69	1	hence	hence	ADV
ejpam-3423	69	2	we	we	PRON
ejpam-3423	69	3	can	can	AUX
ejpam-3423	69	4	write	write	VERB
ejpam-3423	69	5	(	(	PUNCT
ejpam-3423	69	6	4	4	NUM
ejpam-3423	69	7	)	)	PUNCT
ejpam-3423	69	8	as	as	ADP
ejpam-3423	69	9	un+1	un+1	NOUN
ejpam-3423	69	10	=	=	SYM
ejpam-3423	69	11	ehaun	ehaun	NOUN
ejpam-3423	70	1	+	+	CCONJ
ejpam-3423	70	2	h(ϕ1(ha)−	h(ϕ1(ha)−	NOUN
ejpam-3423	70	3	ϕ2(ha))bun1	ϕ2(ha))bun1	NOUN
ejpam-3423	71	1	+	+	CCONJ
ejpam-3423	71	2	hϕ2(ha)bun2	hϕ2(ha)bun2	ADJ
ejpam-3423	71	3	+	+	CCONJ
ejpam-3423	71	4	h(ϕ1(ha)−	h(ϕ1(ha)−	NOUN
ejpam-3423	71	5	ϕ2(ha))g(tn	ϕ2(ha))g(tn	NOUN
ejpam-3423	71	6	)	)	PUNCT
ejpam-3423	72	1	+	+	CCONJ
ejpam-3423	72	2	hϕ2(ha)g(tn	hϕ2(ha)g(tn	PROPN
ejpam-3423	72	3	+	+	NUM
ejpam-3423	72	4	h	h	NOUN
ejpam-3423	72	5	)	)	PUNCT
ejpam-3423	72	6	,	,	PUNCT
ejpam-3423	72	7	(	(	PUNCT
ejpam-3423	72	8	6	6	X
ejpam-3423	72	9	)	)	PUNCT
ejpam-3423	72	10	un1	un1	NOUN
ejpam-3423	73	1	=	=	SYM
ejpam-3423	73	2	un	un	PROPN
ejpam-3423	73	3	,	,	PUNCT
ejpam-3423	73	4	un2	un2	ADJ
ejpam-3423	73	5	=	=	NOUN
ejpam-3423	73	6	ehaun	ehaun	NOUN
ejpam-3423	74	1	+	+	CCONJ
ejpam-3423	74	2	hϕ1(ha	hϕ1(ha	NOUN
ejpam-3423	74	3	)	)	PUNCT
ejpam-3423	74	4	(	(	PUNCT
ejpam-3423	74	5	bun1	bun1	NOUN
ejpam-3423	74	6	+	+	CCONJ
ejpam-3423	74	7	g(tn	g(tn	NOUN
ejpam-3423	74	8	)	)	PUNCT
ejpam-3423	74	9	)	)	PUNCT
ejpam-3423	74	10	.	.	PUNCT
ejpam-3423	75	1	(	(	PUNCT
ejpam-3423	75	2	7	7	X
ejpam-3423	75	3	)	)	PUNCT
ejpam-3423	75	4	this	this	PRON
ejpam-3423	75	5	is	be	AUX
ejpam-3423	75	6	the	the	DET
ejpam-3423	75	7	exponential	exponential	ADJ
ejpam-3423	75	8	runge	runge	NOUN
ejpam-3423	75	9	–	–	PUNCT
ejpam-3423	75	10	kutta	kutta	NOUN
ejpam-3423	75	11	method	method	NOUN
ejpam-3423	75	12	of	of	ADP
ejpam-3423	75	13	order	order	NOUN
ejpam-3423	75	14	two	two	NUM
ejpam-3423	75	15	for	for	ADP
ejpam-3423	75	16	the	the	DET
ejpam-3423	75	17	problem	problem	NOUN
ejpam-3423	75	18	(	(	PUNCT
ejpam-3423	75	19	1	1	NUM
ejpam-3423	75	20	)	)	PUNCT
ejpam-3423	75	21	.	.	PUNCT
ejpam-3423	76	1	m.	m.	NOUN
ejpam-3423	76	2	a.	a.	PROPN
ejpam-3423	76	3	gondal	gondal	PROPN
ejpam-3423	76	4	,	,	PUNCT
ejpam-3423	76	5	i.	i.	PROPN
ejpam-3423	76	6	rehman	rehman	PROPN
ejpam-3423	76	7	,	,	PUNCT
ejpam-3423	76	8	a.	a.	NOUN
ejpam-3423	76	9	razzaque	razzaque	NOUN
ejpam-3423	76	10	/	/	SYM
ejpam-3423	76	11	eur	eur	NOUN
ejpam-3423	76	12	.	.	PUNCT
ejpam-3423	77	1	j.	j.	PROPN
ejpam-3423	77	2	pure	pure	PROPN
ejpam-3423	77	3	appl	appl	PROPN
ejpam-3423	77	4	.	.	PROPN
ejpam-3423	77	5	math	math	PROPN
ejpam-3423	77	6	,	,	PUNCT
ejpam-3423	77	7	12	12	NUM
ejpam-3423	77	8	(	(	PUNCT
ejpam-3423	77	9	3	3	NUM
ejpam-3423	77	10	)	)	PUNCT
ejpam-3423	77	11	(	(	PUNCT
ejpam-3423	77	12	2019	2019	NUM
ejpam-3423	77	13	)	)	PUNCT
ejpam-3423	77	14	,	,	PUNCT
ejpam-3423	77	15	1215	1215	NUM
ejpam-3423	77	16	-	-	SYM
ejpam-3423	77	17	1230	1230	NUM
ejpam-3423	77	18	1218	1218	NUM
ejpam-3423	77	19	3	3	NUM
ejpam-3423	77	20	.	.	X
ejpam-3423	77	21	convergence	convergence	NOUN
ejpam-3423	77	22	of	of	ADP
ejpam-3423	77	23	an	an	DET
ejpam-3423	77	24	exponential	exponential	ADJ
ejpam-3423	77	25	runge	runge	NOUN
ejpam-3423	77	26	–	–	PUNCT
ejpam-3423	77	27	kutta	kutta	NOUN
ejpam-3423	77	28	method	method	NOUN
ejpam-3423	77	29	of	of	ADP
ejpam-3423	77	30	order	order	NOUN
ejpam-3423	77	31	two	two	NUM
ejpam-3423	77	32	for	for	ADP
ejpam-3423	77	33	non	non	ADJ
ejpam-3423	77	34	-	-	ADJ
ejpam-3423	77	35	smooth	smooth	ADJ
ejpam-3423	77	36	initial	initial	ADJ
ejpam-3423	77	37	data	datum	NOUN
ejpam-3423	77	38	in	in	ADP
ejpam-3423	77	39	this	this	DET
ejpam-3423	77	40	section	section	NOUN
ejpam-3423	77	41	we	we	PRON
ejpam-3423	77	42	study	study	VERB
ejpam-3423	77	43	an	an	DET
ejpam-3423	77	44	exponential	exponential	ADJ
ejpam-3423	77	45	runge	runge	NOUN
ejpam-3423	77	46	–	–	PUNCT
ejpam-3423	77	47	kutta	kutta	NOUN
ejpam-3423	77	48	method	method	NOUN
ejpam-3423	77	49	of	of	ADP
ejpam-3423	77	50	order	order	NOUN
ejpam-3423	77	51	two	two	NUM
ejpam-3423	77	52	for	for	ADP
ejpam-3423	77	53	discretizing	discretize	VERB
ejpam-3423	77	54	an	an	DET
ejpam-3423	77	55	abstract	abstract	ADJ
ejpam-3423	77	56	problem	problem	NOUN
ejpam-3423	77	57	(	(	PUNCT
ejpam-3423	77	58	1	1	X
ejpam-3423	77	59	)	)	PUNCT
ejpam-3423	77	60	in	in	ADP
ejpam-3423	77	61	time	time	NOUN
ejpam-3423	77	62	.	.	PUNCT
ejpam-3423	78	1	on	on	ADP
ejpam-3423	78	2	a	a	DET
ejpam-3423	78	3	,	,	PUNCT
ejpam-3423	78	4	b	b	NOUN
ejpam-3423	78	5	and	and	CCONJ
ejpam-3423	78	6	g	g	NOUN
ejpam-3423	78	7	,	,	PUNCT
ejpam-3423	78	8	our	our	PRON
ejpam-3423	78	9	assumptions	assumption	NOUN
ejpam-3423	78	10	are	be	AUX
ejpam-3423	78	11	the	the	DET
ejpam-3423	78	12	same	same	ADJ
ejpam-3423	78	13	as	as	SCONJ
ejpam-3423	78	14	given	give	VERB
ejpam-3423	78	15	in	in	ADP
ejpam-3423	78	16	gondal	gondal	NOUN
ejpam-3423	78	17	[	[	X
ejpam-3423	78	18	5	5	NUM
ejpam-3423	78	19	]	]	PUNCT
ejpam-3423	78	20	.	.	PUNCT
ejpam-3423	79	1	now	now	ADV
ejpam-3423	79	2	first	first	ADV
ejpam-3423	79	3	we	we	PRON
ejpam-3423	79	4	are	be	AUX
ejpam-3423	79	5	going	go	VERB
ejpam-3423	79	6	to	to	PART
ejpam-3423	79	7	prove	prove	VERB
ejpam-3423	79	8	vital	vital	ADJ
ejpam-3423	79	9	properties	property	NOUN
ejpam-3423	79	10	of	of	ADP
ejpam-3423	79	11	the	the	DET
ejpam-3423	79	12	exact	exact	ADJ
ejpam-3423	79	13	solution	solution	NOUN
ejpam-3423	79	14	and	and	CCONJ
ejpam-3423	79	15	then	then	ADV
ejpam-3423	79	16	we	we	PRON
ejpam-3423	79	17	will	will	AUX
ejpam-3423	79	18	move	move	VERB
ejpam-3423	79	19	to	to	ADP
ejpam-3423	79	20	the	the	DET
ejpam-3423	79	21	numerical	numerical	ADJ
ejpam-3423	79	22	solution	solution	NOUN
ejpam-3423	79	23	.	.	PUNCT
ejpam-3423	80	1	lemma	lemma	PROPN
ejpam-3423	80	2	1	1	X
ejpam-3423	80	3	.	.	PUNCT
ejpam-3423	80	4	assume	assume	VERB
ejpam-3423	80	5	that	that	SCONJ
ejpam-3423	80	6	problem	problem	NOUN
ejpam-3423	80	7	(	(	PUNCT
ejpam-3423	80	8	1	1	X
ejpam-3423	80	9	)	)	PUNCT
ejpam-3423	80	10	fulfill	fulfill	VERB
ejpam-3423	80	11	the	the	DET
ejpam-3423	80	12	hypotheses	hypothesis	NOUN
ejpam-3423	80	13	of	of	ADP
ejpam-3423	80	14	lemma	lemma	PROPN
ejpam-3423	80	15	2	2	NUM
ejpam-3423	80	16	given	give	VERB
ejpam-3423	80	17	in	in	ADP
ejpam-3423	80	18	gondal	gondal	NOUN
ejpam-3423	80	19	[	[	X
ejpam-3423	80	20	5	5	NUM
ejpam-3423	80	21	]	]	PUNCT
ejpam-3423	80	22	.	.	PUNCT
ejpam-3423	81	1	then	then	ADV
ejpam-3423	81	2	the	the	DET
ejpam-3423	81	3	bounds	bound	NOUN
ejpam-3423	81	4	‖l−1u′′(t)‖	‖l−1u′′(t)‖	VERB
ejpam-3423	81	5	≤	≤	NUM
ejpam-3423	81	6	c	c	PROPN
ejpam-3423	81	7	t	t	PROPN
ejpam-3423	81	8	,	,	PUNCT
ejpam-3423	81	9	on	on	ADP
ejpam-3423	81	10	(	(	PUNCT
ejpam-3423	81	11	0	0	NUM
ejpam-3423	81	12	,	,	PUNCT
ejpam-3423	81	13	t	t	X
ejpam-3423	81	14	]	]	PUNCT
ejpam-3423	81	15	,	,	PUNCT
ejpam-3423	81	16	(	(	PUNCT
ejpam-3423	81	17	8)	8)	NUM
ejpam-3423	81	18	hold	hold	VERB
ejpam-3423	81	19	uniformly	uniformly	ADV
ejpam-3423	81	20	on	on	ADP
ejpam-3423	81	21	0	0	NUM
ejpam-3423	81	22	≤	≤	NUM
ejpam-3423	81	23	t	t	PROPN
ejpam-3423	81	24	≤	≤	X
ejpam-3423	81	25	t	t	PROPN
ejpam-3423	81	26	for	for	ADP
ejpam-3423	81	27	non	non	ADJ
ejpam-3423	81	28	-	-	ADJ
ejpam-3423	81	29	smooth	smooth	ADJ
ejpam-3423	81	30	initial	initial	ADJ
ejpam-3423	81	31	data	datum	NOUN
ejpam-3423	81	32	.	.	PUNCT
ejpam-3423	82	1	proof	proof	NOUN
ejpam-3423	82	2	.	.	PUNCT
ejpam-3423	83	1	from	from	ADP
ejpam-3423	83	2	lemma	lemma	PROPN
ejpam-3423	83	3	2	2	NUM
ejpam-3423	83	4	given	give	VERB
ejpam-3423	83	5	in	in	ADP
ejpam-3423	83	6	gondal	gondal	NOUN
ejpam-3423	83	7	[	[	X
ejpam-3423	83	8	5	5	NUM
ejpam-3423	83	9	]	]	PUNCT
ejpam-3423	83	10	we	we	PRON
ejpam-3423	83	11	use	use	VERB
ejpam-3423	83	12	equation	equation	NOUN
ejpam-3423	83	13	(	(	PUNCT
ejpam-3423	83	14	18	18	NUM
ejpam-3423	83	15	)	)	PUNCT
ejpam-3423	83	16	given	give	VERB
ejpam-3423	83	17	in	in	ADP
ejpam-3423	83	18	gondal	gondal	NOUN
ejpam-3423	83	19	[	[	X
ejpam-3423	83	20	5	5	NUM
ejpam-3423	83	21	]	]	PUNCT
ejpam-3423	83	22	in	in	ADP
ejpam-3423	83	23	u′′(t	u′′(t	NOUN
ejpam-3423	83	24	)	)	PUNCT
ejpam-3423	83	25	=	=	SYM
ejpam-3423	83	26	lu′(t	lu′(t	NOUN
ejpam-3423	83	27	)	)	PUNCT
ejpam-3423	84	1	+	+	PUNCT
ejpam-3423	84	2	g′(t	g′(t	PROPN
ejpam-3423	84	3	)	)	PUNCT
ejpam-3423	84	4	,	,	PUNCT
ejpam-3423	84	5	(	(	PUNCT
ejpam-3423	84	6	9	9	X
ejpam-3423	84	7	)	)	PUNCT
ejpam-3423	84	8	and	and	CCONJ
ejpam-3423	84	9	we	we	PRON
ejpam-3423	84	10	get	get	VERB
ejpam-3423	84	11	u′′(t	u′′(t	VERB
ejpam-3423	84	12	)	)	PUNCT
ejpam-3423	84	13	=	=	PUNCT
ejpam-3423	84	14	l2etlu0	l2etlu0	NOUN
ejpam-3423	84	15	+	+	NUM
ejpam-3423	84	16	letlg(0	letlg(0	NOUN
ejpam-3423	84	17	)	)	PUNCT
ejpam-3423	85	1	+	+	NUM
ejpam-3423	85	2	etlg′(0	etlg′(0	NOUN
ejpam-3423	85	3	)	)	PUNCT
ejpam-3423	86	1	+	+	CCONJ
ejpam-3423	86	2	tϕ1(tl)g′′(0	tϕ1(tl)g′′(0	NOUN
ejpam-3423	86	3	)	)	PUNCT
ejpam-3423	86	4	+	+	CCONJ
ejpam-3423	86	5	.	.	PUNCT
ejpam-3423	86	6	.	.	PUNCT
ejpam-3423	86	7	.	.	PUNCT
ejpam-3423	86	8	.	.	PUNCT
ejpam-3423	87	1	(	(	PUNCT
ejpam-3423	87	2	10	10	X
ejpam-3423	87	3	)	)	PUNCT
ejpam-3423	87	4	premultiplying	premultiplye	VERB
ejpam-3423	87	5	with	with	ADP
ejpam-3423	87	6	l−1	l−1	PROPN
ejpam-3423	87	7	on	on	ADP
ejpam-3423	87	8	both	both	DET
ejpam-3423	87	9	sides	side	NOUN
ejpam-3423	87	10	of	of	ADP
ejpam-3423	87	11	(	(	PUNCT
ejpam-3423	87	12	10	10	NUM
ejpam-3423	87	13	)	)	PUNCT
ejpam-3423	87	14	,	,	PUNCT
ejpam-3423	87	15	we	we	PRON
ejpam-3423	87	16	get	get	VERB
ejpam-3423	87	17	l−1u′′(t	l−1u′′(t	PROPN
ejpam-3423	87	18	)	)	PUNCT
ejpam-3423	87	19	=	=	PUNCT
ejpam-3423	88	1	letlu0	letlu0	NOUN
ejpam-3423	88	2	+	+	PUNCT
ejpam-3423	88	3	etlg(0	etlg(0	NOUN
ejpam-3423	88	4	)	)	PUNCT
ejpam-3423	88	5	+	+	CCONJ
ejpam-3423	88	6	.	.	PUNCT
ejpam-3423	88	7	.	.	PUNCT
ejpam-3423	89	1	.	.	PUNCT
ejpam-3423	90	1	(	(	PUNCT
ejpam-3423	90	2	11	11	NUM
ejpam-3423	90	3	)	)	PUNCT
ejpam-3423	90	4	now	now	ADV
ejpam-3423	90	5	multiplying	multiply	VERB
ejpam-3423	90	6	with	with	ADP
ejpam-3423	90	7	t	t	PROPN
ejpam-3423	90	8	on	on	ADP
ejpam-3423	90	9	both	both	DET
ejpam-3423	90	10	sides	side	NOUN
ejpam-3423	90	11	of	of	ADP
ejpam-3423	90	12	(	(	PUNCT
ejpam-3423	90	13	11	11	NUM
ejpam-3423	90	14	)	)	PUNCT
ejpam-3423	90	15	,	,	PUNCT
ejpam-3423	90	16	we	we	PRON
ejpam-3423	90	17	have	have	VERB
ejpam-3423	90	18	tl−1u′′(t	tl−1u′′(t	NOUN
ejpam-3423	90	19	)	)	PUNCT
ejpam-3423	90	20	=	=	PUNCT
ejpam-3423	91	1	tletlu0	tletlu0	NOUN
ejpam-3423	92	1	+	+	PUNCT
ejpam-3423	92	2	tetlg(0	tetlg(0	NOUN
ejpam-3423	92	3	)	)	PUNCT
ejpam-3423	93	1	+	+	CCONJ
ejpam-3423	93	2	.	.	PUNCT
ejpam-3423	93	3	.	.	PUNCT
ejpam-3423	94	1	.	.	PUNCT
ejpam-3423	95	1	(	(	PUNCT
ejpam-3423	95	2	12	12	NUM
ejpam-3423	95	3	)	)	PUNCT
ejpam-3423	95	4	therefore	therefore	ADV
ejpam-3423	95	5	‖tl−1u′′(t)‖	‖tl−1u′′(t)‖	NOUN
ejpam-3423	95	6	≤	≤	PROPN
ejpam-3423	95	7	c	c	NOUN
ejpam-3423	95	8	or	or	CCONJ
ejpam-3423	95	9	‖l−1u′′(t)‖	‖l−1u′′(t)‖	PROPN
ejpam-3423	95	10	≤	≤	NUM
ejpam-3423	95	11	c	c	PROPN
ejpam-3423	95	12	t	t	PROPN
ejpam-3423	95	13	.	.	PUNCT
ejpam-3423	96	1	(	(	PUNCT
ejpam-3423	96	2	13	13	NUM
ejpam-3423	96	3	)	)	PUNCT
ejpam-3423	96	4	in	in	ADP
ejpam-3423	96	5	this	this	DET
ejpam-3423	96	6	section	section	NOUN
ejpam-3423	96	7	,	,	PUNCT
ejpam-3423	96	8	we	we	PRON
ejpam-3423	96	9	will	will	AUX
ejpam-3423	96	10	also	also	ADV
ejpam-3423	96	11	derive	derive	VERB
ejpam-3423	96	12	error	error	NOUN
ejpam-3423	96	13	bounds	bound	NOUN
ejpam-3423	96	14	for	for	ADP
ejpam-3423	96	15	exponential	exponential	ADJ
ejpam-3423	96	16	runge	runge	NOUN
ejpam-3423	96	17	–	–	PUNCT
ejpam-3423	96	18	kutta	kutta	NOUN
ejpam-3423	96	19	discretizations	discretization	NOUN
ejpam-3423	96	20	of	of	ADP
ejpam-3423	96	21	(	(	PUNCT
ejpam-3423	96	22	1	1	NUM
ejpam-3423	96	23	)	)	PUNCT
ejpam-3423	96	24	.	.	PUNCT
ejpam-3423	97	1	the	the	DET
ejpam-3423	97	2	exponential	exponential	ADJ
ejpam-3423	97	3	runge	runge	NOUN
ejpam-3423	97	4	–	–	PUNCT
ejpam-3423	97	5	kutta	kutta	NOUN
ejpam-3423	97	6	method	method	NOUN
ejpam-3423	97	7	of	of	ADP
ejpam-3423	97	8	order	order	NOUN
ejpam-3423	97	9	two	two	NUM
ejpam-3423	97	10	for	for	ADP
ejpam-3423	97	11	given	give	VERB
ejpam-3423	97	12	problem	problem	NOUN
ejpam-3423	97	13	is	be	AUX
ejpam-3423	97	14	(	(	PUNCT
ejpam-3423	97	15	6	6	NUM
ejpam-3423	97	16	)	)	PUNCT
ejpam-3423	97	17	.	.	PUNCT
ejpam-3423	98	1	to	to	PART
ejpam-3423	98	2	analyze	analyze	VERB
ejpam-3423	98	3	(	(	PUNCT
ejpam-3423	98	4	6	6	NUM
ejpam-3423	98	5	)	)	PUNCT
ejpam-3423	98	6	,	,	PUNCT
ejpam-3423	98	7	one	one	PRON
ejpam-3423	98	8	can	can	AUX
ejpam-3423	98	9	write	write	VERB
ejpam-3423	98	10	the	the	DET
ejpam-3423	98	11	exact	exact	ADJ
ejpam-3423	98	12	solution	solution	NOUN
ejpam-3423	98	13	of	of	ADP
ejpam-3423	98	14	(	(	PUNCT
ejpam-3423	98	15	1	1	NUM
ejpam-3423	98	16	)	)	PUNCT
ejpam-3423	98	17	as	as	ADP
ejpam-3423	98	18	u(tn+1	u(tn+1	VERB
ejpam-3423	98	19	)	)	PUNCT
ejpam-3423	98	20	=	=	SYM
ejpam-3423	98	21	ehau(tn	ehau(tn	PROPN
ejpam-3423	98	22	)	)	PUNCT
ejpam-3423	98	23	+	+	NUM
ejpam-3423	98	24	∫	∫	PROPN
ejpam-3423	98	25	tn+1	tn+1	PROPN
ejpam-3423	98	26	tn	tn	PROPN
ejpam-3423	98	27	e(tn+1−τ)abu(τ)dτ	e(tn+1−τ)abu(τ)dτ	PROPN
ejpam-3423	99	1	+	+	CCONJ
ejpam-3423	99	2	∫	∫	PROPN
ejpam-3423	99	3	tn+1	tn+1	PROPN
ejpam-3423	99	4	tn	tn	PROPN
ejpam-3423	99	5	e(tn+1−τ)ag(τ)dτ	e(tn+1−τ)ag(τ)dτ	PROPN
ejpam-3423	99	6	.	.	PUNCT
ejpam-3423	100	1	(	(	PUNCT
ejpam-3423	100	2	14	14	NUM
ejpam-3423	100	3	)	)	PUNCT
ejpam-3423	100	4	to	to	PART
ejpam-3423	100	5	write	write	VERB
ejpam-3423	100	6	(	(	PUNCT
ejpam-3423	100	7	14	14	NUM
ejpam-3423	100	8	)	)	PUNCT
ejpam-3423	100	9	in	in	ADP
ejpam-3423	100	10	the	the	DET
ejpam-3423	100	11	form	form	NOUN
ejpam-3423	100	12	of	of	ADP
ejpam-3423	100	13	(	(	PUNCT
ejpam-3423	100	14	6	6	NUM
ejpam-3423	100	15	)	)	PUNCT
ejpam-3423	100	16	,	,	PUNCT
ejpam-3423	100	17	below	below	ADP
ejpam-3423	100	18	result	result	NOUN
ejpam-3423	100	19	will	will	AUX
ejpam-3423	100	20	be	be	AUX
ejpam-3423	100	21	helpful∫	helpful∫	NOUN
ejpam-3423	100	22	tn+1	tn+1	NUM
ejpam-3423	100	23	tn	tn	NOUN
ejpam-3423	100	24	e(tn+1−τ)adτ	e(tn+1−τ)adτ	PROPN
ejpam-3423	101	1	=	=	SYM
ejpam-3423	101	2	∫	∫	PROPN
ejpam-3423	101	3	h	h	NOUN
ejpam-3423	101	4	0	0	NUM
ejpam-3423	101	5	e(h−s)ads	e(h−s)ad	NOUN
ejpam-3423	101	6	=	=	SYM
ejpam-3423	101	7	hϕ1(ha	hϕ1(ha	NOUN
ejpam-3423	101	8	)	)	PUNCT
ejpam-3423	101	9	(	(	PUNCT
ejpam-3423	101	10	15	15	NUM
ejpam-3423	101	11	)	)	PUNCT
ejpam-3423	101	12	∫	∫	PROPN
ejpam-3423	101	13	tn+1	tn+1	PROPN
ejpam-3423	101	14	tn	tn	NOUN
ejpam-3423	102	1	e(tn+1−τ)a(τ	e(tn+1−τ)a(τ	PROPN
ejpam-3423	102	2	−	−	PROPN
ejpam-3423	103	1	tn)dτ	tn)dτ	PUNCT
ejpam-3423	104	1	=	=	SYM
ejpam-3423	104	2	∫	∫	PROPN
ejpam-3423	105	1	h	h	NOUN
ejpam-3423	105	2	0	0	X
ejpam-3423	106	1	e(h−s)asds	e(h−s)asds	PROPN
ejpam-3423	106	2	=	=	PUNCT
ejpam-3423	106	3	h2ϕ2(ha	h2ϕ2(ha	PROPN
ejpam-3423	106	4	)	)	PUNCT
ejpam-3423	106	5	.	.	PUNCT
ejpam-3423	107	1	(	(	PUNCT
ejpam-3423	107	2	16	16	NUM
ejpam-3423	107	3	)	)	PUNCT
ejpam-3423	107	4	m.	m.	NOUN
ejpam-3423	107	5	a.	a.	NOUN
ejpam-3423	107	6	gondal	gondal	PROPN
ejpam-3423	107	7	,	,	PUNCT
ejpam-3423	107	8	i.	i.	PROPN
ejpam-3423	107	9	rehman	rehman	PROPN
ejpam-3423	107	10	,	,	PUNCT
ejpam-3423	107	11	a.	a.	NOUN
ejpam-3423	107	12	razzaque	razzaque	NOUN
ejpam-3423	107	13	/	/	SYM
ejpam-3423	107	14	eur	eur	NOUN
ejpam-3423	107	15	.	.	PUNCT
ejpam-3423	108	1	j.	j.	PROPN
ejpam-3423	108	2	pure	pure	PROPN
ejpam-3423	108	3	appl	appl	PROPN
ejpam-3423	108	4	.	.	PROPN
ejpam-3423	108	5	math	math	PROPN
ejpam-3423	108	6	,	,	PUNCT
ejpam-3423	108	7	12	12	NUM
ejpam-3423	108	8	(	(	PUNCT
ejpam-3423	108	9	3	3	NUM
ejpam-3423	108	10	)	)	PUNCT
ejpam-3423	108	11	(	(	PUNCT
ejpam-3423	108	12	2019	2019	NUM
ejpam-3423	108	13	)	)	PUNCT
ejpam-3423	108	14	,	,	PUNCT
ejpam-3423	108	15	1215	1215	NUM
ejpam-3423	108	16	-	-	SYM
ejpam-3423	108	17	1230	1230	NUM
ejpam-3423	108	18	1219	1219	NUM
ejpam-3423	108	19	theorem	theorem	NOUN
ejpam-3423	108	20	1	1	NUM
ejpam-3423	108	21	.	.	PUNCT
ejpam-3423	108	22	assume	assume	VERB
ejpam-3423	108	23	that	that	SCONJ
ejpam-3423	108	24	problem	problem	NOUN
ejpam-3423	108	25	(	(	PUNCT
ejpam-3423	108	26	1	1	X
ejpam-3423	108	27	)	)	PUNCT
ejpam-3423	108	28	fulfill	fulfill	VERB
ejpam-3423	108	29	the	the	DET
ejpam-3423	108	30	hypotheses	hypothesis	NOUN
ejpam-3423	108	31	of	of	ADP
ejpam-3423	108	32	lemma	lemma	PROPN
ejpam-3423	108	33	2	2	NUM
ejpam-3423	108	34	given	give	VERB
ejpam-3423	108	35	in	in	ADP
ejpam-3423	108	36	gondal	gondal	NOUN
ejpam-3423	108	37	[	[	X
ejpam-3423	108	38	5	5	NUM
ejpam-3423	108	39	]	]	PUNCT
ejpam-3423	108	40	and	and	CCONJ
ejpam-3423	108	41	that	that	SCONJ
ejpam-3423	108	42	ab	ab	PROPN
ejpam-3423	108	43	=	=	SYM
ejpam-3423	108	44	ba	ba	PROPN
ejpam-3423	108	45	.	.	PUNCT
ejpam-3423	109	1	for	for	ADP
ejpam-3423	109	2	the	the	DET
ejpam-3423	109	3	numerical	numerical	ADJ
ejpam-3423	109	4	solution	solution	NOUN
ejpam-3423	109	5	,	,	PUNCT
ejpam-3423	109	6	we	we	PRON
ejpam-3423	109	7	consider	consider	VERB
ejpam-3423	109	8	the	the	DET
ejpam-3423	109	9	exponential	exponential	ADJ
ejpam-3423	109	10	rungekutta	rungekutta	NOUN
ejpam-3423	109	11	method	method	NOUN
ejpam-3423	109	12	(	(	PUNCT
ejpam-3423	109	13	6	6	NUM
ejpam-3423	109	14	)	)	PUNCT
ejpam-3423	109	15	.	.	PUNCT
ejpam-3423	110	1	also	also	ADV
ejpam-3423	110	2	suppose	suppose	VERB
ejpam-3423	110	3	that	that	SCONJ
ejpam-3423	110	4	g′	g′	NOUN
ejpam-3423	110	5	,	,	PUNCT
ejpam-3423	110	6	g′′	g′′	PROPN
ejpam-3423	110	7	are	be	AUX
ejpam-3423	110	8	bounded	bound	VERB
ejpam-3423	110	9	and	and	CCONJ
ejpam-3423	110	10	g	g	NOUN
ejpam-3423	110	11	:	:	PUNCT
ejpam-3423	111	1	[	[	X
ejpam-3423	111	2	0	0	NUM
ejpam-3423	111	3	,	,	PUNCT
ejpam-3423	111	4	t	t	X
ejpam-3423	111	5	]	]	PUNCT
ejpam-3423	111	6	→	→	SYM
ejpam-3423	111	7	x	x	PUNCT
ejpam-3423	111	8	is	be	AUX
ejpam-3423	111	9	differentiable	differentiable	ADJ
ejpam-3423	111	10	.	.	PUNCT
ejpam-3423	112	1	then	then	ADV
ejpam-3423	112	2	the	the	DET
ejpam-3423	112	3	following	follow	VERB
ejpam-3423	112	4	error	error	NOUN
ejpam-3423	112	5	bound	bind	VERB
ejpam-3423	112	6	‖u(tn)−	‖u(tn)−	ADV
ejpam-3423	112	7	un‖	un‖	NOUN
ejpam-3423	112	8	≤	≤	NUM
ejpam-3423	112	9	ch2	ch2	PROPN
ejpam-3423	112	10	tn	tn	PROPN
ejpam-3423	112	11	(	(	PUNCT
ejpam-3423	112	12	|	|	ADV
ejpam-3423	112	13	log	log	VERB
ejpam-3423	113	1	h	h	NOUN
ejpam-3423	114	1	|	|	ADV
ejpam-3423	114	2	+1	+1	PROPN
ejpam-3423	114	3	)	)	PUNCT
ejpam-3423	114	4	(	(	PUNCT
ejpam-3423	114	5	17	17	NUM
ejpam-3423	114	6	)	)	PUNCT
ejpam-3423	114	7	holds	hold	VERB
ejpam-3423	114	8	uniformly	uniformly	ADV
ejpam-3423	114	9	in	in	ADP
ejpam-3423	114	10	0	0	NUM
ejpam-3423	114	11	≤	≤	NUM
ejpam-3423	114	12	tn	tn	NOUN
ejpam-3423	114	13	≤	≤	PROPN
ejpam-3423	114	14	t	t	PROPN
ejpam-3423	114	15	for	for	ADP
ejpam-3423	114	16	non	non	ADJ
ejpam-3423	114	17	-	-	ADJ
ejpam-3423	114	18	smooth	smooth	ADJ
ejpam-3423	114	19	initial	initial	ADJ
ejpam-3423	114	20	data	datum	NOUN
ejpam-3423	114	21	.	.	PUNCT
ejpam-3423	115	1	proof	proof	NOUN
ejpam-3423	115	2	.	.	PUNCT
ejpam-3423	116	1	we	we	PRON
ejpam-3423	116	2	can	can	AUX
ejpam-3423	116	3	write	write	VERB
ejpam-3423	116	4	the	the	DET
ejpam-3423	116	5	difference	difference	NOUN
ejpam-3423	116	6	between	between	ADP
ejpam-3423	116	7	the	the	DET
ejpam-3423	116	8	g	g	PROPN
ejpam-3423	116	9	terms	term	NOUN
ejpam-3423	116	10	from	from	ADP
ejpam-3423	116	11	(	(	PUNCT
ejpam-3423	116	12	14	14	NUM
ejpam-3423	116	13	)	)	PUNCT
ejpam-3423	116	14	and	and	CCONJ
ejpam-3423	116	15	(	(	PUNCT
ejpam-3423	116	16	6	6	NUM
ejpam-3423	116	17	)	)	PUNCT
ejpam-3423	116	18	as	as	ADP
ejpam-3423	116	19	εn+1(g	εn+1(g	PROPN
ejpam-3423	116	20	)	)	PUNCT
ejpam-3423	117	1	=	=	SYM
ejpam-3423	117	2	∫	∫	PROPN
ejpam-3423	117	3	tn+1	tn+1	PROPN
ejpam-3423	117	4	tn	tn	PROPN
ejpam-3423	117	5	e(tn+1−τ)ag(τ)dτ	e(tn+1−τ)ag(τ)dτ	PROPN
ejpam-3423	117	6	−	−	NOUN
ejpam-3423	117	7	h(ϕ1(ha)−	h(ϕ1(ha)−	NOUN
ejpam-3423	117	8	ϕ2(ha))g(tn	ϕ2(ha))g(tn	NOUN
ejpam-3423	117	9	)	)	PUNCT
ejpam-3423	117	10	−	−	ADP
ejpam-3423	118	1	hϕ2(ha)g(tn	hϕ2(ha)g(tn	PROPN
ejpam-3423	118	2	+	+	NUM
ejpam-3423	118	3	h	h	NOUN
ejpam-3423	118	4	)	)	PUNCT
ejpam-3423	118	5	.	.	PUNCT
ejpam-3423	119	1	(	(	PUNCT
ejpam-3423	119	2	18	18	NUM
ejpam-3423	119	3	)	)	PUNCT
ejpam-3423	119	4	using	use	VERB
ejpam-3423	119	5	results	result	NOUN
ejpam-3423	119	6	(	(	PUNCT
ejpam-3423	119	7	15	15	NUM
ejpam-3423	119	8	)	)	PUNCT
ejpam-3423	119	9	and	and	CCONJ
ejpam-3423	119	10	(	(	PUNCT
ejpam-3423	119	11	16	16	NUM
ejpam-3423	119	12	)	)	PUNCT
ejpam-3423	119	13	in	in	ADP
ejpam-3423	119	14	(	(	PUNCT
ejpam-3423	119	15	18	18	NUM
ejpam-3423	119	16	)	)	PUNCT
ejpam-3423	119	17	and	and	CCONJ
ejpam-3423	119	18	simplifying	simplify	VERB
ejpam-3423	119	19	,	,	PUNCT
ejpam-3423	119	20	we	we	PRON
ejpam-3423	119	21	get	get	VERB
ejpam-3423	119	22	εn+1(g	εn+1(g	PROPN
ejpam-3423	119	23	)	)	PUNCT
ejpam-3423	120	1	=	=	SYM
ejpam-3423	120	2	∫	∫	PROPN
ejpam-3423	120	3	tn+1	tn+1	PROPN
ejpam-3423	120	4	tn	tn	PROPN
ejpam-3423	120	5	e(tn+1−τ)a	e(tn+1−τ)a	PROPN
ejpam-3423	121	1	(	(	PUNCT
ejpam-3423	121	2	g(τ)−	g(τ)−	PROPN
ejpam-3423	121	3	g(tn	g(tn	PROPN
ejpam-3423	121	4	)	)	PUNCT
ejpam-3423	122	1	+	+	CCONJ
ejpam-3423	122	2	τ	τ	PROPN
ejpam-3423	122	3	−	−	PROPN
ejpam-3423	122	4	tn	tn	PROPN
ejpam-3423	122	5	h	h	NOUN
ejpam-3423	122	6	g(tn	g(tn	PROPN
ejpam-3423	122	7	)	)	PUNCT
ejpam-3423	123	1	−	−	PROPN
ejpam-3423	124	1	τ	τ	PROPN
ejpam-3423	124	2	−	−	PROPN
ejpam-3423	124	3	tn	tn	PROPN
ejpam-3423	124	4	h	h	NOUN
ejpam-3423	124	5	g(tn	g(tn	PROPN
ejpam-3423	124	6	+	+	CCONJ
ejpam-3423	124	7	h	h	NOUN
ejpam-3423	124	8	)	)	PUNCT
ejpam-3423	124	9	)	)	PUNCT
ejpam-3423	125	1	dτ	dτ	PROPN
ejpam-3423	125	2	.	.	PROPN
ejpam-3423	126	1	(	(	PUNCT
ejpam-3423	126	2	19	19	NUM
ejpam-3423	126	3	)	)	PUNCT
ejpam-3423	126	4	by	by	ADP
ejpam-3423	126	5	using	use	VERB
ejpam-3423	126	6	taylor	taylor	PROPN
ejpam-3423	126	7	series	series	NOUN
ejpam-3423	126	8	we	we	PRON
ejpam-3423	126	9	can	can	AUX
ejpam-3423	126	10	write	write	VERB
ejpam-3423	126	11	g(τ	g(τ	PROPN
ejpam-3423	126	12	)	)	PUNCT
ejpam-3423	126	13	=	=	SYM
ejpam-3423	126	14	g(tn	g(tn	NOUN
ejpam-3423	126	15	)	)	PUNCT
ejpam-3423	126	16	+	+	CCONJ
ejpam-3423	126	17	(	(	PUNCT
ejpam-3423	126	18	τ	τ	PROPN
ejpam-3423	126	19	−	−	PROPN
ejpam-3423	126	20	tn)g′(tn	tn)g′(tn	NOUN
ejpam-3423	126	21	)	)	PUNCT
ejpam-3423	127	1	+	+	CCONJ
ejpam-3423	127	2	1	1	NUM
ejpam-3423	127	3	2	2	NUM
ejpam-3423	127	4	(	(	PUNCT
ejpam-3423	127	5	τ	τ	NOUN
ejpam-3423	127	6	−	−	NUM
ejpam-3423	127	7	tn)2g′′(tn	tn)2g′′(tn	NUM
ejpam-3423	127	8	)	)	PUNCT
ejpam-3423	127	9	+	+	CCONJ
ejpam-3423	127	10	.	.	PUNCT
ejpam-3423	127	11	.	.	PUNCT
ejpam-3423	127	12	.	.	PUNCT
ejpam-3423	128	1	(	(	PUNCT
ejpam-3423	128	2	20	20	NUM
ejpam-3423	128	3	)	)	PUNCT
ejpam-3423	128	4	g(tn	g(tn	NOUN
ejpam-3423	128	5	+	+	CCONJ
ejpam-3423	128	6	h	h	NOUN
ejpam-3423	128	7	)	)	PUNCT
ejpam-3423	128	8	=	=	SYM
ejpam-3423	128	9	g(tn	g(tn	NOUN
ejpam-3423	128	10	)	)	PUNCT
ejpam-3423	128	11	+	+	NUM
ejpam-3423	128	12	hg′(tn	hg′(tn	X
ejpam-3423	128	13	)	)	PUNCT
ejpam-3423	129	1	+	+	CCONJ
ejpam-3423	129	2	h2	h2	NOUN
ejpam-3423	129	3	2	2	NUM
ejpam-3423	129	4	g′′(tn	g′′(tn	NOUN
ejpam-3423	129	5	)	)	PUNCT
ejpam-3423	130	1	+	+	CCONJ
ejpam-3423	130	2	.	.	PUNCT
ejpam-3423	130	3	.	.	PUNCT
ejpam-3423	130	4	.	.	PUNCT
ejpam-3423	131	1	.	.	PUNCT
ejpam-3423	132	1	(	(	PUNCT
ejpam-3423	132	2	21	21	NUM
ejpam-3423	132	3	)	)	PUNCT
ejpam-3423	132	4	now	now	ADV
ejpam-3423	132	5	substitute	substitute	PROPN
ejpam-3423	132	6	g(τ	g(τ	PROPN
ejpam-3423	132	7	)	)	PUNCT
ejpam-3423	132	8	and	and	CCONJ
ejpam-3423	132	9	g(tn	g(tn	PROPN
ejpam-3423	132	10	+	+	CCONJ
ejpam-3423	132	11	h	h	NOUN
ejpam-3423	132	12	)	)	PUNCT
ejpam-3423	132	13	from	from	ADP
ejpam-3423	132	14	equations	equation	NOUN
ejpam-3423	132	15	(	(	PUNCT
ejpam-3423	132	16	20	20	NUM
ejpam-3423	132	17	)	)	PUNCT
ejpam-3423	132	18	and	and	CCONJ
ejpam-3423	132	19	(	(	PUNCT
ejpam-3423	132	20	21	21	NUM
ejpam-3423	132	21	)	)	PUNCT
ejpam-3423	132	22	in	in	ADP
ejpam-3423	132	23	(	(	PUNCT
ejpam-3423	132	24	19	19	NUM
ejpam-3423	132	25	)	)	PUNCT
ejpam-3423	132	26	and	and	CCONJ
ejpam-3423	132	27	after	after	ADP
ejpam-3423	132	28	simplification	simplification	NOUN
ejpam-3423	132	29	and	and	CCONJ
ejpam-3423	132	30	neglecting	neglect	VERB
ejpam-3423	132	31	higher	high	ADJ
ejpam-3423	132	32	order	order	NOUN
ejpam-3423	132	33	terms	term	NOUN
ejpam-3423	132	34	we	we	PRON
ejpam-3423	132	35	can	can	AUX
ejpam-3423	132	36	write	write	VERB
ejpam-3423	132	37	(	(	PUNCT
ejpam-3423	132	38	19	19	NUM
ejpam-3423	132	39	)	)	PUNCT
ejpam-3423	132	40	as	as	ADP
ejpam-3423	132	41	εn+1(g	εn+1(g	PROPN
ejpam-3423	132	42	)	)	PUNCT
ejpam-3423	133	1	=	=	SYM
ejpam-3423	133	2	∫	∫	PROPN
ejpam-3423	133	3	tn+1	tn+1	PROPN
ejpam-3423	133	4	tn	tn	PROPN
ejpam-3423	133	5	e(tn+1−τ)a	e(tn+1−τ)a	PROPN
ejpam-3423	133	6	(	(	PUNCT
ejpam-3423	133	7	(	(	PUNCT
ejpam-3423	133	8	τ	τ	PROPN
ejpam-3423	133	9	−	−	PROPN
ejpam-3423	134	1	tn)2	tn)2	PROPN
ejpam-3423	134	2	2	2	NUM
ejpam-3423	134	3	−	−	PROPN
ejpam-3423	134	4	h(τ	h(τ	PROPN
ejpam-3423	134	5	−	−	PROPN
ejpam-3423	134	6	tn	tn	PROPN
ejpam-3423	134	7	)	)	PUNCT
ejpam-3423	134	8	2	2	NUM
ejpam-3423	134	9	)	)	PUNCT
ejpam-3423	134	10	g′′(tn)dτ	g′′(tn)dτ	PROPN
ejpam-3423	134	11	,	,	PUNCT
ejpam-3423	134	12	‖εn+1(g)‖	‖εn+1(g)‖	PROPN
ejpam-3423	134	13	≤	≤	NUM
ejpam-3423	134	14	∫	∫	PROPN
ejpam-3423	134	15	tn+1	tn+1	PROPN
ejpam-3423	134	16	tn	tn	PROPN
ejpam-3423	134	17	‖e(tn+1−τ)a‖	‖e(tn+1−τ)a‖	PROPN
ejpam-3423	134	18	∥∥∥(τ	∥∥∥(τ	SYM
ejpam-3423	134	19	−	−	NOUN
ejpam-3423	135	1	tn)2	tn)2	NOUN
ejpam-3423	135	2	2	2	NUM
ejpam-3423	135	3	−	−	PROPN
ejpam-3423	135	4	h(τ	h(τ	PROPN
ejpam-3423	135	5	−	−	PROPN
ejpam-3423	135	6	tn	tn	NOUN
ejpam-3423	135	7	)	)	PUNCT
ejpam-3423	135	8	2	2	NUM
ejpam-3423	135	9	∥∥∥‖g′′(tn)‖dτ	∥∥∥‖g′′(tn)‖dτ	PROPN
ejpam-3423	135	10	,	,	PUNCT
ejpam-3423	135	11	≤	≤	NUM
ejpam-3423	135	12	c	c	PROPN
ejpam-3423	135	13	∫	∫	PROPN
ejpam-3423	135	14	tn+1	tn+1	PROPN
ejpam-3423	135	15	tn	tn	PROPN
ejpam-3423	135	16	(	(	PUNCT
ejpam-3423	135	17	(	(	PUNCT
ejpam-3423	135	18	τ	τ	PROPN
ejpam-3423	135	19	−	−	PROPN
ejpam-3423	136	1	tn)2	tn)2	PROPN
ejpam-3423	136	2	2	2	NUM
ejpam-3423	136	3	+	+	CCONJ
ejpam-3423	136	4	h(τ	h(τ	PROPN
ejpam-3423	136	5	−	−	PROPN
ejpam-3423	136	6	tn	tn	PROPN
ejpam-3423	136	7	)	)	PUNCT
ejpam-3423	136	8	2	2	NUM
ejpam-3423	136	9	)	)	PUNCT
ejpam-3423	136	10	dτ	dτ	NOUN
ejpam-3423	136	11	,	,	PUNCT
ejpam-3423	136	12	≤	≤	NUM
ejpam-3423	136	13	c	c	NOUN
ejpam-3423	136	14	h3	h3	NOUN
ejpam-3423	136	15	6	6	NUM
ejpam-3423	136	16	+	+	NOUN
ejpam-3423	136	17	c	c	NOUN
ejpam-3423	136	18	h3	h3	NOUN
ejpam-3423	136	19	2	2	NUM
ejpam-3423	136	20	=	=	SYM
ejpam-3423	136	21	ch3	ch3	PROPN
ejpam-3423	136	22	.	.	PUNCT
ejpam-3423	137	1	(	(	PUNCT
ejpam-3423	137	2	22	22	NUM
ejpam-3423	137	3	)	)	PUNCT
ejpam-3423	137	4	this	this	PRON
ejpam-3423	137	5	is	be	AUX
ejpam-3423	137	6	the	the	DET
ejpam-3423	137	7	one	one	NUM
ejpam-3423	137	8	way	way	NOUN
ejpam-3423	137	9	to	to	PART
ejpam-3423	137	10	solve	solve	VERB
ejpam-3423	137	11	(	(	PUNCT
ejpam-3423	137	12	19	19	NUM
ejpam-3423	137	13	)	)	PUNCT
ejpam-3423	137	14	in	in	ADP
ejpam-3423	137	15	which	which	PRON
ejpam-3423	137	16	we	we	PRON
ejpam-3423	137	17	have	have	VERB
ejpam-3423	137	18	to	to	PART
ejpam-3423	137	19	assume	assume	VERB
ejpam-3423	137	20	that	that	SCONJ
ejpam-3423	137	21	all	all	DET
ejpam-3423	137	22	higher	high	ADJ
ejpam-3423	137	23	order	order	NOUN
ejpam-3423	137	24	derivatives	derivative	NOUN
ejpam-3423	137	25	of	of	ADP
ejpam-3423	137	26	g	g	PROPN
ejpam-3423	137	27	are	be	AUX
ejpam-3423	137	28	bounded	bound	VERB
ejpam-3423	137	29	.	.	PUNCT
ejpam-3423	138	1	there	there	PRON
ejpam-3423	138	2	is	be	VERB
ejpam-3423	138	3	another	another	DET
ejpam-3423	138	4	good	good	ADJ
ejpam-3423	138	5	and	and	CCONJ
ejpam-3423	138	6	tricky	tricky	ADJ
ejpam-3423	138	7	way	way	NOUN
ejpam-3423	138	8	to	to	PART
ejpam-3423	138	9	prove	prove	VERB
ejpam-3423	138	10	that	that	SCONJ
ejpam-3423	138	11	εn+1(g	εn+1(g	PROPN
ejpam-3423	138	12	)	)	PUNCT
ejpam-3423	138	13	is	be	AUX
ejpam-3423	138	14	bounded	bound	VERB
ejpam-3423	138	15	.	.	PUNCT
ejpam-3423	139	1	for	for	ADP
ejpam-3423	139	2	this	this	DET
ejpam-3423	139	3	trick	trick	NOUN
ejpam-3423	139	4	,	,	PUNCT
ejpam-3423	139	5	we	we	PRON
ejpam-3423	139	6	can	can	AUX
ejpam-3423	139	7	write	write	VERB
ejpam-3423	139	8	g(τ	g(τ	PROPN
ejpam-3423	139	9	)	)	PUNCT
ejpam-3423	139	10	instead	instead	ADV
ejpam-3423	139	11	of	of	ADP
ejpam-3423	139	12	taylor	taylor	PROPN
ejpam-3423	139	13	series	series	PROPN
ejpam-3423	139	14	in	in	ADP
ejpam-3423	139	15	the	the	DET
ejpam-3423	139	16	following	follow	VERB
ejpam-3423	139	17	form	form	NOUN
ejpam-3423	139	18	g(τ	g(τ	PROPN
ejpam-3423	139	19	)	)	PUNCT
ejpam-3423	139	20	=	=	SYM
ejpam-3423	139	21	g(tn	g(tn	NOUN
ejpam-3423	139	22	)	)	PUNCT
ejpam-3423	140	1	+	+	CCONJ
ejpam-3423	140	2	∫	∫	PROPN
ejpam-3423	140	3	τ	τ	PROPN
ejpam-3423	140	4	tn	tn	PROPN
ejpam-3423	140	5	g′(s)ds	g′(s)ds	PROPN
ejpam-3423	140	6	,	,	PUNCT
ejpam-3423	140	7	m.	m.	NOUN
ejpam-3423	140	8	a.	a.	NOUN
ejpam-3423	140	9	gondal	gondal	PROPN
ejpam-3423	140	10	,	,	PUNCT
ejpam-3423	140	11	i.	i.	PROPN
ejpam-3423	140	12	rehman	rehman	PROPN
ejpam-3423	140	13	,	,	PUNCT
ejpam-3423	140	14	a.	a.	NOUN
ejpam-3423	140	15	razzaque	razzaque	NOUN
ejpam-3423	140	16	/	/	SYM
ejpam-3423	140	17	eur	eur	NOUN
ejpam-3423	140	18	.	.	PUNCT
ejpam-3423	141	1	j.	j.	PROPN
ejpam-3423	141	2	pure	pure	PROPN
ejpam-3423	141	3	appl	appl	PROPN
ejpam-3423	141	4	.	.	PROPN
ejpam-3423	141	5	math	math	PROPN
ejpam-3423	141	6	,	,	PUNCT
ejpam-3423	141	7	12	12	NUM
ejpam-3423	141	8	(	(	PUNCT
ejpam-3423	141	9	3	3	NUM
ejpam-3423	141	10	)	)	PUNCT
ejpam-3423	141	11	(	(	PUNCT
ejpam-3423	141	12	2019	2019	NUM
ejpam-3423	141	13	)	)	PUNCT
ejpam-3423	141	14	,	,	PUNCT
ejpam-3423	141	15	1215	1215	NUM
ejpam-3423	141	16	-	-	SYM
ejpam-3423	141	17	1230	1230	NUM
ejpam-3423	141	18	1220	1220	NUM
ejpam-3423	141	19	=	=	SYM
ejpam-3423	141	20	g(tn	g(tn	NOUN
ejpam-3423	141	21	)	)	PUNCT
ejpam-3423	141	22	+	+	CCONJ
ejpam-3423	142	1	∫	∫	PROPN
ejpam-3423	142	2	τ	τ	PROPN
ejpam-3423	142	3	tn	tn	PROPN
ejpam-3423	142	4	1	1	NUM
ejpam-3423	142	5	·	·	PUNCT
ejpam-3423	142	6	g′(s)ds	g′(s)ds	PROPN
ejpam-3423	142	7	,	,	PUNCT
ejpam-3423	142	8	=	=	SYM
ejpam-3423	142	9	g(tn	g(tn	X
ejpam-3423	142	10	)	)	PUNCT
ejpam-3423	142	11	+	+	CCONJ
ejpam-3423	142	12	∫	∫	PROPN
ejpam-3423	142	13	τ	τ	PROPN
ejpam-3423	142	14	tn	tn	PROPN
ejpam-3423	142	15	(	(	PUNCT
ejpam-3423	142	16	s−	s−	PROPN
ejpam-3423	142	17	τ)′g′(s)ds	τ)′g′(s)ds	PROPN
ejpam-3423	142	18	.	.	PUNCT
ejpam-3423	143	1	(	(	PUNCT
ejpam-3423	143	2	23	23	NUM
ejpam-3423	143	3	)	)	PUNCT
ejpam-3423	143	4	integration	integration	NOUN
ejpam-3423	143	5	by	by	ADP
ejpam-3423	143	6	part	part	NOUN
ejpam-3423	143	7	yields	yield	NOUN
ejpam-3423	143	8	g(τ	g(τ	PROPN
ejpam-3423	143	9	)	)	PUNCT
ejpam-3423	143	10	=	=	SYM
ejpam-3423	143	11	g(tn	g(tn	NOUN
ejpam-3423	143	12	)	)	PUNCT
ejpam-3423	144	1	+	+	CCONJ
ejpam-3423	144	2	(	(	PUNCT
ejpam-3423	144	3	τ	τ	PROPN
ejpam-3423	144	4	−	−	PROPN
ejpam-3423	144	5	tn)g′(tn	tn)g′(tn	NOUN
ejpam-3423	144	6	)	)	PUNCT
ejpam-3423	145	1	+	+	CCONJ
ejpam-3423	145	2	∫	∫	PROPN
ejpam-3423	145	3	τ	τ	PROPN
ejpam-3423	145	4	tn	tn	PROPN
ejpam-3423	145	5	(	(	PUNCT
ejpam-3423	145	6	τ	τ	PROPN
ejpam-3423	145	7	−	−	PROPN
ejpam-3423	145	8	s)g′′(s)ds	s)g′′(s)ds	PROPN
ejpam-3423	145	9	,	,	PUNCT
ejpam-3423	145	10	(	(	PUNCT
ejpam-3423	145	11	24	24	NUM
ejpam-3423	145	12	)	)	PUNCT
ejpam-3423	145	13	since	since	SCONJ
ejpam-3423	145	14	we	we	PRON
ejpam-3423	145	15	can	can	AUX
ejpam-3423	145	16	write	write	VERB
ejpam-3423	145	17	g(tn	g(tn	NOUN
ejpam-3423	145	18	+	+	CCONJ
ejpam-3423	145	19	h	h	NOUN
ejpam-3423	145	20	)	)	PUNCT
ejpam-3423	145	21	=	=	SYM
ejpam-3423	145	22	g(tn+1	g(tn+1	PROPN
ejpam-3423	145	23	)	)	PUNCT
ejpam-3423	145	24	.	.	PUNCT
ejpam-3423	146	1	using	use	VERB
ejpam-3423	146	2	(	(	PUNCT
ejpam-3423	146	3	24	24	NUM
ejpam-3423	146	4	)	)	PUNCT
ejpam-3423	146	5	,	,	PUNCT
ejpam-3423	146	6	one	one	PRON
ejpam-3423	146	7	can	can	AUX
ejpam-3423	146	8	write	write	VERB
ejpam-3423	146	9	g(tn+1	g(tn+1	PROPN
ejpam-3423	146	10	)	)	PUNCT
ejpam-3423	146	11	=	=	SYM
ejpam-3423	146	12	g(tn	g(tn	NOUN
ejpam-3423	146	13	)	)	PUNCT
ejpam-3423	146	14	+	+	CCONJ
ejpam-3423	146	15	(	(	PUNCT
ejpam-3423	146	16	tn+1	tn+1	NUM
ejpam-3423	146	17	−	−	NOUN
ejpam-3423	146	18	tn)g′(tn	tn)g′(tn	NOUN
ejpam-3423	146	19	)	)	PUNCT
ejpam-3423	147	1	+	+	CCONJ
ejpam-3423	147	2	∫	∫	PROPN
ejpam-3423	147	3	tn+1	tn+1	PROPN
ejpam-3423	147	4	tn	tn	PROPN
ejpam-3423	147	5	(	(	PUNCT
ejpam-3423	147	6	tn+1	tn+1	PROPN
ejpam-3423	147	7	−	−	PROPN
ejpam-3423	147	8	s)g′′(s)ds	s)g′′(s)ds	PROPN
ejpam-3423	147	9	,	,	PUNCT
ejpam-3423	147	10	=	=	SYM
ejpam-3423	147	11	g(tn	g(tn	NOUN
ejpam-3423	147	12	)	)	PUNCT
ejpam-3423	147	13	+	+	NUM
ejpam-3423	147	14	hg′(tn	hg′(tn	X
ejpam-3423	147	15	)	)	PUNCT
ejpam-3423	148	1	+	+	CCONJ
ejpam-3423	148	2	∫	∫	PROPN
ejpam-3423	148	3	tn+1	tn+1	PROPN
ejpam-3423	148	4	tn	tn	PROPN
ejpam-3423	148	5	(	(	PUNCT
ejpam-3423	148	6	tn+1	tn+1	NOUN
ejpam-3423	148	7	−	−	PROPN
ejpam-3423	148	8	s)g′′(s)ds	s)g′′(s)ds	PROPN
ejpam-3423	148	9	.	.	PUNCT
ejpam-3423	149	1	(	(	PUNCT
ejpam-3423	149	2	25	25	NUM
ejpam-3423	149	3	)	)	PUNCT
ejpam-3423	149	4	now	now	ADV
ejpam-3423	149	5	we	we	PRON
ejpam-3423	149	6	can	can	AUX
ejpam-3423	149	7	use	use	VERB
ejpam-3423	149	8	expressions	expression	NOUN
ejpam-3423	149	9	(	(	PUNCT
ejpam-3423	149	10	24	24	NUM
ejpam-3423	149	11	)	)	PUNCT
ejpam-3423	149	12	and	and	CCONJ
ejpam-3423	149	13	(	(	PUNCT
ejpam-3423	149	14	25	25	NUM
ejpam-3423	149	15	)	)	PUNCT
ejpam-3423	149	16	for	for	ADP
ejpam-3423	149	17	g(τ	g(τ	PROPN
ejpam-3423	149	18	)	)	PUNCT
ejpam-3423	149	19	and	and	CCONJ
ejpam-3423	149	20	g(tn	g(tn	PROPN
ejpam-3423	149	21	+	+	CCONJ
ejpam-3423	149	22	h	h	NOUN
ejpam-3423	149	23	)	)	PUNCT
ejpam-3423	149	24	instead	instead	ADV
ejpam-3423	149	25	of	of	ADP
ejpam-3423	149	26	(	(	PUNCT
ejpam-3423	149	27	20	20	NUM
ejpam-3423	149	28	)	)	PUNCT
ejpam-3423	149	29	and	and	CCONJ
ejpam-3423	149	30	(	(	PUNCT
ejpam-3423	149	31	21	21	NUM
ejpam-3423	149	32	)	)	PUNCT
ejpam-3423	149	33	in	in	ADP
ejpam-3423	149	34	(	(	PUNCT
ejpam-3423	149	35	19	19	NUM
ejpam-3423	149	36	)	)	PUNCT
ejpam-3423	149	37	to	to	PART
ejpam-3423	149	38	get	get	VERB
ejpam-3423	149	39	(	(	PUNCT
ejpam-3423	149	40	22	22	NUM
ejpam-3423	149	41	)	)	PUNCT
ejpam-3423	149	42	.	.	PUNCT
ejpam-3423	150	1	in	in	ADP
ejpam-3423	150	2	this	this	DET
ejpam-3423	150	3	case	case	NOUN
ejpam-3423	150	4	we	we	PRON
ejpam-3423	150	5	only	only	ADV
ejpam-3423	150	6	assume	assume	VERB
ejpam-3423	150	7	that	that	SCONJ
ejpam-3423	150	8	first	first	ADJ
ejpam-3423	150	9	and	and	CCONJ
ejpam-3423	150	10	second	second	ADJ
ejpam-3423	150	11	order	order	NOUN
ejpam-3423	150	12	derivatives	derivative	NOUN
ejpam-3423	150	13	of	of	ADP
ejpam-3423	150	14	g	g	PROPN
ejpam-3423	150	15	are	be	AUX
ejpam-3423	150	16	bounded	bound	VERB
ejpam-3423	150	17	.	.	PUNCT
ejpam-3423	151	1	now	now	ADV
ejpam-3423	151	2	in	in	ADP
ejpam-3423	151	3	the	the	DET
ejpam-3423	151	4	same	same	ADJ
ejpam-3423	151	5	way	way	NOUN
ejpam-3423	151	6	we	we	PRON
ejpam-3423	151	7	can	can	AUX
ejpam-3423	151	8	write	write	VERB
ejpam-3423	151	9	the	the	DET
ejpam-3423	151	10	difference	difference	NOUN
ejpam-3423	151	11	between	between	ADP
ejpam-3423	151	12	the	the	DET
ejpam-3423	151	13	u	u	NOUN
ejpam-3423	151	14	terms	term	NOUN
ejpam-3423	151	15	from	from	ADP
ejpam-3423	151	16	(	(	PUNCT
ejpam-3423	151	17	14	14	NUM
ejpam-3423	151	18	)	)	PUNCT
ejpam-3423	151	19	and	and	CCONJ
ejpam-3423	151	20	(	(	PUNCT
ejpam-3423	151	21	6	6	NUM
ejpam-3423	151	22	)	)	PUNCT
ejpam-3423	151	23	as	as	ADP
ejpam-3423	151	24	εn+1(u	εn+1(u	NOUN
ejpam-3423	151	25	)	)	PUNCT
ejpam-3423	152	1	=	=	SYM
ejpam-3423	152	2	∫	∫	PROPN
ejpam-3423	152	3	tn+1	tn+1	PROPN
ejpam-3423	152	4	tn	tn	PROPN
ejpam-3423	152	5	e(tn+1−τ)abu(τ)dτ	e(tn+1−τ)abu(τ)dτ	PROPN
ejpam-3423	153	1	−	−	PROPN
ejpam-3423	154	1	h(ϕ1(ha)−	h(ϕ1(ha)−	NOUN
ejpam-3423	155	1	ϕ2(ha))bun	ϕ2(ha))bun	PROPN
ejpam-3423	156	1	−	−	PROPN
ejpam-3423	156	2	hϕ2(ha)bun2	hϕ2(ha)bun2	PROPN
ejpam-3423	156	3	,	,	PUNCT
ejpam-3423	156	4	(	(	PUNCT
ejpam-3423	156	5	26	26	NUM
ejpam-3423	156	6	)	)	PUNCT
ejpam-3423	156	7	since	since	SCONJ
ejpam-3423	156	8	we	we	PRON
ejpam-3423	156	9	can	can	AUX
ejpam-3423	156	10	write∫	write∫	VERB
ejpam-3423	156	11	tn+1	tn+1	PROPN
ejpam-3423	156	12	tn	tn	NOUN
ejpam-3423	156	13	e(tn+1−τ)abu(τ)dτ	e(tn+1−τ)abu(τ)dτ	PUNCT
ejpam-3423	157	1	=	=	SYM
ejpam-3423	157	2	∫	∫	PROPN
ejpam-3423	157	3	tn+1	tn+1	PROPN
ejpam-3423	157	4	tn	tn	PROPN
ejpam-3423	157	5	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	158	1	(	(	PUNCT
ejpam-3423	158	2	u(tn)−	u(tn)−	X
ejpam-3423	158	3	τ	τ	PROPN
ejpam-3423	158	4	−	−	PROPN
ejpam-3423	158	5	tn	tn	PROPN
ejpam-3423	158	6	h	h	PROPN
ejpam-3423	158	7	u(tn	u(tn	PROPN
ejpam-3423	158	8	)	)	PUNCT
ejpam-3423	159	1	+	+	CCONJ
ejpam-3423	159	2	τ	τ	PROPN
ejpam-3423	159	3	−	−	PROPN
ejpam-3423	159	4	tn	tn	PROPN
ejpam-3423	159	5	h	h	PROPN
ejpam-3423	159	6	u(tn	u(tn	PROPN
ejpam-3423	159	7	+	+	CCONJ
ejpam-3423	159	8	h	h	NOUN
ejpam-3423	159	9	)	)	PUNCT
ejpam-3423	159	10	+	+	CCONJ
ejpam-3423	159	11	u(τ)−	u(τ)−	PROPN
ejpam-3423	159	12	u(tn	u(tn	PROPN
ejpam-3423	159	13	)	)	PUNCT
ejpam-3423	159	14	+	+	CCONJ
ejpam-3423	159	15	τ	τ	PROPN
ejpam-3423	159	16	−	−	PROPN
ejpam-3423	159	17	tn	tn	PROPN
ejpam-3423	159	18	h	h	PROPN
ejpam-3423	159	19	u(tn)−	u(tn)−	X
ejpam-3423	159	20	τ	τ	PROPN
ejpam-3423	159	21	−	−	PROPN
ejpam-3423	159	22	tn	tn	PROPN
ejpam-3423	159	23	h	h	PROPN
ejpam-3423	159	24	u(tn	u(tn	PROPN
ejpam-3423	160	1	+	+	CCONJ
ejpam-3423	161	1	h	h	NOUN
ejpam-3423	161	2	)	)	PUNCT
ejpam-3423	161	3	)	)	PUNCT
ejpam-3423	161	4	dτ	dτ	NOUN
ejpam-3423	162	1	=	=	NOUN
ejpam-3423	162	2	hϕ1(ha)bu(tn)−	hϕ1(ha)bu(tn)−	NOUN
ejpam-3423	162	3	hϕ2(ha)bu(tn	hϕ2(ha)bu(tn	NOUN
ejpam-3423	162	4	)	)	PUNCT
ejpam-3423	162	5	+	+	NUM
ejpam-3423	162	6	hϕ2(ha)bu(tn+1	hϕ2(ha)bu(tn+1	NUM
ejpam-3423	162	7	)	)	PUNCT
ejpam-3423	163	1	+	+	CCONJ
ejpam-3423	163	2	∫	∫	PROPN
ejpam-3423	163	3	tn+1	tn+1	X
ejpam-3423	163	4	tn	tn	PROPN
ejpam-3423	163	5	e(tn+1−τ)ab	e(tn+1−τ)ab	PROPN
ejpam-3423	164	1	(	(	PUNCT
ejpam-3423	164	2	u(τ)−	u(τ)−	PROPN
ejpam-3423	164	3	u(tn	u(tn	PROPN
ejpam-3423	164	4	)	)	PUNCT
ejpam-3423	165	1	+	+	CCONJ
ejpam-3423	165	2	τ	τ	PROPN
ejpam-3423	165	3	−	−	PROPN
ejpam-3423	165	4	tn	tn	PROPN
ejpam-3423	165	5	h	h	PROPN
ejpam-3423	165	6	u(tn	u(tn	PROPN
ejpam-3423	165	7	)	)	PUNCT
ejpam-3423	165	8	−	−	PROPN
ejpam-3423	166	1	τ	τ	PROPN
ejpam-3423	166	2	−	−	PROPN
ejpam-3423	166	3	tn	tn	PROPN
ejpam-3423	166	4	h	h	PROPN
ejpam-3423	166	5	u(tn	u(tn	PROPN
ejpam-3423	167	1	+	+	CCONJ
ejpam-3423	167	2	h	h	NOUN
ejpam-3423	167	3	)	)	PUNCT
ejpam-3423	168	1	)	)	PUNCT
ejpam-3423	168	2	dτ	dτ	PROPN
ejpam-3423	168	3	.	.	PROPN
ejpam-3423	169	1	(	(	PUNCT
ejpam-3423	169	2	27	27	NUM
ejpam-3423	169	3	)	)	PUNCT
ejpam-3423	169	4	using	use	VERB
ejpam-3423	169	5	(	(	PUNCT
ejpam-3423	169	6	27	27	NUM
ejpam-3423	169	7	)	)	PUNCT
ejpam-3423	169	8	in	in	ADP
ejpam-3423	169	9	(	(	PUNCT
ejpam-3423	169	10	26	26	NUM
ejpam-3423	169	11	)	)	PUNCT
ejpam-3423	169	12	and	and	CCONJ
ejpam-3423	169	13	simplifying	simplify	VERB
ejpam-3423	169	14	,	,	PUNCT
ejpam-3423	169	15	we	we	PRON
ejpam-3423	169	16	have	have	VERB
ejpam-3423	169	17	εn+1(u	εn+1(u	NOUN
ejpam-3423	169	18	)	)	PUNCT
ejpam-3423	170	1	=	=	SYM
ejpam-3423	170	2	hϕ1(ha)b(u(tn)−	hϕ1(ha)b(u(tn)−	NOUN
ejpam-3423	170	3	un)−	un)−	NUM
ejpam-3423	170	4	hϕ2(ha)b(u(tn)−	hϕ2(ha)b(u(tn)−	NOUN
ejpam-3423	170	5	un	un	ADJ
ejpam-3423	170	6	)	)	PUNCT
ejpam-3423	171	1	+	+	CCONJ
ejpam-3423	171	2	hϕ2(ha)b(u(tn+1)−	hϕ2(ha)b(u(tn+1)−	ADJ
ejpam-3423	171	3	un2	un2	NOUN
ejpam-3423	171	4	)	)	PUNCT
ejpam-3423	172	1	+	+	NOUN
ejpam-3423	172	2	rn+1	rn+1	ADJ
ejpam-3423	172	3	,	,	PUNCT
ejpam-3423	172	4	‖εn+1(u)‖	‖εn+1(u)‖	PROPN
ejpam-3423	172	5	≤	≤	NUM
ejpam-3423	172	6	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	172	7	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	172	8	ch‖u(tn+1)−	ch‖u(tn+1)−	ADJ
ejpam-3423	172	9	un2‖+	un2‖+	NOUN
ejpam-3423	172	10	‖rn+1‖	‖rn+1‖	PROPN
ejpam-3423	172	11	,	,	PUNCT
ejpam-3423	172	12	(	(	PUNCT
ejpam-3423	172	13	28	28	NUM
ejpam-3423	172	14	)	)	PUNCT
ejpam-3423	172	15	m.	m.	NOUN
ejpam-3423	172	16	a.	a.	NOUN
ejpam-3423	172	17	gondal	gondal	PROPN
ejpam-3423	172	18	,	,	PUNCT
ejpam-3423	172	19	i.	i.	PROPN
ejpam-3423	172	20	rehman	rehman	PROPN
ejpam-3423	172	21	,	,	PUNCT
ejpam-3423	172	22	a.	a.	NOUN
ejpam-3423	172	23	razzaque	razzaque	NOUN
ejpam-3423	172	24	/	/	SYM
ejpam-3423	172	25	eur	eur	NOUN
ejpam-3423	172	26	.	.	PUNCT
ejpam-3423	173	1	j.	j.	PROPN
ejpam-3423	173	2	pure	pure	PROPN
ejpam-3423	173	3	appl	appl	PROPN
ejpam-3423	173	4	.	.	PROPN
ejpam-3423	173	5	math	math	PROPN
ejpam-3423	173	6	,	,	PUNCT
ejpam-3423	173	7	12	12	NUM
ejpam-3423	173	8	(	(	PUNCT
ejpam-3423	173	9	3	3	NUM
ejpam-3423	173	10	)	)	PUNCT
ejpam-3423	173	11	(	(	PUNCT
ejpam-3423	173	12	2019	2019	NUM
ejpam-3423	173	13	)	)	PUNCT
ejpam-3423	173	14	,	,	PUNCT
ejpam-3423	173	15	1215	1215	NUM
ejpam-3423	173	16	-	-	SYM
ejpam-3423	173	17	1230	1230	NUM
ejpam-3423	173	18	1221	1221	NUM
ejpam-3423	173	19	where	where	SCONJ
ejpam-3423	173	20	rn+1	rn+1	ADV
ejpam-3423	173	21	=	=	SYM
ejpam-3423	173	22	∫	∫	PROPN
ejpam-3423	173	23	tn+1	tn+1	PROPN
ejpam-3423	173	24	tn	tn	PROPN
ejpam-3423	173	25	e(tn+1−τ)ab	e(tn+1−τ)ab	PROPN
ejpam-3423	174	1	(	(	PUNCT
ejpam-3423	174	2	u(τ)−	u(τ)−	PROPN
ejpam-3423	174	3	u(tn	u(tn	PROPN
ejpam-3423	174	4	)	)	PUNCT
ejpam-3423	175	1	+	+	CCONJ
ejpam-3423	175	2	τ	τ	PROPN
ejpam-3423	175	3	−	−	PROPN
ejpam-3423	175	4	tn	tn	PROPN
ejpam-3423	175	5	h	h	PROPN
ejpam-3423	175	6	u(tn)−	u(tn)−	X
ejpam-3423	175	7	τ	τ	PROPN
ejpam-3423	175	8	−	−	PROPN
ejpam-3423	175	9	tn	tn	PROPN
ejpam-3423	175	10	h	h	PROPN
ejpam-3423	175	11	u(tn	u(tn	PROPN
ejpam-3423	176	1	+	+	CCONJ
ejpam-3423	176	2	h	h	NOUN
ejpam-3423	176	3	)	)	PUNCT
ejpam-3423	177	1	)	)	PUNCT
ejpam-3423	177	2	dτ	dτ	PROPN
ejpam-3423	177	3	.	.	PROPN
ejpam-3423	177	4	(	(	PUNCT
ejpam-3423	177	5	29	29	NUM
ejpam-3423	177	6	)	)	PUNCT
ejpam-3423	177	7	since	since	SCONJ
ejpam-3423	177	8	one	one	PRON
ejpam-3423	177	9	can	can	AUX
ejpam-3423	177	10	write	write	VERB
ejpam-3423	177	11	u(tn+1)−	u(tn+1)−	PROPN
ejpam-3423	177	12	un2	un2	NOUN
ejpam-3423	178	1	=	=	SYM
ejpam-3423	178	2	u(tn+1)−	u(tn+1)−	PROPN
ejpam-3423	179	1	un+1	un+1	PROPN
ejpam-3423	180	1	+	+	CCONJ
ejpam-3423	180	2	un+1	un+1	ADV
ejpam-3423	180	3	−	−	PROPN
ejpam-3423	180	4	un2	un2	NOUN
ejpam-3423	180	5	,	,	PUNCT
ejpam-3423	180	6	=	=	SYM
ejpam-3423	180	7	εn+1	εn+1	PROPN
ejpam-3423	180	8	+	+	CCONJ
ejpam-3423	180	9	un+1	un+1	ADV
ejpam-3423	180	10	−	−	PROPN
ejpam-3423	180	11	un2	un2	ADJ
ejpam-3423	180	12	,	,	PUNCT
ejpam-3423	180	13	‖u(tn+1)−	‖u(tn+1)−	PROPN
ejpam-3423	180	14	un2‖	un2‖	PROPN
ejpam-3423	180	15	≤	≤	PROPN
ejpam-3423	180	16	‖εn+1‖+	‖εn+1‖+	NOUN
ejpam-3423	180	17	‖un+1	‖un+1	PUNCT
ejpam-3423	180	18	−	−	PROPN
ejpam-3423	180	19	un2‖.	un2‖.	PROPN
ejpam-3423	180	20	(	(	PUNCT
ejpam-3423	180	21	30	30	NUM
ejpam-3423	180	22	)	)	PUNCT
ejpam-3423	180	23	now	now	ADV
ejpam-3423	180	24	from	from	ADP
ejpam-3423	180	25	(	(	PUNCT
ejpam-3423	180	26	6	6	NUM
ejpam-3423	180	27	)	)	PUNCT
ejpam-3423	180	28	and	and	CCONJ
ejpam-3423	180	29	(	(	PUNCT
ejpam-3423	180	30	7	7	X
ejpam-3423	180	31	)	)	PUNCT
ejpam-3423	180	32	we	we	PRON
ejpam-3423	180	33	can	can	AUX
ejpam-3423	180	34	write	write	VERB
ejpam-3423	180	35	un+1	un+1	ADV
ejpam-3423	180	36	−	−	PROPN
ejpam-3423	180	37	un2	un2	NOUN
ejpam-3423	180	38	=	=	SYM
ejpam-3423	180	39	hϕ2(ha)b(un2	hϕ2(ha)b(un2	CCONJ
ejpam-3423	180	40	−	−	PROPN
ejpam-3423	180	41	un	un	PROPN
ejpam-3423	180	42	)	)	PUNCT
ejpam-3423	180	43	+	+	CCONJ
ejpam-3423	180	44	hϕ2(ha)(g(tn	hϕ2(ha)(g(tn	PROPN
ejpam-3423	180	45	+	+	NUM
ejpam-3423	180	46	h)−	h)−	PROPN
ejpam-3423	180	47	g(tn	g(tn	NOUN
ejpam-3423	180	48	)	)	PUNCT
ejpam-3423	180	49	)	)	PUNCT
ejpam-3423	180	50	,	,	PUNCT
ejpam-3423	180	51	‖un+1	‖un+1	PUNCT
ejpam-3423	180	52	−	−	DET
ejpam-3423	180	53	un2‖	un2‖	NOUN
ejpam-3423	180	54	≤	≤	PROPN
ejpam-3423	180	55	ch‖un2	ch‖un2	NUM
ejpam-3423	180	56	−	−	PROPN
ejpam-3423	180	57	un‖+	un‖+	PROPN
ejpam-3423	180	58	ch2	ch2	PROPN
ejpam-3423	180	59	,	,	PUNCT
ejpam-3423	180	60	(	(	PUNCT
ejpam-3423	180	61	31	31	NUM
ejpam-3423	180	62	)	)	PUNCT
ejpam-3423	180	63	since	since	SCONJ
ejpam-3423	180	64	‖g(tn	‖g(tn	PROPN
ejpam-3423	180	65	+	+	CCONJ
ejpam-3423	180	66	h)−	h)−	PROPN
ejpam-3423	180	67	g(tn)‖	g(tn)‖	NOUN
ejpam-3423	180	68	≤	≤	NUM
ejpam-3423	180	69	ch	ch	NOUN
ejpam-3423	180	70	.	.	PUNCT
ejpam-3423	181	1	now	now	ADV
ejpam-3423	181	2	un2	un2	VERB
ejpam-3423	181	3	−	−	PROPN
ejpam-3423	181	4	un	un	PROPN
ejpam-3423	181	5	=	=	PROPN
ejpam-3423	181	6	un2	un2	PROPN
ejpam-3423	182	1	−	−	PROPN
ejpam-3423	182	2	un+1	un+1	PROPN
ejpam-3423	183	1	+	+	CCONJ
ejpam-3423	183	2	un+1	un+1	PROPN
ejpam-3423	183	3	−	−	PROPN
ejpam-3423	183	4	un	un	PROPN
ejpam-3423	183	5	,	,	PUNCT
ejpam-3423	183	6	‖un2	‖un2	PROPN
ejpam-3423	183	7	−	−	PUNCT
ejpam-3423	183	8	un‖	un‖	PROPN
ejpam-3423	183	9	≤	≤	PROPN
ejpam-3423	183	10	‖un+1	‖un+1	PUNCT
ejpam-3423	183	11	−	−	PROPN
ejpam-3423	183	12	un‖+	un‖+	PROPN
ejpam-3423	183	13	‖un+1	‖un+1	SYM
ejpam-3423	183	14	−	−	PROPN
ejpam-3423	183	15	un2‖.	un2‖.	PROPN
ejpam-3423	183	16	(	(	PUNCT
ejpam-3423	183	17	32	32	NUM
ejpam-3423	183	18	)	)	PUNCT
ejpam-3423	183	19	from	from	ADP
ejpam-3423	183	20	(	(	PUNCT
ejpam-3423	183	21	6	6	NUM
ejpam-3423	183	22	)	)	PUNCT
ejpam-3423	183	23	we	we	PRON
ejpam-3423	183	24	can	can	AUX
ejpam-3423	183	25	write	write	VERB
ejpam-3423	183	26	for	for	ADP
ejpam-3423	183	27	n	n	X
ejpam-3423	183	28	≥	≥	NUM
ejpam-3423	183	29	1	1	NUM
ejpam-3423	183	30	un+1	un+1	PROPN
ejpam-3423	183	31	−	−	PROPN
ejpam-3423	183	32	un	un	PROPN
ejpam-3423	183	33	=	=	PROPN
ejpam-3423	183	34	ehaun	ehaun	PROPN
ejpam-3423	183	35	−	−	PROPN
ejpam-3423	183	36	un	un	PROPN
ejpam-3423	184	1	+	+	ADJ
ejpam-3423	184	2	o(h	o(h	PROPN
ejpam-3423	184	3	)	)	PUNCT
ejpam-3423	184	4	,	,	PUNCT
ejpam-3423	185	1	=	=	SYM
ejpam-3423	185	2	(	(	PUNCT
ejpam-3423	185	3	eha	eha	PROPN
ejpam-3423	185	4	−	−	PROPN
ejpam-3423	185	5	1)un	1)un	PROPN
ejpam-3423	186	1	+	+	NOUN
ejpam-3423	186	2	o(h	o(h	ADJ
ejpam-3423	186	3	)	)	PUNCT
ejpam-3423	186	4	=	=	NOUN
ejpam-3423	186	5	haϕ1(ha)un	haϕ1(ha)un	VERB
ejpam-3423	187	1	+	+	ADJ
ejpam-3423	187	2	o(h	o(h	ADJ
ejpam-3423	187	3	)	)	PUNCT
ejpam-3423	187	4	,	,	PUNCT
ejpam-3423	187	5	=	=	SYM
ejpam-3423	187	6	haϕ1(ha)(u(tn	haϕ1(ha)(u(tn	X
ejpam-3423	187	7	)	)	PUNCT
ejpam-3423	188	1	+	+	CCONJ
ejpam-3423	188	2	εn	εn	ADJ
ejpam-3423	188	3	)	)	PUNCT
ejpam-3423	188	4	+	+	NOUN
ejpam-3423	188	5	o(h	o(h	ADJ
ejpam-3423	188	6	)	)	PUNCT
ejpam-3423	188	7	,	,	PUNCT
ejpam-3423	188	8	since	since	SCONJ
ejpam-3423	188	9	un	un	PROPN
ejpam-3423	188	10	=	=	PROPN
ejpam-3423	188	11	u(tn	u(tn	PROPN
ejpam-3423	188	12	)	)	PUNCT
ejpam-3423	189	1	+	+	CCONJ
ejpam-3423	189	2	un	un	PROPN
ejpam-3423	189	3	−	−	PROPN
ejpam-3423	189	4	u(tn	u(tn	PROPN
ejpam-3423	189	5	)	)	PUNCT
ejpam-3423	189	6	,	,	PUNCT
ejpam-3423	189	7	=	=	SYM
ejpam-3423	190	1	haϕ1(ha)εn	haϕ1(ha)εn	NOUN
ejpam-3423	191	1	+	+	X
ejpam-3423	191	2	haϕ1(ha)u(tn	haϕ1(ha)u(tn	X
ejpam-3423	191	3	)	)	PUNCT
ejpam-3423	192	1	+	+	NOUN
ejpam-3423	192	2	o(h	o(h	ADJ
ejpam-3423	192	3	)	)	PUNCT
ejpam-3423	192	4	,	,	PUNCT
ejpam-3423	192	5	‖un+1	‖un+1	PUNCT
ejpam-3423	193	1	−	−	NOUN
ejpam-3423	193	2	un‖	un‖	PROPN
ejpam-3423	193	3	≤	≤	NOUN
ejpam-3423	193	4	c‖εn‖+	c‖εn‖+	NOUN
ejpam-3423	193	5	ch	ch	NOUN
ejpam-3423	193	6	·	·	PUNCT
ejpam-3423	193	7	c	c	X
ejpam-3423	193	8	tn	tn	PUNCT
ejpam-3423	194	1	+	+	NOUN
ejpam-3423	194	2	o(h	o(h	ADJ
ejpam-3423	194	3	)	)	PUNCT
ejpam-3423	194	4	,	,	PUNCT
ejpam-3423	194	5	(	(	PUNCT
ejpam-3423	194	6	33	33	NUM
ejpam-3423	194	7	)	)	PUNCT
ejpam-3423	194	8	since	since	SCONJ
ejpam-3423	194	9	‖au(tn)‖	‖au(tn)‖	NUM
ejpam-3423	194	10	≤	≤	NUM
ejpam-3423	194	11	c	c	NOUN
ejpam-3423	194	12	tn	tn	PROPN
ejpam-3423	194	13	for	for	ADP
ejpam-3423	194	14	n	n	X
ejpam-3423	194	15	>	>	X
ejpam-3423	194	16	0	0	X
ejpam-3423	194	17	.	.	PUNCT
ejpam-3423	195	1	using	use	VERB
ejpam-3423	195	2	(	(	PUNCT
ejpam-3423	195	3	33	33	NUM
ejpam-3423	195	4	)	)	PUNCT
ejpam-3423	195	5	in	in	ADP
ejpam-3423	195	6	(	(	PUNCT
ejpam-3423	195	7	32	32	NUM
ejpam-3423	195	8	)	)	PUNCT
ejpam-3423	195	9	and	and	CCONJ
ejpam-3423	195	10	then	then	ADV
ejpam-3423	195	11	(	(	PUNCT
ejpam-3423	195	12	32	32	NUM
ejpam-3423	195	13	)	)	PUNCT
ejpam-3423	195	14	in	in	ADP
ejpam-3423	195	15	(	(	PUNCT
ejpam-3423	195	16	31	31	NUM
ejpam-3423	195	17	)	)	PUNCT
ejpam-3423	195	18	yields	yield	NOUN
ejpam-3423	195	19	‖un+1	‖un+1	PUNCT
ejpam-3423	195	20	−	−	PRON
ejpam-3423	195	21	un2‖	un2‖	NOUN
ejpam-3423	195	22	≤	≤	NOUN
ejpam-3423	195	23	ch‖un+1	ch‖un+1	VERB
ejpam-3423	195	24	−	−	NOUN
ejpam-3423	195	25	un2‖+	un2‖+	ADV
ejpam-3423	195	26	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	195	27	ch2	ch2	NOUN
ejpam-3423	195	28	·	·	PUNCT
ejpam-3423	195	29	1	1	NUM
ejpam-3423	195	30	tn	tn	PROPN
ejpam-3423	195	31	+	+	CCONJ
ejpam-3423	195	32	ch2	ch2	PROPN
ejpam-3423	195	33	,	,	PUNCT
ejpam-3423	195	34	(	(	PUNCT
ejpam-3423	195	35	1−	1−	NUM
ejpam-3423	195	36	hc)‖un+1	hc)‖un+1	PROPN
ejpam-3423	195	37	−	−	PROPN
ejpam-3423	195	38	un2‖	un2‖	NOUN
ejpam-3423	195	39	≤	≤	NOUN
ejpam-3423	196	1	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	196	2	ch2	ch2	NOUN
ejpam-3423	196	3	·	·	PUNCT
ejpam-3423	196	4	1	1	NUM
ejpam-3423	196	5	tn	tn	PROPN
ejpam-3423	196	6	+	+	CCONJ
ejpam-3423	196	7	ch2	ch2	PROPN
ejpam-3423	196	8	,	,	PUNCT
ejpam-3423	196	9	(	(	PUNCT
ejpam-3423	196	10	34	34	NUM
ejpam-3423	196	11	)	)	PUNCT
ejpam-3423	196	12	h	h	NOUN
ejpam-3423	196	13	small	small	ADJ
ejpam-3423	196	14	gives	give	VERB
ejpam-3423	196	15	that	that	PRON
ejpam-3423	196	16	hc	hc	PRON
ejpam-3423	196	17	≤	≤	ADV
ejpam-3423	196	18	1	1	NUM
ejpam-3423	196	19	2	2	NUM
ejpam-3423	196	20	,	,	PUNCT
ejpam-3423	196	21	therefore	therefore	ADV
ejpam-3423	196	22	for	for	ADP
ejpam-3423	196	23	n	n	PROPN
ejpam-3423	196	24	≥	≥	NUM
ejpam-3423	196	25	1	1	NUM
ejpam-3423	196	26	‖un+1	‖un+1	SYM
ejpam-3423	196	27	−	−	NOUN
ejpam-3423	196	28	un2‖	un2‖	NOUN
ejpam-3423	196	29	≤	≤	NOUN
ejpam-3423	197	1	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	197	2	ch2	ch2	PROPN
ejpam-3423	197	3	tn	tn	PROPN
ejpam-3423	197	4	+	+	CCONJ
ejpam-3423	197	5	ch2	ch2	PROPN
ejpam-3423	197	6	.	.	PUNCT
ejpam-3423	198	1	(	(	PUNCT
ejpam-3423	198	2	35	35	NUM
ejpam-3423	198	3	)	)	PUNCT
ejpam-3423	198	4	using	use	VERB
ejpam-3423	198	5	(	(	PUNCT
ejpam-3423	198	6	35	35	NUM
ejpam-3423	198	7	)	)	PUNCT
ejpam-3423	198	8	in	in	ADP
ejpam-3423	198	9	(	(	PUNCT
ejpam-3423	198	10	30	30	NUM
ejpam-3423	198	11	)	)	PUNCT
ejpam-3423	198	12	and	and	CCONJ
ejpam-3423	198	13	then	then	ADV
ejpam-3423	198	14	(	(	PUNCT
ejpam-3423	198	15	30	30	NUM
ejpam-3423	198	16	)	)	PUNCT
ejpam-3423	198	17	in	in	ADP
ejpam-3423	198	18	(	(	PUNCT
ejpam-3423	198	19	28	28	NUM
ejpam-3423	198	20	)	)	PUNCT
ejpam-3423	198	21	gives	give	VERB
ejpam-3423	198	22	for	for	ADP
ejpam-3423	198	23	n	n	PRON
ejpam-3423	198	24	≥	≥	NUM
ejpam-3423	198	25	1	1	NUM
ejpam-3423	198	26	‖εn+1(u)‖	‖εn+1(u)‖	PROPN
ejpam-3423	198	27	≤	≤	PUNCT
ejpam-3423	198	28	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	198	29	ch‖εn+1‖+	ch‖εn+1‖+	PROPN
ejpam-3423	198	30	ch2‖εn‖+	ch2‖εn‖+	PROPN
ejpam-3423	198	31	ch3	ch3	PROPN
ejpam-3423	198	32	tn	tn	PROPN
ejpam-3423	198	33	m.	m.	PROPN
ejpam-3423	198	34	a.	a.	PROPN
ejpam-3423	198	35	gondal	gondal	PROPN
ejpam-3423	198	36	,	,	PUNCT
ejpam-3423	198	37	i.	i.	PROPN
ejpam-3423	198	38	rehman	rehman	PROPN
ejpam-3423	198	39	,	,	PUNCT
ejpam-3423	198	40	a.	a.	NOUN
ejpam-3423	198	41	razzaque	razzaque	NOUN
ejpam-3423	198	42	/	/	SYM
ejpam-3423	198	43	eur	eur	NOUN
ejpam-3423	198	44	.	.	PUNCT
ejpam-3423	199	1	j.	j.	PROPN
ejpam-3423	199	2	pure	pure	PROPN
ejpam-3423	199	3	appl	appl	PROPN
ejpam-3423	199	4	.	.	PROPN
ejpam-3423	199	5	math	math	PROPN
ejpam-3423	199	6	,	,	PUNCT
ejpam-3423	199	7	12	12	NUM
ejpam-3423	199	8	(	(	PUNCT
ejpam-3423	199	9	3	3	NUM
ejpam-3423	199	10	)	)	PUNCT
ejpam-3423	199	11	(	(	PUNCT
ejpam-3423	199	12	2019	2019	NUM
ejpam-3423	199	13	)	)	PUNCT
ejpam-3423	199	14	,	,	PUNCT
ejpam-3423	199	15	1215	1215	NUM
ejpam-3423	199	16	-	-	SYM
ejpam-3423	199	17	1230	1230	NUM
ejpam-3423	199	18	1222	1222	NUM
ejpam-3423	199	19	+	+	CCONJ
ejpam-3423	199	20	ch3	ch3	PROPN
ejpam-3423	199	21	+	+	PROPN
ejpam-3423	199	22	‖rn+1‖.	‖rn+1‖.	PROPN
ejpam-3423	199	23	(	(	PUNCT
ejpam-3423	199	24	36	36	NUM
ejpam-3423	199	25	)	)	PUNCT
ejpam-3423	199	26	for	for	ADP
ejpam-3423	199	27	n	n	NOUN
ejpam-3423	199	28	=	=	SYM
ejpam-3423	199	29	0	0	NUM
ejpam-3423	199	30	,	,	PUNCT
ejpam-3423	199	31	we	we	PRON
ejpam-3423	199	32	use	use	VERB
ejpam-3423	199	33	(	(	PUNCT
ejpam-3423	199	34	30	30	NUM
ejpam-3423	199	35	)	)	PUNCT
ejpam-3423	199	36	to	to	PART
ejpam-3423	199	37	obtain	obtain	VERB
ejpam-3423	199	38	‖u(t1)−	‖u(t1)−	PROPN
ejpam-3423	199	39	u02‖	u02‖	PUNCT
ejpam-3423	199	40	≤	≤	PROPN
ejpam-3423	199	41	‖ε1‖+	‖ε1‖+	NUM
ejpam-3423	199	42	‖u1	‖u1	NOUN
ejpam-3423	199	43	−	−	NOUN
ejpam-3423	199	44	u02‖	u02‖	X
ejpam-3423	199	45	,	,	PUNCT
ejpam-3423	199	46	with	with	ADP
ejpam-3423	199	47	the	the	DET
ejpam-3423	199	48	help	help	NOUN
ejpam-3423	199	49	of	of	ADP
ejpam-3423	199	50	(	(	PUNCT
ejpam-3423	199	51	31	31	NUM
ejpam-3423	199	52	)	)	PUNCT
ejpam-3423	200	1	,	,	PUNCT
ejpam-3423	200	2	we	we	PRON
ejpam-3423	200	3	get	get	VERB
ejpam-3423	200	4	‖u(t1)−	‖u(t1)−	PROPN
ejpam-3423	200	5	u02‖	u02‖	X
ejpam-3423	200	6	≤	≤	PROPN
ejpam-3423	200	7	‖ε1‖+	‖ε1‖+	NUM
ejpam-3423	200	8	ch‖u02	ch‖u02	X
ejpam-3423	200	9	−	−	X
ejpam-3423	200	10	u0‖+	u0‖+	ADJ
ejpam-3423	200	11	ch2	ch2	PROPN
ejpam-3423	200	12	.	.	PUNCT
ejpam-3423	201	1	(	(	PUNCT
ejpam-3423	201	2	37	37	NUM
ejpam-3423	201	3	)	)	PUNCT
ejpam-3423	201	4	as	as	ADP
ejpam-3423	201	5	‖u02	‖u02	X
ejpam-3423	201	6	−	−	PROPN
ejpam-3423	201	7	u0‖	u0‖	PROPN
ejpam-3423	201	8	≤	≤	NUM
ejpam-3423	201	9	‖u02‖+	‖u02‖+	NUM
ejpam-3423	201	10	‖u0‖	‖u0‖	NOUN
ejpam-3423	201	11	≤	≤	NUM
ejpam-3423	201	12	c	c	NOUN
ejpam-3423	201	13	,	,	PUNCT
ejpam-3423	201	14	(	(	PUNCT
ejpam-3423	201	15	38	38	NUM
ejpam-3423	201	16	)	)	PUNCT
ejpam-3423	201	17	using	use	VERB
ejpam-3423	201	18	(	(	PUNCT
ejpam-3423	201	19	38	38	NUM
ejpam-3423	201	20	)	)	PUNCT
ejpam-3423	201	21	in	in	ADP
ejpam-3423	201	22	(	(	PUNCT
ejpam-3423	201	23	37	37	NUM
ejpam-3423	201	24	)	)	PUNCT
ejpam-3423	201	25	and	and	CCONJ
ejpam-3423	201	26	then	then	ADV
ejpam-3423	201	27	(	(	PUNCT
ejpam-3423	201	28	37	37	NUM
ejpam-3423	201	29	)	)	PUNCT
ejpam-3423	201	30	in	in	ADP
ejpam-3423	201	31	(	(	PUNCT
ejpam-3423	201	32	28	28	NUM
ejpam-3423	201	33	)	)	PUNCT
ejpam-3423	201	34	gives	give	VERB
ejpam-3423	201	35	for	for	ADP
ejpam-3423	201	36	n	n	NOUN
ejpam-3423	201	37	=	=	SYM
ejpam-3423	201	38	0	0	SYM
ejpam-3423	201	39	‖ε1(u)‖	‖ε1(u)‖	PROPN
ejpam-3423	201	40	≤	≤	PROPN
ejpam-3423	201	41	ch‖ε0‖+	ch‖ε0‖+	ADJ
ejpam-3423	201	42	ch‖ε1‖+	ch‖ε1‖+	NOUN
ejpam-3423	201	43	ch2	ch2	NOUN
ejpam-3423	201	44	+	+	CCONJ
ejpam-3423	201	45	ch3	ch3	PROPN
ejpam-3423	201	46	+	+	SYM
ejpam-3423	201	47	‖r1‖.	‖r1‖.	PROPN
ejpam-3423	201	48	(	(	PUNCT
ejpam-3423	201	49	39	39	NUM
ejpam-3423	201	50	)	)	PUNCT
ejpam-3423	201	51	now	now	ADV
ejpam-3423	201	52	we	we	PRON
ejpam-3423	201	53	want	want	VERB
ejpam-3423	201	54	to	to	PART
ejpam-3423	201	55	prove	prove	VERB
ejpam-3423	201	56	‖rn+1‖	‖rn+1‖	PROPN
ejpam-3423	201	57	is	be	AUX
ejpam-3423	201	58	bounded	bound	VERB
ejpam-3423	201	59	.	.	PUNCT
ejpam-3423	202	1	for	for	ADP
ejpam-3423	202	2	this	this	PRON
ejpam-3423	202	3	,	,	PUNCT
ejpam-3423	202	4	we	we	PRON
ejpam-3423	202	5	can	can	AUX
ejpam-3423	202	6	write	write	VERB
ejpam-3423	202	7	u(τ	u(τ	ADV
ejpam-3423	202	8	)	)	PUNCT
ejpam-3423	202	9	and	and	CCONJ
ejpam-3423	202	10	u(tn+1	u(tn+1	VERB
ejpam-3423	202	11	)	)	PUNCT
ejpam-3423	202	12	by	by	ADP
ejpam-3423	202	13	using	use	VERB
ejpam-3423	202	14	the	the	DET
ejpam-3423	202	15	same	same	ADJ
ejpam-3423	202	16	concept	concept	NOUN
ejpam-3423	202	17	of	of	ADP
ejpam-3423	202	18	(	(	PUNCT
ejpam-3423	202	19	24	24	NUM
ejpam-3423	202	20	)	)	PUNCT
ejpam-3423	202	21	in	in	ADP
ejpam-3423	202	22	the	the	DET
ejpam-3423	202	23	following	follow	VERB
ejpam-3423	202	24	form	form	NOUN
ejpam-3423	202	25	u(τ	u(τ	NUM
ejpam-3423	202	26	)	)	PUNCT
ejpam-3423	202	27	=	=	SYM
ejpam-3423	202	28	u(tn	u(tn	NOUN
ejpam-3423	202	29	)	)	PUNCT
ejpam-3423	203	1	+	+	CCONJ
ejpam-3423	203	2	(	(	PUNCT
ejpam-3423	203	3	τ	τ	PROPN
ejpam-3423	203	4	−	−	PROPN
ejpam-3423	203	5	tn)u′(tn	tn)u′(tn	PROPN
ejpam-3423	203	6	)	)	PUNCT
ejpam-3423	204	1	+	+	CCONJ
ejpam-3423	204	2	∫	∫	PROPN
ejpam-3423	204	3	τ	τ	PROPN
ejpam-3423	204	4	tn	tn	PROPN
ejpam-3423	204	5	(	(	PUNCT
ejpam-3423	204	6	τ	τ	PROPN
ejpam-3423	204	7	−	−	PROPN
ejpam-3423	204	8	s)u′′(s)ds	s)u′′(s)ds	PROPN
ejpam-3423	204	9	,	,	PUNCT
ejpam-3423	204	10	(	(	PUNCT
ejpam-3423	204	11	40	40	NUM
ejpam-3423	204	12	)	)	PUNCT
ejpam-3423	204	13	u(tn+1	u(tn+1	PROPN
ejpam-3423	204	14	)	)	PUNCT
ejpam-3423	204	15	=	=	SYM
ejpam-3423	204	16	u(tn	u(tn	NOUN
ejpam-3423	204	17	)	)	PUNCT
ejpam-3423	204	18	+	+	NUM
ejpam-3423	204	19	hu′(tn	hu′(tn	X
ejpam-3423	204	20	)	)	PUNCT
ejpam-3423	205	1	+	+	CCONJ
ejpam-3423	205	2	∫	∫	PROPN
ejpam-3423	205	3	tn+1	tn+1	PROPN
ejpam-3423	205	4	tn	tn	PROPN
ejpam-3423	205	5	(	(	PUNCT
ejpam-3423	205	6	tn+1	tn+1	PROPN
ejpam-3423	205	7	−	−	PROPN
ejpam-3423	205	8	s)u′′(s)ds	s)u′′(s)ds	PROPN
ejpam-3423	205	9	.	.	PUNCT
ejpam-3423	206	1	(	(	PUNCT
ejpam-3423	206	2	41	41	NUM
ejpam-3423	206	3	)	)	PUNCT
ejpam-3423	206	4	substituting	substitute	VERB
ejpam-3423	206	5	the	the	DET
ejpam-3423	206	6	expressions	expression	NOUN
ejpam-3423	206	7	for	for	ADP
ejpam-3423	206	8	u(τ	u(τ	PROPN
ejpam-3423	206	9	)	)	PUNCT
ejpam-3423	206	10	from	from	ADP
ejpam-3423	206	11	(	(	PUNCT
ejpam-3423	206	12	40	40	NUM
ejpam-3423	206	13	)	)	PUNCT
ejpam-3423	206	14	and	and	CCONJ
ejpam-3423	206	15	u(tn	u(tn	PROPN
ejpam-3423	206	16	+	+	CCONJ
ejpam-3423	206	17	h	h	NOUN
ejpam-3423	206	18	)	)	PUNCT
ejpam-3423	206	19	=	=	PUNCT
ejpam-3423	206	20	u(tn+1	u(tn+1	PROPN
ejpam-3423	206	21	)	)	PUNCT
ejpam-3423	206	22	from	from	ADP
ejpam-3423	206	23	(	(	PUNCT
ejpam-3423	206	24	41	41	NUM
ejpam-3423	206	25	)	)	PUNCT
ejpam-3423	206	26	in	in	ADP
ejpam-3423	206	27	(	(	PUNCT
ejpam-3423	206	28	29	29	NUM
ejpam-3423	206	29	)	)	PUNCT
ejpam-3423	206	30	and	and	CCONJ
ejpam-3423	206	31	simplifying	simplify	VERB
ejpam-3423	206	32	,	,	PUNCT
ejpam-3423	206	33	we	we	PRON
ejpam-3423	206	34	get	get	VERB
ejpam-3423	206	35	rn+1	rn+1	PRON
ejpam-3423	206	36	=	=	SYM
ejpam-3423	206	37	∫	∫	PROPN
ejpam-3423	206	38	tn+1	tn+1	PROPN
ejpam-3423	206	39	tn	tn	PROPN
ejpam-3423	206	40	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	207	1	(	(	PUNCT
ejpam-3423	207	2	∫	∫	PROPN
ejpam-3423	207	3	τ	τ	PROPN
ejpam-3423	207	4	tn	tn	PROPN
ejpam-3423	207	5	(	(	PUNCT
ejpam-3423	207	6	τ	τ	X
ejpam-3423	207	7	−	−	PROPN
ejpam-3423	207	8	s)u′′(s)ds−	s)u′′(s)ds−	PROPN
ejpam-3423	207	9	τ	τ	PROPN
ejpam-3423	207	10	−	−	PROPN
ejpam-3423	207	11	tn	tn	PROPN
ejpam-3423	208	1	h	h	PROPN
ejpam-3423	208	2	∫	∫	PROPN
ejpam-3423	208	3	tn+1	tn+1	PROPN
ejpam-3423	208	4	tn	tn	PROPN
ejpam-3423	208	5	(	(	PUNCT
ejpam-3423	208	6	tn+1−	tn+1−	PROPN
ejpam-3423	208	7	s)u′′(s)ds	s)u′′(s)ds	PROPN
ejpam-3423	208	8	)	)	PUNCT
ejpam-3423	208	9	dτ	dτ	PROPN
ejpam-3423	208	10	.	.	PROPN
ejpam-3423	208	11	(	(	PUNCT
ejpam-3423	208	12	42	42	NUM
ejpam-3423	208	13	)	)	PUNCT
ejpam-3423	208	14	rn+1	rn+1	PROPN
ejpam-3423	209	1	=	=	SYM
ejpam-3423	209	2	rn+1,2	rn+1,2	PROPN
ejpam-3423	209	3	+	+	PROPN
ejpam-3423	209	4	rn+1,3	rn+1,3	PROPN
ejpam-3423	209	5	(	(	PUNCT
ejpam-3423	209	6	43	43	NUM
ejpam-3423	209	7	)	)	PUNCT
ejpam-3423	209	8	where	where	SCONJ
ejpam-3423	209	9	rn+1,2	rn+1,2	ADJ
ejpam-3423	209	10	=	=	PUNCT
ejpam-3423	209	11	∫	∫	PROPN
ejpam-3423	209	12	tn+1	tn+1	PROPN
ejpam-3423	209	13	tn	tn	PROPN
ejpam-3423	209	14	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	210	1	(	(	PUNCT
ejpam-3423	210	2	∫	∫	PROPN
ejpam-3423	210	3	τ	τ	PROPN
ejpam-3423	210	4	tn	tn	PROPN
ejpam-3423	210	5	(	(	PUNCT
ejpam-3423	210	6	τ	τ	PROPN
ejpam-3423	210	7	−	−	PROPN
ejpam-3423	210	8	s)u′′(s)ds	s)u′′(s)ds	PROPN
ejpam-3423	210	9	)	)	PUNCT
ejpam-3423	210	10	dτ	dτ	PROPN
ejpam-3423	210	11	,	,	PUNCT
ejpam-3423	210	12	(	(	PUNCT
ejpam-3423	210	13	44	44	NUM
ejpam-3423	210	14	)	)	PUNCT
ejpam-3423	210	15	and	and	CCONJ
ejpam-3423	210	16	rn+1,3	rn+1,3	PROPN
ejpam-3423	210	17	=	=	SYM
ejpam-3423	210	18	∫	∫	PROPN
ejpam-3423	210	19	tn+1	tn+1	PROPN
ejpam-3423	210	20	tn	tn	PROPN
ejpam-3423	210	21	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	211	1	(	(	PUNCT
ejpam-3423	211	2	τ	τ	X
ejpam-3423	211	3	−	−	PROPN
ejpam-3423	211	4	tn	tn	PROPN
ejpam-3423	211	5	h	h	PROPN
ejpam-3423	211	6	∫	∫	PROPN
ejpam-3423	211	7	tn+1	tn+1	PROPN
ejpam-3423	211	8	tn	tn	PROPN
ejpam-3423	211	9	(	(	PUNCT
ejpam-3423	211	10	tn+1	tn+1	PROPN
ejpam-3423	211	11	−	−	PROPN
ejpam-3423	211	12	s)u′′(s)ds	s)u′′(s)ds	PROPN
ejpam-3423	211	13	)	)	PUNCT
ejpam-3423	211	14	dτ	dτ	PROPN
ejpam-3423	211	15	.	.	PROPN
ejpam-3423	211	16	(	(	PUNCT
ejpam-3423	211	17	45	45	NUM
ejpam-3423	211	18	)	)	PUNCT
ejpam-3423	211	19	now	now	ADV
ejpam-3423	211	20	to	to	PART
ejpam-3423	211	21	solve	solve	VERB
ejpam-3423	211	22	(	(	PUNCT
ejpam-3423	211	23	44	44	NUM
ejpam-3423	211	24	)	)	PUNCT
ejpam-3423	211	25	,	,	PUNCT
ejpam-3423	211	26	we	we	PRON
ejpam-3423	211	27	use	use	VERB
ejpam-3423	211	28	the	the	DET
ejpam-3423	211	29	identity	identity	NOUN
ejpam-3423	211	30	aa−1	aa−1	NOUN
ejpam-3423	211	31	=	=	PUNCT
ejpam-3423	212	1	i	i	PRON
ejpam-3423	212	2	and	and	CCONJ
ejpam-3423	212	3	get	get	VERB
ejpam-3423	212	4	rn+1,2	rn+1,2	PROPN
ejpam-3423	212	5	=	=	PUNCT
ejpam-3423	212	6	∫	∫	PROPN
ejpam-3423	212	7	tn+1	tn+1	PROPN
ejpam-3423	212	8	tn	tn	PROPN
ejpam-3423	212	9	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	213	1	(	(	PUNCT
ejpam-3423	213	2	∫	∫	PROPN
ejpam-3423	213	3	τ	τ	PROPN
ejpam-3423	213	4	tn	tn	PROPN
ejpam-3423	213	5	(	(	PUNCT
ejpam-3423	213	6	τ	τ	PROPN
ejpam-3423	213	7	−	−	PROPN
ejpam-3423	213	8	s)aa−1u′′(s)ds	s)aa−1u′′(s)ds	PUNCT
ejpam-3423	213	9	)	)	PUNCT
ejpam-3423	213	10	dτ	dτ	NOUN
ejpam-3423	213	11	.	.	PROPN
ejpam-3423	213	12	(	(	PUNCT
ejpam-3423	213	13	46	46	NUM
ejpam-3423	213	14	)	)	PUNCT
ejpam-3423	213	15	a	a	PRON
ejpam-3423	213	16	can	can	AUX
ejpam-3423	213	17	commute	commute	VERB
ejpam-3423	213	18	with	with	ADP
ejpam-3423	213	19	b	b	PROPN
ejpam-3423	213	20	,	,	PUNCT
ejpam-3423	213	21	i.e.	i.e.	X
ejpam-3423	213	22	,	,	PUNCT
ejpam-3423	213	23	ab	ab	PROPN
ejpam-3423	213	24	=	=	SYM
ejpam-3423	213	25	ba	ba	PROPN
ejpam-3423	213	26	,	,	PUNCT
ejpam-3423	213	27	if	if	SCONJ
ejpam-3423	213	28	b	b	NOUN
ejpam-3423	213	29	is	be	AUX
ejpam-3423	213	30	a	a	DET
ejpam-3423	213	31	convolution	convolution	NOUN
ejpam-3423	213	32	integral	integral	ADJ
ejpam-3423	213	33	,	,	PUNCT
ejpam-3423	213	34	therefore	therefore	ADV
ejpam-3423	213	35	rn+1,2	rn+1,2	PROPN
ejpam-3423	213	36	=	=	PROPN
ejpam-3423	213	37	a	a	DET
ejpam-3423	213	38	∫	∫	PROPN
ejpam-3423	213	39	tn+1	tn+1	NOUN
ejpam-3423	213	40	tn	tn	PROPN
ejpam-3423	213	41	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	214	1	(	(	PUNCT
ejpam-3423	214	2	∫	∫	PROPN
ejpam-3423	214	3	τ	τ	PROPN
ejpam-3423	214	4	tn	tn	PROPN
ejpam-3423	214	5	(	(	PUNCT
ejpam-3423	214	6	τ	τ	X
ejpam-3423	214	7	−	−	PROPN
ejpam-3423	214	8	s)a−1u′′(s)ds	s)a−1u′′(s)ds	CCONJ
ejpam-3423	214	9	)	)	PUNCT
ejpam-3423	214	10	dτ	dτ	NOUN
ejpam-3423	214	11	.	.	PROPN
ejpam-3423	214	12	(	(	PUNCT
ejpam-3423	214	13	47	47	NUM
ejpam-3423	214	14	)	)	PUNCT
ejpam-3423	214	15	m.	m.	NOUN
ejpam-3423	214	16	a.	a.	NOUN
ejpam-3423	214	17	gondal	gondal	PROPN
ejpam-3423	214	18	,	,	PUNCT
ejpam-3423	214	19	i.	i.	PROPN
ejpam-3423	214	20	rehman	rehman	PROPN
ejpam-3423	214	21	,	,	PUNCT
ejpam-3423	214	22	a.	a.	NOUN
ejpam-3423	214	23	razzaque	razzaque	NOUN
ejpam-3423	214	24	/	/	SYM
ejpam-3423	214	25	eur	eur	NOUN
ejpam-3423	214	26	.	.	PUNCT
ejpam-3423	215	1	j.	j.	PROPN
ejpam-3423	215	2	pure	pure	PROPN
ejpam-3423	215	3	appl	appl	PROPN
ejpam-3423	215	4	.	.	PROPN
ejpam-3423	215	5	math	math	PROPN
ejpam-3423	215	6	,	,	PUNCT
ejpam-3423	215	7	12	12	NUM
ejpam-3423	215	8	(	(	PUNCT
ejpam-3423	215	9	3	3	NUM
ejpam-3423	215	10	)	)	PUNCT
ejpam-3423	215	11	(	(	PUNCT
ejpam-3423	215	12	2019	2019	NUM
ejpam-3423	215	13	)	)	PUNCT
ejpam-3423	215	14	,	,	PUNCT
ejpam-3423	215	15	1215	1215	NUM
ejpam-3423	215	16	-	-	SYM
ejpam-3423	215	17	1230	1230	NUM
ejpam-3423	215	18	1223	1223	NUM
ejpam-3423	215	19	since	since	SCONJ
ejpam-3423	215	20	εn+1	εn+1	NUM
ejpam-3423	215	21	=	=	SYM
ejpam-3423	215	22	u(tn+1)−	u(tn+1)−	PROPN
ejpam-3423	215	23	un+1	un+1	PROPN
ejpam-3423	215	24	,	,	PUNCT
ejpam-3423	215	25	=	=	SYM
ejpam-3423	215	26	ehaεn	ehaεn	NOUN
ejpam-3423	215	27	+	+	CCONJ
ejpam-3423	215	28	εn+1(g	εn+1(g	PROPN
ejpam-3423	215	29	)	)	PUNCT
ejpam-3423	216	1	+	+	CCONJ
ejpam-3423	216	2	εn+1(u	εn+1(u	NOUN
ejpam-3423	216	3	)	)	PUNCT
ejpam-3423	216	4	.	.	PUNCT
ejpam-3423	217	1	(	(	PUNCT
ejpam-3423	217	2	48	48	NUM
ejpam-3423	217	3	)	)	PUNCT
ejpam-3423	217	4	therefore	therefore	ADV
ejpam-3423	217	5	from	from	ADP
ejpam-3423	217	6	(	(	PUNCT
ejpam-3423	217	7	6	6	NUM
ejpam-3423	217	8	)	)	PUNCT
ejpam-3423	217	9	and	and	CCONJ
ejpam-3423	217	10	(	(	PUNCT
ejpam-3423	217	11	14	14	NUM
ejpam-3423	217	12	)	)	PUNCT
ejpam-3423	217	13	and	and	CCONJ
ejpam-3423	217	14	using	use	VERB
ejpam-3423	217	15	(	(	PUNCT
ejpam-3423	217	16	22	22	NUM
ejpam-3423	217	17	)	)	PUNCT
ejpam-3423	217	18	and	and	CCONJ
ejpam-3423	217	19	(	(	PUNCT
ejpam-3423	217	20	36	36	NUM
ejpam-3423	217	21	)	)	PUNCT
ejpam-3423	217	22	,	,	PUNCT
ejpam-3423	217	23	we	we	PRON
ejpam-3423	217	24	get	get	VERB
ejpam-3423	217	25	from	from	ADP
ejpam-3423	217	26	(	(	PUNCT
ejpam-3423	217	27	48	48	NUM
ejpam-3423	217	28	)	)	PUNCT
ejpam-3423	217	29	the	the	DET
ejpam-3423	217	30	error	error	NOUN
ejpam-3423	217	31	recursion	recursion	NOUN
ejpam-3423	217	32	for	for	ADP
ejpam-3423	217	33	n	n	X
ejpam-3423	217	34	>	>	X
ejpam-3423	217	35	0	0	PUNCT
ejpam-3423	218	1	‖εn+1‖	‖εn+1‖	ADJ
ejpam-3423	218	2	≤	≤	ADJ
ejpam-3423	218	3	‖eha‖‖εn‖+	‖eha‖‖εn‖+	PROPN
ejpam-3423	218	4	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	218	5	ch‖εn+1‖+	ch‖εn+1‖+	PROPN
ejpam-3423	218	6	ch2‖εn‖	ch2‖εn‖	NOUN
ejpam-3423	218	7	+	+	CCONJ
ejpam-3423	218	8	ch3	ch3	PROPN
ejpam-3423	218	9	tn	tn	PROPN
ejpam-3423	218	10	+	+	PROPN
ejpam-3423	218	11	ch3	ch3	PROPN
ejpam-3423	218	12	+	+	PROPN
ejpam-3423	218	13	‖rn+1‖.	‖rn+1‖.	PROPN
ejpam-3423	218	14	(	(	PUNCT
ejpam-3423	218	15	49	49	NUM
ejpam-3423	218	16	)	)	PUNCT
ejpam-3423	218	17	h	h	NOUN
ejpam-3423	218	18	small	small	ADJ
ejpam-3423	218	19	gives	give	VERB
ejpam-3423	218	20	that	that	PRON
ejpam-3423	218	21	hc	hc	PRON
ejpam-3423	218	22	≤	≤	ADV
ejpam-3423	218	23	1	1	NUM
ejpam-3423	218	24	2	2	NUM
ejpam-3423	218	25	,	,	PUNCT
ejpam-3423	218	26	therefore	therefore	ADV
ejpam-3423	218	27	‖εn+1‖	‖εn+1‖	VERB
ejpam-3423	218	28	≤	≤	ADJ
ejpam-3423	218	29	c‖eha‖‖εn‖+	c‖eha‖‖εn‖+	ADP
ejpam-3423	218	30	ch‖εn‖+	ch‖εn‖+	PROPN
ejpam-3423	218	31	ch2‖εn‖+	ch2‖εn‖+	PROPN
ejpam-3423	218	32	ch3	ch3	PROPN
ejpam-3423	218	33	tn	tn	PROPN
ejpam-3423	218	34	+	+	PROPN
ejpam-3423	218	35	ch3	ch3	PROPN
ejpam-3423	218	36	+	+	CCONJ
ejpam-3423	218	37	‖rn+1‖	‖rn+1‖	PROPN
ejpam-3423	218	38	,	,	PUNCT
ejpam-3423	218	39	...	...	PUNCT
ejpam-3423	219	1	‖εn‖	‖εn‖	PROPN
ejpam-3423	219	2	≤	≤	NOUN
ejpam-3423	219	3	c‖enha‖‖ε0‖+	c‖enha‖‖ε0‖+	NUM
ejpam-3423	219	4	ch	ch	NOUN
ejpam-3423	219	5	n−1∑	n−1∑	PROPN
ejpam-3423	219	6	j=0	j=0	PROPN
ejpam-3423	219	7	‖e(n−j−1)ha‖‖εj‖+	‖e(n−j−1)ha‖‖εj‖+	SYM
ejpam-3423	219	8	ch2	ch2	PROPN
ejpam-3423	219	9	n−1∑	n−1∑	PROPN
ejpam-3423	219	10	j=0	j=0	PROPN
ejpam-3423	219	11	‖e(n−j−1)ha‖‖εj‖	‖e(n−j−1)ha‖‖εj‖	PUNCT
ejpam-3423	220	1	+	+	CCONJ
ejpam-3423	220	2	ch3	ch3	PROPN
ejpam-3423	220	3	n−1∑	n−1∑	ADJ
ejpam-3423	220	4	j=1	j=1	NOUN
ejpam-3423	220	5	‖e(n−j−1)ha‖‖	‖e(n−j−1)ha‖‖	NUM
ejpam-3423	220	6	1	1	NUM
ejpam-3423	220	7	tj	tj	NOUN
ejpam-3423	220	8	‖+	‖+	PROPN
ejpam-3423	220	9	ch2	ch2	PROPN
ejpam-3423	220	10	+	+	CCONJ
ejpam-3423	220	11	ch3	ch3	PROPN
ejpam-3423	220	12	n−1∑	n−1∑	PROPN
ejpam-3423	220	13	j=0	j=0	PROPN
ejpam-3423	220	14	‖e(n−j−1)ha‖	‖e(n−j−1)ha‖	PROPN
ejpam-3423	220	15	+	+	CCONJ
ejpam-3423	220	16	n−1∑	n−1∑	PROPN
ejpam-3423	220	17	j=0	j=0	PROPN
ejpam-3423	220	18	‖e(n−j−1)harj+1‖.	‖e(n−j−1)harj+1‖.	PROPN
ejpam-3423	220	19	(	(	PUNCT
ejpam-3423	220	20	50	50	NUM
ejpam-3423	220	21	)	)	PUNCT
ejpam-3423	220	22	using	use	VERB
ejpam-3423	220	23	lemma	lemma	PROPN
ejpam-3423	220	24	2	2	NUM
ejpam-3423	220	25	given	give	VERB
ejpam-3423	220	26	in	in	ADP
ejpam-3423	220	27	gondal	gondal	NOUN
ejpam-3423	220	28	[	[	X
ejpam-3423	220	29	5	5	NUM
ejpam-3423	220	30	]	]	PUNCT
ejpam-3423	220	31	and	and	CCONJ
ejpam-3423	220	32	fact	fact	NOUN
ejpam-3423	220	33	that	that	SCONJ
ejpam-3423	220	34	tj	tj	PROPN
ejpam-3423	220	35	=	=	SYM
ejpam-3423	220	36	jh	jh	PROPN
ejpam-3423	220	37	,	,	PUNCT
ejpam-3423	220	38	we	we	PRON
ejpam-3423	220	39	get	get	VERB
ejpam-3423	220	40	‖εn‖	‖εn‖	ADJ
ejpam-3423	220	41	≤	≤	NOUN
ejpam-3423	220	42	c‖ε0‖+	c‖ε0‖+	NUM
ejpam-3423	220	43	ch	ch	NOUN
ejpam-3423	220	44	n−1∑	n−1∑	PROPN
ejpam-3423	220	45	j=0	j=0	PROPN
ejpam-3423	220	46	‖εj‖+	‖εj‖+	PROPN
ejpam-3423	220	47	ch2	ch2	PROPN
ejpam-3423	220	48	n−1∑	n−1∑	NUM
ejpam-3423	220	49	j=0	j=0	PROPN
ejpam-3423	220	50	‖εj‖+	‖εj‖+	PROPN
ejpam-3423	220	51	ch2	ch2	PROPN
ejpam-3423	220	52	n−1∑	n−1∑	PROPN
ejpam-3423	220	53	j=1	j=1	PROPN
ejpam-3423	220	54	‖1	‖1	PROPN
ejpam-3423	220	55	j	j	PROPN
ejpam-3423	220	56	‖+	‖+	PROPN
ejpam-3423	220	57	ch3	ch3	PROPN
ejpam-3423	220	58	·	·	PUNCT
ejpam-3423	220	59	n	n	PROPN
ejpam-3423	220	60	+	+	CCONJ
ejpam-3423	220	61	n−1∑	n−1∑	NUM
ejpam-3423	220	62	j=0	j=0	PROPN
ejpam-3423	220	63	‖e(n−j−1)harj+1‖.	‖e(n−j−1)harj+1‖.	PROPN
ejpam-3423	220	64	(	(	PUNCT
ejpam-3423	220	65	51	51	NUM
ejpam-3423	220	66	)	)	PUNCT
ejpam-3423	220	67	since	since	SCONJ
ejpam-3423	220	68	we	we	PRON
ejpam-3423	220	69	know	know	VERB
ejpam-3423	220	70	that	that	DET
ejpam-3423	220	71	nh	nh	PROPN
ejpam-3423	220	72	=	=	SYM
ejpam-3423	220	73	t	t	PROPN
ejpam-3423	220	74	,	,	PUNCT
ejpam-3423	220	75	ch2	ch2	PROPN
ejpam-3423	220	76	≤	≤	PROPN
ejpam-3423	220	77	ch	ch	NOUN
ejpam-3423	220	78	and	and	CCONJ
ejpam-3423	220	79	‖ε0‖	‖ε0‖	NOUN
ejpam-3423	220	80	=	=	PUNCT
ejpam-3423	220	81	0	0	NUM
ejpam-3423	220	82	,	,	PUNCT
ejpam-3423	220	83	and	and	CCONJ
ejpam-3423	220	84	using	use	VERB
ejpam-3423	220	85	the	the	DET
ejpam-3423	220	86	result	result	NOUN
ejpam-3423	220	87	(	(	PUNCT
ejpam-3423	220	88	?	?	PUNCT
ejpam-3423	220	89	?	?	PUNCT
ejpam-3423	220	90	)	)	PUNCT
ejpam-3423	220	91	,	,	PUNCT
ejpam-3423	220	92	we	we	PRON
ejpam-3423	220	93	get	get	VERB
ejpam-3423	220	94	‖εn‖	‖εn‖	ADJ
ejpam-3423	220	95	≤	≤	NUM
ejpam-3423	220	96	ch	ch	NOUN
ejpam-3423	220	97	n−1∑	n−1∑	PROPN
ejpam-3423	220	98	j=0	j=0	PROPN
ejpam-3423	220	99	‖εj‖+	‖εj‖+	PROPN
ejpam-3423	220	100	ch2(c+	ch2(c+	NUM
ejpam-3423	220	101	|	|	ADV
ejpam-3423	220	102	log	log	NOUN
ejpam-3423	220	103	h	h	NOUN
ejpam-3423	220	104	|	|	NOUN
ejpam-3423	220	105	)	)	PUNCT
ejpam-3423	221	1	+	+	CCONJ
ejpam-3423	221	2	ch2	ch2	PROPN
ejpam-3423	221	3	t	t	NOUN
ejpam-3423	221	4	+	+	CCONJ
ejpam-3423	221	5	n−1∑	n−1∑	PROPN
ejpam-3423	221	6	j=0	j=0	PROPN
ejpam-3423	221	7	‖e(n−j−1)harj+1‖	‖e(n−j−1)harj+1‖	PROPN
ejpam-3423	221	8	,	,	PUNCT
ejpam-3423	221	9	≤	≤	NUM
ejpam-3423	221	10	ch	ch	NOUN
ejpam-3423	221	11	n−1∑	n−1∑	PROPN
ejpam-3423	221	12	j=0	j=0	PROPN
ejpam-3423	221	13	‖εj‖+	‖εj‖+	PROPN
ejpam-3423	221	14	ch2(1	ch2(1	NOUN
ejpam-3423	221	15	+	+	CCONJ
ejpam-3423	221	16	|	|	ADV
ejpam-3423	221	17	log	log	NOUN
ejpam-3423	221	18	h	h	NOUN
ejpam-3423	221	19	|	|	NOUN
ejpam-3423	221	20	)	)	PUNCT
ejpam-3423	222	1	+	+	CCONJ
ejpam-3423	223	1	n−1∑	n−1∑	NUM
ejpam-3423	223	2	j=0	j=0	PROPN
ejpam-3423	223	3	‖e(n−j−1)harj+1‖.	‖e(n−j−1)harj+1‖.	PROPN
ejpam-3423	223	4	(	(	PUNCT
ejpam-3423	223	5	52	52	NUM
ejpam-3423	223	6	)	)	PUNCT
ejpam-3423	223	7	m.	m.	NOUN
ejpam-3423	223	8	a.	a.	NOUN
ejpam-3423	223	9	gondal	gondal	PROPN
ejpam-3423	223	10	,	,	PUNCT
ejpam-3423	223	11	i.	i.	PROPN
ejpam-3423	223	12	rehman	rehman	PROPN
ejpam-3423	223	13	,	,	PUNCT
ejpam-3423	223	14	a.	a.	NOUN
ejpam-3423	223	15	razzaque	razzaque	NOUN
ejpam-3423	223	16	/	/	SYM
ejpam-3423	223	17	eur	eur	NOUN
ejpam-3423	223	18	.	.	PUNCT
ejpam-3423	224	1	j.	j.	PROPN
ejpam-3423	224	2	pure	pure	PROPN
ejpam-3423	224	3	appl	appl	PROPN
ejpam-3423	224	4	.	.	PROPN
ejpam-3423	224	5	math	math	PROPN
ejpam-3423	224	6	,	,	PUNCT
ejpam-3423	224	7	12	12	NUM
ejpam-3423	224	8	(	(	PUNCT
ejpam-3423	224	9	3	3	NUM
ejpam-3423	224	10	)	)	PUNCT
ejpam-3423	224	11	(	(	PUNCT
ejpam-3423	224	12	2019	2019	NUM
ejpam-3423	224	13	)	)	PUNCT
ejpam-3423	224	14	,	,	PUNCT
ejpam-3423	224	15	1215	1215	NUM
ejpam-3423	224	16	-	-	SYM
ejpam-3423	224	17	1230	1230	NUM
ejpam-3423	224	18	1224	1224	NUM
ejpam-3423	224	19	after	after	ADP
ejpam-3423	224	20	this	this	PRON
ejpam-3423	224	21	,	,	PUNCT
ejpam-3423	224	22	it	it	PRON
ejpam-3423	224	23	is	be	AUX
ejpam-3423	224	24	proved	prove	VERB
ejpam-3423	224	25	that	that	SCONJ
ejpam-3423	224	26	∑n−1	∑n−1	SCONJ
ejpam-3423	224	27	j=0	j=0	PROPN
ejpam-3423	224	28	‖e(n−j−1)harj+1‖	‖e(n−j−1)harj+1‖	PROPN
ejpam-3423	224	29	is	be	AUX
ejpam-3423	224	30	bounded	bound	VERB
ejpam-3423	224	31	.	.	PUNCT
ejpam-3423	225	1	for	for	ADP
ejpam-3423	225	2	this	this	PRON
ejpam-3423	225	3	we	we	PRON
ejpam-3423	225	4	can	can	AUX
ejpam-3423	225	5	write	write	VERB
ejpam-3423	225	6	n−1∑	n−1∑	PROPN
ejpam-3423	225	7	j=0	j=0	PROPN
ejpam-3423	225	8	‖e(n−j−1)harj+1‖	‖e(n−j−1)harj+1‖	PROPN
ejpam-3423	225	9	=	=	PROPN
ejpam-3423	225	10	n−1∑	n−1∑	PROPN
ejpam-3423	225	11	j=0	j=0	VERB
ejpam-3423	225	12	‖e(n−j−1)harj+1,2‖+	‖e(n−j−1)harj+1,2‖+	PROPN
ejpam-3423	225	13	n−1∑	n−1∑	NUM
ejpam-3423	225	14	j=0	j=0	PROPN
ejpam-3423	225	15	‖e(n−j−1)harj+1,3‖.	‖e(n−j−1)harj+1,3‖.	PROPN
ejpam-3423	225	16	(	(	PUNCT
ejpam-3423	225	17	53	53	NUM
ejpam-3423	225	18	)	)	PUNCT
ejpam-3423	225	19	from	from	ADP
ejpam-3423	225	20	(	(	PUNCT
ejpam-3423	225	21	47	47	NUM
ejpam-3423	225	22	)	)	PUNCT
ejpam-3423	225	23	we	we	PRON
ejpam-3423	225	24	can	can	AUX
ejpam-3423	225	25	write∑n−2	write∑n−2	VERB
ejpam-3423	225	26	j=1	j=1	PROPN
ejpam-3423	225	27	‖e(n−j−1)harj+1,2‖	‖e(n−j−1)harj+1,2‖	NOUN
ejpam-3423	225	28	(	(	PUNCT
ejpam-3423	225	29	54	54	NUM
ejpam-3423	225	30	)	)	PUNCT
ejpam-3423	225	31	=	=	SYM
ejpam-3423	225	32	n−2∑	n−2∑	PROPN
ejpam-3423	225	33	j=1	j=1	PROPN
ejpam-3423	225	34	‖ae(n−j−1)ha	‖ae(n−j−1)ha	ADJ
ejpam-3423	225	35	∫	∫	PROPN
ejpam-3423	225	36	tj+1	tj+1	PROPN
ejpam-3423	225	37	tj	tj	X
ejpam-3423	225	38	e(tj+1−τ)ab	e(tj+1−τ)ab	PROPN
ejpam-3423	225	39	(	(	PUNCT
ejpam-3423	225	40	∫	∫	PROPN
ejpam-3423	225	41	τ	τ	PROPN
ejpam-3423	225	42	tj	tj	PROPN
ejpam-3423	225	43	(	(	PUNCT
ejpam-3423	225	44	τ	τ	PROPN
ejpam-3423	225	45	−	−	PROPN
ejpam-3423	225	46	s)a−1u′′(s)ds	s)a−1u′′(s)ds	CCONJ
ejpam-3423	225	47	)	)	PUNCT
ejpam-3423	225	48	dτ‖	dτ‖	PROPN
ejpam-3423	225	49	,	,	PUNCT
ejpam-3423	225	50	≤	≤	NUM
ejpam-3423	225	51	n−2∑	n−2∑	NUM
ejpam-3423	225	52	j=1	j=1	PROPN
ejpam-3423	225	53	‖ae(n−j−1)ha‖	‖ae(n−j−1)ha‖	PROPN
ejpam-3423	225	54	∫	∫	PROPN
ejpam-3423	225	55	tj+1	tj+1	PROPN
ejpam-3423	225	56	tj	tj	PROPN
ejpam-3423	225	57	‖e(tj+1−τ)a‖‖b‖	‖e(tj+1−τ)a‖‖b‖	PROPN
ejpam-3423	225	58	(	(	PUNCT
ejpam-3423	225	59	∫	∫	PROPN
ejpam-3423	225	60	τ	τ	PROPN
ejpam-3423	225	61	tj	tj	PROPN
ejpam-3423	225	62	(	(	PUNCT
ejpam-3423	225	63	τ	τ	PROPN
ejpam-3423	225	64	−	−	PROPN
ejpam-3423	225	65	s)‖a−1u′′(s)‖ds	s)‖a−1u′′(s)‖ds	PROPN
ejpam-3423	225	66	)	)	PUNCT
ejpam-3423	225	67	dτ	dτ	NOUN
ejpam-3423	225	68	.	.	PROPN
ejpam-3423	225	69	using	use	VERB
ejpam-3423	225	70	lemma	lemma	PROPN
ejpam-3423	225	71	2	2	NUM
ejpam-3423	225	72	given	give	VERB
ejpam-3423	225	73	in	in	ADP
ejpam-3423	225	74	gondal	gondal	NOUN
ejpam-3423	225	75	[	[	X
ejpam-3423	225	76	5	5	NUM
ejpam-3423	225	77	]	]	PUNCT
ejpam-3423	225	78	and	and	CCONJ
ejpam-3423	225	79	lemma	lemma	PROPN
ejpam-3423	225	80	1	1	NUM
ejpam-3423	225	81	in	in	ADP
ejpam-3423	225	82	above	above	ADP
ejpam-3423	225	83	equation	equation	NOUN
ejpam-3423	225	84	and	and	CCONJ
ejpam-3423	225	85	then	then	ADV
ejpam-3423	225	86	integrating	integrate	VERB
ejpam-3423	225	87	,	,	PUNCT
ejpam-3423	225	88	we	we	PRON
ejpam-3423	225	89	get	get	VERB
ejpam-3423	225	90	n−1∑	n−1∑	NUM
ejpam-3423	225	91	j=0	j=0	VERB
ejpam-3423	225	92	‖e(n−j−1)harj+1,2‖	‖e(n−j−1)harj+1,2‖	VERB
ejpam-3423	225	93	≤	≤	NOUN
ejpam-3423	226	1	n−2∑	n−2∑	NUM
ejpam-3423	226	2	j=1	j=1	PROPN
ejpam-3423	226	3	c	c	PROPN
ejpam-3423	226	4	tn−j−1	tn−j−1	PRON
ejpam-3423	226	5	·	·	PUNCT
ejpam-3423	226	6	h3	h3	NOUN
ejpam-3423	226	7	·	·	PUNCT
ejpam-3423	226	8	c	c	NOUN
ejpam-3423	226	9	tj	tj	NOUN
ejpam-3423	226	10	+	+	NUM
ejpam-3423	226	11	term	term	NOUN
ejpam-3423	226	12	for	for	ADP
ejpam-3423	226	13	j	j	PROPN
ejpam-3423	226	14	=	=	SYM
ejpam-3423	226	15	0	0	PUNCT
ejpam-3423	226	16	+	+	NUM
ejpam-3423	226	17	term	term	NOUN
ejpam-3423	226	18	for	for	ADP
ejpam-3423	226	19	j	j	PROPN
ejpam-3423	226	20	=	=	SYM
ejpam-3423	226	21	n−	n−	PROPN
ejpam-3423	226	22	1	1	NUM
ejpam-3423	226	23	,	,	PUNCT
ejpam-3423	226	24	=	=	SYM
ejpam-3423	226	25	ch3	ch3	PROPN
ejpam-3423	226	26	n−2∑	n−2∑	PROPN
ejpam-3423	226	27	j=1	j=1	PROPN
ejpam-3423	226	28	1	1	NUM
ejpam-3423	226	29	tn−j−1tj	tn−j−1tj	NOUN
ejpam-3423	226	30	+	+	CCONJ
ejpam-3423	226	31	‖e(n−1)har1,2‖+	‖e(n−1)har1,2‖+	PROPN
ejpam-3423	226	32	‖rn,2‖	‖rn,2‖	NOUN
ejpam-3423	226	33	,	,	PUNCT
ejpam-3423	226	34	=	=	SYM
ejpam-3423	226	35	ch3	ch3	PROPN
ejpam-3423	227	1	[	[	X
ejpam-3423	227	2	n/2]∑	n/2]∑	VERB
ejpam-3423	227	3	j=1	j=1	ADJ
ejpam-3423	227	4	1	1	NUM
ejpam-3423	227	5	tn−j−1tj	tn−j−1tj	NOUN
ejpam-3423	227	6	+	+	CCONJ
ejpam-3423	227	7	ch3	ch3	PROPN
ejpam-3423	227	8	n−2∑	n−2∑	PROPN
ejpam-3423	227	9	j=[n/2]+1	j=[n/2]+1	NOUN
ejpam-3423	227	10	1	1	NUM
ejpam-3423	227	11	tn−j−1tj	tn−j−1tj	NOUN
ejpam-3423	227	12	+	+	CCONJ
ejpam-3423	227	13	‖e(n−1)har1,2‖	‖e(n−1)har1,2‖	X
ejpam-3423	227	14	+	+	CCONJ
ejpam-3423	227	15	‖rn,2‖	‖rn,2‖	ADJ
ejpam-3423	227	16	,	,	PUNCT
ejpam-3423	227	17	≤	≤	PROPN
ejpam-3423	227	18	ch3	ch3	PROPN
ejpam-3423	227	19	tn	tn	PUNCT
ejpam-3423	228	1	[	[	X
ejpam-3423	228	2	n/2]∑	n/2]∑	VERB
ejpam-3423	228	3	j=1	j=1	NOUN
ejpam-3423	228	4	1	1	NUM
ejpam-3423	228	5	tj	tj	NOUN
ejpam-3423	228	6	+	+	PROPN
ejpam-3423	228	7	ch3	ch3	PROPN
ejpam-3423	228	8	tn	tn	PROPN
ejpam-3423	228	9	n−2∑	n−2∑	PROPN
ejpam-3423	228	10	j=[n/2]+1	j=[n/2]+1	NOUN
ejpam-3423	228	11	1	1	NUM
ejpam-3423	228	12	tn−j−1	tn−j−1	NOUN
ejpam-3423	228	13	+	+	CCONJ
ejpam-3423	229	1	‖e(n−1)har1,2‖+	‖e(n−1)har1,2‖+	PROPN
ejpam-3423	229	2	‖rn,2‖	‖rn,2‖	NOUN
ejpam-3423	229	3	,	,	PUNCT
ejpam-3423	229	4	≤	≤	NUM
ejpam-3423	229	5	2ch2	2ch2	NUM
ejpam-3423	229	6	tn	tn	NOUN
ejpam-3423	229	7	(	(	PUNCT
ejpam-3423	229	8	1	1	X
ejpam-3423	229	9	+	+	CCONJ
ejpam-3423	229	10	|	|	ADV
ejpam-3423	229	11	log	log	NOUN
ejpam-3423	229	12	h	h	NOUN
ejpam-3423	229	13	|	|	NOUN
ejpam-3423	229	14	)	)	PUNCT
ejpam-3423	230	1	+	+	CCONJ
ejpam-3423	230	2	‖e(n−1)har1,2‖+	‖e(n−1)har1,2‖+	PROPN
ejpam-3423	230	3	‖rn,2‖.	‖rn,2‖.	PROPN
ejpam-3423	230	4	(	(	PUNCT
ejpam-3423	230	5	55	55	NUM
ejpam-3423	230	6	)	)	PUNCT
ejpam-3423	230	7	now	now	ADV
ejpam-3423	230	8	for	for	ADP
ejpam-3423	230	9	j	j	PROPN
ejpam-3423	230	10	=	=	SYM
ejpam-3423	230	11	0	0	PROPN
ejpam-3423	230	12	and	and	CCONJ
ejpam-3423	230	13	j	j	PROPN
ejpam-3423	231	1	=	=	SYM
ejpam-3423	231	2	n−	n−	NOUN
ejpam-3423	231	3	1	1	NUM
ejpam-3423	231	4	term	term	NOUN
ejpam-3423	231	5	,	,	PUNCT
ejpam-3423	231	6	we	we	PRON
ejpam-3423	231	7	first	first	ADV
ejpam-3423	231	8	rewrite	rewrite	VERB
ejpam-3423	231	9	rn+1	rn+1	VERB
ejpam-3423	231	10	by	by	ADP
ejpam-3423	231	11	using	use	VERB
ejpam-3423	231	12	u(τ	u(τ	ADV
ejpam-3423	231	13	)	)	PUNCT
ejpam-3423	231	14	=	=	SYM
ejpam-3423	231	15	u(tn	u(tn	NOUN
ejpam-3423	231	16	)	)	PUNCT
ejpam-3423	232	1	+	+	CCONJ
ejpam-3423	232	2	∫	∫	PROPN
ejpam-3423	232	3	τ	τ	PROPN
ejpam-3423	232	4	tn	tn	PROPN
ejpam-3423	232	5	u′(s)ds	u′(s)ds	PROPN
ejpam-3423	232	6	,	,	PUNCT
ejpam-3423	232	7	(	(	PUNCT
ejpam-3423	232	8	56	56	NUM
ejpam-3423	232	9	)	)	PUNCT
ejpam-3423	232	10	and	and	CCONJ
ejpam-3423	232	11	u(tn+1	u(tn+1	ADJ
ejpam-3423	232	12	)	)	PUNCT
ejpam-3423	232	13	=	=	SYM
ejpam-3423	232	14	u(tn	u(tn	NOUN
ejpam-3423	232	15	)	)	PUNCT
ejpam-3423	233	1	+	+	CCONJ
ejpam-3423	233	2	∫	∫	PROPN
ejpam-3423	233	3	tn+1	tn+1	PROPN
ejpam-3423	233	4	tn	tn	PROPN
ejpam-3423	233	5	u′(s)ds	u′(s)ds	PROPN
ejpam-3423	233	6	,	,	PUNCT
ejpam-3423	233	7	(	(	PUNCT
ejpam-3423	233	8	57	57	NUM
ejpam-3423	233	9	)	)	PUNCT
ejpam-3423	233	10	in	in	ADP
ejpam-3423	233	11	(	(	PUNCT
ejpam-3423	233	12	29	29	NUM
ejpam-3423	233	13	)	)	PUNCT
ejpam-3423	233	14	and	and	CCONJ
ejpam-3423	233	15	simplifying	simplify	VERB
ejpam-3423	233	16	,	,	PUNCT
ejpam-3423	233	17	we	we	PRON
ejpam-3423	233	18	get	get	VERB
ejpam-3423	233	19	rn+1	rn+1	PRON
ejpam-3423	233	20	=	=	SYM
ejpam-3423	233	21	∫	∫	PROPN
ejpam-3423	233	22	tn+1	tn+1	PROPN
ejpam-3423	233	23	tn	tn	PROPN
ejpam-3423	233	24	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	234	1	(	(	PUNCT
ejpam-3423	234	2	∫	∫	PROPN
ejpam-3423	234	3	τ	τ	PROPN
ejpam-3423	234	4	tn	tn	PROPN
ejpam-3423	234	5	u′(s)ds−	u′(s)ds−	NOUN
ejpam-3423	234	6	τ	τ	PROPN
ejpam-3423	234	7	−	−	PROPN
ejpam-3423	234	8	tn	tn	PROPN
ejpam-3423	235	1	h	h	PROPN
ejpam-3423	235	2	∫	∫	PROPN
ejpam-3423	235	3	tn+1	tn+1	PROPN
ejpam-3423	235	4	tn	tn	PROPN
ejpam-3423	235	5	u′(s)ds	u′(s)ds	PROPN
ejpam-3423	235	6	)	)	PUNCT
ejpam-3423	235	7	dτ	dτ	NOUN
ejpam-3423	235	8	.	.	PROPN
ejpam-3423	236	1	(	(	PUNCT
ejpam-3423	236	2	58	58	NUM
ejpam-3423	236	3	)	)	PUNCT
ejpam-3423	236	4	m.	m.	NOUN
ejpam-3423	236	5	a.	a.	NOUN
ejpam-3423	236	6	gondal	gondal	PROPN
ejpam-3423	236	7	,	,	PUNCT
ejpam-3423	236	8	i.	i.	PROPN
ejpam-3423	236	9	rehman	rehman	PROPN
ejpam-3423	236	10	,	,	PUNCT
ejpam-3423	236	11	a.	a.	NOUN
ejpam-3423	236	12	razzaque	razzaque	NOUN
ejpam-3423	236	13	/	/	SYM
ejpam-3423	236	14	eur	eur	NOUN
ejpam-3423	236	15	.	.	PUNCT
ejpam-3423	237	1	j.	j.	PROPN
ejpam-3423	237	2	pure	pure	PROPN
ejpam-3423	237	3	appl	appl	PROPN
ejpam-3423	237	4	.	.	PROPN
ejpam-3423	237	5	math	math	PROPN
ejpam-3423	237	6	,	,	PUNCT
ejpam-3423	237	7	12	12	NUM
ejpam-3423	237	8	(	(	PUNCT
ejpam-3423	237	9	3	3	NUM
ejpam-3423	237	10	)	)	PUNCT
ejpam-3423	237	11	(	(	PUNCT
ejpam-3423	237	12	2019	2019	NUM
ejpam-3423	237	13	)	)	PUNCT
ejpam-3423	237	14	,	,	PUNCT
ejpam-3423	237	15	1215	1215	NUM
ejpam-3423	237	16	-	-	SYM
ejpam-3423	237	17	1230	1230	NUM
ejpam-3423	237	18	1225	1225	NUM
ejpam-3423	237	19	rn+1	rn+1	PROPN
ejpam-3423	238	1	=	=	SYM
ejpam-3423	238	2	rn+1,2	rn+1,2	PROPN
ejpam-3423	238	3	+	+	PROPN
ejpam-3423	238	4	rn+1,3	rn+1,3	PROPN
ejpam-3423	238	5	(	(	PUNCT
ejpam-3423	238	6	59	59	NUM
ejpam-3423	238	7	)	)	PUNCT
ejpam-3423	238	8	where	where	SCONJ
ejpam-3423	238	9	rn+1,2	rn+1,2	ADJ
ejpam-3423	238	10	=	=	PUNCT
ejpam-3423	238	11	∫	∫	PROPN
ejpam-3423	238	12	tn+1	tn+1	PROPN
ejpam-3423	238	13	tn	tn	PROPN
ejpam-3423	238	14	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	239	1	(	(	PUNCT
ejpam-3423	239	2	∫	∫	PROPN
ejpam-3423	239	3	τ	τ	PROPN
ejpam-3423	239	4	tn	tn	PROPN
ejpam-3423	239	5	u′(s)ds	u′(s)ds	PROPN
ejpam-3423	239	6	)	)	PUNCT
ejpam-3423	239	7	dτ	dτ	NOUN
ejpam-3423	239	8	,	,	PUNCT
ejpam-3423	239	9	(	(	PUNCT
ejpam-3423	239	10	60	60	NUM
ejpam-3423	239	11	)	)	PUNCT
ejpam-3423	239	12	and	and	CCONJ
ejpam-3423	239	13	rn+1,3	rn+1,3	PROPN
ejpam-3423	239	14	=	=	SYM
ejpam-3423	240	1	∫	∫	PROPN
ejpam-3423	240	2	tn+1	tn+1	PROPN
ejpam-3423	240	3	tn	tn	PROPN
ejpam-3423	240	4	e(tn+1−τ)ab	e(tn+1−τ)ab	PUNCT
ejpam-3423	241	1	(	(	PUNCT
ejpam-3423	241	2	τ	τ	X
ejpam-3423	241	3	−	−	PROPN
ejpam-3423	241	4	tn	tn	PROPN
ejpam-3423	241	5	h	h	PROPN
ejpam-3423	241	6	∫	∫	PROPN
ejpam-3423	241	7	tn+1	tn+1	PROPN
ejpam-3423	241	8	tn	tn	PROPN
ejpam-3423	241	9	u′(s)ds	u′(s)ds	PROPN
ejpam-3423	241	10	)	)	PUNCT
ejpam-3423	241	11	dτ	dτ	PROPN
ejpam-3423	241	12	.	.	PROPN
ejpam-3423	241	13	(	(	PUNCT
ejpam-3423	241	14	61	61	NUM
ejpam-3423	241	15	)	)	PUNCT
ejpam-3423	241	16	by	by	ADP
ejpam-3423	241	17	substituting	substitute	VERB
ejpam-3423	241	18	n	n	PROPN
ejpam-3423	241	19	=	=	SYM
ejpam-3423	241	20	0	0	NUM
ejpam-3423	241	21	in	in	ADP
ejpam-3423	241	22	(	(	PUNCT
ejpam-3423	241	23	60	60	NUM
ejpam-3423	241	24	)	)	PUNCT
ejpam-3423	241	25	and	and	CCONJ
ejpam-3423	241	26	using	use	VERB
ejpam-3423	241	27	the	the	DET
ejpam-3423	241	28	identity	identity	NOUN
ejpam-3423	241	29	aa−1	aa−1	NOUN
ejpam-3423	241	30	=	=	PUNCT
ejpam-3423	242	1	i	i	PROPN
ejpam-3423	242	2	and	and	CCONJ
ejpam-3423	242	3	ab	ab	PROPN
ejpam-3423	242	4	=	=	SYM
ejpam-3423	242	5	ba	ba	PROPN
ejpam-3423	242	6	,	,	PUNCT
ejpam-3423	242	7	we	we	PRON
ejpam-3423	242	8	can	can	AUX
ejpam-3423	242	9	write	write	VERB
ejpam-3423	242	10	the	the	DET
ejpam-3423	242	11	term	term	NOUN
ejpam-3423	242	12	for	for	ADP
ejpam-3423	242	13	j	j	PROPN
ejpam-3423	242	14	=	=	SYM
ejpam-3423	242	15	0	0	PUNCT
ejpam-3423	243	1	as	as	ADP
ejpam-3423	243	2	‖e(n−1)har1,2‖	‖e(n−1)har1,2‖	ADV
ejpam-3423	243	3	=	=	SYM
ejpam-3423	243	4	‖ae(n−1)ha	‖ae(n−1)ha	SYM
ejpam-3423	243	5	∫	∫	PROPN
ejpam-3423	243	6	h	h	PROPN
ejpam-3423	243	7	0	0	PROPN
ejpam-3423	244	1	e(h−τ)ab	e(h−τ)ab	PROPN
ejpam-3423	244	2	(	(	PUNCT
ejpam-3423	244	3	∫	∫	PROPN
ejpam-3423	244	4	τ	τ	PROPN
ejpam-3423	244	5	0	0	PROPN
ejpam-3423	244	6	a−1u′(s)ds	a−1u′(s)ds	PROPN
ejpam-3423	244	7	)	)	PUNCT
ejpam-3423	244	8	dτ‖	dτ‖	PROPN
ejpam-3423	244	9	,	,	PUNCT
ejpam-3423	244	10	≤	≤	NUM
ejpam-3423	244	11	‖ae(n−1)ha‖	‖ae(n−1)ha‖	PROPN
ejpam-3423	244	12	∫	∫	PROPN
ejpam-3423	244	13	h	h	PROPN
ejpam-3423	244	14	0	0	PROPN
ejpam-3423	244	15	‖e(h−τ)a‖‖b‖	‖e(h−τ)a‖‖b‖	X
ejpam-3423	244	16	(	(	PUNCT
ejpam-3423	244	17	∫	∫	PROPN
ejpam-3423	244	18	τ	τ	PROPN
ejpam-3423	244	19	0	0	NUM
ejpam-3423	244	20	‖a−1u′(s)‖ds	‖a−1u′(s)‖ds	NOUN
ejpam-3423	244	21	)	)	PUNCT
ejpam-3423	244	22	dτ	dτ	NOUN
ejpam-3423	244	23	,	,	PUNCT
ejpam-3423	244	24	≤	≤	NUM
ejpam-3423	244	25	c	c	X
ejpam-3423	244	26	tn−1	tn−1	PROPN
ejpam-3423	244	27	·	·	PUNCT
ejpam-3423	244	28	c	c	PROPN
ejpam-3423	244	29	·	·	PUNCT
ejpam-3423	244	30	h2	h2	NOUN
ejpam-3423	244	31	,	,	PUNCT
ejpam-3423	244	32	≤	≤	NUM
ejpam-3423	244	33	ch2	ch2	PROPN
ejpam-3423	244	34	tn−1	tn−1	PROPN
ejpam-3423	244	35	.	.	PUNCT
ejpam-3423	245	1	(	(	PUNCT
ejpam-3423	245	2	62	62	NUM
ejpam-3423	245	3	)	)	PUNCT
ejpam-3423	245	4	now	now	ADV
ejpam-3423	245	5	by	by	ADP
ejpam-3423	245	6	substituting	substitute	VERB
ejpam-3423	245	7	n	n	NOUN
ejpam-3423	245	8	=	=	PUNCT
ejpam-3423	245	9	n−	n−	NOUN
ejpam-3423	245	10	1	1	NUM
ejpam-3423	245	11	in	in	ADP
ejpam-3423	245	12	(	(	PUNCT
ejpam-3423	245	13	60	60	NUM
ejpam-3423	245	14	)	)	PUNCT
ejpam-3423	245	15	we	we	PRON
ejpam-3423	245	16	can	can	AUX
ejpam-3423	245	17	write	write	VERB
ejpam-3423	245	18	the	the	DET
ejpam-3423	245	19	term	term	NOUN
ejpam-3423	245	20	for	for	ADP
ejpam-3423	245	21	j	j	PROPN
ejpam-3423	245	22	=	=	PUNCT
ejpam-3423	245	23	n−	n−	NOUN
ejpam-3423	245	24	1	1	NUM
ejpam-3423	245	25	as	as	ADP
ejpam-3423	245	26	‖rn,2‖	‖rn,2‖	ADJ
ejpam-3423	245	27	=	=	SYM
ejpam-3423	245	28	‖	‖	PROPN
ejpam-3423	245	29	∫	∫	PROPN
ejpam-3423	245	30	tn	tn	PROPN
ejpam-3423	245	31	tn−1	tn−1	PROPN
ejpam-3423	245	32	e(tn−τ)ab	e(tn−τ)ab	PROPN
ejpam-3423	245	33	(	(	PUNCT
ejpam-3423	245	34	∫	∫	PROPN
ejpam-3423	245	35	τ	τ	PROPN
ejpam-3423	245	36	tn−1	tn−1	PROPN
ejpam-3423	245	37	u′(s)ds	u′(s)ds	PROPN
ejpam-3423	245	38	)	)	PUNCT
ejpam-3423	245	39	dτ‖	dτ‖	PROPN
ejpam-3423	245	40	,	,	PUNCT
ejpam-3423	245	41	≤	≤	NUM
ejpam-3423	245	42	∫	∫	PROPN
ejpam-3423	245	43	tn	tn	PROPN
ejpam-3423	245	44	tn−1	tn−1	PROPN
ejpam-3423	245	45	‖e(tn−1−τ)a‖‖b‖	‖e(tn−1−τ)a‖‖b‖	NOUN
ejpam-3423	245	46	(	(	PUNCT
ejpam-3423	245	47	∫	∫	PROPN
ejpam-3423	245	48	τ	τ	PROPN
ejpam-3423	245	49	tn−1	tn−1	PROPN
ejpam-3423	245	50	‖u′(s)‖ds	‖u′(s)‖ds	PROPN
ejpam-3423	245	51	)	)	PUNCT
ejpam-3423	245	52	dτ	dτ	PROPN
ejpam-3423	245	53	,	,	PUNCT
ejpam-3423	245	54	≤	≤	NUM
ejpam-3423	245	55	ch2	ch2	PROPN
ejpam-3423	245	56	tn−1	tn−1	PROPN
ejpam-3423	245	57	.	.	PUNCT
ejpam-3423	246	1	(	(	PUNCT
ejpam-3423	246	2	63	63	NUM
ejpam-3423	246	3	)	)	PUNCT
ejpam-3423	246	4	substituting	substituting	NOUN
ejpam-3423	246	5	(	(	PUNCT
ejpam-3423	246	6	62	62	NUM
ejpam-3423	246	7	)	)	PUNCT
ejpam-3423	246	8	and	and	CCONJ
ejpam-3423	246	9	(	(	PUNCT
ejpam-3423	246	10	63	63	NUM
ejpam-3423	246	11	)	)	PUNCT
ejpam-3423	246	12	in	in	ADP
ejpam-3423	246	13	(	(	PUNCT
ejpam-3423	246	14	55	55	NUM
ejpam-3423	246	15	)	)	PUNCT
ejpam-3423	246	16	,	,	PUNCT
ejpam-3423	246	17	we	we	PRON
ejpam-3423	246	18	get	get	VERB
ejpam-3423	246	19	n−1∑	n−1∑	NUM
ejpam-3423	246	20	j=0	j=0	VERB
ejpam-3423	246	21	‖e(n−j−1)harj+1,2‖	‖e(n−j−1)harj+1,2‖	VERB
ejpam-3423	246	22	≤	≤	NUM
ejpam-3423	246	23	ch2	ch2	PROPN
ejpam-3423	246	24	tn	tn	PROPN
ejpam-3423	246	25	(	(	PUNCT
ejpam-3423	246	26	1	1	X
ejpam-3423	246	27	+	+	CCONJ
ejpam-3423	246	28	|	|	ADV
ejpam-3423	246	29	log	log	NOUN
ejpam-3423	246	30	h	h	NOUN
ejpam-3423	246	31	|	|	NOUN
ejpam-3423	246	32	)	)	PUNCT
ejpam-3423	247	1	+	+	CCONJ
ejpam-3423	247	2	ch2	ch2	PROPN
ejpam-3423	247	3	tn−1	tn−1	PROPN
ejpam-3423	247	4	.	.	PUNCT
ejpam-3423	248	1	(	(	PUNCT
ejpam-3423	248	2	64	64	NUM
ejpam-3423	248	3	)	)	PUNCT
ejpam-3423	248	4	similarly	similarly	ADV
ejpam-3423	248	5	we	we	PRON
ejpam-3423	248	6	can	can	AUX
ejpam-3423	248	7	prove	prove	VERB
ejpam-3423	248	8	that	that	SCONJ
ejpam-3423	248	9	∑n−1	∑n−1	SCONJ
ejpam-3423	248	10	j=0	j=0	PROPN
ejpam-3423	248	11	‖e(n−j−1)harj+1,3‖	‖e(n−j−1)harj+1,3‖	PROPN
ejpam-3423	248	12	is	be	AUX
ejpam-3423	248	13	bounded	bound	VERB
ejpam-3423	248	14	and	and	CCONJ
ejpam-3423	248	15	we	we	PRON
ejpam-3423	248	16	get	get	VERB
ejpam-3423	248	17	n−1∑	n−1∑	NUM
ejpam-3423	248	18	j=0	j=0	PROPN
ejpam-3423	248	19	‖e(n−j−1)harj+1,3‖	‖e(n−j−1)harj+1,3‖	NOUN
ejpam-3423	248	20	≤	≤	NUM
ejpam-3423	248	21	ch2	ch2	PROPN
ejpam-3423	248	22	tn	tn	PROPN
ejpam-3423	249	1	(	(	PUNCT
ejpam-3423	249	2	1	1	X
ejpam-3423	249	3	+	+	CCONJ
ejpam-3423	249	4	|	|	ADV
ejpam-3423	249	5	log	log	NOUN
ejpam-3423	249	6	h	h	NOUN
ejpam-3423	249	7	|	|	NOUN
ejpam-3423	249	8	)	)	PUNCT
ejpam-3423	250	1	+	+	CCONJ
ejpam-3423	250	2	ch2	ch2	PROPN
ejpam-3423	250	3	tn−1	tn−1	PROPN
ejpam-3423	250	4	.	.	PUNCT
ejpam-3423	251	1	(	(	PUNCT
ejpam-3423	251	2	65	65	NUM
ejpam-3423	251	3	)	)	PUNCT
ejpam-3423	251	4	now	now	ADV
ejpam-3423	251	5	substitute	substitute	NOUN
ejpam-3423	251	6	(	(	PUNCT
ejpam-3423	251	7	64	64	NUM
ejpam-3423	251	8	)	)	PUNCT
ejpam-3423	251	9	and	and	CCONJ
ejpam-3423	251	10	(	(	PUNCT
ejpam-3423	251	11	65	65	NUM
ejpam-3423	251	12	)	)	PUNCT
ejpam-3423	251	13	in	in	ADP
ejpam-3423	251	14	(	(	PUNCT
ejpam-3423	251	15	53	53	NUM
ejpam-3423	251	16	)	)	PUNCT
ejpam-3423	251	17	and	and	CCONJ
ejpam-3423	251	18	then	then	ADV
ejpam-3423	251	19	(	(	PUNCT
ejpam-3423	251	20	53	53	NUM
ejpam-3423	251	21	)	)	PUNCT
ejpam-3423	251	22	in	in	ADP
ejpam-3423	251	23	(	(	PUNCT
ejpam-3423	251	24	52	52	NUM
ejpam-3423	251	25	)	)	PUNCT
ejpam-3423	251	26	and	and	CCONJ
ejpam-3423	251	27	simplifying	simplify	VERB
ejpam-3423	251	28	we	we	PRON
ejpam-3423	251	29	get	get	VERB
ejpam-3423	251	30	‖εn‖	‖εn‖	ADJ
ejpam-3423	251	31	≤	≤	NUM
ejpam-3423	251	32	ch	ch	NOUN
ejpam-3423	251	33	n−1∑	n−1∑	PROPN
ejpam-3423	251	34	j=0	j=0	PROPN
ejpam-3423	251	35	‖εj‖+	‖εj‖+	PROPN
ejpam-3423	251	36	ch2	ch2	PROPN
ejpam-3423	251	37	tn	tn	PROPN
ejpam-3423	251	38	(	(	PUNCT
ejpam-3423	251	39	1	1	X
ejpam-3423	251	40	+	+	CCONJ
ejpam-3423	251	41	|	|	ADV
ejpam-3423	251	42	log	log	NOUN
ejpam-3423	251	43	h	h	NOUN
ejpam-3423	251	44	|	|	NOUN
ejpam-3423	251	45	)	)	PUNCT
ejpam-3423	252	1	+	+	CCONJ
ejpam-3423	252	2	ch2	ch2	PROPN
ejpam-3423	252	3	tn−1	tn−1	PROPN
ejpam-3423	252	4	.	.	PUNCT
ejpam-3423	253	1	(	(	PUNCT
ejpam-3423	253	2	66	66	NUM
ejpam-3423	253	3	)	)	PUNCT
ejpam-3423	253	4	m.	m.	NOUN
ejpam-3423	253	5	a.	a.	NOUN
ejpam-3423	253	6	gondal	gondal	PROPN
ejpam-3423	253	7	,	,	PUNCT
ejpam-3423	253	8	i.	i.	PROPN
ejpam-3423	253	9	rehman	rehman	PROPN
ejpam-3423	253	10	,	,	PUNCT
ejpam-3423	253	11	a.	a.	NOUN
ejpam-3423	253	12	razzaque	razzaque	NOUN
ejpam-3423	253	13	/	/	SYM
ejpam-3423	253	14	eur	eur	NOUN
ejpam-3423	253	15	.	.	PUNCT
ejpam-3423	254	1	j.	j.	PROPN
ejpam-3423	254	2	pure	pure	PROPN
ejpam-3423	254	3	appl	appl	PROPN
ejpam-3423	254	4	.	.	PROPN
ejpam-3423	254	5	math	math	PROPN
ejpam-3423	254	6	,	,	PUNCT
ejpam-3423	254	7	12	12	NUM
ejpam-3423	254	8	(	(	PUNCT
ejpam-3423	254	9	3	3	NUM
ejpam-3423	254	10	)	)	PUNCT
ejpam-3423	254	11	(	(	PUNCT
ejpam-3423	254	12	2019	2019	NUM
ejpam-3423	254	13	)	)	PUNCT
ejpam-3423	254	14	,	,	PUNCT
ejpam-3423	254	15	1215	1215	NUM
ejpam-3423	254	16	-	-	SYM
ejpam-3423	254	17	1230	1230	NUM
ejpam-3423	254	18	1226	1226	NUM
ejpam-3423	254	19	note	note	NOUN
ejpam-3423	254	20	that	that	PRON
ejpam-3423	254	21	ch2	ch2	PROPN
ejpam-3423	254	22	tn−1	tn−1	PROPN
ejpam-3423	254	23	=	=	PUNCT
ejpam-3423	254	24	ch2	ch2	PROPN
ejpam-3423	254	25	tn	tn	PROPN
ejpam-3423	254	26	·	·	PUNCT
ejpam-3423	255	1	tn−1+h	tn−1+h	AUX
ejpam-3423	255	2	tn−1	tn−1	ADJ
ejpam-3423	255	3	≤	≤	NUM
ejpam-3423	255	4	ch2	ch2	PROPN
ejpam-3423	255	5	tn	tn	PROPN
ejpam-3423	255	6	.	.	PUNCT
ejpam-3423	256	1	by	by	ADP
ejpam-3423	256	2	using	use	VERB
ejpam-3423	256	3	the	the	DET
ejpam-3423	256	4	lemma	lemma	PROPN
ejpam-3423	256	5	6.2(gronwall	6.2(gronwall	PROPN
ejpam-3423	256	6	lemma	lemma	PROPN
ejpam-3423	256	7	)	)	PUNCT
ejpam-3423	256	8	given	give	VERB
ejpam-3423	256	9	in	in	ADP
ejpam-3423	256	10	[	[	X
ejpam-3423	256	11	15	15	NUM
ejpam-3423	256	12	]	]	PUNCT
ejpam-3423	256	13	,	,	PUNCT
ejpam-3423	256	14	we	we	PRON
ejpam-3423	256	15	get	get	VERB
ejpam-3423	256	16	‖εn‖	‖εn‖	PROPN
ejpam-3423	256	17	≤	≤	NUM
ejpam-3423	256	18	ch2	ch2	PROPN
ejpam-3423	256	19	tn	tn	PROPN
ejpam-3423	257	1	(	(	PUNCT
ejpam-3423	257	2	|	|	ADV
ejpam-3423	257	3	log	log	VERB
ejpam-3423	257	4	h	h	NOUN
ejpam-3423	258	1	|	|	ADV
ejpam-3423	258	2	+1	+1	PROPN
ejpam-3423	258	3	)	)	PUNCT
ejpam-3423	258	4	.	.	PUNCT
ejpam-3423	259	1	(	(	PUNCT
ejpam-3423	259	2	67	67	NUM
ejpam-3423	259	3	)	)	SYM
ejpam-3423	259	4	4	4	NUM
ejpam-3423	259	5	.	.	PUNCT
ejpam-3423	259	6	numerical	numerical	ADJ
ejpam-3423	259	7	experiments	experiment	NOUN
ejpam-3423	259	8	this	this	DET
ejpam-3423	259	9	section	section	NOUN
ejpam-3423	259	10	deals	deal	VERB
ejpam-3423	259	11	with	with	ADP
ejpam-3423	259	12	the	the	DET
ejpam-3423	259	13	numerical	numerical	ADJ
ejpam-3423	259	14	experiments	experiment	NOUN
ejpam-3423	259	15	for	for	ADP
ejpam-3423	259	16	the	the	DET
ejpam-3423	259	17	verification	verification	NOUN
ejpam-3423	259	18	of	of	ADP
ejpam-3423	259	19	our	our	PRON
ejpam-3423	259	20	calculated	calculate	VERB
ejpam-3423	259	21	error	error	NOUN
ejpam-3423	259	22	bounds	bound	NOUN
ejpam-3423	259	23	.	.	PUNCT
ejpam-3423	260	1	lets	let	VERB
ejpam-3423	260	2	assume	assume	VERB
ejpam-3423	260	3	the	the	DET
ejpam-3423	260	4	linear	linear	PROPN
ejpam-3423	260	5	parabolic	parabolic	PROPN
ejpam-3423	260	6	problem	problem	NOUN
ejpam-3423	260	7	,	,	PUNCT
ejpam-3423	260	8	called	call	VERB
ejpam-3423	260	9	as	as	ADP
ejpam-3423	260	10	partial	partial	ADJ
ejpam-3423	260	11	integrodifferential	integrodifferential	ADJ
ejpam-3423	260	12	equations	equation	NOUN
ejpam-3423	260	13	,	,	PUNCT
ejpam-3423	260	14	that	that	PRON
ejpam-3423	260	15	arise	arise	VERB
ejpam-3423	260	16	in	in	ADP
ejpam-3423	260	17	financial	financial	ADJ
ejpam-3423	260	18	mathematics	mathematic	NOUN
ejpam-3423	260	19	.	.	PUNCT
ejpam-3423	261	1	this	this	PRON
ejpam-3423	261	2	was	be	AUX
ejpam-3423	261	3	studied	study	VERB
ejpam-3423	261	4	by	by	ADP
ejpam-3423	261	5	tangman	tangman	NOUN
ejpam-3423	261	6	,	,	PUNCT
ejpam-3423	261	7	gopaul	gopaul	NOUN
ejpam-3423	261	8	,	,	PUNCT
ejpam-3423	261	9	&	&	CCONJ
ejpam-3423	261	10	bhuruth	bhuruth	PROPN
ejpam-3423	261	11	[	[	X
ejpam-3423	261	12	3	3	NUM
ejpam-3423	261	13	]	]	X
ejpam-3423	261	14	∂u	∂u	PROPN
ejpam-3423	261	15	∂τ	∂τ	NOUN
ejpam-3423	261	16	=	=	SYM
ejpam-3423	261	17	1	1	NUM
ejpam-3423	261	18	2	2	NUM
ejpam-3423	261	19	σ2	σ2	PROPN
ejpam-3423	261	20	∂2u	∂2u	ADJ
ejpam-3423	261	21	∂x2	∂x2	PROPN
ejpam-3423	262	1	+	+	CCONJ
ejpam-3423	262	2	(	(	PUNCT
ejpam-3423	262	3	r	r	NOUN
ejpam-3423	262	4	−	−	NUM
ejpam-3423	262	5	1	1	NUM
ejpam-3423	262	6	2	2	NUM
ejpam-3423	262	7	σ2	σ2	NOUN
ejpam-3423	262	8	−	−	PROPN
ejpam-3423	262	9	λκ	λκ	NOUN
ejpam-3423	262	10	)	)	PUNCT
ejpam-3423	263	1	∂u	∂u	PROPN
ejpam-3423	263	2	∂x	∂x	NOUN
ejpam-3423	263	3	−	−	NOUN
ejpam-3423	264	1	(	(	PUNCT
ejpam-3423	264	2	r	r	NOUN
ejpam-3423	264	3	+	+	SYM
ejpam-3423	264	4	λ)u+	λ)u+	NOUN
ejpam-3423	264	5	λ	λ	X
ejpam-3423	264	6	∫	∫	NOUN
ejpam-3423	264	7	r	r	NOUN
ejpam-3423	264	8	b(x−	b(x−	NOUN
ejpam-3423	264	9	y)u(y	y)u(y	NOUN
ejpam-3423	264	10	,	,	PUNCT
ejpam-3423	264	11	τ)dy	τ)dy	PROPN
ejpam-3423	264	12	.	.	PUNCT
ejpam-3423	264	13	(	(	PUNCT
ejpam-3423	264	14	68	68	NUM
ejpam-3423	264	15	)	)	PUNCT
ejpam-3423	264	16	with	with	ADP
ejpam-3423	264	17	b(z	b(z	NOUN
ejpam-3423	264	18	)	)	PUNCT
ejpam-3423	264	19	=	=	SYM
ejpam-3423	264	20	1√	1√	NUM
ejpam-3423	264	21	2πγ	2πγ	NOUN
ejpam-3423	264	22	e−(z−µ	e−(z−µ	NOUN
ejpam-3423	264	23	)	)	PUNCT
ejpam-3423	264	24	2/(2γ2	2/(2γ2	NOUN
ejpam-3423	264	25	)	)	PUNCT
ejpam-3423	264	26	.	.	PUNCT
ejpam-3423	265	1	(	(	PUNCT
ejpam-3423	265	2	69	69	NUM
ejpam-3423	265	3	)	)	PUNCT
ejpam-3423	265	4	where	where	SCONJ
ejpam-3423	265	5	we	we	PRON
ejpam-3423	265	6	considers	consider	VERB
ejpam-3423	265	7	parameters	parameter	NOUN
ejpam-3423	265	8	r	r	NOUN
ejpam-3423	265	9	,	,	PUNCT
ejpam-3423	265	10	σ	σ	PROPN
ejpam-3423	265	11	,	,	PUNCT
ejpam-3423	265	12	λ	λ	PROPN
ejpam-3423	265	13	,	,	PUNCT
ejpam-3423	265	14	γ	γ	PROPN
ejpam-3423	265	15	,	,	PUNCT
ejpam-3423	265	16	κ	κ	PROPN
ejpam-3423	265	17	,	,	PUNCT
ejpam-3423	265	18	µ.	µ.	ADJ
ejpam-3423	265	19	equation	equation	NOUN
ejpam-3423	265	20	(	(	PUNCT
ejpam-3423	265	21	68	68	NUM
ejpam-3423	265	22	)	)	PUNCT
ejpam-3423	265	23	indicates	indicate	VERB
ejpam-3423	265	24	the	the	DET
ejpam-3423	265	25	european	european	ADJ
ejpam-3423	265	26	option	option	NOUN
ejpam-3423	265	27	pricing	pricing	NOUN
ejpam-3423	265	28	problem	problem	NOUN
ejpam-3423	265	29	in	in	ADP
ejpam-3423	265	30	mertons	merton	NOUN
ejpam-3423	265	31	jump	jump	NOUN
ejpam-3423	265	32	-	-	PUNCT
ejpam-3423	265	33	diffusion	diffusion	NOUN
ejpam-3423	265	34	model	model	NOUN
ejpam-3423	265	35	.	.	PUNCT
ejpam-3423	266	1	the	the	DET
ejpam-3423	266	2	initial	initial	ADJ
ejpam-3423	266	3	condition	condition	NOUN
ejpam-3423	266	4	associated	associate	VERB
ejpam-3423	266	5	with	with	ADP
ejpam-3423	266	6	the	the	DET
ejpam-3423	266	7	european	european	ADJ
ejpam-3423	266	8	call	call	NOUN
ejpam-3423	266	9	option	option	NOUN
ejpam-3423	266	10	price	price	NOUN
ejpam-3423	266	11	u(x	u(x	NOUN
ejpam-3423	266	12	,	,	PUNCT
ejpam-3423	266	13	0	0	NUM
ejpam-3423	266	14	)	)	PUNCT
ejpam-3423	266	15	=	=	SYM
ejpam-3423	266	16	max(eex	max(eex	NOUN
ejpam-3423	266	17	−	−	PROPN
ejpam-3423	266	18	e	e	NOUN
ejpam-3423	266	19	,	,	PUNCT
ejpam-3423	266	20	0	0	NUM
ejpam-3423	266	21	)	)	PUNCT
ejpam-3423	266	22	(	(	PUNCT
ejpam-3423	266	23	70	70	NUM
ejpam-3423	266	24	)	)	PUNCT
ejpam-3423	266	25	and	and	CCONJ
ejpam-3423	266	26	boundary	boundary	ADJ
ejpam-3423	266	27	conditions	condition	NOUN
ejpam-3423	266	28	suggested	suggest	VERB
ejpam-3423	266	29	in	in	ADP
ejpam-3423	266	30	[	[	X
ejpam-3423	266	31	3	3	NUM
ejpam-3423	266	32	]	]	PUNCT
ejpam-3423	266	33	are	be	AUX
ejpam-3423	266	34	uτ	uτ	INTJ
ejpam-3423	266	35	(	(	PUNCT
ejpam-3423	266	36	x	x	NOUN
ejpam-3423	266	37	,	,	PUNCT
ejpam-3423	266	38	τ	τ	X
ejpam-3423	266	39	)	)	PUNCT
ejpam-3423	266	40	=	=	PUNCT
ejpam-3423	267	1	−ru(x	−ru(x	PROPN
ejpam-3423	267	2	,	,	PUNCT
ejpam-3423	267	3	τ	τ	PROPN
ejpam-3423	267	4	)	)	PUNCT
ejpam-3423	267	5	,	,	PUNCT
ejpam-3423	267	6	x→	x→	PUNCT
ejpam-3423	268	1	−∞	−∞	NOUN
ejpam-3423	268	2	,	,	PUNCT
ejpam-3423	268	3	(	(	PUNCT
ejpam-3423	268	4	71	71	NUM
ejpam-3423	268	5	)	)	PUNCT
ejpam-3423	268	6	uxx(x	uxx(x	PROPN
ejpam-3423	268	7	,	,	PUNCT
ejpam-3423	268	8	τ	τ	X
ejpam-3423	268	9	)	)	PUNCT
ejpam-3423	268	10	=	=	SYM
ejpam-3423	268	11	ux(x	ux(x	PROPN
ejpam-3423	268	12	,	,	PUNCT
ejpam-3423	268	13	τ	τ	PROPN
ejpam-3423	268	14	)	)	PUNCT
ejpam-3423	268	15	,	,	PUNCT
ejpam-3423	268	16	x→∞.	x→∞.	PUNCT
ejpam-3423	268	17	(	(	PUNCT
ejpam-3423	268	18	72	72	NUM
ejpam-3423	268	19	)	)	PUNCT
ejpam-3423	268	20	4.1	4.1	NUM
ejpam-3423	268	21	.	.	PUNCT
ejpam-3423	269	1	space	space	NOUN
ejpam-3423	269	2	discretization	discretization	NOUN
ejpam-3423	269	3	the	the	DET
ejpam-3423	269	4	discretization	discretization	NOUN
ejpam-3423	269	5	for	for	ADP
ejpam-3423	269	6	the	the	DET
ejpam-3423	269	7	problem	problem	NOUN
ejpam-3423	269	8	(	(	PUNCT
ejpam-3423	269	9	68	68	NUM
ejpam-3423	269	10	)	)	PUNCT
ejpam-3423	269	11	,	,	PUNCT
ejpam-3423	269	12	using	use	VERB
ejpam-3423	269	13	finite	finite	ADJ
ejpam-3423	269	14	difference	difference	NOUN
ejpam-3423	269	15	schemes	scheme	NOUN
ejpam-3423	269	16	will	will	AUX
ejpam-3423	269	17	be	be	AUX
ejpam-3423	269	18	given	give	VERB
ejpam-3423	269	19	here	here	ADV
ejpam-3423	269	20	.	.	PUNCT
ejpam-3423	270	1	we	we	PRON
ejpam-3423	270	2	require	require	VERB
ejpam-3423	270	3	to	to	PART
ejpam-3423	270	4	truncate	truncate	VERB
ejpam-3423	270	5	the	the	DET
ejpam-3423	270	6	infinite	infinite	ADJ
ejpam-3423	270	7	x	x	NOUN
ejpam-3423	270	8	-	-	NOUN
ejpam-3423	270	9	domain	domain	NOUN
ejpam-3423	270	10	to	to	PART
ejpam-3423	270	11	finite	finite	VERB
ejpam-3423	270	12	x	x	NOUN
ejpam-3423	270	13	-	-	NOUN
ejpam-3423	270	14	domain	domain	NOUN
ejpam-3423	270	15	,	,	PUNCT
ejpam-3423	270	16	for	for	ADP
ejpam-3423	270	17	instance	instance	NOUN
ejpam-3423	270	18	,	,	PUNCT
ejpam-3423	270	19	xmin	xmin	PROPN
ejpam-3423	270	20	≤	≤	PROPN
ejpam-3423	270	21	x	x	PUNCT
ejpam-3423	270	22	≤	≤	NUM
ejpam-3423	270	23	xmax	xmax	NOUN
ejpam-3423	270	24	for	for	ADP
ejpam-3423	270	25	a	a	DET
ejpam-3423	270	26	finite	finite	ADJ
ejpam-3423	270	27	difference	difference	NOUN
ejpam-3423	270	28	discretization	discretization	NOUN
ejpam-3423	270	29	of	of	ADP
ejpam-3423	270	30	the	the	DET
ejpam-3423	270	31	spatial	spatial	ADJ
ejpam-3423	270	32	derivatives	derivative	NOUN
ejpam-3423	270	33	.	.	PUNCT
ejpam-3423	271	1	hence	hence	ADV
ejpam-3423	271	2	−1.5	−1.5	PROPN
ejpam-3423	271	3	=	=	SYM
ejpam-3423	271	4	xmin	xmin	PROPN
ejpam-3423	272	1	=	=	PUNCT
ejpam-3423	272	2	x0	x0	PROPN
ejpam-3423	272	3	<	<	X
ejpam-3423	273	1	x1	x1	X
ejpam-3423	273	2	<	<	X
ejpam-3423	273	3	x2	x2	X
ejpam-3423	273	4	<	<	X
ejpam-3423	273	5	x3	x3	X
ejpam-3423	273	6	<	<	X
ejpam-3423	273	7	.	.	PUNCT
ejpam-3423	273	8	.	.	PUNCT
ejpam-3423	273	9	.	.	PUNCT
ejpam-3423	274	1	<	<	X
ejpam-3423	274	2	xm	xm	X
ejpam-3423	274	3	<	<	X
ejpam-3423	274	4	xm+1	xm+1	PROPN
ejpam-3423	274	5	=	=	SYM
ejpam-3423	274	6	xmax	xmax	PROPN
ejpam-3423	274	7	=	=	NOUN
ejpam-3423	274	8	1.5	1.5	NUM
ejpam-3423	274	9	,	,	PUNCT
ejpam-3423	274	10	with	with	ADP
ejpam-3423	274	11	grid	grid	NOUN
ejpam-3423	274	12	points	point	NOUN
ejpam-3423	274	13	xi	xi	X
ejpam-3423	274	14	=	=	PUNCT
ejpam-3423	274	15	xi−1	xi−1	PROPN
ejpam-3423	275	1	+	+	CCONJ
ejpam-3423	275	2	δxi	δxi	ADJ
ejpam-3423	275	3	and	and	CCONJ
ejpam-3423	275	4	δxi	δxi	ADJ
ejpam-3423	275	5	=	=	SYM
ejpam-3423	275	6	xi	xi	ADP
ejpam-3423	275	7	−	−	PROPN
ejpam-3423	275	8	xi−1	xi−1	PROPN
ejpam-3423	275	9	here	here	ADV
ejpam-3423	275	10	we	we	PRON
ejpam-3423	275	11	need	need	VERB
ejpam-3423	275	12	the	the	DET
ejpam-3423	275	13	first	first	ADJ
ejpam-3423	275	14	-	-	PUNCT
ejpam-3423	275	15	order	order	NOUN
ejpam-3423	275	16	and	and	CCONJ
ejpam-3423	275	17	second	second	ADJ
ejpam-3423	275	18	-	-	PUNCT
ejpam-3423	275	19	order	order	NOUN
ejpam-3423	275	20	finite	finite	ADJ
ejpam-3423	275	21	difference	difference	NOUN
ejpam-3423	275	22	approximations	approximation	NOUN
ejpam-3423	275	23	for	for	ADP
ejpam-3423	275	24	the	the	DET
ejpam-3423	275	25	discretization	discretization	NOUN
ejpam-3423	275	26	of	of	ADP
ejpam-3423	275	27	(	(	PUNCT
ejpam-3423	275	28	68	68	NUM
ejpam-3423	275	29	)	)	PUNCT
ejpam-3423	275	30	on	on	ADP
ejpam-3423	275	31	a	a	DET
ejpam-3423	275	32	non	non	ADJ
ejpam-3423	275	33	-	-	ADJ
ejpam-3423	275	34	equidistant	equidistant	ADJ
ejpam-3423	275	35	grid	grid	NOUN
ejpam-3423	275	36	,	,	PUNCT
ejpam-3423	275	37	which	which	PRON
ejpam-3423	275	38	are	be	AUX
ejpam-3423	275	39	given	give	VERB
ejpam-3423	275	40	as	as	ADP
ejpam-3423	275	41	∂u	∂u	PROPN
ejpam-3423	275	42	∂x	∂x	PROPN
ejpam-3423	275	43	(	(	PUNCT
ejpam-3423	275	44	xi	xi	NOUN
ejpam-3423	275	45	)	)	PUNCT
ejpam-3423	275	46	∼=	∼=	PROPN
ejpam-3423	275	47	u(xi+1)−	u(xi+1)−	PROPN
ejpam-3423	275	48	u(xi−1	u(xi−1	PROPN
ejpam-3423	275	49	)	)	PUNCT
ejpam-3423	275	50	δxi	δxi	ADP
ejpam-3423	275	51	+	+	CCONJ
ejpam-3423	275	52	δxi+1	δxi+1	NOUN
ejpam-3423	275	53	,	,	PUNCT
ejpam-3423	275	54	(	(	PUNCT
ejpam-3423	275	55	73	73	NUM
ejpam-3423	275	56	)	)	PUNCT
ejpam-3423	275	57	m.	m.	NOUN
ejpam-3423	275	58	a.	a.	NOUN
ejpam-3423	275	59	gondal	gondal	PROPN
ejpam-3423	275	60	,	,	PUNCT
ejpam-3423	275	61	i.	i.	PROPN
ejpam-3423	275	62	rehman	rehman	PROPN
ejpam-3423	275	63	,	,	PUNCT
ejpam-3423	275	64	a.	a.	NOUN
ejpam-3423	275	65	razzaque	razzaque	NOUN
ejpam-3423	275	66	/	/	SYM
ejpam-3423	275	67	eur	eur	NOUN
ejpam-3423	275	68	.	.	PUNCT
ejpam-3423	276	1	j.	j.	PROPN
ejpam-3423	276	2	pure	pure	PROPN
ejpam-3423	276	3	appl	appl	PROPN
ejpam-3423	276	4	.	.	PROPN
ejpam-3423	276	5	math	math	PROPN
ejpam-3423	276	6	,	,	PUNCT
ejpam-3423	276	7	12	12	NUM
ejpam-3423	276	8	(	(	PUNCT
ejpam-3423	276	9	3	3	NUM
ejpam-3423	276	10	)	)	PUNCT
ejpam-3423	276	11	(	(	PUNCT
ejpam-3423	276	12	2019	2019	NUM
ejpam-3423	276	13	)	)	PUNCT
ejpam-3423	276	14	,	,	PUNCT
ejpam-3423	276	15	1215	1215	NUM
ejpam-3423	276	16	-	-	SYM
ejpam-3423	276	17	1230	1230	NUM
ejpam-3423	276	18	1227	1227	NUM
ejpam-3423	276	19	∂2u	∂2u	ADJ
ejpam-3423	276	20	∂x2	∂x2	PROPN
ejpam-3423	276	21	(	(	PUNCT
ejpam-3423	276	22	xi	xi	NOUN
ejpam-3423	276	23	)	)	PUNCT
ejpam-3423	276	24	∼=	∼=	PROPN
ejpam-3423	276	25	2u(xi+1	2u(xi+1	NUM
ejpam-3423	276	26	)	)	PUNCT
ejpam-3423	276	27	δxi+1(δxi+1	δxi+1(δxi+1	NOUN
ejpam-3423	277	1	+	+	CCONJ
ejpam-3423	277	2	δxi	δxi	ADJ
ejpam-3423	277	3	)	)	PUNCT
ejpam-3423	277	4	−	−	PROPN
ejpam-3423	277	5	2u(xi	2u(xi	NUM
ejpam-3423	277	6	)	)	PUNCT
ejpam-3423	278	1	δxiδxi+1	δxiδxi+1	ADP
ejpam-3423	278	2	+	+	ADJ
ejpam-3423	278	3	2u(xi−1	2u(xi−1	NUM
ejpam-3423	278	4	)	)	PUNCT
ejpam-3423	278	5	δxi(δxi	δxi(δxi	NOUN
ejpam-3423	278	6	+	+	CCONJ
ejpam-3423	278	7	δxi+1	δxi+1	NOUN
ejpam-3423	278	8	)	)	PUNCT
ejpam-3423	278	9	,	,	PUNCT
ejpam-3423	278	10	(	(	PUNCT
ejpam-3423	278	11	74	74	X
ejpam-3423	278	12	)	)	PUNCT
ejpam-3423	278	13	the	the	DET
ejpam-3423	278	14	integral	integral	ADJ
ejpam-3423	278	15	term	term	NOUN
ejpam-3423	278	16	in	in	ADP
ejpam-3423	278	17	(	(	PUNCT
ejpam-3423	278	18	68	68	NUM
ejpam-3423	278	19	)	)	PUNCT
ejpam-3423	278	20	is	be	AUX
ejpam-3423	278	21	discretized	discretize	VERB
ejpam-3423	278	22	in	in	ADP
ejpam-3423	278	23	such	such	DET
ejpam-3423	278	24	a	a	DET
ejpam-3423	278	25	way	way	NOUN
ejpam-3423	278	26	that	that	PRON
ejpam-3423	278	27	the	the	DET
ejpam-3423	278	28	infinite	infinite	ADJ
ejpam-3423	278	29	integral	integral	ADJ
ejpam-3423	278	30	will	will	AUX
ejpam-3423	278	31	split	split	VERB
ejpam-3423	278	32	into	into	ADP
ejpam-3423	278	33	three	three	NUM
ejpam-3423	278	34	parts	part	NOUN
ejpam-3423	278	35	.	.	PUNCT
ejpam-3423	279	1	see	see	VERB
ejpam-3423	279	2	[	[	X
ejpam-3423	279	3	1	1	NUM
ejpam-3423	279	4	]	]	PUNCT
ejpam-3423	279	5	.	.	PUNCT
ejpam-3423	280	1	∫	∫	PROPN
ejpam-3423	281	1	∞	∞	PROPN
ejpam-3423	282	1	−∞	−∞	X
ejpam-3423	282	2	b(x−y)u(y	b(x−y)u(y	NOUN
ejpam-3423	282	3	,	,	PUNCT
ejpam-3423	282	4	t)dy	t)dy	PROPN
ejpam-3423	282	5	=	=	SYM
ejpam-3423	282	6	∫	∫	PROPN
ejpam-3423	282	7	a	a	DET
ejpam-3423	282	8	−∞	−∞	X
ejpam-3423	282	9	b(x−y)u(y	b(x−y)u(y	NOUN
ejpam-3423	282	10	,	,	PUNCT
ejpam-3423	282	11	t)dy+	t)dy+	NUM
ejpam-3423	283	1	∫	∫	PROPN
ejpam-3423	283	2	c	c	PROPN
ejpam-3423	283	3	a	a	DET
ejpam-3423	283	4	b(x−y)u(y	b(x−y)u(y	NOUN
ejpam-3423	283	5	,	,	PUNCT
ejpam-3423	283	6	t)dy+	t)dy+	NUM
ejpam-3423	283	7	∫	∫	PROPN
ejpam-3423	283	8	∞	∞	PROPN
ejpam-3423	283	9	c	c	PROPN
ejpam-3423	283	10	b(x−y)u(y	b(x−y)u(y	NOUN
ejpam-3423	283	11	,	,	PUNCT
ejpam-3423	283	12	t)dy	t)dy	PROPN
ejpam-3423	283	13	,	,	PUNCT
ejpam-3423	283	14	(	(	PUNCT
ejpam-3423	283	15	75	75	NUM
ejpam-3423	283	16	)	)	PUNCT
ejpam-3423	283	17	in	in	ADP
ejpam-3423	283	18	above	above	ADP
ejpam-3423	283	19	equation	equation	NOUN
ejpam-3423	283	20	[	[	X
ejpam-3423	283	21	a	a	X
ejpam-3423	283	22	,	,	PUNCT
ejpam-3423	283	23	c	c	NOUN
ejpam-3423	283	24	]	]	X
ejpam-3423	283	25	=	=	PUNCT
ejpam-3423	284	1	[	[	X
ejpam-3423	284	2	ymin	ymin	X
ejpam-3423	284	3	,	,	PUNCT
ejpam-3423	284	4	ymax	ymax	NUM
ejpam-3423	284	5	]	]	PUNCT
ejpam-3423	284	6	and	and	CCONJ
ejpam-3423	284	7	ymin	ymin	NOUN
ejpam-3423	284	8	=	=	SYM
ejpam-3423	284	9	xmin	xmin	NOUN
ejpam-3423	284	10	,	,	PUNCT
ejpam-3423	284	11	ymax	ymax	X
ejpam-3423	284	12	=	=	SYM
ejpam-3423	284	13	xmax	xmax	PROPN
ejpam-3423	284	14	.	.	PUNCT
ejpam-3423	285	1	with	with	ADP
ejpam-3423	285	2	the	the	DET
ejpam-3423	285	3	help	help	NOUN
ejpam-3423	285	4	of	of	ADP
ejpam-3423	285	5	the	the	DET
ejpam-3423	285	6	composite	composite	ADJ
ejpam-3423	285	7	trapezoidal	trapezoidal	ADJ
ejpam-3423	285	8	rule	rule	NOUN
ejpam-3423	285	9	,	,	PUNCT
ejpam-3423	285	10	one	one	PRON
ejpam-3423	285	11	can	can	AUX
ejpam-3423	285	12	write	write	VERB
ejpam-3423	285	13	∫	∫	PROPN
ejpam-3423	285	14	c	c	PROPN
ejpam-3423	285	15	a	a	DET
ejpam-3423	285	16	b(x−	b(x−	NOUN
ejpam-3423	285	17	y)u(y	y)u(y	NOUN
ejpam-3423	285	18	,	,	PUNCT
ejpam-3423	285	19	t)dy	t)dy	PROPN
ejpam-3423	285	20	in	in	ADP
ejpam-3423	285	21	the	the	DET
ejpam-3423	285	22	form	form	NOUN
ejpam-3423	285	23	of	of	ADP
ejpam-3423	285	24	bu(t	bu(t	NOUN
ejpam-3423	285	25	)	)	PUNCT
ejpam-3423	286	1	≈	≈	PROPN
ejpam-3423	286	2	λ	λ	PROPN
ejpam-3423	286	3	∫	∫	PROPN
ejpam-3423	286	4	c	c	PROPN
ejpam-3423	286	5	a	a	DET
ejpam-3423	286	6	b(xi	b(xi	PROPN
ejpam-3423	286	7	−	−	NOUN
ejpam-3423	286	8	y)u(y	y)u(y	NOUN
ejpam-3423	286	9	,	,	PUNCT
ejpam-3423	286	10	t)dy	t)dy	PROPN
ejpam-3423	286	11	,	,	PUNCT
ejpam-3423	286	12	≈	≈	PROPN
ejpam-3423	286	13	λ	λ	PROPN
ejpam-3423	287	1	[	[	X
ejpam-3423	287	2	1	1	NUM
ejpam-3423	287	3	2	2	NUM
ejpam-3423	287	4	δx1b(xi	δx1b(xi	NOUN
ejpam-3423	287	5	−	−	PROPN
ejpam-3423	287	6	y1)u(y1	y1)u(y1	PROPN
ejpam-3423	287	7	,	,	PUNCT
ejpam-3423	287	8	t	t	PROPN
ejpam-3423	287	9	)	)	PUNCT
ejpam-3423	287	10	+	+	CCONJ
ejpam-3423	287	11	1	1	NUM
ejpam-3423	287	12	2	2	NUM
ejpam-3423	287	13	δxm−1b(xi	δxm−1b(xi	PROPN
ejpam-3423	287	14	−	−	PROPN
ejpam-3423	288	1	ym	ym	INTJ
ejpam-3423	288	2	)	)	PUNCT
ejpam-3423	289	1	u(ym	u(ym	PROPN
ejpam-3423	289	2	,	,	PUNCT
ejpam-3423	289	3	t	t	PROPN
ejpam-3423	289	4	)	)	PUNCT
ejpam-3423	290	1	+	+	CCONJ
ejpam-3423	290	2	m−1∑	m−1∑	NUM
ejpam-3423	290	3	j=2	j=2	PROPN
ejpam-3423	290	4	δxj	δxj	NOUN
ejpam-3423	290	5	+	+	CCONJ
ejpam-3423	290	6	δxj−1	δxj−1	PROPN
ejpam-3423	290	7	2	2	NUM
ejpam-3423	290	8	b(xi	b(xi	VERB
ejpam-3423	290	9	−	−	PROPN
ejpam-3423	290	10	yj)u(yj	yj)u(yj	NOUN
ejpam-3423	290	11	,	,	PUNCT
ejpam-3423	290	12	t	t	PROPN
ejpam-3423	290	13	)	)	PUNCT
ejpam-3423	290	14	]	]	PUNCT
ejpam-3423	290	15	.	.	PUNCT
ejpam-3423	291	1	(	(	PUNCT
ejpam-3423	291	2	76	76	NUM
ejpam-3423	291	3	)	)	PUNCT
ejpam-3423	291	4	to	to	PART
ejpam-3423	291	5	compute	compute	VERB
ejpam-3423	291	6	a	a	DET
ejpam-3423	291	7	european	european	ADJ
ejpam-3423	291	8	call	call	NOUN
ejpam-3423	291	9	option	option	NOUN
ejpam-3423	291	10	,	,	PUNCT
ejpam-3423	291	11	[	[	X
ejpam-3423	291	12	1	1	X
ejpam-3423	291	13	]	]	PUNCT
ejpam-3423	291	14	proposed	propose	VERB
ejpam-3423	291	15	the	the	DET
ejpam-3423	291	16	replacement	replacement	NOUN
ejpam-3423	291	17	of	of	ADP
ejpam-3423	291	18	the	the	DET
ejpam-3423	291	19	integrand	integrand	NOUN
ejpam-3423	291	20	u(x	u(x	NOUN
ejpam-3423	291	21	,	,	PUNCT
ejpam-3423	291	22	τ	τ	X
ejpam-3423	291	23	)	)	PUNCT
ejpam-3423	291	24	over	over	ADP
ejpam-3423	291	25	(	(	PUNCT
ejpam-3423	291	26	−∞	−∞	NOUN
ejpam-3423	291	27	,	,	PUNCT
ejpam-3423	291	28	a	a	PRON
ejpam-3423	291	29	)	)	PUNCT
ejpam-3423	291	30	and	and	CCONJ
ejpam-3423	291	31	(	(	PUNCT
ejpam-3423	291	32	c,∞	c,∞	PROPN
ejpam-3423	291	33	)	)	PUNCT
ejpam-3423	291	34	by	by	ADP
ejpam-3423	291	35	using	use	VERB
ejpam-3423	291	36	the	the	DET
ejpam-3423	291	37	following	follow	VERB
ejpam-3423	291	38	approximations	approximation	NOUN
ejpam-3423	291	39	u(x	u(x	NOUN
ejpam-3423	291	40	,	,	PUNCT
ejpam-3423	291	41	τ)→	τ)→	NOUN
ejpam-3423	291	42	eex	eex	PROPN
ejpam-3423	291	43	−	−	PROPN
ejpam-3423	291	44	ee−rτ	ee−rτ	ADJ
ejpam-3423	291	45	,	,	PUNCT
ejpam-3423	291	46	as	as	ADP
ejpam-3423	291	47	x→	x→	PUNCT
ejpam-3423	291	48	+	+	NOUN
ejpam-3423	291	49	∞	∞	PROPN
ejpam-3423	291	50	,	,	PUNCT
ejpam-3423	291	51	u(x	u(x	NOUN
ejpam-3423	291	52	,	,	PUNCT
ejpam-3423	291	53	τ)→	τ)→	PROPN
ejpam-3423	291	54	0	0	NUM
ejpam-3423	291	55	,	,	PUNCT
ejpam-3423	291	56	as	as	SCONJ
ejpam-3423	291	57	x→	x→	X
ejpam-3423	291	58	−∞.	−∞.	PROPN
ejpam-3423	291	59	hence	hence	ADV
ejpam-3423	291	60	,	,	PUNCT
ejpam-3423	291	61	the	the	DET
ejpam-3423	291	62	other	other	ADJ
ejpam-3423	291	63	part	part	NOUN
ejpam-3423	291	64	of	of	ADP
ejpam-3423	291	65	integral	integral	ADJ
ejpam-3423	291	66	can	can	AUX
ejpam-3423	291	67	be	be	AUX
ejpam-3423	291	68	written	write	VERB
ejpam-3423	291	69	as	as	ADP
ejpam-3423	291	70	g(t	g(t	PROPN
ejpam-3423	291	71	)	)	PUNCT
ejpam-3423	292	1	=	=	PUNCT
ejpam-3423	293	1	λ	λ	X
ejpam-3423	293	2	∫	∫	PROPN
ejpam-3423	293	3	a	a	PRON
ejpam-3423	293	4	−∞	−∞	ADP
ejpam-3423	293	5	b(x−	b(x−	NOUN
ejpam-3423	293	6	y)u(y	y)u(y	NOUN
ejpam-3423	293	7	,	,	PUNCT
ejpam-3423	293	8	t)dy	t)dy	PROPN
ejpam-3423	293	9	+	+	NUM
ejpam-3423	293	10	λ	λ	PROPN
ejpam-3423	293	11	∫	∫	PROPN
ejpam-3423	293	12	∞	∞	PROPN
ejpam-3423	293	13	c	c	NOUN
ejpam-3423	293	14	b(x−	b(x−	NOUN
ejpam-3423	293	15	y)u(y	y)u(y	NOUN
ejpam-3423	293	16	,	,	PUNCT
ejpam-3423	293	17	t)dy	t)dy	PROPN
ejpam-3423	293	18	=	=	PUNCT
ejpam-3423	293	19	λeex+µ+	λeex+µ+	CCONJ
ejpam-3423	293	20	γ2	γ2	PROPN
ejpam-3423	293	21	2	2	NUM
ejpam-3423	293	22	φ	φ	NOUN
ejpam-3423	293	23	(	(	PUNCT
ejpam-3423	293	24	xi	xi	PROPN
ejpam-3423	293	25	−	−	PROPN
ejpam-3423	293	26	xmax	xmax	PROPN
ejpam-3423	294	1	+	+	CCONJ
ejpam-3423	294	2	µ+	µ+	PUNCT
ejpam-3423	294	3	γ2	γ2	PROPN
ejpam-3423	294	4	γ	γ	X
ejpam-3423	294	5	)	)	PUNCT
ejpam-3423	294	6	−	−	PROPN
ejpam-3423	295	1	λee−rtφ	λee−rtφ	ADJ
ejpam-3423	295	2	(	(	PUNCT
ejpam-3423	295	3	xi	xi	X
ejpam-3423	295	4	−	−	PROPN
ejpam-3423	295	5	xmax	xmax	PROPN
ejpam-3423	295	6	+	+	X
ejpam-3423	295	7	µ	µ	X
ejpam-3423	295	8	γ	γ	X
ejpam-3423	295	9	)	)	PUNCT
ejpam-3423	295	10	,	,	PUNCT
ejpam-3423	295	11	(	(	PUNCT
ejpam-3423	295	12	77	77	NUM
ejpam-3423	295	13	)	)	PUNCT
ejpam-3423	295	14	with	with	ADP
ejpam-3423	295	15	φ(y	φ(y	NOUN
ejpam-3423	295	16	)	)	PUNCT
ejpam-3423	295	17	=	=	SYM
ejpam-3423	296	1	1√	1√	NUM
ejpam-3423	296	2	2π	2π	NUM
ejpam-3423	296	3	∫	∫	NOUN
ejpam-3423	296	4	y	y	PROPN
ejpam-3423	296	5	−∞	−∞	ADP
ejpam-3423	296	6	e	e	PROPN
ejpam-3423	296	7	−β2γ	−β2γ	PROPN
ejpam-3423	296	8	2	2	NUM
ejpam-3423	296	9	dβ	dβ	NOUN
ejpam-3423	296	10	.	.	PUNCT
ejpam-3423	297	1	we	we	PRON
ejpam-3423	297	2	can	can	AUX
ejpam-3423	297	3	write	write	VERB
ejpam-3423	297	4	equation	equation	NOUN
ejpam-3423	297	5	(	(	PUNCT
ejpam-3423	297	6	68	68	NUM
ejpam-3423	297	7	)	)	PUNCT
ejpam-3423	297	8	in	in	ADP
ejpam-3423	297	9	abstract	abstract	ADJ
ejpam-3423	297	10	form	form	NOUN
ejpam-3423	297	11	as	as	ADP
ejpam-3423	297	12	u′(t	u′(t	NOUN
ejpam-3423	297	13	)	)	PUNCT
ejpam-3423	297	14	=	=	SYM
ejpam-3423	297	15	au(t	au(t	X
ejpam-3423	297	16	)	)	PUNCT
ejpam-3423	297	17	+	+	NOUN
ejpam-3423	297	18	bu(t	bu(t	X
ejpam-3423	297	19	)	)	PUNCT
ejpam-3423	297	20	+	+	NUM
ejpam-3423	297	21	g(t	g(t	PROPN
ejpam-3423	297	22	)	)	PUNCT
ejpam-3423	297	23	.	.	PUNCT
ejpam-3423	298	1	(	(	PUNCT
ejpam-3423	298	2	78	78	NUM
ejpam-3423	298	3	)	)	PUNCT
ejpam-3423	298	4	above	above	ADP
ejpam-3423	298	5	equation	equation	NOUN
ejpam-3423	298	6	(	(	PUNCT
ejpam-3423	298	7	78	78	NUM
ejpam-3423	298	8	)	)	PUNCT
ejpam-3423	298	9	is	be	AUX
ejpam-3423	298	10	a	a	DET
ejpam-3423	298	11	parabolic	parabolic	ADJ
ejpam-3423	298	12	equation	equation	NOUN
ejpam-3423	298	13	with	with	ADP
ejpam-3423	298	14	a	a	DET
ejpam-3423	298	15	=	=	PUNCT
ejpam-3423	298	16	a4	a4	NOUN
ejpam-3423	298	17	+	+	NOUN
ejpam-3423	298	18	a3	a3	NOUN
ejpam-3423	298	19	+	+	SYM
ejpam-3423	298	20	a2	a2	NOUN
ejpam-3423	298	21	,	,	PUNCT
ejpam-3423	298	22	m.	m.	NOUN
ejpam-3423	298	23	a.	a.	NOUN
ejpam-3423	298	24	gondal	gondal	PROPN
ejpam-3423	298	25	,	,	PUNCT
ejpam-3423	298	26	i.	i.	PROPN
ejpam-3423	298	27	rehman	rehman	PROPN
ejpam-3423	298	28	,	,	PUNCT
ejpam-3423	298	29	a.	a.	NOUN
ejpam-3423	298	30	razzaque	razzaque	NOUN
ejpam-3423	298	31	/	/	SYM
ejpam-3423	298	32	eur	eur	NOUN
ejpam-3423	298	33	.	.	PUNCT
ejpam-3423	299	1	j.	j.	PROPN
ejpam-3423	299	2	pure	pure	PROPN
ejpam-3423	299	3	appl	appl	PROPN
ejpam-3423	299	4	.	.	PROPN
ejpam-3423	299	5	math	math	PROPN
ejpam-3423	299	6	,	,	PUNCT
ejpam-3423	299	7	12	12	NUM
ejpam-3423	299	8	(	(	PUNCT
ejpam-3423	299	9	3	3	NUM
ejpam-3423	299	10	)	)	PUNCT
ejpam-3423	299	11	(	(	PUNCT
ejpam-3423	299	12	2019	2019	NUM
ejpam-3423	299	13	)	)	PUNCT
ejpam-3423	299	14	,	,	PUNCT
ejpam-3423	299	15	1215	1215	NUM
ejpam-3423	299	16	-	-	SYM
ejpam-3423	299	17	1230	1230	NUM
ejpam-3423	299	18	1228	1228	NUM
ejpam-3423	299	19	10	10	NUM
ejpam-3423	299	20	−3	−3	PROPN
ejpam-3423	299	21	10	10	NUM
ejpam-3423	299	22	−2	−2	NOUN
ejpam-3423	299	23	10	10	NUM
ejpam-3423	299	24	−1	−1	NOUN
ejpam-3423	299	25	10	10	NUM
ejpam-3423	299	26	−6	−6	NOUN
ejpam-3423	299	27	10	10	NUM
ejpam-3423	299	28	−4	−4	NUM
ejpam-3423	299	29	10	10	NUM
ejpam-3423	299	30	−2	−2	NOUN
ejpam-3423	299	31	10	10	NUM
ejpam-3423	299	32	0	0	NUM
ejpam-3423	299	33	step	step	NOUN
ejpam-3423	299	34	size	size	NOUN
ejpam-3423	299	35	e	e	NOUN
ejpam-3423	299	36	rr	rr	NOUN
ejpam-3423	299	37	o	o	NOUN
ejpam-3423	299	38	r	r	NOUN
ejpam-3423	299	39	exponential	exponential	NOUN
ejpam-3423	299	40	euler	euler	NOUN
ejpam-3423	299	41	exponential	exponential	NOUN
ejpam-3423	299	42	runge−−kutta	runge−−kutta	ADJ
ejpam-3423	299	43	exact	exact	ADJ
ejpam-3423	299	44	order1	order1	NOUN
ejpam-3423	299	45	exact	exact	ADJ
ejpam-3423	299	46	order2	order2	NOUN
ejpam-3423	299	47	figure	figure	NOUN
ejpam-3423	299	48	1	1	NUM
ejpam-3423	299	49	:	:	PUNCT
ejpam-3423	299	50	the	the	DET
ejpam-3423	299	51	error	error	NOUN
ejpam-3423	299	52	of	of	ADP
ejpam-3423	299	53	the	the	DET
ejpam-3423	299	54	exponential	exponential	NOUN
ejpam-3423	299	55	euler	euler	NOUN
ejpam-3423	299	56	method	method	NOUN
ejpam-3423	299	57	of	of	ADP
ejpam-3423	299	58	order	order	NOUN
ejpam-3423	299	59	one	one	NUM
ejpam-3423	299	60	and	and	CCONJ
ejpam-3423	299	61	the	the	DET
ejpam-3423	299	62	exponential	exponential	ADJ
ejpam-3423	299	63	runge	runge	NOUN
ejpam-3423	299	64	–	–	PUNCT
ejpam-3423	299	65	kutta	kutta	NOUN
ejpam-3423	299	66	method	method	NOUN
ejpam-3423	299	67	of	of	ADP
ejpam-3423	299	68	order	order	NOUN
ejpam-3423	299	69	two	two	NUM
ejpam-3423	299	70	when	when	SCONJ
ejpam-3423	299	71	applied	apply	VERB
ejpam-3423	299	72	to	to	ADP
ejpam-3423	299	73	(	(	PUNCT
ejpam-3423	299	74	68	68	NUM
ejpam-3423	299	75	)	)	PUNCT
ejpam-3423	299	76	with	with	ADP
ejpam-3423	299	77	200	200	NUM
ejpam-3423	299	78	grid	grid	NOUN
ejpam-3423	299	79	points	point	NOUN
ejpam-3423	299	80	.	.	PUNCT
ejpam-3423	300	1	for	for	ADP
ejpam-3423	300	2	comparison	comparison	NOUN
ejpam-3423	300	3	,	,	PUNCT
ejpam-3423	300	4	we	we	PRON
ejpam-3423	300	5	added	add	VERB
ejpam-3423	300	6	lines	line	NOUN
ejpam-3423	300	7	with	with	ADP
ejpam-3423	300	8	slope	slope	NOUN
ejpam-3423	300	9	one	one	NUM
ejpam-3423	300	10	and	and	CCONJ
ejpam-3423	300	11	two	two	NUM
ejpam-3423	300	12	.	.	PUNCT
ejpam-3423	301	1	table	table	NOUN
ejpam-3423	301	2	1	1	NUM
ejpam-3423	301	3	:	:	PUNCT
ejpam-3423	301	4	the	the	DET
ejpam-3423	301	5	table	table	NOUN
ejpam-3423	301	6	clearly	clearly	ADV
ejpam-3423	301	7	demonstrates	demonstrate	VERB
ejpam-3423	301	8	the	the	DET
ejpam-3423	301	9	numerically	numerically	ADV
ejpam-3423	301	10	observed	observe	VERB
ejpam-3423	301	11	temporal	temporal	ADJ
ejpam-3423	301	12	orders	order	NOUN
ejpam-3423	301	13	of	of	ADP
ejpam-3423	301	14	convergence	convergence	NOUN
ejpam-3423	301	15	in	in	ADP
ejpam-3423	301	16	the	the	DET
ejpam-3423	301	17	l2	l2	NOUN
ejpam-3423	301	18	norm	norm	NOUN
ejpam-3423	301	19	with	with	ADP
ejpam-3423	301	20	m	m	PROPN
ejpam-3423	301	21	grid	grid	NOUN
ejpam-3423	301	22	points	point	NOUN
ejpam-3423	301	23	and	and	CCONJ
ejpam-3423	301	24	h	h	NOUN
ejpam-3423	301	25	=	=	NOUN
ejpam-3423	301	26	1/128	1/128	NUM
ejpam-3423	301	27	.	.	PUNCT
ejpam-3423	302	1	here	here	ADV
ejpam-3423	302	2	r	r	NOUN
ejpam-3423	302	3	=	=	NOUN
ejpam-3423	302	4	0.05	0.05	NUM
ejpam-3423	302	5	,	,	PUNCT
ejpam-3423	302	6	e	e	NOUN
ejpam-3423	302	7	=	=	SYM
ejpam-3423	302	8	100	100	NUM
ejpam-3423	302	9	,	,	PUNCT
ejpam-3423	302	10	σ	σ	NOUN
ejpam-3423	302	11	=	=	NUM
ejpam-3423	302	12	0.2	0.2	NUM
ejpam-3423	302	13	,	,	PUNCT
ejpam-3423	302	14	λ	λ	X
ejpam-3423	302	15	=	=	SYM
ejpam-3423	302	16	2	2	NUM
ejpam-3423	302	17	,	,	PUNCT
ejpam-3423	302	18	t	t	NOUN
ejpam-3423	302	19	=	=	SYM
ejpam-3423	302	20	1	1	X
ejpam-3423	302	21	.	.	X
ejpam-3423	302	22	m	m	PROPN
ejpam-3423	302	23	exponential	exponential	PROPN
ejpam-3423	302	24	euler	euler	NOUN
ejpam-3423	302	25	method	method	NOUN
ejpam-3423	302	26	exponential	exponential	ADJ
ejpam-3423	302	27	runge	runge	NOUN
ejpam-3423	302	28	–	–	PUNCT
ejpam-3423	302	29	kutta	kutta	NOUN
ejpam-3423	302	30	method	method	VERB
ejpam-3423	302	31	50	50	NUM
ejpam-3423	302	32	1.0660	1.0660	NUM
ejpam-3423	302	33	2.0159	2.0159	NUM
ejpam-3423	302	34	100	100	NUM
ejpam-3423	302	35	1.0650	1.0650	NUM
ejpam-3423	302	36	2.0149	2.0149	NUM
ejpam-3423	302	37	200	200	NUM
ejpam-3423	302	38	1.0644	1.0644	NUM
ejpam-3423	302	39	2.0141	2.0141	NUM
ejpam-3423	302	40	where	where	SCONJ
ejpam-3423	302	41	a4	a4	NOUN
ejpam-3423	302	42	=	=	SYM
ejpam-3423	302	43	1	1	NUM
ejpam-3423	302	44	2	2	NUM
ejpam-3423	302	45	σ2	σ2	PROPN
ejpam-3423	302	46	∂2u	∂2u	PROPN
ejpam-3423	302	47	∂x2	∂x2	PROPN
ejpam-3423	302	48	,	,	PUNCT
ejpam-3423	302	49	a3	a3	NOUN
ejpam-3423	302	50	=	=	SYM
ejpam-3423	303	1	(	(	PUNCT
ejpam-3423	303	2	r	r	NOUN
ejpam-3423	303	3	−	−	NUM
ejpam-3423	303	4	1	1	NUM
ejpam-3423	303	5	2	2	NUM
ejpam-3423	303	6	σ2	σ2	NOUN
ejpam-3423	303	7	−	−	PROPN
ejpam-3423	303	8	λκ	λκ	NOUN
ejpam-3423	303	9	)	)	PUNCT
ejpam-3423	303	10	∂u	∂u	PROPN
ejpam-3423	303	11	∂x	∂x	PROPN
ejpam-3423	303	12	,	,	PUNCT
ejpam-3423	303	13	a2	a2	PROPN
ejpam-3423	303	14	=	=	PUNCT
ejpam-3423	303	15	−(r	−(r	PROPN
ejpam-3423	303	16	+	+	X
ejpam-3423	303	17	λ)u	λ)u	PROPN
ejpam-3423	303	18	.	.	PUNCT
ejpam-3423	304	1	figure	figure	NOUN
ejpam-3423	304	2	1	1	NUM
ejpam-3423	304	3	clearly	clearly	ADV
ejpam-3423	304	4	elucidates	elucidate	VERB
ejpam-3423	304	5	the	the	DET
ejpam-3423	304	6	convergence	convergence	NOUN
ejpam-3423	304	7	of	of	ADP
ejpam-3423	304	8	computed	compute	VERB
ejpam-3423	304	9	first	first	ADJ
ejpam-3423	304	10	order	order	NOUN
ejpam-3423	304	11	exponential	exponential	NOUN
ejpam-3423	304	12	euler	euler	NOUN
ejpam-3423	304	13	method	method	NOUN
ejpam-3423	304	14	and	and	CCONJ
ejpam-3423	304	15	second	second	ADJ
ejpam-3423	304	16	order	order	NOUN
ejpam-3423	304	17	exponential	exponential	ADJ
ejpam-3423	304	18	runge	runge	NOUN
ejpam-3423	304	19	-	-	PUNCT
ejpam-3423	304	20	kutta	kutta	NOUN
ejpam-3423	304	21	method	method	NOUN
ejpam-3423	304	22	for	for	ADP
ejpam-3423	304	23	constant	constant	ADJ
ejpam-3423	304	24	time	time	NOUN
ejpam-3423	304	25	steps	step	NOUN
ejpam-3423	304	26	with	with	ADP
ejpam-3423	304	27	200	200	NUM
ejpam-3423	304	28	grid	grid	NOUN
ejpam-3423	304	29	points	point	NOUN
ejpam-3423	304	30	.	.	PUNCT
ejpam-3423	305	1	the	the	DET
ejpam-3423	305	2	computed	compute	VERB
ejpam-3423	305	3	solution	solution	NOUN
ejpam-3423	305	4	for	for	ADP
ejpam-3423	305	5	exponential	exponential	ADJ
ejpam-3423	305	6	euler	euler	NOUN
ejpam-3423	305	7	converges	converge	NOUN
ejpam-3423	305	8	at	at	ADP
ejpam-3423	305	9	a	a	DET
ejpam-3423	305	10	first	first	ADJ
ejpam-3423	305	11	-	-	PUNCT
ejpam-3423	305	12	order	order	NOUN
ejpam-3423	305	13	and	and	CCONJ
ejpam-3423	305	14	for	for	ADP
ejpam-3423	305	15	exponential	exponential	ADJ
ejpam-3423	305	16	rungekutta	rungekutta	NOUN
ejpam-3423	305	17	as	as	ADP
ejpam-3423	305	18	second	second	ADJ
ejpam-3423	305	19	-	-	PUNCT
ejpam-3423	305	20	order	order	NOUN
ejpam-3423	305	21	rate	rate	NOUN
ejpam-3423	305	22	,	,	PUNCT
ejpam-3423	305	23	as	as	SCONJ
ejpam-3423	305	24	one	one	PRON
ejpam-3423	305	25	can	can	AUX
ejpam-3423	305	26	see	see	VERB
ejpam-3423	305	27	undoubtedly	undoubtedly	ADV
ejpam-3423	305	28	from	from	ADP
ejpam-3423	305	29	figure	figure	NOUN
ejpam-3423	305	30	1	1	NUM
ejpam-3423	305	31	.	.	PUNCT
ejpam-3423	306	1	the	the	DET
ejpam-3423	306	2	errors	error	NOUN
ejpam-3423	306	3	are	be	AUX
ejpam-3423	306	4	measured	measure	VERB
ejpam-3423	306	5	in	in	ADP
ejpam-3423	306	6	the	the	DET
ejpam-3423	306	7	l2	l2	NOUN
ejpam-3423	306	8	norm	norm	NOUN
ejpam-3423	306	9	.	.	PUNCT
ejpam-3423	307	1	for	for	ADP
ejpam-3423	307	2	comparison	comparison	NOUN
ejpam-3423	307	3	,	,	PUNCT
ejpam-3423	307	4	we	we	PRON
ejpam-3423	307	5	added	add	VERB
ejpam-3423	307	6	the	the	DET
ejpam-3423	307	7	lines	line	NOUN
ejpam-3423	307	8	with	with	ADP
ejpam-3423	307	9	slope	slope	NOUN
ejpam-3423	307	10	one	one	NUM
ejpam-3423	307	11	and	and	CCONJ
ejpam-3423	307	12	slope	slope	NOUN
ejpam-3423	307	13	two	two	NUM
ejpam-3423	307	14	.	.	PUNCT
ejpam-3423	308	1	the	the	DET
ejpam-3423	308	2	numerical	numerical	ADJ
ejpam-3423	308	3	values	value	NOUN
ejpam-3423	308	4	are	be	AUX
ejpam-3423	308	5	shown	show	VERB
ejpam-3423	308	6	in	in	ADP
ejpam-3423	308	7	table	table	NOUN
ejpam-3423	308	8	1	1	NUM
ejpam-3423	308	9	for	for	ADP
ejpam-3423	308	10	the	the	DET
ejpam-3423	308	11	temporal	temporal	ADJ
ejpam-3423	308	12	orders	order	NOUN
ejpam-3423	308	13	of	of	ADP
ejpam-3423	308	14	convergence	convergence	NOUN
ejpam-3423	308	15	in	in	ADP
ejpam-3423	308	16	the	the	DET
ejpam-3423	308	17	l2	l2	NOUN
ejpam-3423	308	18	norm	norm	NOUN
ejpam-3423	308	19	with	with	ADP
ejpam-3423	308	20	m	m	PROPN
ejpam-3423	308	21	grid	grid	NOUN
ejpam-3423	308	22	points	point	NOUN
ejpam-3423	308	23	and	and	CCONJ
ejpam-3423	308	24	h	h	NOUN
ejpam-3423	308	25	=	=	NOUN
ejpam-3423	308	26	1/128	1/128	NUM
ejpam-3423	308	27	,	,	PUNCT
ejpam-3423	308	28	for	for	ADP
ejpam-3423	308	29	the	the	DET
ejpam-3423	308	30	pide	pide	NOUN
ejpam-3423	308	31	(	(	PUNCT
ejpam-3423	308	32	68	68	NUM
ejpam-3423	308	33	)	)	PUNCT
ejpam-3423	308	34	in	in	ADP
ejpam-3423	308	35	case	case	NOUN
ejpam-3423	308	36	of	of	ADP
ejpam-3423	308	37	the	the	DET
ejpam-3423	308	38	exponential	exponential	NOUN
ejpam-3423	308	39	euler	euler	NOUN
ejpam-3423	308	40	method	method	NOUN
ejpam-3423	308	41	of	of	ADP
ejpam-3423	308	42	order	order	NOUN
ejpam-3423	308	43	one	one	NUM
ejpam-3423	308	44	and	and	CCONJ
ejpam-3423	308	45	the	the	DET
ejpam-3423	308	46	exponential	exponential	ADJ
ejpam-3423	308	47	rungekutta	rungekutta	NOUN
ejpam-3423	308	48	method	method	NOUN
ejpam-3423	308	49	of	of	ADP
ejpam-3423	308	50	order	order	NOUN
ejpam-3423	308	51	two	two	NUM
ejpam-3423	308	52	.	.	PUNCT
ejpam-3423	309	1	5	5	X
ejpam-3423	309	2	.	.	X
ejpam-3423	309	3	concluding	conclude	VERB
ejpam-3423	309	4	remarks	remark	VERB
ejpam-3423	309	5	the	the	DET
ejpam-3423	309	6	current	current	ADJ
ejpam-3423	309	7	paper	paper	NOUN
ejpam-3423	309	8	deals	deal	NOUN
ejpam-3423	309	9	with	with	ADP
ejpam-3423	309	10	the	the	DET
ejpam-3423	309	11	convergence	convergence	NOUN
ejpam-3423	309	12	analysis	analysis	NOUN
ejpam-3423	309	13	for	for	ADP
ejpam-3423	309	14	the	the	DET
ejpam-3423	309	15	non	non	ADJ
ejpam-3423	309	16	-	-	ADJ
ejpam-3423	309	17	smooth	smooth	ADJ
ejpam-3423	309	18	initial	initial	ADJ
ejpam-3423	309	19	data	datum	NOUN
ejpam-3423	309	20	,	,	PUNCT
ejpam-3423	309	21	discussed	discuss	VERB
ejpam-3423	309	22	in	in	ADP
ejpam-3423	309	23	sections	section	NOUN
ejpam-3423	309	24	3	3	NUM
ejpam-3423	309	25	.	.	PUNCT
ejpam-3423	310	1	for	for	ADP
ejpam-3423	310	2	the	the	DET
ejpam-3423	310	3	applications	application	NOUN
ejpam-3423	310	4	in	in	ADP
ejpam-3423	310	5	financial	financial	ADJ
ejpam-3423	310	6	mathematics	mathematic	NOUN
ejpam-3423	310	7	,	,	PUNCT
ejpam-3423	310	8	these	these	DET
ejpam-3423	310	9	error	error	NOUN
ejpam-3423	310	10	estimates	estimate	NOUN
ejpam-3423	310	11	are	be	AUX
ejpam-3423	310	12	very	very	ADV
ejpam-3423	310	13	essential	essential	ADJ
ejpam-3423	310	14	.	.	PUNCT
ejpam-3423	311	1	optimal	optimal	ADJ
ejpam-3423	311	2	convergence	convergence	NOUN
ejpam-3423	311	3	is	be	AUX
ejpam-3423	311	4	proved	prove	VERB
ejpam-3423	311	5	for	for	ADP
ejpam-3423	311	6	second	second	ADJ
ejpam-3423	311	7	order	order	NOUN
ejpam-3423	311	8	convergence	convergence	NOUN
ejpam-3423	311	9	for	for	ADP
ejpam-3423	311	10	a	a	DET
ejpam-3423	311	11	two	two	NUM
ejpam-3423	311	12	stage	stage	NOUN
ejpam-3423	311	13	explicit	explicit	ADJ
ejpam-3423	311	14	exponential	exponential	ADJ
ejpam-3423	311	15	runge	runge	NOUN
ejpam-3423	311	16	-	-	PUNCT
ejpam-3423	311	17	kutta	kutta	NOUN
ejpam-3423	311	18	method	method	NOUN
ejpam-3423	311	19	(	(	PUNCT
ejpam-3423	311	20	theorem	theorem	NOUN
ejpam-3423	311	21	1	1	NUM
ejpam-3423	311	22	)	)	PUNCT
ejpam-3423	311	23	.	.	PUNCT
ejpam-3423	312	1	the	the	DET
ejpam-3423	312	2	results	result	NOUN
ejpam-3423	312	3	added	add	VERB
ejpam-3423	312	4	significantly	significantly	ADV
ejpam-3423	312	5	to	to	ADP
ejpam-3423	312	6	the	the	DET
ejpam-3423	312	7	known	know	VERB
ejpam-3423	312	8	convergence	convergence	NOUN
ejpam-3423	312	9	results	result	NOUN
ejpam-3423	312	10	of	of	ADP
ejpam-3423	312	11	exponential	exponential	ADJ
ejpam-3423	312	12	integrators	integrator	NOUN
ejpam-3423	312	13	.	.	PUNCT
ejpam-3423	313	1	the	the	DET
ejpam-3423	313	2	error	error	NOUN
ejpam-3423	313	3	bounds	bound	NOUN
ejpam-3423	313	4	were	be	AUX
ejpam-3423	313	5	previously	previously	ADV
ejpam-3423	313	6	experienced	experience	VERB
ejpam-3423	313	7	only	only	ADV
ejpam-3423	313	8	for	for	ADP
ejpam-3423	313	9	runge	runge	NOUN
ejpam-3423	313	10	-	-	PUNCT
ejpam-3423	313	11	kutta	kutta	NOUN
ejpam-3423	313	12	methods	method	NOUN
ejpam-3423	313	13	and	and	CCONJ
ejpam-3423	313	14	rosenbrock	rosenbrock	NOUN
ejpam-3423	313	15	methods	method	NOUN
ejpam-3423	313	16	.	.	PUNCT
ejpam-3423	314	1	we	we	PRON
ejpam-3423	314	2	developed	develop	VERB
ejpam-3423	314	3	them	they	PRON
ejpam-3423	314	4	for	for	ADP
ejpam-3423	314	5	their	their	PRON
ejpam-3423	314	6	exponential	exponential	ADJ
ejpam-3423	314	7	counterparts	counterpart	NOUN
ejpam-3423	314	8	for	for	ADP
ejpam-3423	314	9	non	non	ADJ
ejpam-3423	314	10	-	-	ADJ
ejpam-3423	314	11	smooth	smooth	ADJ
ejpam-3423	314	12	initial	initial	ADJ
ejpam-3423	314	13	data	datum	NOUN
ejpam-3423	314	14	.	.	PUNCT
ejpam-3423	315	1	references	reference	NOUN
ejpam-3423	315	2	1229	1229	NUM
ejpam-3423	315	3	references	reference	NOUN
ejpam-3423	315	4	[	[	X
ejpam-3423	315	5	1	1	NUM
ejpam-3423	315	6	]	]	PUNCT
ejpam-3423	315	7	a.	a.	NOUN
ejpam-3423	315	8	almendral	almendral	ADJ
ejpam-3423	315	9	and	and	CCONJ
ejpam-3423	315	10	c.w	c.w	PROPN
ejpam-3423	315	11	.	.	PROPN
ejpam-3423	315	12	oosterlee	oosterlee	PROPN
ejpam-3423	315	13	.	.	PUNCT
ejpam-3423	316	1	numerical	numerical	PROPN
ejpam-3423	316	2	valuation	valuation	PROPN
ejpam-3423	316	3	of	of	ADP
ejpam-3423	316	4	options	option	NOUN
ejpam-3423	316	5	with	with	ADP
ejpam-3423	316	6	jumps	jump	NOUN
ejpam-3423	316	7	in	in	ADP
ejpam-3423	316	8	the	the	DET
ejpam-3423	316	9	underlying	underlying	ADJ
ejpam-3423	316	10	.	.	PUNCT
ejpam-3423	316	11	appl	appl	PROPN
ejpam-3423	316	12	.	.	PUNCT
ejpam-3423	317	1	numer	numer	PROPN
ejpam-3423	317	2	.	.	PUNCT
ejpam-3423	317	3	math	math	PROPN
ejpam-3423	317	4	.	.	PUNCT
ejpam-3423	317	5	,	,	PUNCT
ejpam-3423	317	6	53:1–18	53:1–18	NUM
ejpam-3423	317	7	,	,	PUNCT
ejpam-3423	317	8	2005	2005	NUM
ejpam-3423	317	9	.	.	PUNCT
ejpam-3423	318	1	[	[	X
ejpam-3423	318	2	2	2	NUM
ejpam-3423	318	3	]	]	PUNCT
ejpam-3423	318	4	r.	r.	PROPN
ejpam-3423	318	5	cont	cont	PROPN
ejpam-3423	318	6	and	and	CCONJ
ejpam-3423	318	7	e.	e.	PROPN
ejpam-3423	318	8	voltchkova	voltchkova	PROPN
ejpam-3423	318	9	.	.	PUNCT
ejpam-3423	319	1	finite	finite	ADJ
ejpam-3423	319	2	difference	difference	NOUN
ejpam-3423	319	3	methods	method	NOUN
ejpam-3423	319	4	for	for	ADP
ejpam-3423	319	5	option	option	NOUN
ejpam-3423	319	6	pricing	pricing	NOUN
ejpam-3423	319	7	in	in	ADP
ejpam-3423	319	8	jumpdiffusion	jumpdiffusion	NOUN
ejpam-3423	319	9	and	and	CCONJ
ejpam-3423	319	10	exponential	exponential	ADJ
ejpam-3423	319	11	lévy	lévy	NUM
ejpam-3423	319	12	models	model	NOUN
ejpam-3423	319	13	.	.	PUNCT
ejpam-3423	320	1	rapport	rapport	NOUN
ejpam-3423	320	2	interne	interne	PROPN
ejpam-3423	320	3	513(september	513(september	NUM
ejpam-3423	320	4	)	)	PUNCT
ejpam-3423	320	5	,	,	PUNCT
ejpam-3423	321	1	cmap	cmap	NOUN
ejpam-3423	321	2	,	,	PUNCT
ejpam-3423	321	3	2003	2003	NUM
ejpam-3423	321	4	.	.	PUNCT
ejpam-3423	322	1	[	[	X
ejpam-3423	322	2	3	3	NUM
ejpam-3423	322	3	]	]	PUNCT
ejpam-3423	322	4	a.	a.	NOUN
ejpam-3423	322	5	gopaul	gopaul	PROPN
ejpam-3423	322	6	d.y	d.y	PROPN
ejpam-3423	322	7	.	.	PROPN
ejpam-3423	322	8	tangman	tangman	PROPN
ejpam-3423	322	9	and	and	CCONJ
ejpam-3423	322	10	m.	m.	NOUN
ejpam-3423	322	11	bhuruth	bhuruth	PROPN
ejpam-3423	322	12	.	.	PUNCT
ejpam-3423	323	1	exponential	exponential	ADJ
ejpam-3423	323	2	time	time	NOUN
ejpam-3423	323	3	integration	integration	NOUN
ejpam-3423	323	4	and	and	CCONJ
ejpam-3423	323	5	chebychev	chebychev	NOUN
ejpam-3423	323	6	discretization	discretization	NOUN
ejpam-3423	323	7	schemes	scheme	NOUN
ejpam-3423	323	8	for	for	ADP
ejpam-3423	323	9	fast	fast	ADJ
ejpam-3423	323	10	pricing	pricing	NOUN
ejpam-3423	323	11	of	of	ADP
ejpam-3423	323	12	options	option	NOUN
ejpam-3423	323	13	.	.	PUNCT
ejpam-3423	324	1	appl	appl	PROPN
ejpam-3423	324	2	.	.	PUNCT
ejpam-3423	325	1	numer	numer	PROPN
ejpam-3423	325	2	.	.	PUNCT
ejpam-3423	325	3	math	math	PROPN
ejpam-3423	325	4	.	.	PUNCT
ejpam-3423	325	5	,	,	PUNCT
ejpam-3423	325	6	58:1309	58:1309	NUM
ejpam-3423	325	7	–	–	PUNCT
ejpam-3423	325	8	1319	1319	NUM
ejpam-3423	325	9	,	,	PUNCT
ejpam-3423	325	10	2008	2008	NUM
ejpam-3423	325	11	.	.	PUNCT
ejpam-3423	326	1	[	[	X
ejpam-3423	326	2	4	4	NUM
ejpam-3423	326	3	]	]	X
ejpam-3423	326	4	m.a	m.a	PROPN
ejpam-3423	326	5	.	.	PROPN
ejpam-3423	326	6	gondal	gondal	NOUN
ejpam-3423	326	7	.	.	PUNCT
ejpam-3423	327	1	exponential	exponential	ADJ
ejpam-3423	327	2	rosenbrock	rosenbrock	NOUN
ejpam-3423	327	3	integrators	integrator	NOUN
ejpam-3423	327	4	for	for	ADP
ejpam-3423	327	5	option	option	NOUN
ejpam-3423	327	6	pricing	pricing	NOUN
ejpam-3423	327	7	.	.	PUNCT
ejpam-3423	328	1	j.	j.	PROPN
ejpam-3423	328	2	comput	comput	PROPN
ejpam-3423	328	3	.	.	PUNCT
ejpam-3423	329	1	appl	appl	PROPN
ejpam-3423	329	2	.	.	PROPN
ejpam-3423	329	3	math	math	PROPN
ejpam-3423	329	4	.	.	PUNCT
ejpam-3423	329	5	,	,	PUNCT
ejpam-3423	329	6	234:1153–1160	234:1153–1160	NUM
ejpam-3423	329	7	,	,	PUNCT
ejpam-3423	329	8	2010	2010	NUM
ejpam-3423	329	9	.	.	PUNCT
ejpam-3423	330	1	[	[	X
ejpam-3423	330	2	5	5	NUM
ejpam-3423	330	3	]	]	X
ejpam-3423	330	4	m.a	m.a	PROPN
ejpam-3423	330	5	.	.	PROPN
ejpam-3423	330	6	gondal	gondal	NOUN
ejpam-3423	330	7	.	.	PUNCT
ejpam-3423	331	1	convergence	convergence	NOUN
ejpam-3423	331	2	of	of	ADP
ejpam-3423	331	3	an	an	DET
ejpam-3423	331	4	exponential	exponential	NOUN
ejpam-3423	331	5	euler	euler	NOUN
ejpam-3423	331	6	method	method	NOUN
ejpam-3423	331	7	for	for	ADP
ejpam-3423	331	8	option	option	NOUN
ejpam-3423	331	9	pricing	pricing	NOUN
ejpam-3423	331	10	.	.	PUNCT
ejpam-3423	332	1	world	world	NOUN
ejpam-3423	332	2	applied	apply	VERB
ejpam-3423	332	3	sciences	science	NOUN
ejpam-3423	332	4	journal	journal	NOUN
ejpam-3423	332	5	,	,	PUNCT
ejpam-3423	332	6	14:1816–1822	14:1816–1822	NUM
ejpam-3423	332	7	,	,	PUNCT
ejpam-3423	332	8	2011	2011	NUM
ejpam-3423	332	9	.	.	PUNCT
ejpam-3423	333	1	[	[	X
ejpam-3423	333	2	6	6	NUM
ejpam-3423	333	3	]	]	X
ejpam-3423	333	4	m.a	m.a	PROPN
ejpam-3423	333	5	.	.	PROPN
ejpam-3423	333	6	gondal	gondal	NOUN
ejpam-3423	333	7	.	.	PUNCT
ejpam-3423	334	1	option	option	NOUN
ejpam-3423	334	2	valuation	valuation	NOUN
ejpam-3423	334	3	in	in	ADP
ejpam-3423	334	4	jump	jump	NOUN
ejpam-3423	334	5	diffusion	diffusion	NOUN
ejpam-3423	334	6	models	model	NOUN
ejpam-3423	334	7	using	use	VERB
ejpam-3423	334	8	the	the	DET
ejpam-3423	334	9	exponential	exponential	ADJ
ejpam-3423	334	10	runge	runge	NOUN
ejpam-3423	334	11	-	-	PUNCT
ejpam-3423	334	12	kutta	kutta	NOUN
ejpam-3423	334	13	methods	method	NOUN
ejpam-3423	334	14	.	.	PUNCT
ejpam-3423	335	1	world	world	NOUN
ejpam-3423	335	2	applied	apply	VERB
ejpam-3423	335	3	sciences	science	NOUN
ejpam-3423	335	4	journal	journal	NOUN
ejpam-3423	335	5	,	,	PUNCT
ejpam-3423	335	6	13:2396–2404	13:2396–2404	NUM
ejpam-3423	335	7	,	,	PUNCT
ejpam-3423	335	8	2011	2011	NUM
ejpam-3423	335	9	.	.	PUNCT
ejpam-3423	336	1	[	[	X
ejpam-3423	336	2	7	7	X
ejpam-3423	336	3	]	]	X
ejpam-3423	336	4	d.	d.	PROPN
ejpam-3423	336	5	henry	henry	PROPN
ejpam-3423	336	6	.	.	PUNCT
ejpam-3423	337	1	geometric	geometric	ADJ
ejpam-3423	337	2	theorey	theorey	NOUN
ejpam-3423	337	3	of	of	ADP
ejpam-3423	337	4	semilinear	semilinear	PROPN
ejpam-3423	337	5	parabolic	parabolic	PROPN
ejpam-3423	337	6	equations	equation	NOUN
ejpam-3423	337	7	,	,	PUNCT
ejpam-3423	337	8	lecture	lecture	NOUN
ejpam-3423	337	9	notes	note	NOUN
ejpam-3423	337	10	in	in	ADP
ejpam-3423	337	11	math.840	math.840	PROPN
ejpam-3423	337	12	.	.	PUNCT
ejpam-3423	338	1	springer	springer	NOUN
ejpam-3423	338	2	,	,	PUNCT
ejpam-3423	338	3	berlin	berlin	PROPN
ejpam-3423	338	4	,	,	PUNCT
ejpam-3423	338	5	heidelberg	heidelberg	PROPN
ejpam-3423	338	6	,	,	PUNCT
ejpam-3423	338	7	1981	1981	NUM
ejpam-3423	338	8	.	.	PUNCT
ejpam-3423	339	1	[	[	X
ejpam-3423	339	2	8	8	NUM
ejpam-3423	339	3	]	]	X
ejpam-3423	339	4	m.	m.	NOUN
ejpam-3423	339	5	hochbruck	hochbruck	NOUN
ejpam-3423	339	6	and	and	CCONJ
ejpam-3423	339	7	a.	a.	NOUN
ejpam-3423	339	8	ostermann	ostermann	PROPN
ejpam-3423	339	9	.	.	PUNCT
ejpam-3423	340	1	explicit	explicit	ADJ
ejpam-3423	340	2	exponential	exponential	ADJ
ejpam-3423	340	3	runge	runge	NOUN
ejpam-3423	340	4	–	–	PUNCT
ejpam-3423	340	5	kutta	kutta	NOUN
ejpam-3423	340	6	methods	method	NOUN
ejpam-3423	340	7	for	for	ADP
ejpam-3423	340	8	semilinear	semilinear	PROPN
ejpam-3423	340	9	parabolic	parabolic	PROPN
ejpam-3423	340	10	problems	problem	NOUN
ejpam-3423	340	11	.	.	PUNCT
ejpam-3423	341	1	siam	siam	PROPN
ejpam-3423	341	2	j.	j.	PROPN
ejpam-3423	341	3	numer	numer	PROPN
ejpam-3423	341	4	.	.	PUNCT
ejpam-3423	342	1	anal	anal	PROPN
ejpam-3423	342	2	.	.	PROPN
ejpam-3423	342	3	,	,	PUNCT
ejpam-3423	342	4	43:1069–1090	43:1069–1090	NUM
ejpam-3423	342	5	,	,	PUNCT
ejpam-3423	342	6	2005	2005	NUM
ejpam-3423	342	7	.	.	PUNCT
ejpam-3423	343	1	[	[	X
ejpam-3423	343	2	9	9	NUM
ejpam-3423	343	3	]	]	PUNCT
ejpam-3423	343	4	m.	m.	NOUN
ejpam-3423	343	5	hochbruck	hochbruck	NOUN
ejpam-3423	343	6	and	and	CCONJ
ejpam-3423	343	7	a.	a.	NOUN
ejpam-3423	343	8	ostermann	ostermann	PROPN
ejpam-3423	343	9	.	.	PUNCT
ejpam-3423	343	10	exponential	exponential	ADJ
ejpam-3423	343	11	runge	runge	NOUN
ejpam-3423	343	12	–	–	PUNCT
ejpam-3423	343	13	kutta	kutta	NOUN
ejpam-3423	343	14	methods	method	NOUN
ejpam-3423	343	15	for	for	ADP
ejpam-3423	343	16	parabolic	parabolic	ADJ
ejpam-3423	343	17	problems	problem	NOUN
ejpam-3423	343	18	.	.	PUNCT
ejpam-3423	344	1	appl	appl	PROPN
ejpam-3423	344	2	.	.	PUNCT
ejpam-3423	345	1	numer	numer	PROPN
ejpam-3423	345	2	.	.	PUNCT
ejpam-3423	345	3	math	math	PROPN
ejpam-3423	345	4	.	.	PUNCT
ejpam-3423	345	5	,	,	PUNCT
ejpam-3423	346	1	53:323–339	53:323–339	PROPN
ejpam-3423	346	2	,	,	PUNCT
ejpam-3423	346	3	2005	2005	NUM
ejpam-3423	346	4	.	.	PUNCT
ejpam-3423	347	1	[	[	X
ejpam-3423	347	2	10	10	NUM
ejpam-3423	347	3	]	]	PUNCT
ejpam-3423	347	4	m.	m.	NOUN
ejpam-3423	347	5	hochbruck	hochbruck	NOUN
ejpam-3423	347	6	and	and	CCONJ
ejpam-3423	347	7	a.	a.	NOUN
ejpam-3423	347	8	ostermann	ostermann	PROPN
ejpam-3423	347	9	.	.	PUNCT
ejpam-3423	348	1	exponential	exponential	ADJ
ejpam-3423	348	2	integrators	integrator	NOUN
ejpam-3423	348	3	.	.	PUNCT
ejpam-3423	349	1	acta	acta	PROPN
ejpam-3423	349	2	numerica	numerica	PROPN
ejpam-3423	349	3	,	,	PUNCT
ejpam-3423	349	4	19:209	19:209	NUM
ejpam-3423	349	5	–	–	PUNCT
ejpam-3423	349	6	286	286	NUM
ejpam-3423	349	7	,	,	PUNCT
ejpam-3423	349	8	2010	2010	NUM
ejpam-3423	349	9	.	.	PUNCT
ejpam-3423	350	1	[	[	X
ejpam-3423	350	2	11	11	NUM
ejpam-3423	350	3	]	]	X
ejpam-3423	350	4	j.loffeld	j.loffeld	NOUN
ejpam-3423	350	5	and	and	CCONJ
ejpam-3423	350	6	m.tokman	m.tokman	NOUN
ejpam-3423	350	7	.	.	PUNCT
ejpam-3423	351	1	comparative	comparative	ADJ
ejpam-3423	351	2	performance	performance	NOUN
ejpam-3423	351	3	of	of	ADP
ejpam-3423	351	4	exponential	exponential	NOUN
ejpam-3423	351	5	,	,	PUNCT
ejpam-3423	351	6	implicit	implicit	ADJ
ejpam-3423	351	7	,	,	PUNCT
ejpam-3423	351	8	and	and	CCONJ
ejpam-3423	351	9	explicit	explicit	ADJ
ejpam-3423	351	10	integrators	integrator	NOUN
ejpam-3423	351	11	for	for	ADP
ejpam-3423	351	12	stiff	stiff	ADJ
ejpam-3423	351	13	systems	system	NOUN
ejpam-3423	351	14	of	of	ADP
ejpam-3423	351	15	odes	ode	NOUN
ejpam-3423	351	16	.	.	PUNCT
ejpam-3423	352	1	journal	journal	NOUN
ejpam-3423	352	2	of	of	ADP
ejpam-3423	352	3	computational	computational	ADJ
ejpam-3423	352	4	and	and	CCONJ
ejpam-3423	352	5	applied	applied	ADJ
ejpam-3423	352	6	mathematics	mathematic	NOUN
ejpam-3423	352	7	,	,	PUNCT
ejpam-3423	352	8	245:45–67	245:45–67	NUM
ejpam-3423	352	9	,	,	PUNCT
ejpam-3423	352	10	2013	2013	NUM
ejpam-3423	352	11	.	.	PUNCT
ejpam-3423	353	1	[	[	X
ejpam-3423	353	2	12	12	NUM
ejpam-3423	353	3	]	]	X
ejpam-3423	353	4	ch	ch	NOUN
ejpam-3423	353	5	.	.	PUNCT
ejpam-3423	353	6	lubich	lubich	PROPN
ejpam-3423	353	7	and	and	CCONJ
ejpam-3423	353	8	a.	a.	NOUN
ejpam-3423	353	9	ostermann	ostermann	PROPN
ejpam-3423	353	10	.	.	PUNCT
ejpam-3423	353	11	runge	runge	PROPN
ejpam-3423	353	12	–	–	PUNCT
ejpam-3423	353	13	kutta	kutta	NOUN
ejpam-3423	353	14	time	time	NOUN
ejpam-3423	353	15	discretization	discretization	NOUN
ejpam-3423	353	16	of	of	ADP
ejpam-3423	353	17	reaction	reaction	NOUN
ejpam-3423	353	18	-	-	PUNCT
ejpam-3423	353	19	diffusion	diffusion	NOUN
ejpam-3423	353	20	and	and	CCONJ
ejpam-3423	353	21	navier	navier	NOUN
ejpam-3423	353	22	–	–	PUNCT
ejpam-3423	353	23	stokes	stokes	PROPN
ejpam-3423	353	24	equations	equation	NOUN
ejpam-3423	353	25	:	:	PUNCT
ejpam-3423	353	26	nonsmooth	nonsmooth	ADJ
ejpam-3423	353	27	-	-	PUNCT
ejpam-3423	353	28	data	datum	NOUN
ejpam-3423	353	29	error	error	NOUN
ejpam-3423	353	30	estimates	estimate	NOUN
ejpam-3423	353	31	and	and	CCONJ
ejpam-3423	353	32	applications	application	NOUN
ejpam-3423	353	33	to	to	ADP
ejpam-3423	353	34	long	long	ADJ
ejpam-3423	353	35	-	-	PUNCT
ejpam-3423	353	36	time	time	NOUN
ejpam-3423	353	37	behaviour	behaviour	NOUN
ejpam-3423	353	38	.	.	PUNCT
ejpam-3423	354	1	appl	appl	PROPN
ejpam-3423	354	2	.	.	PUNCT
ejpam-3423	355	1	numer	numer	PROPN
ejpam-3423	355	2	.	.	PUNCT
ejpam-3423	355	3	math	math	PROPN
ejpam-3423	355	4	.	.	PUNCT
ejpam-3423	355	5	,	,	PUNCT
ejpam-3423	355	6	22:279–292	22:279–292	NUM
ejpam-3423	355	7	,	,	PUNCT
ejpam-3423	355	8	1996	1996	NUM
ejpam-3423	355	9	.	.	PUNCT
ejpam-3423	356	1	[	[	X
ejpam-3423	356	2	13	13	NUM
ejpam-3423	356	3	]	]	X
ejpam-3423	356	4	c.la	c.la	PROPN
ejpam-3423	356	5	chioma	chioma	PROPN
ejpam-3423	356	6	m.	m.	NOUN
ejpam-3423	356	7	briani	briani	PROPN
ejpam-3423	356	8	and	and	CCONJ
ejpam-3423	356	9	r.	r.	PROPN
ejpam-3423	356	10	natalini	natalini	PROPN
ejpam-3423	356	11	.	.	PUNCT
ejpam-3423	357	1	convergence	convergence	NOUN
ejpam-3423	357	2	of	of	ADP
ejpam-3423	357	3	numerical	numerical	ADJ
ejpam-3423	357	4	schemes	scheme	NOUN
ejpam-3423	357	5	for	for	ADP
ejpam-3423	357	6	viscosity	viscosity	NOUN
ejpam-3423	357	7	solutions	solution	NOUN
ejpam-3423	357	8	to	to	ADP
ejpam-3423	357	9	integro	integro	ADJ
ejpam-3423	357	10	-	-	PUNCT
ejpam-3423	357	11	differential	differential	ADJ
ejpam-3423	357	12	degenerate	degenerate	ADJ
ejpam-3423	357	13	parabolic	parabolic	NOUN
ejpam-3423	357	14	problems	problem	NOUN
ejpam-3423	357	15	arising	arise	VERB
ejpam-3423	357	16	in	in	ADP
ejpam-3423	357	17	financial	financial	ADJ
ejpam-3423	357	18	theory	theory	NOUN
ejpam-3423	357	19	.	.	PUNCT
ejpam-3423	358	1	numerische	numerische	PROPN
ejpam-3423	358	2	mathematik	mathematik	PROPN
ejpam-3423	358	3	,	,	PUNCT
ejpam-3423	358	4	98:607–646	98:607–646	NUM
ejpam-3423	358	5	,	,	PUNCT
ejpam-3423	358	6	2004	2004	NUM
ejpam-3423	358	7	.	.	PUNCT
ejpam-3423	359	1	[	[	X
ejpam-3423	359	2	14	14	NUM
ejpam-3423	359	3	]	]	X
ejpam-3423	359	4	r.c	r.c	PROPN
ejpam-3423	359	5	.	.	PROPN
ejpam-3423	359	6	merton	merton	PROPN
ejpam-3423	359	7	.	.	PUNCT
ejpam-3423	360	1	option	option	NOUN
ejpam-3423	360	2	pricing	pricing	NOUN
ejpam-3423	360	3	when	when	SCONJ
ejpam-3423	360	4	the	the	DET
ejpam-3423	360	5	underlying	underlie	VERB
ejpam-3423	360	6	stocks	stock	NOUN
ejpam-3423	360	7	are	be	AUX
ejpam-3423	360	8	discontinuous	discontinuous	ADJ
ejpam-3423	360	9	.	.	PUNCT
ejpam-3423	361	1	j.	j.	PROPN
ejpam-3423	361	2	finance	finance	PROPN
ejpam-3423	361	3	.	.	PUNCT
ejpam-3423	362	1	econ	econ	PROPN
ejpam-3423	362	2	.	.	PROPN
ejpam-3423	362	3	,	,	PUNCT
ejpam-3423	363	1	5:125–144	5:125–144	NUM
ejpam-3423	363	2	,	,	PUNCT
ejpam-3423	363	3	1976	1976	NUM
ejpam-3423	363	4	.	.	PUNCT
ejpam-3423	364	1	references	reference	NOUN
ejpam-3423	364	2	1230	1230	NUM
ejpam-3423	364	3	[	[	X
ejpam-3423	364	4	15	15	NUM
ejpam-3423	364	5	]	]	X
ejpam-3423	364	6	a.	a.	NOUN
ejpam-3423	364	7	ostermann	ostermann	PROPN
ejpam-3423	364	8	and	and	CCONJ
ejpam-3423	364	9	m.	m.	PROPN
ejpam-3423	364	10	thalhammer	thalhammer	PROPN
ejpam-3423	364	11	.	.	PUNCT
ejpam-3423	365	1	non	non	ADJ
ejpam-3423	365	2	-	-	ADJ
ejpam-3423	365	3	smooth	smooth	ADJ
ejpam-3423	365	4	data	datum	NOUN
ejpam-3423	365	5	error	error	NOUN
ejpam-3423	365	6	estimates	estimate	NOUN
ejpam-3423	365	7	for	for	ADP
ejpam-3423	365	8	linearly	linearly	ADV
ejpam-3423	365	9	implicit	implicit	ADJ
ejpam-3423	365	10	runge	runge	NOUN
ejpam-3423	365	11	–	–	PUNCT
ejpam-3423	365	12	kutta	kutta	NOUN
ejpam-3423	365	13	methods	method	NOUN
ejpam-3423	365	14	.	.	PUNCT
ejpam-3423	366	1	i	i	PRON
ejpam-3423	366	2	m	m	VERB
ejpam-3423	366	3	a	a	PROPN
ejpam-3423	366	4	j.	j.	PROPN
ejpam-3423	366	5	numer	numer	PROPN
ejpam-3423	366	6	.	.	PUNCT
ejpam-3423	367	1	anal	anal	PROPN
ejpam-3423	367	2	,	,	PUNCT
ejpam-3423	367	3	20:167–184	20:167–184	PROPN
ejpam-3423	367	4	,	,	PUNCT
ejpam-3423	367	5	2000	2000	NUM
ejpam-3423	367	6	.	.	PUNCT
ejpam-3423	368	1	[	[	X
ejpam-3423	368	2	16	16	NUM
ejpam-3423	368	3	]	]	PUNCT
ejpam-3423	368	4	a.	a.	NOUN
ejpam-3423	368	5	pazy	pazy	NOUN
ejpam-3423	368	6	.	.	PUNCT
ejpam-3423	369	1	semigroups	semigroup	NOUN
ejpam-3423	369	2	of	of	ADP
ejpam-3423	369	3	linear	linear	PROPN
ejpam-3423	369	4	operators	operator	NOUN
ejpam-3423	369	5	and	and	CCONJ
ejpam-3423	369	6	applications	application	NOUN
ejpam-3423	369	7	to	to	ADP
ejpam-3423	369	8	partial	partial	ADJ
ejpam-3423	369	9	differential	differential	NOUN
ejpam-3423	369	10	equations	equation	NOUN
ejpam-3423	369	11	.	.	PUNCT
ejpam-3423	370	1	springer	springer	NOUN
ejpam-3423	370	2	,	,	PUNCT
ejpam-3423	370	3	new	new	PROPN
ejpam-3423	370	4	york	york	PROPN
ejpam-3423	370	5	,	,	PUNCT
ejpam-3423	370	6	1983	1983	NUM
ejpam-3423	370	7	.	.	PUNCT
ejpam-3423	371	1	[	[	X
ejpam-3423	371	2	17	17	NUM
ejpam-3423	371	3	]	]	X
ejpam-3423	371	4	m.n	m.n	PROPN
ejpam-3423	371	5	.	.	PROPN
ejpam-3423	371	6	le	le	PROPN
ejpam-3423	371	7	roux	roux	VERB
ejpam-3423	371	8	.	.	PUNCT
ejpam-3423	372	1	semidiscretization	semidiscretization	NOUN
ejpam-3423	372	2	in	in	ADP
ejpam-3423	372	3	time	time	NOUN
ejpam-3423	372	4	for	for	ADP
ejpam-3423	372	5	parabolic	parabolic	ADJ
ejpam-3423	372	6	problems	problem	NOUN
ejpam-3423	372	7	.	.	PUNCT
ejpam-3423	373	1	math	math	NOUN
ejpam-3423	373	2	.	.	PUNCT
ejpam-3423	374	1	comput	comput	NOUN
ejpam-3423	374	2	.	.	PUNCT
ejpam-3423	374	3	,	,	PUNCT
ejpam-3423	374	4	33:919–931	33:919–931	NUM
ejpam-3423	374	5	,	,	PUNCT
ejpam-3423	374	6	1979	1979	NUM
ejpam-3423	374	7	.	.	PUNCT
ejpam-3423	375	1	[	[	X
ejpam-3423	375	2	18	18	NUM
ejpam-3423	375	3	]	]	X
ejpam-3423	375	4	j.a	j.a	PROPN
ejpam-3423	375	5	.	.	PROPN
ejpam-3423	375	6	pudykiewicz	pudykiewicz	PROPN
ejpam-3423	375	7	v.t	v.t	PROPN
ejpam-3423	375	8	.	.	PROPN
ejpam-3423	375	9	luan	luan	PROPN
ejpam-3423	375	10	and	and	CCONJ
ejpam-3423	375	11	d.r	d.r	PROPN
ejpam-3423	375	12	.	.	PROPN
ejpam-3423	375	13	reynolds	reynolds	PROPN
ejpam-3423	375	14	.	.	PUNCT
ejpam-3423	376	1	further	further	ADJ
ejpam-3423	376	2	development	development	NOUN
ejpam-3423	376	3	of	of	ADP
ejpam-3423	376	4	efficient	efficient	ADJ
ejpam-3423	376	5	and	and	CCONJ
ejpam-3423	376	6	accurate	accurate	ADJ
ejpam-3423	376	7	time	time	NOUN
ejpam-3423	376	8	integration	integration	NOUN
ejpam-3423	376	9	schemes	scheme	NOUN
ejpam-3423	376	10	for	for	ADP
ejpam-3423	376	11	meteorological	meteorological	ADJ
ejpam-3423	376	12	models	model	NOUN
ejpam-3423	376	13	.	.	PUNCT
ejpam-3423	377	1	journal	journal	NOUN
ejpam-3423	377	2	of	of	ADP
ejpam-3423	377	3	computational	computational	ADJ
ejpam-3423	377	4	physics	physics	NOUN
ejpam-3423	377	5	,	,	PUNCT
ejpam-3423	377	6	376:817–837	376:817–837	NUM
ejpam-3423	377	7	,	,	PUNCT
ejpam-3423	377	8	2019	2019	NUM
ejpam-3423	377	9	.	.	PUNCT
ejpam-3423	378	1	[	[	X
ejpam-3423	378	2	19	19	NUM
ejpam-3423	378	3	]	]	PUNCT
ejpam-3423	378	4	b.	b.	PROPN
ejpam-3423	378	5	wang	wang	PROPN
ejpam-3423	378	6	and	and	CCONJ
ejpam-3423	378	7	x.	x.	PROPN
ejpam-3423	378	8	wu	wu	PROPN
ejpam-3423	378	9	.	.	PUNCT
ejpam-3423	378	10	exponential	exponential	ADJ
ejpam-3423	378	11	collocation	collocation	NOUN
ejpam-3423	378	12	methods	method	NOUN
ejpam-3423	378	13	for	for	ADP
ejpam-3423	378	14	conservative	conservative	ADJ
ejpam-3423	378	15	or	or	CCONJ
ejpam-3423	378	16	dissipative	dissipative	ADJ
ejpam-3423	378	17	systems	system	NOUN
ejpam-3423	378	18	.	.	PUNCT
ejpam-3423	379	1	journal	journal	NOUN
ejpam-3423	379	2	of	of	ADP
ejpam-3423	379	3	computational	computational	ADJ
ejpam-3423	379	4	and	and	CCONJ
ejpam-3423	379	5	applied	applied	ADJ
ejpam-3423	379	6	mathematics	mathematic	NOUN
ejpam-3423	379	7	,	,	PUNCT
ejpam-3423	379	8	360:99–116	360:99–116	NUM
ejpam-3423	379	9	,	,	PUNCT
ejpam-3423	379	10	2019	2019	NUM
ejpam-3423	379	11	.	.	PUNCT