id	sid	tid	token	lemma	pos
ap-9228	1	1	acta	acta	PROPN
ap-9228	1	2	polytechnica	polytechnica	PROPN
ap-9228	1	3	https://doi.org/10.14311/ap.2024.64.0087	https://doi.org/10.14311/ap.2024.64.0087	PROPN
ap-9228	1	4	acta	acta	PROPN
ap-9228	1	5	polytechnica	polytechnica	PROPN
ap-9228	1	6	64(2):87–102	64(2):87–102	NUM
ap-9228	1	7	,	,	PUNCT
ap-9228	1	8	2024	2024	NUM
ap-9228	1	9	©	©	ADP
ap-9228	1	10	2024	2024	NUM
ap-9228	1	11	the	the	DET
ap-9228	1	12	author(s	author(s	NOUN
ap-9228	1	13	)	)	PUNCT
ap-9228	1	14	.	.	PUNCT
ap-9228	2	1	licensed	license	VERB
ap-9228	2	2	under	under	ADP
ap-9228	2	3	a	a	DET
ap-9228	2	4	cc	cc	NOUN
ap-9228	2	5	-	-	PUNCT
ap-9228	2	6	by	by	ADP
ap-9228	2	7	4.0	4.0	NUM
ap-9228	2	8	licence	licence	NOUN
ap-9228	2	9	published	publish	VERB
ap-9228	2	10	by	by	ADP
ap-9228	2	11	the	the	DET
ap-9228	2	12	czech	czech	PROPN
ap-9228	2	13	technical	technical	PROPN
ap-9228	2	14	university	university	PROPN
ap-9228	2	15	in	in	ADP
ap-9228	2	16	prague	prague	PROPN
ap-9228	2	17	an	an	DET
ap-9228	2	18	innovative	innovative	ADJ
ap-9228	2	19	iterative	iterative	NOUN
ap-9228	2	20	approach	approach	NOUN
ap-9228	2	21	to	to	ADP
ap-9228	2	22	solving	solve	VERB
ap-9228	2	23	volterra	volterra	NOUN
ap-9228	2	24	integral	integral	ADJ
ap-9228	2	25	equations	equation	NOUN
ap-9228	2	26	of	of	ADP
ap-9228	2	27	second	second	ADJ
ap-9228	2	28	kind	kind	NOUN
ap-9228	2	29	mohammed	mohammed	PROPN
ap-9228	2	30	abdulshareef	abdulshareef	PROPN
ap-9228	2	31	husseina	husseina	PROPN
ap-9228	2	32	,	,	PUNCT
ap-9228	2	33	b,∗	b,∗	PROPN
ap-9228	2	34	,	,	PUNCT
ap-9228	2	35	hassan	hassan	PROPN
ap-9228	2	36	kamil	kamil	PROPN
ap-9228	2	37	jassimc	jassimc	PROPN
ap-9228	2	38	,	,	PUNCT
ap-9228	2	39	ali	ali	PROPN
ap-9228	2	40	kareem	kareem	PROPN
ap-9228	2	41	jassimb	jassimb	PROPN
ap-9228	2	42	,	,	PUNCT
ap-9228	2	43	d	d	PROPN
ap-9228	2	44	a	a	DET
ap-9228	2	45	al	al	PROPN
ap-9228	2	46	-	-	PUNCT
ap-9228	2	47	ayen	ayen	PROPN
ap-9228	2	48	university	university	NOUN
ap-9228	2	49	,	,	PUNCT
ap-9228	2	50	scientific	scientific	ADJ
ap-9228	2	51	research	research	NOUN
ap-9228	2	52	center	center	NOUN
ap-9228	2	53	,	,	PUNCT
ap-9228	2	54	64001	64001	NUM
ap-9228	2	55	nasiriyah	nasiriyah	NOUN
ap-9228	2	56	,	,	PUNCT
ap-9228	2	57	iraq	iraq	PROPN
ap-9228	2	58	b	b	PROPN
ap-9228	2	59	ministry	ministry	PROPN
ap-9228	2	60	of	of	ADP
ap-9228	2	61	education	education	PROPN
ap-9228	2	62	,	,	PUNCT
ap-9228	2	63	education	education	NOUN
ap-9228	2	64	directorate	directorate	NOUN
ap-9228	2	65	of	of	ADP
ap-9228	2	66	thi	thi	PROPN
ap-9228	2	67	-	-	PUNCT
ap-9228	2	68	qar	qar	PROPN
ap-9228	2	69	,	,	PUNCT
ap-9228	2	70	64001	64001	NUM
ap-9228	2	71	nasiriyah	nasiriyah	NOUN
ap-9228	2	72	,	,	PUNCT
ap-9228	2	73	iraq	iraq	PROPN
ap-9228	2	74	c	c	PROPN
ap-9228	2	75	university	university	PROPN
ap-9228	2	76	of	of	ADP
ap-9228	2	77	thi	thi	PROPN
ap-9228	2	78	-	-	PUNCT
ap-9228	2	79	qar	qar	PROPN
ap-9228	2	80	,	,	PUNCT
ap-9228	2	81	department	department	NOUN
ap-9228	2	82	of	of	ADP
ap-9228	2	83	mathematics	mathematic	NOUN
ap-9228	2	84	,	,	PUNCT
ap-9228	2	85	64001	64001	NUM
ap-9228	2	86	nasiriyah	nasiriyah	NOUN
ap-9228	2	87	,	,	PUNCT
ap-9228	2	88	iraq	iraq	PROPN
ap-9228	2	89	d	d	PROPN
ap-9228	2	90	national	national	PROPN
ap-9228	2	91	university	university	PROPN
ap-9228	2	92	of	of	ADP
ap-9228	2	93	science	science	NOUN
ap-9228	2	94	and	and	CCONJ
ap-9228	2	95	technology	technology	NOUN
ap-9228	2	96	,	,	PUNCT
ap-9228	2	97	college	college	NOUN
ap-9228	2	98	of	of	ADP
ap-9228	2	99	technical	technical	ADJ
ap-9228	2	100	engineering	engineering	NOUN
ap-9228	2	101	,	,	PUNCT
ap-9228	2	102	64001	64001	NUM
ap-9228	2	103	nasiriyah	nasiriyah	NOUN
ap-9228	2	104	,	,	PUNCT
ap-9228	2	105	iraq	iraq	PROPN
ap-9228	2	106	∗	∗	VERB
ap-9228	2	107	corresponding	correspond	VERB
ap-9228	2	108	author	author	NOUN
ap-9228	2	109	:	:	PUNCT
ap-9228	2	110	mshirq@utq.edu.iq	mshirq@utq.edu.iq	ADJ
ap-9228	2	111	abstract	abstract	NOUN
ap-9228	2	112	.	.	PUNCT
ap-9228	3	1	many	many	ADJ
ap-9228	3	2	scientists	scientist	NOUN
ap-9228	3	3	have	have	AUX
ap-9228	3	4	shown	show	VERB
ap-9228	3	5	great	great	ADJ
ap-9228	3	6	interest	interest	NOUN
ap-9228	3	7	in	in	ADP
ap-9228	3	8	exploring	explore	VERB
ap-9228	3	9	the	the	DET
ap-9228	3	10	realm	realm	NOUN
ap-9228	3	11	of	of	ADP
ap-9228	3	12	second	second	ADJ
ap-9228	3	13	-	-	PUNCT
ap-9228	3	14	kind	kind	NOUN
ap-9228	3	15	integral	integral	ADJ
ap-9228	3	16	equations	equation	NOUN
ap-9228	3	17	,	,	PUNCT
ap-9228	3	18	offering	offer	VERB
ap-9228	3	19	many	many	ADJ
ap-9228	3	20	techniques	technique	NOUN
ap-9228	3	21	for	for	ADP
ap-9228	3	22	solving	solve	VERB
ap-9228	3	23	them	they	PRON
ap-9228	3	24	,	,	PUNCT
ap-9228	3	25	including	include	VERB
ap-9228	3	26	exact	exact	ADJ
ap-9228	3	27	,	,	PUNCT
ap-9228	3	28	approximate	approximate	ADJ
ap-9228	3	29	,	,	PUNCT
ap-9228	3	30	and	and	CCONJ
ap-9228	3	31	numerical	numerical	ADJ
ap-9228	3	32	methods	method	NOUN
ap-9228	3	33	.	.	PUNCT
ap-9228	4	1	this	this	DET
ap-9228	4	2	paper	paper	NOUN
ap-9228	4	3	introduces	introduce	VERB
ap-9228	4	4	the	the	DET
ap-9228	4	5	hussein	hussein	NOUN
ap-9228	4	6	-	-	PUNCT
ap-9228	4	7	jassim	jassim	NOUN
ap-9228	4	8	method	method	NOUN
ap-9228	4	9	(	(	PUNCT
ap-9228	4	10	hj	hj	NOUN
ap-9228	4	11	-	-	PUNCT
ap-9228	4	12	method	method	NOUN
ap-9228	4	13	)	)	PUNCT
ap-9228	4	14	for	for	ADP
ap-9228	4	15	solving	solve	VERB
ap-9228	4	16	volterra	volterra	NOUN
ap-9228	4	17	integral	integral	ADJ
ap-9228	4	18	equations	equation	NOUN
ap-9228	4	19	of	of	ADP
ap-9228	4	20	the	the	DET
ap-9228	4	21	second	second	ADJ
ap-9228	4	22	kind	kind	NOUN
ap-9228	4	23	(	(	PUNCT
ap-9228	4	24	viesks	viesk	NOUN
ap-9228	4	25	)	)	PUNCT
ap-9228	4	26	.	.	PUNCT
ap-9228	5	1	the	the	DET
ap-9228	5	2	foundation	foundation	NOUN
ap-9228	5	3	of	of	ADP
ap-9228	5	4	this	this	DET
ap-9228	5	5	approach	approach	NOUN
ap-9228	5	6	lies	lie	VERB
ap-9228	5	7	in	in	ADP
ap-9228	5	8	the	the	DET
ap-9228	5	9	principle	principle	NOUN
ap-9228	5	10	of	of	ADP
ap-9228	5	11	maclaurin	maclaurin	NOUN
ap-9228	5	12	expansion	expansion	NOUN
ap-9228	5	13	.	.	PUNCT
ap-9228	6	1	the	the	DET
ap-9228	6	2	algorithm	algorithm	NOUN
ap-9228	6	3	of	of	ADP
ap-9228	6	4	the	the	DET
ap-9228	6	5	method	method	NOUN
ap-9228	6	6	was	be	AUX
ap-9228	6	7	derived	derive	VERB
ap-9228	6	8	,	,	PUNCT
ap-9228	6	9	and	and	CCONJ
ap-9228	6	10	its	its	PRON
ap-9228	6	11	convergence	convergence	NOUN
ap-9228	6	12	was	be	AUX
ap-9228	6	13	analysed	analyse	VERB
ap-9228	6	14	.	.	PUNCT
ap-9228	7	1	furthermore	furthermore	ADV
ap-9228	7	2	,	,	PUNCT
ap-9228	7	3	the	the	DET
ap-9228	7	4	method	method	NOUN
ap-9228	7	5	was	be	AUX
ap-9228	7	6	applied	apply	VERB
ap-9228	7	7	to	to	ADP
ap-9228	7	8	various	various	ADJ
ap-9228	7	9	volterra	volterra	NOUN
ap-9228	7	10	integral	integral	ADJ
ap-9228	7	11	equations	equation	NOUN
ap-9228	7	12	,	,	PUNCT
ap-9228	7	13	encompassing	encompass	VERB
ap-9228	7	14	linear	linear	NOUN
ap-9228	7	15	,	,	PUNCT
ap-9228	7	16	nonlinear	nonlinear	ADJ
ap-9228	7	17	,	,	PUNCT
ap-9228	7	18	homogeneous	homogeneous	ADJ
ap-9228	7	19	,	,	PUNCT
ap-9228	7	20	and	and	CCONJ
ap-9228	7	21	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	7	22	cases	case	NOUN
ap-9228	7	23	.	.	PUNCT
ap-9228	8	1	ultimately	ultimately	ADV
ap-9228	8	2	,	,	PUNCT
ap-9228	8	3	the	the	DET
ap-9228	8	4	proposed	propose	VERB
ap-9228	8	5	method	method	NOUN
ap-9228	8	6	successfully	successfully	ADV
ap-9228	8	7	addressed	address	VERB
ap-9228	8	8	these	these	DET
ap-9228	8	9	equations	equation	NOUN
ap-9228	8	10	,	,	PUNCT
ap-9228	8	11	with	with	ADP
ap-9228	8	12	the	the	DET
ap-9228	8	13	approximate	approximate	ADJ
ap-9228	8	14	solutions	solution	NOUN
ap-9228	8	15	converging	converge	VERB
ap-9228	8	16	toward	toward	ADP
ap-9228	8	17	the	the	DET
ap-9228	8	18	exact	exact	ADJ
ap-9228	8	19	solution	solution	NOUN
ap-9228	8	20	.	.	PUNCT
ap-9228	9	1	keywords	keyword	NOUN
ap-9228	9	2	:	:	PUNCT
ap-9228	9	3	volterra	volterra	PROPN
ap-9228	9	4	integral	integral	ADJ
ap-9228	9	5	equations	equation	NOUN
ap-9228	9	6	,	,	PUNCT
ap-9228	9	7	new	new	ADJ
ap-9228	9	8	iterative	iterative	NOUN
ap-9228	9	9	method	method	NOUN
ap-9228	9	10	,	,	PUNCT
ap-9228	9	11	taylor	taylor	PROPN
ap-9228	9	12	series	series	PROPN
ap-9228	9	13	,	,	PUNCT
ap-9228	9	14	hussein	hussein	PROPN
ap-9228	9	15	jassim	jassim	PROPN
ap-9228	9	16	method	method	NOUN
ap-9228	9	17	.	.	PUNCT
ap-9228	10	1	1	1	X
ap-9228	10	2	.	.	X
ap-9228	10	3	introduction	introduction	NOUN
ap-9228	10	4	in	in	ADP
ap-9228	10	5	the	the	DET
ap-9228	10	6	realm	realm	NOUN
ap-9228	10	7	of	of	ADP
ap-9228	10	8	quantum	quantum	ADJ
ap-9228	10	9	mechanics	mechanic	NOUN
ap-9228	10	10	,	,	PUNCT
ap-9228	10	11	volterra	volterra	NOUN
ap-9228	10	12	integral	integral	ADJ
ap-9228	10	13	equations	equation	NOUN
ap-9228	10	14	of	of	ADP
ap-9228	10	15	the	the	DET
ap-9228	10	16	second	second	ADJ
ap-9228	10	17	kind	kind	NOUN
ap-9228	10	18	are	be	AUX
ap-9228	10	19	employed	employ	VERB
ap-9228	10	20	to	to	PART
ap-9228	10	21	describe	describe	VERB
ap-9228	10	22	scenarios	scenario	NOUN
ap-9228	10	23	involving	involve	VERB
ap-9228	10	24	quantum	quantum	ADJ
ap-9228	10	25	interference	interference	NOUN
ap-9228	10	26	and	and	CCONJ
ap-9228	10	27	fusion	fusion	NOUN
ap-9228	10	28	of	of	ADP
ap-9228	10	29	systems	system	NOUN
ap-9228	10	30	.	.	PUNCT
ap-9228	11	1	in	in	ADP
ap-9228	11	2	addition	addition	NOUN
ap-9228	11	3	,	,	PUNCT
ap-9228	11	4	integral	integral	ADJ
ap-9228	11	5	equations	equation	NOUN
ap-9228	11	6	find	find	VERB
ap-9228	11	7	applications	application	NOUN
ap-9228	11	8	in	in	ADP
ap-9228	11	9	studying	study	VERB
ap-9228	11	10	electric	electric	ADJ
ap-9228	11	11	current	current	ADJ
ap-9228	11	12	,	,	PUNCT
ap-9228	11	13	electric	electric	ADJ
ap-9228	11	14	charge	charge	NOUN
ap-9228	11	15	,	,	PUNCT
ap-9228	11	16	and	and	CCONJ
ap-9228	11	17	the	the	DET
ap-9228	11	18	distribution	distribution	NOUN
ap-9228	11	19	of	of	ADP
ap-9228	11	20	electromagnetic	electromagnetic	ADJ
ap-9228	11	21	fields	field	NOUN
ap-9228	11	22	.	.	PUNCT
ap-9228	12	1	in	in	ADP
ap-9228	12	2	engineering	engineering	NOUN
ap-9228	12	3	,	,	PUNCT
ap-9228	12	4	integral	integral	ADJ
ap-9228	12	5	equations	equation	NOUN
ap-9228	12	6	are	be	AUX
ap-9228	12	7	extensively	extensively	ADV
ap-9228	12	8	used	use	VERB
ap-9228	12	9	.	.	PUNCT
ap-9228	13	1	in	in	ADP
ap-9228	13	2	structural	structural	ADJ
ap-9228	13	3	analysis	analysis	NOUN
ap-9228	13	4	,	,	PUNCT
ap-9228	13	5	they	they	PRON
ap-9228	13	6	provide	provide	VERB
ap-9228	13	7	insights	insight	NOUN
ap-9228	13	8	into	into	ADP
ap-9228	13	9	the	the	DET
ap-9228	13	10	behaviour	behaviour	NOUN
ap-9228	13	11	of	of	ADP
ap-9228	13	12	structures	structure	NOUN
ap-9228	13	13	under	under	ADP
ap-9228	13	14	external	external	ADJ
ap-9228	13	15	forces	force	NOUN
ap-9228	13	16	and	and	CCONJ
ap-9228	13	17	load	load	NOUN
ap-9228	13	18	distribution	distribution	NOUN
ap-9228	13	19	.	.	PUNCT
ap-9228	14	1	additionally	additionally	ADV
ap-9228	14	2	,	,	PUNCT
ap-9228	14	3	in	in	ADP
ap-9228	14	4	the	the	DET
ap-9228	14	5	field	field	NOUN
ap-9228	14	6	of	of	ADP
ap-9228	14	7	electromagnetic	electromagnetic	ADJ
ap-9228	14	8	fields	field	NOUN
ap-9228	14	9	,	,	PUNCT
ap-9228	14	10	integral	integral	ADJ
ap-9228	14	11	equations	equation	NOUN
ap-9228	14	12	are	be	AUX
ap-9228	14	13	valuable	valuable	ADJ
ap-9228	14	14	tools	tool	NOUN
ap-9228	14	15	for	for	ADP
ap-9228	14	16	analysing	analyse	VERB
ap-9228	14	17	the	the	DET
ap-9228	14	18	interaction	interaction	NOUN
ap-9228	14	19	between	between	ADP
ap-9228	14	20	electric	electric	ADJ
ap-9228	14	21	currents	current	NOUN
ap-9228	14	22	and	and	CCONJ
ap-9228	14	23	electromagnetic	electromagnetic	ADJ
ap-9228	14	24	fields	field	NOUN
ap-9228	14	25	[	[	X
ap-9228	14	26	1	1	NUM
ap-9228	14	27	,	,	PUNCT
ap-9228	14	28	2	2	NUM
ap-9228	14	29	]	]	PUNCT
ap-9228	14	30	.	.	PUNCT
ap-9228	15	1	integral	integral	ADJ
ap-9228	15	2	equations	equation	NOUN
ap-9228	15	3	have	have	AUX
ap-9228	15	4	proven	prove	VERB
ap-9228	15	5	their	their	PRON
ap-9228	15	6	value	value	NOUN
ap-9228	15	7	in	in	ADP
ap-9228	15	8	fields	field	NOUN
ap-9228	15	9	as	as	ADV
ap-9228	15	10	diverse	diverse	ADJ
ap-9228	15	11	as	as	ADP
ap-9228	15	12	mathematical	mathematical	ADJ
ap-9228	15	13	modelling	modelling	NOUN
ap-9228	15	14	,	,	PUNCT
ap-9228	15	15	image	image	NOUN
ap-9228	15	16	processing	processing	NOUN
ap-9228	15	17	,	,	PUNCT
ap-9228	15	18	and	and	CCONJ
ap-9228	15	19	signal	signal	VERB
ap-9228	15	20	analysis	analysis	NOUN
ap-9228	15	21	.	.	PUNCT
ap-9228	16	1	offering	offer	VERB
ap-9228	16	2	a	a	DET
ap-9228	16	3	versatile	versatile	ADJ
ap-9228	16	4	framework	framework	NOUN
ap-9228	16	5	,	,	PUNCT
ap-9228	16	6	these	these	DET
ap-9228	16	7	equations	equation	NOUN
ap-9228	16	8	prove	prove	VERB
ap-9228	16	9	instrumental	instrumental	ADJ
ap-9228	16	10	in	in	ADP
ap-9228	16	11	solving	solve	VERB
ap-9228	16	12	complex	complex	ADJ
ap-9228	16	13	problems	problem	NOUN
ap-9228	16	14	involving	involve	VERB
ap-9228	16	15	boundary	boundary	ADJ
ap-9228	16	16	or	or	CCONJ
ap-9228	16	17	initial	initial	ADJ
ap-9228	16	18	conditions	condition	NOUN
ap-9228	16	19	.	.	PUNCT
ap-9228	17	1	in	in	ADP
ap-9228	17	2	the	the	DET
ap-9228	17	3	field	field	NOUN
ap-9228	17	4	of	of	ADP
ap-9228	17	5	mathematical	mathematical	ADJ
ap-9228	17	6	modelling	modelling	NOUN
ap-9228	17	7	,	,	PUNCT
ap-9228	17	8	integral	integral	ADJ
ap-9228	17	9	equations	equation	NOUN
ap-9228	17	10	empower	empower	VERB
ap-9228	17	11	researchers	researcher	NOUN
ap-9228	17	12	to	to	PART
ap-9228	17	13	articulate	articulate	VERB
ap-9228	17	14	and	and	CCONJ
ap-9228	17	15	comprehend	comprehend	VERB
ap-9228	17	16	real	real	ADJ
ap-9228	17	17	-	-	PUNCT
ap-9228	17	18	world	world	NOUN
ap-9228	17	19	phenomena	phenomenon	NOUN
ap-9228	17	20	through	through	ADP
ap-9228	17	21	mathematical	mathematical	ADJ
ap-9228	17	22	representations	representation	NOUN
ap-9228	17	23	.	.	PUNCT
ap-9228	18	1	in	in	ADP
ap-9228	18	2	image	image	NOUN
ap-9228	18	3	processing	processing	NOUN
ap-9228	18	4	,	,	PUNCT
ap-9228	18	5	they	they	PRON
ap-9228	18	6	find	find	VERB
ap-9228	18	7	application	application	NOUN
ap-9228	18	8	in	in	ADP
ap-9228	18	9	tasks	task	NOUN
ap-9228	18	10	such	such	ADJ
ap-9228	18	11	as	as	ADP
ap-9228	18	12	image	image	NOUN
ap-9228	18	13	reconstruction	reconstruction	NOUN
ap-9228	18	14	and	and	CCONJ
ap-9228	18	15	denoising	denoising	NOUN
ap-9228	18	16	.	.	PUNCT
ap-9228	19	1	in	in	ADP
ap-9228	19	2	addition	addition	NOUN
ap-9228	19	3	,	,	PUNCT
ap-9228	19	4	integral	integral	ADJ
ap-9228	19	5	equations	equation	NOUN
ap-9228	19	6	play	play	VERB
ap-9228	19	7	a	a	DET
ap-9228	19	8	pivotal	pivotal	ADJ
ap-9228	19	9	role	role	NOUN
ap-9228	19	10	in	in	ADP
ap-9228	19	11	signal	signal	ADJ
ap-9228	19	12	analysis	analysis	NOUN
ap-9228	19	13	,	,	PUNCT
ap-9228	19	14	where	where	SCONJ
ap-9228	19	15	they	they	PRON
ap-9228	19	16	are	be	AUX
ap-9228	19	17	used	use	VERB
ap-9228	19	18	for	for	ADP
ap-9228	19	19	tasks	task	NOUN
ap-9228	19	20	such	such	ADJ
ap-9228	19	21	as	as	ADP
ap-9228	19	22	signal	signal	ADJ
ap-9228	19	23	deconvolution	deconvolution	NOUN
ap-9228	19	24	and	and	CCONJ
ap-9228	19	25	system	system	NOUN
ap-9228	19	26	identification	identification	NOUN
ap-9228	19	27	[	[	X
ap-9228	19	28	3–7	3–7	NOUN
ap-9228	19	29	]	]	PUNCT
ap-9228	19	30	.	.	PUNCT
ap-9228	20	1	the	the	DET
ap-9228	20	2	applications	application	NOUN
ap-9228	20	3	of	of	ADP
ap-9228	20	4	integral	integral	ADJ
ap-9228	20	5	equations	equation	NOUN
ap-9228	20	6	are	be	AUX
ap-9228	20	7	vast	vast	ADJ
ap-9228	20	8	and	and	CCONJ
ap-9228	20	9	varied	varied	ADJ
ap-9228	20	10	,	,	PUNCT
ap-9228	20	11	spanning	span	VERB
ap-9228	20	12	fields	field	NOUN
ap-9228	20	13	including	include	VERB
ap-9228	20	14	physics	physics	NOUN
ap-9228	20	15	,	,	PUNCT
ap-9228	20	16	engineering	engineering	NOUN
ap-9228	20	17	,	,	PUNCT
ap-9228	20	18	mathematics	mathematic	NOUN
ap-9228	20	19	,	,	PUNCT
ap-9228	20	20	image	image	NOUN
ap-9228	20	21	processing	processing	NOUN
ap-9228	20	22	,	,	PUNCT
ap-9228	20	23	and	and	CCONJ
ap-9228	20	24	signal	signal	VERB
ap-9228	20	25	analysis	analysis	NOUN
ap-9228	20	26	.	.	PUNCT
ap-9228	21	1	their	their	PRON
ap-9228	21	2	versatility	versatility	NOUN
ap-9228	21	3	in	in	ADP
ap-9228	21	4	handling	handle	VERB
ap-9228	21	5	complex	complex	ADJ
ap-9228	21	6	problems	problem	NOUN
ap-9228	21	7	and	and	CCONJ
ap-9228	21	8	incorporating	incorporate	VERB
ap-9228	21	9	boundary	boundary	ADJ
ap-9228	21	10	or	or	CCONJ
ap-9228	21	11	initial	initial	ADJ
ap-9228	21	12	conditions	condition	NOUN
ap-9228	21	13	renders	render	VERB
ap-9228	21	14	them	they	PRON
ap-9228	21	15	a	a	DET
ap-9228	21	16	valuable	valuable	ADJ
ap-9228	21	17	tool	tool	NOUN
ap-9228	21	18	for	for	ADP
ap-9228	21	19	modelling	modelling	NOUN
ap-9228	21	20	and	and	CCONJ
ap-9228	21	21	understanding	understand	VERB
ap-9228	21	22	diverse	diverse	ADJ
ap-9228	21	23	phenomena	phenomenon	NOUN
ap-9228	21	24	across	across	ADP
ap-9228	21	25	different	different	ADJ
ap-9228	21	26	disciplines	discipline	NOUN
ap-9228	21	27	[	[	X
ap-9228	21	28	8	8	NUM
ap-9228	21	29	]	]	PUNCT
ap-9228	21	30	.	.	PUNCT
ap-9228	22	1	volterra	volterra	PROPN
ap-9228	22	2	integral	integral	ADJ
ap-9228	22	3	equations	equation	NOUN
ap-9228	22	4	of	of	ADP
ap-9228	22	5	the	the	DET
ap-9228	22	6	second	second	ADJ
ap-9228	22	7	kind	kind	NOUN
ap-9228	22	8	are	be	AUX
ap-9228	22	9	a	a	DET
ap-9228	22	10	set	set	NOUN
ap-9228	22	11	of	of	ADP
ap-9228	22	12	integral	integral	ADJ
ap-9228	22	13	equations	equation	NOUN
ap-9228	22	14	with	with	ADP
ap-9228	22	15	the	the	DET
ap-9228	22	16	general	general	ADJ
ap-9228	22	17	form	form	NOUN
ap-9228	22	18	:	:	PUNCT
ap-9228	22	19	u(x	u(x	NOUN
ap-9228	22	20	)	)	PUNCT
ap-9228	22	21	=	=	SYM
ap-9228	22	22	ψ(x	ψ(x	NOUN
ap-9228	22	23	)	)	PUNCT
ap-9228	23	1	+	+	CCONJ
ap-9228	23	2	σ	σ	NUM
ap-9228	23	3	∫	∫	PROPN
ap-9228	23	4	x	x	SYM
ap-9228	23	5	0	0	PUNCT
ap-9228	23	6	k(x	k(x	PROPN
ap-9228	23	7	,	,	PUNCT
ap-9228	23	8	t)n(u(t	t)n(u(t	NOUN
ap-9228	23	9	)	)	PUNCT
ap-9228	23	10	)	)	PUNCT
ap-9228	24	1	dt	dt	PROPN
ap-9228	24	2	,	,	PUNCT
ap-9228	24	3	where	where	SCONJ
ap-9228	24	4	u(x	u(x	NOUN
ap-9228	24	5	)	)	PUNCT
ap-9228	24	6	is	be	AUX
ap-9228	24	7	the	the	DET
ap-9228	24	8	unknown	unknown	ADJ
ap-9228	24	9	function	function	NOUN
ap-9228	24	10	to	to	PART
ap-9228	24	11	be	be	AUX
ap-9228	24	12	determined	determine	VERB
ap-9228	24	13	,	,	PUNCT
ap-9228	24	14	ψ(x	ψ(x	PROPN
ap-9228	24	15	)	)	PUNCT
ap-9228	24	16	is	be	AUX
ap-9228	24	17	the	the	DET
ap-9228	24	18	known	know	VERB
ap-9228	24	19	function	function	NOUN
ap-9228	24	20	,	,	PUNCT
ap-9228	24	21	σ	σ	PROPN
ap-9228	24	22	is	be	AUX
ap-9228	24	23	a	a	DET
ap-9228	24	24	real	real	ADJ
ap-9228	24	25	number	number	NOUN
ap-9228	24	26	,	,	PUNCT
ap-9228	24	27	k(x	k(x	PROPN
ap-9228	24	28	,	,	PUNCT
ap-9228	24	29	t	t	PROPN
ap-9228	24	30	)	)	PUNCT
ap-9228	24	31	is	be	AUX
ap-9228	24	32	the	the	DET
ap-9228	24	33	integral	integral	ADJ
ap-9228	24	34	kernel	kernel	NOUN
ap-9228	24	35	that	that	PRON
ap-9228	24	36	determines	determine	VERB
ap-9228	24	37	the	the	DET
ap-9228	24	38	interaction	interaction	NOUN
ap-9228	24	39	between	between	ADP
ap-9228	24	40	the	the	DET
ap-9228	24	41	variables	variable	NOUN
ap-9228	24	42	x	x	PUNCT
ap-9228	24	43	and	and	CCONJ
ap-9228	24	44	t	t	PROPN
ap-9228	24	45	,	,	PUNCT
ap-9228	24	46	and	and	CCONJ
ap-9228	24	47	n	n	PRON
ap-9228	24	48	is	be	AUX
ap-9228	24	49	a	a	DET
ap-9228	24	50	nonlinear	nonlinear	ADJ
ap-9228	24	51	operator	operator	NOUN
ap-9228	24	52	.	.	PUNCT
ap-9228	25	1	many	many	ADJ
ap-9228	25	2	complex	complex	ADJ
ap-9228	25	3	problems	problem	NOUN
ap-9228	25	4	in	in	ADP
ap-9228	25	5	mathematics	mathematic	NOUN
ap-9228	25	6	,	,	PUNCT
ap-9228	25	7	chemistry	chemistry	NOUN
ap-9228	25	8	,	,	PUNCT
ap-9228	25	9	biology	biology	NOUN
ap-9228	25	10	,	,	PUNCT
ap-9228	25	11	astrophysics	astrophysic	NOUN
ap-9228	25	12	,	,	PUNCT
ap-9228	25	13	and	and	CCONJ
ap-9228	25	14	mechanics	mechanic	NOUN
ap-9228	25	15	,	,	PUNCT
ap-9228	25	16	such	such	ADJ
ap-9228	25	17	as	as	ADP
ap-9228	25	18	the	the	DET
ap-9228	25	19	problem	problem	NOUN
ap-9228	25	20	of	of	ADP
ap-9228	25	21	radiative	radiative	ADJ
ap-9228	25	22	energy	energy	NOUN
ap-9228	25	23	transfer	transfer	NOUN
ap-9228	25	24	,	,	PUNCT
ap-9228	25	25	the	the	DET
ap-9228	25	26	oscillation	oscillation	NOUN
ap-9228	25	27	problems	problem	NOUN
ap-9228	25	28	of	of	ADP
ap-9228	25	29	string	string	NOUN
ap-9228	25	30	and	and	CCONJ
ap-9228	25	31	membrane	membrane	NOUN
ap-9228	25	32	,	,	PUNCT
ap-9228	25	33	and	and	CCONJ
ap-9228	25	34	the	the	DET
ap-9228	25	35	problem	problem	NOUN
ap-9228	25	36	of	of	ADP
ap-9228	25	37	momentum	momentum	NOUN
ap-9228	25	38	representation	representation	NOUN
ap-9228	25	39	in	in	ADP
ap-9228	25	40	quantum	quantum	ADJ
ap-9228	25	41	mechanics	mechanic	NOUN
ap-9228	25	42	,	,	PUNCT
ap-9228	25	43	can	can	AUX
ap-9228	25	44	be	be	AUX
ap-9228	25	45	expressed	express	VERB
ap-9228	25	46	in	in	ADP
ap-9228	25	47	terms	term	NOUN
ap-9228	25	48	of	of	ADP
ap-9228	25	49	the	the	DET
ap-9228	25	50	volterra	volterra	NOUN
ap-9228	25	51	integral	integral	ADJ
ap-9228	25	52	equation	equation	NOUN
ap-9228	25	53	.	.	PUNCT
ap-9228	26	1	solving	solve	VERB
ap-9228	26	2	these	these	DET
ap-9228	26	3	equations	equation	NOUN
ap-9228	26	4	poses	pose	VERB
ap-9228	26	5	a	a	DET
ap-9228	26	6	challenge	challenge	NOUN
ap-9228	26	7	due	due	ADP
ap-9228	26	8	to	to	ADP
ap-9228	26	9	their	their	PRON
ap-9228	26	10	nonlinearity	nonlinearity	NOUN
ap-9228	26	11	and	and	CCONJ
ap-9228	26	12	complex	complex	ADJ
ap-9228	26	13	mathematical	mathematical	ADJ
ap-9228	26	14	implications	implication	NOUN
ap-9228	26	15	.	.	PUNCT
ap-9228	27	1	solving	solve	VERB
ap-9228	27	2	volterra	volterra	NOUN
ap-9228	27	3	integral	integral	ADJ
ap-9228	27	4	equations	equation	NOUN
ap-9228	27	5	of	of	ADP
ap-9228	27	6	the	the	DET
ap-9228	27	7	second	second	ADJ
ap-9228	27	8	kind	kind	NOUN
ap-9228	27	9	requires	require	VERB
ap-9228	27	10	specialised	specialised	ADJ
ap-9228	27	11	techniques	technique	NOUN
ap-9228	27	12	,	,	PUNCT
ap-9228	27	13	such	such	ADJ
ap-9228	27	14	as	as	ADP
ap-9228	27	15	laplace	laplace	NOUN
ap-9228	27	16	transformations	transformation	NOUN
ap-9228	27	17	,	,	PUNCT
ap-9228	27	18	numerical	numerical	ADJ
ap-9228	27	19	analysis	analysis	NOUN
ap-9228	27	20	,	,	PUNCT
ap-9228	27	21	and	and	CCONJ
ap-9228	27	22	numerical	numerical	ADJ
ap-9228	27	23	approximation	approximation	NOUN
ap-9228	27	24	methods	method	NOUN
ap-9228	27	25	to	to	PART
ap-9228	27	26	obtain	obtain	VERB
ap-9228	27	27	accurate	accurate	ADJ
ap-9228	27	28	and	and	CCONJ
ap-9228	27	29	reliable	reliable	ADJ
ap-9228	27	30	results	result	NOUN
ap-9228	27	31	[	[	X
ap-9228	27	32	9–13	9–13	NOUN
ap-9228	27	33	]	]	PUNCT
ap-9228	27	34	.	.	PUNCT
ap-9228	28	1	in	in	ADP
ap-9228	28	2	recent	recent	ADJ
ap-9228	28	3	times	time	NOUN
ap-9228	28	4	,	,	PUNCT
ap-9228	28	5	numerous	numerous	ADJ
ap-9228	28	6	researchers	researcher	NOUN
ap-9228	28	7	have	have	AUX
ap-9228	28	8	directed	direct	VERB
ap-9228	28	9	their	their	PRON
ap-9228	28	10	attention	attention	NOUN
ap-9228	28	11	towards	towards	ADP
ap-9228	28	12	investigating	investigate	VERB
ap-9228	28	13	solutions	solution	NOUN
ap-9228	28	14	for	for	ADP
ap-9228	28	15	ndes	nde	NOUN
ap-9228	28	16	,	,	PUNCT
ap-9228	28	17	employing	employ	VERB
ap-9228	28	18	diverse	diverse	ADJ
ap-9228	28	19	techniques	technique	NOUN
ap-9228	28	20	as	as	SCONJ
ap-9228	28	21	exemplified	exemplify	VERB
ap-9228	28	22	by	by	ADP
ap-9228	28	23	their	their	PRON
ap-9228	28	24	work	work	NOUN
ap-9228	28	25	,	,	PUNCT
ap-9228	28	26	adomian	adomian	NOUN
ap-9228	28	27	decomposition	decomposition	NOUN
ap-9228	28	28	technique	technique	NOUN
ap-9228	28	29	(	(	PUNCT
ap-9228	28	30	adm	adm	PROPN
ap-9228	28	31	)	)	PUNCT
ap-9228	29	1	[	[	X
ap-9228	29	2	14	14	NUM
ap-9228	29	3	]	]	X
ap-9228	29	4	,	,	PUNCT
ap-9228	29	5	87	87	NUM
ap-9228	29	6	https://doi.org/10.14311/ap.2024.64.0087	https://doi.org/10.14311/ap.2024.64.0087	NOUN
ap-9228	29	7	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
ap-9228	29	8	https://www.cvut.cz/en	https://www.cvut.cz/en	PROPN
ap-9228	29	9	m.	m.	NOUN
ap-9228	29	10	a.	a.	PROPN
ap-9228	29	11	hussein	hussein	PROPN
ap-9228	29	12	,	,	PUNCT
ap-9228	29	13	h.	h.	PROPN
ap-9228	29	14	k.	k.	PROPN
ap-9228	29	15	jassim	jassim	PROPN
ap-9228	29	16	,	,	PUNCT
ap-9228	29	17	a.	a.	PROPN
ap-9228	29	18	k.	k.	PROPN
ap-9228	29	19	jassim	jassim	PROPN
ap-9228	29	20	acta	acta	PROPN
ap-9228	29	21	polytechnica	polytechnica	PROPN
ap-9228	29	22	variational	variational	ADJ
ap-9228	29	23	iteration	iteration	NOUN
ap-9228	29	24	technique	technique	NOUN
ap-9228	29	25	(	(	PUNCT
ap-9228	29	26	vim	vim	NOUN
ap-9228	29	27	)	)	PUNCT
ap-9228	30	1	[	[	X
ap-9228	30	2	15	15	NUM
ap-9228	30	3	]	]	PUNCT
ap-9228	30	4	,	,	PUNCT
ap-9228	30	5	homotopy	homotopy	VERB
ap-9228	30	6	perturbation	perturbation	NOUN
ap-9228	30	7	technique	technique	NOUN
ap-9228	30	8	(	(	PUNCT
ap-9228	30	9	hpm	hpm	NOUN
ap-9228	30	10	)	)	PUNCT
ap-9228	31	1	[	[	X
ap-9228	31	2	16	16	NUM
ap-9228	31	3	]	]	PUNCT
ap-9228	31	4	and	and	CCONJ
ap-9228	31	5	daftardar	daftardar	ADV
ap-9228	31	6	-	-	PUNCT
ap-9228	31	7	jafari	jafari	ADJ
ap-9228	31	8	technique	technique	NOUN
ap-9228	31	9	(	(	PUNCT
ap-9228	31	10	djm	djm	PROPN
ap-9228	31	11	)	)	PUNCT
ap-9228	32	1	[	[	X
ap-9228	32	2	17	17	NUM
ap-9228	32	3	]	]	PUNCT
ap-9228	32	4	.	.	PUNCT
ap-9228	33	1	this	this	DET
ap-9228	33	2	article	article	NOUN
ap-9228	33	3	presents	present	VERB
ap-9228	33	4	a	a	DET
ap-9228	33	5	new	new	ADJ
ap-9228	33	6	iterative	iterative	NOUN
ap-9228	33	7	technique	technique	NOUN
ap-9228	33	8	for	for	ADP
ap-9228	33	9	solving	solve	VERB
ap-9228	33	10	volterra	volterra	NOUN
ap-9228	33	11	integral	integral	ADJ
ap-9228	33	12	equations	equation	NOUN
ap-9228	33	13	of	of	ADP
ap-9228	33	14	the	the	DET
ap-9228	33	15	second	second	ADJ
ap-9228	33	16	kind	kind	NOUN
ap-9228	33	17	.	.	PUNCT
ap-9228	34	1	this	this	DET
ap-9228	34	2	technique	technique	NOUN
ap-9228	34	3	is	be	AUX
ap-9228	34	4	mainly	mainly	ADV
ap-9228	34	5	based	base	VERB
ap-9228	34	6	on	on	ADP
ap-9228	34	7	taylor	taylor	PROPN
ap-9228	34	8	’s	’s	PART
ap-9228	34	9	expansion	expansion	NOUN
ap-9228	34	10	.	.	PUNCT
ap-9228	35	1	in	in	ADP
ap-9228	35	2	order	order	NOUN
ap-9228	35	3	to	to	PART
ap-9228	35	4	introduce	introduce	VERB
ap-9228	35	5	the	the	DET
ap-9228	35	6	technique	technique	NOUN
ap-9228	35	7	,	,	PUNCT
ap-9228	35	8	we	we	PRON
ap-9228	35	9	need	need	VERB
ap-9228	35	10	to	to	PART
ap-9228	35	11	present	present	VERB
ap-9228	35	12	the	the	DET
ap-9228	35	13	definition	definition	NOUN
ap-9228	35	14	of	of	ADP
ap-9228	35	15	the	the	DET
ap-9228	35	16	maclaurin	maclaurin	NOUN
ap-9228	35	17	series	series	NOUN
ap-9228	35	18	.	.	PUNCT
ap-9228	36	1	a	a	DET
ap-9228	36	2	maclaurin	maclaurin	NOUN
ap-9228	36	3	series	series	NOUN
ap-9228	36	4	is	be	AUX
ap-9228	36	5	a	a	DET
ap-9228	36	6	series	series	NOUN
ap-9228	36	7	expansion	expansion	NOUN
ap-9228	36	8	of	of	ADP
ap-9228	36	9	a	a	DET
ap-9228	36	10	function	function	NOUN
ap-9228	36	11	about	about	ADP
ap-9228	36	12	the	the	DET
ap-9228	36	13	origin	origin	NOUN
ap-9228	36	14	point	point	NOUN
ap-9228	36	15	.	.	PUNCT
ap-9228	37	1	a	a	DET
ap-9228	37	2	one	one	NUM
ap-9228	37	3	-	-	PUNCT
ap-9228	37	4	dimensional	dimensional	ADJ
ap-9228	37	5	maclaurin	maclaurin	NOUN
ap-9228	37	6	series	series	NOUN
ap-9228	37	7	is	be	AUX
ap-9228	37	8	an	an	DET
ap-9228	37	9	expansion	expansion	NOUN
ap-9228	37	10	of	of	ADP
ap-9228	37	11	a	a	DET
ap-9228	37	12	real	real	ADJ
ap-9228	37	13	function	function	NOUN
ap-9228	37	14	f(x	f(x	PROPN
ap-9228	37	15	)	)	PUNCT
ap-9228	37	16	about	about	ADV
ap-9228	37	17	point	point	NOUN
ap-9228	37	18	x	x	SYM
ap-9228	37	19	=	=	SYM
ap-9228	37	20	0	0	NUM
ap-9228	37	21	,	,	PUNCT
ap-9228	37	22	given	give	VERB
ap-9228	37	23	by	by	ADP
ap-9228	37	24	[	[	X
ap-9228	37	25	18	18	NUM
ap-9228	37	26	]	]	X
ap-9228	37	27	:	:	PUNCT
ap-9228	37	28	f(x	f(x	PROPN
ap-9228	37	29	)	)	PUNCT
ap-9228	37	30	=	=	SYM
ap-9228	37	31	f(0	f(0	NOUN
ap-9228	37	32	)	)	PUNCT
ap-9228	37	33	+	+	NUM
ap-9228	37	34	xf	xf	PROPN
ap-9228	37	35	′(0	′(0	PROPN
ap-9228	37	36	)	)	PUNCT
ap-9228	38	1	+	+	CCONJ
ap-9228	38	2	x2	x2	PROPN
ap-9228	38	3	2	2	NUM
ap-9228	38	4	!	!	PUNCT
ap-9228	38	5	f	f	X
ap-9228	39	1	′′(0	′′(0	PUNCT
ap-9228	39	2	)	)	PUNCT
ap-9228	39	3	+	+	NUM
ap-9228	39	4	·	·	PUNCT
ap-9228	39	5	·	·	PUNCT
ap-9228	39	6	·	·	PUNCT
ap-9228	40	1	+	+	NUM
ap-9228	40	2	xn	xn	PROPN
ap-9228	40	3	n	n	CCONJ
ap-9228	40	4	!	!	PUNCT
ap-9228	40	5	f	f	PROPN
ap-9228	40	6	(	(	PUNCT
ap-9228	40	7	n)(0	n)(0	NUM
ap-9228	40	8	)	)	PUNCT
ap-9228	40	9	+	+	CCONJ
ap-9228	40	10	.	.	PUNCT
ap-9228	40	11	.	.	PUNCT
ap-9228	40	12	.	.	PUNCT
ap-9228	40	13	.	.	PUNCT
ap-9228	41	1	the	the	DET
ap-9228	41	2	hussein	hussein	PROPN
ap-9228	41	3	-	-	PUNCT
ap-9228	41	4	jassim	jassim	PROPN
ap-9228	41	5	method	method	NOUN
ap-9228	41	6	is	be	AUX
ap-9228	41	7	considered	consider	VERB
ap-9228	41	8	as	as	ADP
ap-9228	41	9	one	one	NUM
ap-9228	41	10	of	of	ADP
ap-9228	41	11	the	the	DET
ap-9228	41	12	latest	late	ADJ
ap-9228	41	13	analytical	analytical	ADJ
ap-9228	41	14	approaches	approach	NOUN
ap-9228	41	15	to	to	ADP
ap-9228	41	16	finding	find	VERB
ap-9228	41	17	approximate	approximate	ADJ
ap-9228	41	18	solutions	solution	NOUN
ap-9228	41	19	to	to	PART
ap-9228	41	20	differential	differential	VERB
ap-9228	41	21	and	and	CCONJ
ap-9228	41	22	integral	integral	ADJ
ap-9228	41	23	equations	equation	NOUN
ap-9228	41	24	.	.	PUNCT
ap-9228	42	1	the	the	DET
ap-9228	42	2	idea	idea	NOUN
ap-9228	42	3	of	of	ADP
ap-9228	42	4	this	this	DET
ap-9228	42	5	method	method	NOUN
ap-9228	42	6	is	be	AUX
ap-9228	42	7	based	base	VERB
ap-9228	42	8	on	on	ADP
ap-9228	42	9	the	the	DET
ap-9228	42	10	use	use	NOUN
ap-9228	42	11	of	of	ADP
ap-9228	42	12	maclaurin	maclaurin	NOUN
ap-9228	42	13	’s	’s	PART
ap-9228	42	14	expansion	expansion	NOUN
ap-9228	42	15	.	.	PUNCT
ap-9228	43	1	it	it	PRON
ap-9228	43	2	has	have	AUX
ap-9228	43	3	been	be	AUX
ap-9228	43	4	successfully	successfully	ADV
ap-9228	43	5	applied	apply	VERB
ap-9228	43	6	to	to	ADP
ap-9228	43	7	various	various	ADJ
ap-9228	43	8	types	type	NOUN
ap-9228	43	9	of	of	ADP
ap-9228	43	10	equations	equation	NOUN
ap-9228	43	11	,	,	PUNCT
ap-9228	43	12	including	include	VERB
ap-9228	43	13	linear	linear	PROPN
ap-9228	43	14	,	,	PUNCT
ap-9228	43	15	non	non	ADJ
ap-9228	43	16	-	-	ADJ
ap-9228	43	17	linear	linear	ADJ
ap-9228	43	18	,	,	PUNCT
ap-9228	43	19	homogeneous	homogeneous	ADJ
ap-9228	43	20	,	,	PUNCT
ap-9228	43	21	and	and	CCONJ
ap-9228	43	22	non	non	ADJ
ap-9228	43	23	-	-	ADJ
ap-9228	43	24	homogeneous	homogeneous	ADJ
ap-9228	43	25	,	,	PUNCT
ap-9228	43	26	as	as	ADV
ap-9228	43	27	well	well	ADV
ap-9228	43	28	as	as	ADP
ap-9228	43	29	to	to	ADP
ap-9228	43	30	both	both	CCONJ
ap-9228	43	31	differential	differential	ADJ
ap-9228	43	32	and	and	CCONJ
ap-9228	43	33	integral	integral	ADJ
ap-9228	43	34	equation	equation	NOUN
ap-9228	43	35	systems	system	NOUN
ap-9228	43	36	.	.	PUNCT
ap-9228	44	1	ultimately	ultimately	ADV
ap-9228	44	2	,	,	PUNCT
ap-9228	44	3	this	this	DET
ap-9228	44	4	method	method	NOUN
ap-9228	44	5	has	have	AUX
ap-9228	44	6	proven	prove	VERB
ap-9228	44	7	to	to	PART
ap-9228	44	8	be	be	AUX
ap-9228	44	9	efficient	efficient	ADJ
ap-9228	44	10	and	and	CCONJ
ap-9228	44	11	effective	effective	ADJ
ap-9228	44	12	in	in	ADP
ap-9228	44	13	solving	solve	VERB
ap-9228	44	14	these	these	DET
ap-9228	44	15	equations	equation	NOUN
ap-9228	44	16	.	.	PUNCT
ap-9228	45	1	following	follow	VERB
ap-9228	45	2	is	be	AUX
ap-9228	45	3	an	an	DET
ap-9228	45	4	outline	outline	NOUN
ap-9228	45	5	of	of	ADP
ap-9228	45	6	the	the	DET
ap-9228	45	7	paper	paper	NOUN
ap-9228	45	8	.	.	PUNCT
ap-9228	46	1	in	in	ADP
ap-9228	46	2	the	the	DET
ap-9228	46	3	second	second	ADJ
ap-9228	46	4	section	section	NOUN
ap-9228	46	5	of	of	ADP
ap-9228	46	6	the	the	DET
ap-9228	46	7	paper	paper	NOUN
ap-9228	46	8	,	,	PUNCT
ap-9228	46	9	we	we	PRON
ap-9228	46	10	derive	derive	VERB
ap-9228	46	11	the	the	DET
ap-9228	46	12	algorithm	algorithm	NOUN
ap-9228	46	13	for	for	ADP
ap-9228	46	14	the	the	DET
ap-9228	46	15	method	method	NOUN
ap-9228	46	16	.	.	PUNCT
ap-9228	47	1	the	the	DET
ap-9228	47	2	third	third	ADJ
ap-9228	47	3	section	section	NOUN
ap-9228	47	4	discusses	discuss	VERB
ap-9228	47	5	the	the	DET
ap-9228	47	6	convergence	convergence	NOUN
ap-9228	47	7	of	of	ADP
ap-9228	47	8	the	the	DET
ap-9228	47	9	approximate	approximate	ADJ
ap-9228	47	10	solution	solution	NOUN
ap-9228	47	11	of	of	ADP
ap-9228	47	12	the	the	DET
ap-9228	47	13	method	method	NOUN
ap-9228	47	14	to	to	ADP
ap-9228	47	15	the	the	DET
ap-9228	47	16	exact	exact	ADJ
ap-9228	47	17	solution	solution	NOUN
ap-9228	47	18	.	.	PUNCT
ap-9228	48	1	in	in	ADP
ap-9228	48	2	the	the	DET
ap-9228	48	3	fourth	fourth	ADJ
ap-9228	48	4	section	section	NOUN
ap-9228	48	5	,	,	PUNCT
ap-9228	48	6	we	we	PRON
ap-9228	48	7	apply	apply	VERB
ap-9228	48	8	the	the	DET
ap-9228	48	9	method	method	NOUN
ap-9228	48	10	to	to	ADP
ap-9228	48	11	several	several	ADJ
ap-9228	48	12	illustrative	illustrative	ADJ
ap-9228	48	13	examples	example	NOUN
ap-9228	48	14	.	.	PUNCT
ap-9228	49	1	the	the	DET
ap-9228	49	2	last	last	ADJ
ap-9228	49	3	section	section	NOUN
ap-9228	49	4	presents	present	VERB
ap-9228	49	5	the	the	DET
ap-9228	49	6	conclusion	conclusion	NOUN
ap-9228	49	7	and	and	CCONJ
ap-9228	49	8	the	the	DET
ap-9228	49	9	results	result	NOUN
ap-9228	49	10	reached	reach	VERB
ap-9228	49	11	.	.	PUNCT
ap-9228	50	1	2	2	X
ap-9228	50	2	.	.	X
ap-9228	50	3	analysis	analysis	NOUN
ap-9228	50	4	of	of	ADP
ap-9228	50	5	the	the	DET
ap-9228	50	6	technique	technique	NOUN
ap-9228	50	7	consider	consider	VERB
ap-9228	50	8	the	the	DET
ap-9228	50	9	following	follow	VERB
ap-9228	50	10	a	a	DET
ap-9228	50	11	nonlinear	nonlinear	ADJ
ap-9228	50	12	volterra	volterra	NOUN
ap-9228	50	13	integral	integral	ADJ
ap-9228	50	14	equation	equation	NOUN
ap-9228	50	15	of	of	ADP
ap-9228	50	16	the	the	DET
ap-9228	50	17	second	second	ADJ
ap-9228	50	18	kind	kind	NOUN
ap-9228	50	19	:	:	PUNCT
ap-9228	50	20	u(x	u(x	X
ap-9228	50	21	)	)	PUNCT
ap-9228	50	22	=	=	SYM
ap-9228	50	23	ψ(x	ψ(x	NOUN
ap-9228	50	24	)	)	PUNCT
ap-9228	51	1	+	+	CCONJ
ap-9228	51	2	σ	σ	NUM
ap-9228	51	3	∫	∫	PROPN
ap-9228	51	4	x	x	SYM
ap-9228	51	5	0	0	PUNCT
ap-9228	51	6	k(x	k(x	PROPN
ap-9228	51	7	,	,	PUNCT
ap-9228	51	8	t)n(u(t	t)n(u(t	NOUN
ap-9228	51	9	)	)	PUNCT
ap-9228	51	10	)	)	PUNCT
ap-9228	52	1	dt	dt	X
ap-9228	52	2	.	.	PUNCT
ap-9228	53	1	(	(	PUNCT
ap-9228	53	2	1	1	X
ap-9228	53	3	)	)	PUNCT
ap-9228	53	4	now	now	ADV
ap-9228	53	5	,	,	PUNCT
ap-9228	53	6	we	we	PRON
ap-9228	53	7	rewrite	rewrite	VERB
ap-9228	53	8	equation	equation	NOUN
ap-9228	53	9	(	(	PUNCT
ap-9228	53	10	1	1	NUM
ap-9228	53	11	)	)	PUNCT
ap-9228	53	12	by	by	ADP
ap-9228	53	13	using	use	VERB
ap-9228	53	14	maclaurin	maclaurin	NOUN
ap-9228	53	15	’s	’s	PART
ap-9228	53	16	expansion	expansion	NOUN
ap-9228	53	17	with	with	ADP
ap-9228	53	18	respect	respect	NOUN
ap-9228	53	19	to	to	ADP
ap-9228	53	20	x	x	NOUN
ap-9228	53	21	:	:	PUNCT
ap-9228	53	22	u(x	u(x	PROPN
ap-9228	53	23	)	)	PUNCT
ap-9228	53	24	=	=	PUNCT
ap-9228	54	1	∞∑	∞∑	NUM
ap-9228	54	2	i=0	i=0	PROPN
ap-9228	54	3	xi	xi	X
ap-9228	54	4	i	i	PROPN
ap-9228	54	5	!	!	PUNCT
ap-9228	55	1	d	d	VERB
ap-9228	55	2	i	i	PRON
ap-9228	55	3	x	x	X
ap-9228	55	4	(	(	PUNCT
ap-9228	55	5	ψ(x	ψ(x	NOUN
ap-9228	55	6	)	)	PUNCT
ap-9228	56	1	+	+	CCONJ
ap-9228	56	2	σ	σ	NUM
ap-9228	56	3	∫	∫	PROPN
ap-9228	56	4	x	x	SYM
ap-9228	56	5	0	0	PUNCT
ap-9228	56	6	k(x	k(x	PROPN
ap-9228	56	7	,	,	PUNCT
ap-9228	56	8	t)n(u(t	t)n(u(t	NOUN
ap-9228	56	9	)	)	PUNCT
ap-9228	56	10	)	)	PUNCT
ap-9228	56	11	dt	dt	NOUN
ap-9228	56	12	)	)	PUNCT
ap-9228	56	13	x=0	x=0	PROPN
ap-9228	56	14	,	,	PUNCT
ap-9228	56	15	(	(	PUNCT
ap-9228	56	16	2	2	X
ap-9228	56	17	)	)	PUNCT
ap-9228	56	18	where	where	SCONJ
ap-9228	56	19	di	di	NOUN
ap-9228	56	20	x	x	VERB
ap-9228	56	21	is	be	AUX
ap-9228	56	22	a	a	DET
ap-9228	56	23	differential	differential	ADJ
ap-9228	56	24	operator	operator	NOUN
ap-9228	56	25	of	of	ADP
ap-9228	56	26	order	order	NOUN
ap-9228	56	27	i.	i.	NOUN
ap-9228	56	28	suppose	suppose	VERB
ap-9228	56	29	that	that	SCONJ
ap-9228	56	30	u(x	u(x	NOUN
ap-9228	56	31	)	)	PUNCT
ap-9228	56	32	is	be	AUX
ap-9228	56	33	a	a	DET
ap-9228	56	34	solution	solution	NOUN
ap-9228	56	35	of	of	ADP
ap-9228	56	36	equation	equation	NOUN
ap-9228	56	37	(	(	PUNCT
ap-9228	56	38	2	2	NUM
ap-9228	56	39	)	)	PUNCT
ap-9228	56	40	,	,	PUNCT
ap-9228	56	41	which	which	PRON
ap-9228	56	42	we	we	PRON
ap-9228	56	43	express	express	VERB
ap-9228	56	44	as	as	ADP
ap-9228	56	45	:	:	PUNCT
ap-9228	56	46	u(x	u(x	PROPN
ap-9228	56	47	)	)	PUNCT
ap-9228	56	48	=	=	PUNCT
ap-9228	57	1	∞∑	∞∑	NUM
ap-9228	57	2	i=0	i=0	ADJ
ap-9228	57	3	ui(x	ui(x	NUM
ap-9228	57	4	)	)	PUNCT
ap-9228	57	5	,	,	PUNCT
ap-9228	57	6	(	(	PUNCT
ap-9228	57	7	3	3	X
ap-9228	57	8	)	)	PUNCT
ap-9228	57	9	by	by	ADP
ap-9228	57	10	substituting	substitute	VERB
ap-9228	57	11	equation	equation	NOUN
ap-9228	57	12	(	(	PUNCT
ap-9228	57	13	3	3	NUM
ap-9228	57	14	)	)	PUNCT
ap-9228	57	15	in	in	ADP
ap-9228	57	16	equation	equation	NOUN
ap-9228	57	17	(	(	PUNCT
ap-9228	57	18	2	2	NUM
ap-9228	57	19	):	):	PUNCT
ap-9228	57	20	∞∑	∞∑	ADJ
ap-9228	57	21	i=0	i=0	ADJ
ap-9228	57	22	ui(x	ui(x	NUM
ap-9228	57	23	)	)	PUNCT
ap-9228	57	24	=	=	PUNCT
ap-9228	58	1	∞∑	∞∑	NUM
ap-9228	58	2	i=0	i=0	PROPN
ap-9228	58	3	xi	xi	X
ap-9228	58	4	i	i	PROPN
ap-9228	58	5	!	!	PUNCT
ap-9228	59	1	d	d	VERB
ap-9228	59	2	i	i	PRON
ap-9228	59	3	x	x	X
ap-9228	59	4	(	(	PUNCT
ap-9228	59	5	ψ(x	ψ(x	NOUN
ap-9228	59	6	)	)	PUNCT
ap-9228	60	1	+	+	CCONJ
ap-9228	60	2	σ	σ	NUM
ap-9228	60	3	∫	∫	PROPN
ap-9228	60	4	x	x	SYM
ap-9228	60	5	0	0	NUM
ap-9228	60	6	k(x	k(x	PROPN
ap-9228	60	7	,	,	PUNCT
ap-9228	60	8	t)n	t)n	NOUN
ap-9228	60	9	(	(	PUNCT
ap-9228	60	10	∞∑	∞∑	NUM
ap-9228	60	11	i=0	i=0	ADJ
ap-9228	60	12	ui(t	ui(t	NOUN
ap-9228	60	13	)	)	PUNCT
ap-9228	60	14	)	)	PUNCT
ap-9228	61	1	dt	dt	X
ap-9228	61	2	)	)	PUNCT
ap-9228	62	1	x=0	x=0	PROPN
ap-9228	62	2	,	,	PUNCT
ap-9228	62	3	(	(	PUNCT
ap-9228	62	4	4	4	NUM
ap-9228	62	5	)	)	PUNCT
ap-9228	62	6	by	by	ADP
ap-9228	62	7	putting	put	VERB
ap-9228	62	8	i	i	PRON
ap-9228	62	9	=	=	PUNCT
ap-9228	62	10	i+	i+	PUNCT
ap-9228	62	11	1	1	NUM
ap-9228	62	12	in	in	ADP
ap-9228	62	13	the	the	DET
ap-9228	62	14	left	left	ADJ
ap-9228	62	15	side	side	NOUN
ap-9228	62	16	of	of	ADP
ap-9228	62	17	equation	equation	NOUN
ap-9228	62	18	(	(	PUNCT
ap-9228	62	19	4	4	NUM
ap-9228	62	20	):	):	PUNCT
ap-9228	62	21	u0(x	u0(x	NUM
ap-9228	62	22	)	)	PUNCT
ap-9228	63	1	+	+	CCONJ
ap-9228	63	2	∞∑	∞∑	DET
ap-9228	63	3	i=0	i=0	ADJ
ap-9228	63	4	ui+1(x	ui+1(x	NOUN
ap-9228	63	5	)	)	PUNCT
ap-9228	63	6	=	=	PUNCT
ap-9228	64	1	∞∑	∞∑	NUM
ap-9228	64	2	i=0	i=0	PROPN
ap-9228	64	3	xi	xi	X
ap-9228	64	4	i	i	PROPN
ap-9228	64	5	!	!	PUNCT
ap-9228	65	1	d	d	VERB
ap-9228	65	2	i	i	PRON
ap-9228	65	3	x	x	X
ap-9228	65	4	(	(	PUNCT
ap-9228	65	5	ψ(x	ψ(x	NOUN
ap-9228	65	6	)	)	PUNCT
ap-9228	66	1	+	+	CCONJ
ap-9228	66	2	σ	σ	NUM
ap-9228	66	3	∫	∫	PROPN
ap-9228	66	4	x	x	SYM
ap-9228	66	5	0	0	NUM
ap-9228	66	6	k(x	k(x	PROPN
ap-9228	66	7	,	,	PUNCT
ap-9228	66	8	t)n	t)n	NOUN
ap-9228	66	9	(	(	PUNCT
ap-9228	66	10	∞∑	∞∑	NUM
ap-9228	66	11	i=0	i=0	ADJ
ap-9228	66	12	ui(t	ui(t	NOUN
ap-9228	66	13	)	)	PUNCT
ap-9228	66	14	)	)	PUNCT
ap-9228	67	1	dt	dt	X
ap-9228	67	2	)	)	PUNCT
ap-9228	68	1	x=0	x=0	PROPN
ap-9228	68	2	,	,	PUNCT
ap-9228	68	3	(	(	PUNCT
ap-9228	68	4	5	5	NUM
ap-9228	68	5	)	)	PUNCT
ap-9228	68	6	by	by	ADP
ap-9228	68	7	comparing	compare	VERB
ap-9228	68	8	the	the	DET
ap-9228	68	9	two	two	NUM
ap-9228	68	10	sides	side	NOUN
ap-9228	68	11	of	of	ADP
ap-9228	68	12	equation	equation	NOUN
ap-9228	68	13	(	(	PUNCT
ap-9228	68	14	5):	5):	NUM
ap-9228	68	15	u0	u0	ADJ
ap-9228	68	16	=	=	PROPN
ap-9228	68	17	u(0	u(0	PROPN
ap-9228	68	18	)	)	PUNCT
ap-9228	68	19	=	=	PUNCT
ap-9228	69	1	ψ(0	ψ(0	NOUN
ap-9228	69	2	)	)	PUNCT
ap-9228	69	3	,	,	PUNCT
ap-9228	69	4	u1(x	u1(x	NOUN
ap-9228	69	5	)	)	PUNCT
ap-9228	69	6	=	=	SYM
ap-9228	69	7	u′(0	u′(0	PROPN
ap-9228	69	8	)	)	PUNCT
ap-9228	69	9	=	=	SYM
ap-9228	69	10	xdx	xdx	PROPN
ap-9228	69	11	(	(	PUNCT
ap-9228	69	12	ψ(x	ψ(x	PROPN
ap-9228	69	13	)	)	PUNCT
ap-9228	70	1	+	+	CCONJ
ap-9228	70	2	σ	σ	NUM
ap-9228	70	3	∫	∫	PROPN
ap-9228	70	4	x	x	SYM
ap-9228	70	5	0	0	PUNCT
ap-9228	70	6	k(x	k(x	PROPN
ap-9228	70	7	,	,	PUNCT
ap-9228	70	8	t)n(u0(t	t)n(u0(t	PROPN
ap-9228	70	9	)	)	PUNCT
ap-9228	70	10	)	)	PUNCT
ap-9228	70	11	dt	dt	NOUN
ap-9228	70	12	)	)	PUNCT
ap-9228	70	13	x=0	x=0	NUM
ap-9228	70	14	,	,	PUNCT
ap-9228	70	15	u2(x	u2(x	X
ap-9228	70	16	)	)	PUNCT
ap-9228	70	17	=	=	SYM
ap-9228	70	18	u′′(0	u′′(0	NOUN
ap-9228	70	19	)	)	PUNCT
ap-9228	70	20	=	=	SYM
ap-9228	70	21	x2	x2	PROPN
ap-9228	71	1	2	2	X
ap-9228	71	2	!	!	PUNCT
ap-9228	72	1	d	d	NOUN
ap-9228	72	2	2	2	NUM
ap-9228	72	3	x	x	SYM
ap-9228	72	4	(	(	PUNCT
ap-9228	72	5	ψ(x	ψ(x	NOUN
ap-9228	72	6	)	)	PUNCT
ap-9228	72	7	+	+	CCONJ
ap-9228	72	8	σ	σ	NUM
ap-9228	72	9	∫	∫	PROPN
ap-9228	72	10	x	x	SYM
ap-9228	72	11	0	0	PUNCT
ap-9228	72	12	k(x	k(x	PROPN
ap-9228	72	13	,	,	PUNCT
ap-9228	72	14	t)n(u0(t	t)n(u0(t	X
ap-9228	72	15	)	)	PUNCT
ap-9228	72	16	+	+	CCONJ
ap-9228	72	17	u1(t	u1(t	PROPN
ap-9228	72	18	)	)	PUNCT
ap-9228	72	19	)	)	PUNCT
ap-9228	72	20	dt	dt	NOUN
ap-9228	72	21	)	)	PUNCT
ap-9228	73	1	x=0	x=0	PROPN
ap-9228	73	2	,	,	PUNCT
ap-9228	73	3	u3(x	u3(x	PROPN
ap-9228	73	4	)	)	PUNCT
ap-9228	73	5	=	=	SYM
ap-9228	73	6	u′′′(0	u′′′(0	NOUN
ap-9228	73	7	)	)	PUNCT
ap-9228	73	8	=	=	PUNCT
ap-9228	74	1	x3	x3	ADJ
ap-9228	75	1	3	3	X
ap-9228	75	2	!	!	PUNCT
ap-9228	76	1	d	d	ADP
ap-9228	76	2	2	2	NUM
ap-9228	76	3	x	x	SYM
ap-9228	76	4	(	(	PUNCT
ap-9228	76	5	ψ(x	ψ(x	NOUN
ap-9228	76	6	)	)	PUNCT
ap-9228	77	1	+	+	CCONJ
ap-9228	77	2	σ	σ	NUM
ap-9228	77	3	∫	∫	PROPN
ap-9228	77	4	x	x	SYM
ap-9228	77	5	0	0	PUNCT
ap-9228	77	6	k(x	k(x	PROPN
ap-9228	77	7	,	,	PUNCT
ap-9228	77	8	t)n(u0(t	t)n(u0(t	X
ap-9228	77	9	)	)	PUNCT
ap-9228	77	10	+	+	CCONJ
ap-9228	77	11	u1(t	u1(t	X
ap-9228	77	12	)	)	PUNCT
ap-9228	77	13	+	+	NUM
ap-9228	77	14	u2(t	u2(t	NOUN
ap-9228	77	15	)	)	PUNCT
ap-9228	77	16	)	)	PUNCT
ap-9228	77	17	dt	dt	NOUN
ap-9228	77	18	)	)	PUNCT
ap-9228	77	19	x=0	x=0	PROPN
ap-9228	77	20	,	,	PUNCT
ap-9228	77	21	...	...	PUNCT
ap-9228	77	22	ui+1(x	ui+1(x	NOUN
ap-9228	77	23	)	)	PUNCT
ap-9228	77	24	=	=	SYM
ap-9228	77	25	ui+1(0	ui+1(0	X
ap-9228	77	26	)	)	PUNCT
ap-9228	77	27	=	=	SYM
ap-9228	77	28	xi+1	xi+1	PROPN
ap-9228	77	29	(	(	PUNCT
ap-9228	77	30	i+	i+	NUM
ap-9228	77	31	1)!d	1)!d	NUM
ap-9228	77	32	2	2	NUM
ap-9228	77	33	x	x	SYM
ap-9228	77	34	ψ(x	ψ(x	PROPN
ap-9228	77	35	)	)	PUNCT
ap-9228	77	36	+	+	PROPN
ap-9228	77	37	σ	σ	NUM
ap-9228	77	38	∫	∫	PROPN
ap-9228	77	39	x	x	SYM
ap-9228	77	40	0	0	NUM
ap-9228	77	41	k(x	k(x	PROPN
ap-9228	77	42	,	,	PUNCT
ap-9228	77	43	t)n	t)n	PUNCT
ap-9228	77	44			PROPN
ap-9228	77	45	i∑	i∑	NUM
ap-9228	77	46	j=0	j=0	PROPN
ap-9228	77	47	uj(t	uj(t	PUNCT
ap-9228	77	48	)	)	PUNCT
ap-9228	78	1			PROPN
ap-9228	78	2	dt	dt	X
ap-9228	78	3			PROPN
ap-9228	78	4	x=0	x=0	PUNCT
ap-9228	78	5	.	.	PUNCT
ap-9228	79	1	(	(	PUNCT
ap-9228	79	2	6	6	NUM
ap-9228	79	3	)	)	SYM
ap-9228	79	4	88	88	NUM
ap-9228	79	5	vol	vol	NOUN
ap-9228	79	6	.	.	PUNCT
ap-9228	80	1	64	64	NUM
ap-9228	80	2	no	no	NOUN
ap-9228	80	3	.	.	PUNCT
ap-9228	81	1	2/2024	2/2024	NUM
ap-9228	81	2	an	an	DET
ap-9228	81	3	innovative	innovative	ADJ
ap-9228	81	4	iterative	iterative	NOUN
ap-9228	81	5	approach	approach	NOUN
ap-9228	81	6	to	to	ADP
ap-9228	81	7	solving	solve	VERB
ap-9228	81	8	volterra	volterra	NOUN
ap-9228	81	9	integral	integral	ADJ
ap-9228	81	10	.	.	PUNCT
ap-9228	81	11	.	.	PUNCT
ap-9228	81	12	.	.	PUNCT
ap-9228	82	1	therefore	therefore	ADV
ap-9228	82	2	,	,	PUNCT
ap-9228	82	3	the	the	DET
ap-9228	82	4	approximate	approximate	ADJ
ap-9228	82	5	solution	solution	NOUN
ap-9228	82	6	can	can	AUX
ap-9228	82	7	be	be	AUX
ap-9228	82	8	formulated	formulate	VERB
ap-9228	82	9	as	as	SCONJ
ap-9228	82	10	follows	follow	VERB
ap-9228	82	11	:	:	PUNCT
ap-9228	82	12	u(x	u(x	NOUN
ap-9228	82	13	)	)	PUNCT
ap-9228	82	14	=	=	PUNCT
ap-9228	82	15	u0	u0	ADJ
ap-9228	82	16	+	+	NUM
ap-9228	82	17	u1	u1	NOUN
ap-9228	82	18	+	+	CCONJ
ap-9228	82	19	u3	u3	NOUN
ap-9228	82	20	+	+	CCONJ
ap-9228	82	21	·	·	PUNCT
ap-9228	82	22	·	·	PUNCT
ap-9228	82	23	·	·	PUNCT
ap-9228	83	1	=	=	PUNCT
ap-9228	83	2	∞∑	∞∑	NUM
ap-9228	83	3	i=0	i=0	PROPN
ap-9228	83	4	ui	ui	PROPN
ap-9228	83	5	.	.	PUNCT
ap-9228	84	1	(	(	PUNCT
ap-9228	84	2	7	7	NUM
ap-9228	84	3	)	)	SYM
ap-9228	84	4	3	3	NUM
ap-9228	84	5	.	.	X
ap-9228	85	1	convergence	convergence	NOUN
ap-9228	85	2	of	of	ADP
ap-9228	85	3	the	the	DET
ap-9228	85	4	technique	technique	NOUN
ap-9228	85	5	theorem	theorem	NOUN
ap-9228	85	6	1	1	NUM
ap-9228	85	7	.	.	PUNCT
ap-9228	86	1	the	the	DET
ap-9228	86	2	new	new	ADJ
ap-9228	86	3	technique	technique	NOUN
ap-9228	86	4	used	use	VERB
ap-9228	86	5	in	in	ADP
ap-9228	86	6	solving	solve	VERB
ap-9228	86	7	equation	equation	NOUN
ap-9228	86	8	(	(	PUNCT
ap-9228	86	9	1	1	X
ap-9228	86	10	)	)	PUNCT
ap-9228	86	11	is	be	AUX
ap-9228	86	12	equivalent	equivalent	ADJ
ap-9228	86	13	to	to	ADP
ap-9228	86	14	determining	determine	VERB
ap-9228	86	15	the	the	DET
ap-9228	86	16	following	follow	VERB
ap-9228	86	17	sequence	sequence	NOUN
ap-9228	86	18	:	:	PUNCT
ap-9228	86	19	εη	εη	ADJ
ap-9228	86	20	=	=	X
ap-9228	86	21	u1	u1	PROPN
ap-9228	86	22	+	+	CCONJ
ap-9228	86	23	u2	u2	PROPN
ap-9228	86	24	+	+	CCONJ
ap-9228	86	25	u3	u3	NOUN
ap-9228	86	26	+	+	CCONJ
ap-9228	86	27	·	·	PUNCT
ap-9228	86	28	·	·	PUNCT
ap-9228	86	29	·	·	PUNCT
ap-9228	87	1	+	+	CCONJ
ap-9228	87	2	uη	uη	PROPN
ap-9228	87	3	,	,	PUNCT
ap-9228	87	4	ε0	ε0	PROPN
ap-9228	87	5	=	=	SYM
ap-9228	87	6	0	0	PUNCT
ap-9228	88	1	(	(	PUNCT
ap-9228	88	2	8)	8)	NUM
ap-9228	88	3	by	by	ADP
ap-9228	88	4	using	use	VERB
ap-9228	88	5	the	the	DET
ap-9228	88	6	iterative	iterative	NOUN
ap-9228	88	7	scheme	scheme	NOUN
ap-9228	88	8	:	:	PUNCT
ap-9228	88	9	εη+1	εη+1	PROPN
ap-9228	88	10	=	=	SYM
ap-9228	88	11	η+1∑	η+1∑	PROPN
ap-9228	89	1	i=1	i=1	X
ap-9228	89	2	xi	xi	X
ap-9228	90	1	i	i	PRON
ap-9228	90	2	!	!	PUNCT
ap-9228	91	1	d	d	INTJ
ap-9228	92	1	i	i	PRON
ap-9228	92	2	t	t	PROPN
ap-9228	92	3	ψ(x	ψ(x	PROPN
ap-9228	92	4	)	)	PUNCT
ap-9228	93	1	+	+	PROPN
ap-9228	93	2	σ	σ	NUM
ap-9228	93	3	∫	∫	PROPN
ap-9228	93	4	x	x	SYM
ap-9228	93	5	0	0	NUM
ap-9228	93	6	k(x	k(x	PROPN
ap-9228	93	7	,	,	PUNCT
ap-9228	93	8	t)n	t)n	NOUN
ap-9228	93	9	i−1∑	i−1∑	VERB
ap-9228	93	10	j=0	j=0	PROPN
ap-9228	93	11	uj(t	uj(t	PUNCT
ap-9228	93	12	)	)	PUNCT
ap-9228	94	1			PROPN
ap-9228	94	2	dt	dt	X
ap-9228	94	3			PROPN
ap-9228	94	4	x=0	x=0	PUNCT
ap-9228	94	5	.	.	PUNCT
ap-9228	95	1	(	(	PUNCT
ap-9228	95	2	9	9	X
ap-9228	95	3	)	)	PUNCT
ap-9228	95	4	proof	proof	NOUN
ap-9228	95	5	.	.	PUNCT
ap-9228	96	1	for	for	ADP
ap-9228	96	2	η	η	PROPN
ap-9228	96	3	=	=	PROPN
ap-9228	96	4	0	0	PROPN
ap-9228	96	5	,	,	PUNCT
ap-9228	96	6	from	from	ADP
ap-9228	96	7	equation	equation	NOUN
ap-9228	96	8	(	(	PUNCT
ap-9228	96	9	9	9	NUM
ap-9228	96	10	)	)	PUNCT
ap-9228	96	11	,	,	PUNCT
ap-9228	96	12	we	we	PRON
ap-9228	96	13	have	have	VERB
ap-9228	96	14	:	:	PUNCT
ap-9228	96	15	ε1	ε1	PROPN
ap-9228	96	16	=	=	SYM
ap-9228	96	17	xdx	xdx	PROPN
ap-9228	96	18	(	(	PUNCT
ap-9228	96	19	ψ(x	ψ(x	PROPN
ap-9228	96	20	)	)	PUNCT
ap-9228	97	1	+	+	CCONJ
ap-9228	97	2	σ	σ	NUM
ap-9228	97	3	∫	∫	PROPN
ap-9228	97	4	x	x	SYM
ap-9228	97	5	0	0	PUNCT
ap-9228	97	6	k(x	k(x	PROPN
ap-9228	97	7	,	,	PUNCT
ap-9228	97	8	t)n(u0(t	t)n(u0(t	PROPN
ap-9228	97	9	)	)	PUNCT
ap-9228	97	10	)	)	PUNCT
ap-9228	97	11	dt	dt	NOUN
ap-9228	97	12	)	)	PUNCT
ap-9228	97	13	x=0	x=0	PROPN
ap-9228	97	14	,	,	PUNCT
ap-9228	97	15	then	then	ADV
ap-9228	97	16	:	:	PUNCT
ap-9228	97	17	u1	u1	PROPN
ap-9228	97	18	=	=	SYM
ap-9228	97	19	xdx	xdx	PROPN
ap-9228	97	20	(	(	PUNCT
ap-9228	97	21	ψ(x	ψ(x	PROPN
ap-9228	97	22	)	)	PUNCT
ap-9228	98	1	+	+	CCONJ
ap-9228	98	2	σ	σ	NUM
ap-9228	98	3	∫	∫	PROPN
ap-9228	98	4	x	x	SYM
ap-9228	98	5	0	0	PUNCT
ap-9228	98	6	k(x	k(x	PROPN
ap-9228	98	7	,	,	PUNCT
ap-9228	98	8	t)n(u0(t	t)n(u0(t	PROPN
ap-9228	98	9	)	)	PUNCT
ap-9228	98	10	)	)	PUNCT
ap-9228	98	11	dt	dt	NOUN
ap-9228	98	12	)	)	PUNCT
ap-9228	98	13	x=0	x=0	PROPN
ap-9228	98	14	.	.	PUNCT
ap-9228	99	1	for	for	ADP
ap-9228	99	2	η	η	PROPN
ap-9228	99	3	=	=	PROPN
ap-9228	99	4	1	1	NUM
ap-9228	99	5	and	and	CCONJ
ap-9228	99	6	by	by	ADP
ap-9228	99	7	equation	equation	NOUN
ap-9228	99	8	(	(	PUNCT
ap-9228	99	9	9	9	NUM
ap-9228	99	10	):	):	PUNCT
ap-9228	99	11	ε2	ε2	NOUN
ap-9228	99	12	=	=	PUNCT
ap-9228	99	13	2∑	2∑	NUM
ap-9228	99	14	i=1	i=1	X
ap-9228	99	15	xi	xi	PROPN
ap-9228	100	1	i	i	PRON
ap-9228	100	2	!	!	PUNCT
ap-9228	101	1	ψ(x	ψ(x	NUM
ap-9228	101	2	)	)	PUNCT
ap-9228	102	1	+	+	PROPN
ap-9228	102	2	σ	σ	NUM
ap-9228	102	3	∫	∫	PROPN
ap-9228	102	4	x	x	SYM
ap-9228	102	5	0	0	NUM
ap-9228	102	6	k(x	k(x	PROPN
ap-9228	102	7	,	,	PUNCT
ap-9228	102	8	t)n	t)n	PUNCT
ap-9228	102	9			PROPN
ap-9228	102	10	i∑	i∑	NUM
ap-9228	102	11	j=0	j=0	PROPN
ap-9228	102	12	uj(t	uj(t	PUNCT
ap-9228	102	13	)	)	PUNCT
ap-9228	103	1			PROPN
ap-9228	103	2	dt	dt	X
ap-9228	104	1			PROPN
ap-9228	104	2	x=0	x=0	PROPN
ap-9228	104	3	=	=	SYM
ap-9228	104	4	xdx	xdx	PROPN
ap-9228	104	5	(	(	PUNCT
ap-9228	104	6	ψ(x	ψ(x	PROPN
ap-9228	104	7	)	)	PUNCT
ap-9228	105	1	+	+	CCONJ
ap-9228	105	2	σ	σ	NUM
ap-9228	105	3	∫	∫	PROPN
ap-9228	105	4	x	x	SYM
ap-9228	105	5	0	0	PUNCT
ap-9228	105	6	k(x	k(x	PROPN
ap-9228	105	7	,	,	PUNCT
ap-9228	105	8	t)n(u0(t	t)n(u0(t	PROPN
ap-9228	105	9	)	)	PUNCT
ap-9228	105	10	)	)	PUNCT
ap-9228	105	11	dt	dt	NOUN
ap-9228	105	12	)	)	PUNCT
ap-9228	106	1	x=0	x=0	PUNCT
ap-9228	107	1	+	+	CCONJ
ap-9228	107	2	x2	x2	PROPN
ap-9228	108	1	2	2	X
ap-9228	108	2	!	!	PUNCT
ap-9228	108	3	d	d	NOUN
ap-9228	108	4	2	2	NUM
ap-9228	108	5	x	x	SYM
ap-9228	108	6	(	(	PUNCT
ap-9228	108	7	ψ(x	ψ(x	NOUN
ap-9228	108	8	)	)	PUNCT
ap-9228	109	1	+	+	CCONJ
ap-9228	109	2	σ	σ	NUM
ap-9228	109	3	∫	∫	PROPN
ap-9228	109	4	x	x	SYM
ap-9228	109	5	0	0	PUNCT
ap-9228	109	6	k(x	k(x	PROPN
ap-9228	109	7	,	,	PUNCT
ap-9228	109	8	t)n(u0(t	t)n(u0(t	X
ap-9228	109	9	)	)	PUNCT
ap-9228	109	10	+	+	CCONJ
ap-9228	109	11	u1(t	u1(t	PROPN
ap-9228	109	12	)	)	PUNCT
ap-9228	109	13	)	)	PUNCT
ap-9228	109	14	dt	dt	PUNCT
ap-9228	109	15	)	)	PUNCT
ap-9228	109	16	x=0	x=0	PUNCT
ap-9228	110	1	=	=	SYM
ap-9228	110	2	u1	u1	PROPN
ap-9228	110	3	+	+	CCONJ
ap-9228	110	4	x2	x2	PROPN
ap-9228	111	1	2	2	NUM
ap-9228	111	2	!	!	PUNCT
ap-9228	111	3	d	d	NOUN
ap-9228	111	4	2	2	NUM
ap-9228	111	5	x	x	SYM
ap-9228	111	6	(	(	PUNCT
ap-9228	111	7	ψ(x	ψ(x	NOUN
ap-9228	111	8	)	)	PUNCT
ap-9228	112	1	+	+	CCONJ
ap-9228	112	2	σ	σ	NUM
ap-9228	112	3	∫	∫	PROPN
ap-9228	112	4	x	x	SYM
ap-9228	112	5	0	0	PUNCT
ap-9228	112	6	k(x	k(x	PROPN
ap-9228	112	7	,	,	PUNCT
ap-9228	112	8	t)n(u0(t	t)n(u0(t	X
ap-9228	112	9	)	)	PUNCT
ap-9228	112	10	+	+	CCONJ
ap-9228	112	11	u1(t	u1(t	PROPN
ap-9228	112	12	)	)	PUNCT
ap-9228	112	13	)	)	PUNCT
ap-9228	112	14	dt	dt	NOUN
ap-9228	112	15	)	)	PUNCT
ap-9228	112	16	x=0	x=0	PROPN
ap-9228	112	17	.	.	PUNCT
ap-9228	113	1	according	accord	VERB
ap-9228	113	2	to	to	ADP
ap-9228	113	3	ε2	ε2	NOUN
ap-9228	113	4	=	=	SYM
ap-9228	113	5	u1	u1	NOUN
ap-9228	113	6	+	+	CCONJ
ap-9228	113	7	u2	u2	NOUN
ap-9228	113	8	,	,	PUNCT
ap-9228	113	9	we	we	PRON
ap-9228	113	10	get	get	VERB
ap-9228	113	11	:	:	PUNCT
ap-9228	113	12	u2	u2	NOUN
ap-9228	113	13	=	=	SYM
ap-9228	113	14	x2	x2	PROPN
ap-9228	114	1	2	2	NUM
ap-9228	114	2	!	!	PUNCT
ap-9228	115	1	d	d	NOUN
ap-9228	115	2	2	2	NUM
ap-9228	115	3	x	x	SYM
ap-9228	115	4	(	(	PUNCT
ap-9228	115	5	ψ(x	ψ(x	NOUN
ap-9228	115	6	)	)	PUNCT
ap-9228	115	7	+	+	CCONJ
ap-9228	115	8	σ	σ	NUM
ap-9228	115	9	∫	∫	PROPN
ap-9228	115	10	x	x	SYM
ap-9228	115	11	0	0	PUNCT
ap-9228	115	12	k(x	k(x	PROPN
ap-9228	115	13	,	,	PUNCT
ap-9228	115	14	t)n(u0(t	t)n(u0(t	X
ap-9228	115	15	)	)	PUNCT
ap-9228	115	16	+	+	CCONJ
ap-9228	115	17	u1(t	u1(t	PROPN
ap-9228	115	18	)	)	PUNCT
ap-9228	115	19	)	)	PUNCT
ap-9228	115	20	dt	dt	NOUN
ap-9228	115	21	)	)	PUNCT
ap-9228	115	22	x=0	x=0	PROPN
ap-9228	115	23	.	.	PUNCT
ap-9228	116	1	this	this	DET
ap-9228	116	2	theorem	theorem	NOUN
ap-9228	116	3	will	will	AUX
ap-9228	116	4	be	be	AUX
ap-9228	116	5	proved	prove	VERB
ap-9228	116	6	by	by	ADP
ap-9228	116	7	strong	strong	ADJ
ap-9228	116	8	induction	induction	NOUN
ap-9228	116	9	.	.	PUNCT
ap-9228	117	1	let	let	VERB
ap-9228	117	2	’s	’s	PRON
ap-9228	117	3	assume	assume	VERB
ap-9228	117	4	that	that	SCONJ
ap-9228	117	5	:	:	PUNCT
ap-9228	117	6	um+1	um+1	PROPN
ap-9228	117	7	=	=	SYM
ap-9228	117	8	xm+1	xm+1	PROPN
ap-9228	117	9	(	(	PUNCT
ap-9228	117	10	m+	m+	NOUN
ap-9228	117	11	1)!d	1)!d	NUM
ap-9228	117	12	m+1	m+1	NUM
ap-9228	117	13	t	t	NOUN
ap-9228	117	14	ψ(x	ψ(x	NOUN
ap-9228	117	15	)	)	PUNCT
ap-9228	118	1	+	+	PROPN
ap-9228	118	2	σ	σ	NUM
ap-9228	118	3	∫	∫	PROPN
ap-9228	118	4	x	x	SYM
ap-9228	118	5	0	0	NUM
ap-9228	118	6	k(x	k(x	PROPN
ap-9228	118	7	,	,	PUNCT
ap-9228	119	1	t)n	t)n	PUNCT
ap-9228	119	2			PROPN
ap-9228	119	3	m∑	m∑	ADV
ap-9228	119	4	j=0	j=0	PROPN
ap-9228	119	5	uj(t	uj(t	X
ap-9228	119	6	)	)	PUNCT
ap-9228	119	7			PROPN
ap-9228	119	8	dt	dt	X
ap-9228	119	9			PROPN
ap-9228	119	10	x=0	x=0	PROPN
ap-9228	119	11	,	,	PUNCT
ap-9228	119	12	where	where	SCONJ
ap-9228	119	13	m	m	VERB
ap-9228	119	14	=	=	SYM
ap-9228	119	15	1	1	NUM
ap-9228	119	16	,	,	PUNCT
ap-9228	119	17	2	2	NUM
ap-9228	119	18	,	,	PUNCT
ap-9228	119	19	3	3	NUM
ap-9228	119	20	,	,	PUNCT
ap-9228	119	21	.	.	PUNCT
ap-9228	119	22	.	.	PUNCT
ap-9228	120	1	.	.	PUNCT
ap-9228	121	1	,	,	PUNCT
ap-9228	121	2	η	η	PROPN
ap-9228	121	3	−	−	PROPN
ap-9228	121	4	1	1	NUM
ap-9228	121	5	,	,	PUNCT
ap-9228	121	6	so	so	ADV
ap-9228	121	7	:	:	PUNCT
ap-9228	121	8	εη+1	εη+1	PROPN
ap-9228	121	9	=	=	SYM
ap-9228	121	10	η+1∑	η+1∑	PROPN
ap-9228	122	1	i=1	i=1	X
ap-9228	122	2	xi	xi	X
ap-9228	123	1	i	i	PRON
ap-9228	123	2	!	!	PUNCT
ap-9228	124	1	d	d	INTJ
ap-9228	125	1	i	i	PRON
ap-9228	125	2	t	t	PROPN
ap-9228	125	3	ψ(x	ψ(x	PROPN
ap-9228	125	4	)	)	PUNCT
ap-9228	126	1	+	+	PROPN
ap-9228	126	2	σ	σ	NUM
ap-9228	126	3	∫	∫	PROPN
ap-9228	126	4	x	x	SYM
ap-9228	126	5	0	0	NUM
ap-9228	126	6	k(x	k(x	PROPN
ap-9228	126	7	,	,	PUNCT
ap-9228	126	8	t)n	t)n	NOUN
ap-9228	126	9	i−1∑	i−1∑	VERB
ap-9228	126	10	j=0	j=0	PROPN
ap-9228	126	11	uj(t	uj(t	PUNCT
ap-9228	126	12	)	)	PUNCT
ap-9228	127	1			PROPN
ap-9228	127	2	dt	dt	X
ap-9228	127	3			PROPN
ap-9228	127	4	x=0	x=0	PROPN
ap-9228	127	5	=	=	SYM
ap-9228	127	6	xdx	xdx	PROPN
ap-9228	127	7	(	(	PUNCT
ap-9228	127	8	ψ(x	ψ(x	PROPN
ap-9228	127	9	)	)	PUNCT
ap-9228	128	1	+	+	CCONJ
ap-9228	128	2	σ	σ	NUM
ap-9228	128	3	∫	∫	PROPN
ap-9228	128	4	x	x	SYM
ap-9228	128	5	0	0	PUNCT
ap-9228	128	6	k(x	k(x	PROPN
ap-9228	128	7	,	,	PUNCT
ap-9228	128	8	t)n(u0(t	t)n(u0(t	PROPN
ap-9228	128	9	)	)	PUNCT
ap-9228	128	10	)	)	PUNCT
ap-9228	128	11	dt	dt	NOUN
ap-9228	128	12	)	)	PUNCT
ap-9228	129	1	x=0	x=0	PUNCT
ap-9228	130	1	+	+	CCONJ
ap-9228	130	2	x2	x2	PROPN
ap-9228	131	1	2	2	X
ap-9228	131	2	!	!	PUNCT
ap-9228	131	3	d	d	NOUN
ap-9228	131	4	2	2	NUM
ap-9228	131	5	x	x	SYM
ap-9228	131	6	(	(	PUNCT
ap-9228	131	7	ψ(x	ψ(x	NOUN
ap-9228	131	8	)	)	PUNCT
ap-9228	132	1	+	+	CCONJ
ap-9228	132	2	σ	σ	NUM
ap-9228	132	3	∫	∫	PROPN
ap-9228	132	4	x	x	SYM
ap-9228	132	5	0	0	PUNCT
ap-9228	132	6	k(x	k(x	PROPN
ap-9228	132	7	,	,	PUNCT
ap-9228	132	8	t)n(u0(t	t)n(u0(t	X
ap-9228	132	9	)	)	PUNCT
ap-9228	132	10	+	+	CCONJ
ap-9228	132	11	u1(t	u1(t	PROPN
ap-9228	132	12	)	)	PUNCT
ap-9228	132	13	)	)	PUNCT
ap-9228	132	14	dt	dt	PUNCT
ap-9228	132	15	)	)	PUNCT
ap-9228	132	16	x=0	x=0	PROPN
ap-9228	133	1	+	+	CCONJ
ap-9228	133	2	·	·	PUNCT
ap-9228	133	3	·	·	PUNCT
ap-9228	133	4	·	·	PUNCT
ap-9228	134	1	+	+	CCONJ
ap-9228	134	2	xη+1	xη+1	PROPN
ap-9228	134	3	(	(	PUNCT
ap-9228	134	4	η	η	X
ap-9228	134	5	+	+	PROPN
ap-9228	134	6	1)!d	1)!d	PROPN
ap-9228	134	7	η+1	η+1	PROPN
ap-9228	134	8	t	t	PROPN
ap-9228	134	9	ψ(x	ψ(x	PROPN
ap-9228	134	10	)	)	PUNCT
ap-9228	135	1	+	+	PROPN
ap-9228	135	2	σ	σ	NUM
ap-9228	135	3	∫	∫	PROPN
ap-9228	135	4	x	x	SYM
ap-9228	135	5	0	0	NUM
ap-9228	135	6	k(x	k(x	PROPN
ap-9228	135	7	,	,	PUNCT
ap-9228	135	8	t)n	t)n	PUNCT
ap-9228	135	9			PROPN
ap-9228	135	10	η∑	η∑	PROPN
ap-9228	135	11	j=0	j=0	PROPN
ap-9228	135	12	uj(t	uj(t	PUNCT
ap-9228	135	13	)	)	PUNCT
ap-9228	136	1			PROPN
ap-9228	136	2	dt	dt	X
ap-9228	136	3			PROPN
ap-9228	136	4	x=0	x=0	PUNCT
ap-9228	136	5	=	=	SYM
ap-9228	136	6	u1	u1	PROPN
ap-9228	136	7	+	+	X
ap-9228	136	8	·	·	PUNCT
ap-9228	136	9	·	·	PUNCT
ap-9228	136	10	·	·	PUNCT
ap-9228	137	1	+	+	CCONJ
ap-9228	137	2	uη	uη	ADJ
ap-9228	137	3	+	+	CCONJ
ap-9228	137	4	xη+1	xη+1	PROPN
ap-9228	137	5	(	(	PUNCT
ap-9228	137	6	η	η	PROPN
ap-9228	137	7	+	+	PROPN
ap-9228	137	8	1)!d	1)!d	PROPN
ap-9228	137	9	η+1	η+1	PROPN
ap-9228	137	10	t	t	PROPN
ap-9228	137	11	ψ(x	ψ(x	PROPN
ap-9228	137	12	)	)	PUNCT
ap-9228	137	13	+	+	PROPN
ap-9228	137	14	σ	σ	NUM
ap-9228	137	15	∫	∫	PROPN
ap-9228	137	16	x	x	SYM
ap-9228	137	17	0	0	NUM
ap-9228	137	18	k(x	k(x	PROPN
ap-9228	137	19	,	,	PUNCT
ap-9228	137	20	t)n	t)n	PUNCT
ap-9228	137	21			PROPN
ap-9228	137	22	η∑	η∑	PROPN
ap-9228	137	23	j=0	j=0	PROPN
ap-9228	137	24	uj(t	uj(t	PUNCT
ap-9228	137	25	)	)	PUNCT
ap-9228	138	1			PROPN
ap-9228	138	2	dt	dt	X
ap-9228	138	3			PROPN
ap-9228	138	4	x=0	x=0	PROPN
ap-9228	138	5	.	.	PUNCT
ap-9228	139	1	89	89	NUM
ap-9228	139	2	m.	m.	NOUN
ap-9228	139	3	a.	a.	PROPN
ap-9228	139	4	hussein	hussein	PROPN
ap-9228	139	5	,	,	PUNCT
ap-9228	139	6	h.	h.	PROPN
ap-9228	139	7	k.	k.	PROPN
ap-9228	139	8	jassim	jassim	PROPN
ap-9228	139	9	,	,	PUNCT
ap-9228	139	10	a.	a.	PROPN
ap-9228	139	11	k.	k.	PROPN
ap-9228	139	12	jassim	jassim	PROPN
ap-9228	139	13	acta	acta	PROPN
ap-9228	139	14	polytechnica	polytechnica	PROPN
ap-9228	139	15	then	then	ADV
ap-9228	139	16	,	,	PUNCT
ap-9228	139	17	from	from	ADP
ap-9228	139	18	equation	equation	NOUN
ap-9228	139	19	(	(	PUNCT
ap-9228	139	20	8)	8)	NUM
ap-9228	139	21	,	,	PUNCT
ap-9228	139	22	it	it	PRON
ap-9228	139	23	can	can	AUX
ap-9228	139	24	be	be	AUX
ap-9228	139	25	derived	derive	VERB
ap-9228	139	26	:	:	PUNCT
ap-9228	139	27	uη+1	uη+1	PROPN
ap-9228	139	28	=	=	SYM
ap-9228	139	29	tη+1	tη+1	PROPN
ap-9228	139	30	(	(	PUNCT
ap-9228	139	31	η	η	PROPN
ap-9228	139	32	+	+	PROPN
ap-9228	139	33	1)!d	1)!d	PROPN
ap-9228	140	1	η+1	η+1	PROPN
ap-9228	140	2	t	t	PROPN
ap-9228	140	3	ψ(x	ψ(x	PROPN
ap-9228	140	4	)	)	PUNCT
ap-9228	141	1	+	+	PROPN
ap-9228	141	2	σ	σ	NUM
ap-9228	141	3	∫	∫	PROPN
ap-9228	141	4	x	x	SYM
ap-9228	141	5	0	0	NUM
ap-9228	141	6	k(x	k(x	PROPN
ap-9228	141	7	,	,	PUNCT
ap-9228	141	8	t)n	t)n	PUNCT
ap-9228	141	9			PROPN
ap-9228	141	10	η∑	η∑	PROPN
ap-9228	141	11	j=0	j=0	PROPN
ap-9228	141	12	uj(t	uj(t	PUNCT
ap-9228	141	13	)	)	PUNCT
ap-9228	142	1			PROPN
ap-9228	142	2	dt	dt	X
ap-9228	142	3			PROPN
ap-9228	142	4	x=0	x=0	PROPN
ap-9228	142	5	,	,	PUNCT
ap-9228	142	6	which	which	PRON
ap-9228	142	7	is	be	AUX
ap-9228	142	8	the	the	DET
ap-9228	142	9	same	same	ADJ
ap-9228	142	10	as	as	ADP
ap-9228	142	11	the	the	DET
ap-9228	142	12	result	result	NOUN
ap-9228	142	13	of	of	ADP
ap-9228	142	14	equation	equation	NOUN
ap-9228	142	15	(	(	PUNCT
ap-9228	142	16	6	6	NUM
ap-9228	142	17	)	)	PUNCT
ap-9228	142	18	,	,	PUNCT
ap-9228	142	19	and	and	CCONJ
ap-9228	142	20	the	the	DET
ap-9228	142	21	theorem	theorem	NOUN
ap-9228	142	22	is	be	AUX
ap-9228	142	23	proved	prove	VERB
ap-9228	142	24	.	.	PUNCT
ap-9228	143	1	theorem	theorem	NOUN
ap-9228	143	2	2	2	NUM
ap-9228	143	3	.	.	PUNCT
ap-9228	144	1	let	let	VERB
ap-9228	144	2	b	b	X
ap-9228	144	3	be	be	AUX
ap-9228	144	4	a	a	DET
ap-9228	144	5	banach	banach	NOUN
ap-9228	144	6	space	space	NOUN
ap-9228	144	7	.	.	PUNCT
ap-9228	145	1	(	(	PUNCT
ap-9228	145	2	i	i	NOUN
ap-9228	145	3	)	)	PUNCT
ap-9228	145	4	∑∞	∑∞	NOUN
ap-9228	145	5	i=0	i=0	PROPN
ap-9228	145	6	ui	ui	NOUN
ap-9228	145	7	obtained	obtain	VERB
ap-9228	145	8	by	by	ADP
ap-9228	145	9	equation	equation	NOUN
ap-9228	145	10	(	(	PUNCT
ap-9228	145	11	6	6	NUM
ap-9228	145	12	)	)	PUNCT
ap-9228	145	13	convergence	convergence	NOUN
ap-9228	145	14	to	to	ADP
ap-9228	145	15	ε	ε	PROPN
ap-9228	145	16	∈	∈	PROPN
ap-9228	145	17	b	b	PROPN
ap-9228	145	18	,	,	PUNCT
ap-9228	145	19	if	if	SCONJ
ap-9228	145	20	:	:	PUNCT
ap-9228	145	21	∃(0	∃(0	NOUN
ap-9228	145	22	≤	≤	NUM
ap-9228	145	23	ϵ	ϵ	ADP
ap-9228	145	24	≤	≤	NUM
ap-9228	145	25	1	1	NUM
ap-9228	145	26	)	)	PUNCT
ap-9228	145	27	,	,	PUNCT
ap-9228	145	28	such	such	ADJ
ap-9228	145	29	that	that	SCONJ
ap-9228	145	30	(	(	PUNCT
ap-9228	145	31	∀η	∀η	X
ap-9228	145	32	∈	∈	PROPN
ap-9228	145	33	n	n	PART
ap-9228	145	34	⇒	⇒	NOUN
ap-9228	145	35	∥uη∥	∥uη∥	VERB
ap-9228	145	36	≤	≤	NUM
ap-9228	145	37	ϵ	ϵ	ADP
ap-9228	145	38	∥uη+1∥	∥uη+1∥	NUM
ap-9228	145	39	)	)	PUNCT
ap-9228	145	40	.	.	PUNCT
ap-9228	146	1	(	(	PUNCT
ap-9228	146	2	10	10	NUM
ap-9228	146	3	)	)	PUNCT
ap-9228	146	4	(	(	PUNCT
ap-9228	146	5	ii	ii	NOUN
ap-9228	146	6	)	)	PUNCT
ap-9228	146	7	ε	ε	PROPN
ap-9228	146	8	=	=	SYM
ap-9228	146	9	∑∞	∑∞	PROPN
ap-9228	146	10	η=1	η=1	X
ap-9228	146	11	uη	uη	PROPN
ap-9228	146	12	,	,	PUNCT
ap-9228	146	13	satisfies	satisfie	NOUN
ap-9228	146	14	in	in	ADP
ap-9228	146	15	:	:	PUNCT
ap-9228	146	16	ε	ε	PROPN
ap-9228	146	17	=	=	PUNCT
ap-9228	147	1	∞∑	∞∑	NUM
ap-9228	147	2	i=1	i=1	PRON
ap-9228	147	3	xi	xi	X
ap-9228	147	4	(	(	PUNCT
ap-9228	147	5	i+	i+	NUM
ap-9228	147	6	1)!d	1)!d	NOUN
ap-9228	148	1	i	i	PRON
ap-9228	148	2	t	t	PROPN
ap-9228	148	3	ψ(x	ψ(x	PROPN
ap-9228	148	4	)	)	PUNCT
ap-9228	149	1	+	+	PROPN
ap-9228	149	2	σ	σ	NUM
ap-9228	149	3	∫	∫	PROPN
ap-9228	149	4	x	x	SYM
ap-9228	149	5	0	0	NUM
ap-9228	149	6	k(x	k(x	PROPN
ap-9228	149	7	,	,	PUNCT
ap-9228	149	8	t)n	t)n	PUNCT
ap-9228	149	9			PROPN
ap-9228	149	10	∞∑	∞∑	NUM
ap-9228	149	11	j=0	j=0	PROPN
ap-9228	149	12	uj(t	uj(t	PUNCT
ap-9228	149	13	)	)	PUNCT
ap-9228	150	1			PROPN
ap-9228	150	2	dt	dt	X
ap-9228	150	3			PROPN
ap-9228	150	4	x=0	x=0	PUNCT
ap-9228	150	5	.	.	PUNCT
ap-9228	151	1	(	(	PUNCT
ap-9228	151	2	11	11	NUM
ap-9228	151	3	)	)	PUNCT
ap-9228	151	4	proof	proof	NOUN
ap-9228	151	5	.	.	PUNCT
ap-9228	152	1	(	(	PUNCT
ap-9228	152	2	i	i	NOUN
ap-9228	152	3	)	)	PUNCT
ap-9228	152	4	from	from	ADP
ap-9228	152	5	equation	equation	NOUN
ap-9228	152	6	(	(	PUNCT
ap-9228	152	7	10	10	NUM
ap-9228	152	8	)	)	PUNCT
ap-9228	152	9	,	,	PUNCT
ap-9228	152	10	we	we	PRON
ap-9228	152	11	prove	prove	VERB
ap-9228	152	12	that	that	SCONJ
ap-9228	152	13	{	{	PUNCT
ap-9228	152	14	ε}∞	ε}∞	ADP
ap-9228	152	15	η=0	η=0	PROPN
ap-9228	152	16	is	be	AUX
ap-9228	152	17	a	a	DET
ap-9228	152	18	cauchy	cauchy	ADJ
ap-9228	152	19	sequence	sequence	NOUN
ap-9228	152	20	in	in	ADP
ap-9228	152	21	b.	b.	PROPN
ap-9228	152	22	∥εη+1	∥εη+1	PROPN
ap-9228	153	1	−	−	PROPN
ap-9228	153	2	εη∥	εη∥	PROPN
ap-9228	153	3	=	=	SYM
ap-9228	153	4	∥∥∥∥∥	∥∥∥∥∥	PROPN
ap-9228	153	5	η+1∑	η+1∑	PROPN
ap-9228	153	6	i=0	i=0	PROPN
ap-9228	153	7	ui	ui	NOUN
ap-9228	154	1	−	−	PROPN
ap-9228	154	2	η∑	η∑	PROPN
ap-9228	154	3	i=0	i=0	PROPN
ap-9228	154	4	ui	ui	NOUN
ap-9228	154	5	∥∥∥∥∥	∥∥∥∥∥	PROPN
ap-9228	155	1	=	=	SYM
ap-9228	156	1	∥uη+1∥	∥uη+1∥	NUM
ap-9228	156	2	≤	≤	NUM
ap-9228	156	3	ϵ	ϵ	ADP
ap-9228	156	4	∥uη∥	∥uη∥	PROPN
ap-9228	156	5	≤	≤	NUM
ap-9228	156	6	ϵ2	ϵ2	ADJ
ap-9228	156	7	∥uη−1∥	∥uη−1∥	NOUN
ap-9228	156	8	≤	≤	NUM
ap-9228	156	9	ϵ3	ϵ3	PROPN
ap-9228	156	10	∥uη−2∥	∥uη−2∥	PRON
ap-9228	156	11	≤	≤	NUM
ap-9228	156	12	·	·	PUNCT
ap-9228	156	13	·	·	PUNCT
ap-9228	156	14	·	·	PUNCT
ap-9228	157	1	≤	≤	NUM
ap-9228	157	2	ϵη+1	ϵη+1	X
ap-9228	157	3	∥u0∥	∥u0∥	PROPN
ap-9228	157	4	.	.	PUNCT
ap-9228	158	1	(	(	PUNCT
ap-9228	158	2	12	12	NUM
ap-9228	158	3	)	)	PUNCT
ap-9228	158	4	for	for	ADP
ap-9228	158	5	all	all	DET
ap-9228	158	6	η	η	PROPN
ap-9228	158	7	,	,	PUNCT
ap-9228	158	8	m	m	PROPN
ap-9228	158	9	∈	∈	PROPN
ap-9228	158	10	n	n	CCONJ
ap-9228	158	11	,	,	PUNCT
ap-9228	158	12	η	η	PROPN
ap-9228	158	13	≥	≥	PROPN
ap-9228	158	14	m	m	PROPN
ap-9228	158	15	:	:	PUNCT
ap-9228	158	16	∥εη	∥εη	VERB
ap-9228	159	1	−	−	PROPN
ap-9228	159	2	εm∥	εm∥	NOUN
ap-9228	159	3	=	=	PUNCT
ap-9228	159	4	∥(εη	∥(εη	NUM
ap-9228	159	5	−	−	NOUN
ap-9228	159	6	εη−1	εη−1	PROPN
ap-9228	159	7	)	)	PUNCT
ap-9228	159	8	+	+	CCONJ
ap-9228	159	9	(	(	PUNCT
ap-9228	159	10	εη−1	εη−1	PROPN
ap-9228	159	11	−	−	PROPN
ap-9228	159	12	εη−2	εη−2	PROPN
ap-9228	159	13	)	)	PUNCT
ap-9228	159	14	+	+	CCONJ
ap-9228	159	15	·	·	PUNCT
ap-9228	159	16	·	·	PUNCT
ap-9228	159	17	·	·	PUNCT
ap-9228	160	1	+	+	CCONJ
ap-9228	160	2	(	(	PUNCT
ap-9228	160	3	εm+1	εm+1	X
ap-9228	160	4	−	−	NOUN
ap-9228	160	5	εm)∥	εm)∥	PROPN
ap-9228	160	6	≤	≤	X
ap-9228	160	7	∥εη	∥εη	PART
ap-9228	160	8	−	−	PROPN
ap-9228	160	9	εη−1∥	εη−1∥	NOUN
ap-9228	160	10	+	+	CCONJ
ap-9228	160	11	∥εη−1	∥εη−1	NUM
ap-9228	160	12	−	−	PROPN
ap-9228	160	13	εη−2∥	εη−2∥	PROPN
ap-9228	160	14	+	+	CCONJ
ap-9228	160	15	·	·	PUNCT
ap-9228	160	16	·	·	PUNCT
ap-9228	160	17	·	·	PUNCT
ap-9228	161	1	+	+	CCONJ
ap-9228	161	2	∥εm+1	∥εm+1	PRON
ap-9228	161	3	−	−	PROPN
ap-9228	161	4	εm∥	εm∥	NOUN
ap-9228	161	5	≤	≤	NUM
ap-9228	161	6	ϵη	ϵη	ADP
ap-9228	161	7	∥u0∥	∥u0∥	PROPN
ap-9228	161	8	≤	≤	PUNCT
ap-9228	161	9	ϵη−1	ϵη−1	PROPN
ap-9228	161	10	∥u0∥	∥u0∥	PROPN
ap-9228	161	11	≤	≤	PUNCT
ap-9228	161	12	ϵη−2	ϵη−2	VERB
ap-9228	161	13	∥u0∥	∥u0∥	PROPN
ap-9228	161	14	≤	≤	NOUN
ap-9228	161	15	·	·	PUNCT
ap-9228	161	16	·	·	PUNCT
ap-9228	161	17	·	·	PUNCT
ap-9228	162	1	≤	≤	NUM
ap-9228	162	2	ϵm+1	ϵm+1	NUM
ap-9228	162	3	∥u0∥	∥u0∥	PROPN
ap-9228	162	4	≤	≤	NOUN
ap-9228	162	5	ϵm+1	ϵm+1	PRON
ap-9228	162	6	∥u0∥	∥u0∥	PROPN
ap-9228	162	7	(	(	PUNCT
ap-9228	162	8	ϵη−m−1	ϵη−m−1	NUM
ap-9228	162	9	+	+	CCONJ
ap-9228	162	10	ϵη−m−2	ϵη−m−2	NUM
ap-9228	162	11	+	+	CCONJ
ap-9228	162	12	·	·	PUNCT
ap-9228	162	13	·	·	PUNCT
ap-9228	162	14	·	·	PUNCT
ap-9228	163	1	+	+	CCONJ
ap-9228	163	2	1	1	X
ap-9228	163	3	)	)	PUNCT
ap-9228	163	4	=	=	SYM
ap-9228	163	5	1	1	NUM
ap-9228	163	6	−	−	PROPN
ap-9228	163	7	ϵn−m	ϵn−m	NOUN
ap-9228	163	8	1	1	NUM
ap-9228	163	9	−	−	NOUN
ap-9228	163	10	ϵ	ϵ	ADP
ap-9228	163	11	ϵm+1	ϵm+1	PROPN
ap-9228	163	12	∥u0∥	∥u0∥	PROPN
ap-9228	163	13	.	.	PUNCT
ap-9228	164	1	(	(	PUNCT
ap-9228	164	2	13	13	NUM
ap-9228	164	3	)	)	PUNCT
ap-9228	164	4	since	since	SCONJ
ap-9228	164	5	(	(	PUNCT
ap-9228	164	6	ϵn−m−1	ϵn−m−1	X
ap-9228	164	7	+	+	X
ap-9228	164	8	ϵn−m−2	ϵn−m−2	PUNCT
ap-9228	164	9	+	+	X
ap-9228	164	10	·	·	PUNCT
ap-9228	164	11	·	·	PUNCT
ap-9228	164	12	·	·	PUNCT
ap-9228	165	1	+	+	CCONJ
ap-9228	165	2	1	1	X
ap-9228	165	3	)	)	PUNCT
ap-9228	165	4	is	be	AUX
ap-9228	165	5	a	a	DET
ap-9228	165	6	geometric	geometric	ADJ
ap-9228	165	7	series	series	NOUN
ap-9228	165	8	,	,	PUNCT
ap-9228	165	9	then	then	ADV
ap-9228	165	10	:	:	PUNCT
ap-9228	165	11	lim	lim	PROPN
ap-9228	165	12	η	η	PROPN
ap-9228	165	13	,	,	PUNCT
ap-9228	165	14	m→∞	m→∞	NOUN
ap-9228	165	15	∥εη	∥εη	NOUN
ap-9228	165	16	−	−	PROPN
ap-9228	165	17	εm∥	εm∥	NOUN
ap-9228	165	18	=	=	NOUN
ap-9228	165	19	0	0	NUM
ap-9228	165	20	.	.	PUNCT
ap-9228	166	1	thus	thus	ADV
ap-9228	166	2	,	,	PUNCT
ap-9228	166	3	{	{	PUNCT
ap-9228	166	4	εη	εη	ADJ
ap-9228	166	5	}	}	PUNCT
ap-9228	166	6	is	be	AUX
ap-9228	166	7	cauchy	cauchy	ADJ
ap-9228	166	8	sequence	sequence	NOUN
ap-9228	166	9	in	in	ADP
ap-9228	166	10	b	b	NOUN
ap-9228	166	11	,	,	PUNCT
ap-9228	166	12	and	and	CCONJ
ap-9228	166	13	it	it	PRON
ap-9228	166	14	is	be	AUX
ap-9228	166	15	convergent	convergent	ADJ
ap-9228	166	16	.	.	PUNCT
ap-9228	167	1	(	(	PUNCT
ap-9228	167	2	ii	ii	NOUN
ap-9228	167	3	)	)	PUNCT
ap-9228	167	4	from	from	ADP
ap-9228	167	5	equation	equation	NOUN
ap-9228	167	6	(	(	PUNCT
ap-9228	167	7	9	9	NUM
ap-9228	167	8	):	):	PUNCT
ap-9228	167	9	lim	lim	PROPN
ap-9228	167	10	η→∞	η→∞	NUM
ap-9228	167	11	εη+1	εη+1	PROPN
ap-9228	167	12	=	=	PROPN
ap-9228	167	13	lim	lim	PROPN
ap-9228	167	14	η→∞	η→∞	NUM
ap-9228	167	15	η+1∑	η+1∑	PROPN
ap-9228	167	16	i=1	i=1	X
ap-9228	167	17	xi	xi	PROPN
ap-9228	168	1	i	i	PRON
ap-9228	168	2	!	!	PUNCT
ap-9228	169	1	d	d	INTJ
ap-9228	170	1	i	i	PRON
ap-9228	170	2	t	t	PROPN
ap-9228	170	3	ψ(x	ψ(x	PROPN
ap-9228	170	4	)	)	PUNCT
ap-9228	171	1	+	+	PROPN
ap-9228	171	2	σ	σ	NUM
ap-9228	171	3	∫	∫	PROPN
ap-9228	171	4	x	x	SYM
ap-9228	171	5	0	0	NUM
ap-9228	171	6	k(x	k(x	PROPN
ap-9228	171	7	,	,	PUNCT
ap-9228	171	8	t)n	t)n	NOUN
ap-9228	171	9	i−1∑	i−1∑	VERB
ap-9228	171	10	j=0	j=0	PROPN
ap-9228	171	11	uj(t	uj(t	PUNCT
ap-9228	171	12	)	)	PUNCT
ap-9228	172	1			PROPN
ap-9228	172	2	dt	dt	X
ap-9228	172	3			PROPN
ap-9228	172	4	x=0	x=0	PUNCT
ap-9228	173	1	=	=	PUNCT
ap-9228	174	1	∞∑	∞∑	NUM
ap-9228	174	2	i=1	i=1	PRON
ap-9228	174	3	xi	xi	X
ap-9228	175	1	i	i	PRON
ap-9228	175	2	!	!	PUNCT
ap-9228	176	1	d	d	INTJ
ap-9228	177	1	i	i	PRON
ap-9228	177	2	t	t	PROPN
ap-9228	177	3	ψ(x	ψ(x	PROPN
ap-9228	177	4	)	)	PUNCT
ap-9228	178	1	+	+	PROPN
ap-9228	178	2	σ	σ	NUM
ap-9228	178	3	∫	∫	PROPN
ap-9228	178	4	x	x	SYM
ap-9228	178	5	0	0	NUM
ap-9228	178	6	k(x	k(x	PROPN
ap-9228	178	7	,	,	PUNCT
ap-9228	178	8	t)n	t)n	NOUN
ap-9228	178	9	i−1∑	i−1∑	VERB
ap-9228	178	10	j=0	j=0	PROPN
ap-9228	178	11	uj(t	uj(t	PUNCT
ap-9228	178	12	)	)	PUNCT
ap-9228	179	1			PROPN
ap-9228	179	2	dt	dt	X
ap-9228	179	3			PROPN
ap-9228	179	4	x=0	x=0	PROPN
ap-9228	179	5	.	.	PUNCT
ap-9228	180	1	since	since	SCONJ
ap-9228	180	2	the	the	DET
ap-9228	180	3	upper	upper	ADJ
ap-9228	180	4	bound	bound	NOUN
ap-9228	180	5	of	of	ADP
ap-9228	180	6	the	the	DET
ap-9228	180	7	sum	sum	NOUN
ap-9228	180	8	is	be	AUX
ap-9228	180	9	approaching	approach	VERB
ap-9228	180	10	infinity	infinity	NOUN
ap-9228	180	11	,	,	PUNCT
ap-9228	180	12	we	we	PRON
ap-9228	180	13	get	get	VERB
ap-9228	180	14	the	the	DET
ap-9228	180	15	following	follow	VERB
ap-9228	180	16	result	result	NOUN
ap-9228	180	17	:	:	PUNCT
ap-9228	181	1	ε	ε	PROPN
ap-9228	181	2	=	=	PUNCT
ap-9228	182	1	∞∑	∞∑	NUM
ap-9228	182	2	i=1	i=1	PRON
ap-9228	182	3	xi	xi	X
ap-9228	182	4	i	i	PRON
ap-9228	182	5	!	!	PUNCT
ap-9228	183	1	d	d	INTJ
ap-9228	184	1	i	i	PRON
ap-9228	184	2	t	t	PROPN
ap-9228	184	3	ψ(x	ψ(x	PROPN
ap-9228	184	4	)	)	PUNCT
ap-9228	185	1	+	+	PROPN
ap-9228	185	2	σ	σ	NUM
ap-9228	185	3	∫	∫	PROPN
ap-9228	185	4	x	x	SYM
ap-9228	185	5	0	0	NUM
ap-9228	185	6	k(x	k(x	PROPN
ap-9228	185	7	,	,	PUNCT
ap-9228	185	8	t)n	t)n	PUNCT
ap-9228	185	9			PROPN
ap-9228	185	10	∞∑	∞∑	NUM
ap-9228	185	11	j=0	j=0	PROPN
ap-9228	185	12	uj(t	uj(t	PUNCT
ap-9228	185	13	)	)	PUNCT
ap-9228	186	1			PROPN
ap-9228	186	2	dt	dt	X
ap-9228	186	3			PROPN
ap-9228	186	4	x=0	x=0	PUNCT
ap-9228	187	1	=	=	PUNCT
ap-9228	188	1	∞∑	∞∑	NUM
ap-9228	188	2	i=1	i=1	PROPN
ap-9228	188	3	ui	ui	PROPN
ap-9228	188	4	.	.	PUNCT
ap-9228	189	1	theorem	theorem	PROPN
ap-9228	189	2	3	3	NUM
ap-9228	189	3	.	.	PUNCT
ap-9228	190	1	the	the	DET
ap-9228	190	2	following	follow	VERB
ap-9228	190	3	equation	equation	NOUN
ap-9228	190	4	(	(	PUNCT
ap-9228	190	5	11	11	NUM
ap-9228	190	6	):	):	PUNCT
ap-9228	190	7	ε	ε	PROPN
ap-9228	190	8	=	=	PUNCT
ap-9228	191	1	∞∑	∞∑	NUM
ap-9228	191	2	i=1	i=1	PRON
ap-9228	191	3	xi	xi	X
ap-9228	191	4	(	(	PUNCT
ap-9228	191	5	i+	i+	NUM
ap-9228	191	6	1)!d	1)!d	NOUN
ap-9228	192	1	i	i	PRON
ap-9228	192	2	t	t	PROPN
ap-9228	192	3	ψ(x	ψ(x	PROPN
ap-9228	192	4	)	)	PUNCT
ap-9228	193	1	+	+	PROPN
ap-9228	193	2	σ	σ	NUM
ap-9228	193	3	∫	∫	PROPN
ap-9228	193	4	x	x	SYM
ap-9228	193	5	0	0	NUM
ap-9228	193	6	k(x	k(x	PROPN
ap-9228	193	7	,	,	PUNCT
ap-9228	193	8	t)n	t)n	PUNCT
ap-9228	193	9			PROPN
ap-9228	193	10	∞∑	∞∑	NUM
ap-9228	193	11	j=0	j=0	PROPN
ap-9228	193	12	uj(t	uj(t	PUNCT
ap-9228	193	13	)	)	PUNCT
ap-9228	194	1			PROPN
ap-9228	194	2	dt	dt	X
ap-9228	194	3			PROPN
ap-9228	194	4	t=0	t=0	PROPN
ap-9228	194	5	,	,	PUNCT
ap-9228	194	6	is	be	AUX
ap-9228	194	7	equivalent	equivalent	ADJ
ap-9228	194	8	to	to	ADP
ap-9228	194	9	equation	equation	NOUN
ap-9228	194	10	(	(	PUNCT
ap-9228	194	11	1	1	NUM
ap-9228	194	12	):	):	PUNCT
ap-9228	194	13	u(x	u(x	NOUN
ap-9228	194	14	)	)	PUNCT
ap-9228	194	15	=	=	SYM
ap-9228	194	16	ψ(x	ψ(x	NOUN
ap-9228	194	17	)	)	PUNCT
ap-9228	195	1	+	+	CCONJ
ap-9228	195	2	σ	σ	NUM
ap-9228	195	3	∫	∫	PROPN
ap-9228	195	4	x	x	SYM
ap-9228	195	5	0	0	PUNCT
ap-9228	195	6	k(x	k(x	PROPN
ap-9228	195	7	,	,	PUNCT
ap-9228	195	8	t)n(u(t	t)n(u(t	NOUN
ap-9228	195	9	)	)	PUNCT
ap-9228	195	10	)	)	PUNCT
ap-9228	196	1	dt	dt	NOUN
ap-9228	196	2	.	.	PUNCT
ap-9228	197	1	90	90	NUM
ap-9228	197	2	vol	vol	NOUN
ap-9228	197	3	.	.	PUNCT
ap-9228	198	1	64	64	NUM
ap-9228	198	2	no	no	NOUN
ap-9228	198	3	.	.	PUNCT
ap-9228	199	1	2/2024	2/2024	NUM
ap-9228	199	2	an	an	DET
ap-9228	199	3	innovative	innovative	ADJ
ap-9228	199	4	iterative	iterative	NOUN
ap-9228	199	5	approach	approach	NOUN
ap-9228	199	6	to	to	ADP
ap-9228	199	7	solving	solve	VERB
ap-9228	199	8	volterra	volterra	NOUN
ap-9228	199	9	integral	integral	ADJ
ap-9228	199	10	.	.	PUNCT
ap-9228	199	11	.	.	PUNCT
ap-9228	199	12	.	.	PUNCT
ap-9228	200	1	proof	proof	NOUN
ap-9228	200	2	.	.	PUNCT
ap-9228	201	1	we	we	PRON
ap-9228	201	2	rewrite	rewrite	VERB
ap-9228	201	3	equation	equation	NOUN
ap-9228	201	4	(	(	PUNCT
ap-9228	201	5	11	11	NUM
ap-9228	201	6	)	)	PUNCT
ap-9228	201	7	as	as	SCONJ
ap-9228	201	8	follows	follow	VERB
ap-9228	201	9	:	:	PUNCT
ap-9228	201	10	ψ(0	ψ(0	X
ap-9228	201	11	)	)	PUNCT
ap-9228	202	1	+	+	NUM
ap-9228	202	2	ε	ε	X
ap-9228	202	3	=	=	SYM
ap-9228	202	4	ψ(0	ψ(0	PROPN
ap-9228	202	5	)	)	PUNCT
ap-9228	203	1	+	+	CCONJ
ap-9228	204	1	∞∑	∞∑	NUM
ap-9228	204	2	i=1	i=1	PRON
ap-9228	204	3	xi	xi	X
ap-9228	204	4	i	i	PRON
ap-9228	204	5	!	!	PUNCT
ap-9228	205	1	d	d	INTJ
ap-9228	206	1	i	i	PRON
ap-9228	206	2	t	t	PROPN
ap-9228	206	3	ψ(x	ψ(x	PROPN
ap-9228	206	4	)	)	PUNCT
ap-9228	207	1	+	+	PROPN
ap-9228	207	2	σ	σ	NUM
ap-9228	207	3	∫	∫	PROPN
ap-9228	207	4	x	x	SYM
ap-9228	207	5	0	0	NUM
ap-9228	207	6	k(x	k(x	PROPN
ap-9228	207	7	,	,	PUNCT
ap-9228	207	8	t)n	t)n	PUNCT
ap-9228	207	9			PROPN
ap-9228	207	10	∞∑	∞∑	NUM
ap-9228	207	11	j=0	j=0	PROPN
ap-9228	207	12	uj(t	uj(t	PUNCT
ap-9228	207	13	)	)	PUNCT
ap-9228	208	1			PROPN
ap-9228	208	2	dt	dt	X
ap-9228	208	3			PROPN
ap-9228	208	4	x=0	x=0	PUNCT
ap-9228	209	1	=	=	PUNCT
ap-9228	209	2	∞∑	∞∑	NUM
ap-9228	209	3	i=0	i=0	PROPN
ap-9228	209	4	xi	xi	X
ap-9228	209	5	i	i	PROPN
ap-9228	209	6	!	!	PUNCT
ap-9228	210	1	d	d	INTJ
ap-9228	211	1	i	i	PRON
ap-9228	211	2	t	t	PROPN
ap-9228	211	3	ψ(x	ψ(x	PROPN
ap-9228	211	4	)	)	PUNCT
ap-9228	212	1	+	+	PROPN
ap-9228	212	2	σ	σ	NUM
ap-9228	212	3	∫	∫	PROPN
ap-9228	212	4	x	x	SYM
ap-9228	212	5	0	0	NUM
ap-9228	212	6	k(x	k(x	PROPN
ap-9228	212	7	,	,	PUNCT
ap-9228	212	8	t)n	t)n	PUNCT
ap-9228	212	9			PROPN
ap-9228	212	10	∞∑	∞∑	NUM
ap-9228	212	11	j=0	j=0	PROPN
ap-9228	212	12	uj(t	uj(t	PUNCT
ap-9228	212	13	)	)	PUNCT
ap-9228	213	1			PROPN
ap-9228	213	2	dt	dt	X
ap-9228	213	3			PROPN
ap-9228	213	4	x=0	x=0	PROPN
ap-9228	213	5	.	.	PUNCT
ap-9228	214	1	thus	thus	ADV
ap-9228	214	2	,	,	PUNCT
ap-9228	214	3	we	we	PRON
ap-9228	214	4	get	get	VERB
ap-9228	214	5	:	:	PUNCT
ap-9228	214	6	u	u	NOUN
ap-9228	214	7	=	=	NOUN
ap-9228	214	8	ε+	ε+	NOUN
ap-9228	214	9	ψ(0	ψ(0	NOUN
ap-9228	214	10	)	)	PUNCT
ap-9228	214	11	=	=	PUNCT
ap-9228	215	1	∞∑	∞∑	NUM
ap-9228	215	2	i=1	i=1	X
ap-9228	215	3	ui	ui	PROPN
ap-9228	216	1	+	+	NUM
ap-9228	216	2	u0	u0	ADJ
ap-9228	216	3	=	=	NOUN
ap-9228	216	4	∞∑	∞∑	NUM
ap-9228	216	5	i=0	i=0	PROPN
ap-9228	216	6	ui	ui	PROPN
ap-9228	216	7	.	.	PUNCT
ap-9228	217	1	now	now	ADV
ap-9228	217	2	,	,	PUNCT
ap-9228	217	3	we	we	PRON
ap-9228	217	4	have	have	VERB
ap-9228	217	5	:	:	PUNCT
ap-9228	217	6	∞∑	∞∑	NUM
ap-9228	217	7	i=0	i=0	PROPN
ap-9228	217	8	ui	ui	NOUN
ap-9228	217	9	=	=	PUNCT
ap-9228	218	1	∞∑	∞∑	NUM
ap-9228	218	2	i=0	i=0	PROPN
ap-9228	218	3	xi	xi	X
ap-9228	218	4	i	i	PROPN
ap-9228	218	5	!	!	PUNCT
ap-9228	219	1	d	d	INTJ
ap-9228	220	1	i	i	PRON
ap-9228	220	2	t	t	PROPN
ap-9228	220	3	ψ(x	ψ(x	PROPN
ap-9228	220	4	)	)	PUNCT
ap-9228	221	1	+	+	PROPN
ap-9228	221	2	σ	σ	NUM
ap-9228	221	3	∫	∫	PROPN
ap-9228	221	4	x	x	SYM
ap-9228	221	5	0	0	NUM
ap-9228	221	6	k(x	k(x	PROPN
ap-9228	221	7	,	,	PUNCT
ap-9228	221	8	t)n	t)n	PUNCT
ap-9228	221	9			PROPN
ap-9228	221	10	∞∑	∞∑	NUM
ap-9228	221	11	j=0	j=0	PROPN
ap-9228	221	12	uj(t	uj(t	PUNCT
ap-9228	221	13	)	)	PUNCT
ap-9228	222	1			PROPN
ap-9228	222	2	dt	dt	X
ap-9228	222	3			PROPN
ap-9228	222	4	x=0	x=0	PUNCT
ap-9228	222	5	.	.	PUNCT
ap-9228	223	1	(	(	PUNCT
ap-9228	223	2	14	14	NUM
ap-9228	223	3	)	)	PUNCT
ap-9228	223	4	it	it	PRON
ap-9228	223	5	seems	seem	VERB
ap-9228	223	6	clear	clear	ADJ
ap-9228	223	7	that	that	SCONJ
ap-9228	223	8	the	the	DET
ap-9228	223	9	right	right	ADJ
ap-9228	223	10	side	side	NOUN
ap-9228	223	11	of	of	ADP
ap-9228	223	12	equation	equation	NOUN
ap-9228	223	13	(	(	PUNCT
ap-9228	223	14	14	14	NUM
ap-9228	223	15	)	)	PUNCT
ap-9228	223	16	is	be	AUX
ap-9228	223	17	maclaurin	maclaurin	NOUN
ap-9228	223	18	’s	’s	PART
ap-9228	223	19	expansion	expansion	NOUN
ap-9228	223	20	,	,	PUNCT
ap-9228	223	21	so	so	CCONJ
ap-9228	223	22	equation	equation	NOUN
ap-9228	223	23	(	(	PUNCT
ap-9228	223	24	14	14	NUM
ap-9228	223	25	)	)	PUNCT
ap-9228	223	26	can	can	AUX
ap-9228	223	27	take	take	VERB
ap-9228	223	28	the	the	DET
ap-9228	223	29	following	follow	VERB
ap-9228	223	30	form	form	NOUN
ap-9228	223	31	:	:	PUNCT
ap-9228	223	32	∞∑	∞∑	NUM
ap-9228	223	33	i=0	i=0	ADJ
ap-9228	223	34	ui(x	ui(x	NUM
ap-9228	223	35	)	)	PUNCT
ap-9228	223	36	=	=	SYM
ap-9228	223	37	ψ(x	ψ(x	NOUN
ap-9228	223	38	)	)	PUNCT
ap-9228	224	1	+	+	CCONJ
ap-9228	224	2	σ	σ	NUM
ap-9228	224	3	∫	∫	PROPN
ap-9228	224	4	x	x	SYM
ap-9228	224	5	0	0	NUM
ap-9228	224	6	k(x	k(x	PROPN
ap-9228	224	7	,	,	PUNCT
ap-9228	224	8	t)n	t)n	PUNCT
ap-9228	224	9			PROPN
ap-9228	224	10	∞∑	∞∑	NUM
ap-9228	224	11	j=0	j=0	PROPN
ap-9228	224	12	uj(t	uj(t	PUNCT
ap-9228	224	13	)	)	PUNCT
ap-9228	225	1			PROPN
ap-9228	225	2	dt	dt	X
ap-9228	225	3	,	,	PUNCT
ap-9228	225	4	(	(	PUNCT
ap-9228	225	5	15	15	NUM
ap-9228	225	6	)	)	PUNCT
ap-9228	225	7	by	by	ADP
ap-9228	225	8	using	use	VERB
ap-9228	225	9	equation	equation	NOUN
ap-9228	225	10	(	(	PUNCT
ap-9228	225	11	3	3	NUM
ap-9228	225	12	)	)	PUNCT
ap-9228	225	13	,	,	PUNCT
ap-9228	225	14	equation	equation	NOUN
ap-9228	225	15	(	(	PUNCT
ap-9228	225	16	15	15	NUM
ap-9228	225	17	)	)	PUNCT
ap-9228	225	18	becomes	become	VERB
ap-9228	225	19	as	as	SCONJ
ap-9228	225	20	follows	follow	VERB
ap-9228	225	21	:	:	PUNCT
ap-9228	225	22	u(x	u(x	NOUN
ap-9228	225	23	)	)	PUNCT
ap-9228	225	24	=	=	SYM
ap-9228	225	25	ψ(x	ψ(x	NOUN
ap-9228	225	26	)	)	PUNCT
ap-9228	226	1	+	+	CCONJ
ap-9228	226	2	σ	σ	NUM
ap-9228	226	3	∫	∫	PROPN
ap-9228	226	4	x	x	SYM
ap-9228	226	5	0	0	PUNCT
ap-9228	226	6	k(x	k(x	PROPN
ap-9228	226	7	,	,	PUNCT
ap-9228	226	8	t)n(u(t	t)n(u(t	NOUN
ap-9228	226	9	)	)	PUNCT
ap-9228	226	10	)	)	PUNCT
ap-9228	227	1	dt	dt	X
ap-9228	227	2	.	.	PUNCT
ap-9228	228	1	(	(	PUNCT
ap-9228	228	2	16	16	NUM
ap-9228	228	3	)	)	PUNCT
ap-9228	228	4	then	then	ADV
ap-9228	228	5	,	,	PUNCT
ap-9228	228	6	the	the	DET
ap-9228	228	7	solution	solution	NOUN
ap-9228	228	8	of	of	ADP
ap-9228	228	9	equation	equation	NOUN
ap-9228	228	10	(	(	PUNCT
ap-9228	228	11	11	11	NUM
ap-9228	228	12	)	)	PUNCT
ap-9228	228	13	is	be	AUX
ap-9228	228	14	the	the	DET
ap-9228	228	15	same	same	ADJ
ap-9228	228	16	as	as	ADP
ap-9228	228	17	the	the	DET
ap-9228	228	18	solution	solution	NOUN
ap-9228	228	19	of	of	ADP
ap-9228	228	20	equation	equation	NOUN
ap-9228	228	21	(	(	PUNCT
ap-9228	228	22	1	1	NUM
ap-9228	228	23	)	)	PUNCT
ap-9228	228	24	.	.	PUNCT
ap-9228	229	1	4	4	X
ap-9228	229	2	.	.	X
ap-9228	229	3	illustrative	illustrative	ADJ
ap-9228	229	4	examples	example	NOUN
ap-9228	229	5	this	this	DET
ap-9228	229	6	section	section	NOUN
ap-9228	229	7	involves	involve	VERB
ap-9228	229	8	the	the	DET
ap-9228	229	9	application	application	NOUN
ap-9228	229	10	of	of	ADP
ap-9228	229	11	the	the	DET
ap-9228	229	12	hussein	hussein	PROPN
ap-9228	229	13	jassim	jassim	PROPN
ap-9228	229	14	method	method	NOUN
ap-9228	229	15	(	(	PUNCT
ap-9228	229	16	hjm	hjm	PROPN
ap-9228	229	17	)	)	PUNCT
ap-9228	229	18	to	to	PART
ap-9228	229	19	solve	solve	VERB
ap-9228	229	20	certain	certain	ADJ
ap-9228	229	21	second	second	ADJ
ap-9228	229	22	-	-	PUNCT
ap-9228	229	23	kind	kind	NOUN
ap-9228	229	24	volterra	volterra	NOUN
ap-9228	229	25	integral	integral	ADJ
ap-9228	229	26	equations	equation	NOUN
ap-9228	229	27	.	.	PUNCT
ap-9228	230	1	we	we	PRON
ap-9228	230	2	will	will	AUX
ap-9228	230	3	examine	examine	VERB
ap-9228	230	4	and	and	CCONJ
ap-9228	230	5	analyse	analyse	VERB
ap-9228	230	6	the	the	DET
ap-9228	230	7	results	result	NOUN
ap-9228	230	8	obtained	obtain	VERB
ap-9228	230	9	using	use	VERB
ap-9228	230	10	this	this	DET
ap-9228	230	11	innovative	innovative	ADJ
ap-9228	230	12	method	method	NOUN
ap-9228	230	13	.	.	PUNCT
ap-9228	231	1	additionally	additionally	ADV
ap-9228	231	2	,	,	PUNCT
ap-9228	231	3	we	we	PRON
ap-9228	231	4	will	will	AUX
ap-9228	231	5	conduct	conduct	VERB
ap-9228	231	6	a	a	DET
ap-9228	231	7	comparative	comparative	ADJ
ap-9228	231	8	analysis	analysis	NOUN
ap-9228	231	9	between	between	ADP
ap-9228	231	10	the	the	DET
ap-9228	231	11	outcomes	outcome	NOUN
ap-9228	231	12	of	of	ADP
ap-9228	231	13	this	this	DET
ap-9228	231	14	method	method	NOUN
ap-9228	231	15	and	and	CCONJ
ap-9228	231	16	those	those	PRON
ap-9228	231	17	of	of	ADP
ap-9228	231	18	the	the	DET
ap-9228	231	19	power	power	NOUN
ap-9228	231	20	series	series	PROPN
ap-9228	231	21	method	method	NOUN
ap-9228	231	22	(	(	PUNCT
ap-9228	231	23	psm	psm	NOUN
ap-9228	231	24	)	)	PUNCT
ap-9228	231	25	,	,	PUNCT
ap-9228	231	26	reduced	reduce	VERB
ap-9228	231	27	differential	differential	ADJ
ap-9228	231	28	transform	transform	NOUN
ap-9228	231	29	method	method	NOUN
ap-9228	231	30	(	(	PUNCT
ap-9228	231	31	rdtm	rdtm	NOUN
ap-9228	231	32	)	)	PUNCT
ap-9228	231	33	,	,	PUNCT
ap-9228	231	34	and	and	CCONJ
ap-9228	231	35	adomian	adomian	NOUN
ap-9228	231	36	decomposition	decomposition	NOUN
ap-9228	231	37	method	method	NOUN
ap-9228	231	38	(	(	PUNCT
ap-9228	231	39	adm	adm	PROPN
ap-9228	231	40	)	)	PUNCT
ap-9228	231	41	.	.	PUNCT
ap-9228	232	1	this	this	DET
ap-9228	232	2	comparison	comparison	NOUN
ap-9228	232	3	will	will	AUX
ap-9228	232	4	be	be	AUX
ap-9228	232	5	carried	carry	VERB
ap-9228	232	6	out	out	ADP
ap-9228	232	7	by	by	ADP
ap-9228	232	8	measuring	measure	VERB
ap-9228	232	9	the	the	DET
ap-9228	232	10	absolute	absolute	ADJ
ap-9228	232	11	error	error	NOUN
ap-9228	232	12	through	through	ADP
ap-9228	232	13	matlab	matlab	PROPN
ap-9228	232	14	.	.	PUNCT
ap-9228	233	1	the	the	DET
ap-9228	233	2	purpose	purpose	NOUN
ap-9228	233	3	of	of	ADP
ap-9228	233	4	this	this	DET
ap-9228	233	5	comparison	comparison	NOUN
ap-9228	233	6	is	be	AUX
ap-9228	233	7	to	to	PART
ap-9228	233	8	provide	provide	VERB
ap-9228	233	9	a	a	DET
ap-9228	233	10	deeper	deep	ADJ
ap-9228	233	11	understanding	understanding	NOUN
ap-9228	233	12	of	of	ADP
ap-9228	233	13	the	the	DET
ap-9228	233	14	effectiveness	effectiveness	NOUN
ap-9228	233	15	of	of	ADP
ap-9228	233	16	the	the	DET
ap-9228	233	17	method	method	NOUN
ap-9228	233	18	used	use	VERB
ap-9228	233	19	compared	compare	VERB
ap-9228	233	20	to	to	ADP
ap-9228	233	21	other	other	ADJ
ap-9228	233	22	approaches	approach	NOUN
ap-9228	233	23	commonly	commonly	ADV
ap-9228	233	24	used	use	VERB
ap-9228	233	25	to	to	PART
ap-9228	233	26	solve	solve	VERB
ap-9228	233	27	this	this	DET
ap-9228	233	28	type	type	NOUN
ap-9228	233	29	of	of	ADP
ap-9228	233	30	equation	equation	NOUN
ap-9228	233	31	.	.	PUNCT
ap-9228	234	1	we	we	PRON
ap-9228	234	2	will	will	AUX
ap-9228	234	3	highlight	highlight	VERB
ap-9228	234	4	the	the	DET
ap-9228	234	5	relative	relative	ADJ
ap-9228	234	6	superiority	superiority	NOUN
ap-9228	234	7	of	of	ADP
ap-9228	234	8	the	the	DET
ap-9228	234	9	hussein	hussein	PROPN
ap-9228	234	10	jassim	jassim	PROPN
ap-9228	234	11	method	method	NOUN
ap-9228	234	12	and	and	CCONJ
ap-9228	234	13	underscore	underscore	VERB
ap-9228	234	14	its	its	PRON
ap-9228	234	15	ability	ability	NOUN
ap-9228	234	16	to	to	PART
ap-9228	234	17	address	address	VERB
ap-9228	234	18	the	the	DET
ap-9228	234	19	complex	complex	ADJ
ap-9228	234	20	challenges	challenge	NOUN
ap-9228	234	21	associated	associate	VERB
ap-9228	234	22	with	with	ADP
ap-9228	234	23	second	second	ADJ
ap-9228	234	24	-	-	PUNCT
ap-9228	234	25	type	type	NOUN
ap-9228	234	26	volterra	volterra	NOUN
ap-9228	234	27	integral	integral	ADJ
ap-9228	234	28	equations	equation	NOUN
ap-9228	234	29	,	,	PUNCT
ap-9228	234	30	whether	whether	SCONJ
ap-9228	234	31	linear	linear	ADJ
ap-9228	234	32	or	or	CCONJ
ap-9228	234	33	nonlinear	nonlinear	ADJ
ap-9228	234	34	.	.	PUNCT
ap-9228	234	35	example	example	NOUN
ap-9228	235	1	1	1	NUM
ap-9228	235	2	.	.	X
ap-9228	235	3	consider	consider	VERB
ap-9228	235	4	the	the	DET
ap-9228	235	5	following	follow	VERB
ap-9228	235	6	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	235	7	linear	linear	PROPN
ap-9228	235	8	viesk	viesk	NOUN
ap-9228	235	9	:	:	PUNCT
ap-9228	235	10	u(x	u(x	PROPN
ap-9228	235	11	)	)	PUNCT
ap-9228	235	12	=	=	SYM
ap-9228	235	13	1	1	NUM
ap-9228	235	14	+	+	NUM
ap-9228	235	15	∫	∫	PROPN
ap-9228	235	16	x	x	SYM
ap-9228	235	17	0	0	PUNCT
ap-9228	235	18	(	(	PUNCT
ap-9228	235	19	t−	t−	PROPN
ap-9228	235	20	x)u(t	x)u(t	PROPN
ap-9228	235	21	)	)	PUNCT
ap-9228	236	1	dt	dt	X
ap-9228	236	2	,	,	PUNCT
ap-9228	236	3	(	(	PUNCT
ap-9228	236	4	17	17	NUM
ap-9228	236	5	)	)	PUNCT
ap-9228	236	6	using	use	VERB
ap-9228	236	7	the	the	DET
ap-9228	236	8	maclaurin	maclaurin	NOUN
ap-9228	236	9	expansion	expansion	NOUN
ap-9228	236	10	with	with	ADP
ap-9228	236	11	equation	equation	NOUN
ap-9228	236	12	(	(	PUNCT
ap-9228	236	13	17	17	NUM
ap-9228	236	14	)	)	PUNCT
ap-9228	236	15	,	,	PUNCT
ap-9228	236	16	we	we	PRON
ap-9228	236	17	obtain	obtain	VERB
ap-9228	236	18	:	:	PUNCT
ap-9228	236	19	u	u	NOUN
ap-9228	236	20	=	=	NOUN
ap-9228	236	21	∞∑	∞∑	NUM
ap-9228	236	22	i=0	i=0	PROPN
ap-9228	236	23	ti	ti	X
ap-9228	236	24	i!d	i!d	PROPN
ap-9228	237	1	i	i	NOUN
ap-9228	237	2	t	t	PROPN
ap-9228	237	3	(	(	PUNCT
ap-9228	237	4	1	1	NUM
ap-9228	238	1	+	+	NUM
ap-9228	238	2	∫	∫	PROPN
ap-9228	238	3	x	x	SYM
ap-9228	238	4	0	0	PUNCT
ap-9228	238	5	(	(	PUNCT
ap-9228	238	6	t−	t−	PROPN
ap-9228	238	7	x)u(t	x)u(t	PROPN
ap-9228	238	8	)	)	PUNCT
ap-9228	238	9	dt	dt	NOUN
ap-9228	238	10	)	)	PUNCT
ap-9228	239	1	x=0	x=0	PROPN
ap-9228	239	2	.	.	PUNCT
ap-9228	240	1	(	(	PUNCT
ap-9228	240	2	18	18	NUM
ap-9228	240	3	)	)	PUNCT
ap-9228	240	4	suppose	suppose	VERB
ap-9228	240	5	that	that	SCONJ
ap-9228	240	6	u(x	u(x	NOUN
ap-9228	240	7	)	)	PUNCT
ap-9228	240	8	is	be	AUX
ap-9228	240	9	a	a	DET
ap-9228	240	10	solution	solution	NOUN
ap-9228	240	11	of	of	ADP
ap-9228	240	12	equation	equation	NOUN
ap-9228	240	13	(	(	PUNCT
ap-9228	240	14	17	17	NUM
ap-9228	240	15	)	)	PUNCT
ap-9228	240	16	,	,	PUNCT
ap-9228	240	17	we	we	PRON
ap-9228	240	18	obtain	obtain	VERB
ap-9228	240	19	:	:	PUNCT
ap-9228	240	20	u	u	NOUN
ap-9228	240	21	=	=	PUNCT
ap-9228	240	22	∞∑	∞∑	NUM
ap-9228	240	23	i=0	i=0	PROPN
ap-9228	240	24	ui	ui	NOUN
ap-9228	240	25	,	,	PUNCT
ap-9228	240	26	(	(	PUNCT
ap-9228	240	27	19	19	NUM
ap-9228	240	28	)	)	PUNCT
ap-9228	240	29	by	by	ADP
ap-9228	240	30	substituting	substitute	VERB
ap-9228	240	31	equation	equation	NOUN
ap-9228	240	32	(	(	PUNCT
ap-9228	240	33	19	19	NUM
ap-9228	240	34	)	)	PUNCT
ap-9228	240	35	in	in	ADP
ap-9228	240	36	equation	equation	NOUN
ap-9228	240	37	(	(	PUNCT
ap-9228	240	38	18	18	NUM
ap-9228	240	39	)	)	PUNCT
ap-9228	240	40	,	,	PUNCT
ap-9228	240	41	we	we	PRON
ap-9228	240	42	obtain	obtain	VERB
ap-9228	240	43	:	:	PUNCT
ap-9228	240	44	∞∑	∞∑	NUM
ap-9228	240	45	i=0	i=0	PROPN
ap-9228	240	46	ui	ui	NOUN
ap-9228	241	1	=	=	PUNCT
ap-9228	242	1	∞∑	∞∑	NUM
ap-9228	242	2	i=0	i=0	PROPN
ap-9228	242	3	xi	xi	X
ap-9228	242	4	i	i	PROPN
ap-9228	242	5	!	!	PUNCT
ap-9228	243	1	d	d	VERB
ap-9228	243	2	i	i	PRON
ap-9228	243	3	x	x	PROPN
ap-9228	243	4	1	1	PROPN
ap-9228	244	1	+	+	CCONJ
ap-9228	244	2	∫	∫	PROPN
ap-9228	244	3	x	x	SYM
ap-9228	244	4	0	0	PUNCT
ap-9228	244	5	(	(	PUNCT
ap-9228	244	6	t−	t−	PROPN
ap-9228	244	7	x	x	SYM
ap-9228	244	8	)	)	PUNCT
ap-9228	244	9	i−1∑	i−1∑	NUM
ap-9228	244	10	j=0	j=0	PROPN
ap-9228	244	11	uj(t	uj(t	PUNCT
ap-9228	244	12	)	)	PUNCT
ap-9228	244	13	dt	dt	X
ap-9228	245	1			PROPN
ap-9228	245	2	x=0	x=0	PUNCT
ap-9228	245	3	.	.	PUNCT
ap-9228	246	1	(	(	PUNCT
ap-9228	246	2	20	20	NUM
ap-9228	246	3	)	)	SYM
ap-9228	246	4	91	91	NUM
ap-9228	246	5	m.	m.	NOUN
ap-9228	246	6	a.	a.	PROPN
ap-9228	246	7	hussein	hussein	PROPN
ap-9228	246	8	,	,	PUNCT
ap-9228	246	9	h.	h.	PROPN
ap-9228	246	10	k.	k.	PROPN
ap-9228	246	11	jassim	jassim	PROPN
ap-9228	246	12	,	,	PUNCT
ap-9228	246	13	a.	a.	PROPN
ap-9228	246	14	k.	k.	PROPN
ap-9228	246	15	jassim	jassim	PROPN
ap-9228	246	16	acta	acta	PROPN
ap-9228	246	17	polytechnica	polytechnica	PROPN
ap-9228	246	18	by	by	ADP
ap-9228	246	19	comparing	compare	VERB
ap-9228	246	20	both	both	DET
ap-9228	246	21	sides	side	NOUN
ap-9228	246	22	of	of	ADP
ap-9228	246	23	equation	equation	NOUN
ap-9228	246	24	(	(	PUNCT
ap-9228	246	25	20	20	NUM
ap-9228	246	26	)	)	PUNCT
ap-9228	246	27	,	,	PUNCT
ap-9228	246	28	we	we	PRON
ap-9228	246	29	obtain:	obtain:	VERB
ap-9228	246	30	u0	u0	NOUN
ap-9228	246	31	=	=	ADJ
ap-9228	246	32	1	1	NUM
ap-9228	246	33	,	,	PUNCT
ap-9228	246	34	u1	u1	NOUN
ap-9228	246	35	=	=	SYM
ap-9228	246	36	xdx	xdx	PROPN
ap-9228	246	37	(	(	PUNCT
ap-9228	246	38	1	1	NUM
ap-9228	246	39	+	+	NUM
ap-9228	246	40	∫	∫	PROPN
ap-9228	246	41	x	x	SYM
ap-9228	246	42	0	0	PUNCT
ap-9228	246	43	(	(	PUNCT
ap-9228	246	44	t−	t−	PROPN
ap-9228	246	45	x	x	SYM
ap-9228	246	46	)	)	PUNCT
ap-9228	246	47	dt	dt	NOUN
ap-9228	246	48	)	)	PUNCT
ap-9228	246	49	x=0	x=0	PUNCT
ap-9228	247	1	=	=	SYM
ap-9228	247	2	0	0	NUM
ap-9228	247	3	,	,	PUNCT
ap-9228	247	4	u2	u2	NOUN
ap-9228	247	5	=	=	SYM
ap-9228	247	6	x2	x2	PROPN
ap-9228	248	1	2	2	NUM
ap-9228	248	2	!	!	PUNCT
ap-9228	248	3	d	d	NOUN
ap-9228	248	4	2	2	NUM
ap-9228	248	5	x	x	SYM
ap-9228	248	6	(	(	PUNCT
ap-9228	248	7	1	1	NUM
ap-9228	248	8	+	+	NUM
ap-9228	248	9	∫	∫	PROPN
ap-9228	248	10	x	x	SYM
ap-9228	248	11	0	0	PUNCT
ap-9228	248	12	(	(	PUNCT
ap-9228	248	13	t−	t−	PROPN
ap-9228	248	14	x	x	SYM
ap-9228	248	15	)	)	PUNCT
ap-9228	248	16	dt	dt	NOUN
ap-9228	248	17	)	)	PUNCT
ap-9228	248	18	x=0	x=0	PUNCT
ap-9228	249	1	=	=	SYM
ap-9228	249	2	−x2	−x2	PROPN
ap-9228	249	3	2	2	NUM
ap-9228	249	4	!	!	NUM
ap-9228	249	5	,	,	PUNCT
ap-9228	249	6	u3	u3	PROPN
ap-9228	249	7	=	=	SYM
ap-9228	249	8	x3	x3	NOUN
ap-9228	249	9	3	3	X
ap-9228	249	10	!	!	PUNCT
ap-9228	250	1	d	d	NOUN
ap-9228	250	2	3	3	NUM
ap-9228	250	3	x	x	SYM
ap-9228	250	4	(	(	PUNCT
ap-9228	250	5	1	1	NUM
ap-9228	250	6	+	+	NUM
ap-9228	250	7	∫	∫	PROPN
ap-9228	250	8	x	x	SYM
ap-9228	250	9	0	0	PUNCT
ap-9228	250	10	(	(	PUNCT
ap-9228	250	11	t−	t−	PROPN
ap-9228	250	12	x	x	SYM
ap-9228	250	13	)	)	PUNCT
ap-9228	250	14	dt	dt	NOUN
ap-9228	250	15	)	)	PUNCT
ap-9228	251	1	x=0	x=0	PUNCT
ap-9228	251	2	=	=	SYM
ap-9228	251	3	0	0	PROPN
ap-9228	251	4	,	,	PUNCT
ap-9228	251	5	...	...	PUNCT
ap-9228	252	1	ui+1	ui+1	X
ap-9228	252	2	=	=	SYM
ap-9228	252	3	xi+1	xi+1	PROPN
ap-9228	252	4	(	(	PUNCT
ap-9228	252	5	i+	i+	NOUN
ap-9228	252	6	1)!d	1)!d	NUM
ap-9228	252	7	i+1	i+1	ADV
ap-9228	252	8	x	x	SYM
ap-9228	252	9	1	1	PROPN
ap-9228	252	10	+	+	CCONJ
ap-9228	252	11	∫	∫	PROPN
ap-9228	252	12	x	x	SYM
ap-9228	252	13	0	0	PUNCT
ap-9228	252	14	(	(	PUNCT
ap-9228	252	15	t−	t−	PROPN
ap-9228	252	16	x	x	SYM
ap-9228	252	17	)	)	PUNCT
ap-9228	252	18	i∑	i∑	PROPN
ap-9228	252	19	j=0	j=0	PROPN
ap-9228	252	20	uj(t	uj(t	PUNCT
ap-9228	252	21	)	)	PUNCT
ap-9228	252	22	dt	dt	X
ap-9228	253	1			PROPN
ap-9228	253	2	x=0	x=0	PROPN
ap-9228	253	3	=	=	SYM
ap-9228	253	4	(	(	PUNCT
ap-9228	253	5	−1	−1	NOUN
ap-9228	253	6	)	)	PUNCT
ap-9228	254	1	i+1	i+1	NUM
ap-9228	254	2	2	2	NUM
ap-9228	254	3	xi+1	xi+1	X
ap-9228	254	4	(	(	PUNCT
ap-9228	254	5	i+	i+	NOUN
ap-9228	254	6	1	1	NUM
ap-9228	254	7	)	)	PUNCT
ap-9228	254	8	!	!	PUNCT
ap-9228	255	1	,	,	PUNCT
ap-9228	255	2	for	for	ADP
ap-9228	255	3	all	all	DET
ap-9228	255	4	i	i	PRON
ap-9228	255	5	odd	odd	ADJ
ap-9228	255	6	number	number	NOUN
ap-9228	255	7	.	.	PUNCT
ap-9228	256	1	thus	thus	ADV
ap-9228	256	2	,	,	PUNCT
ap-9228	256	3	the	the	DET
ap-9228	256	4	approximate	approximate	ADJ
ap-9228	256	5	solution	solution	NOUN
ap-9228	256	6	of	of	ADP
ap-9228	256	7	equation	equation	NOUN
ap-9228	256	8	(	(	PUNCT
ap-9228	256	9	17	17	NUM
ap-9228	256	10	)	)	PUNCT
ap-9228	256	11	is	be	AUX
ap-9228	256	12	:	:	PUNCT
ap-9228	256	13	ua	ua	PROPN
ap-9228	256	14	(	(	PUNCT
ap-9228	256	15	x	x	X
ap-9228	256	16	)	)	PUNCT
ap-9228	256	17	=	=	SYM
ap-9228	257	1	1	1	NUM
ap-9228	257	2	−	−	NOUN
ap-9228	257	3	x2	x2	INTJ
ap-9228	257	4	2	2	NUM
ap-9228	257	5	!	!	PUNCT
ap-9228	258	1	+	+	CCONJ
ap-9228	258	2	x4	x4	PROPN
ap-9228	258	3	4	4	NUM
ap-9228	258	4	!	!	PUNCT
ap-9228	258	5	−	−	PROPN
ap-9228	258	6	·	·	PUNCT
ap-9228	258	7	·	·	PUNCT
ap-9228	258	8	·	·	PUNCT
ap-9228	259	1	=	=	PUNCT
ap-9228	259	2	∞∑	∞∑	NUM
ap-9228	259	3	i=0	i=0	PROPN
ap-9228	259	4	(	(	PUNCT
ap-9228	259	5	−1)i	−1)i	X
ap-9228	259	6	x2i	x2i	X
ap-9228	259	7	(	(	PUNCT
ap-9228	259	8	2i	2i	NUM
ap-9228	259	9	)	)	PUNCT
ap-9228	259	10	!	!	PUNCT
ap-9228	260	1	,	,	PUNCT
ap-9228	260	2	(	(	PUNCT
ap-9228	260	3	21	21	NUM
ap-9228	260	4	)	)	PUNCT
ap-9228	260	5	so	so	ADV
ap-9228	260	6	,	,	PUNCT
ap-9228	260	7	the	the	DET
ap-9228	260	8	exact	exact	ADJ
ap-9228	260	9	solution	solution	NOUN
ap-9228	260	10	of	of	ADP
ap-9228	260	11	equation	equation	NOUN
ap-9228	260	12	(	(	PUNCT
ap-9228	260	13	17	17	NUM
ap-9228	260	14	)	)	PUNCT
ap-9228	260	15	is	be	AUX
ap-9228	260	16	:	:	PUNCT
ap-9228	260	17	ue(x	ue(x	X
ap-9228	260	18	)	)	PUNCT
ap-9228	261	1	=	=	SYM
ap-9228	261	2	cosx	cosx	X
ap-9228	261	3	.	.	PUNCT
ap-9228	261	4	absolute	absolute	ADJ
ap-9228	261	5	error	error	NOUN
ap-9228	261	6	x	x	X
ap-9228	261	7	ua	ua	PROPN
ap-9228	261	8	ue	ue	PROPN
ap-9228	261	9	hjm	hjm	PROPN
ap-9228	261	10	rdtm	rdtm	PROPN
ap-9228	261	11	psm	psm	PROPN
ap-9228	261	12	adm	adm	PROPN
ap-9228	261	13	0.1000	0.1000	NUM
ap-9228	261	14	0.9950	0.9950	NUM
ap-9228	261	15	0.9950	0.9950	NUM
ap-9228	261	16	0.0000	0.0000	NUM
ap-9228	261	17	0.0000	0.0000	NUM
ap-9228	261	18	0.0000	0.0000	NUM
ap-9228	261	19	0.0000	0.0000	NUM
ap-9228	261	20	0.2000	0.2000	NUM
ap-9228	261	21	0.9801	0.9801	NUM
ap-9228	261	22	0.9801	0.9801	NUM
ap-9228	261	23	0.0000	0.0000	NUM
ap-9228	261	24	0.0003	0.0003	NUM
ap-9228	261	25	0.0000	0.0000	NUM
ap-9228	261	26	0.0000	0.0000	NUM
ap-9228	261	27	0.3000	0.3000	NUM
ap-9228	261	28	0.9553	0.9553	NUM
ap-9228	261	29	0.9553	0.9553	NUM
ap-9228	261	30	0.0000	0.0000	NUM
ap-9228	261	31	0.0017	0.0017	NUM
ap-9228	261	32	0.0000	0.0000	NUM
ap-9228	261	33	0.0000	0.0000	NUM
ap-9228	261	34	0.4000	0.4000	NUM
ap-9228	261	35	0.9211	0.9211	NUM
ap-9228	261	36	0.9211	0.9211	NUM
ap-9228	261	37	0.0000	0.0000	NUM
ap-9228	261	38	0.0053	0.0053	NUM
ap-9228	261	39	0.0000	0.0000	NUM
ap-9228	261	40	0.0000	0.0000	NUM
ap-9228	261	41	0.5000	0.5000	NUM
ap-9228	262	1	0.8776	0.8776	NUM
ap-9228	262	2	0.8776	0.8776	NUM
ap-9228	262	3	0.0000	0.0000	NUM
ap-9228	262	4	0.0130	0.0130	NUM
ap-9228	262	5	0.0000	0.0000	NUM
ap-9228	262	6	0.0000	0.0000	NUM
ap-9228	262	7	0.6000	0.6000	NUM
ap-9228	262	8	0.8254	0.8254	NUM
ap-9228	262	9	0.8253	0.8253	NUM
ap-9228	262	10	0.0001	0.0001	NUM
ap-9228	262	11	0.0271	0.0271	NUM
ap-9228	262	12	0.0001	0.0001	NUM
ap-9228	262	13	0.0000	0.0000	NUM
ap-9228	262	14	0.7000	0.7000	NUM
ap-9228	262	15	0.7650	0.7650	NUM
ap-9228	262	16	0.7648	0.7648	NUM
ap-9228	263	1	0.0002	0.0002	NUM
ap-9228	263	2	0.0502	0.0502	NUM
ap-9228	263	3	0.0002	0.0002	NUM
ap-9228	263	4	0.0000	0.0000	NUM
ap-9228	263	5	0.8000	0.8000	NUM
ap-9228	263	6	0.6971	0.6971	NUM
ap-9228	263	7	0.6967	0.6967	NUM
ap-9228	263	8	0.0004	0.0004	NUM
ap-9228	263	9	0.0857	0.0857	NUM
ap-9228	263	10	0.0004	0.0004	NUM
ap-9228	263	11	0.0000	0.0000	NUM
ap-9228	263	12	0.9000	0.9000	NUM
ap-9228	263	13	0.6223	0.6223	NUM
ap-9228	263	14	0.6216	0.6216	NUM
ap-9228	263	15	0.0007	0.0007	NUM
ap-9228	263	16	0.1374	0.1374	NUM
ap-9228	263	17	0.0007	0.0007	NUM
ap-9228	263	18	0.0000	0.0000	NUM
ap-9228	263	19	1.0000	1.0000	NUM
ap-9228	263	20	0.5417	0.5417	NUM
ap-9228	263	21	0.5403	0.5403	NUM
ap-9228	263	22	0.0014	0.0014	NUM
ap-9228	263	23	0.2097	0.2097	NUM
ap-9228	263	24	0.0014	0.0014	NUM
ap-9228	263	25	0.0000	0.0000	NUM
ap-9228	263	26	table	table	NOUN
ap-9228	263	27	1	1	NUM
ap-9228	263	28	.	.	PUNCT
ap-9228	264	1	the	the	DET
ap-9228	264	2	values	value	NOUN
ap-9228	264	3	of	of	ADP
ap-9228	264	4	the	the	DET
ap-9228	264	5	approximate	approximate	ADJ
ap-9228	264	6	and	and	CCONJ
ap-9228	264	7	exact	exact	ADJ
ap-9228	264	8	solutions	solution	NOUN
ap-9228	264	9	of	of	ADP
ap-9228	264	10	equation	equation	NOUN
ap-9228	264	11	(	(	PUNCT
ap-9228	264	12	17	17	NUM
ap-9228	264	13	)	)	PUNCT
ap-9228	264	14	for	for	ADP
ap-9228	264	15	different	different	ADJ
ap-9228	264	16	values	value	NOUN
ap-9228	264	17	of	of	ADP
ap-9228	264	18	x.	x.	NOUN
ap-9228	264	19	figure	figure	NOUN
ap-9228	264	20	1	1	NUM
ap-9228	264	21	.	.	PUNCT
ap-9228	265	1	the	the	DET
ap-9228	265	2	plot	plot	NOUN
ap-9228	265	3	of	of	ADP
ap-9228	265	4	approximate	approximate	ADJ
ap-9228	265	5	solution	solution	NOUN
ap-9228	265	6	,	,	PUNCT
ap-9228	265	7	exact	exact	ADJ
ap-9228	265	8	solution	solution	NOUN
ap-9228	265	9	,	,	PUNCT
ap-9228	265	10	and	and	CCONJ
ap-9228	265	11	absolute	absolute	ADJ
ap-9228	265	12	error	error	NOUN
ap-9228	265	13	of	of	ADP
ap-9228	265	14	equation	equation	NOUN
ap-9228	265	15	(	(	PUNCT
ap-9228	265	16	17	17	NUM
ap-9228	265	17	)	)	PUNCT
ap-9228	265	18	.	.	PUNCT
ap-9228	266	1	92	92	NUM
ap-9228	266	2	vol	vol	NOUN
ap-9228	266	3	.	.	PUNCT
ap-9228	266	4	64	64	NUM
ap-9228	266	5	no	no	NOUN
ap-9228	266	6	.	.	PUNCT
ap-9228	267	1	2/2024	2/2024	NUM
ap-9228	267	2	an	an	DET
ap-9228	267	3	innovative	innovative	ADJ
ap-9228	267	4	iterative	iterative	NOUN
ap-9228	267	5	approach	approach	NOUN
ap-9228	267	6	to	to	ADP
ap-9228	267	7	solving	solve	VERB
ap-9228	267	8	volterra	volterra	NOUN
ap-9228	267	9	integral	integral	ADJ
ap-9228	267	10	.	.	PUNCT
ap-9228	267	11	.	.	PUNCT
ap-9228	268	1	.	.	PUNCT
ap-9228	269	1	example	example	NOUN
ap-9228	270	1	2	2	NUM
ap-9228	270	2	.	.	X
ap-9228	270	3	assume	assume	VERB
ap-9228	270	4	the	the	DET
ap-9228	270	5	following	follow	VERB
ap-9228	270	6	linear	linear	PROPN
ap-9228	270	7	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	270	8	viesk	viesk	NOUN
ap-9228	270	9	:	:	PUNCT
ap-9228	270	10	u(x	u(x	PROPN
ap-9228	270	11	)	)	PUNCT
ap-9228	270	12	=	=	SYM
ap-9228	271	1	1	1	NUM
ap-9228	271	2	−	−	NOUN
ap-9228	271	3	x−	x−	PROPN
ap-9228	271	4	1	1	NUM
ap-9228	271	5	2x	2x	NUM
ap-9228	271	6	2	2	NUM
ap-9228	271	7	−	−	NOUN
ap-9228	271	8	∫	∫	PROPN
ap-9228	271	9	x	x	X
ap-9228	271	10	0	0	PUNCT
ap-9228	271	11	(	(	PUNCT
ap-9228	271	12	t−	t−	PROPN
ap-9228	271	13	x)u(t	x)u(t	PROPN
ap-9228	271	14	)	)	PUNCT
ap-9228	271	15	dt	dt	PROPN
ap-9228	271	16	.	.	PUNCT
ap-9228	272	1	(	(	PUNCT
ap-9228	272	2	22	22	NUM
ap-9228	272	3	)	)	PUNCT
ap-9228	272	4	by	by	ADP
ap-9228	272	5	applying	apply	VERB
ap-9228	272	6	the	the	DET
ap-9228	272	7	algorithm	algorithm	NOUN
ap-9228	272	8	of	of	ADP
ap-9228	272	9	the	the	DET
ap-9228	272	10	new	new	ADJ
ap-9228	272	11	technique	technique	NOUN
ap-9228	272	12	to	to	ADP
ap-9228	272	13	equation	equation	NOUN
ap-9228	272	14	(	(	PUNCT
ap-9228	272	15	22	22	NUM
ap-9228	272	16	)	)	PUNCT
ap-9228	272	17	,	,	PUNCT
ap-9228	272	18	we	we	PRON
ap-9228	272	19	obtain	obtain	VERB
ap-9228	272	20	:	:	PUNCT
ap-9228	272	21	∞∑	∞∑	NUM
ap-9228	272	22	i=0	i=0	PROPN
ap-9228	272	23	ui	ui	NOUN
ap-9228	273	1	=	=	PUNCT
ap-9228	273	2	∞∑	∞∑	NUM
ap-9228	273	3	i=0	i=0	PROPN
ap-9228	273	4	ti	ti	X
ap-9228	273	5	i!d	i!d	PROPN
ap-9228	274	1	i	i	NOUN
ap-9228	274	2	t	t	PROPN
ap-9228	274	3	1	1	PROPN
ap-9228	274	4	−	−	PROPN
ap-9228	274	5	x−	x−	PROPN
ap-9228	274	6	1	1	NUM
ap-9228	274	7	2x	2x	NUM
ap-9228	274	8	2	2	NUM
ap-9228	274	9	−	−	NOUN
ap-9228	274	10	∫	∫	PROPN
ap-9228	274	11	x	x	X
ap-9228	274	12	0	0	PUNCT
ap-9228	274	13	(	(	PUNCT
ap-9228	274	14	t−	t−	PROPN
ap-9228	274	15	x	x	SYM
ap-9228	274	16	)	)	PUNCT
ap-9228	274	17	i−1∑	i−1∑	NUM
ap-9228	274	18	j=0	j=0	PROPN
ap-9228	274	19	uj(t	uj(t	PUNCT
ap-9228	274	20	)	)	PUNCT
ap-9228	275	1	dt	dt	PUNCT
ap-9228	276	1			PROPN
ap-9228	276	2	t=0	t=0	PROPN
ap-9228	276	3	.	.	PUNCT
ap-9228	277	1	(	(	PUNCT
ap-9228	277	2	23	23	NUM
ap-9228	277	3	)	)	PUNCT
ap-9228	277	4	by	by	ADP
ap-9228	277	5	comparing	compare	VERB
ap-9228	277	6	both	both	DET
ap-9228	277	7	sides	side	NOUN
ap-9228	277	8	of	of	ADP
ap-9228	277	9	equation	equation	NOUN
ap-9228	277	10	(	(	PUNCT
ap-9228	277	11	23	23	NUM
ap-9228	277	12	)	)	PUNCT
ap-9228	277	13	,	,	PUNCT
ap-9228	277	14	we	we	PRON
ap-9228	277	15	obtain	obtain	VERB
ap-9228	277	16	:	:	PUNCT
ap-9228	277	17			NOUN
ap-9228	277	18	u0	u0	NOUN
ap-9228	277	19	=	=	NOUN
ap-9228	277	20	1	1	NUM
ap-9228	277	21	,	,	PUNCT
ap-9228	277	22	u1	u1	NOUN
ap-9228	277	23	=	=	SYM
ap-9228	277	24	xdx	xdx	PROPN
ap-9228	277	25	(	(	PUNCT
ap-9228	277	26	1	1	NUM
ap-9228	277	27	−	−	NOUN
ap-9228	277	28	x−	x−	PROPN
ap-9228	277	29	1	1	NUM
ap-9228	277	30	2x	2x	NUM
ap-9228	277	31	2	2	NUM
ap-9228	277	32	−	−	NOUN
ap-9228	277	33	∫	∫	PROPN
ap-9228	277	34	x	x	X
ap-9228	277	35	0	0	PUNCT
ap-9228	277	36	(	(	PUNCT
ap-9228	277	37	t−	t−	PROPN
ap-9228	277	38	x	x	SYM
ap-9228	277	39	)	)	PUNCT
ap-9228	277	40	dt	dt	NOUN
ap-9228	277	41	)	)	PUNCT
ap-9228	277	42	x=0	x=0	PUNCT
ap-9228	278	1	=	=	SYM
ap-9228	278	2	−x	−x	NOUN
ap-9228	278	3	,	,	PUNCT
ap-9228	278	4	u2	u2	NOUN
ap-9228	278	5	=	=	SYM
ap-9228	278	6	x2	x2	PROPN
ap-9228	278	7	2	2	NUM
ap-9228	278	8	!	!	PUNCT
ap-9228	279	1	d	d	NOUN
ap-9228	279	2	2	2	NUM
ap-9228	279	3	x	x	SYM
ap-9228	279	4	(	(	PUNCT
ap-9228	279	5	1	1	NUM
ap-9228	279	6	−	−	NOUN
ap-9228	279	7	x−	x−	PROPN
ap-9228	279	8	1	1	NUM
ap-9228	279	9	2x	2x	NUM
ap-9228	279	10	2	2	NUM
ap-9228	279	11	−	−	NOUN
ap-9228	279	12	∫	∫	PROPN
ap-9228	279	13	x	x	X
ap-9228	279	14	0	0	PUNCT
ap-9228	279	15	(	(	PUNCT
ap-9228	279	16	t−	t−	NOUN
ap-9228	279	17	x)(1	x)(1	X
ap-9228	279	18	−	−	PROPN
ap-9228	279	19	t	t	PROPN
ap-9228	279	20	)	)	PUNCT
ap-9228	279	21	dt	dt	NOUN
ap-9228	279	22	)	)	PUNCT
ap-9228	279	23	x=0	x=0	PUNCT
ap-9228	280	1	=	=	SYM
ap-9228	280	2	0	0	NUM
ap-9228	280	3	,	,	PUNCT
ap-9228	280	4	u3	u3	NOUN
ap-9228	280	5	=	=	SYM
ap-9228	280	6	x3	x3	NOUN
ap-9228	281	1	3	3	X
ap-9228	281	2	!	!	PUNCT
ap-9228	282	1	d	d	NOUN
ap-9228	282	2	3	3	NUM
ap-9228	282	3	x	x	SYM
ap-9228	282	4	(	(	PUNCT
ap-9228	282	5	1	1	NUM
ap-9228	282	6	−	−	NOUN
ap-9228	282	7	x−	x−	PROPN
ap-9228	282	8	1	1	NUM
ap-9228	282	9	2x	2x	NUM
ap-9228	282	10	2	2	NUM
ap-9228	282	11	−	−	NOUN
ap-9228	282	12	∫	∫	PROPN
ap-9228	282	13	x	x	X
ap-9228	282	14	0	0	PUNCT
ap-9228	282	15	(	(	PUNCT
ap-9228	282	16	t−	t−	NOUN
ap-9228	282	17	x)(1	x)(1	X
ap-9228	282	18	−	−	PROPN
ap-9228	282	19	t	t	PROPN
ap-9228	282	20	)	)	PUNCT
ap-9228	282	21	dt	dt	NOUN
ap-9228	282	22	)	)	PUNCT
ap-9228	283	1	x=0	x=0	PUNCT
ap-9228	284	1	=	=	SYM
ap-9228	284	2	−x3	−x3	PROPN
ap-9228	284	3	3	3	NUM
ap-9228	284	4	!	!	NUM
ap-9228	284	5	,	,	PUNCT
ap-9228	284	6	...	...	PUNCT
ap-9228	285	1	ui+1	ui+1	X
ap-9228	285	2	=	=	SYM
ap-9228	285	3	xi+1	xi+1	PROPN
ap-9228	285	4	(	(	PUNCT
ap-9228	285	5	i+	i+	NUM
ap-9228	285	6	1)!d	1)!d	NOUN
ap-9228	286	1	i	i	PRON
ap-9228	286	2	t	t	PROPN
ap-9228	286	3	1	1	PROPN
ap-9228	286	4	−	−	PROPN
ap-9228	286	5	x−	x−	PROPN
ap-9228	286	6	1	1	NUM
ap-9228	286	7	2x	2x	NUM
ap-9228	286	8	2	2	NUM
ap-9228	286	9	−	−	NOUN
ap-9228	286	10	∫	∫	PROPN
ap-9228	286	11	x	x	X
ap-9228	286	12	0	0	PUNCT
ap-9228	286	13	(	(	PUNCT
ap-9228	286	14	t−	t−	PROPN
ap-9228	286	15	x	x	SYM
ap-9228	286	16	)	)	PUNCT
ap-9228	286	17	i∑	i∑	PROPN
ap-9228	286	18	j=0	j=0	PROPN
ap-9228	286	19	uj(t	uj(t	PUNCT
ap-9228	286	20	)	)	PUNCT
ap-9228	286	21	dt	dt	X
ap-9228	287	1			PROPN
ap-9228	287	2	x=0	x=0	PROPN
ap-9228	287	3	=	=	SYM
ap-9228	288	1	−	−	PROPN
ap-9228	288	2	xi+1	xi+1	SYM
ap-9228	288	3	(	(	PUNCT
ap-9228	288	4	i+	i+	NOUN
ap-9228	288	5	1	1	NUM
ap-9228	288	6	)	)	PUNCT
ap-9228	288	7	!	!	PUNCT
ap-9228	288	8	.	.	PUNCT
ap-9228	289	1	thus	thus	ADV
ap-9228	289	2	,	,	PUNCT
ap-9228	289	3	the	the	DET
ap-9228	289	4	approximate	approximate	ADJ
ap-9228	289	5	solution	solution	NOUN
ap-9228	289	6	of	of	ADP
ap-9228	289	7	equation	equation	NOUN
ap-9228	289	8	(	(	PUNCT
ap-9228	289	9	22	22	NUM
ap-9228	289	10	)	)	PUNCT
ap-9228	289	11	is	be	AUX
ap-9228	289	12	:	:	PUNCT
ap-9228	289	13	ua(x	ua(x	X
ap-9228	289	14	)	)	PUNCT
ap-9228	289	15	=	=	SYM
ap-9228	289	16	1	1	NUM
ap-9228	289	17	−	−	NOUN
ap-9228	289	18	x−	x−	PROPN
ap-9228	289	19	x3	x3	PROPN
ap-9228	289	20	3	3	X
ap-9228	289	21	!	!	PUNCT
ap-9228	289	22	−	−	PROPN
ap-9228	289	23	.	.	PUNCT
ap-9228	289	24	.	.	PUNCT
ap-9228	289	25	.	.	PUNCT
ap-9228	290	1	=	=	SYM
ap-9228	291	1	1	1	NUM
ap-9228	291	2	−	−	PROPN
ap-9228	291	3	∞∑	∞∑	PROPN
ap-9228	291	4	i=0	i=0	PROPN
ap-9228	291	5	x2i+1	x2i+1	PROPN
ap-9228	291	6	(	(	PUNCT
ap-9228	291	7	2i+	2i+	NUM
ap-9228	291	8	1	1	NUM
ap-9228	291	9	)	)	PUNCT
ap-9228	291	10	!	!	PUNCT
ap-9228	292	1	,	,	PUNCT
ap-9228	292	2	for	for	ADP
ap-9228	292	3	all	all	PRON
ap-9228	292	4	i	i	PRON
ap-9228	292	5	even	even	ADV
ap-9228	292	6	number	number	NOUN
ap-9228	292	7	.	.	PUNCT
ap-9228	293	1	therefore	therefore	ADV
ap-9228	293	2	,	,	PUNCT
ap-9228	293	3	the	the	DET
ap-9228	293	4	exact	exact	ADJ
ap-9228	293	5	solution	solution	NOUN
ap-9228	293	6	of	of	ADP
ap-9228	293	7	equation	equation	NOUN
ap-9228	293	8	(	(	PUNCT
ap-9228	293	9	22	22	NUM
ap-9228	293	10	)	)	PUNCT
ap-9228	293	11	is	be	AUX
ap-9228	293	12	:	:	PUNCT
ap-9228	293	13	ue(x	ue(x	X
ap-9228	293	14	)	)	PUNCT
ap-9228	293	15	=	=	SYM
ap-9228	294	1	1	1	NUM
ap-9228	294	2	−	−	NUM
ap-9228	294	3	sinh	sinh	NOUN
ap-9228	294	4	x.	x.	NOUN
ap-9228	295	1	absolute	absolute	ADJ
ap-9228	295	2	error	error	NOUN
ap-9228	295	3	x	x	X
ap-9228	295	4	ua	ua	PROPN
ap-9228	295	5	ue	ue	PROPN
ap-9228	296	1	hjm	hjm	PROPN
ap-9228	296	2	rdtm	rdtm	PROPN
ap-9228	296	3	psm	psm	PROPN
ap-9228	296	4	adm	adm	PROPN
ap-9228	296	5	0.1000	0.1000	NUM
ap-9228	297	1	0.8998	0.8998	NUM
ap-9228	297	2	0.8998	0.8998	NUM
ap-9228	297	3	0.0000	0.0000	NUM
ap-9228	297	4	0.0002	0.0002	NUM
ap-9228	297	5	0.0000	0.0000	NUM
ap-9228	297	6	0.0000	0.0000	NUM
ap-9228	297	7	0.2000	0.2000	NUM
ap-9228	297	8	0.7987	0.7987	NUM
ap-9228	297	9	0.7987	0.7987	NUM
ap-9228	297	10	0.0000	0.0000	NUM
ap-9228	297	11	0.0013	0.0013	NUM
ap-9228	297	12	0.0000	0.0000	NUM
ap-9228	297	13	0.0000	0.0000	NUM
ap-9228	297	14	0.3000	0.3000	NUM
ap-9228	297	15	0.6955	0.6955	NUM
ap-9228	297	16	0.6955	0.6955	NUM
ap-9228	297	17	0.0000	0.0000	NUM
ap-9228	297	18	0.0045	0.0045	NUM
ap-9228	297	19	0.0000	0.0000	NUM
ap-9228	297	20	0.0000	0.0000	NUM
ap-9228	297	21	0.4000	0.4000	NUM
ap-9228	298	1	0.5893	0.5893	NUM
ap-9228	298	2	0.5892	0.5892	NUM
ap-9228	298	3	0.0001	0.0001	NUM
ap-9228	298	4	0.0106	0.0106	NUM
ap-9228	298	5	0.0001	0.0001	NUM
ap-9228	298	6	0.0000	0.0000	NUM
ap-9228	298	7	0.5000	0.5000	NUM
ap-9228	298	8	0.4792	0.4792	NUM
ap-9228	298	9	0.4789	0.4789	NUM
ap-9228	298	10	0.0003	0.0003	NUM
ap-9228	298	11	0.0206	0.0206	NUM
ap-9228	298	12	0.0003	0.0003	NUM
ap-9228	298	13	0.0000	0.0000	NUM
ap-9228	298	14	0.6000	0.6000	NUM
ap-9228	298	15	0.3640	0.3640	NUM
ap-9228	298	16	0.3633	0.3633	NUM
ap-9228	298	17	0.0007	0.0007	NUM
ap-9228	298	18	0.0353	0.0353	NUM
ap-9228	298	19	0.0007	0.0007	NUM
ap-9228	298	20	0.0000	0.0000	NUM
ap-9228	298	21	0.7000	0.7000	NUM
ap-9228	298	22	0.2428	0.2428	NUM
ap-9228	298	23	0.2414	0.2414	NUM
ap-9228	298	24	0.0014	0.0014	NUM
ap-9228	298	25	0.0557	0.0557	NUM
ap-9228	298	26	0.0014	0.0014	NUM
ap-9228	298	27	0.0000	0.0000	NUM
ap-9228	298	28	0.8000	0.8000	NUM
ap-9228	298	29	0.1147	0.1147	NUM
ap-9228	298	30	0.1119	0.1119	NUM
ap-9228	298	31	0.0028	0.0028	NUM
ap-9228	298	32	0.0826	0.0826	NUM
ap-9228	298	33	0.0028	0.0028	NUM
ap-9228	298	34	0.0000	0.0000	NUM
ap-9228	298	35	0.9000	0.9000	NUM
ap-9228	298	36	−0.0215	−0.0215	X
ap-9228	298	37	−0.0265	−0.0265	X
ap-9228	298	38	0.0050	0.0050	NUM
ap-9228	298	39	0.1165	0.1165	NUM
ap-9228	298	40	0.0050	0.0050	NUM
ap-9228	298	41	0.0001	0.0001	NUM
ap-9228	298	42	1.0000	1.0000	NUM
ap-9228	298	43	−0.1667	−0.1667	X
ap-9228	298	44	−0.1752	−0.1752	NUM
ap-9228	298	45	0.0085	0.0085	NUM
ap-9228	298	46	0.1581	0.1581	NUM
ap-9228	298	47	0.0085	0.0085	NUM
ap-9228	298	48	0.0002	0.0002	NUM
ap-9228	298	49	table	table	NOUN
ap-9228	298	50	2	2	NUM
ap-9228	298	51	.	.	PUNCT
ap-9228	299	1	the	the	DET
ap-9228	299	2	values	value	NOUN
ap-9228	299	3	of	of	ADP
ap-9228	299	4	the	the	DET
ap-9228	299	5	approximate	approximate	ADJ
ap-9228	299	6	and	and	CCONJ
ap-9228	299	7	exact	exact	ADJ
ap-9228	299	8	solutions	solution	NOUN
ap-9228	299	9	of	of	ADP
ap-9228	299	10	equation	equation	NOUN
ap-9228	299	11	(	(	PUNCT
ap-9228	299	12	22	22	NUM
ap-9228	299	13	)	)	PUNCT
ap-9228	299	14	for	for	ADP
ap-9228	299	15	different	different	ADJ
ap-9228	299	16	values	value	NOUN
ap-9228	299	17	of	of	ADP
ap-9228	299	18	x.	x.	PROPN
ap-9228	299	19	93	93	NUM
ap-9228	299	20	m.	m.	NOUN
ap-9228	299	21	a.	a.	PROPN
ap-9228	299	22	hussein	hussein	PROPN
ap-9228	299	23	,	,	PUNCT
ap-9228	299	24	h.	h.	PROPN
ap-9228	299	25	k.	k.	PROPN
ap-9228	299	26	jassim	jassim	PROPN
ap-9228	299	27	,	,	PUNCT
ap-9228	299	28	a.	a.	PROPN
ap-9228	299	29	k.	k.	PROPN
ap-9228	299	30	jassim	jassim	PROPN
ap-9228	299	31	acta	acta	PROPN
ap-9228	299	32	polytechnica	polytechnica	PROPN
ap-9228	299	33	figure	figure	NOUN
ap-9228	299	34	2	2	NUM
ap-9228	299	35	.	.	PUNCT
ap-9228	300	1	the	the	DET
ap-9228	300	2	plot	plot	NOUN
ap-9228	300	3	of	of	ADP
ap-9228	300	4	approximate	approximate	ADJ
ap-9228	300	5	solution	solution	NOUN
ap-9228	300	6	,	,	PUNCT
ap-9228	300	7	exact	exact	ADJ
ap-9228	300	8	solution	solution	NOUN
ap-9228	300	9	,	,	PUNCT
ap-9228	300	10	and	and	CCONJ
ap-9228	300	11	absolute	absolute	ADJ
ap-9228	300	12	error	error	NOUN
ap-9228	300	13	of	of	ADP
ap-9228	300	14	equation	equation	NOUN
ap-9228	300	15	(	(	PUNCT
ap-9228	300	16	22	22	NUM
ap-9228	300	17	)	)	PUNCT
ap-9228	300	18	.	.	PUNCT
ap-9228	301	1	example	example	NOUN
ap-9228	302	1	3	3	X
ap-9228	302	2	.	.	PUNCT
ap-9228	302	3	suppose	suppose	VERB
ap-9228	302	4	that	that	SCONJ
ap-9228	302	5	the	the	DET
ap-9228	302	6	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	302	7	nonlinear	nonlinear	ADJ
ap-9228	302	8	viesk	viesk	NOUN
ap-9228	302	9	:	:	PUNCT
ap-9228	302	10	u(x	u(x	PROPN
ap-9228	302	11	)	)	PUNCT
ap-9228	302	12	=	=	PUNCT
ap-9228	302	13	x+	x+	NUM
ap-9228	302	14	∫	∫	PROPN
ap-9228	302	15	x	x	SYM
ap-9228	302	16	0	0	NUM
ap-9228	302	17	u2(t	u2(t	NOUN
ap-9228	302	18	)	)	PUNCT
ap-9228	302	19	dt	dt	NOUN
ap-9228	302	20	,	,	PUNCT
ap-9228	302	21	(	(	PUNCT
ap-9228	302	22	24	24	NUM
ap-9228	302	23	)	)	PUNCT
ap-9228	302	24	by	by	ADP
ap-9228	302	25	applying	apply	VERB
ap-9228	302	26	the	the	DET
ap-9228	302	27	algorithm	algorithm	NOUN
ap-9228	302	28	of	of	ADP
ap-9228	302	29	the	the	DET
ap-9228	302	30	technique	technique	NOUN
ap-9228	302	31	to	to	ADP
ap-9228	302	32	equation	equation	NOUN
ap-9228	302	33	(	(	PUNCT
ap-9228	302	34	24	24	NUM
ap-9228	302	35	)	)	PUNCT
ap-9228	302	36	,	,	PUNCT
ap-9228	302	37	we	we	PRON
ap-9228	302	38	obtain	obtain	VERB
ap-9228	302	39	:	:	PUNCT
ap-9228	302	40	∞∑	∞∑	NUM
ap-9228	302	41	i=0	i=0	PROPN
ap-9228	302	42	ui	ui	NOUN
ap-9228	303	1	=	=	PUNCT
ap-9228	304	1	∞∑	∞∑	NUM
ap-9228	304	2	i=0	i=0	PROPN
ap-9228	304	3	xi	xi	X
ap-9228	304	4	i	i	PROPN
ap-9228	304	5	!	!	PUNCT
ap-9228	305	1	d	d	VERB
ap-9228	305	2	i	i	PRON
ap-9228	305	3	x	x	X
ap-9228	305	4	x+	x+	NUM
ap-9228	305	5	∫	∫	PROPN
ap-9228	305	6	x	x	SYM
ap-9228	305	7	0	0	NUM
ap-9228	305	8	i−1∑	i−1∑	VERB
ap-9228	305	9	j=0	j=0	PROPN
ap-9228	305	10	uj	uj	PROPN
ap-9228	305	11	2	2	PROPN
ap-9228	305	12	dt	dt	PROPN
ap-9228	305	13			PROPN
ap-9228	305	14	x=0	x=0	PROPN
ap-9228	305	15	.	.	PUNCT
ap-9228	306	1	(	(	PUNCT
ap-9228	306	2	25	25	NUM
ap-9228	306	3	)	)	PUNCT
ap-9228	306	4	by	by	ADP
ap-9228	306	5	comparing	compare	VERB
ap-9228	306	6	both	both	DET
ap-9228	306	7	sides	side	NOUN
ap-9228	306	8	of	of	ADP
ap-9228	306	9	equation	equation	NOUN
ap-9228	306	10	(	(	PUNCT
ap-9228	306	11	25	25	NUM
ap-9228	306	12	)	)	PUNCT
ap-9228	306	13	,	,	PUNCT
ap-9228	306	14	we	we	PRON
ap-9228	306	15	obtain	obtain	VERB
ap-9228	306	16	:	:	PUNCT
ap-9228	306	17			NUM
ap-9228	306	18	u0	u0	ADJ
ap-9228	306	19	=	=	SYM
ap-9228	306	20	0	0	NUM
ap-9228	306	21	,	,	PUNCT
ap-9228	306	22	u1	u1	NOUN
ap-9228	306	23	=	=	SYM
ap-9228	306	24	xdx	xdx	PROPN
ap-9228	306	25	(	(	PUNCT
ap-9228	306	26	x+	x+	PROPN
ap-9228	306	27	∫	∫	PROPN
ap-9228	306	28	x	x	SYM
ap-9228	306	29	0	0	NUM
ap-9228	307	1	(	(	PUNCT
ap-9228	307	2	0)2	0)2	NUM
ap-9228	307	3	dt	dt	NOUN
ap-9228	307	4	)	)	PUNCT
ap-9228	308	1	x=0	x=0	PUNCT
ap-9228	309	1	=	=	SYM
ap-9228	309	2	x	x	PROPN
ap-9228	309	3	,	,	PUNCT
ap-9228	309	4	u2	u2	PROPN
ap-9228	309	5	=	=	SYM
ap-9228	309	6	x2	x2	PROPN
ap-9228	309	7	2	2	NUM
ap-9228	309	8	!	!	PUNCT
ap-9228	310	1	d	d	NOUN
ap-9228	310	2	2	2	NUM
ap-9228	310	3	x	x	SYM
ap-9228	310	4	(	(	PUNCT
ap-9228	310	5	x+	x+	NUM
ap-9228	310	6	∫	∫	PROPN
ap-9228	310	7	x	x	SYM
ap-9228	310	8	0	0	NUM
ap-9228	310	9	t2	t2	PROPN
ap-9228	310	10	dt	dt	NOUN
ap-9228	310	11	)	)	PUNCT
ap-9228	310	12	x=0	x=0	PUNCT
ap-9228	311	1	=	=	SYM
ap-9228	311	2	0	0	NUM
ap-9228	311	3	,	,	PUNCT
ap-9228	311	4	u3	u3	NOUN
ap-9228	311	5	=	=	SYM
ap-9228	311	6	x3	x3	NOUN
ap-9228	312	1	3	3	X
ap-9228	312	2	!	!	PUNCT
ap-9228	313	1	d	d	NOUN
ap-9228	313	2	3	3	NUM
ap-9228	313	3	x	x	SYM
ap-9228	313	4	(	(	PUNCT
ap-9228	313	5	x+	x+	NUM
ap-9228	313	6	∫	∫	PROPN
ap-9228	313	7	x	x	SYM
ap-9228	313	8	0	0	NUM
ap-9228	313	9	t2	t2	PROPN
ap-9228	313	10	dt	dt	NOUN
ap-9228	313	11	)	)	PUNCT
ap-9228	313	12	x=0	x=0	PUNCT
ap-9228	314	1	=	=	PUNCT
ap-9228	314	2	x3	x3	NOUN
ap-9228	314	3	3	3	NUM
ap-9228	314	4	,	,	PUNCT
ap-9228	314	5	...	...	PUNCT
ap-9228	314	6	thus	thus	ADV
ap-9228	314	7	,	,	PUNCT
ap-9228	314	8	the	the	DET
ap-9228	314	9	approximate	approximate	ADJ
ap-9228	314	10	solution	solution	NOUN
ap-9228	314	11	of	of	ADP
ap-9228	314	12	equation	equation	NOUN
ap-9228	314	13	(	(	PUNCT
ap-9228	314	14	24	24	NUM
ap-9228	314	15	)	)	PUNCT
ap-9228	314	16	is	be	AUX
ap-9228	314	17	:	:	PUNCT
ap-9228	314	18	ua(x	ua(x	X
ap-9228	314	19	)	)	PUNCT
ap-9228	315	1	=	=	SYM
ap-9228	315	2	x+	x+	PUNCT
ap-9228	316	1	x3	x3	ADJ
ap-9228	316	2	3	3	NUM
ap-9228	317	1	+	+	NUM
ap-9228	317	2	2x5	2x5	NUM
ap-9228	317	3	15	15	NUM
ap-9228	317	4	+	+	NUM
ap-9228	317	5	.	.	PUNCT
ap-9228	317	6	.	.	PUNCT
ap-9228	318	1	.	.	PUNCT
ap-9228	319	1	.	.	PUNCT
ap-9228	320	1	therefore	therefore	ADV
ap-9228	320	2	,	,	PUNCT
ap-9228	320	3	the	the	DET
ap-9228	320	4	exact	exact	ADJ
ap-9228	320	5	solution	solution	NOUN
ap-9228	320	6	of	of	ADP
ap-9228	320	7	equation	equation	NOUN
ap-9228	320	8	(	(	PUNCT
ap-9228	320	9	24	24	NUM
ap-9228	320	10	)	)	PUNCT
ap-9228	320	11	is	be	AUX
ap-9228	320	12	:	:	PUNCT
ap-9228	320	13	ue(x	ue(x	PUNCT
ap-9228	320	14	)	)	PUNCT
ap-9228	321	1	=	=	SYM
ap-9228	321	2	tan	tan	ADJ
ap-9228	321	3	x.	x.	NOUN
ap-9228	321	4	94	94	NUM
ap-9228	321	5	vol	vol	NOUN
ap-9228	321	6	.	.	PROPN
ap-9228	322	1	64	64	NUM
ap-9228	322	2	no	no	NOUN
ap-9228	322	3	.	.	PUNCT
ap-9228	323	1	2/2024	2/2024	NUM
ap-9228	323	2	an	an	DET
ap-9228	323	3	innovative	innovative	ADJ
ap-9228	323	4	iterative	iterative	NOUN
ap-9228	323	5	approach	approach	NOUN
ap-9228	323	6	to	to	ADP
ap-9228	323	7	solving	solve	VERB
ap-9228	323	8	volterra	volterra	NOUN
ap-9228	323	9	integral	integral	ADJ
ap-9228	323	10	.	.	PUNCT
ap-9228	323	11	.	.	PUNCT
ap-9228	323	12	.	.	PUNCT
ap-9228	324	1	absolute	absolute	ADJ
ap-9228	324	2	error	error	NOUN
ap-9228	324	3	x	x	X
ap-9228	324	4	ua	ua	PROPN
ap-9228	324	5	ue	ue	PROPN
ap-9228	324	6	hjm	hjm	PROPN
ap-9228	324	7	rdtm	rdtm	PROPN
ap-9228	324	8	psm	psm	PROPN
ap-9228	324	9	adm	adm	PROPN
ap-9228	324	10	0.1000	0.1000	NUM
ap-9228	324	11	0.1003	0.1003	NUM
ap-9228	324	12	0.1003	0.1003	NUM
ap-9228	324	13	0.0000	0.0000	NUM
ap-9228	324	14	0.0007	0.0007	NUM
ap-9228	324	15	0.0000	0.0000	NUM
ap-9228	324	16	0.0000	0.0000	NUM
ap-9228	324	17	0.2000	0.2000	NUM
ap-9228	324	18	0.2027	0.2027	NUM
ap-9228	324	19	0.2027	0.2027	NUM
ap-9228	324	20	0.0000	0.0000	NUM
ap-9228	324	21	0.0053	0.0053	NUM
ap-9228	324	22	0.0000	0.0000	NUM
ap-9228	324	23	0.0000	0.0000	NUM
ap-9228	324	24	0.3000	0.3000	NUM
ap-9228	324	25	0.3090	0.3090	NUM
ap-9228	324	26	0.3093	0.3093	NUM
ap-9228	324	27	0.0003	0.0003	NUM
ap-9228	324	28	0.0177	0.0177	NUM
ap-9228	324	29	0.0003	0.0003	NUM
ap-9228	324	30	0.0000	0.0000	NUM
ap-9228	324	31	0.4000	0.4000	NUM
ap-9228	324	32	0.4213	0.4213	NUM
ap-9228	324	33	0.4228	0.4228	NUM
ap-9228	324	34	0.0015	0.0015	NUM
ap-9228	324	35	0.0412	0.0412	NUM
ap-9228	324	36	0.0015	0.0015	NUM
ap-9228	324	37	0.0001	0.0001	NUM
ap-9228	324	38	0.5000	0.5000	NUM
ap-9228	324	39	0.5417	0.5417	NUM
ap-9228	324	40	0.5463	0.5463	NUM
ap-9228	324	41	0.0046	0.0046	NUM
ap-9228	324	42	0.0787	0.0787	NUM
ap-9228	324	43	0.0046	0.0046	NUM
ap-9228	324	44	0.0005	0.0005	NUM
ap-9228	324	45	0.6000	0.6000	NUM
ap-9228	324	46	0.6720	0.6720	NUM
ap-9228	324	47	0.6841	0.6841	NUM
ap-9228	324	48	0.0121	0.0121	NUM
ap-9228	324	49	0.1319	0.1319	NUM
ap-9228	324	50	0.0121	0.0121	NUM
ap-9228	324	51	0.0018	0.0018	NUM
ap-9228	324	52	0.7000	0.7000	NUM
ap-9228	324	53	0.8143	0.8143	NUM
ap-9228	324	54	0.8423	0.8423	NUM
ap-9228	324	55	0.0280	0.0280	NUM
ap-9228	324	56	0.2007	0.2007	NUM
ap-9228	324	57	0.0280	0.0280	NUM
ap-9228	324	58	0.0055	0.0055	NUM
ap-9228	324	59	0.8000	0.8000	NUM
ap-9228	324	60	0.9707	0.9707	NUM
ap-9228	324	61	1.0296	1.0296	NUM
ap-9228	324	62	0.0590	0.0590	NUM
ap-9228	324	63	0.2824	0.2824	NUM
ap-9228	324	64	0.0590	0.0590	NUM
ap-9228	324	65	0.0153	0.0153	NUM
ap-9228	324	66	0.9000	0.9000	NUM
ap-9228	324	67	1.1430	1.1430	NUM
ap-9228	324	68	1.2602	1.2602	NUM
ap-9228	324	69	0.1172	0.1172	NUM
ap-9228	324	70	0.3688	0.3688	NUM
ap-9228	324	71	0.1172	0.1172	NUM
ap-9228	324	72	0.0384	0.0384	NUM
ap-9228	324	73	1.0000	1.0000	NUM
ap-9228	324	74	1.3333	1.3333	NUM
ap-9228	324	75	1.5574	1.5574	NUM
ap-9228	324	76	0.2241	0.2241	NUM
ap-9228	324	77	0.4426	0.4426	NUM
ap-9228	324	78	0.2241	0.2241	NUM
ap-9228	324	79	0.0907	0.0907	NUM
ap-9228	324	80	table	table	NOUN
ap-9228	324	81	3	3	NUM
ap-9228	324	82	.	.	PUNCT
ap-9228	325	1	the	the	DET
ap-9228	325	2	values	value	NOUN
ap-9228	325	3	of	of	ADP
ap-9228	325	4	the	the	DET
ap-9228	325	5	approximate	approximate	ADJ
ap-9228	325	6	and	and	CCONJ
ap-9228	325	7	exact	exact	ADJ
ap-9228	325	8	solutions	solution	NOUN
ap-9228	325	9	of	of	ADP
ap-9228	325	10	equation	equation	NOUN
ap-9228	325	11	(	(	PUNCT
ap-9228	325	12	24	24	NUM
ap-9228	325	13	)	)	PUNCT
ap-9228	325	14	for	for	ADP
ap-9228	325	15	different	different	ADJ
ap-9228	325	16	values	value	NOUN
ap-9228	325	17	of	of	ADP
ap-9228	325	18	x.	x.	NOUN
ap-9228	325	19	figure	figure	NOUN
ap-9228	325	20	3	3	NUM
ap-9228	325	21	.	.	PUNCT
ap-9228	326	1	the	the	DET
ap-9228	326	2	plot	plot	NOUN
ap-9228	326	3	of	of	ADP
ap-9228	326	4	approximate	approximate	ADJ
ap-9228	326	5	solution	solution	NOUN
ap-9228	326	6	,	,	PUNCT
ap-9228	326	7	exact	exact	ADJ
ap-9228	326	8	solution	solution	NOUN
ap-9228	326	9	,	,	PUNCT
ap-9228	326	10	and	and	CCONJ
ap-9228	326	11	absolute	absolute	ADJ
ap-9228	326	12	error	error	NOUN
ap-9228	326	13	of	of	ADP
ap-9228	326	14	equation	equation	NOUN
ap-9228	326	15	(	(	PUNCT
ap-9228	326	16	24	24	NUM
ap-9228	326	17	)	)	PUNCT
ap-9228	326	18	.	.	PUNCT
ap-9228	327	1	example	example	NOUN
ap-9228	328	1	4	4	NUM
ap-9228	328	2	.	.	PUNCT
ap-9228	328	3	suppose	suppose	VERB
ap-9228	328	4	that	that	SCONJ
ap-9228	328	5	the	the	DET
ap-9228	328	6	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	328	7	nonlinear	nonlinear	ADJ
ap-9228	328	8	viesk	viesk	NOUN
ap-9228	328	9	:	:	PUNCT
ap-9228	328	10	u(x	u(x	PROPN
ap-9228	328	11	)	)	PUNCT
ap-9228	328	12	=	=	PUNCT
ap-9228	328	13	ex	ex	X
ap-9228	329	1	+	+	NOUN
ap-9228	329	2	1	1	NUM
ap-9228	329	3	3x(1	3x(1	NUM
ap-9228	329	4	−	−	NOUN
ap-9228	329	5	e3x	e3x	PROPN
ap-9228	329	6	)	)	PUNCT
ap-9228	329	7	+	+	CCONJ
ap-9228	329	8	∫	∫	PROPN
ap-9228	329	9	x	x	SYM
ap-9228	329	10	0	0	NUM
ap-9228	329	11	xu3(t	xu3(t	PROPN
ap-9228	329	12	)	)	PUNCT
ap-9228	329	13	dt	dt	X
ap-9228	329	14	,	,	PUNCT
ap-9228	329	15	(	(	PUNCT
ap-9228	329	16	26	26	NUM
ap-9228	329	17	)	)	PUNCT
ap-9228	329	18	by	by	ADP
ap-9228	329	19	applying	apply	VERB
ap-9228	329	20	the	the	DET
ap-9228	329	21	algorithm	algorithm	NOUN
ap-9228	329	22	of	of	ADP
ap-9228	329	23	the	the	DET
ap-9228	329	24	new	new	ADJ
ap-9228	329	25	technique	technique	NOUN
ap-9228	329	26	to	to	ADP
ap-9228	329	27	equation	equation	NOUN
ap-9228	329	28	(	(	PUNCT
ap-9228	329	29	26	26	NUM
ap-9228	329	30	)	)	PUNCT
ap-9228	329	31	,	,	PUNCT
ap-9228	329	32	we	we	PRON
ap-9228	329	33	obtain	obtain	VERB
ap-9228	329	34	:	:	PUNCT
ap-9228	329	35	∞∑	∞∑	NUM
ap-9228	329	36	i=0	i=0	PROPN
ap-9228	329	37	ui	ui	NOUN
ap-9228	330	1	=	=	PUNCT
ap-9228	331	1	∞∑	∞∑	NUM
ap-9228	331	2	i=0	i=0	PROPN
ap-9228	331	3	xi	xi	X
ap-9228	331	4	i	i	PROPN
ap-9228	331	5	!	!	PUNCT
ap-9228	332	1	d	d	VERB
ap-9228	332	2	i	i	PRON
ap-9228	332	3	x	x	VERB
ap-9228	332	4	ex	ex	VERB
ap-9228	332	5	+	+	CCONJ
ap-9228	332	6	1	1	NUM
ap-9228	332	7	3x(1	3x(1	NOUN
ap-9228	332	8	−	−	NOUN
ap-9228	332	9	e3x	e3x	PROPN
ap-9228	332	10	)	)	PUNCT
ap-9228	333	1	+	+	CCONJ
ap-9228	333	2	∫	∫	PROPN
ap-9228	333	3	x	x	SYM
ap-9228	333	4	0	0	NUM
ap-9228	333	5	x	x	X
ap-9228	333	6	i−1∑	i−1∑	VERB
ap-9228	333	7	j=0	j=0	PROPN
ap-9228	333	8	uj	uj	PROPN
ap-9228	333	9	3	3	PRON
ap-9228	333	10	dt	dt	PROPN
ap-9228	333	11			PROPN
ap-9228	333	12	x=0	x=0	PROPN
ap-9228	333	13	.	.	PUNCT
ap-9228	334	1	(	(	PUNCT
ap-9228	334	2	27	27	NUM
ap-9228	334	3	)	)	PUNCT
ap-9228	334	4	by	by	ADP
ap-9228	334	5	comparing	compare	VERB
ap-9228	334	6	both	both	DET
ap-9228	334	7	sides	side	NOUN
ap-9228	334	8	of	of	ADP
ap-9228	334	9	equation	equation	NOUN
ap-9228	334	10	(	(	PUNCT
ap-9228	334	11	27	27	NUM
ap-9228	334	12	)	)	PUNCT
ap-9228	334	13	,	,	PUNCT
ap-9228	334	14	we	we	PRON
ap-9228	334	15	obtain:	obtain:	VERB
ap-9228	334	16	u0	u0	NOUN
ap-9228	334	17	=	=	PROPN
ap-9228	334	18	1	1	NUM
ap-9228	334	19	,	,	PUNCT
ap-9228	334	20	u1	u1	NOUN
ap-9228	334	21	=	=	SYM
ap-9228	334	22	xdx	xdx	PROPN
ap-9228	334	23	(	(	PUNCT
ap-9228	334	24	ex	ex	X
ap-9228	335	1	+	+	NOUN
ap-9228	335	2	1	1	NUM
ap-9228	335	3	3x(1	3x(1	NUM
ap-9228	335	4	−	−	NOUN
ap-9228	335	5	e3x	e3x	PROPN
ap-9228	335	6	)	)	PUNCT
ap-9228	335	7	+	+	CCONJ
ap-9228	335	8	∫	∫	PROPN
ap-9228	335	9	x	x	SYM
ap-9228	335	10	0	0	PUNCT
ap-9228	335	11	x	x	SYM
ap-9228	335	12	dt	dt	NOUN
ap-9228	335	13	)	)	PUNCT
ap-9228	335	14	x=0	x=0	PUNCT
ap-9228	336	1	=	=	SYM
ap-9228	336	2	x	x	PROPN
ap-9228	336	3	,	,	PUNCT
ap-9228	336	4	u2	u2	PROPN
ap-9228	336	5	=	=	SYM
ap-9228	336	6	x2	x2	PROPN
ap-9228	336	7	2	2	NUM
ap-9228	336	8	!	!	PUNCT
ap-9228	337	1	d	d	NOUN
ap-9228	337	2	2	2	NUM
ap-9228	337	3	x	x	X
ap-9228	337	4	(	(	PUNCT
ap-9228	337	5	ex	ex	X
ap-9228	337	6	+	+	NOUN
ap-9228	337	7	1	1	NUM
ap-9228	337	8	3x(1	3x(1	NUM
ap-9228	337	9	−	−	NOUN
ap-9228	337	10	e3x	e3x	PROPN
ap-9228	337	11	)	)	PUNCT
ap-9228	337	12	+	+	CCONJ
ap-9228	337	13	∫	∫	PROPN
ap-9228	337	14	x	x	SYM
ap-9228	337	15	0	0	PUNCT
ap-9228	338	1	x(1	x(1	PROPN
ap-9228	338	2	+	+	CCONJ
ap-9228	338	3	t)3	t)3	NOUN
ap-9228	338	4	dt	dt	NOUN
ap-9228	338	5	)	)	PUNCT
ap-9228	338	6	x=0	x=0	PUNCT
ap-9228	339	1	=	=	SYM
ap-9228	339	2	x2	x2	PROPN
ap-9228	339	3	2	2	X
ap-9228	339	4	!	!	NUM
ap-9228	339	5	,	,	PUNCT
ap-9228	339	6	u3	u3	PROPN
ap-9228	339	7	=	=	SYM
ap-9228	339	8	x3	x3	NOUN
ap-9228	339	9	3	3	X
ap-9228	339	10	!	!	PUNCT
ap-9228	340	1	d	d	NOUN
ap-9228	340	2	3	3	NUM
ap-9228	340	3	x	x	SYM
ap-9228	340	4	(	(	PUNCT
ap-9228	340	5	ex	ex	X
ap-9228	340	6	+	+	NOUN
ap-9228	340	7	1	1	NUM
ap-9228	340	8	3x(1	3x(1	NUM
ap-9228	340	9	−	−	NOUN
ap-9228	340	10	e3x	e3x	PROPN
ap-9228	340	11	)	)	PUNCT
ap-9228	340	12	+	+	CCONJ
ap-9228	340	13	∫	∫	PROPN
ap-9228	340	14	x	x	SYM
ap-9228	340	15	0	0	PUNCT
ap-9228	341	1	x(1	x(1	PROPN
ap-9228	341	2	+	+	CCONJ
ap-9228	341	3	t)3	t)3	NOUN
ap-9228	341	4	dt	dt	NOUN
ap-9228	341	5	)	)	PUNCT
ap-9228	341	6	x=0	x=0	PUNCT
ap-9228	342	1	=	=	SYM
ap-9228	342	2	x3	x3	NOUN
ap-9228	342	3	3	3	NUM
ap-9228	342	4	!	!	NUM
ap-9228	342	5	,	,	PUNCT
ap-9228	342	6	...	...	PUNCT
ap-9228	343	1	ui+1	ui+1	X
ap-9228	343	2	=	=	SYM
ap-9228	343	3	xi+1	xi+1	PROPN
ap-9228	343	4	(	(	PUNCT
ap-9228	343	5	i+	i+	NOUN
ap-9228	343	6	1)!d	1)!d	NUM
ap-9228	343	7	i+1	i+1	NOUN
ap-9228	343	8	x	x	PUNCT
ap-9228	343	9	ex	ex	PROPN
ap-9228	343	10	+	+	CCONJ
ap-9228	343	11	1	1	NUM
ap-9228	343	12	3x(1	3x(1	NOUN
ap-9228	343	13	−	−	NOUN
ap-9228	343	14	e3x	e3x	PROPN
ap-9228	343	15	)	)	PUNCT
ap-9228	343	16	+	+	CCONJ
ap-9228	343	17	∫	∫	PROPN
ap-9228	343	18	x	x	SYM
ap-9228	343	19	0	0	NUM
ap-9228	343	20	x	x	SYM
ap-9228	343	21			PROPN
ap-9228	343	22	i∑	i∑	NUM
ap-9228	343	23	j=0	j=0	PROPN
ap-9228	343	24	uj	uj	PROPN
ap-9228	343	25	3	3	PRON
ap-9228	343	26	dt	dt	PROPN
ap-9228	343	27			PROPN
ap-9228	343	28	x=0	x=0	PROPN
ap-9228	343	29	=	=	SYM
ap-9228	343	30	xi+1	xi+1	PROPN
ap-9228	343	31	(	(	PUNCT
ap-9228	343	32	i+	i+	NOUN
ap-9228	343	33	1	1	NUM
ap-9228	343	34	)	)	PUNCT
ap-9228	343	35	!	!	PUNCT
ap-9228	343	36	.	.	PUNCT
ap-9228	344	1	95	95	NUM
ap-9228	344	2	m.	m.	NOUN
ap-9228	344	3	a.	a.	PROPN
ap-9228	344	4	hussein	hussein	PROPN
ap-9228	344	5	,	,	PUNCT
ap-9228	344	6	h.	h.	PROPN
ap-9228	344	7	k.	k.	PROPN
ap-9228	344	8	jassim	jassim	PROPN
ap-9228	344	9	,	,	PUNCT
ap-9228	344	10	a.	a.	PROPN
ap-9228	344	11	k.	k.	PROPN
ap-9228	344	12	jassim	jassim	PROPN
ap-9228	344	13	acta	acta	PROPN
ap-9228	344	14	polytechnica	polytechnica	PROPN
ap-9228	344	15	thus	thus	ADV
ap-9228	344	16	,	,	PUNCT
ap-9228	344	17	the	the	DET
ap-9228	344	18	approximate	approximate	ADJ
ap-9228	344	19	solution	solution	NOUN
ap-9228	344	20	of	of	ADP
ap-9228	344	21	equation	equation	NOUN
ap-9228	344	22	(	(	PUNCT
ap-9228	344	23	26	26	NUM
ap-9228	344	24	)	)	PUNCT
ap-9228	344	25	is	be	AUX
ap-9228	344	26	:	:	PUNCT
ap-9228	344	27	ua(x	ua(x	X
ap-9228	344	28	,	,	PUNCT
ap-9228	344	29	t	t	PROPN
ap-9228	344	30	)	)	PUNCT
ap-9228	344	31	=	=	SYM
ap-9228	344	32	1	1	NUM
ap-9228	345	1	+	+	CCONJ
ap-9228	345	2	x+	x+	ADJ
ap-9228	345	3	x2	x2	NOUN
ap-9228	345	4	2	2	NUM
ap-9228	345	5	!	!	PUNCT
ap-9228	346	1	+	+	CCONJ
ap-9228	346	2	.	.	PUNCT
ap-9228	346	3	.	.	PUNCT
ap-9228	346	4	.	.	PUNCT
ap-9228	347	1	=	=	PUNCT
ap-9228	348	1	∞∑	∞∑	NUM
ap-9228	348	2	i=0	i=0	PROPN
ap-9228	348	3	xi	xi	X
ap-9228	348	4	i	i	PROPN
ap-9228	348	5	!	!	PUNCT
ap-9228	348	6	.	.	PUNCT
ap-9228	349	1	therefore	therefore	ADV
ap-9228	349	2	,	,	PUNCT
ap-9228	349	3	the	the	DET
ap-9228	349	4	exact	exact	ADJ
ap-9228	349	5	solution	solution	NOUN
ap-9228	349	6	of	of	ADP
ap-9228	349	7	equation	equation	NOUN
ap-9228	349	8	(	(	PUNCT
ap-9228	349	9	26	26	NUM
ap-9228	349	10	)	)	PUNCT
ap-9228	349	11	is	be	AUX
ap-9228	349	12	:	:	PUNCT
ap-9228	349	13	ue(x	ue(x	NUM
ap-9228	349	14	,	,	PUNCT
ap-9228	349	15	t	t	PROPN
ap-9228	349	16	)	)	PUNCT
ap-9228	349	17	=	=	SYM
ap-9228	350	1	ex	ex	X
ap-9228	350	2	.	.	NOUN
ap-9228	350	3	absolute	absolute	ADJ
ap-9228	350	4	error	error	NOUN
ap-9228	350	5	x	x	X
ap-9228	350	6	ua	ua	PROPN
ap-9228	350	7	ue	ue	PROPN
ap-9228	350	8	hjm	hjm	PROPN
ap-9228	350	9	rdtm	rdtm	PROPN
ap-9228	350	10	psm	psm	PROPN
ap-9228	350	11	adm	adm	PROPN
ap-9228	350	12	0.1000	0.1000	NUM
ap-9228	351	1	1.1052	1.1052	NUM
ap-9228	352	1	1.1052	1.1052	NUM
ap-9228	352	2	0.0000	0.0000	NUM
ap-9228	352	3	0.0000	0.0000	NUM
ap-9228	352	4	0.0000	0.0000	NUM
ap-9228	352	5	0.0000	0.0000	NUM
ap-9228	352	6	0.2000	0.2000	NUM
ap-9228	352	7	1.2213	1.2213	NUM
ap-9228	352	8	1.2214	1.2214	NUM
ap-9228	352	9	0.0001	0.0001	NUM
ap-9228	352	10	0.0001	0.0001	NUM
ap-9228	352	11	0.0001	0.0001	NUM
ap-9228	352	12	0.0001	0.0001	NUM
ap-9228	352	13	0.3000	0.3000	NUM
ap-9228	352	14	1.3495	1.3495	NUM
ap-9228	352	15	1.3499	1.3499	NUM
ap-9228	352	16	0.0004	0.0004	NUM
ap-9228	352	17	0.0004	0.0004	NUM
ap-9228	352	18	0.0004	0.0004	NUM
ap-9228	352	19	0.0004	0.0004	NUM
ap-9228	352	20	0.4000	0.4000	NUM
ap-9228	352	21	1.4907	1.4907	NUM
ap-9228	352	22	1.4918	1.4918	NUM
ap-9228	352	23	0.0012	0.0012	NUM
ap-9228	352	24	0.0012	0.0012	NUM
ap-9228	352	25	0.0012	0.0012	NUM
ap-9228	352	26	0.0012	0.0012	NUM
ap-9228	352	27	0.5000	0.5000	NUM
ap-9228	352	28	1.6458	1.6458	NUM
ap-9228	352	29	1.6487	1.6487	NUM
ap-9228	352	30	0.0029	0.0029	NUM
ap-9228	352	31	0.0029	0.0029	NUM
ap-9228	352	32	0.0029	0.0029	NUM
ap-9228	352	33	0.0029	0.0029	NUM
ap-9228	352	34	0.6000	0.6000	NUM
ap-9228	352	35	1.8160	1.8160	NUM
ap-9228	352	36	1.8221	1.8221	NUM
ap-9228	352	37	0.0061	0.0061	NUM
ap-9228	352	38	0.0061	0.0061	NUM
ap-9228	352	39	0.0061	0.0061	NUM
ap-9228	352	40	0.0061	0.0061	NUM
ap-9228	352	41	0.7000	0.7000	NUM
ap-9228	352	42	2.0022	2.0022	NUM
ap-9228	352	43	2.0138	2.0138	NUM
ap-9228	352	44	0.0116	0.0116	NUM
ap-9228	352	45	0.0116	0.0116	NUM
ap-9228	352	46	0.0116	0.0116	NUM
ap-9228	352	47	0.0116	0.0116	NUM
ap-9228	352	48	0.8000	0.8000	NUM
ap-9228	352	49	2.2053	2.2053	NUM
ap-9228	352	50	2.2255	2.2255	NUM
ap-9228	353	1	0.0202	0.0202	NUM
ap-9228	353	2	0.0202	0.0202	NUM
ap-9228	353	3	0.0202	0.0202	NUM
ap-9228	354	1	0.0202	0.0202	NUM
ap-9228	354	2	0.9000	0.9000	NUM
ap-9228	354	3	2.4265	2.4265	NUM
ap-9228	354	4	2.4596	2.4596	NUM
ap-9228	354	5	0.0331	0.0331	NUM
ap-9228	354	6	0.0331	0.0331	NUM
ap-9228	354	7	0.0331	0.0331	NUM
ap-9228	354	8	0.0331	0.0331	NUM
ap-9228	354	9	1.0000	1.0000	NUM
ap-9228	354	10	2.6667	2.6667	NUM
ap-9228	354	11	2.7183	2.7183	NUM
ap-9228	354	12	0.0516	0.0516	NUM
ap-9228	354	13	0.0516	0.0516	NUM
ap-9228	354	14	0.0516	0.0516	NUM
ap-9228	354	15	0.0516	0.0516	NUM
ap-9228	354	16	table	table	NOUN
ap-9228	354	17	4	4	NUM
ap-9228	354	18	.	.	PUNCT
ap-9228	355	1	the	the	DET
ap-9228	355	2	values	value	NOUN
ap-9228	355	3	of	of	ADP
ap-9228	355	4	the	the	DET
ap-9228	355	5	approximate	approximate	ADJ
ap-9228	355	6	and	and	CCONJ
ap-9228	355	7	exact	exact	ADJ
ap-9228	355	8	solutions	solution	NOUN
ap-9228	355	9	of	of	ADP
ap-9228	355	10	equation	equation	NOUN
ap-9228	355	11	(	(	PUNCT
ap-9228	355	12	26	26	NUM
ap-9228	355	13	)	)	PUNCT
ap-9228	355	14	for	for	ADP
ap-9228	355	15	different	different	ADJ
ap-9228	355	16	values	value	NOUN
ap-9228	355	17	of	of	ADP
ap-9228	355	18	x.	x.	NOUN
ap-9228	355	19	figure	figure	NOUN
ap-9228	355	20	4	4	NUM
ap-9228	355	21	.	.	PUNCT
ap-9228	356	1	the	the	DET
ap-9228	356	2	plot	plot	NOUN
ap-9228	356	3	of	of	ADP
ap-9228	356	4	approximate	approximate	ADJ
ap-9228	356	5	solution	solution	NOUN
ap-9228	356	6	,	,	PUNCT
ap-9228	356	7	exact	exact	ADJ
ap-9228	356	8	solution	solution	NOUN
ap-9228	356	9	,	,	PUNCT
ap-9228	356	10	and	and	CCONJ
ap-9228	356	11	absolute	absolute	ADJ
ap-9228	356	12	error	error	NOUN
ap-9228	356	13	of	of	ADP
ap-9228	356	14	equation	equation	NOUN
ap-9228	356	15	(	(	PUNCT
ap-9228	356	16	26	26	NUM
ap-9228	356	17	)	)	PUNCT
ap-9228	356	18	.	.	PUNCT
ap-9228	357	1	example	example	NOUN
ap-9228	358	1	5	5	NUM
ap-9228	358	2	.	.	PUNCT
ap-9228	358	3	consider	consider	VERB
ap-9228	358	4	the	the	DET
ap-9228	358	5	linear	linear	ADJ
ap-9228	358	6	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	358	7	system	system	NOUN
ap-9228	358	8	volterra	volterra	NOUN
ap-9228	358	9	integral	integral	ADJ
ap-9228	358	10	equation	equation	NOUN
ap-9228	358	11	of	of	ADP
ap-9228	358	12	the	the	DET
ap-9228	358	13	second	second	ADJ
ap-9228	358	14	kind:	kind:	PROPN
ap-9228	358	15	u(x	u(x	PROPN
ap-9228	358	16	)	)	PUNCT
ap-9228	359	1	=	=	SYM
ap-9228	359	2	ex	ex	PRON
ap-9228	359	3	−	−	NOUN
ap-9228	360	1	2x+	2x+	NUM
ap-9228	360	2	∫	∫	NOUN
ap-9228	360	3	x	x	SYM
ap-9228	360	4	0	0	NUM
ap-9228	360	5	e−tu(t	e−tu(t	PROPN
ap-9228	360	6	)	)	PUNCT
ap-9228	360	7	+	+	CCONJ
ap-9228	360	8	etv(t	etv(t	PROPN
ap-9228	360	9	)	)	PUNCT
ap-9228	360	10	dt	dt	NOUN
ap-9228	360	11	,	,	PUNCT
ap-9228	360	12	v(x	v(x	PROPN
ap-9228	360	13	)	)	PUNCT
ap-9228	360	14	=	=	SYM
ap-9228	361	1	e−x	e−x	PROPN
ap-9228	361	2	−	−	PROPN
ap-9228	361	3	sinh	sinh	NOUN
ap-9228	361	4	2x+	2x+	NUM
ap-9228	361	5	∫	∫	NOUN
ap-9228	361	6	x	x	SYM
ap-9228	361	7	0	0	NUM
ap-9228	361	8	etu(t	etu(t	PROPN
ap-9228	361	9	)	)	PUNCT
ap-9228	361	10	+	+	NUM
ap-9228	361	11	e−tv(t	e−tv(t	NOUN
ap-9228	361	12	)	)	PUNCT
ap-9228	361	13	dt	dt	NOUN
ap-9228	361	14	,	,	PUNCT
ap-9228	361	15	(	(	PUNCT
ap-9228	361	16	28	28	NUM
ap-9228	361	17	)	)	PUNCT
ap-9228	361	18	by	by	ADP
ap-9228	361	19	applying	apply	VERB
ap-9228	361	20	the	the	DET
ap-9228	361	21	algorithm	algorithm	NOUN
ap-9228	361	22	of	of	ADP
ap-9228	361	23	the	the	DET
ap-9228	361	24	new	new	ADJ
ap-9228	361	25	technique	technique	NOUN
ap-9228	361	26	to	to	ADP
ap-9228	361	27	equation	equation	NOUN
ap-9228	361	28	(	(	PUNCT
ap-9228	361	29	28	28	NUM
ap-9228	361	30	)	)	PUNCT
ap-9228	361	31	,	,	PUNCT
ap-9228	361	32	we	we	PRON
ap-9228	361	33	obtain:	obtain:	VERB
ap-9228	361	34	∞∑	∞∑	NUM
ap-9228	361	35	i=0	i=0	ADJ
ap-9228	361	36	u(x	u(x	PROPN
ap-9228	361	37	)	)	PUNCT
ap-9228	361	38	=	=	SYM
ap-9228	362	1	ex	ex	PRON
ap-9228	363	1	−	−	NOUN
ap-9228	363	2	2x+	2x+	NUM
ap-9228	363	3	∫	∫	NOUN
ap-9228	363	4	x	x	SYM
ap-9228	363	5	0	0	NUM
ap-9228	363	6	e−t	e−t	NOUN
ap-9228	363	7	∞∑	∞∑	NUM
ap-9228	363	8	i=0	i=0	ADJ
ap-9228	363	9	u(t	u(t	NOUN
ap-9228	363	10	)	)	PUNCT
ap-9228	364	1	+	+	NUM
ap-9228	364	2	et	et	NOUN
ap-9228	364	3	∞∑	∞∑	PRON
ap-9228	364	4	i=0	i=0	PROPN
ap-9228	364	5	v(t	v(t	NUM
ap-9228	364	6	)	)	PUNCT
ap-9228	364	7	dt	dt	PROPN
ap-9228	364	8	,	,	PUNCT
ap-9228	364	9	∞∑	∞∑	ADJ
ap-9228	364	10	i=0	i=0	ADJ
ap-9228	364	11	v(x	v(x	PROPN
ap-9228	364	12	)	)	PUNCT
ap-9228	364	13	=	=	SYM
ap-9228	365	1	e−x	e−x	PROPN
ap-9228	365	2	−	−	PROPN
ap-9228	365	3	sinh	sinh	NOUN
ap-9228	365	4	2x+	2x+	NUM
ap-9228	365	5	∫	∫	NOUN
ap-9228	365	6	x	x	SYM
ap-9228	365	7	0	0	NUM
ap-9228	365	8	et	et	NOUN
ap-9228	365	9	∞∑	∞∑	PRON
ap-9228	365	10	i=0	i=0	ADJ
ap-9228	365	11	u(t	u(t	NOUN
ap-9228	365	12	)	)	PUNCT
ap-9228	366	1	+	+	NUM
ap-9228	366	2	e−t	e−t	NOUN
ap-9228	366	3	∞∑	∞∑	PRON
ap-9228	366	4	i=0	i=0	PROPN
ap-9228	366	5	v(t	v(t	NUM
ap-9228	366	6	)	)	PUNCT
ap-9228	366	7	dt	dt	PROPN
ap-9228	366	8	.	.	PUNCT
ap-9228	367	1	(	(	PUNCT
ap-9228	367	2	29	29	NUM
ap-9228	367	3	)	)	SYM
ap-9228	367	4	96	96	NUM
ap-9228	367	5	vol	vol	NOUN
ap-9228	367	6	.	.	PROPN
ap-9228	368	1	64	64	NUM
ap-9228	368	2	no	no	NOUN
ap-9228	368	3	.	.	PUNCT
ap-9228	369	1	2/2024	2/2024	NUM
ap-9228	369	2	an	an	DET
ap-9228	369	3	innovative	innovative	ADJ
ap-9228	369	4	iterative	iterative	NOUN
ap-9228	369	5	approach	approach	NOUN
ap-9228	369	6	to	to	ADP
ap-9228	369	7	solving	solve	VERB
ap-9228	369	8	volterra	volterra	NOUN
ap-9228	369	9	integral	integral	ADJ
ap-9228	369	10	.	.	PUNCT
ap-9228	369	11	.	.	PUNCT
ap-9228	369	12	.	.	PUNCT
ap-9228	370	1	by	by	ADP
ap-9228	370	2	comparing	compare	VERB
ap-9228	370	3	both	both	DET
ap-9228	370	4	sides	side	NOUN
ap-9228	370	5	of	of	ADP
ap-9228	370	6	equation	equation	NOUN
ap-9228	370	7	(	(	PUNCT
ap-9228	370	8	29	29	NUM
ap-9228	370	9	)	)	PUNCT
ap-9228	370	10	,	,	PUNCT
ap-9228	370	11	we	we	PRON
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ap-9228	371	9	+	+	CCONJ
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ap-9228	371	11	dt	dt	NOUN
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ap-9228	372	9	−	−	PROPN
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ap-9228	373	3	,	,	PUNCT
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ap-9228	375	1	+	+	NUM
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ap-9228	376	5	,	,	PUNCT
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ap-9228	376	7	=	=	SYM
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ap-9228	377	7	e−x	e−x	PROPN
ap-9228	377	8	−	−	PROPN
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ap-9228	377	13	0	0	NUM
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ap-9228	383	5	,	,	PUNCT
ap-9228	383	6	v3	v3	PROPN
ap-9228	383	7	=	=	PUNCT
ap-9228	383	8	x3	x3	ADJ
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ap-9228	385	5	e−x	e−x	PROPN
ap-9228	385	6	−	−	PROPN
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ap-9228	385	8	2x+	2x+	NUM
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ap-9228	385	10	x	x	SYM
ap-9228	385	11	0	0	NUM
ap-9228	385	12	et	et	NOUN
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ap-9228	385	14	1	1	NUM
ap-9228	385	15	+	+	NUM
ap-9228	385	16	t+	t+	NOUN
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ap-9228	386	3	(	(	PUNCT
ap-9228	386	4	1	1	NUM
ap-9228	386	5	−	−	NOUN
ap-9228	386	6	t+	t+	PUNCT
ap-9228	386	7	t2	t2	NOUN
ap-9228	386	8	2	2	NUM
ap-9228	386	9	)	)	PUNCT
ap-9228	386	10	dt	dt	NOUN
ap-9228	386	11	)	)	PUNCT
ap-9228	386	12	x=0	x=0	PUNCT
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ap-9228	387	3	3	3	NUM
ap-9228	387	4	!	!	NUM
ap-9228	387	5	,	,	PUNCT
ap-9228	387	6	...	...	PUNCT
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ap-9228	388	2	=	=	SYM
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ap-9228	388	4	(	(	PUNCT
ap-9228	388	5	i+	i+	PROPN
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ap-9228	388	11	2x+	2x+	NUM
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ap-9228	388	17	i∑	i∑	NUM
ap-9228	388	18	j=0	j=0	PROPN
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ap-9228	388	20	+	+	PROPN
ap-9228	388	21	et	et	PROPN
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ap-9228	388	26			PROPN
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ap-9228	388	37	,	,	PUNCT
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ap-9228	389	5	i+	i+	NOUN
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ap-9228	389	7	i+1	i+1	NOUN
ap-9228	390	1	x	x	SYM
ap-9228	391	1	e−x	e−x	ADV
ap-9228	392	1	−	−	PROPN
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ap-9228	393	2	2x+	2x+	NUM
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ap-9228	393	5	0	0	PUNCT
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ap-9228	394	2	i∑	i∑	NUM
ap-9228	394	3	j=0	j=0	PROPN
ap-9228	394	4	uj	uj	VERB
ap-9228	394	5	+	+	PROPN
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ap-9228	394	7			PROPN
ap-9228	394	8	i∑	i∑	PROPN
ap-9228	394	9	j=0	j=0	PROPN
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ap-9228	394	12	dt	dt	X
ap-9228	394	13			PROPN
ap-9228	394	14	x=0	x=0	PUNCT
ap-9228	395	1	=	=	SYM
ap-9228	395	2	(	(	PUNCT
ap-9228	395	3	−1)i	−1)i	X
ap-9228	395	4	xi+1	xi+1	PROPN
ap-9228	395	5	(	(	PUNCT
ap-9228	395	6	i+	i+	NOUN
ap-9228	395	7	1	1	NUM
ap-9228	395	8	)	)	PUNCT
ap-9228	395	9	!	!	PUNCT
ap-9228	395	10	.	.	PUNCT
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ap-9228	396	2	,	,	PUNCT
ap-9228	396	3	the	the	DET
ap-9228	396	4	approximate	approximate	ADJ
ap-9228	396	5	solution	solution	NOUN
ap-9228	396	6	of	of	ADP
ap-9228	396	7	equation	equation	NOUN
ap-9228	396	8	(	(	PUNCT
ap-9228	396	9	28	28	NUM
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ap-9228	396	12	ua(x	ua(x	X
ap-9228	396	13	,	,	PUNCT
ap-9228	396	14	t	t	PROPN
ap-9228	396	15	)	)	PUNCT
ap-9228	396	16	=	=	SYM
ap-9228	396	17	1	1	NUM
ap-9228	396	18	+	+	CCONJ
ap-9228	396	19	x+	x+	ADJ
ap-9228	396	20	x2	x2	NOUN
ap-9228	396	21	2	2	NUM
ap-9228	396	22	!	!	PUNCT
ap-9228	397	1	+	+	CCONJ
ap-9228	397	2	.	.	PUNCT
ap-9228	397	3	.	.	PUNCT
ap-9228	397	4	.	.	PUNCT
ap-9228	398	1	=	=	PUNCT
ap-9228	399	1	∞∑	∞∑	NUM
ap-9228	399	2	i=0	i=0	PROPN
ap-9228	399	3	xi	xi	X
ap-9228	399	4	i	i	PROPN
ap-9228	399	5	!	!	PUNCT
ap-9228	399	6	,	,	PUNCT
ap-9228	399	7	va(x	va(x	NOUN
ap-9228	399	8	,	,	PUNCT
ap-9228	399	9	t	t	PROPN
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ap-9228	399	13	−	−	NOUN
ap-9228	399	14	x+	x+	PUNCT
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ap-9228	399	17	!	!	PUNCT
ap-9228	399	18	+	+	CCONJ
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ap-9228	399	20	.	.	PUNCT
ap-9228	399	21	.	.	PUNCT
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ap-9228	401	1	∞∑	∞∑	NUM
ap-9228	401	2	i=0	i=0	PROPN
ap-9228	401	3	(	(	PUNCT
ap-9228	401	4	−1)ix	−1)ix	PROPN
ap-9228	401	5	i	i	PRON
ap-9228	401	6	i	i	PROPN
ap-9228	401	7	!	!	PUNCT
ap-9228	401	8	.	.	PUNCT
ap-9228	402	1	therefore	therefore	ADV
ap-9228	402	2	,	,	PUNCT
ap-9228	402	3	the	the	DET
ap-9228	402	4	exact	exact	ADJ
ap-9228	402	5	solution	solution	NOUN
ap-9228	402	6	of	of	ADP
ap-9228	402	7	equation	equation	NOUN
ap-9228	402	8	(	(	PUNCT
ap-9228	402	9	28	28	NUM
ap-9228	402	10	)	)	PUNCT
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ap-9228	402	12	:	:	PUNCT
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ap-9228	402	15	,	,	PUNCT
ap-9228	402	16	t	t	PROPN
ap-9228	402	17	)	)	PUNCT
ap-9228	402	18	=	=	SYM
ap-9228	402	19	ex	ex	NOUN
ap-9228	402	20	,	,	PUNCT
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ap-9228	402	22	,	,	PUNCT
ap-9228	402	23	t	t	PROPN
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ap-9228	402	25	=	=	SYM
ap-9228	403	1	e−x	e−x	NOUN
ap-9228	403	2	.	.	PUNCT
ap-9228	404	1	absolute	absolute	ADJ
ap-9228	404	2	error	error	NOUN
ap-9228	404	3	x	x	X
ap-9228	404	4	ua	ua	PROPN
ap-9228	404	5	ue	ue	PROPN
ap-9228	404	6	hjm	hjm	PROPN
ap-9228	404	7	rdtm	rdtm	PROPN
ap-9228	404	8	psm	psm	PROPN
ap-9228	404	9	adm	adm	PROPN
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ap-9228	404	11	1.1052	1.1052	NUM
ap-9228	405	1	1.1052	1.1052	NUM
ap-9228	405	2	0.0000	0.0000	NUM
ap-9228	405	3	0.0000	0.0000	NUM
ap-9228	405	4	0.0000	0.0000	NUM
ap-9228	405	5	0.0000	0.0000	NUM
ap-9228	405	6	0.2000	0.2000	NUM
ap-9228	405	7	1.2213	1.2213	NUM
ap-9228	405	8	1.2214	1.2214	NUM
ap-9228	405	9	0.0001	0.0001	NUM
ap-9228	405	10	0.0001	0.0001	NUM
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ap-9228	405	13	0.3000	0.3000	NUM
ap-9228	405	14	1.3495	1.3495	NUM
ap-9228	405	15	1.3499	1.3499	NUM
ap-9228	405	16	0.0004	0.0004	NUM
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ap-9228	405	22	1.4918	1.4918	NUM
ap-9228	405	23	0.0012	0.0012	NUM
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ap-9228	405	25	0.0012	0.0012	NUM
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ap-9228	405	36	1.8221	1.8221	NUM
ap-9228	405	37	0.0061	0.0061	NUM
ap-9228	405	38	0.0061	0.0061	NUM
ap-9228	405	39	0.0061	0.0061	NUM
ap-9228	405	40	0.0061	0.0061	NUM
ap-9228	405	41	0.7000	0.7000	NUM
ap-9228	405	42	2.0022	2.0022	NUM
ap-9228	405	43	2.0138	2.0138	NUM
ap-9228	405	44	0.0116	0.0116	NUM
ap-9228	405	45	0.0116	0.0116	NUM
ap-9228	405	46	0.0116	0.0116	NUM
ap-9228	405	47	0.0116	0.0116	NUM
ap-9228	405	48	0.8000	0.8000	NUM
ap-9228	405	49	2.2053	2.2053	NUM
ap-9228	405	50	2.2255	2.2255	NUM
ap-9228	406	1	0.0202	0.0202	NUM
ap-9228	406	2	0.0202	0.0202	NUM
ap-9228	406	3	0.0202	0.0202	NUM
ap-9228	407	1	0.0202	0.0202	NUM
ap-9228	407	2	0.9000	0.9000	NUM
ap-9228	407	3	2.4265	2.4265	NUM
ap-9228	407	4	2.4596	2.4596	NUM
ap-9228	407	5	0.0331	0.0331	NUM
ap-9228	407	6	0.0331	0.0331	NUM
ap-9228	407	7	0.0331	0.0331	NUM
ap-9228	407	8	0.0331	0.0331	NUM
ap-9228	407	9	1.0000	1.0000	NUM
ap-9228	407	10	2.6667	2.6667	NUM
ap-9228	407	11	2.7183	2.7183	NUM
ap-9228	407	12	0.0516	0.0516	NUM
ap-9228	407	13	0.0516	0.0516	NUM
ap-9228	407	14	0.0516	0.0516	NUM
ap-9228	407	15	0.0516	0.0516	NUM
ap-9228	407	16	table	table	NOUN
ap-9228	407	17	5	5	NUM
ap-9228	407	18	.	.	PUNCT
ap-9228	408	1	the	the	DET
ap-9228	408	2	values	value	NOUN
ap-9228	408	3	of	of	ADP
ap-9228	408	4	approximate	approximate	ADJ
ap-9228	408	5	and	and	CCONJ
ap-9228	408	6	exact	exact	ADJ
ap-9228	408	7	solutions	solution	NOUN
ap-9228	408	8	of	of	ADP
ap-9228	408	9	equation	equation	NOUN
ap-9228	408	10	(	(	PUNCT
ap-9228	408	11	28	28	NUM
ap-9228	408	12	)	)	PUNCT
ap-9228	408	13	of	of	ADP
ap-9228	408	14	u	u	NOUN
ap-9228	408	15	for	for	ADP
ap-9228	408	16	different	different	ADJ
ap-9228	408	17	values	value	NOUN
ap-9228	408	18	of	of	ADP
ap-9228	408	19	x.	x.	PROPN
ap-9228	408	20	97	97	NUM
ap-9228	408	21	m.	m.	NOUN
ap-9228	408	22	a.	a.	PROPN
ap-9228	408	23	hussein	hussein	PROPN
ap-9228	408	24	,	,	PUNCT
ap-9228	408	25	h.	h.	PROPN
ap-9228	408	26	k.	k.	PROPN
ap-9228	408	27	jassim	jassim	PROPN
ap-9228	408	28	,	,	PUNCT
ap-9228	408	29	a.	a.	PROPN
ap-9228	408	30	k.	k.	PROPN
ap-9228	408	31	jassim	jassim	PROPN
ap-9228	408	32	acta	acta	PROPN
ap-9228	408	33	polytechnica	polytechnica	PROPN
ap-9228	408	34	absolute	absolute	ADJ
ap-9228	408	35	error	error	NOUN
ap-9228	408	36	x	x	X
ap-9228	408	37	va	va	PROPN
ap-9228	408	38	ve	ve	AUX
ap-9228	408	39	hjm	hjm	VERB
ap-9228	408	40	rdtm	rdtm	PROPN
ap-9228	408	41	psm	psm	PROPN
ap-9228	408	42	adm	adm	PROPN
ap-9228	408	43	0.1000	0.1000	NUM
ap-9228	408	44	0.9048	0.9048	NUM
ap-9228	408	45	0.9048	0.9048	NUM
ap-9228	408	46	0.0000	0.0000	NUM
ap-9228	408	47	0.0000	0.0000	NUM
ap-9228	408	48	0.0000	0.0000	NUM
ap-9228	408	49	0.0000	0.0000	NUM
ap-9228	408	50	0.2000	0.2000	NUM
ap-9228	408	51	0.8187	0.8187	NUM
ap-9228	408	52	0.8187	0.8187	NUM
ap-9228	408	53	0.0001	0.0001	NUM
ap-9228	408	54	0.0001	0.0001	NUM
ap-9228	408	55	0.0001	0.0001	NUM
ap-9228	408	56	0.0001	0.0001	NUM
ap-9228	408	57	0.3000	0.3000	NUM
ap-9228	408	58	0.7405	0.7405	NUM
ap-9228	408	59	0.7408	0.7408	NUM
ap-9228	408	60	0.0003	0.0003	NUM
ap-9228	408	61	0.0003	0.0003	NUM
ap-9228	408	62	0.0003	0.0003	NUM
ap-9228	408	63	0.0003	0.0003	NUM
ap-9228	408	64	0.4000	0.4000	NUM
ap-9228	408	65	0.6693	0.6693	NUM
ap-9228	408	66	0.6703	0.6703	NUM
ap-9228	408	67	0.0010	0.0010	NUM
ap-9228	408	68	0.0010	0.0010	NUM
ap-9228	408	69	0.0010	0.0010	NUM
ap-9228	408	70	0.0010	0.0010	NUM
ap-9228	408	71	0.5000	0.5000	NUM
ap-9228	408	72	0.6042	0.6042	NUM
ap-9228	408	73	0.6065	0.6065	NUM
ap-9228	408	74	0.0024	0.0024	NUM
ap-9228	408	75	0.0024	0.0024	NUM
ap-9228	408	76	0.0024	0.0024	NUM
ap-9228	408	77	0.0024	0.0024	NUM
ap-9228	408	78	0.6000	0.6000	NUM
ap-9228	408	79	0.5440	0.5440	NUM
ap-9228	408	80	0.5488	0.5488	NUM
ap-9228	408	81	0.0048	0.0048	NUM
ap-9228	408	82	0.0048	0.0048	NUM
ap-9228	408	83	0.0048	0.0048	NUM
ap-9228	408	84	0.0048	0.0048	NUM
ap-9228	408	85	0.7000	0.7000	NUM
ap-9228	408	86	0.4878	0.4878	NUM
ap-9228	408	87	0.4966	0.4966	NUM
ap-9228	408	88	0.0088	0.0088	NUM
ap-9228	408	89	0.0088	0.0088	NUM
ap-9228	408	90	0.0088	0.0088	NUM
ap-9228	408	91	0.0088	0.0088	NUM
ap-9228	408	92	0.8000	0.8000	NUM
ap-9228	408	93	0.4347	0.4347	NUM
ap-9228	408	94	0.4493	0.4493	NUM
ap-9228	408	95	0.0147	0.0147	NUM
ap-9228	408	96	0.0147	0.0147	NUM
ap-9228	409	1	0.0147	0.0147	NUM
ap-9228	409	2	0.0147	0.0147	NUM
ap-9228	409	3	0.9000	0.9000	NUM
ap-9228	409	4	0.3835	0.3835	NUM
ap-9228	409	5	0.4066	0.4066	NUM
ap-9228	409	6	0.0231	0.0231	NUM
ap-9228	409	7	0.0231	0.0231	NUM
ap-9228	409	8	0.0231	0.0231	NUM
ap-9228	409	9	0.0231	0.0231	NUM
ap-9228	409	10	1.0000	1.0000	NUM
ap-9228	409	11	0.3333	0.3333	NUM
ap-9228	409	12	0.3679	0.3679	NUM
ap-9228	409	13	0.0345	0.0345	NUM
ap-9228	409	14	0.0345	0.0345	NUM
ap-9228	409	15	0.0345	0.0345	NUM
ap-9228	409	16	0.0345	0.0345	NUM
ap-9228	409	17	table	table	NOUN
ap-9228	409	18	6	6	NUM
ap-9228	409	19	.	.	PUNCT
ap-9228	410	1	the	the	DET
ap-9228	410	2	values	value	NOUN
ap-9228	410	3	of	of	ADP
ap-9228	410	4	approximate	approximate	ADJ
ap-9228	410	5	and	and	CCONJ
ap-9228	410	6	exact	exact	ADJ
ap-9228	410	7	solutions	solution	NOUN
ap-9228	410	8	of	of	ADP
ap-9228	410	9	equation	equation	NOUN
ap-9228	410	10	(	(	PUNCT
ap-9228	410	11	28	28	NUM
ap-9228	410	12	)	)	PUNCT
ap-9228	410	13	of	of	ADP
ap-9228	410	14	v	v	NOUN
ap-9228	410	15	for	for	ADP
ap-9228	410	16	different	different	ADJ
ap-9228	410	17	values	value	NOUN
ap-9228	410	18	of	of	ADP
ap-9228	410	19	x.	x.	NOUN
ap-9228	410	20	figure	figure	NOUN
ap-9228	410	21	5	5	NUM
ap-9228	410	22	.	.	PUNCT
ap-9228	411	1	the	the	DET
ap-9228	411	2	plot	plot	NOUN
ap-9228	411	3	of	of	ADP
ap-9228	411	4	approximate	approximate	ADJ
ap-9228	411	5	solution	solution	NOUN
ap-9228	411	6	,	,	PUNCT
ap-9228	411	7	exact	exact	ADJ
ap-9228	411	8	solution	solution	NOUN
ap-9228	411	9	,	,	PUNCT
ap-9228	411	10	and	and	CCONJ
ap-9228	411	11	absolute	absolute	ADJ
ap-9228	411	12	error	error	NOUN
ap-9228	411	13	of	of	ADP
ap-9228	411	14	u	u	NOUN
ap-9228	411	15	of	of	ADP
ap-9228	411	16	equation	equation	NOUN
ap-9228	411	17	(	(	PUNCT
ap-9228	411	18	28	28	NUM
ap-9228	411	19	)	)	PUNCT
ap-9228	411	20	.	.	PUNCT
ap-9228	412	1	figure	figure	VERB
ap-9228	412	2	6	6	NUM
ap-9228	412	3	.	.	PUNCT
ap-9228	413	1	the	the	DET
ap-9228	413	2	plot	plot	NOUN
ap-9228	413	3	of	of	ADP
ap-9228	413	4	approximate	approximate	ADJ
ap-9228	413	5	solution	solution	NOUN
ap-9228	413	6	,	,	PUNCT
ap-9228	413	7	exact	exact	ADJ
ap-9228	413	8	solution	solution	NOUN
ap-9228	413	9	,	,	PUNCT
ap-9228	413	10	and	and	CCONJ
ap-9228	413	11	absolute	absolute	ADJ
ap-9228	413	12	error	error	NOUN
ap-9228	413	13	of	of	ADP
ap-9228	413	14	v	v	NUM
ap-9228	413	15	of	of	ADP
ap-9228	413	16	equation	equation	NOUN
ap-9228	413	17	(	(	PUNCT
ap-9228	413	18	28	28	NUM
ap-9228	413	19	)	)	PUNCT
ap-9228	413	20	.	.	PUNCT
ap-9228	414	1	98	98	NUM
ap-9228	414	2	vol	vol	NOUN
ap-9228	414	3	.	.	PUNCT
ap-9228	415	1	64	64	NUM
ap-9228	415	2	no	no	NOUN
ap-9228	415	3	.	.	PUNCT
ap-9228	416	1	2/2024	2/2024	NUM
ap-9228	416	2	an	an	DET
ap-9228	416	3	innovative	innovative	ADJ
ap-9228	416	4	iterative	iterative	NOUN
ap-9228	416	5	approach	approach	NOUN
ap-9228	416	6	to	to	ADP
ap-9228	416	7	solving	solve	VERB
ap-9228	416	8	volterra	volterra	NOUN
ap-9228	416	9	integral	integral	ADJ
ap-9228	416	10	.	.	PUNCT
ap-9228	416	11	.	.	PUNCT
ap-9228	417	1	.	.	PUNCT
ap-9228	418	1	example	example	NOUN
ap-9228	419	1	6	6	NUM
ap-9228	419	2	.	.	PUNCT
ap-9228	419	3	consider	consider	VERB
ap-9228	419	4	the	the	DET
ap-9228	419	5	linear	linear	ADJ
ap-9228	419	6	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	419	7	system	system	NOUN
ap-9228	419	8	viesk:	viesk:	PROPN
ap-9228	419	9	u(x	u(x	NOUN
ap-9228	419	10	)	)	PUNCT
ap-9228	419	11	=	=	SYM
ap-9228	420	1	1	1	NUM
ap-9228	420	2	−	−	NUM
ap-9228	421	1	x2	x2	PROPN
ap-9228	422	1	+	+	CCONJ
ap-9228	422	2	x3	x3	ADJ
ap-9228	422	3	+	+	CCONJ
ap-9228	422	4	∫	∫	PROPN
ap-9228	422	5	x	x	SYM
ap-9228	422	6	0	0	PUNCT
ap-9228	422	7	(	(	PUNCT
ap-9228	422	8	x−	x−	NOUN
ap-9228	422	9	t)u(t	t)u(t	PROPN
ap-9228	422	10	)	)	PUNCT
ap-9228	423	1	+	+	CCONJ
ap-9228	423	2	(	(	PUNCT
ap-9228	423	3	x−	x−	PROPN
ap-9228	423	4	t)v(t	t)v(t	PROPN
ap-9228	423	5	)	)	PUNCT
ap-9228	423	6	dt	dt	PROPN
ap-9228	423	7	,	,	PUNCT
ap-9228	423	8	v(x	v(x	PROPN
ap-9228	423	9	)	)	PUNCT
ap-9228	423	10	=	=	SYM
ap-9228	424	1	1	1	NUM
ap-9228	424	2	−	−	NUM
ap-9228	424	3	x3	x3	ADJ
ap-9228	424	4	−	−	PROPN
ap-9228	424	5	1	1	NUM
ap-9228	424	6	10x	10x	NUM
ap-9228	424	7	5	5	NUM
ap-9228	424	8	+	+	CCONJ
ap-9228	424	9	∫	∫	PROPN
ap-9228	424	10	x	x	SYM
ap-9228	424	11	0	0	PUNCT
ap-9228	424	12	(	(	PUNCT
ap-9228	424	13	x−	x−	PROPN
ap-9228	424	14	t)u(t	t)u(t	PROPN
ap-9228	424	15	)	)	PUNCT
ap-9228	424	16	−	−	PROPN
ap-9228	424	17	(	(	PUNCT
ap-9228	424	18	x−	x−	PROPN
ap-9228	424	19	t)v(t	t)v(t	PROPN
ap-9228	424	20	)	)	PUNCT
ap-9228	424	21	dt	dt	NOUN
ap-9228	424	22	,	,	PUNCT
ap-9228	424	23	(	(	PUNCT
ap-9228	424	24	30	30	NUM
ap-9228	424	25	)	)	PUNCT
ap-9228	424	26	by	by	ADP
ap-9228	424	27	applying	apply	VERB
ap-9228	424	28	the	the	DET
ap-9228	424	29	algorithm	algorithm	NOUN
ap-9228	424	30	of	of	ADP
ap-9228	424	31	the	the	DET
ap-9228	424	32	new	new	ADJ
ap-9228	424	33	technique	technique	NOUN
ap-9228	424	34	to	to	ADP
ap-9228	424	35	equation	equation	NOUN
ap-9228	424	36	(	(	PUNCT
ap-9228	424	37	30	30	NUM
ap-9228	424	38	)	)	PUNCT
ap-9228	424	39	,	,	PUNCT
ap-9228	424	40	we	we	PRON
ap-9228	424	41	obtain:	obtain:	VERB
ap-9228	424	42	∞∑	∞∑	NUM
ap-9228	424	43	i=0	i=0	ADJ
ap-9228	424	44	u(x	u(x	PROPN
ap-9228	424	45	)	)	PUNCT
ap-9228	424	46	=	=	SYM
ap-9228	425	1	1	1	NUM
ap-9228	425	2	−	−	NUM
ap-9228	426	1	x2	x2	PROPN
ap-9228	427	1	+	+	CCONJ
ap-9228	427	2	x3	x3	ADJ
ap-9228	427	3	+	+	CCONJ
ap-9228	427	4	∫	∫	PROPN
ap-9228	427	5	x	x	SYM
ap-9228	427	6	0	0	NUM
ap-9228	427	7	(	(	PUNCT
ap-9228	427	8	x−	x−	PROPN
ap-9228	427	9	t	t	PROPN
ap-9228	427	10	)	)	PUNCT
ap-9228	427	11	∞∑	∞∑	PRON
ap-9228	427	12	i=0	i=0	ADJ
ap-9228	427	13	u(t	u(t	NOUN
ap-9228	427	14	)	)	PUNCT
ap-9228	427	15	+	+	CCONJ
ap-9228	427	16	(	(	PUNCT
ap-9228	427	17	x−	x−	PROPN
ap-9228	427	18	t	t	PROPN
ap-9228	427	19	)	)	PUNCT
ap-9228	427	20	∞∑	∞∑	PRON
ap-9228	427	21	i=0	i=0	PROPN
ap-9228	427	22	v(t	v(t	NUM
ap-9228	427	23	)	)	PUNCT
ap-9228	427	24	dt	dt	PROPN
ap-9228	427	25	,	,	PUNCT
ap-9228	427	26	∞∑	∞∑	ADJ
ap-9228	427	27	i=0	i=0	ADJ
ap-9228	427	28	v(x	v(x	PROPN
ap-9228	427	29	)	)	PUNCT
ap-9228	427	30	=	=	SYM
ap-9228	428	1	1	1	NUM
ap-9228	428	2	−	−	NUM
ap-9228	429	1	x3	x3	ADJ
ap-9228	429	2	−	−	PROPN
ap-9228	429	3	1	1	NUM
ap-9228	429	4	10x	10x	NUM
ap-9228	429	5	5	5	NUM
ap-9228	429	6	+	+	CCONJ
ap-9228	429	7	∫	∫	PROPN
ap-9228	429	8	x	x	SYM
ap-9228	429	9	0	0	NUM
ap-9228	429	10	(	(	PUNCT
ap-9228	429	11	x−	x−	PROPN
ap-9228	429	12	t	t	PROPN
ap-9228	429	13	)	)	PUNCT
ap-9228	429	14	∞∑	∞∑	PRON
ap-9228	429	15	i=0	i=0	ADJ
ap-9228	429	16	u(t	u(t	NOUN
ap-9228	429	17	)	)	PUNCT
ap-9228	429	18	−	−	PROPN
ap-9228	429	19	(	(	PUNCT
ap-9228	429	20	x−	x−	PROPN
ap-9228	429	21	t	t	PROPN
ap-9228	429	22	)	)	PUNCT
ap-9228	429	23	∞∑	∞∑	PRON
ap-9228	429	24	i=0	i=0	PROPN
ap-9228	429	25	v(t	v(t	NUM
ap-9228	429	26	)	)	PUNCT
ap-9228	429	27	dt	dt	PROPN
ap-9228	429	28	.	.	PUNCT
ap-9228	430	1	(	(	PUNCT
ap-9228	430	2	31	31	NUM
ap-9228	430	3	)	)	PUNCT
ap-9228	430	4	by	by	ADP
ap-9228	430	5	comparing	compare	VERB
ap-9228	430	6	both	both	DET
ap-9228	430	7	sides	side	NOUN
ap-9228	430	8	of	of	ADP
ap-9228	430	9	equation	equation	NOUN
ap-9228	430	10	(	(	PUNCT
ap-9228	430	11	31	31	NUM
ap-9228	430	12	)	)	PUNCT
ap-9228	430	13	,	,	PUNCT
ap-9228	431	1	we	we	PRON
ap-9228	431	2	obtain:	obtain:	VERB
ap-9228	431	3	u0	u0	NOUN
ap-9228	431	4	=	=	ADJ
ap-9228	431	5	1	1	NUM
ap-9228	431	6	,	,	PUNCT
ap-9228	431	7	v0	v0	NOUN
ap-9228	431	8	=	=	SYM
ap-9228	431	9	1	1	NUM
ap-9228	431	10	,	,	PUNCT
ap-9228	431	11	u1	u1	NOUN
ap-9228	431	12	=	=	SYM
ap-9228	431	13	xdx	xdx	PROPN
ap-9228	431	14	(	(	PUNCT
ap-9228	431	15	1	1	NUM
ap-9228	431	16	−	−	PROPN
ap-9228	431	17	x2	x2	PROPN
ap-9228	432	1	+	+	CCONJ
ap-9228	432	2	x3	x3	ADJ
ap-9228	432	3	+	+	CCONJ
ap-9228	432	4	∫	∫	PROPN
ap-9228	432	5	x	x	SYM
ap-9228	432	6	0	0	PROPN
ap-9228	432	7	2(x−	2(x−	NUM
ap-9228	432	8	t	t	PROPN
ap-9228	432	9	)	)	PUNCT
ap-9228	432	10	dt	dt	NOUN
ap-9228	432	11	)	)	PUNCT
ap-9228	433	1	x=0	x=0	PUNCT
ap-9228	434	1	=	=	SYM
ap-9228	434	2	0	0	NUM
ap-9228	434	3	,	,	PUNCT
ap-9228	434	4	v1	v1	PROPN
ap-9228	434	5	=	=	SYM
ap-9228	434	6	xdx	xdx	PROPN
ap-9228	434	7	(	(	PUNCT
ap-9228	434	8	1	1	NUM
ap-9228	434	9	−	−	NUM
ap-9228	434	10	x3	x3	ADJ
ap-9228	434	11	−	−	PROPN
ap-9228	434	12	1	1	NUM
ap-9228	434	13	10x	10x	NUM
ap-9228	434	14	5	5	NUM
ap-9228	434	15	+	+	CCONJ
ap-9228	434	16	∫	∫	PROPN
ap-9228	434	17	x	x	SYM
ap-9228	434	18	0	0	NUM
ap-9228	434	19	(	(	PUNCT
ap-9228	434	20	0	0	NUM
ap-9228	434	21	)	)	PUNCT
ap-9228	434	22	dt	dt	NOUN
ap-9228	434	23	)	)	PUNCT
ap-9228	435	1	x=0	x=0	PUNCT
ap-9228	435	2	=	=	SYM
ap-9228	435	3	0	0	NUM
ap-9228	435	4	,	,	PUNCT
ap-9228	435	5	u2	u2	NOUN
ap-9228	435	6	=	=	SYM
ap-9228	435	7	x2	x2	PROPN
ap-9228	436	1	2	2	NUM
ap-9228	436	2	!	!	PUNCT
ap-9228	436	3	d	d	NOUN
ap-9228	436	4	2	2	NUM
ap-9228	436	5	x	x	SYM
ap-9228	436	6	(	(	PUNCT
ap-9228	436	7	1	1	NUM
ap-9228	436	8	−	−	NOUN
ap-9228	436	9	x3	x3	ADJ
ap-9228	436	10	−	−	PROPN
ap-9228	436	11	1	1	NUM
ap-9228	436	12	10x	10x	NUM
ap-9228	436	13	5	5	NUM
ap-9228	436	14	+	+	CCONJ
ap-9228	436	15	∫	∫	PROPN
ap-9228	436	16	x	x	SYM
ap-9228	436	17	0	0	PROPN
ap-9228	436	18	2(x−	2(x−	NUM
ap-9228	436	19	t	t	PROPN
ap-9228	436	20	)	)	PUNCT
ap-9228	436	21	dt	dt	NOUN
ap-9228	436	22	)	)	PUNCT
ap-9228	436	23	x=0	x=0	PUNCT
ap-9228	437	1	=	=	SYM
ap-9228	437	2	0	0	NUM
ap-9228	437	3	,	,	PUNCT
ap-9228	437	4	v2	v2	PROPN
ap-9228	437	5	=	=	SYM
ap-9228	437	6	x2	x2	PROPN
ap-9228	438	1	2	2	X
ap-9228	438	2	!	!	PUNCT
ap-9228	438	3	d	d	NOUN
ap-9228	438	4	2	2	NUM
ap-9228	438	5	x	x	SYM
ap-9228	438	6	(	(	PUNCT
ap-9228	438	7	1	1	NUM
ap-9228	438	8	−	−	NOUN
ap-9228	438	9	x3	x3	ADJ
ap-9228	438	10	−	−	PROPN
ap-9228	438	11	1	1	NUM
ap-9228	438	12	10x	10x	NUM
ap-9228	438	13	5	5	NUM
ap-9228	438	14	+	+	CCONJ
ap-9228	438	15	∫	∫	PROPN
ap-9228	438	16	x	x	SYM
ap-9228	438	17	0	0	NUM
ap-9228	438	18	(	(	PUNCT
ap-9228	438	19	0	0	NUM
ap-9228	438	20	)	)	PUNCT
ap-9228	438	21	dt	dt	NOUN
ap-9228	438	22	)	)	PUNCT
ap-9228	439	1	x=0	x=0	PUNCT
ap-9228	439	2	=	=	SYM
ap-9228	439	3	0	0	NUM
ap-9228	439	4	,	,	PUNCT
ap-9228	439	5	u3	u3	NOUN
ap-9228	439	6	=	=	SYM
ap-9228	439	7	x3	x3	NOUN
ap-9228	440	1	3	3	X
ap-9228	440	2	!	!	PUNCT
ap-9228	441	1	d	d	NOUN
ap-9228	441	2	3	3	NUM
ap-9228	441	3	x	x	SYM
ap-9228	441	4	(	(	PUNCT
ap-9228	441	5	1	1	NUM
ap-9228	441	6	−	−	NOUN
ap-9228	441	7	x2	x2	PROPN
ap-9228	442	1	+	+	CCONJ
ap-9228	442	2	x3	x3	ADJ
ap-9228	442	3	+	+	CCONJ
ap-9228	442	4	∫	∫	PROPN
ap-9228	442	5	x	x	SYM
ap-9228	442	6	0	0	PROPN
ap-9228	442	7	2(x−	2(x−	NUM
ap-9228	442	8	t	t	PROPN
ap-9228	442	9	)	)	PUNCT
ap-9228	442	10	dt	dt	NOUN
ap-9228	442	11	)	)	PUNCT
ap-9228	442	12	x=0	x=0	PUNCT
ap-9228	443	1	=	=	SYM
ap-9228	443	2	0	0	PROPN
ap-9228	443	3	,	,	PUNCT
ap-9228	443	4	v3	v3	PROPN
ap-9228	443	5	=	=	PUNCT
ap-9228	444	1	x3	x3	ADJ
ap-9228	445	1	3	3	X
ap-9228	445	2	!	!	PUNCT
ap-9228	446	1	d	d	NOUN
ap-9228	446	2	3	3	NUM
ap-9228	446	3	x	x	SYM
ap-9228	446	4	(	(	PUNCT
ap-9228	446	5	1	1	NUM
ap-9228	446	6	−	−	NOUN
ap-9228	446	7	x3	x3	ADJ
ap-9228	446	8	−	−	PROPN
ap-9228	446	9	1	1	NUM
ap-9228	446	10	10x	10x	NUM
ap-9228	446	11	5	5	NUM
ap-9228	446	12	+	+	CCONJ
ap-9228	446	13	∫	∫	PROPN
ap-9228	446	14	x	x	SYM
ap-9228	446	15	0	0	NUM
ap-9228	446	16	(	(	PUNCT
ap-9228	446	17	0	0	NUM
ap-9228	446	18	)	)	PUNCT
ap-9228	446	19	dt	dt	NOUN
ap-9228	446	20	)	)	PUNCT
ap-9228	447	1	x=0	x=0	PUNCT
ap-9228	447	2	=	=	SYM
ap-9228	447	3	0	0	PROPN
ap-9228	447	4	,	,	PUNCT
ap-9228	447	5	u4	u4	PROPN
ap-9228	447	6	=	=	NOUN
ap-9228	447	7	x3	x3	PROPN
ap-9228	448	1	3	3	X
ap-9228	448	2	!	!	PUNCT
ap-9228	449	1	d	d	NOUN
ap-9228	449	2	3	3	NUM
ap-9228	449	3	x	x	SYM
ap-9228	449	4	(	(	PUNCT
ap-9228	449	5	1	1	NUM
ap-9228	449	6	−	−	NOUN
ap-9228	449	7	x2	x2	PROPN
ap-9228	450	1	+	+	CCONJ
ap-9228	450	2	x3	x3	ADJ
ap-9228	450	3	+	+	CCONJ
ap-9228	450	4	∫	∫	PROPN
ap-9228	450	5	x	x	SYM
ap-9228	450	6	0	0	PROPN
ap-9228	450	7	2(x−	2(x−	NUM
ap-9228	450	8	t	t	PROPN
ap-9228	450	9	)	)	PUNCT
ap-9228	450	10	dt	dt	NOUN
ap-9228	450	11	)	)	PUNCT
ap-9228	451	1	x=0	x=0	PUNCT
ap-9228	452	1	=	=	SYM
ap-9228	452	2	x3	x3	PROPN
ap-9228	452	3	,	,	PUNCT
ap-9228	452	4	v4	v4	PROPN
ap-9228	452	5	=	=	SYM
ap-9228	452	6	x3	x3	NOUN
ap-9228	452	7	3	3	X
ap-9228	452	8	!	!	PUNCT
ap-9228	453	1	d	d	NOUN
ap-9228	453	2	3	3	NUM
ap-9228	453	3	x	x	SYM
ap-9228	453	4	(	(	PUNCT
ap-9228	453	5	1	1	NUM
ap-9228	453	6	−	−	NOUN
ap-9228	453	7	x3	x3	ADJ
ap-9228	453	8	−	−	PROPN
ap-9228	453	9	1	1	NUM
ap-9228	453	10	10x	10x	NUM
ap-9228	453	11	5	5	NUM
ap-9228	453	12	+	+	CCONJ
ap-9228	453	13	∫	∫	PROPN
ap-9228	453	14	x	x	SYM
ap-9228	453	15	0	0	NUM
ap-9228	453	16	(	(	PUNCT
ap-9228	453	17	0	0	NUM
ap-9228	453	18	)	)	PUNCT
ap-9228	453	19	dt	dt	NOUN
ap-9228	453	20	)	)	PUNCT
ap-9228	454	1	x=0	x=0	PUNCT
ap-9228	454	2	=	=	SYM
ap-9228	454	3	−x3	−x3	PROPN
ap-9228	454	4	,	,	PUNCT
ap-9228	454	5	...	...	PUNCT
ap-9228	455	1	ui+1	ui+1	X
ap-9228	455	2	=	=	SYM
ap-9228	455	3	xi+1	xi+1	PROPN
ap-9228	455	4	(	(	PUNCT
ap-9228	455	5	i+	i+	PROPN
ap-9228	455	6	1)!d	1)!d	NUM
ap-9228	455	7	i+1	i+1	ADV
ap-9228	455	8	x	x	SYM
ap-9228	455	9	ex	ex	NOUN
ap-9228	455	10	−	−	NUM
ap-9228	455	11	2x+	2x+	NUM
ap-9228	455	12	∫	∫	NOUN
ap-9228	455	13	x	x	SYM
ap-9228	455	14	0	0	NUM
ap-9228	455	15	e−t	e−t	PROPN
ap-9228	455	16	(	(	PUNCT
ap-9228	455	17	i∑	i∑	PROPN
ap-9228	455	18	j=0	j=0	PROPN
ap-9228	455	19	uj	uj	PROPN
ap-9228	455	20	)	)	PUNCT
ap-9228	455	21	+	+	NUM
ap-9228	455	22	et	et	NOUN
ap-9228	455	23	(	(	PUNCT
ap-9228	455	24	i∑	i∑	PROPN
ap-9228	455	25	j=0	j=0	PROPN
ap-9228	455	26	vj	vj	PROPN
ap-9228	455	27	)	)	PUNCT
ap-9228	455	28	dt	dt	NOUN
ap-9228	456	1			PROPN
ap-9228	456	2	x=0	x=0	PUNCT
ap-9228	456	3	=	=	SYM
ap-9228	456	4	0	0	NUM
ap-9228	456	5	∀i	∀i	NOUN
ap-9228	456	6	>	>	X
ap-9228	456	7	3	3	NUM
ap-9228	456	8	,	,	PUNCT
ap-9228	456	9	vi+1	vi+1	NOUN
ap-9228	456	10	=	=	SYM
ap-9228	456	11	xi+1	xi+1	PROPN
ap-9228	456	12	(	(	PUNCT
ap-9228	456	13	i+	i+	PROPN
ap-9228	456	14	1)!d	1)!d	NUM
ap-9228	456	15	i+1	i+1	ADV
ap-9228	456	16	x	x	SYM
ap-9228	456	17	ex	ex	NOUN
ap-9228	456	18	−	−	NUM
ap-9228	456	19	2x+	2x+	NUM
ap-9228	456	20	∫	∫	NOUN
ap-9228	456	21	x	x	SYM
ap-9228	456	22	0	0	NUM
ap-9228	456	23	et	et	NOUN
ap-9228	456	24	(	(	PUNCT
ap-9228	456	25	i∑	i∑	PROPN
ap-9228	456	26	j=0	j=0	PROPN
ap-9228	456	27	uj	uj	PROPN
ap-9228	456	28	)	)	PUNCT
ap-9228	456	29	+	+	NUM
ap-9228	456	30	e−t	e−t	PROPN
ap-9228	456	31	(	(	PUNCT
ap-9228	456	32	i∑	i∑	PROPN
ap-9228	456	33	j=0	j=0	PROPN
ap-9228	456	34	vj	vj	PROPN
ap-9228	456	35	)	)	PUNCT
ap-9228	456	36	dt	dt	NOUN
ap-9228	457	1			PROPN
ap-9228	457	2	x=0	x=0	PUNCT
ap-9228	457	3	=	=	SYM
ap-9228	457	4	0	0	NUM
ap-9228	457	5	∀i	∀i	NOUN
ap-9228	457	6	>	>	X
ap-9228	457	7	3	3	NUM
ap-9228	457	8	.	.	PUNCT
ap-9228	458	1	thus	thus	ADV
ap-9228	458	2	,	,	PUNCT
ap-9228	458	3	the	the	DET
ap-9228	458	4	approximate	approximate	ADJ
ap-9228	458	5	solution	solution	NOUN
ap-9228	458	6	of	of	ADP
ap-9228	458	7	equation	equation	NOUN
ap-9228	458	8	(	(	PUNCT
ap-9228	458	9	30	30	NUM
ap-9228	458	10	)	)	PUNCT
ap-9228	458	11	is	be	AUX
ap-9228	458	12	:	:	PUNCT
ap-9228	458	13	{	{	PUNCT
ap-9228	458	14	ua(x	ua(x	X
ap-9228	458	15	,	,	PUNCT
ap-9228	458	16	t	t	PROPN
ap-9228	458	17	)	)	PUNCT
ap-9228	458	18	=	=	SYM
ap-9228	458	19	1	1	NUM
ap-9228	458	20	+	+	CCONJ
ap-9228	458	21	x3	x3	ADJ
ap-9228	458	22	,	,	PUNCT
ap-9228	458	23	va(x	va(x	NOUN
ap-9228	458	24	,	,	PUNCT
ap-9228	458	25	t	t	PROPN
ap-9228	458	26	)	)	PUNCT
ap-9228	458	27	=	=	SYM
ap-9228	459	1	1	1	NUM
ap-9228	459	2	−	−	NUM
ap-9228	459	3	x3	x3	ADJ
ap-9228	459	4	.	.	PUNCT
ap-9228	460	1	therefore	therefore	ADV
ap-9228	460	2	,	,	PUNCT
ap-9228	460	3	the	the	DET
ap-9228	460	4	exact	exact	ADJ
ap-9228	460	5	solution	solution	NOUN
ap-9228	460	6	to	to	ADP
ap-9228	460	7	equation	equation	NOUN
ap-9228	460	8	(	(	PUNCT
ap-9228	460	9	30	30	NUM
ap-9228	460	10	)	)	PUNCT
ap-9228	460	11	is	be	AUX
ap-9228	460	12	:	:	PUNCT
ap-9228	460	13	{	{	PUNCT
ap-9228	460	14	ue(x	ue(x	PROPN
ap-9228	460	15	,	,	PUNCT
ap-9228	460	16	t	t	PROPN
ap-9228	460	17	)	)	PUNCT
ap-9228	460	18	=	=	SYM
ap-9228	460	19	1	1	NUM
ap-9228	460	20	+	+	CCONJ
ap-9228	460	21	x3	x3	ADJ
ap-9228	460	22	,	,	PUNCT
ap-9228	460	23	ve(x	ve(x	NOUN
ap-9228	460	24	,	,	PUNCT
ap-9228	460	25	t	t	PROPN
ap-9228	460	26	)	)	PUNCT
ap-9228	460	27	=	=	SYM
ap-9228	461	1	1	1	NUM
ap-9228	461	2	−	−	NUM
ap-9228	461	3	x3	x3	ADJ
ap-9228	461	4	.	.	PUNCT
ap-9228	462	1	99	99	NUM
ap-9228	462	2	m.	m.	NOUN
ap-9228	462	3	a.	a.	PROPN
ap-9228	462	4	hussein	hussein	PROPN
ap-9228	462	5	,	,	PUNCT
ap-9228	462	6	h.	h.	PROPN
ap-9228	462	7	k.	k.	PROPN
ap-9228	462	8	jassim	jassim	PROPN
ap-9228	462	9	,	,	PUNCT
ap-9228	462	10	a.	a.	PROPN
ap-9228	462	11	k.	k.	PROPN
ap-9228	462	12	jassim	jassim	PROPN
ap-9228	462	13	acta	acta	PROPN
ap-9228	462	14	polytechnica	polytechnica	PROPN
ap-9228	462	15	absolute	absolute	ADJ
ap-9228	462	16	error	error	NOUN
ap-9228	462	17	x	x	X
ap-9228	462	18	ua	ua	PROPN
ap-9228	462	19	ue	ue	PROPN
ap-9228	462	20	hjm	hjm	PROPN
ap-9228	462	21	rdtm	rdtm	PROPN
ap-9228	462	22	psm	psm	PROPN
ap-9228	462	23	adm	adm	PROPN
ap-9228	462	24	0.1000	0.1000	NUM
ap-9228	462	25	1.0010	1.0010	NUM
ap-9228	463	1	1.0010	1.0010	NUM
ap-9228	463	2	0.0000	0.0000	NUM
ap-9228	463	3	0.0000	0.0000	NUM
ap-9228	463	4	0.0000	0.0000	NUM
ap-9228	463	5	0.0000	0.0000	NUM
ap-9228	463	6	0.2000	0.2000	NUM
ap-9228	463	7	1.0080	1.0080	NUM
ap-9228	463	8	1.0080	1.0080	NUM
ap-9228	463	9	0.0000	0.0000	NUM
ap-9228	463	10	0.0000	0.0000	NUM
ap-9228	463	11	0.0000	0.0000	NUM
ap-9228	463	12	0.0000	0.0000	NUM
ap-9228	463	13	0.3000	0.3000	NUM
ap-9228	463	14	1.0270	1.0270	NUM
ap-9228	463	15	1.0270	1.0270	NUM
ap-9228	463	16	0.0000	0.0000	NUM
ap-9228	463	17	0.0000	0.0000	NUM
ap-9228	463	18	0.0000	0.0000	NUM
ap-9228	463	19	0.0000	0.0000	NUM
ap-9228	463	20	0.4000	0.4000	NUM
ap-9228	464	1	1.0640	1.0640	NUM
ap-9228	464	2	1.0640	1.0640	NUM
ap-9228	464	3	0.0000	0.0000	NUM
ap-9228	464	4	0.0000	0.0000	NUM
ap-9228	464	5	0.0000	0.0000	NUM
ap-9228	464	6	0.0000	0.0000	NUM
ap-9228	464	7	0.5000	0.5000	NUM
ap-9228	465	1	1.1250	1.1250	NUM
ap-9228	465	2	1.1250	1.1250	NUM
ap-9228	465	3	0.0000	0.0000	NUM
ap-9228	465	4	0.0000	0.0000	NUM
ap-9228	465	5	0.0000	0.0000	NUM
ap-9228	465	6	0.0000	0.0000	NUM
ap-9228	465	7	0.6000	0.6000	NUM
ap-9228	465	8	1.2160	1.2160	NUM
ap-9228	466	1	1.2160	1.2160	NUM
ap-9228	466	2	0.0000	0.0000	NUM
ap-9228	466	3	0.0000	0.0000	NUM
ap-9228	466	4	0.0000	0.0000	NUM
ap-9228	466	5	0.0000	0.0000	NUM
ap-9228	466	6	0.7000	0.7000	NUM
ap-9228	466	7	1.3430	1.3430	NUM
ap-9228	466	8	1.3430	1.3430	NUM
ap-9228	466	9	0.0000	0.0000	NUM
ap-9228	466	10	0.0000	0.0000	NUM
ap-9228	466	11	0.0000	0.0000	NUM
ap-9228	466	12	0.0000	0.0000	NUM
ap-9228	466	13	0.8000	0.8000	NUM
ap-9228	467	1	1.5120	1.5120	NUM
ap-9228	467	2	1.5120	1.5120	NUM
ap-9228	467	3	0.0000	0.0000	NUM
ap-9228	467	4	0.0000	0.0000	NUM
ap-9228	467	5	0.0000	0.0000	NUM
ap-9228	467	6	0.0000	0.0000	NUM
ap-9228	467	7	0.9000	0.9000	NUM
ap-9228	467	8	1.7290	1.7290	NUM
ap-9228	467	9	1.7290	1.7290	NUM
ap-9228	467	10	0.0000	0.0000	NUM
ap-9228	467	11	0.0000	0.0000	NUM
ap-9228	467	12	0.0000	0.0000	NUM
ap-9228	467	13	0.0000	0.0000	NUM
ap-9228	467	14	1.0000	1.0000	NUM
ap-9228	467	15	2.0000	2.0000	NUM
ap-9228	467	16	2.0000	2.0000	NUM
ap-9228	467	17	0.0000	0.0000	NUM
ap-9228	467	18	0.0000	0.0000	NUM
ap-9228	467	19	0.0000	0.0000	NUM
ap-9228	467	20	0.0000	0.0000	NUM
ap-9228	467	21	table	table	NOUN
ap-9228	467	22	7	7	NUM
ap-9228	467	23	.	.	PUNCT
ap-9228	468	1	the	the	DET
ap-9228	468	2	values	value	NOUN
ap-9228	468	3	of	of	ADP
ap-9228	468	4	approximate	approximate	ADJ
ap-9228	468	5	and	and	CCONJ
ap-9228	468	6	exact	exact	ADJ
ap-9228	468	7	solutions	solution	NOUN
ap-9228	468	8	of	of	ADP
ap-9228	468	9	equation	equation	NOUN
ap-9228	468	10	(	(	PUNCT
ap-9228	468	11	30	30	NUM
ap-9228	468	12	)	)	PUNCT
ap-9228	468	13	of	of	ADP
ap-9228	468	14	u	u	NOUN
ap-9228	468	15	for	for	ADP
ap-9228	468	16	different	different	ADJ
ap-9228	468	17	values	value	NOUN
ap-9228	468	18	of	of	ADP
ap-9228	468	19	x.	x.	NOUN
ap-9228	468	20	absolute	absolute	ADJ
ap-9228	468	21	error	error	NOUN
ap-9228	468	22	x	x	X
ap-9228	468	23	va	va	PROPN
ap-9228	468	24	ve	ve	AUX
ap-9228	468	25	hjm	hjm	VERB
ap-9228	468	26	rdtm	rdtm	PROPN
ap-9228	468	27	psm	psm	PROPN
ap-9228	468	28	adm	adm	PROPN
ap-9228	468	29	0.1000	0.1000	NUM
ap-9228	468	30	0.9990	0.9990	NUM
ap-9228	468	31	0.9990	0.9990	NUM
ap-9228	468	32	0.0000	0.0000	NUM
ap-9228	468	33	0.0000	0.0000	NUM
ap-9228	468	34	0.0000	0.0000	NUM
ap-9228	468	35	0.0000	0.0000	NUM
ap-9228	468	36	0.2000	0.2000	NUM
ap-9228	468	37	0.9920	0.9920	NUM
ap-9228	468	38	0.9920	0.9920	NUM
ap-9228	468	39	0.0000	0.0000	NUM
ap-9228	468	40	0.0000	0.0000	NUM
ap-9228	468	41	0.0000	0.0000	NUM
ap-9228	468	42	0.0000	0.0000	NUM
ap-9228	468	43	0.3000	0.3000	NUM
ap-9228	468	44	0.9730	0.9730	NUM
ap-9228	468	45	0.9730	0.9730	NUM
ap-9228	468	46	0.0000	0.0000	NUM
ap-9228	468	47	0.0000	0.0000	NUM
ap-9228	468	48	0.0000	0.0000	NUM
ap-9228	468	49	0.0000	0.0000	NUM
ap-9228	468	50	0.4000	0.4000	NUM
ap-9228	468	51	0.9360	0.9360	NUM
ap-9228	468	52	0.9360	0.9360	NUM
ap-9228	468	53	0.0000	0.0000	NUM
ap-9228	468	54	0.0000	0.0000	NUM
ap-9228	468	55	0.0000	0.0000	NUM
ap-9228	468	56	0.0000	0.0000	NUM
ap-9228	468	57	0.5000	0.5000	NUM
ap-9228	468	58	0.8750	0.8750	NUM
ap-9228	468	59	0.8750	0.8750	NUM
ap-9228	468	60	0.0000	0.0000	NUM
ap-9228	468	61	0.0000	0.0000	NUM
ap-9228	468	62	0.0000	0.0000	NUM
ap-9228	468	63	0.0000	0.0000	NUM
ap-9228	468	64	0.6000	0.6000	NUM
ap-9228	468	65	0.7840	0.7840	NUM
ap-9228	468	66	0.7840	0.7840	NUM
ap-9228	468	67	0.0000	0.0000	NUM
ap-9228	468	68	0.0000	0.0000	NUM
ap-9228	468	69	0.0000	0.0000	NUM
ap-9228	468	70	0.0000	0.0000	NUM
ap-9228	468	71	0.7000	0.7000	NUM
ap-9228	468	72	0.6570	0.6570	NUM
ap-9228	468	73	0.6570	0.6570	NUM
ap-9228	468	74	0.0000	0.0000	NUM
ap-9228	468	75	0.0000	0.0000	NUM
ap-9228	468	76	0.0000	0.0000	NUM
ap-9228	468	77	0.0000	0.0000	NUM
ap-9228	468	78	0.8000	0.8000	NUM
ap-9228	468	79	0.4880	0.4880	NUM
ap-9228	468	80	0.4880	0.4880	NUM
ap-9228	468	81	0.0000	0.0000	NUM
ap-9228	468	82	0.0000	0.0000	NUM
ap-9228	469	1	0.0000	0.0000	NUM
ap-9228	469	2	0.0000	0.0000	NUM
ap-9228	469	3	0.9000	0.9000	NUM
ap-9228	469	4	0.2710	0.2710	NUM
ap-9228	469	5	0.2710	0.2710	NUM
ap-9228	470	1	0.0000	0.0000	NUM
ap-9228	470	2	0.0000	0.0000	NUM
ap-9228	470	3	0.0000	0.0000	NUM
ap-9228	470	4	0.0000	0.0000	NUM
ap-9228	470	5	1.0000	1.0000	NUM
ap-9228	470	6	0.0000	0.0000	NUM
ap-9228	470	7	0.0000	0.0000	NUM
ap-9228	470	8	0.0000	0.0000	NUM
ap-9228	470	9	0.0000	0.0000	NUM
ap-9228	470	10	0.0000	0.0000	NUM
ap-9228	470	11	0.0000	0.0000	NUM
ap-9228	470	12	table	table	NOUN
ap-9228	470	13	8	8	NUM
ap-9228	470	14	.	.	PUNCT
ap-9228	471	1	the	the	DET
ap-9228	471	2	values	value	NOUN
ap-9228	471	3	of	of	ADP
ap-9228	471	4	approximate	approximate	ADJ
ap-9228	471	5	and	and	CCONJ
ap-9228	471	6	exact	exact	ADJ
ap-9228	471	7	solutions	solution	NOUN
ap-9228	471	8	of	of	ADP
ap-9228	471	9	equation	equation	NOUN
ap-9228	471	10	(	(	PUNCT
ap-9228	471	11	30	30	NUM
ap-9228	471	12	)	)	PUNCT
ap-9228	471	13	of	of	ADP
ap-9228	471	14	v	v	NOUN
ap-9228	471	15	for	for	ADP
ap-9228	471	16	different	different	ADJ
ap-9228	471	17	values	value	NOUN
ap-9228	471	18	of	of	ADP
ap-9228	471	19	x.	x.	NOUN
ap-9228	471	20	figure	figure	NOUN
ap-9228	471	21	7	7	NUM
ap-9228	471	22	.	.	PUNCT
ap-9228	472	1	the	the	DET
ap-9228	472	2	plot	plot	NOUN
ap-9228	472	3	of	of	ADP
ap-9228	472	4	approximate	approximate	ADJ
ap-9228	472	5	solution	solution	NOUN
ap-9228	472	6	,	,	PUNCT
ap-9228	472	7	exact	exact	ADJ
ap-9228	472	8	solution	solution	NOUN
ap-9228	472	9	,	,	PUNCT
ap-9228	472	10	and	and	CCONJ
ap-9228	472	11	absolute	absolute	ADJ
ap-9228	472	12	error	error	NOUN
ap-9228	472	13	of	of	ADP
ap-9228	472	14	u	u	NOUN
ap-9228	472	15	of	of	ADP
ap-9228	472	16	equation	equation	NOUN
ap-9228	472	17	(	(	PUNCT
ap-9228	472	18	30	30	NUM
ap-9228	472	19	)	)	PUNCT
ap-9228	472	20	.	.	PUNCT
ap-9228	473	1	100	100	NUM
ap-9228	473	2	vol	vol	NOUN
ap-9228	473	3	.	.	PUNCT
ap-9228	474	1	64	64	NUM
ap-9228	474	2	no	no	NOUN
ap-9228	474	3	.	.	PUNCT
ap-9228	475	1	2/2024	2/2024	NUM
ap-9228	475	2	an	an	DET
ap-9228	475	3	innovative	innovative	ADJ
ap-9228	475	4	iterative	iterative	NOUN
ap-9228	475	5	approach	approach	NOUN
ap-9228	475	6	to	to	ADP
ap-9228	475	7	solving	solve	VERB
ap-9228	475	8	volterra	volterra	NOUN
ap-9228	475	9	integral	integral	ADJ
ap-9228	475	10	.	.	PUNCT
ap-9228	475	11	.	.	PUNCT
ap-9228	475	12	.	.	PUNCT
ap-9228	476	1	figure	figure	VERB
ap-9228	476	2	8	8	NUM
ap-9228	476	3	.	.	PUNCT
ap-9228	477	1	the	the	DET
ap-9228	477	2	plot	plot	NOUN
ap-9228	477	3	of	of	ADP
ap-9228	477	4	approximate	approximate	ADJ
ap-9228	477	5	solution	solution	NOUN
ap-9228	477	6	,	,	PUNCT
ap-9228	477	7	exact	exact	ADJ
ap-9228	477	8	solution	solution	NOUN
ap-9228	477	9	,	,	PUNCT
ap-9228	477	10	and	and	CCONJ
ap-9228	477	11	absolute	absolute	ADJ
ap-9228	477	12	error	error	NOUN
ap-9228	477	13	of	of	ADP
ap-9228	477	14	v	v	NUM
ap-9228	477	15	of	of	ADP
ap-9228	477	16	equation	equation	NOUN
ap-9228	477	17	(	(	PUNCT
ap-9228	477	18	30	30	NUM
ap-9228	477	19	)	)	PUNCT
ap-9228	477	20	.	.	PUNCT
ap-9228	478	1	5	5	X
ap-9228	478	2	.	.	NOUN
ap-9228	478	3	results	result	NOUN
ap-9228	478	4	and	and	CCONJ
ap-9228	478	5	conclusion	conclusion	NOUN
ap-9228	478	6	in	in	ADP
ap-9228	478	7	conclusion	conclusion	NOUN
ap-9228	478	8	,	,	PUNCT
ap-9228	478	9	the	the	DET
ap-9228	478	10	development	development	NOUN
ap-9228	478	11	of	of	ADP
ap-9228	478	12	an	an	DET
ap-9228	478	13	innovative	innovative	ADJ
ap-9228	478	14	approximation	approximation	NOUN
ap-9228	478	15	method	method	NOUN
ap-9228	478	16	for	for	ADP
ap-9228	478	17	solving	solve	VERB
ap-9228	478	18	second	second	ADJ
ap-9228	478	19	-	-	PUNCT
ap-9228	478	20	kind	kind	NOUN
ap-9228	478	21	volterra	volterra	NOUN
ap-9228	478	22	integral	integral	ADJ
ap-9228	478	23	equations	equation	NOUN
ap-9228	478	24	,	,	PUNCT
ap-9228	478	25	including	include	VERB
ap-9228	478	26	linear	linear	PROPN
ap-9228	478	27	,	,	PUNCT
ap-9228	478	28	nonlinear	nonlinear	ADJ
ap-9228	478	29	,	,	PUNCT
ap-9228	478	30	homogeneous	homogeneous	ADJ
ap-9228	478	31	,	,	PUNCT
ap-9228	478	32	and	and	CCONJ
ap-9228	478	33	nonhomogeneous	nonhomogeneous	ADJ
ap-9228	478	34	cases	case	NOUN
ap-9228	478	35	,	,	PUNCT
ap-9228	478	36	marks	mark	VERB
ap-9228	478	37	a	a	DET
ap-9228	478	38	significant	significant	ADJ
ap-9228	478	39	breakthrough	breakthrough	NOUN
ap-9228	478	40	in	in	ADP
ap-9228	478	41	integral	integral	ADJ
ap-9228	478	42	equation	equation	NOUN
ap-9228	478	43	solving	solving	NOUN
ap-9228	478	44	.	.	PUNCT
ap-9228	479	1	this	this	DET
ap-9228	479	2	method	method	NOUN
ap-9228	479	3	introduces	introduce	VERB
ap-9228	479	4	a	a	DET
ap-9228	479	5	versatile	versatile	ADJ
ap-9228	479	6	and	and	CCONJ
ap-9228	479	7	robust	robust	ADJ
ap-9228	479	8	approach	approach	NOUN
ap-9228	479	9	that	that	PRON
ap-9228	479	10	effectively	effectively	ADV
ap-9228	479	11	addresses	address	VERB
ap-9228	479	12	a	a	DET
ap-9228	479	13	wide	wide	ADJ
ap-9228	479	14	range	range	NOUN
ap-9228	479	15	of	of	ADP
ap-9228	479	16	volterra	volterra	PROPN
ap-9228	479	17	integral	integral	ADJ
ap-9228	479	18	equations	equation	NOUN
ap-9228	479	19	.	.	PUNCT
ap-9228	480	1	incorporating	incorporate	VERB
ap-9228	480	2	innovative	innovative	ADJ
ap-9228	480	3	techniques	technique	NOUN
ap-9228	480	4	and	and	CCONJ
ap-9228	480	5	numerical	numerical	ADJ
ap-9228	480	6	algorithms	algorithm	NOUN
ap-9228	480	7	,	,	PUNCT
ap-9228	480	8	this	this	DET
ap-9228	480	9	novel	novel	ADJ
ap-9228	480	10	approach	approach	NOUN
ap-9228	480	11	accurately	accurately	ADV
ap-9228	480	12	estimates	estimate	VERB
ap-9228	480	13	solutions	solution	NOUN
ap-9228	480	14	to	to	ADP
ap-9228	480	15	both	both	CCONJ
ap-9228	480	16	linear	linear	ADJ
ap-9228	480	17	and	and	CCONJ
ap-9228	480	18	nonlinear	nonlinear	PROPN
ap-9228	480	19	volterra	volterra	PROPN
ap-9228	480	20	equations	equation	NOUN
ap-9228	480	21	,	,	PUNCT
ap-9228	480	22	providing	provide	VERB
ap-9228	480	23	a	a	DET
ap-9228	480	24	comprehensive	comprehensive	ADJ
ap-9228	480	25	solution	solution	NOUN
ap-9228	480	26	framework	framework	NOUN
ap-9228	480	27	capable	capable	ADJ
ap-9228	480	28	of	of	ADP
ap-9228	480	29	handling	handle	VERB
ap-9228	480	30	different	different	ADJ
ap-9228	480	31	scenarios	scenario	NOUN
ap-9228	480	32	.	.	PUNCT
ap-9228	481	1	effectively	effectively	ADV
ap-9228	481	2	navigating	navigate	VERB
ap-9228	481	3	the	the	DET
ap-9228	481	4	complexities	complexity	NOUN
ap-9228	481	5	inherent	inherent	ADJ
ap-9228	481	6	to	to	PART
ap-9228	481	7	volterra	volterra	VERB
ap-9228	481	8	integral	integral	ADJ
ap-9228	481	9	equations	equation	NOUN
ap-9228	481	10	,	,	PUNCT
ap-9228	481	11	including	include	VERB
ap-9228	481	12	nonlinearity	nonlinearity	NOUN
ap-9228	481	13	and	and	CCONJ
ap-9228	481	14	diverse	diverse	ADJ
ap-9228	481	15	nature	nature	NOUN
ap-9228	481	16	,	,	PUNCT
ap-9228	481	17	this	this	DET
ap-9228	481	18	method	method	NOUN
ap-9228	481	19	opens	open	VERB
ap-9228	481	20	avenues	avenue	NOUN
ap-9228	481	21	for	for	ADP
ap-9228	481	22	solving	solve	VERB
ap-9228	481	23	real	real	ADJ
ap-9228	481	24	-	-	PUNCT
ap-9228	481	25	world	world	NOUN
ap-9228	481	26	problems	problem	NOUN
ap-9228	481	27	across	across	ADP
ap-9228	481	28	disciplines	discipline	NOUN
ap-9228	481	29	,	,	PUNCT
ap-9228	481	30	such	such	ADJ
ap-9228	481	31	as	as	ADP
ap-9228	481	32	physics	physics	NOUN
ap-9228	481	33	,	,	PUNCT
ap-9228	481	34	engineering	engineering	NOUN
ap-9228	481	35	,	,	PUNCT
ap-9228	481	36	biology	biology	NOUN
ap-9228	481	37	,	,	PUNCT
ap-9228	481	38	and	and	CCONJ
ap-9228	481	39	economics	economic	NOUN
ap-9228	481	40	.	.	PUNCT
ap-9228	482	1	researchers	researcher	NOUN
ap-9228	482	2	and	and	CCONJ
ap-9228	482	3	practitioners	practitioner	NOUN
ap-9228	482	4	now	now	ADV
ap-9228	482	5	have	have	VERB
ap-9228	482	6	a	a	DET
ap-9228	482	7	powerful	powerful	ADJ
ap-9228	482	8	tool	tool	NOUN
ap-9228	482	9	with	with	ADP
ap-9228	482	10	a	a	DET
ap-9228	482	11	wide	wide	ADJ
ap-9228	482	12	range	range	NOUN
ap-9228	482	13	of	of	ADP
ap-9228	482	14	applications	application	NOUN
ap-9228	482	15	.	.	PUNCT
ap-9228	483	1	while	while	SCONJ
ap-9228	483	2	further	further	ADJ
ap-9228	483	3	research	research	NOUN
ap-9228	483	4	and	and	CCONJ
ap-9228	483	5	validation	validation	NOUN
ap-9228	483	6	is	be	AUX
ap-9228	483	7	needed	need	VERB
ap-9228	483	8	to	to	PART
ap-9228	483	9	fully	fully	ADV
ap-9228	483	10	explore	explore	VERB
ap-9228	483	11	the	the	DET
ap-9228	483	12	capabilities	capability	NOUN
ap-9228	483	13	and	and	CCONJ
ap-9228	483	14	limitations	limitation	NOUN
ap-9228	483	15	of	of	ADP
ap-9228	483	16	the	the	DET
ap-9228	483	17	method	method	NOUN
ap-9228	483	18	,	,	PUNCT
ap-9228	483	19	its	its	PRON
ap-9228	483	20	potential	potential	NOUN
ap-9228	483	21	to	to	PART
ap-9228	483	22	revolutionise	revolutionise	VERB
ap-9228	483	23	the	the	DET
ap-9228	483	24	field	field	NOUN
ap-9228	483	25	of	of	ADP
ap-9228	483	26	integral	integral	ADJ
ap-9228	483	27	equations	equation	NOUN
ap-9228	483	28	and	and	CCONJ
ap-9228	483	29	improve	improve	VERB
ap-9228	483	30	our	our	PRON
ap-9228	483	31	understanding	understanding	NOUN
ap-9228	483	32	of	of	ADP
ap-9228	483	33	complex	complex	ADJ
ap-9228	483	34	systems	system	NOUN
ap-9228	483	35	is	be	AUX
ap-9228	483	36	evident	evident	ADJ
ap-9228	483	37	.	.	PUNCT
ap-9228	484	1	in	in	ADP
ap-9228	484	2	summary	summary	NOUN
ap-9228	484	3	,	,	PUNCT
ap-9228	484	4	the	the	DET
ap-9228	484	5	analysis	analysis	NOUN
ap-9228	484	6	of	of	ADP
ap-9228	484	7	the	the	DET
ap-9228	484	8	results	result	NOUN
ap-9228	484	9	in	in	ADP
ap-9228	484	10	figures	figure	NOUN
ap-9228	484	11	1–8	1–8	NUM
ap-9228	484	12	and	and	CCONJ
ap-9228	484	13	tables	table	NOUN
ap-9228	484	14	1–8	1–8	PRON
ap-9228	484	15	shows	show	VERB
ap-9228	484	16	that	that	SCONJ
ap-9228	484	17	the	the	DET
ap-9228	484	18	hussein	hussein	PROPN
ap-9228	484	19	jassim	jassim	PROPN
ap-9228	484	20	method	method	PROPN
ap-9228	484	21	(	(	PUNCT
ap-9228	484	22	hjm	hjm	NOUN
ap-9228	484	23	)	)	PUNCT
ap-9228	484	24	outperforms	outperform	VERB
ap-9228	484	25	the	the	DET
ap-9228	484	26	reduced	reduce	VERB
ap-9228	484	27	differential	differential	ADJ
ap-9228	484	28	transform	transform	NOUN
ap-9228	484	29	method	method	NOUN
ap-9228	484	30	(	(	PUNCT
ap-9228	484	31	rdtm	rdtm	VERB
ap-9228	484	32	)	)	PUNCT
ap-9228	484	33	in	in	ADP
ap-9228	484	34	terms	term	NOUN
ap-9228	484	35	of	of	ADP
ap-9228	484	36	accuracy	accuracy	NOUN
ap-9228	484	37	and	and	CCONJ
ap-9228	484	38	has	have	VERB
ap-9228	484	39	a	a	DET
ap-9228	484	40	lower	low	ADJ
ap-9228	484	41	absolute	absolute	ADJ
ap-9228	484	42	error	error	NOUN
ap-9228	484	43	.	.	PUNCT
ap-9228	485	1	in	in	ADP
ap-9228	485	2	addition	addition	NOUN
ap-9228	485	3	,	,	PUNCT
ap-9228	485	4	accuracy	accuracy	NOUN
ap-9228	485	5	of	of	ADP
ap-9228	485	6	the	the	DET
ap-9228	485	7	hjm	hjm	NOUN
ap-9228	485	8	is	be	AUX
ap-9228	485	9	comparable	comparable	ADJ
ap-9228	485	10	to	to	ADP
ap-9228	485	11	that	that	PRON
ap-9228	485	12	of	of	ADP
ap-9228	485	13	the	the	DET
ap-9228	485	14	power	power	NOUN
ap-9228	485	15	series	series	PROPN
ap-9228	485	16	method	method	NOUN
ap-9228	485	17	(	(	PUNCT
ap-9228	485	18	psm	psm	PROPN
ap-9228	485	19	)	)	PUNCT
ap-9228	485	20	,	,	PUNCT
ap-9228	485	21	highlighting	highlight	VERB
ap-9228	485	22	its	its	PRON
ap-9228	485	23	outstanding	outstanding	ADJ
ap-9228	485	24	effectiveness	effectiveness	NOUN
ap-9228	485	25	.	.	PUNCT
ap-9228	486	1	however	however	ADV
ap-9228	486	2	,	,	PUNCT
ap-9228	486	3	the	the	DET
ap-9228	486	4	hjm	hjm	NOUN
ap-9228	486	5	exhibits	exhibit	VERB
ap-9228	486	6	a	a	DET
ap-9228	486	7	higher	high	ADJ
ap-9228	486	8	absolute	absolute	ADJ
ap-9228	486	9	error	error	NOUN
ap-9228	486	10	as	as	SCONJ
ap-9228	486	11	compared	compare	VERB
ap-9228	486	12	to	to	ADP
ap-9228	486	13	the	the	DET
ap-9228	486	14	adomian	adomian	NOUN
ap-9228	486	15	decomposition	decomposition	NOUN
ap-9228	486	16	method	method	NOUN
ap-9228	486	17	(	(	PUNCT
ap-9228	486	18	adm	adm	PROPN
ap-9228	486	19	)	)	PUNCT
ap-9228	486	20	.	.	PUNCT
ap-9228	487	1	these	these	DET
ap-9228	487	2	findings	finding	NOUN
ap-9228	487	3	contribute	contribute	VERB
ap-9228	487	4	to	to	ADP
ap-9228	487	5	a	a	DET
ap-9228	487	6	deeper	deep	ADJ
ap-9228	487	7	understanding	understanding	NOUN
ap-9228	487	8	of	of	ADP
ap-9228	487	9	the	the	DET
ap-9228	487	10	hjm	hjm	PROPN
ap-9228	487	11	’s	’s	PART
ap-9228	487	12	efficacy	efficacy	NOUN
ap-9228	487	13	,	,	PUNCT
ap-9228	487	14	emphasizing	emphasize	VERB
ap-9228	487	15	its	its	PRON
ap-9228	487	16	relative	relative	ADJ
ap-9228	487	17	superiority	superiority	NOUN
ap-9228	487	18	in	in	ADP
ap-9228	487	19	solving	solve	VERB
ap-9228	487	20	second	second	ADJ
ap-9228	487	21	-	-	PUNCT
ap-9228	487	22	kind	kind	NOUN
ap-9228	487	23	volterra	volterra	NOUN
ap-9228	487	24	integral	integral	ADJ
ap-9228	487	25	equations	equation	NOUN
ap-9228	487	26	,	,	PUNCT
ap-9228	487	27	whether	whether	SCONJ
ap-9228	487	28	linear	linear	ADJ
ap-9228	487	29	or	or	CCONJ
ap-9228	487	30	nonlinear	nonlinear	ADJ
ap-9228	487	31	.	.	PUNCT
ap-9228	488	1	references	reference	NOUN
ap-9228	488	2	[	[	X
ap-9228	488	3	1	1	NUM
ap-9228	488	4	]	]	PUNCT
ap-9228	488	5	m.	m.	NOUN
ap-9228	488	6	higazy	higazy	NOUN
ap-9228	488	7	,	,	PUNCT
ap-9228	488	8	s.	s.	PROPN
ap-9228	488	9	aggarwal	aggarwal	PROPN
ap-9228	488	10	,	,	PUNCT
ap-9228	488	11	t.	t.	PROPN
ap-9228	488	12	a.	a.	NOUN
ap-9228	488	13	nofal	nofal	PROPN
ap-9228	488	14	.	.	PUNCT
ap-9228	489	1	sawi	sawi	ADJ
ap-9228	489	2	decomposition	decomposition	NOUN
ap-9228	489	3	method	method	NOUN
ap-9228	489	4	for	for	ADP
ap-9228	489	5	volterra	volterra	NOUN
ap-9228	489	6	integral	integral	ADJ
ap-9228	489	7	equation	equation	NOUN
ap-9228	489	8	with	with	ADP
ap-9228	489	9	application	application	NOUN
ap-9228	489	10	.	.	PUNCT
ap-9228	490	1	journal	journal	NOUN
ap-9228	490	2	of	of	ADP
ap-9228	490	3	mathematics	mathematics	PROPN
ap-9228	490	4	2020:6687134	2020:6687134	NUM
ap-9228	490	5	,	,	PUNCT
ap-9228	490	6	2020	2020	NUM
ap-9228	490	7	.	.	PUNCT
ap-9228	491	1	https://doi.org/10.1155/2020/6687134	https://doi.org/10.1155/2020/6687134	PROPN
ap-9228	492	1	[	[	X
ap-9228	492	2	2	2	NUM
ap-9228	492	3	]	]	PUNCT
ap-9228	492	4	h.	h.	PROPN
ap-9228	492	5	k.	k.	PROPN
ap-9228	492	6	jassim	jassim	PROPN
ap-9228	492	7	,	,	PUNCT
ap-9228	492	8	m.	m.	NOUN
ap-9228	492	9	a.	a.	PROPN
ap-9228	492	10	hussein	hussein	PROPN
ap-9228	492	11	.	.	PUNCT
ap-9228	493	1	a	a	DET
ap-9228	493	2	new	new	ADJ
ap-9228	493	3	approach	approach	NOUN
ap-9228	493	4	for	for	ADP
ap-9228	493	5	solving	solve	VERB
ap-9228	493	6	nonlinear	nonlinear	ADJ
ap-9228	493	7	fractional	fractional	ADJ
ap-9228	493	8	ordinary	ordinary	ADJ
ap-9228	493	9	differential	differential	ADJ
ap-9228	493	10	equations	equation	NOUN
ap-9228	493	11	.	.	PUNCT
ap-9228	494	1	mathematics	mathematic	NOUN
ap-9228	494	2	11(7):1565	11(7):1565	NUM
ap-9228	494	3	,	,	PUNCT
ap-9228	494	4	2023	2023	NUM
ap-9228	494	5	.	.	PUNCT
ap-9228	495	1	https://doi.org/10.3390/math11071565	https://doi.org/10.3390/math11071565	PROPN
ap-9228	495	2	[	[	X
ap-9228	495	3	3	3	X
ap-9228	495	4	]	]	X
ap-9228	495	5	h.	h.	PROPN
ap-9228	495	6	brunner	brunner	PROPN
ap-9228	495	7	,	,	PUNCT
ap-9228	495	8	m.	m.	PROPN
ap-9228	495	9	d.	d.	PROPN
ap-9228	495	10	evans	evans	PROPN
ap-9228	495	11	.	.	PUNCT
ap-9228	496	1	piecewise	piecewise	NOUN
ap-9228	496	2	polynomial	polynomial	ADJ
ap-9228	496	3	collocation	collocation	NOUN
ap-9228	496	4	for	for	ADP
ap-9228	496	5	volterra	volterra	NOUN
ap-9228	496	6	-	-	PUNCT
ap-9228	496	7	type	type	NOUN
ap-9228	496	8	integral	integral	ADJ
ap-9228	496	9	equations	equation	NOUN
ap-9228	496	10	of	of	ADP
ap-9228	496	11	the	the	DET
ap-9228	496	12	second	second	ADJ
ap-9228	496	13	kind	kind	NOUN
ap-9228	496	14	.	.	PUNCT
ap-9228	497	1	i	i	PRON
ap-9228	497	2	m	m	VERB
ap-9228	497	3	a	a	DET
ap-9228	497	4	journal	journal	NOUN
ap-9228	497	5	of	of	ADP
ap-9228	497	6	applied	apply	VERB
ap-9228	497	7	mathematics	mathematic	NOUN
ap-9228	497	8	20(4):415–423	20(4):415–423	NUM
ap-9228	497	9	,	,	PUNCT
ap-9228	497	10	1977	1977	NUM
ap-9228	497	11	.	.	PUNCT
ap-9228	498	1	https://doi.org/10.1093/imamat/20.4.415	https://doi.org/10.1093/imamat/20.4.415	PUNCT
ap-9228	498	2	[	[	X
ap-9228	498	3	4	4	NUM
ap-9228	498	4	]	]	PUNCT
ap-9228	498	5	h.	h.	PROPN
ap-9228	498	6	k.	k.	PROPN
ap-9228	498	7	jassim	jassim	PROPN
ap-9228	498	8	,	,	PUNCT
ap-9228	498	9	m.	m.	NOUN
ap-9228	498	10	a.	a.	PROPN
ap-9228	498	11	hussein	hussein	PROPN
ap-9228	498	12	,	,	PUNCT
ap-9228	498	13	m.	m.	PROPN
ap-9228	498	14	r.	r.	PROPN
ap-9228	498	15	ali	ali	PROPN
ap-9228	498	16	.	.	PUNCT
ap-9228	499	1	an	an	DET
ap-9228	499	2	efficient	efficient	ADJ
ap-9228	499	3	homotopy	homotopy	NOUN
ap-9228	499	4	permutation	permutation	NOUN
ap-9228	499	5	technique	technique	NOUN
ap-9228	499	6	for	for	ADP
ap-9228	499	7	solving	solve	VERB
ap-9228	499	8	fractional	fractional	ADJ
ap-9228	499	9	differential	differential	ADJ
ap-9228	499	10	equations	equation	NOUN
ap-9228	499	11	using	use	VERB
ap-9228	499	12	atangana	atangana	PROPN
ap-9228	499	13	-	-	PUNCT
ap-9228	499	14	baleanu	baleanu	PROPN
ap-9228	499	15	-	-	PUNCT
ap-9228	499	16	caputo	caputo	NOUN
ap-9228	499	17	operator	operator	NOUN
ap-9228	499	18	.	.	PUNCT
ap-9228	500	1	aip	aip	PROPN
ap-9228	500	2	conference	conference	NOUN
ap-9228	500	3	proceedings	proceeding	NOUN
ap-9228	500	4	2845:060008	2845:060008	NUM
ap-9228	500	5	,	,	PUNCT
ap-9228	500	6	2023	2023	NUM
ap-9228	500	7	.	.	PUNCT
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ap-9228	502	2	5	5	NUM
ap-9228	502	3	]	]	PUNCT
ap-9228	502	4	p.	p.	NOUN
ap-9228	502	5	j.	j.	PROPN
ap-9228	502	6	van	van	PROPN
ap-9228	502	7	der	der	PROPN
ap-9228	502	8	houwen	houwen	NOUN
ap-9228	502	9	,	,	PUNCT
ap-9228	502	10	p.	p.	PROPN
ap-9228	502	11	h.	h.	PROPN
ap-9228	502	12	m.	m.	PROPN
ap-9228	502	13	wolkenfelt	wolkenfelt	PROPN
ap-9228	502	14	,	,	PUNCT
ap-9228	502	15	c.	c.	PROPN
ap-9228	502	16	t.	t.	PROPN
ap-9228	502	17	h.	h.	PROPN
ap-9228	502	18	baker	baker	PROPN
ap-9228	502	19	.	.	PUNCT
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ap-9228	503	2	and	and	CCONJ
ap-9228	503	3	stability	stability	NOUN
ap-9228	503	4	analysis	analysis	NOUN
ap-9228	503	5	for	for	ADP
ap-9228	503	6	modified	modified	ADJ
ap-9228	503	7	runge	runge	NOUN
ap-9228	503	8	-	-	PUNCT
ap-9228	503	9	kutta	kutta	NOUN
ap-9228	503	10	methods	method	NOUN
ap-9228	503	11	in	in	ADP
ap-9228	503	12	the	the	DET
ap-9228	503	13	numerical	numerical	ADJ
ap-9228	503	14	treatment	treatment	NOUN
ap-9228	503	15	of	of	ADP
ap-9228	503	16	second	second	ADJ
ap-9228	503	17	-	-	PUNCT
ap-9228	503	18	kind	kind	NOUN
ap-9228	503	19	volterra	volterra	NOUN
ap-9228	503	20	integral	integral	ADJ
ap-9228	503	21	equations	equation	NOUN
ap-9228	503	22	.	.	PUNCT
ap-9228	504	1	i	i	PRON
ap-9228	504	2	m	m	VERB
ap-9228	504	3	a	a	DET
ap-9228	504	4	journal	journal	NOUN
ap-9228	504	5	of	of	ADP
ap-9228	504	6	numerical	numerical	ADJ
ap-9228	504	7	analysis	analysis	NOUN
ap-9228	504	8	1(3):303–328	1(3):303–328	NUM
ap-9228	504	9	,	,	PUNCT
ap-9228	504	10	1981	1981	NUM
ap-9228	504	11	.	.	PUNCT
ap-9228	505	1	https://doi.org/10.1093/imanum/1.3.303	https://doi.org/10.1093/imanum/1.3.303	X
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ap-9228	506	2	6	6	NUM
ap-9228	506	3	]	]	PUNCT
ap-9228	506	4	m.	m.	NOUN
ap-9228	506	5	a.	a.	PROPN
ap-9228	506	6	hussein	hussein	PROPN
ap-9228	506	7	,	,	PUNCT
ap-9228	506	8	h.	h.	PROPN
ap-9228	506	9	k.	k.	PROPN
ap-9228	506	10	jassim	jassim	PROPN
ap-9228	506	11	.	.	PUNCT
ap-9228	507	1	analysis	analysis	NOUN
ap-9228	507	2	of	of	ADP
ap-9228	507	3	fractional	fractional	ADJ
ap-9228	507	4	differential	differential	ADJ
ap-9228	507	5	equations	equation	NOUN
ap-9228	507	6	with	with	ADP
ap-9228	507	7	antagana	antagana	PROPN
ap-9228	507	8	-	-	PUNCT
ap-9228	507	9	baleanu	baleanu	ADJ
ap-9228	507	10	fractional	fractional	ADJ
ap-9228	507	11	operator	operator	NOUN
ap-9228	507	12	.	.	PUNCT
ap-9228	508	1	progress	progress	NOUN
ap-9228	508	2	in	in	ADP
ap-9228	508	3	fractional	fractional	ADJ
ap-9228	508	4	differentiation	differentiation	NOUN
ap-9228	508	5	and	and	CCONJ
ap-9228	508	6	applications	application	NOUN
ap-9228	508	7	9(4):681–686	9(4):681–686	NUM
ap-9228	508	8	,	,	PUNCT
ap-9228	508	9	2023	2023	NUM
ap-9228	508	10	.	.	PUNCT
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ap-9228	510	1	[	[	X
ap-9228	510	2	7	7	NUM
ap-9228	510	3	]	]	X
ap-9228	510	4	a.-m	a.-m	NOUN
ap-9228	510	5	.	.	PUNCT
ap-9228	511	1	wazwaz	wazwaz	NOUN
ap-9228	511	2	.	.	PUNCT
ap-9228	512	1	linear	linear	ADJ
ap-9228	512	2	and	and	CCONJ
ap-9228	512	3	nonlinear	nonlinear	ADJ
ap-9228	512	4	integral	integral	ADJ
ap-9228	512	5	equations	equation	NOUN
ap-9228	512	6	.	.	PUNCT
ap-9228	513	1	springer	springer	NOUN
ap-9228	513	2	,	,	PUNCT
ap-9228	513	3	1st	1st	ADJ
ap-9228	513	4	edn	edn	PROPN
ap-9228	513	5	.	.	PUNCT
ap-9228	513	6	,	,	PUNCT
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ap-9228	513	8	.	.	PUNCT
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ap-9228	514	2	978	978	NUM
ap-9228	514	3	-	-	SYM
ap-9228	514	4	3	3	NUM
ap-9228	514	5	-	-	PUNCT
ap-9228	514	6	642	642	NUM
ap-9228	514	7	-	-	PUNCT
ap-9228	514	8	21449	21449	NUM
ap-9228	514	9	-	-	PUNCT
ap-9228	514	10	3	3	NUM
ap-9228	514	11	.	.	PUNCT
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ap-9228	515	11	a.	a.	PROPN
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ap-9228	515	14	h.	h.	PROPN
ap-9228	515	15	k.	k.	PROPN
ap-9228	515	16	jassim	jassim	PROPN
ap-9228	515	17	,	,	PUNCT
ap-9228	515	18	a.	a.	PROPN
ap-9228	515	19	k.	k.	PROPN
ap-9228	515	20	jassim	jassim	PROPN
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ap-9228	516	2	8	8	NUM
ap-9228	516	3	]	]	X
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ap-9228	516	9	a.	a.	PROPN
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ap-9228	516	11	.	.	PUNCT
ap-9228	517	1	a	a	DET
ap-9228	517	2	novel	novel	ADJ
ap-9228	517	3	formulation	formulation	NOUN
ap-9228	517	4	of	of	ADP
ap-9228	517	5	the	the	DET
ap-9228	517	6	fractional	fractional	ADJ
ap-9228	517	7	derivative	derivative	NOUN
ap-9228	517	8	with	with	ADP
ap-9228	517	9	the	the	DET
ap-9228	517	10	order	order	NOUN
ap-9228	517	11	α	α	PRON
ap-9228	517	12	≥	≥	NOUN
ap-9228	517	13	0	0	NUM
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ap-9228	517	16	the	the	DET
ap-9228	517	17	singular	singular	ADJ
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ap-9228	519	3	9	9	X
ap-9228	519	4	]	]	X
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ap-9228	520	3	see	see	NOUN
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ap-9228	520	6	-	-	PUNCT
ap-9228	520	7	emad	emad	PROPN
ap-9228	520	8	-	-	PUNCT
ap-9228	520	9	eman	eman	NOUN
ap-9228	520	10	)	)	PUNCT
ap-9228	520	11	integral	integral	ADJ
ap-9228	520	12	transform	transform	NOUN
ap-9228	520	13	for	for	ADP
ap-9228	520	14	solving	solve	VERB
ap-9228	520	15	2nd	2nd	ADJ
ap-9228	520	16	kind	kind	ADJ
ap-9228	520	17	linear	linear	PROPN
ap-9228	520	18	volterra	volterra	PROPN
ap-9228	520	19	integral	integral	ADJ
ap-9228	520	20	equations	equation	NOUN
ap-9228	520	21	.	.	PUNCT
ap-9228	521	1	international	international	ADJ
ap-9228	521	2	journal	journal	NOUN
ap-9228	521	3	of	of	ADP
ap-9228	521	4	all	all	DET
ap-9228	521	5	research	research	NOUN
ap-9228	521	6	education	education	NOUN
ap-9228	521	7	and	and	CCONJ
ap-9228	521	8	scientific	scientific	ADJ
ap-9228	521	9	methods	method	NOUN
ap-9228	521	10	10(12):2029–2034	10(12):2029–2034	NUM
ap-9228	521	11	,	,	PUNCT
ap-9228	521	12	2022	2022	NUM
ap-9228	521	13	.	.	PUNCT
ap-9228	522	1	[	[	X
ap-9228	522	2	10	10	NUM
ap-9228	522	3	]	]	X
ap-9228	522	4	s.	s.	PROPN
ap-9228	522	5	aggarwal	aggarwal	PROPN
ap-9228	522	6	,	,	PUNCT
ap-9228	522	7	s.	s.	PROPN
ap-9228	522	8	d.	d.	PROPN
ap-9228	522	9	sharma	sharma	PROPN
ap-9228	522	10	,	,	PUNCT
ap-9228	522	11	r.	r.	PROPN
ap-9228	522	12	chaudhary	chaudhary	PROPN
ap-9228	522	13	.	.	PUNCT
ap-9228	523	1	method	method	PROPN
ap-9228	523	2	of	of	ADP
ap-9228	523	3	taylor	taylor	PROPN
ap-9228	523	4	’s	’s	PART
ap-9228	523	5	series	series	NOUN
ap-9228	523	6	for	for	ADP
ap-9228	523	7	non	non	ADJ
ap-9228	523	8	-	-	ADJ
ap-9228	523	9	linear	linear	ADJ
ap-9228	523	10	second	second	ADJ
ap-9228	523	11	kind	kind	NOUN
ap-9228	523	12	non	non	ADJ
ap-9228	523	13	-	-	ADJ
ap-9228	523	14	homogeneous	homogeneous	ADJ
ap-9228	523	15	volterra	volterra	NOUN
ap-9228	523	16	integral	integral	ADJ
ap-9228	523	17	equations	equation	NOUN
ap-9228	523	18	.	.	PUNCT
ap-9228	524	1	international	international	ADJ
ap-9228	524	2	journal	journal	PROPN
ap-9228	524	3	of	of	ADP
ap-9228	524	4	research	research	NOUN
ap-9228	524	5	and	and	CCONJ
ap-9228	524	6	innovation	innovation	NOUN
ap-9228	524	7	in	in	ADP
ap-9228	524	8	applied	apply	VERB
ap-9228	524	9	science	science	NOUN
ap-9228	524	10	5(5):40–43	5(5):40–43	NUM
ap-9228	524	11	,	,	PUNCT
ap-9228	524	12	2020	2020	NUM
ap-9228	524	13	.	.	PUNCT
ap-9228	525	1	[	[	X
ap-9228	525	2	11	11	NUM
ap-9228	525	3	]	]	X
ap-9228	525	4	r.	r.	PROPN
ap-9228	525	5	chauhan	chauhan	PROPN
ap-9228	525	6	,	,	PUNCT
ap-9228	525	7	s.	s.	PROPN
ap-9228	525	8	aggarwal	aggarwal	PROPN
ap-9228	525	9	.	.	PUNCT
ap-9228	526	1	laplace	laplace	PROPN
ap-9228	526	2	transform	transform	VERB
ap-9228	526	3	for	for	ADP
ap-9228	526	4	convolution	convolution	NOUN
ap-9228	526	5	type	type	NOUN
ap-9228	526	6	linear	linear	PROPN
ap-9228	526	7	volterra	volterra	NOUN
ap-9228	526	8	integral	integral	ADJ
ap-9228	526	9	equation	equation	NOUN
ap-9228	526	10	of	of	ADP
ap-9228	526	11	second	second	ADJ
ap-9228	526	12	kind	kind	NOUN
ap-9228	526	13	.	.	PUNCT
ap-9228	527	1	journal	journal	NOUN
ap-9228	527	2	of	of	ADP
ap-9228	527	3	advanced	advanced	ADJ
ap-9228	527	4	research	research	NOUN
ap-9228	527	5	in	in	ADP
ap-9228	527	6	applied	apply	VERB
ap-9228	527	7	mathematics	mathematic	NOUN
ap-9228	527	8	and	and	CCONJ
ap-9228	527	9	statistics	statistic	NOUN
ap-9228	527	10	4(3	4(3	NUM
ap-9228	527	11	&	&	CCONJ
ap-9228	527	12	4):1–7	4):1–7	PROPN
ap-9228	527	13	,	,	PUNCT
ap-9228	527	14	2019	2019	NUM
ap-9228	527	15	.	.	PUNCT
ap-9228	528	1	https://doi.org/10.24321/2455.7021.202304	https://doi.org/10.24321/2455.7021.202304	PROPN
ap-9228	529	1	[	[	X
ap-9228	529	2	12	12	NUM
ap-9228	529	3	]	]	X
ap-9228	529	4	s.	s.	PROPN
ap-9228	529	5	aggarwal	aggarwal	PROPN
ap-9228	529	6	,	,	PUNCT
ap-9228	529	7	k.	k.	PROPN
ap-9228	529	8	bhatnagar	bhatnagar	PROPN
ap-9228	529	9	,	,	PUNCT
ap-9228	529	10	a.	a.	PROPN
ap-9228	529	11	dua	dua	PROPN
ap-9228	529	12	.	.	PROPN
ap-9228	529	13	method	method	PROPN
ap-9228	529	14	of	of	ADP
ap-9228	529	15	taylor	taylor	PROPN
ap-9228	529	16	’s	’s	PART
ap-9228	529	17	series	series	NOUN
ap-9228	529	18	for	for	ADP
ap-9228	529	19	the	the	DET
ap-9228	529	20	primitive	primitive	NOUN
ap-9228	529	21	of	of	ADP
ap-9228	529	22	linear	linear	ADJ
ap-9228	529	23	second	second	ADJ
ap-9228	529	24	kind	kind	NOUN
ap-9228	529	25	non	non	ADJ
ap-9228	529	26	-	-	ADJ
ap-9228	529	27	homogeneous	homogeneous	ADJ
ap-9228	529	28	volterra	volterra	NOUN
ap-9228	529	29	integral	integral	ADJ
ap-9228	529	30	equations	equation	NOUN
ap-9228	529	31	.	.	PUNCT
ap-9228	530	1	international	international	ADJ
ap-9228	530	2	journal	journal	PROPN
ap-9228	530	3	of	of	ADP
ap-9228	530	4	research	research	NOUN
ap-9228	530	5	and	and	CCONJ
ap-9228	530	6	innovation	innovation	NOUN
ap-9228	530	7	in	in	ADP
ap-9228	530	8	applied	apply	VERB
ap-9228	530	9	science	science	NOUN
ap-9228	530	10	5(5):32–35	5(5):32–35	NUM
ap-9228	530	11	,	,	PUNCT
ap-9228	530	12	2020	2020	NUM
ap-9228	530	13	.	.	PUNCT
ap-9228	531	1	[	[	X
ap-9228	531	2	13	13	NUM
ap-9228	531	3	]	]	PUNCT
ap-9228	531	4	h.	h.	PROPN
ap-9228	531	5	k.	k.	PROPN
ap-9228	531	6	jassim	jassim	PROPN
ap-9228	531	7	,	,	PUNCT
ap-9228	531	8	m.	m.	NOUN
ap-9228	531	9	a.	a.	PROPN
ap-9228	531	10	s.	s.	PROPN
ap-9228	531	11	hussain	hussain	PROPN
ap-9228	531	12	.	.	PUNCT
ap-9228	532	1	on	on	ADP
ap-9228	532	2	approximate	approximate	ADJ
ap-9228	532	3	solutions	solution	NOUN
ap-9228	532	4	for	for	ADP
ap-9228	532	5	fractional	fractional	ADJ
ap-9228	532	6	system	system	NOUN
ap-9228	532	7	of	of	ADP
ap-9228	532	8	differential	differential	ADJ
ap-9228	532	9	equations	equation	NOUN
ap-9228	532	10	with	with	ADP
ap-9228	532	11	caputo	caputo	PROPN
ap-9228	532	12	-	-	PUNCT
ap-9228	532	13	fabrizio	fabrizio	PROPN
ap-9228	532	14	fractional	fractional	ADJ
ap-9228	532	15	operator	operator	NOUN
ap-9228	532	16	.	.	PUNCT
ap-9228	533	1	journal	journal	PROPN
ap-9228	533	2	of	of	ADP
ap-9228	533	3	mathematics	mathematic	NOUN
ap-9228	533	4	and	and	CCONJ
ap-9228	533	5	computer	computer	NOUN
ap-9228	533	6	science	science	NOUN
ap-9228	533	7	23(1):58–66	23(1):58–66	NUM
ap-9228	533	8	,	,	PUNCT
ap-9228	533	9	2020	2020	NUM
ap-9228	533	10	.	.	PUNCT
ap-9228	534	1	https://doi.org/10.22436/jmcs.023.01.06	https://doi.org/10.22436/jmcs.023.01.06	NUM
ap-9228	535	1	[	[	X
ap-9228	535	2	14	14	NUM
ap-9228	535	3	]	]	X
ap-9228	535	4	g.	g.	PROPN
ap-9228	535	5	adomian	adomian	PROPN
ap-9228	535	6	.	.	PUNCT
ap-9228	536	1	a	a	DET
ap-9228	536	2	new	new	ADJ
ap-9228	536	3	approach	approach	NOUN
ap-9228	536	4	to	to	ADP
ap-9228	536	5	nonlinear	nonlinear	ADJ
ap-9228	536	6	partial	partial	ADJ
ap-9228	536	7	differential	differential	NOUN
ap-9228	536	8	equations	equation	NOUN
ap-9228	536	9	.	.	PUNCT
ap-9228	537	1	journal	journal	PROPN
ap-9228	537	2	of	of	ADP
ap-9228	537	3	mathematical	mathematical	ADJ
ap-9228	537	4	analysis	analysis	NOUN
ap-9228	537	5	and	and	CCONJ
ap-9228	537	6	applications	application	NOUN
ap-9228	537	7	102(2):420–434	102(2):420–434	NUM
ap-9228	537	8	,	,	PUNCT
ap-9228	537	9	1984	1984	NUM
ap-9228	537	10	.	.	PUNCT
ap-9228	538	1	https://doi.org/10.1016/0022-247x(84)90182-3	https://doi.org/10.1016/0022-247x(84)90182-3	PROPN
ap-9228	539	1	[	[	X
ap-9228	539	2	15	15	NUM
ap-9228	539	3	]	]	X
ap-9228	539	4	j.	j.	PROPN
ap-9228	539	5	he	he	PROPN
ap-9228	539	6	.	.	PUNCT
ap-9228	540	1	a	a	DET
ap-9228	540	2	new	new	ADJ
ap-9228	540	3	approach	approach	NOUN
ap-9228	540	4	to	to	ADP
ap-9228	540	5	nonlinear	nonlinear	ADJ
ap-9228	540	6	partial	partial	ADJ
ap-9228	540	7	differential	differential	NOUN
ap-9228	540	8	equations	equation	NOUN
ap-9228	540	9	.	.	PUNCT
ap-9228	541	1	communications	communication	NOUN
ap-9228	541	2	in	in	ADP
ap-9228	541	3	nonlinear	nonlinear	ADJ
ap-9228	541	4	science	science	NOUN
ap-9228	541	5	and	and	CCONJ
ap-9228	541	6	numerical	numerical	PROPN
ap-9228	541	7	simulation	simulation	PROPN
ap-9228	541	8	2(4):230–235	2(4):230–235	PROPN
ap-9228	541	9	,	,	PUNCT
ap-9228	541	10	1997	1997	NUM
ap-9228	541	11	.	.	PUNCT
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ap-9228	543	1	[	[	X
ap-9228	543	2	16	16	NUM
ap-9228	543	3	]	]	X
ap-9228	543	4	j.-h	j.-h	NOUN
ap-9228	543	5	.	.	PUNCT
ap-9228	544	1	he	he	PRON
ap-9228	544	2	.	.	PUNCT
ap-9228	545	1	homotopy	homotopy	VERB
ap-9228	545	2	perturbation	perturbation	NOUN
ap-9228	545	3	method	method	NOUN
ap-9228	545	4	:	:	PUNCT
ap-9228	545	5	a	a	DET
ap-9228	545	6	new	new	ADJ
ap-9228	545	7	nonlinear	nonlinear	ADJ
ap-9228	545	8	analytical	analytical	ADJ
ap-9228	545	9	technique	technique	NOUN
ap-9228	545	10	.	.	PUNCT
ap-9228	546	1	applied	apply	VERB
ap-9228	546	2	mathematics	mathematic	NOUN
ap-9228	546	3	and	and	CCONJ
ap-9228	546	4	computation	computation	NOUN
ap-9228	546	5	135(1):73–79	135(1):73–79	NUM
ap-9228	546	6	,	,	PUNCT
ap-9228	546	7	2003	2003	NUM
ap-9228	546	8	.	.	PUNCT
ap-9228	547	1	https://doi.org/10.1016/s0096-3003(01)00312-5	https://doi.org/10.1016/s0096-3003(01)00312-5	NOUN
ap-9228	548	1	[	[	X
ap-9228	548	2	17	17	NUM
ap-9228	548	3	]	]	X
ap-9228	548	4	v.	v.	ADP
ap-9228	548	5	daftardar	daftardar	NOUN
ap-9228	548	6	-	-	PUNCT
ap-9228	548	7	gejji	gejji	NOUN
ap-9228	548	8	,	,	PUNCT
ap-9228	548	9	h.	h.	PROPN
ap-9228	548	10	jafari	jafari	PROPN
ap-9228	548	11	.	.	PUNCT
ap-9228	549	1	an	an	DET
ap-9228	549	2	iterative	iterative	NOUN
ap-9228	549	3	method	method	NOUN
ap-9228	549	4	for	for	ADP
ap-9228	549	5	solving	solve	VERB
ap-9228	549	6	nonlinear	nonlinear	ADJ
ap-9228	549	7	functional	functional	ADJ
ap-9228	549	8	equations	equation	NOUN
ap-9228	549	9	.	.	PUNCT
ap-9228	550	1	journal	journal	PROPN
ap-9228	550	2	of	of	ADP
ap-9228	550	3	mathematical	mathematical	ADJ
ap-9228	550	4	analysis	analysis	NOUN
ap-9228	550	5	and	and	CCONJ
ap-9228	550	6	applications	application	NOUN
ap-9228	550	7	316(2):753–763	316(2):753–763	NUM
ap-9228	550	8	,	,	PUNCT
ap-9228	550	9	2006	2006	NUM
ap-9228	550	10	.	.	PUNCT
ap-9228	551	1	https://doi.org/10.1016/j.jmaa.2005.05.009	https://doi.org/10.1016/j.jmaa.2005.05.009	NUM
ap-9228	551	2	[	[	X
ap-9228	551	3	18	18	NUM
ap-9228	551	4	]	]	PUNCT
ap-9228	551	5	m.	m.	NOUN
ap-9228	551	6	abramowitz	abramowitz	PROPN
ap-9228	551	7	,	,	PUNCT
ap-9228	551	8	i.	i.	PROPN
ap-9228	551	9	a.	a.	PROPN
ap-9228	551	10	stegun	stegun	PROPN
ap-9228	551	11	,	,	PUNCT
ap-9228	551	12	r.	r.	PROPN
ap-9228	551	13	h.	h.	PROPN
ap-9228	551	14	romer	romer	PROPN
ap-9228	551	15	.	.	PUNCT
ap-9228	552	1	handbook	handbook	NOUN
ap-9228	552	2	of	of	ADP
ap-9228	552	3	mathematical	mathematical	ADJ
ap-9228	552	4	functions	function	NOUN
ap-9228	552	5	with	with	ADP
ap-9228	552	6	formulas	formula	NOUN
ap-9228	552	7	,	,	PUNCT
ap-9228	552	8	graphs	graph	NOUN
ap-9228	552	9	,	,	PUNCT
ap-9228	552	10	and	and	CCONJ
ap-9228	552	11	mathematical	mathematical	ADJ
ap-9228	552	12	tables	table	NOUN
ap-9228	552	13	.	.	PUNCT
ap-9228	553	1	american	american	ADJ
ap-9228	553	2	journal	journal	PROPN
ap-9228	553	3	of	of	ADP
ap-9228	553	4	physics	physics	PROPN
ap-9228	553	5	56(10):958–958	56(10):958–958	NUM
ap-9228	553	6	,	,	PUNCT
ap-9228	553	7	1988	1988	NUM
ap-9228	553	8	.	.	PUNCT
ap-9228	554	1	https://doi.org/10.1119/1.15378	https://doi.org/10.1119/1.15378	PRON
ap-9228	554	2	102	102	NUM
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ap-9228	554	4	https://doi.org/10.24321/2455.7021.202304	https://doi.org/10.24321/2455.7021.202304	PROPN
ap-9228	554	5	https://doi.org/10.22436/jmcs.023.01.06	https://doi.org/10.22436/jmcs.023.01.06	NOUN
ap-9228	555	1	https://doi.org/10.1016/0022-247x(84)90182-3	https://doi.org/10.1016/0022-247x(84)90182-3	PROPN
ap-9228	555	2	https://doi.org/10.1016/s1007-5704(97)90007-1	https://doi.org/10.1016/s1007-5704(97)90007-1	PROPN
ap-9228	555	3	https://doi.org/10.1016/s0096-3003(01)00312-5	https://doi.org/10.1016/s0096-3003(01)00312-5	ADJ
ap-9228	555	4	https://doi.org/10.1016/j.jmaa.2005.05.009	https://doi.org/10.1016/j.jmaa.2005.05.009	PROPN
ap-9228	555	5	https://doi.org/10.1119/1.15378	https://doi.org/10.1119/1.15378	PROPN
ap-9228	555	6	acta	acta	PROPN
ap-9228	555	7	polytechnica	polytechnica	PROPN
ap-9228	555	8	64(2):87–102	64(2):87–102	NUM
ap-9228	555	9	,	,	PUNCT
ap-9228	555	10	2024	2024	NUM
ap-9228	555	11	1	1	NUM
ap-9228	555	12	introduction	introduction	NOUN
ap-9228	555	13	2	2	NUM
ap-9228	555	14	analysis	analysis	NOUN
ap-9228	555	15	of	of	ADP
ap-9228	555	16	the	the	DET
ap-9228	555	17	technique	technique	NOUN
ap-9228	555	18	3	3	NUM
ap-9228	555	19	convergence	convergence	NOUN
ap-9228	555	20	of	of	ADP
ap-9228	555	21	the	the	DET
ap-9228	555	22	technique	technique	NOUN
ap-9228	555	23	4	4	NUM
ap-9228	555	24	illustrative	illustrative	ADJ
ap-9228	555	25	examples	example	NOUN
ap-9228	555	26	5	5	NUM
ap-9228	555	27	results	result	NOUN
ap-9228	555	28	and	and	CCONJ
ap-9228	555	29	conclusion	conclusion	NOUN
ap-9228	555	30	references	reference	NOUN