id	sid	tid	token	lemma	pos
iajs-2860	1	1	ihjpas	ihjpas	PROPN
iajs-2860	1	2	.	.	PUNCT
iajs-2860	2	1	36(1)2023	36(1)2023	NUM
iajs-2860	2	2	407	407	NUM
iajs-2860	2	3	this	this	DET
iajs-2860	2	4	work	work	NOUN
iajs-2860	2	5	is	be	AUX
iajs-2860	2	6	licensed	license	VERB
iajs-2860	2	7	under	under	ADP
iajs-2860	2	8	a	a	DET
iajs-2860	2	9	creative	creative	ADJ
iajs-2860	2	10	commons	common	NOUN
iajs-2860	2	11	attribution	attribution	NOUN
iajs-2860	2	12	4.0	4.0	NUM
iajs-2860	2	13	international	international	ADJ
iajs-2860	2	14	license	license	NOUN
iajs-2860	2	15	approximation	approximation	NOUN
iajs-2860	2	16	solution	solution	NOUN
iajs-2860	2	17	of	of	ADP
iajs-2860	2	18	fuzzy	fuzzy	ADJ
iajs-2860	2	19	singular	singular	PROPN
iajs-2860	2	20	volterra	volterra	PROPN
iajs-2860	2	21	integral	integral	ADJ
iajs-2860	2	22	equation	equation	NOUN
iajs-2860	2	23	by	by	ADP
iajs-2860	2	24	non	non	ADJ
iajs-2860	2	25	-	-	ADJ
iajs-2860	2	26	polynomial	polynomial	ADJ
iajs-2860	2	27	spline	spline	NOUN
iajs-2860	2	28	abstract	abstract	NOUN
iajs-2860	2	29	a	a	DET
iajs-2860	2	30	non	non	ADJ
iajs-2860	2	31	-	-	ADJ
iajs-2860	2	32	polynomial	polynomial	ADJ
iajs-2860	2	33	spline	spline	NOUN
iajs-2860	2	34	(	(	PUNCT
iajs-2860	2	35	nps	nps	PROPN
iajs-2860	2	36	)	)	PUNCT
iajs-2860	2	37	is	be	AUX
iajs-2860	2	38	an	an	DET
iajs-2860	2	39	approximation	approximation	NOUN
iajs-2860	2	40	method	method	NOUN
iajs-2860	2	41	that	that	PRON
iajs-2860	2	42	relies	rely	VERB
iajs-2860	2	43	on	on	ADP
iajs-2860	2	44	the	the	DET
iajs-2860	2	45	triangular	triangular	NOUN
iajs-2860	2	46	and	and	CCONJ
iajs-2860	2	47	polynomial	polynomial	ADJ
iajs-2860	2	48	parts	part	NOUN
iajs-2860	2	49	,	,	PUNCT
iajs-2860	2	50	so	so	SCONJ
iajs-2860	2	51	the	the	DET
iajs-2860	2	52	method	method	NOUN
iajs-2860	2	53	has	have	AUX
iajs-2860	2	54	infinite	infinite	ADJ
iajs-2860	2	55	derivatives	derivative	NOUN
iajs-2860	2	56	of	of	ADP
iajs-2860	2	57	the	the	DET
iajs-2860	2	58	triangular	triangular	NOUN
iajs-2860	2	59	part	part	NOUN
iajs-2860	2	60	of	of	ADP
iajs-2860	2	61	the	the	DET
iajs-2860	2	62	nps	nps	PROPN
iajs-2860	2	63	to	to	PART
iajs-2860	2	64	compensate	compensate	VERB
iajs-2860	2	65	for	for	ADP
iajs-2860	2	66	the	the	DET
iajs-2860	2	67	loss	loss	NOUN
iajs-2860	2	68	of	of	ADP
iajs-2860	2	69	smoothness	smoothness	NOUN
iajs-2860	2	70	inherited	inherit	VERB
iajs-2860	2	71	by	by	ADP
iajs-2860	2	72	the	the	DET
iajs-2860	2	73	polynomial	polynomial	NOUN
iajs-2860	2	74	.	.	PUNCT
iajs-2860	3	1	in	in	ADP
iajs-2860	3	2	this	this	DET
iajs-2860	3	3	paper	paper	NOUN
iajs-2860	3	4	,	,	PUNCT
iajs-2860	3	5	we	we	PRON
iajs-2860	3	6	propose	propose	VERB
iajs-2860	3	7	polynomial	polynomial	ADJ
iajs-2860	3	8	-	-	PUNCT
iajs-2860	3	9	free	free	ADJ
iajs-2860	3	10	linear	linear	NOUN
iajs-2860	3	11	and	and	CCONJ
iajs-2860	3	12	quadratic	quadratic	ADJ
iajs-2860	3	13	spline	spline	NOUN
iajs-2860	3	14	types	type	NOUN
iajs-2860	3	15	to	to	PART
iajs-2860	3	16	solve	solve	VERB
iajs-2860	3	17	fuzzy	fuzzy	ADJ
iajs-2860	3	18	volterra	volterra	PROPN
iajs-2860	3	19	integral	integral	ADJ
iajs-2860	3	20	equations	equation	NOUN
iajs-2860	3	21	(	(	PUNCT
iajs-2860	3	22	fvie	fvie	ADJ
iajs-2860	3	23	)	)	PUNCT
iajs-2860	3	24	of	of	ADP
iajs-2860	3	25	the	the	DET
iajs-2860	3	26	2nd	2nd	ADJ
iajs-2860	3	27	kind	kind	NOUN
iajs-2860	3	28	with	with	ADP
iajs-2860	3	29	the	the	DET
iajs-2860	3	30	weakly	weakly	ADJ
iajs-2860	3	31	singular	singular	ADJ
iajs-2860	3	32	kernel	kernel	NOUN
iajs-2860	3	33	(	(	PUNCT
iajs-2860	3	34	fviewsk	fviewsk	PROPN
iajs-2860	3	35	)	)	PUNCT
iajs-2860	3	36	and	and	CCONJ
iajs-2860	3	37	abel	abel	PROPN
iajs-2860	3	38	's	's	PART
iajs-2860	3	39	type	type	NOUN
iajs-2860	3	40	kernel	kernel	NOUN
iajs-2860	3	41	.	.	PUNCT
iajs-2860	4	1	the	the	DET
iajs-2860	4	2	linear	linear	ADJ
iajs-2860	4	3	type	type	NOUN
iajs-2860	4	4	algorithm	algorithm	NOUN
iajs-2860	4	5	gives	give	VERB
iajs-2860	4	6	four	four	NUM
iajs-2860	4	7	parameters	parameter	NOUN
iajs-2860	4	8	to	to	PART
iajs-2860	4	9	form	form	VERB
iajs-2860	4	10	a	a	DET
iajs-2860	4	11	linear	linear	ADJ
iajs-2860	4	12	spline	spline	NOUN
iajs-2860	4	13	.	.	PUNCT
iajs-2860	5	1	in	in	ADP
iajs-2860	5	2	comparison	comparison	NOUN
iajs-2860	5	3	,	,	PUNCT
iajs-2860	5	4	the	the	DET
iajs-2860	5	5	quadratic	quadratic	ADJ
iajs-2860	5	6	type	type	NOUN
iajs-2860	5	7	algorithm	algorithm	NOUN
iajs-2860	5	8	gives	give	VERB
iajs-2860	5	9	five	five	NUM
iajs-2860	5	10	parameters	parameter	NOUN
iajs-2860	5	11	to	to	PART
iajs-2860	5	12	create	create	VERB
iajs-2860	5	13	a	a	DET
iajs-2860	5	14	quadratic	quadratic	ADJ
iajs-2860	5	15	spline	spline	NOUN
iajs-2860	5	16	,	,	PUNCT
iajs-2860	5	17	which	which	PRON
iajs-2860	5	18	is	be	AUX
iajs-2860	5	19	more	more	ADJ
iajs-2860	5	20	of	of	ADP
iajs-2860	5	21	a	a	DET
iajs-2860	5	22	credit	credit	NOUN
iajs-2860	5	23	for	for	ADP
iajs-2860	5	24	the	the	DET
iajs-2860	5	25	exact	exact	ADJ
iajs-2860	5	26	solution	solution	NOUN
iajs-2860	5	27	.	.	PUNCT
iajs-2860	6	1	these	these	DET
iajs-2860	6	2	algorithms	algorithms	NOUN
iajs-2860	6	3	process	process	NOUN
iajs-2860	6	4	kernel	kernel	NOUN
iajs-2860	6	5	singularities	singularity	NOUN
iajs-2860	6	6	with	with	ADP
iajs-2860	6	7	a	a	DET
iajs-2860	6	8	simple	simple	ADJ
iajs-2860	6	9	technique	technique	NOUN
iajs-2860	6	10	.	.	PUNCT
iajs-2860	7	1	illustrative	illustrative	ADJ
iajs-2860	7	2	examples	example	NOUN
iajs-2860	7	3	use	use	VERB
iajs-2860	7	4	mathcad	mathcad	PROPN
iajs-2860	7	5	software	software	PROPN
iajs-2860	7	6	to	to	PART
iajs-2860	7	7	deal	deal	VERB
iajs-2860	7	8	with	with	ADP
iajs-2860	7	9	upper	upper	ADJ
iajs-2860	7	10	and	and	CCONJ
iajs-2860	7	11	lower	lower	ADV
iajs-2860	7	12	-	-	PUNCT
iajs-2860	7	13	bound	bind	VERB
iajs-2860	7	14	solutions	solution	NOUN
iajs-2860	7	15	to	to	ADP
iajs-2860	7	16	fuzzy	fuzzy	ADJ
iajs-2860	7	17	problems	problem	NOUN
iajs-2860	7	18	.	.	PUNCT
iajs-2860	8	1	the	the	DET
iajs-2860	8	2	method	method	NOUN
iajs-2860	8	3	provides	provide	VERB
iajs-2860	8	4	a	a	DET
iajs-2860	8	5	reliable	reliable	ADJ
iajs-2860	8	6	way	way	NOUN
iajs-2860	8	7	to	to	PART
iajs-2860	8	8	ensure	ensure	VERB
iajs-2860	8	9	that	that	SCONJ
iajs-2860	8	10	an	an	DET
iajs-2860	8	11	exact	exact	ADJ
iajs-2860	8	12	solution	solution	NOUN
iajs-2860	8	13	is	be	AUX
iajs-2860	8	14	approximated	approximate	VERB
iajs-2860	8	15	.	.	PUNCT
iajs-2860	9	1	also	also	ADV
iajs-2860	9	2	,	,	PUNCT
iajs-2860	9	3	figures	figure	NOUN
iajs-2860	9	4	and	and	CCONJ
iajs-2860	9	5	tables	table	NOUN
iajs-2860	9	6	show	show	VERB
iajs-2860	9	7	the	the	DET
iajs-2860	9	8	potential	potential	NOUN
iajs-2860	9	9	of	of	ADP
iajs-2860	9	10	the	the	DET
iajs-2860	9	11	method	method	NOUN
iajs-2860	9	12	.	.	PUNCT
iajs-2860	10	1	keywords	keyword	NOUN
iajs-2860	10	2	:	:	PUNCT
iajs-2860	10	3	fuzzy	fuzzy	ADJ
iajs-2860	10	4	volterra	volterra	PROPN
iajs-2860	10	5	integral	integral	ADJ
iajs-2860	10	6	equation	equation	NOUN
iajs-2860	10	7	,	,	PUNCT
iajs-2860	10	8	weakly	weakly	ADJ
iajs-2860	10	9	singular	singular	ADJ
iajs-2860	10	10	kernel	kernel	NOUN
iajs-2860	10	11	,	,	PUNCT
iajs-2860	10	12	non	non	ADJ
iajs-2860	10	13	-	-	ADJ
iajs-2860	10	14	polynomial	polynomial	ADJ
iajs-2860	10	15	spline	spline	NOUN
iajs-2860	10	16	.	.	PUNCT
iajs-2860	11	1	1	1	X
iajs-2860	11	2	.	.	X
iajs-2860	11	3	introduction	introduction	NOUN
iajs-2860	11	4	the	the	DET
iajs-2860	11	5	fuzzy	fuzzy	ADJ
iajs-2860	11	6	set	set	NOUN
iajs-2860	11	7	theory	theory	NOUN
iajs-2860	11	8	is	be	AUX
iajs-2860	11	9	one	one	NUM
iajs-2860	11	10	of	of	ADP
iajs-2860	11	11	the	the	DET
iajs-2860	11	12	essential	essential	ADJ
iajs-2860	11	13	theories	theory	NOUN
iajs-2860	11	14	introduced	introduce	VERB
iajs-2860	11	15	by	by	ADP
iajs-2860	11	16	'	'	PUNCT
iajs-2860	11	17	[	[	X
iajs-2860	11	18	1	1	NUM
iajs-2860	11	19	-	-	SYM
iajs-2860	11	20	2	2	NUM
iajs-2860	11	21	]	]	PUNCT
iajs-2860	11	22	solving	solve	VERB
iajs-2860	11	23	linear	linear	NOUN
iajs-2860	11	24	and	and	CCONJ
iajs-2860	11	25	nonlinear	nonlinear	ADJ
iajs-2860	11	26	abel	abel	PROPN
iajs-2860	11	27	fuzzy	fuzzy	ADJ
iajs-2860	11	28	integral	integral	ADJ
iajs-2860	11	29	equations	equation	NOUN
iajs-2860	11	30	using	use	VERB
iajs-2860	11	31	laplace	laplace	NOUN
iajs-2860	11	32	transforms	transform	VERB
iajs-2860	11	33	.	.	PUNCT
iajs-2860	12	1	[	[	X
iajs-2860	12	2	3	3	X
iajs-2860	12	3	]	]	PUNCT
iajs-2860	12	4	found	find	VERB
iajs-2860	12	5	the	the	DET
iajs-2860	12	6	solution	solution	NOUN
iajs-2860	12	7	of	of	ADP
iajs-2860	12	8	the	the	DET
iajs-2860	12	9	fuzzy	fuzzy	ADJ
iajs-2860	12	10	volterra	volterra	NOUN
iajs-2860	12	11	integral	integral	ADJ
iajs-2860	12	12	equation	equation	NOUN
iajs-2860	12	13	of	of	ADP
iajs-2860	12	14	2nd	2nd	ADJ
iajs-2860	12	15	kind	kind	NOUN
iajs-2860	12	16	with	with	ADP
iajs-2860	12	17	abel	abel	PROPN
iajs-2860	12	18	's	's	PART
iajs-2860	12	19	type	type	NOUN
iajs-2860	12	20	kernel	kernel	NOUN
iajs-2860	12	21	by	by	ADP
iajs-2860	12	22	applying	apply	VERB
iajs-2860	12	23	laplace	laplace	NOUN
iajs-2860	12	24	adomain	adomain	NOUN
iajs-2860	12	25	decomposition	decomposition	NOUN
iajs-2860	12	26	method	method	NOUN
iajs-2860	12	27	.	.	PUNCT
iajs-2860	13	1	[	[	X
iajs-2860	13	2	4	4	X
iajs-2860	13	3	]	]	PUNCT
iajs-2860	13	4	proposed	propose	VERB
iajs-2860	13	5	a	a	DET
iajs-2860	13	6	piecewise	piecewise	NOUN
iajs-2860	13	7	spline	spline	NOUN
iajs-2860	13	8	collocations	collocation	NOUN
iajs-2860	13	9	method	method	NOUN
iajs-2860	13	10	with	with	ADP
iajs-2860	13	11	gradient	gradient	ADJ
iajs-2860	13	12	meshes	mesh	NOUN
iajs-2860	13	13	for	for	ADP
iajs-2860	13	14	fviewsk	fviewsk	NOUN
iajs-2860	13	15	,	,	PUNCT
iajs-2860	13	16	showing	show	VERB
iajs-2860	13	17	that	that	SCONJ
iajs-2860	13	18	the	the	DET
iajs-2860	13	19	gradient	gradient	NOUN
iajs-2860	13	20	meshes	mesh	NOUN
iajs-2860	13	21	are	be	AUX
iajs-2860	13	22	superior	superior	ADJ
iajs-2860	13	23	to	to	ADP
iajs-2860	13	24	uniform	uniform	ADJ
iajs-2860	13	25	ones	one	NOUN
iajs-2860	13	26	for	for	ADP
iajs-2860	13	27	this	this	DET
iajs-2860	13	28	problem	problem	NOUN
iajs-2860	13	29	.	.	PUNCT
iajs-2860	14	1	[	[	X
iajs-2860	14	2	5	5	NUM
iajs-2860	14	3	]	]	PUNCT
iajs-2860	14	4	solved	solve	VERB
iajs-2860	14	5	linear	linear	PROPN
iajs-2860	14	6	volterra	volterra	PROPN
iajs-2860	14	7	integral	integral	ADJ
iajs-2860	14	8	equations	equation	NOUN
iajs-2860	14	9	of	of	ADP
iajs-2860	14	10	the	the	DET
iajs-2860	14	11	second	second	ADJ
iajs-2860	14	12	kind	kind	NOUN
iajs-2860	14	13	with	with	ADP
iajs-2860	14	14	a	a	DET
iajs-2860	14	15	weakly	weakly	ADJ
iajs-2860	14	16	singular	singular	ADJ
iajs-2860	14	17	kernel	kernel	NOUN
iajs-2860	14	18	by	by	ADP
iajs-2860	14	19	using	use	VERB
iajs-2860	14	20	the	the	DET
iajs-2860	14	21	sixth	sixth	ADJ
iajs-2860	14	22	order	order	NOUN
iajs-2860	14	23	of	of	ADP
iajs-2860	14	24	non	non	ADJ
iajs-2860	14	25	-	-	ADJ
iajs-2860	14	26	polynomial	polynomial	ADJ
iajs-2860	14	27	spline	spline	NOUN
iajs-2860	14	28	functions	function	NOUN
iajs-2860	14	29	.	.	PUNCT
iajs-2860	15	1	in	in	ADP
iajs-2860	15	2	this	this	DET
iajs-2860	15	3	work	work	NOUN
iajs-2860	15	4	,	,	PUNCT
iajs-2860	15	5	we	we	PRON
iajs-2860	15	6	drive	drive	VERB
iajs-2860	15	7	linear	linear	ADJ
iajs-2860	15	8	and	and	CCONJ
iajs-2860	15	9	quadratic	quadratic	ADJ
iajs-2860	15	10	nps	nps	PROPN
iajs-2860	15	11	to	to	PART
iajs-2860	15	12	solve	solve	VERB
iajs-2860	15	13	fviewsk	fviewsk	NOUN
iajs-2860	15	14	and	and	CCONJ
iajs-2860	15	15	numerically	numerically	ADV
iajs-2860	15	16	drive	drive	VERB
iajs-2860	15	17	linear	linear	PROPN
iajs-2860	15	18	nps	nps	PROPN
iajs-2860	15	19	of	of	ADP
iajs-2860	15	20	fvie	fvie	PROPN
iajs-2860	15	21	with	with	ADP
iajs-2860	15	22	abel	abel	PROPN
iajs-2860	15	23	's	's	PART
iajs-2860	15	24	types	type	NOUN
iajs-2860	15	25	kernel	kernel	VERB
iajs-2860	15	26	.	.	PUNCT
iajs-2860	16	1	this	this	DET
iajs-2860	16	2	paper	paper	NOUN
iajs-2860	16	3	is	be	AUX
iajs-2860	16	4	organized	organize	VERB
iajs-2860	16	5	as	as	SCONJ
iajs-2860	16	6	follows	follow	VERB
iajs-2860	16	7	:	:	PUNCT
iajs-2860	16	8	in	in	ADP
iajs-2860	16	9	section	section	NOUN
iajs-2860	16	10	2	2	NUM
iajs-2860	16	11	,	,	PUNCT
iajs-2860	16	12	some	some	DET
iajs-2860	16	13	preliminaries	preliminary	NOUN
iajs-2860	16	14	are	be	AUX
iajs-2860	16	15	given	give	VERB
iajs-2860	16	16	about	about	ADP
iajs-2860	16	17	basic	basic	ADJ
iajs-2860	16	18	definitions	definition	NOUN
iajs-2860	16	19	.	.	PUNCT
iajs-2860	17	1	section	section	NOUN
iajs-2860	17	2	3	3	NUM
iajs-2860	17	3	is	be	AUX
iajs-2860	17	4	about	about	ADP
iajs-2860	17	5	nps	nps	PROPN
iajs-2860	17	6	'	'	PUNCT
iajs-2860	17	7	functions	function	NOUN
iajs-2860	17	8	in	in	ADP
iajs-2860	17	9	linear	linear	ADJ
iajs-2860	17	10	and	and	CCONJ
iajs-2860	17	11	quadratic	quadratic	ADJ
iajs-2860	17	12	.	.	PUNCT
iajs-2860	18	1	section	section	NOUN
iajs-2860	18	2	4	4	NUM
iajs-2860	18	3	considers	consider	VERB
iajs-2860	18	4	the	the	DET
iajs-2860	18	5	linear	linear	ADJ
iajs-2860	18	6	doi.org/10.30526/36.1.2860	doi.org/10.30526/36.1.2860	NOUN
iajs-2860	18	7	article	article	NOUN
iajs-2860	18	8	history	history	NOUN
iajs-2860	18	9	:	:	PUNCT
iajs-2860	18	10	received	receive	VERB
iajs-2860	18	11	22	22	NUM
iajs-2860	18	12	july	july	PROPN
iajs-2860	18	13	2022	2022	NUM
iajs-2860	18	14	,	,	PUNCT
iajs-2860	18	15	accepted	accept	VERB
iajs-2860	18	16	21	21	NUM
iajs-2860	18	17	augest	augest	NOUN
iajs-2860	18	18	2022	2022	NUM
iajs-2860	18	19	,	,	PUNCT
iajs-2860	18	20	published	publish	VERB
iajs-2860	18	21	in	in	ADP
iajs-2860	18	22	january	january	PROPN
iajs-2860	18	23	2023	2023	NUM
iajs-2860	18	24	.	.	PUNCT
iajs-2860	19	1	ibn	ibn	PROPN
iajs-2860	19	2	al	al	PROPN
iajs-2860	19	3	-	-	PUNCT
iajs-2860	19	4	haitham	haitham	PROPN
iajs-2860	19	5	journal	journal	PROPN
iajs-2860	19	6	for	for	ADP
iajs-2860	19	7	pure	pure	ADJ
iajs-2860	19	8	and	and	CCONJ
iajs-2860	19	9	applied	applied	ADJ
iajs-2860	19	10	sciences	sciences	PROPN
iajs-2860	19	11	journal	journal	PROPN
iajs-2860	19	12	homepage	homepage	NOUN
iajs-2860	19	13	:	:	PUNCT
iajs-2860	19	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	NOUN
iajs-2860	19	15	zahraa	zahraa	PROPN
iajs-2860	19	16	a.	a.	PROPN
iajs-2860	19	17	ibrahim	ibrahim	PROPN
iajs-2860	19	18	department	department	PROPN
iajs-2860	19	19	of	of	ADP
iajs-2860	19	20	mathematics	mathematics	PROPN
iajs-2860	19	21	,	,	PUNCT
iajs-2860	19	22	college	college	NOUN
iajs-2860	19	23	of	of	ADP
iajs-2860	19	24	sciences	sciences	PROPN
iajs-2860	19	25	,	,	PUNCT
iajs-2860	19	26	mustansiriyah	mustansiriyah	NOUN
iajs-2860	19	27	university	university	NOUN
iajs-2860	19	28	,	,	PUNCT
iajs-2860	19	29	baghdad	baghdad	PROPN
iajs-2860	19	30	,	,	PUNCT
iajs-2860	19	31	iraq	iraq	PROPN
iajs-2860	19	32	.	.	PUNCT
iajs-2860	20	1	zara.ali1431@gmail.com	zara.ali1431@gmail.com	PROPN
iajs-2860	20	2	nabaa	nabaa	PROPN
iajs-2860	20	3	n.	n.	PROPN
iajs-2860	20	4	hasan	hasan	PROPN
iajs-2860	20	5	department	department	PROPN
iajs-2860	20	6	of	of	ADP
iajs-2860	20	7	mathematics	mathematics	PROPN
iajs-2860	20	8	,	,	PUNCT
iajs-2860	20	9	college	college	NOUN
iajs-2860	20	10	of	of	ADP
iajs-2860	20	11	sciences	sciences	PROPN
iajs-2860	20	12	,	,	PUNCT
iajs-2860	20	13	mustansiriyah	mustansiriyah	NOUN
iajs-2860	20	14	university	university	NOUN
iajs-2860	20	15	,	,	PUNCT
iajs-2860	20	16	baghdad	baghdad	PROPN
iajs-2860	20	17	,	,	PUNCT
iajs-2860	20	18	iraq	iraq	PROPN
iajs-2860	20	19	.	.	PUNCT
iajs-2860	21	1	alzaer1972@uomustansiriyah.edu.iq	alzaer1972@uomustansiriyah.edu.iq	ADJ
iajs-2860	21	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2860	21	3	mailto:zara.ali1431@gmail.com	mailto:zara.ali1431@gmail.com	PROPN
iajs-2860	21	4	mailto:alzaer1972@uomustansiriyah.edu.iq	mailto:alzaer1972@uomustansiriyah.edu.iq	PROPN
iajs-2860	21	5	ihjpas	ihjpas	PROPN
iajs-2860	21	6	.	.	PUNCT
iajs-2860	22	1	36(1)2023	36(1)2023	NUM
iajs-2860	22	2	408	408	NUM
iajs-2860	22	3	and	and	CCONJ
iajs-2860	22	4	quadratic	quadratic	ADJ
iajs-2860	22	5	methods	method	NOUN
iajs-2860	22	6	of	of	ADP
iajs-2860	22	7	fviewsk	fviewsk	NOUN
iajs-2860	22	8	and	and	CCONJ
iajs-2860	22	9	section	section	NOUN
iajs-2860	22	10	5	5	NUM
iajs-2860	22	11	nps	nps	PROPN
iajs-2860	22	12	of	of	ADP
iajs-2860	22	13	fvie	fvie	PROPN
iajs-2860	22	14	with	with	ADP
iajs-2860	22	15	abel	abel	NOUN
iajs-2860	22	16	's	's	PART
iajs-2860	22	17	type	type	NOUN
iajs-2860	22	18	kernel	kernel	NOUN
iajs-2860	22	19	.	.	PUNCT
iajs-2860	23	1	furthermore	furthermore	ADV
iajs-2860	23	2	,	,	PUNCT
iajs-2860	23	3	in	in	ADP
iajs-2860	23	4	section	section	NOUN
iajs-2860	23	5	6	6	NUM
iajs-2860	23	6	,	,	PUNCT
iajs-2860	23	7	some	some	DET
iajs-2860	23	8	illustrated	illustrated	ADJ
iajs-2860	23	9	examples	example	NOUN
iajs-2860	23	10	are	be	AUX
iajs-2860	23	11	given	give	VERB
iajs-2860	23	12	,	,	PUNCT
iajs-2860	23	13	showing	show	VERB
iajs-2860	23	14	the	the	DET
iajs-2860	23	15	method	method	NOUN
iajs-2860	23	16	's	's	PART
iajs-2860	23	17	accuracy	accuracy	NOUN
iajs-2860	23	18	.	.	PUNCT
iajs-2860	24	1	finally	finally	ADV
iajs-2860	24	2	,	,	PUNCT
iajs-2860	24	3	conclusions	conclusion	NOUN
iajs-2860	24	4	are	be	AUX
iajs-2860	24	5	given	give	VERB
iajs-2860	24	6	in	in	ADP
iajs-2860	24	7	section	section	NOUN
iajs-2860	24	8	7	7	NUM
iajs-2860	24	9	.	.	SYM
iajs-2860	24	10	2	2	NUM
iajs-2860	24	11	.	.	PUNCT
iajs-2860	24	12	preliminaries	preliminary	NOUN
iajs-2860	24	13	concepts	concept	NOUN
iajs-2860	24	14	:	:	PUNCT
iajs-2860	24	15	in	in	ADP
iajs-2860	24	16	this	this	DET
iajs-2860	24	17	section	section	NOUN
iajs-2860	24	18	some	some	DET
iajs-2860	24	19	concepts	concept	NOUN
iajs-2860	24	20	and	and	CCONJ
iajs-2860	24	21	definitions	definition	NOUN
iajs-2860	24	22	of	of	ADP
iajs-2860	24	23	fuzzy	fuzzy	ADJ
iajs-2860	24	24	set	set	NOUN
iajs-2860	24	25	theory	theory	NOUN
iajs-2860	24	26	and	and	CCONJ
iajs-2860	24	27	kinds	kind	NOUN
iajs-2860	24	28	of	of	ADP
iajs-2860	24	29	singular	singular	ADJ
iajs-2860	24	30	kernels	kernel	NOUN
iajs-2860	24	31	are	be	AUX
iajs-2860	24	32	given	give	VERB
iajs-2860	24	33	.	.	PUNCT
iajs-2860	25	1	definition	definition	NOUN
iajs-2860	25	2	2.1[4	2.1[4	NUM
iajs-2860	25	3	]	]	X
iajs-2860	25	4	:	:	PUNCT
iajs-2860	25	5	a	a	DET
iajs-2860	25	6	fuzzy	fuzzy	ADJ
iajs-2860	25	7	numbers	number	NOUN
iajs-2860	25	8	is	be	AUX
iajs-2860	25	9	a	a	DET
iajs-2860	25	10	fuzzy	fuzzy	ADJ
iajs-2860	25	11	set	set	VERB
iajs-2860	25	12	𝐴	𝐴	PROPN
iajs-2860	25	13	:	:	PUNCT
iajs-2860	25	14	𝑅	𝑅	PROPN
iajs-2860	25	15	→	→	SYM
iajs-2860	25	16	[	[	X
iajs-2860	25	17	0	0	NUM
iajs-2860	25	18	,	,	PUNCT
iajs-2860	25	19	1	1	NUM
iajs-2860	25	20	]	]	PUNCT
iajs-2860	25	21	such	such	ADJ
iajs-2860	25	22	that	that	SCONJ
iajs-2860	25	23	:	:	PUNCT
iajs-2860	25	24	(	(	PUNCT
iajs-2860	25	25	𝑖	𝑖	X
iajs-2860	25	26	)	)	PUNCT
iajs-2860	25	27	𝐴	𝐴	PROPN
iajs-2860	25	28	is	be	AUX
iajs-2860	25	29	upper	upper	ADJ
iajs-2860	25	30	semi	semi	ADV
iajs-2860	25	31	continuous	continuous	ADJ
iajs-2860	25	32	,	,	PUNCT
iajs-2860	25	33	(	(	PUNCT
iajs-2860	25	34	𝑖𝑖	𝑖𝑖	NOUN
iajs-2860	25	35	)	)	PUNCT
iajs-2860	25	36	𝐴(𝑥	𝐴(𝑥	NOUN
iajs-2860	25	37	)	)	PUNCT
iajs-2860	25	38	=	=	SYM
iajs-2860	25	39	0	0	NUM
iajs-2860	25	40	outside	outside	ADP
iajs-2860	25	41	some	some	DET
iajs-2860	25	42	interval	interval	NOUN
iajs-2860	26	1	[	[	X
iajs-2860	26	2	𝑎	𝑎	X
iajs-2860	26	3	,	,	PUNCT
iajs-2860	26	4	d	d	X
iajs-2860	26	5	]	]	X
iajs-2860	26	6	,	,	PUNCT
iajs-2860	26	7	(	(	PUNCT
iajs-2860	26	8	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
iajs-2860	26	9	)	)	PUNCT
iajs-2860	26	10	there	there	PRON
iajs-2860	26	11	are	be	VERB
iajs-2860	26	12	real	real	ADJ
iajs-2860	26	13	'	'	PUNCT
iajs-2860	26	14	numbers	number	NOUN
iajs-2860	26	15	𝑏	𝑏	NOUN
iajs-2860	26	16	,	,	PUNCT
iajs-2860	26	17	:	:	PUNCT
iajs-2860	26	18	𝑎	𝑎	X
iajs-2860	26	19	≤	≤	NUM
iajs-2860	26	20	𝑏	𝑏	DET
iajs-2860	26	21	≤	≤	NUM
iajs-2860	26	22	𝑐	𝑐	NOUN
iajs-2860	26	23	≤	≤	NUM
iajs-2860	26	24	𝑑	𝑑	VERB
iajs-2860	26	25	for	for	ADP
iajs-2860	26	26	which	which	PRON
iajs-2860	26	27	(	(	PUNCT
iajs-2860	26	28	1	1	X
iajs-2860	26	29	)	)	PUNCT
iajs-2860	26	30	𝐴(𝑥	𝐴(𝑥	NOUN
iajs-2860	26	31	)	)	PUNCT
iajs-2860	26	32	is	be	AUX
iajs-2860	26	33	monotonically	monotonically	ADV
iajs-2860	26	34	'	'	PUNCT
iajs-2860	26	35	increasing	increase	VERB
iajs-2860	26	36	on	on	ADP
iajs-2860	26	37	[	[	X
iajs-2860	26	38	𝑎	𝑎	X
iajs-2860	26	39	,	,	PUNCT
iajs-2860	26	40	𝑏	𝑏	NOUN
iajs-2860	26	41	]	]	PUNCT
iajs-2860	26	42	,	,	PUNCT
iajs-2860	26	43	(	(	PUNCT
iajs-2860	26	44	2	2	X
iajs-2860	26	45	)	)	PUNCT
iajs-2860	26	46	𝐴(𝑥	𝐴(𝑥	NOUN
iajs-2860	26	47	)	)	PUNCT
iajs-2860	26	48	is	be	AUX
iajs-2860	26	49	'	'	AUX
iajs-2860	26	50	monotonically	monotonically	ADV
iajs-2860	26	51	decreasing	decrease	VERB
iajs-2860	26	52	on	on	ADP
iajs-2860	26	53	[	[	X
iajs-2860	26	54	𝑐	𝑐	X
iajs-2860	26	55	,	,	PUNCT
iajs-2860	26	56	𝑑	𝑑	NOUN
iajs-2860	26	57	]	]	X
iajs-2860	26	58	,	,	PUNCT
iajs-2860	26	59	(	(	PUNCT
iajs-2860	26	60	3	3	X
iajs-2860	26	61	)	)	PUNCT
iajs-2860	26	62	a	a	DET
iajs-2860	26	63	(	(	PUNCT
iajs-2860	26	64	𝑥	𝑥	NOUN
iajs-2860	26	65	)	)	PUNCT
iajs-2860	26	66	=	=	SYM
iajs-2860	26	67	1	1	NUM
iajs-2860	26	68	,	,	PUNCT
iajs-2860	26	69	𝑏	𝑏	PROPN
iajs-2860	26	70	≤	≤	NUM
iajs-2860	26	71	𝑥	𝑥	PRON
iajs-2860	26	72	≤	≤	PROPN
iajs-2860	26	73	𝑐	𝑐	PROPN
iajs-2860	26	74	definition	definition	NOUN
iajs-2860	26	75	2.2	2.2	NUM
iajs-2860	27	1	[	[	X
iajs-2860	27	2	6	6	NUM
iajs-2860	27	3	]	]	PUNCT
iajs-2860	27	4	:	:	PUNCT
iajs-2860	27	5	let	let	VERB
iajs-2860	27	6	x	x	PRON
iajs-2860	27	7	be	be	AUX
iajs-2860	27	8	a	a	DET
iajs-2860	27	9	cartesian	cartesian	ADJ
iajs-2860	27	10	'	'	PUNCT
iajs-2860	27	11	product	product	NOUN
iajs-2860	27	12	of	of	ADP
iajs-2860	27	13	universes	universe	NOUN
iajs-2860	27	14	𝑋1	𝑋1	PROPN
iajs-2860	27	15	,	,	PUNCT
iajs-2860	27	16	𝑋2	𝑋2	VERB
iajs-2860	27	17	,	,	PUNCT
iajs-2860	27	18	…	…	PUNCT
iajs-2860	27	19	,	,	PUNCT
iajs-2860	27	20	𝑋𝑟	𝑋𝑟	PROPN
iajs-2860	27	21	and	and	CCONJ
iajs-2860	27	22	�	�	PROPN
iajs-2860	27	23	̃	̃	PROPN
iajs-2860	27	24	�	�	NOUN
iajs-2860	27	25	1	1	NUM
iajs-2860	27	26	,	,	PUNCT
iajs-2860	27	27	�	�	PROPN
iajs-2860	27	28	̃	̃	NOUN
iajs-2860	27	29	�	�	NOUN
iajs-2860	27	30	2	2	NUM
iajs-2860	27	31	,	,	PUNCT
iajs-2860	27	32	.	.	PUNCT
iajs-2860	27	33	.	.	PUNCT
iajs-2860	28	1	.	.	PUNCT
iajs-2860	29	1	,	,	PUNCT
iajs-2860	29	2	�	�	PROPN
iajs-2860	29	3	̃	̃	PROPN
iajs-2860	29	4	�	�	NOUN
iajs-2860	29	5	𝑟	𝑟	NOUN
iajs-2860	29	6	be	be	VERB
iajs-2860	29	7	r	r	NOUN
iajs-2860	29	8	-	-	PUNCT
iajs-2860	29	9	fuzzy	fuzzy	ADJ
iajs-2860	29	10	subsets	subset	NOUN
iajs-2860	29	11	of	of	ADP
iajs-2860	29	12	𝑋1	𝑋1	PROPN
iajs-2860	29	13	,	,	PUNCT
iajs-2860	29	14	𝑋2	𝑋2	VERB
iajs-2860	29	15	,	,	PUNCT
iajs-2860	29	16	…	…	PUNCT
iajs-2860	29	17	,	,	PUNCT
iajs-2860	29	18	𝑋𝑟	𝑋𝑟	PROPN
iajs-2860	29	19	,	,	PUNCT
iajs-2860	29	20	respectively	respectively	ADV
iajs-2860	29	21	,	,	PUNCT
iajs-2860	29	22	𝑓	𝑓	PROPN
iajs-2860	29	23	=	=	X
iajs-2860	29	24	x	x	SYM
iajs-2860	29	25	→	→	SYM
iajs-2860	29	26	𝑌	𝑌	PROPN
iajs-2860	29	27	,	,	PUNCT
iajs-2860	29	28	𝑦	𝑦	NOUN
iajs-2860	29	29	=	=	SYM
iajs-2860	29	30	𝑓(𝑥1	𝑓(𝑥1	ADJ
iajs-2860	29	31	,	,	PUNCT
iajs-2860	29	32	𝑥2	𝑥2	NOUN
iajs-2860	29	33	,	,	PUNCT
iajs-2860	29	34	…	…	PUNCT
iajs-2860	29	35	,	,	PUNCT
iajs-2860	29	36	𝑥𝑟	𝑥𝑟	NOUN
iajs-2860	29	37	)	)	PUNCT
iajs-2860	29	38	,	,	PUNCT
iajs-2860	29	39	then	then	ADV
iajs-2860	29	40	the	the	DET
iajs-2860	29	41	extensions	extension	NOUN
iajs-2860	29	42	principles	principle	NOUN
iajs-2860	29	43	allow	allow	VERB
iajs-2860	29	44	us	we	PRON
iajs-2860	29	45	to	to	PART
iajs-2860	29	46	define	define	VERB
iajs-2860	29	47	a	a	DET
iajs-2860	29	48	fuzzy	fuzzy	ADJ
iajs-2860	29	49	set	set	VERB
iajs-2860	29	50	�	�	PROPN
iajs-2860	29	51	̃	̃	PROPN
iajs-2860	29	52	�	�	PROPN
iajs-2860	29	53	in	in	ADP
iajs-2860	29	54	y	y	NOUN
iajs-2860	29	55	by	by	ADP
iajs-2860	29	56	:	:	PUNCT
iajs-2860	29	57	�	�	PROPN
iajs-2860	29	58	̃	̃	PROPN
iajs-2860	29	59	�	�	NOUN
iajs-2860	29	60	=	=	SYM
iajs-2860	29	61	{	{	PUNCT
iajs-2860	29	62	(	(	PUNCT
iajs-2860	29	63	𝑦	𝑦	NOUN
iajs-2860	29	64	,	,	PUNCT
iajs-2860	29	65	𝜇	𝜇	ADP
iajs-2860	29	66	�	�	PROPN
iajs-2860	29	67	̃	̃	NOUN
iajs-2860	29	68	�	�	NOUN
iajs-2860	29	69	(𝑦))|	(𝑦))|	X
iajs-2860	29	70	𝑦	𝑦	NOUN
iajs-2860	29	71	=	=	X
iajs-2860	29	72	𝑓(𝑥1	𝑓(𝑥1	ADJ
iajs-2860	29	73	,	,	PUNCT
iajs-2860	29	74	𝑥2	𝑥2	NOUN
iajs-2860	29	75	,	,	PUNCT
iajs-2860	29	76	…	…	PUNCT
iajs-2860	29	77	,	,	PUNCT
iajs-2860	29	78	𝑥𝑟	𝑥𝑟	NOUN
iajs-2860	29	79	)	)	PUNCT
iajs-2860	29	80	,	,	PUNCT
iajs-2860	29	81	𝑥1	𝑥1	NOUN
iajs-2860	29	82	,	,	PUNCT
iajs-2860	29	83	𝑥2	𝑥2	NOUN
iajs-2860	29	84	,	,	PUNCT
iajs-2860	29	85	…	…	PUNCT
iajs-2860	29	86	,	,	PUNCT
iajs-2860	29	87	𝑥𝑟	𝑥𝑟	X
iajs-2860	29	88	∈	∈	NOUN
iajs-2860	29	89	x	x	X
iajs-2860	29	90	}	}	PUNCT
iajs-2860	29	91	where	where	SCONJ
iajs-2860	29	92	𝜇	𝜇	ADP
iajs-2860	29	93	�	�	PROPN
iajs-2860	29	94	̃	̃	NOUN
iajs-2860	29	95	�	�	NOUN
iajs-2860	29	96	(𝑦	(𝑦	NOUN
iajs-2860	29	97	)	)	PUNCT
iajs-2860	29	98	=	=	PRON
iajs-2860	29	99	{	{	PUNCT
iajs-2860	29	100	sup	sup	NOUN
iajs-2860	29	101	𝑀𝑖𝑛{𝜇𝐴1̃	𝑀𝑖𝑛{𝜇𝐴1̃	X
iajs-2860	29	102	(	(	PUNCT
iajs-2860	29	103	𝑥1	𝑥1	NOUN
iajs-2860	29	104	)	)	PUNCT
iajs-2860	29	105	,	,	PUNCT
iajs-2860	29	106	…	…	PUNCT
iajs-2860	29	107	,	,	PUNCT
iajs-2860	29	108	𝜇𝐴	𝜇𝐴	ADP
iajs-2860	29	109	�	�	NOUN
iajs-2860	29	110	̃	̃	NOUN
iajs-2860	29	111	�	�	PROPN
iajs-2860	29	112	(	(	PUNCT
iajs-2860	29	113	𝑥𝑟	𝑥𝑟	NOUN
iajs-2860	29	114	)	)	PUNCT
iajs-2860	29	115	}	}	PUNCT
iajs-2860	29	116	,	,	PUNCT
iajs-2860	29	117	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
iajs-2860	29	118	)	)	PUNCT
iajs-2860	29	119	≠	≠	PROPN
iajs-2860	29	120	∅	∅	NOUN
iajs-2860	29	121	(	(	PUNCT
iajs-2860	29	122	𝑥1	𝑥1	NOUN
iajs-2860	29	123	,	,	PUNCT
iajs-2860	29	124	𝑥2	𝑥2	NOUN
iajs-2860	29	125	,	,	PUNCT
iajs-2860	29	126	…	…	PUNCT
iajs-2860	29	127	,	,	PUNCT
iajs-2860	29	128	𝑥𝑟	𝑥𝑟	NOUN
iajs-2860	29	129	)	)	PUNCT
iajs-2860	29	130	∈	∈	PROPN
iajs-2860	29	131	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
iajs-2860	29	132	)	)	PUNCT
iajs-2860	29	133	0	0	NUM
iajs-2860	29	134	,	,	PUNCT
iajs-2860	29	135	𝑜𝑡ℎ𝑒𝑟	𝑜𝑡ℎ𝑒𝑟	PROPN
iajs-2860	29	136	𝑤𝑖𝑠𝑒	𝑤𝑖𝑠𝑒	PROPN
iajs-2860	29	137	and	and	CCONJ
iajs-2860	29	138	𝑓−1	𝑓−1	NUM
iajs-2860	29	139	is	be	AUX
iajs-2860	29	140	the	the	DET
iajs-2860	29	141	inverse	inverse	ADJ
iajs-2860	29	142	image	image	NOUN
iajs-2860	29	143	of	of	ADP
iajs-2860	29	144	𝑓	𝑓	PRON
iajs-2860	29	145	for	for	ADP
iajs-2860	29	146	𝑟	𝑟	NOUN
iajs-2860	29	147	=	=	SYM
iajs-2860	29	148	1	1	NUM
iajs-2860	29	149	,	,	PUNCT
iajs-2860	29	150	the	the	DET
iajs-2860	29	151	fuzzy	fuzzy	ADJ
iajs-2860	29	152	extension	extension	NOUN
iajs-2860	29	153	principles	principle	NOUN
iajs-2860	29	154	,	,	PUNCT
iajs-2860	29	155	of	of	ADP
iajs-2860	29	156	course	course	NOUN
iajs-2860	29	157	reduces	reduce	VERB
iajs-2860	29	158	to	to	ADP
iajs-2860	29	159	�	�	NOUN
iajs-2860	29	160	̃	̃	NOUN
iajs-2860	29	161	�	�	PROPN
iajs-2860	29	162	=	=	SYM
iajs-2860	29	163	𝑓(	𝑓(	PROPN
iajs-2860	29	164	�	�	PROPN
iajs-2860	29	165	̃	̃	PROPN
iajs-2860	29	166	�	�	PROPN
iajs-2860	29	167	)	)	PUNCT
iajs-2860	30	1	=	=	SYM
iajs-2860	30	2	𝑓({(𝑦	𝑓({(𝑦	NOUN
iajs-2860	30	3	,	,	PUNCT
iajs-2860	30	4	𝜇	𝜇	ADP
iajs-2860	30	5	�	�	PROPN
iajs-2860	30	6	̃	̃	NOUN
iajs-2860	30	7	�	�	NOUN
iajs-2860	30	8	(𝑦))|	(𝑦))|	SYM
iajs-2860	30	9	𝑦	𝑦	NOUN
iajs-2860	30	10	=	=	SYM
iajs-2860	30	11	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	30	12	)	)	PUNCT
iajs-2860	30	13	,	,	PUNCT
iajs-2860	30	14	𝑥	𝑥	PRON
iajs-2860	30	15	∈	∈	PROPN
iajs-2860	30	16	x	x	X
iajs-2860	30	17	}	}	PUNCT
iajs-2860	30	18	where	where	SCONJ
iajs-2860	30	19	𝜇	𝜇	ADP
iajs-2860	30	20	�	�	PROPN
iajs-2860	30	21	̃	̃	NOUN
iajs-2860	30	22	�	�	NOUN
iajs-2860	30	23	(𝑦	(𝑦	NOUN
iajs-2860	30	24	)	)	PUNCT
iajs-2860	30	25	=	=	PRON
iajs-2860	30	26	{	{	PUNCT
iajs-2860	30	27	sup	sup	PROPN
iajs-2860	30	28	𝜇	𝜇	ADP
iajs-2860	30	29	�	�	PROPN
iajs-2860	30	30	̃	̃	NOUN
iajs-2860	30	31	�	�	NOUN
iajs-2860	30	32	(𝑥	(𝑥	NOUN
iajs-2860	30	33	)	)	PUNCT
iajs-2860	30	34	,	,	PUNCT
iajs-2860	30	35	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
iajs-2860	30	36	)	)	PUNCT
iajs-2860	30	37	≠	≠	PROPN
iajs-2860	30	38	∅	∅	VERB
iajs-2860	30	39	𝑥	𝑥	DET
iajs-2860	30	40	∈	∈	PROPN
iajs-2860	30	41	𝑓−1(𝑦	𝑓−1(𝑦	PROPN
iajs-2860	30	42	)	)	PUNCT
iajs-2860	30	43	0	0	NUM
iajs-2860	30	44	,	,	PUNCT
iajs-2860	30	45	𝑜𝑡ℎ𝑒𝑟	𝑜𝑡ℎ𝑒𝑟	PROPN
iajs-2860	30	46	𝑤𝑖𝑠𝑒	𝑤𝑖𝑠𝑒	PROPN
iajs-2860	30	47	which	which	PRON
iajs-2860	30	48	is	be	AUX
iajs-2860	30	49	the	the	DET
iajs-2860	30	50	definition	definition	NOUN
iajs-2860	30	51	of	of	ADP
iajs-2860	30	52	the	the	DET
iajs-2860	30	53	fuzzy	fuzzy	ADJ
iajs-2860	30	54	mapping	mapping	NOUN
iajs-2860	30	55	.	.	PUNCT
iajs-2860	31	1	definition	definition	NOUN
iajs-2860	31	2	2.3	2.3	NUM
iajs-2860	31	3	crisp	crisp	ADJ
iajs-2860	31	4	number	number	NOUN
iajs-2860	31	5	[	[	X
iajs-2860	31	6	6	6	NUM
iajs-2860	31	7	]	]	X
iajs-2860	31	8	:	:	PUNCT
iajs-2860	31	9	a	a	DET
iajs-2860	31	10	crisp	crisp	ADJ
iajs-2860	31	11	number	number	NOUN
iajs-2860	31	12	a	a	PRON
iajs-2860	31	13	is	be	AUX
iajs-2860	31	14	represented	represent	VERB
iajs-2860	31	15	by	by	ADP
iajs-2860	31	16	:	:	PUNCT
iajs-2860	31	17	𝐴(𝑥	𝐴(𝑥	NUM
iajs-2860	31	18	)	)	PUNCT
iajs-2860	31	19	=	=	PRON
iajs-2860	31	20	{	{	PUNCT
iajs-2860	32	1	1	1	NUM
iajs-2860	32	2	𝑖𝑓	𝑖𝑓	ADP
iajs-2860	32	3	𝑥	𝑥	NOUN
iajs-2860	32	4	=	=	PUNCT
iajs-2860	33	1	𝑎	𝑎	X
iajs-2860	33	2	0	0	NUM
iajs-2860	33	3	𝑖𝑓𝑥	𝑖𝑓𝑥	NOUN
iajs-2860	33	4	≠	≠	PROPN
iajs-2860	33	5	𝑎	𝑎	ADP
iajs-2860	33	6	a	a	DET
iajs-2860	33	7	crisp	crisp	ADJ
iajs-2860	33	8	interval	interval	NOUN
iajs-2860	33	9	[	[	X
iajs-2860	33	10	c	c	X
iajs-2860	33	11	,	,	PUNCT
iajs-2860	33	12	d	d	X
iajs-2860	33	13	]	]	X
iajs-2860	33	14	is	be	AUX
iajs-2860	33	15	represented	represent	VERB
iajs-2860	33	16	by	by	ADP
iajs-2860	33	17	a	a	DET
iajs-2860	33	18	fuzzy	fuzzy	ADJ
iajs-2860	33	19	set	set	VERB
iajs-2860	33	20	𝐵(𝑥	𝐵(𝑥	PRON
iajs-2860	33	21	)	)	PUNCT
iajs-2860	33	22	=	=	PRON
iajs-2860	33	23	{	{	PUNCT
iajs-2860	33	24	1	1	NUM
iajs-2860	33	25	if	if	SCONJ
iajs-2860	33	26	x	x	X
iajs-2860	33	27	∈	∈	PROPN
iajs-2860	34	1	[	[	X
iajs-2860	34	2	c	c	X
iajs-2860	34	3	,	,	PUNCT
iajs-2860	34	4	d	d	X
iajs-2860	34	5	]	]	X
iajs-2860	34	6	0	0	PUNCT
iajs-2860	35	1	if	if	SCONJ
iajs-2860	35	2	x	x	PROPN
iajs-2860	35	3	∉	∉	PROPN
iajs-2860	35	4	[	[	X
iajs-2860	35	5	c	c	X
iajs-2860	35	6	,	,	PUNCT
iajs-2860	35	7	d	d	X
iajs-2860	35	8	]	]	X
iajs-2860	35	9	definition	definition	NOUN
iajs-2860	35	10	2.4	2.4	NUM
iajs-2860	35	11	some	some	DET
iajs-2860	35	12	types	type	NOUN
iajs-2860	35	13	of	of	ADP
iajs-2860	35	14	kernels	kernel	NOUN
iajs-2860	36	1	[	[	X
iajs-2860	36	2	7	7	NUM
iajs-2860	36	3	]	]	PUNCT
iajs-2860	36	4	:	:	PUNCT
iajs-2860	36	5	i.	i.	PROPN
iajs-2860	36	6	cauchy	cauchy	PROPN
iajs-2860	36	7	kernel	kernel	PROPN
iajs-2860	36	8	if	if	SCONJ
iajs-2860	36	9	the	the	DET
iajs-2860	36	10	kernel	kernel	PROPN
iajs-2860	36	11	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	36	12	,	,	PUNCT
iajs-2860	36	13	𝑡	𝑡	PROPN
iajs-2860	36	14	)	)	PUNCT
iajs-2860	36	15	is	be	AUX
iajs-2860	36	16	of	of	ADP
iajs-2860	36	17	the	the	DET
iajs-2860	36	18	form	form	NOUN
iajs-2860	36	19	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	36	20	,	,	PUNCT
iajs-2860	36	21	𝑡	𝑡	X
iajs-2860	36	22	)	)	PUNCT
iajs-2860	36	23	=	=	SYM
iajs-2860	36	24	𝐻(𝑥,𝑡	𝐻(𝑥,𝑡	NOUN
iajs-2860	36	25	)	)	PUNCT
iajs-2860	36	26	𝑥−𝑡	𝑥−𝑡	NOUN
iajs-2860	36	27	where	where	SCONJ
iajs-2860	36	28	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	36	29	,	,	PUNCT
iajs-2860	36	30	𝑡	𝑡	X
iajs-2860	36	31	)	)	PUNCT
iajs-2860	36	32	is	be	AUX
iajs-2860	36	33	differentiable	differentiable	ADJ
iajs-2860	36	34	function	function	NOUN
iajs-2860	36	35	of	of	ADP
iajs-2860	36	36	(	(	PUNCT
iajs-2860	36	37	𝑥	𝑥	PROPN
iajs-2860	36	38	,	,	PUNCT
iajs-2860	36	39	𝑡	𝑡	NOUN
iajs-2860	36	40	)	)	PUNCT
iajs-2860	36	41	with	with	ADP
iajs-2860	36	42	𝐻(𝑥	𝐻(𝑥	ADJ
iajs-2860	36	43	,	,	PUNCT
iajs-2860	36	44	𝑡	𝑡	NOUN
iajs-2860	36	45	)	)	PUNCT
iajs-2860	36	46	≠	≠	PROPN
iajs-2860	36	47	0	0	NUM
iajs-2860	36	48	,	,	PUNCT
iajs-2860	36	49	then	then	ADV
iajs-2860	36	50	the	the	DET
iajs-2860	36	51	integral	integral	ADJ
iajs-2860	36	52	,	,	PUNCT
iajs-2860	36	53	.equation	.equation	NUM
iajs-2860	36	54	is	be	AUX
iajs-2860	36	55	said	say	VERB
iajs-2860	36	56	to	to	PART
iajs-2860	36	57	be	be	AUX
iajs-2860	36	58	a	a	DET
iajs-2860	36	59	singular	singular	ADJ
iajs-2860	36	60	equation	equation	NOUN
iajs-2860	36	61	with	with	ADP
iajs-2860	36	62	cauchy	cauchy	PROPN
iajs-2860	36	63	kernels	kernel	NOUN
iajs-2860	36	64	.	.	PUNCT
iajs-2860	37	1	ii	ii	PROPN
iajs-2860	37	2	.	.	PUNCT
iajs-2860	38	1	weakly	weakly	ADJ
iajs-2860	38	2	singular	singular	ADJ
iajs-2860	38	3	kernel	kernel	NOUN
iajs-2860	38	4	if	if	SCONJ
iajs-2860	38	5	the	the	DET
iajs-2860	38	6	kernel	kernel	PROPN
iajs-2860	38	7	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	38	8	,	,	PUNCT
iajs-2860	38	9	𝑡	𝑡	PROPN
iajs-2860	38	10	)	)	PUNCT
iajs-2860	38	11	is	be	AUX
iajs-2860	38	12	of	of	ADP
iajs-2860	38	13	the	the	DET
iajs-2860	38	14	form	form	NOUN
iajs-2860	38	15	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	38	16	,	,	PUNCT
iajs-2860	38	17	𝑡	𝑡	X
iajs-2860	38	18	)	)	PUNCT
iajs-2860	38	19	=	=	PUNCT
iajs-2860	38	20	𝐻(𝑥,𝑡	𝐻(𝑥,𝑡	NOUN
iajs-2860	38	21	)	)	PUNCT
iajs-2860	38	22	|𝑥−𝑡|𝛼	|𝑥−𝑡|𝛼	PROPN
iajs-2860	38	23	where	where	SCONJ
iajs-2860	38	24	0	0	X
iajs-2860	38	25	<	<	X
iajs-2860	38	26	𝛼	𝛼	X
iajs-2860	38	27	<	<	X
iajs-2860	38	28	1	1	NUM
iajs-2860	38	29	and	and	CCONJ
iajs-2860	38	30	𝐻(𝑥	𝐻(𝑥	NUM
iajs-2860	38	31	,	,	PUNCT
iajs-2860	38	32	𝑡	𝑡	PROPN
iajs-2860	38	33	)	)	PUNCT
iajs-2860	38	34	is	be	AUX
iajs-2860	38	35	a	a	DET
iajs-2860	38	36	differentiable	differentiable	ADJ
iajs-2860	38	37	and	and	CCONJ
iajs-2860	38	38	continuous	continuous	ADJ
iajs-2860	38	39	function	function	NOUN
iajs-2860	38	40	with	with	ADP
iajs-2860	38	41	𝐻(𝑥	𝐻(𝑥	ADJ
iajs-2860	38	42	,	,	PUNCT
iajs-2860	38	43	𝑡	𝑡	NOUN
iajs-2860	38	44	)	)	PUNCT
iajs-2860	38	45	≠	≠	PROPN
iajs-2860	38	46	0	0	NUM
iajs-2860	38	47	,	,	PUNCT
iajs-2860	38	48	ihjpas	ihjpa	VERB
iajs-2860	38	49	.	.	PUNCT
iajs-2860	39	1	36(1)2023	36(1)2023	NUM
iajs-2860	39	2	409	409	NUM
iajs-2860	39	3	then	then	ADV
iajs-2860	39	4	the	the	DET
iajs-2860	39	5	integral	integral	ADJ
iajs-2860	39	6	equation	equation	NOUN
iajs-2860	39	7	is	be	AUX
iajs-2860	39	8	said	say	VERB
iajs-2860	39	9	to	to	PART
iajs-2860	39	10	be	be	AUX
iajs-2860	39	11	weakly	weakly	ADJ
iajs-2860	39	12	singular	singular	ADJ
iajs-2860	39	13	kernel	kernel	NOUN
iajs-2860	39	14	.	.	PUNCT
iajs-2860	40	1	iii	iii	X
iajs-2860	40	2	.	.	PUNCT
iajs-2860	40	3	strongly	strongly	ADV
iajs-2860	40	4	singular	singular	ADJ
iajs-2860	40	5	kernel	kernel	NOUN
iajs-2860	40	6	if	if	SCONJ
iajs-2860	40	7	the	the	DET
iajs-2860	40	8	kernel	kernel	PROPN
iajs-2860	40	9	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	40	10	,	,	PUNCT
iajs-2860	40	11	𝑡	𝑡	PROPN
iajs-2860	40	12	)	)	PUNCT
iajs-2860	40	13	is	be	AUX
iajs-2860	40	14	of	of	ADP
iajs-2860	40	15	the	the	DET
iajs-2860	40	16	form	form	NOUN
iajs-2860	40	17	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	40	18	,	,	PUNCT
iajs-2860	40	19	𝑡	𝑡	X
iajs-2860	40	20	)	)	PUNCT
iajs-2860	40	21	=	=	PUNCT
iajs-2860	40	22	𝐻(𝑥,𝑡	𝐻(𝑥,𝑡	NOUN
iajs-2860	40	23	)	)	PUNCT
iajs-2860	41	1	|𝑥−𝑡|𝛼	|𝑥−𝑡|𝛼	PROPN
iajs-2860	41	2	𝛼	𝛼	X
iajs-2860	41	3	≥	≥	NOUN
iajs-2860	41	4	2	2	NUM
iajs-2860	41	5	where	where	SCONJ
iajs-2860	41	6	𝐻(𝑥	𝐻(𝑥	ADP
iajs-2860	41	7	,	,	PUNCT
iajs-2860	41	8	𝑡	𝑡	NOUN
iajs-2860	41	9	)	)	PUNCT
iajs-2860	41	10	is	be	AUX
iajs-2860	41	11	a	a	DET
iajs-2860	41	12	differentiable	differentiable	ADJ
iajs-2860	41	13	function	function	NOUN
iajs-2860	41	14	of	of	ADP
iajs-2860	41	15	(	(	PUNCT
iajs-2860	41	16	𝑥	𝑥	PROPN
iajs-2860	41	17	,	,	PUNCT
iajs-2860	41	18	𝑡	𝑡	NOUN
iajs-2860	41	19	)	)	PUNCT
iajs-2860	41	20	with	with	ADP
iajs-2860	41	21	𝐻(𝑥	𝐻(𝑥	ADJ
iajs-2860	41	22	,	,	PUNCT
iajs-2860	41	23	𝑡	𝑡	NOUN
iajs-2860	41	24	)	)	PUNCT
iajs-2860	41	25	≠	≠	PROPN
iajs-2860	41	26	0	0	NUM
iajs-2860	41	27	,	,	PUNCT
iajs-2860	41	28	then	then	ADV
iajs-2860	41	29	the	the	DET
iajs-2860	41	30	integral	integral	ADJ
iajs-2860	41	31	equation	equation	NOUN
iajs-2860	41	32	is	be	AUX
iajs-2860	41	33	said	say	VERB
iajs-2860	41	34	to	to	PART
iajs-2860	41	35	be	be	AUX
iajs-2860	41	36	strongly	strongly	ADV
iajs-2860	41	37	singular	singular	ADJ
iajs-2860	41	38	kernel	kernel	NOUN
iajs-2860	41	39	.	.	PUNCT
iajs-2860	42	1	3	3	X
iajs-2860	42	2	.	.	X
iajs-2860	42	3	nps	nps	PROPN
iajs-2860	42	4	functions	function	VERB
iajs-2860	42	5	the	the	DET
iajs-2860	42	6	standard	standard	ADJ
iajs-2860	42	7	form	form	NOUN
iajs-2860	42	8	of	of	ADP
iajs-2860	42	9	the	the	DET
iajs-2860	42	10	2nd	2nd	ADJ
iajs-2860	42	11	fvie	fvie	NOUN
iajs-2860	42	12	is	be	AUX
iajs-2860	42	13	defined	define	VERB
iajs-2860	42	14	below	below	ADP
iajs-2860	42	15	[	[	X
iajs-2860	42	16	8	8	NUM
iajs-2860	42	17	]	]	SYM
iajs-2860	42	18	:	:	PUNCT
iajs-2860	42	19	𝑢(𝑥	𝑢(𝑥	NUM
iajs-2860	42	20	,	,	PUNCT
iajs-2860	42	21	𝑟	𝑟	NOUN
iajs-2860	42	22	)	)	PUNCT
iajs-2860	42	23	=	=	SYM
iajs-2860	42	24	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	42	25	,	,	PUNCT
iajs-2860	42	26	𝑟	𝑟	NOUN
iajs-2860	42	27	)	)	PUNCT
iajs-2860	43	1	+	+	CCONJ
iajs-2860	43	2	𝜆	𝜆	X
iajs-2860	43	3	;	;	PUNCT
iajs-2860	43	4	∫	∫	PROPN
iajs-2860	43	5	𝑘(𝑥	𝑘(𝑥	PROPN
iajs-2860	43	6	,	,	PUNCT
iajs-2860	43	7	𝑡	𝑡	PROPN
iajs-2860	43	8	,	,	PUNCT
iajs-2860	43	9	𝑢(𝑡	𝑢(𝑡	NOUN
iajs-2860	43	10	,	,	PUNCT
iajs-2860	43	11	𝑟	𝑟	NOUN
iajs-2860	43	12	)	)	PUNCT
iajs-2860	43	13	,	,	PUNCT
iajs-2860	44	1	𝑟)𝑑𝑡	𝑟)𝑑𝑡	PROPN
iajs-2860	44	2	𝑥	𝑥	NOUN
iajs-2860	44	3	0	0	PUNCT
iajs-2860	44	4	(	(	PUNCT
iajs-2860	44	5	1	1	X
iajs-2860	44	6	)	)	PUNCT
iajs-2860	44	7	consider	consider	VERB
iajs-2860	44	8	the	the	DET
iajs-2860	44	9	partition	partition	NOUN
iajs-2860	44	10	∆=	∆=	NOUN
iajs-2860	44	11	{	{	PUNCT
iajs-2860	44	12	𝑟0	𝑟0	NOUN
iajs-2860	44	13	,	,	PUNCT
iajs-2860	44	14	𝑟1	𝑟1	NOUN
iajs-2860	44	15	,	,	PUNCT
iajs-2860	44	16	𝑟2	𝑟2	NOUN
iajs-2860	44	17	,	,	PUNCT
iajs-2860	44	18	…	…	PUNCT
iajs-2860	44	19	,	,	PUNCT
iajs-2860	44	20	𝑟𝑛	𝑟𝑛	ADP
iajs-2860	44	21	}	}	PUNCT
iajs-2860	44	22	of	of	ADP
iajs-2860	44	23	[	[	X
iajs-2860	44	24	𝑎	𝑎	X
iajs-2860	44	25	,	,	PUNCT
iajs-2860	44	26	𝑏	𝑏	NOUN
iajs-2860	44	27	]	]	X
iajs-2860	44	28	⊂	⊂	X
iajs-2860	44	29	𝑅′	𝑅′	PROPN
iajs-2860	44	30	,	,	PUNCT
iajs-2860	44	31	let	let	VERB
iajs-2860	44	32	𝑆(∆)′and	𝑆(∆)′and	PRON
iajs-2860	44	33	indicate	indicate	VERB
iajs-2860	44	34	the	the	DET
iajs-2860	44	35	arrangement	arrangement	NOUN
iajs-2860	44	36	of	of	ADP
iajs-2860	44	37	piecewise	piecewise	NOUN
iajs-2860	44	38	polynomial	polynomial	NOUN
iajs-2860	44	39	on	on	ADP
iajs-2860	44	40	subinterval	subinterval	NOUN
iajs-2860	45	1	[	[	X
iajs-2860	45	2	9	9	NUM
iajs-2860	45	3	]	]	X
iajs-2860	45	4	(	(	PUNCT
iajs-2860	45	5	𝐼𝑖	𝐼𝑖	NOUN
iajs-2860	45	6	=	=	PUNCT
iajs-2860	46	1	[	[	X
iajs-2860	46	2	𝑟𝑖	𝑟𝑖	X
iajs-2860	46	3	,	,	PUNCT
iajs-2860	46	4	𝑟𝑖+1	𝑟𝑖+1	NUM
iajs-2860	46	5	]	]	PUNCT
iajs-2860	46	6	of	of	ADP
iajs-2860	46	7	segment	segment	NOUN
iajs-2860	46	8	∆.	∆.	PROPN
iajs-2860	46	9	let	let	VERB
iajs-2860	46	10	𝑢(𝑟	𝑢(𝑟	NOUN
iajs-2860	46	11	)	)	PUNCT
iajs-2860	46	12	be	be	AUX
iajs-2860	46	13	the	the	DET
iajs-2860	46	14	exact	exact	ADJ
iajs-2860	46	15	solution	solution	NOUN
iajs-2860	46	16	.	.	PUNCT
iajs-2860	47	1	the	the	DET
iajs-2860	47	2	nps	nps	PROPN
iajs-2860	47	3	of	of	ADP
iajs-2860	47	4	n	n	PRON
iajs-2860	47	5	order	order	NOUN
iajs-2860	47	6	𝑆𝑖(𝑟	𝑆𝑖(𝑟	PROPN
iajs-2860	47	7	)	)	PUNCT
iajs-2860	47	8	at	at	ADP
iajs-2860	47	9	the	the	DET
iajs-2860	47	10	form	form	NOUN
iajs-2860	47	11	is	be	AUX
iajs-2860	47	12	:	:	PUNCT
iajs-2860	47	13	𝑆𝑖(𝑟)′	𝑆𝑖(𝑟)′	PROPN
iajs-2860	47	14	=	=	SYM
iajs-2860	47	15	𝑎𝑖	𝑎𝑖	PRON
iajs-2860	47	16	sin′	sin′	NOUN
iajs-2860	47	17	𝑘(𝑟	𝑘(𝑟	PROPN
iajs-2860	47	18	−	−	PROPN
iajs-2860	47	19	𝑟𝑖	𝑟𝑖	PART
iajs-2860	47	20	)	)	PUNCT
iajs-2860	48	1	+	+	CCONJ
iajs-2860	48	2	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	48	3	cos′	cos′	PROPN
iajs-2860	48	4	𝑘(𝑟	𝑘(𝑟	PROPN
iajs-2860	48	5	−	−	PROPN
iajs-2860	48	6	𝑟𝑖	𝑟𝑖	PART
iajs-2860	48	7	)	)	PUNCT
iajs-2860	49	1	+	+	CCONJ
iajs-2860	49	2	⋯	⋯	VERB
iajs-2860	49	3	+	+	CCONJ
iajs-2860	49	4	𝑦𝑖(𝑟	𝑦𝑖(𝑟	NUM
iajs-2860	49	5	−	−	PROPN
iajs-2860	49	6	𝑟𝑖	𝑟𝑖	NOUN
iajs-2860	49	7	)	)	PUNCT
iajs-2860	49	8	𝑛−1	𝑛−1	PROPN
iajs-2860	49	9	+	+	NUM
iajs-2860	49	10	𝑧𝑖.	𝑧𝑖.	NOUN
iajs-2860	49	11	(	(	PUNCT
iajs-2860	49	12	2	2	NUM
iajs-2860	49	13	)	)	PUNCT
iajs-2860	49	14	where	where	SCONJ
iajs-2860	49	15	𝑎𝑖	𝑎𝑖	INTJ
iajs-2860	49	16	,	,	PUNCT
iajs-2860	49	17	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	49	18	,	,	PUNCT
iajs-2860	49	19	…	…	PUNCT
iajs-2860	49	20	,	,	PUNCT
iajs-2860	49	21	𝑦𝑖	𝑦𝑖	PROPN
iajs-2860	49	22	and	and	CCONJ
iajs-2860	49	23	𝑧𝑖	𝑧𝑖	NOUN
iajs-2860	49	24	constants	constant	NOUN
iajs-2860	49	25	.	.	PUNCT
iajs-2860	50	1	in	in	ADP
iajs-2860	50	2	this	this	DET
iajs-2860	50	3	section	section	NOUN
iajs-2860	50	4	,	,	PUNCT
iajs-2860	50	5	we	we	PRON
iajs-2860	50	6	introduce	introduce	VERB
iajs-2860	50	7	different	different	ADJ
iajs-2860	50	8	types	type	NOUN
iajs-2860	50	9	of	of	ADP
iajs-2860	50	10	nps	nps	PROPN
iajs-2860	50	11	functions	function	NOUN
iajs-2860	50	12	,	,	PUNCT
iajs-2860	50	13	linear	linear	PROPN
iajs-2860	50	14	nps	nps	PROPN
iajs-2860	50	15	functions	function	NOUN
iajs-2860	50	16	and	and	CCONJ
iajs-2860	50	17	quadratic	quadratic	ADJ
iajs-2860	50	18	nps	nps	PROPN
iajs-2860	50	19	functions	function	NOUN
iajs-2860	50	20	.	.	PUNCT
iajs-2860	51	1	3.1	3.1	NUM
iajs-2860	51	2	linear	linear	ADJ
iajs-2860	51	3	non	non	ADJ
iajs-2860	51	4	-	-	ADJ
iajs-2860	51	5	polynomial	polynomial	ADJ
iajs-2860	51	6	spline	spline	NOUN
iajs-2860	51	7	(	(	PUNCT
iajs-2860	51	8	lnps	lnps	PROPN
iajs-2860	51	9	):	):	PUNCT
iajs-2860	51	10	we	we	PRON
iajs-2860	51	11	consider	consider	VERB
iajs-2860	51	12	the	the	DET
iajs-2860	51	13	lnps	lnps	ADJ
iajs-2860	51	14	method	method	NOUN
iajs-2860	51	15	for	for	ADP
iajs-2860	51	16	finding	find	VERB
iajs-2860	51	17	an	an	DET
iajs-2860	51	18	approximate	approximate	ADJ
iajs-2860	51	19	solution	solution	NOUN
iajs-2860	51	20	of	of	ADP
iajs-2860	51	21	fvie	fvie	NOUN
iajs-2860	51	22	of	of	ADP
iajs-2860	51	23	the	the	DET
iajs-2860	51	24	second	second	ADJ
iajs-2860	51	25	kind	kind	NOUN
iajs-2860	51	26	[	[	X
iajs-2860	51	27	9	9	NUM
iajs-2860	51	28	]	]	PUNCT
iajs-2860	51	29	consider	consider	VERB
iajs-2860	51	30	the	the	DET
iajs-2860	51	31	grid	grid	NOUN
iajs-2860	51	32	point	point	NOUN
iajs-2860	51	33	𝑟𝑖	𝑟𝑖	ADV
iajs-2860	51	34	on	on	ADP
iajs-2860	51	35	the	the	DET
iajs-2860	51	36	interval	interval	NOUN
iajs-2860	51	37	[	[	X
iajs-2860	51	38	𝑎	𝑎	X
iajs-2860	51	39	,	,	PUNCT
iajs-2860	51	40	𝑏	𝑏	NOUN
iajs-2860	51	41	]	]	X
iajs-2860	51	42	,	,	PUNCT
iajs-2860	51	43	as	as	SCONJ
iajs-2860	51	44	follows	follow	VERB
iajs-2860	51	45	:	:	PUNCT
iajs-2860	51	46	𝑎	𝑎	PROPN
iajs-2860	51	47	=	=	PUNCT
iajs-2860	51	48	𝑟0	𝑟0	NOUN
iajs-2860	51	49	<	<	X
iajs-2860	51	50	𝑟1	𝑟1	PROPN
iajs-2860	51	51	<	<	X
iajs-2860	51	52	𝑟2	𝑟2	NOUN
iajs-2860	51	53	<	<	X
iajs-2860	51	54	⋯	⋯	X
iajs-2860	51	55	<	<	X
iajs-2860	51	56	𝑟𝑛	𝑟𝑛	PROPN
iajs-2860	51	57	=	=	SYM
iajs-2860	51	58	𝑏	𝑏	PROPN
iajs-2860	51	59	(	(	PUNCT
iajs-2860	51	60	3	3	NUM
iajs-2860	51	61	)	)	PUNCT
iajs-2860	51	62	𝑟𝑖	𝑟𝑖	VERB
iajs-2860	52	1	=	=	SYM
iajs-2860	52	2	𝑟0	𝑟0	PROPN
iajs-2860	52	3	+	+	CCONJ
iajs-2860	52	4	𝑖ℎ	𝑖ℎ	PRON
iajs-2860	52	5	,	,	PUNCT
iajs-2860	52	6	𝑖	𝑖	PROPN
iajs-2860	52	7	=	=	NOUN
iajs-2860	52	8	0,1,2	0,1,2	NUM
iajs-2860	52	9	,	,	PUNCT
iajs-2860	52	10	…	…	PUNCT
iajs-2860	52	11	,	,	PUNCT
iajs-2860	52	12	𝑛	𝑛	X
iajs-2860	52	13	(	(	PUNCT
iajs-2860	52	14	4	4	NUM
iajs-2860	52	15	)	)	PUNCT
iajs-2860	52	16	ℎ	ℎ	PROPN
iajs-2860	52	17	=	=	SYM
iajs-2860	53	1	𝑏−𝑎	𝑏−𝑎	PROPN
iajs-2860	53	2	𝑛	𝑛	PROPN
iajs-2860	53	3	(	(	PUNCT
iajs-2860	53	4	5	5	NUM
iajs-2860	53	5	)	)	PUNCT
iajs-2860	53	6	where	where	SCONJ
iajs-2860	53	7	n	n	PRON
iajs-2860	53	8	is	be	AUX
iajs-2860	53	9	an	an	DET
iajs-2860	53	10	appositive	appositive	ADJ
iajs-2860	53	11	integer	integer	NOUN
iajs-2860	53	12	.	.	PUNCT
iajs-2860	54	1	let	let	VERB
iajs-2860	54	2	𝑢(𝑟	𝑢(𝑟	NOUN
iajs-2860	54	3	)	)	PUNCT
iajs-2860	54	4	be	be	AUX
iajs-2860	54	5	the	the	DET
iajs-2860	54	6	exact	exact	ADJ
iajs-2860	54	7	solution	solution	NOUN
iajs-2860	54	8	of	of	ADP
iajs-2860	54	9	equation	equation	NOUN
iajs-2860	54	10	(	(	PUNCT
iajs-2860	54	11	1	1	NUM
iajs-2860	54	12	)	)	PUNCT
iajs-2860	54	13	and	and	CCONJ
iajs-2860	54	14	𝑆𝑖(𝑟	𝑆𝑖(𝑟	NUM
iajs-2860	54	15	)	)	PUNCT
iajs-2860	54	16	be	be	AUX
iajs-2860	54	17	an	an	DET
iajs-2860	54	18	approximation	approximation	NOUN
iajs-2860	54	19	to	to	ADP
iajs-2860	54	20	𝑢𝑖	𝑢𝑖	NOUN
iajs-2860	54	21	=	=	SYM
iajs-2860	54	22	𝑢(𝑟𝑖	𝑢(𝑟𝑖	NOUN
iajs-2860	54	23	)	)	PUNCT
iajs-2860	54	24	obtained	obtain	VERB
iajs-2860	54	25	by	by	ADP
iajs-2860	54	26	the	the	DET
iajs-2860	54	27	segment	segment	NOUN
iajs-2860	54	28	𝑠𝑖(𝑟	𝑠𝑖(𝑟	NUM
iajs-2860	54	29	)	)	PUNCT
iajs-2860	54	30	.	.	PUNCT
iajs-2860	55	1	each	each	DET
iajs-2860	55	2	nps	nps	PROPN
iajs-2860	55	3	segment	segment	NOUN
iajs-2860	55	4	𝑠𝑖(𝑟	𝑠𝑖(𝑟	NUM
iajs-2860	55	5	)	)	PUNCT
iajs-2860	55	6	has	have	VERB
iajs-2860	55	7	the	the	DET
iajs-2860	55	8	form	form	NOUN
iajs-2860	55	9	:	:	PUNCT
iajs-2860	55	10	𝑆𝑖(𝑟)′	𝑆𝑖(𝑟)′	PROPN
iajs-2860	55	11	=	=	SYM
iajs-2860	55	12	𝑎𝑖	𝑎𝑖	PRON
iajs-2860	55	13	sin′	sin′	NOUN
iajs-2860	55	14	𝑘(𝑟	𝑘(𝑟	PROPN
iajs-2860	55	15	−	−	PROPN
iajs-2860	55	16	𝑟𝑖	𝑟𝑖	PART
iajs-2860	55	17	)	)	PUNCT
iajs-2860	56	1	+	+	CCONJ
iajs-2860	56	2	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	56	3	cos′	cos′	PROPN
iajs-2860	56	4	𝑘(𝑟	𝑘(𝑟	PROPN
iajs-2860	56	5	−	−	PROPN
iajs-2860	56	6	𝑟𝑖	𝑟𝑖	PART
iajs-2860	56	7	)	)	PUNCT
iajs-2860	57	1	+	+	CCONJ
iajs-2860	57	2	𝑐𝑖(𝑟	𝑐𝑖(𝑟	CCONJ
iajs-2860	57	3	−	−	NOUN
iajs-2860	57	4	𝑟𝑖	𝑟𝑖	PUNCT
iajs-2860	57	5	)	)	PUNCT
iajs-2860	58	1	+	+	CCONJ
iajs-2860	58	2	𝑑𝑖.	𝑑𝑖.	ADV
iajs-2860	58	3	(	(	PUNCT
iajs-2860	58	4	6	6	NUM
iajs-2860	58	5	)	)	PUNCT
iajs-2860	58	6	where	where	SCONJ
iajs-2860	58	7	𝑎𝑖	𝑎𝑖	ADV
iajs-2860	58	8	,	,	PUNCT
iajs-2860	58	9	𝑏𝑖	𝑏𝑖	PROPN
iajs-2860	58	10	,	,	PUNCT
iajs-2860	58	11	𝑐𝑖	𝑐𝑖	NOUN
iajs-2860	58	12	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2860	58	13	𝑑𝑖	𝑑𝑖	PROPN
iajs-2860	58	14	are	be	AUX
iajs-2860	58	15	constant	constant	ADJ
iajs-2860	58	16	and	and	CCONJ
iajs-2860	58	17	k	k	PROPN
iajs-2860	58	18	is	be	AUX
iajs-2860	58	19	the	the	DET
iajs-2860	58	20	frequency	frequency	NOUN
iajs-2860	58	21	of	of	ADP
iajs-2860	58	22	the	the	DET
iajs-2860	58	23	trigonometric	trigonometric	ADJ
iajs-2860	58	24	function	function	NOUN
iajs-2860	58	25	which	which	PRON
iajs-2860	58	26	will	will	AUX
iajs-2860	58	27	be	be	AUX
iajs-2860	58	28	used	use	VERB
iajs-2860	58	29	to	to	PART
iajs-2860	58	30	raise	raise	VERB
iajs-2860	58	31	the	the	DET
iajs-2860	58	32	accuracy	accuracy	NOUN
iajs-2860	58	33	'	'	PUNCT
iajs-2860	58	34	of	of	ADP
iajs-2860	58	35	the	the	DET
iajs-2860	58	36	method	method	NOUN
iajs-2860	58	37	.	.	PUNCT
iajs-2860	59	1	we	we	PRON
iajs-2860	59	2	consider	consider	VERB
iajs-2860	59	3	the	the	DET
iajs-2860	59	4	following	follow	VERB
iajs-2860	59	5	relations	relation	NOUN
iajs-2860	59	6	:	:	PUNCT
iajs-2860	59	7	𝑆𝑖(𝑟𝑖	𝑆𝑖(𝑟𝑖	NOUN
iajs-2860	59	8	)	)	PUNCT
iajs-2860	59	9	=	=	SYM
iajs-2860	59	10	𝑢(𝑟𝑖	𝑢(𝑟𝑖	X
iajs-2860	59	11	)	)	PUNCT
iajs-2860	59	12	𝑆′	𝑆′	NOUN
iajs-2860	59	13	𝑖′(𝑟𝑖	𝑖′(𝑟𝑖	PROPN
iajs-2860	59	14	)	)	PUNCT
iajs-2860	59	15	=	=	PUNCT
iajs-2860	60	1	𝑘𝑏𝑖	𝑘𝑏𝑖	NOUN
iajs-2860	61	1	+	+	CCONJ
iajs-2860	61	2	𝑐𝑖	𝑐𝑖	VERB
iajs-2860	61	3	≈	≈	NUM
iajs-2860	61	4	𝑢′	𝑢′	ADP
iajs-2860	61	5	𝑖(𝑟𝑖)𝑠	𝑖(𝑟𝑖)𝑠	NUM
iajs-2860	61	6	𝑆′′	𝑆′′	NOUN
iajs-2860	61	7	𝑖;(𝑟𝑖	𝑖;(𝑟𝑖	PROPN
iajs-2860	61	8	)	)	PUNCT
iajs-2860	61	9	=	=	PUNCT
iajs-2860	62	1	−𝑘2𝑏𝑖	−𝑘2𝑏𝑖	NUM
iajs-2860	62	2	≈	≈	PROPN
iajs-2860	62	3	𝑢′′	𝑢′′	PROPN
iajs-2860	62	4	𝑖(𝑟𝑖)𝑠	𝑖(𝑟𝑖)𝑠	PROPN
iajs-2860	62	5	𝑆′′′	𝑆′′′	ADJ
iajs-2860	62	6	𝑖𝑠(𝑟𝑖	𝑖𝑠(𝑟𝑖	NOUN
iajs-2860	62	7	)	)	PUNCT
iajs-2860	62	8	=	=	SYM
iajs-2860	62	9	−𝑘3𝑎𝑖	−𝑘3𝑎𝑖	NUM
iajs-2860	63	1	≈	≈	NUM
iajs-2860	63	2	𝑢′′′	𝑢′′′	PROPN
iajs-2860	63	3	𝑖(𝑟𝑖)𝑠	𝑖(𝑟𝑖)𝑠	NOUN
iajs-2860	63	4	now	now	ADV
iajs-2860	63	5	,	,	PUNCT
iajs-2860	63	6	we	we	PRON
iajs-2860	63	7	can	can	AUX
iajs-2860	63	8	obtain	obtain	VERB
iajs-2860	63	9	the	the	DET
iajs-2860	63	10	values	value	NOUN
iajs-2860	63	11	of	of	ADP
iajs-2860	63	12	𝑎𝑖	𝑎𝑖	PROPN
iajs-2860	63	13	,	,	PUNCT
iajs-2860	63	14	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	63	15	,	,	PUNCT
iajs-2860	63	16	𝑐𝑖	𝑐𝑖	PROPN
iajs-2860	63	17	and	and	CCONJ
iajs-2860	63	18	𝑑𝑖	𝑑𝑖	PROPN
iajs-2860	63	19	as	as	SCONJ
iajs-2860	63	20	follows	follow	VERB
iajs-2860	63	21	:	:	PUNCT
iajs-2860	63	22	𝑎𝑖	𝑎𝑖	ADP
iajs-2860	63	23	=	=	SYM
iajs-2860	63	24	−1	−1	NOUN
iajs-2860	63	25	𝑘3	𝑘3	PROPN
iajs-2860	63	26	𝑢′′′(𝑟𝑖)𝑠	𝑢′′′(𝑟𝑖)𝑠	NOUN
iajs-2860	63	27	(	(	PUNCT
iajs-2860	63	28	7	7	NUM
iajs-2860	63	29	)	)	PUNCT
iajs-2860	63	30	𝑏𝑖	𝑏𝑖	NOUN
iajs-2860	63	31	=	=	PUNCT
iajs-2860	63	32	−1	−1	NOUN
iajs-2860	63	33	𝑘2	𝑘2	PROPN
iajs-2860	63	34	u′′(𝑟𝑖)s	u′′(𝑟𝑖)s	ADV
iajs-2860	63	35	(	(	PUNCT
iajs-2860	63	36	8)	8)	NUM
iajs-2860	63	37	𝑐𝑖	𝑐𝑖	NOUN
iajs-2860	63	38	=	=	SYM
iajs-2860	63	39	𝑢′(𝑟𝑖	𝑢′(𝑟𝑖	NOUN
iajs-2860	63	40	)	)	PUNCT
iajs-2860	64	1	+	+	CCONJ
iajs-2860	64	2	𝑠	𝑠	PROPN
iajs-2860	64	3	1	1	NUM
iajs-2860	64	4	𝑘	𝑘	PRON
iajs-2860	64	5	u′′(𝑟𝑖)𝑠	u′′(𝑟𝑖)𝑠	PROPN
iajs-2860	64	6	(	(	PUNCT
iajs-2860	64	7	9	9	NUM
iajs-2860	64	8	)	)	PUNCT
iajs-2860	64	9	𝑑𝑖	𝑑𝑖	PART
iajs-2860	64	10	=	=	SYM
iajs-2860	64	11	𝑢(𝑟𝑖	𝑢(𝑟𝑖	PROPN
iajs-2860	64	12	)	)	PUNCT
iajs-2860	64	13	+	+	CCONJ
iajs-2860	64	14	1	1	NUM
iajs-2860	64	15	𝑘	𝑘	PRON
iajs-2860	64	16	u′′(𝑟𝑖)𝑠	u′′(𝑟𝑖)𝑠	PROPN
iajs-2860	64	17	(	(	PUNCT
iajs-2860	64	18	10	10	NUM
iajs-2860	64	19	)	)	PUNCT
iajs-2860	64	20	for	for	ADP
iajs-2860	64	21	i=0	i=0	PROPN
iajs-2860	64	22	,	,	PUNCT
iajs-2860	64	23	1	1	NUM
iajs-2860	64	24	,	,	PUNCT
iajs-2860	64	25	2,	2,	NUM
iajs-2860	64	26	…	…	PUNCT
iajs-2860	64	27	,n	,n	PUNCT
iajs-2860	64	28	we	we	PRON
iajs-2860	64	29	differentiate	differentiate	VERB
iajs-2860	64	30	equation	equation	NOUN
iajs-2860	64	31	(	(	PUNCT
iajs-2860	64	32	1	1	NUM
iajs-2860	64	33	)	)	PUNCT
iajs-2860	64	34	three	three	NUM
iajs-2860	64	35	times	time	NOUN
iajs-2860	64	36	,	,	PUNCT
iajs-2860	64	37	with	with	ADP
iajs-2860	64	38	respect	respect	NOUN
iajs-2860	64	39	to	to	ADP
iajs-2860	64	40	r	r	NOUN
iajs-2860	64	41	,	,	PUNCT
iajs-2860	64	42	then	then	ADV
iajs-2860	64	43	put	put	VERB
iajs-2860	64	44	𝑟	𝑟	NOUN
iajs-2860	64	45	=	=	PUNCT
iajs-2860	64	46	a	a	PRON
iajs-2860	64	47	to	to	PART
iajs-2860	64	48	get	get	VERB
iajs-2860	64	49	:	:	PUNCT
iajs-2860	64	50	𝑢0	𝑢0	PROPN
iajs-2860	64	51	=	=	SYM
iajs-2860	64	52	𝑢(𝑎	𝑢(𝑎	X
iajs-2860	64	53	)	)	PUNCT
iajs-2860	64	54	=	=	SYM
iajs-2860	64	55	𝑓(𝑎	𝑓(𝑎	NOUN
iajs-2860	64	56	)	)	PUNCT
iajs-2860	64	57	(	(	PUNCT
iajs-2860	64	58	11	11	NUM
iajs-2860	64	59	)	)	PUNCT
iajs-2860	64	60	ihjpas	ihjpa	NOUN
iajs-2860	64	61	.	.	PUNCT
iajs-2860	65	1	36(1)2023	36(1)2023	NUM
iajs-2860	65	2	410	410	NUM
iajs-2860	65	3	𝑢0	𝑢0	PROPN
iajs-2860	65	4	=	=	SYM
iajs-2860	65	5	́	́	PROPN
iajs-2860	65	6	𝑢	𝑢	X
iajs-2860	65	7	́	́	PROPN
iajs-2860	65	8	(	(	PUNCT
iajs-2860	65	9	𝑎	𝑎	NOUN
iajs-2860	65	10	)	)	PUNCT
iajs-2860	65	11	=	=	SYM
iajs-2860	65	12	𝑓	𝑓	PROPN
iajs-2860	65	13	́	́	PROPN
iajs-2860	65	14	(	(	PUNCT
iajs-2860	65	15	𝑎	𝑎	NOUN
iajs-2860	65	16	)	)	PUNCT
iajs-2860	65	17	+	+	CCONJ
iajs-2860	65	18	𝑘(𝑎	𝑘(𝑎	ADJ
iajs-2860	65	19	,	,	PUNCT
iajs-2860	65	20	𝑎)𝑢(𝑎	𝑎)𝑢(𝑎	PROPN
iajs-2860	65	21	)	)	PUNCT
iajs-2860	65	22	(	(	PUNCT
iajs-2860	65	23	12	12	NUM
iajs-2860	65	24	)	)	PUNCT
iajs-2860	65	25	let	let	VERB
iajs-2860	65	26	𝐸′(𝑥	𝐸′(𝑥	ADJ
iajs-2860	65	27	,	,	PUNCT
iajs-2860	65	28	𝑡	𝑡	NOUN
iajs-2860	65	29	,	,	PUNCT
iajs-2860	65	30	𝑢(𝑡	𝑢(𝑡	NOUN
iajs-2860	65	31	,	,	PUNCT
iajs-2860	65	32	𝑟	𝑟	NOUN
iajs-2860	65	33	)	)	PUNCT
iajs-2860	65	34	;	;	PUNCT
iajs-2860	65	35	𝑟	𝑟	X
iajs-2860	65	36	)	)	PUNCT
iajs-2860	65	37	=	=	SYM
iajs-2860	65	38	𝜕𝑘(𝑥,𝑡	𝜕𝑘(𝑥,𝑡	NOUN
iajs-2860	65	39	𝑢(𝑡,𝑟))′′	𝑢(𝑡,𝑟))′′	PROPN
iajs-2860	65	40	𝜕𝑟	𝜕𝑟	ADV
iajs-2860	65	41	𝑢′′(𝑟	𝑢′′(𝑟	ADJ
iajs-2860	65	42	)	)	PUNCT
iajs-2860	65	43	=	=	SYM
iajs-2860	65	44	𝑓0	𝑓0	NOUN
iajs-2860	65	45	′′(𝑟	′′(𝑟	NOUN
iajs-2860	65	46	)	)	PUNCT
iajs-2860	66	1	+	+	CCONJ
iajs-2860	66	2	∫	∫	PROPN
iajs-2860	66	3	𝜕𝐸′(𝑥,𝑡,𝑢(𝑡,𝑟);𝑟)′′	𝜕𝐸′(𝑥,𝑡,𝑢(𝑡,𝑟);𝑟)′′	PROPN
iajs-2860	66	4	𝜕𝑟	𝜕𝑟	VERB
iajs-2860	66	5	𝑥	𝑥	ADP
iajs-2860	66	6	0	0	NUM
iajs-2860	66	7	𝑑𝑡	𝑑𝑡	ADP
iajs-2860	66	8	+	+	PROPN
iajs-2860	66	9	2e(x	2e(x	PROPN
iajs-2860	66	10	,	,	PUNCT
iajs-2860	66	11	t	t	PROPN
iajs-2860	66	12	,	,	PUNCT
iajs-2860	66	13	u(t	u(t	NOUN
iajs-2860	66	14	,	,	PUNCT
iajs-2860	66	15	r	r	NOUN
iajs-2860	66	16	):	):	PUNCT
iajs-2860	66	17	r)′′	r)′′	PROPN
iajs-2860	66	18	𝑢0	𝑢0	PROPN
iajs-2860	66	19	′′(𝑎	′′(𝑎	PROPN
iajs-2860	66	20	)	)	PUNCT
iajs-2860	66	21	=	=	SYM
iajs-2860	66	22	𝑢′′(𝑎	𝑢′′(𝑎	ADJ
iajs-2860	66	23	)	)	PUNCT
iajs-2860	66	24	=	=	SYM
iajs-2860	66	25	𝑓0	𝑓0	PROPN
iajs-2860	66	26	′′(𝑎	′′(𝑎	NOUN
iajs-2860	66	27	)	)	PUNCT
iajs-2860	67	1	+	+	X
iajs-2860	67	2	2𝐸′(𝑎	2𝐸′(𝑎	NOUN
iajs-2860	67	3	,	,	PUNCT
iajs-2860	67	4	𝑎	𝑎	NOUN
iajs-2860	67	5	,	,	PUNCT
iajs-2860	67	6	𝑢(𝑎	𝑢(𝑎	PROPN
iajs-2860	67	7	)	)	PUNCT
iajs-2860	67	8	)	)	PUNCT
iajs-2860	67	9	(	(	PUNCT
iajs-2860	67	10	13	13	X
iajs-2860	67	11	)	)	PUNCT
iajs-2860	67	12	let	let	VERB
iajs-2860	67	13	f	f	PROPN
iajs-2860	67	14	(	(	PUNCT
iajs-2860	67	15	x	x	PROPN
iajs-2860	67	16	,	,	PUNCT
iajs-2860	67	17	t	t	PROPN
iajs-2860	67	18	,	,	PUNCT
iajs-2860	67	19	u(t	u(t	PROPN
iajs-2860	67	20	,	,	PUNCT
iajs-2860	67	21	r	r	NOUN
iajs-2860	67	22	)	)	PUNCT
iajs-2860	67	23	;	;	PUNCT
iajs-2860	67	24	r	r	X
iajs-2860	67	25	)	)	PUNCT
iajs-2860	67	26	=	=	PUNCT
iajs-2860	68	1	𝜕𝐸(𝑥,𝑡,𝑢(𝑡,𝑟);𝑟)′	𝜕𝐸(𝑥,𝑡,𝑢(𝑡,𝑟);𝑟)′	X
iajs-2860	68	2	𝜕𝑟	𝜕𝑟	ADV
iajs-2860	68	3	u′′′(𝑟	u′′′(𝑟	ADJ
iajs-2860	68	4	)	)	PUNCT
iajs-2860	68	5	=	=	SYM
iajs-2860	68	6	𝑓′′′(r	𝑓′′′(r	PROPN
iajs-2860	68	7	)	)	PUNCT
iajs-2860	68	8	+	+	NUM
iajs-2860	68	9	∫	∫	PROPN
iajs-2860	68	10	∂f	∂f	PROPN
iajs-2860	68	11	(	(	PUNCT
iajs-2860	68	12	x	x	PROPN
iajs-2860	68	13	,	,	PUNCT
iajs-2860	68	14	t	t	PROPN
iajs-2860	68	15	,	,	PUNCT
iajs-2860	68	16	u(t	u(t	PROPN
iajs-2860	68	17	,	,	PUNCT
iajs-2860	68	18	r);r)′	r);r)′	VERB
iajs-2860	68	19	𝜕𝑟	𝜕𝑟	PROPN
iajs-2860	68	20	𝑥	𝑥	ADP
iajs-2860	68	21	0	0	NUM
iajs-2860	68	22	𝑑𝑡	𝑑𝑡	ADP
iajs-2860	68	23	+	+	ADJ
iajs-2860	68	24	3f	3f	X
iajs-2860	68	25	(	(	PUNCT
iajs-2860	68	26	x	x	PROPN
iajs-2860	68	27	,	,	PUNCT
iajs-2860	68	28	t	t	PROPN
iajs-2860	68	29	,	,	PUNCT
iajs-2860	68	30	u(t	u(t	PROPN
iajs-2860	68	31	,	,	PUNCT
iajs-2860	68	32	r	r	NOUN
iajs-2860	68	33	)	)	PUNCT
iajs-2860	68	34	;	;	PUNCT
iajs-2860	68	35	r	r	X
iajs-2860	68	36	)	)	PUNCT
iajs-2860	68	37	𝑢0	𝑢0	PROPN
iajs-2860	68	38	′′′(a	′′′(a	PROPN
iajs-2860	68	39	)	)	PUNCT
iajs-2860	68	40	=	=	SYM
iajs-2860	68	41	𝑓′′′(𝑎	𝑓′′′(𝑎	PROPN
iajs-2860	68	42	)	)	PUNCT
iajs-2860	69	1	+	+	NUM
iajs-2860	69	2	3f′	3f′	NUM
iajs-2860	69	3	(	(	PUNCT
iajs-2860	69	4	a	a	PROPN
iajs-2860	69	5	,	,	PUNCT
iajs-2860	69	6	a	a	PRON
iajs-2860	69	7	,	,	PUNCT
iajs-2860	69	8	a	a	PRON
iajs-2860	69	9	,	,	PUNCT
iajs-2860	69	10	u(a	u(a	PROPN
iajs-2860	69	11	)	)	PUNCT
iajs-2860	69	12	)	)	PUNCT
iajs-2860	69	13	(	(	PUNCT
iajs-2860	69	14	14	14	NUM
iajs-2860	69	15	)	)	PUNCT
iajs-2860	69	16	3.2	3.2	NUM
iajs-2860	69	17	quadratic	quadratic	ADJ
iajs-2860	69	18	non	non	ADJ
iajs-2860	69	19	-	-	ADJ
iajs-2860	69	20	polynomial	polynomial	ADJ
iajs-2860	69	21	splines	spline	NOUN
iajs-2860	69	22	(	(	PUNCT
iajs-2860	69	23	qnps	qnps	ADJ
iajs-2860	69	24	):	):	PUNCT
iajs-2860	69	25	we	we	PRON
iajs-2860	69	26	consider	consider	VERB
iajs-2860	69	27	the	the	DET
iajs-2860	69	28	qnps	qnps	ADJ
iajs-2860	69	29	method	method	NOUN
iajs-2860	69	30	for	for	ADP
iajs-2860	69	31	finding	find	VERB
iajs-2860	69	32	an	an	DET
iajs-2860	69	33	approximate	approximate	ADJ
iajs-2860	69	34	solution	solution	NOUN
iajs-2860	69	35	of	of	ADP
iajs-2860	69	36	fvie	fvie	NOUN
iajs-2860	69	37	of	of	ADP
iajs-2860	69	38	the	the	DET
iajs-2860	69	39	second	second	ADJ
iajs-2860	69	40	kind	kind	NOUN
iajs-2860	69	41	on	on	ADP
iajs-2860	69	42	the	the	DET
iajs-2860	69	43	interval	interval	NOUN
iajs-2860	69	44	[	[	X
iajs-2860	69	45	𝑎	𝑎	X
iajs-2860	69	46	,	,	PUNCT
iajs-2860	69	47	𝑏	𝑏	NOUN
iajs-2860	69	48	]	]	PUNCT
iajs-2860	69	49	as	as	ADP
iajs-2860	69	50	the	the	DET
iajs-2860	69	51	above	above	ADJ
iajs-2860	69	52	equations	equation	NOUN
iajs-2860	69	53	(	(	PUNCT
iajs-2860	69	54	3),(4	3),(4	PROPN
iajs-2860	69	55	)	)	PUNCT
iajs-2860	69	56	and	and	CCONJ
iajs-2860	69	57	(	(	PUNCT
iajs-2860	69	58	5	5	X
iajs-2860	69	59	)	)	PUNCT
iajs-2860	69	60	the	the	DET
iajs-2860	69	61	form	form	NOUN
iajs-2860	69	62	of	of	ADP
iajs-2860	69	63	qnps	qnps	ADJ
iajs-2860	69	64	function	function	NOUN
iajs-2860	69	65	[	[	X
iajs-2860	69	66	10	10	NUM
iajs-2860	69	67	]	]	PUNCT
iajs-2860	69	68	is	be	AUX
iajs-2860	69	69	𝑄𝑖(𝑥	𝑄𝑖(𝑥	PROPN
iajs-2860	69	70	,	,	PUNCT
iajs-2860	69	71	𝑟)′	𝑟)′	PROPN
iajs-2860	69	72	=	=	PUNCT
iajs-2860	69	73	𝑎𝑖	𝑎𝑖	X
iajs-2860	69	74	sin′	sin′	NOUN
iajs-2860	69	75	𝑘(𝑟	𝑘(𝑟	PROPN
iajs-2860	69	76	−	−	PROPN
iajs-2860	69	77	𝑟𝑖	𝑟𝑖	PART
iajs-2860	69	78	)	)	PUNCT
iajs-2860	70	1	+	+	CCONJ
iajs-2860	70	2	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	70	3	cos′	cos′	PROPN
iajs-2860	70	4	𝑘(𝑟	𝑘(𝑟	PROPN
iajs-2860	70	5	−	−	PROPN
iajs-2860	70	6	𝑟𝑖	𝑟𝑖	PART
iajs-2860	70	7	)	)	PUNCT
iajs-2860	71	1	+	+	CCONJ
iajs-2860	71	2	𝑐𝑖(𝑟	𝑐𝑖(𝑟	CCONJ
iajs-2860	71	3	−	−	NOUN
iajs-2860	71	4	𝑟𝑖	𝑟𝑖	PUNCT
iajs-2860	71	5	)	)	PUNCT
iajs-2860	71	6	+	+	CCONJ
iajs-2860	71	7	𝑑𝑖(𝑟	𝑑𝑖(𝑟	VERB
iajs-2860	71	8	−	−	NUM
iajs-2860	71	9	𝑟𝑖	𝑟𝑖	NOUN
iajs-2860	71	10	)	)	PUNCT
iajs-2860	71	11	2	2	NUM
iajs-2860	71	12	+	+	CCONJ
iajs-2860	71	13	𝑒𝑖	𝑒𝑖	X
iajs-2860	71	14	(	(	PUNCT
iajs-2860	71	15	15	15	NUM
iajs-2860	71	16	)	)	PUNCT
iajs-2860	71	17	where	where	SCONJ
iajs-2860	71	18	𝑎𝑖	𝑎𝑖	ADP
iajs-2860	71	19	,	,	PUNCT
iajs-2860	71	20	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	71	21	,	,	PUNCT
iajs-2860	71	22	𝑐𝑖	𝑐𝑖	NOUN
iajs-2860	71	23	,	,	PUNCT
iajs-2860	71	24	𝑑𝑖	𝑑𝑖	PROPN
iajs-2860	71	25	and	and	CCONJ
iajs-2860	71	26	𝑒𝑖	𝑒𝑖	PROPN
iajs-2860	71	27	are	be	AUX
iajs-2860	71	28	constants	constant	NOUN
iajs-2860	71	29	.	.	PUNCT
iajs-2860	72	1	we	we	PRON
iajs-2860	72	2	differentiate	differentiate	VERB
iajs-2860	72	3	equation	equation	NOUN
iajs-2860	72	4	(	(	PUNCT
iajs-2860	72	5	15	15	NUM
iajs-2860	72	6	)	)	PUNCT
iajs-2860	72	7	four	four	NUM
iajs-2860	72	8	times	time	NOUN
iajs-2860	72	9	with	with	ADP
iajs-2860	72	10	.respect	.respect	ADP
iajs-2860	72	11	to	to	ADP
iajs-2860	72	12	r	r	NOUN
iajs-2860	72	13	and	and	CCONJ
iajs-2860	72	14	put	put	VERB
iajs-2860	72	15	r	r	NOUN
iajs-2860	72	16	=	=	NOUN
iajs-2860	72	17	a	a	PRON
iajs-2860	72	18	then	then	ADV
iajs-2860	72	19	replace	replace	VERB
iajs-2860	72	20	r	r	NOUN
iajs-2860	72	21	by	by	ADP
iajs-2860	72	22	𝑟𝑖	𝑟𝑖	NOUN
iajs-2860	72	23	in	in	ADP
iajs-2860	72	24	the	the	DET
iajs-2860	72	25	relation	relation	NOUN
iajs-2860	72	26	equation	equation	NOUN
iajs-2860	72	27	(	(	PUNCT
iajs-2860	72	28	15	15	NUM
iajs-2860	72	29	)	)	PUNCT
iajs-2860	72	30	to	to	PART
iajs-2860	72	31	yield	yield	VERB
iajs-2860	72	32	:	:	PUNCT
iajs-2860	72	33	𝑄𝑖(𝑥	𝑄𝑖(𝑥	PROPN
iajs-2860	72	34	,	,	PUNCT
iajs-2860	72	35	𝑟)′	𝑟)′	PROPN
iajs-2860	72	36	=	=	PUNCT
iajs-2860	72	37	𝑎𝑖	𝑎𝑖	PROPN
iajs-2860	72	38	+	+	PROPN
iajs-2860	72	39	𝑒𝑖	𝑒𝑖	ADP
iajs-2860	72	40	𝑄′𝑖(𝑥	𝑄′𝑖(𝑥	NOUN
iajs-2860	72	41	,	,	PUNCT
iajs-2860	72	42	𝑟)′	𝑟)′	PROPN
iajs-2860	72	43	=	=	PUNCT
iajs-2860	73	1	𝑘𝑏𝑖	𝑘𝑏𝑖	PROPN
iajs-2860	73	2	+	+	CCONJ
iajs-2860	73	3	𝑐𝑖	𝑐𝑖	NOUN
iajs-2860	73	4	𝑄′′𝑖(𝑥	𝑄′′𝑖(𝑥	PROPN
iajs-2860	73	5	,	,	PUNCT
iajs-2860	73	6	𝑟)′	𝑟)′	PROPN
iajs-2860	73	7	=	=	SYM
iajs-2860	73	8	−𝑘2𝑎𝑖	−𝑘2𝑎𝑖	PROPN
iajs-2860	73	9	+	+	CCONJ
iajs-2860	73	10	2𝑑𝑖	2𝑑𝑖	NOUN
iajs-2860	73	11	𝑄′′′𝑖(𝑥	𝑄′′′𝑖(𝑥	ADJ
iajs-2860	73	12	,	,	PUNCT
iajs-2860	73	13	𝑟)′	𝑟)′	PROPN
iajs-2860	73	14	=	=	PROPN
iajs-2860	73	15	−𝑘3𝑏𝑖	−𝑘3𝑏𝑖	NUM
iajs-2860	73	16	𝑄𝑖	𝑄𝑖	PROPN
iajs-2860	73	17	(	(	PUNCT
iajs-2860	73	18	4)(𝑥	4)(𝑥	NUM
iajs-2860	73	19	,	,	PUNCT
iajs-2860	73	20	𝑟	𝑟	NOUN
iajs-2860	73	21	)	)	PUNCT
iajs-2860	73	22	=	=	VERB
iajs-2860	74	1	𝑘(4)𝑎𝑖	𝑘(4)𝑎𝑖	INTJ
iajs-2860	74	2	we	we	PRON
iajs-2860	74	3	obtain	obtain	VERB
iajs-2860	74	4	the	the	DET
iajs-2860	74	5	values	value	NOUN
iajs-2860	74	6	of	of	ADP
iajs-2860	74	7	𝑎𝑖	𝑎𝑖	PROPN
iajs-2860	74	8	,	,	PUNCT
iajs-2860	74	9	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	74	10	,	,	PUNCT
iajs-2860	74	11	𝑐𝑖	𝑐𝑖	NOUN
iajs-2860	74	12	,	,	PUNCT
iajs-2860	74	13	𝑑𝑖	𝑑𝑖	PROPN
iajs-2860	74	14	and	and	CCONJ
iajs-2860	74	15	𝑒𝑖	𝑒𝑖	X
iajs-2860	74	16	from	from	ADP
iajs-2860	74	17	the	the	DET
iajs-2860	74	18	above	above	ADJ
iajs-2860	74	19	relation	relation	NOUN
iajs-2860	74	20	,	,	PUNCT
iajs-2860	74	21	as	as	SCONJ
iajs-2860	74	22	follows	follow	VERB
iajs-2860	74	23	𝑎𝑖	𝑎𝑖	X
iajs-2860	74	24	=	=	PROPN
iajs-2860	74	25	𝑄𝑖	𝑄𝑖	PROPN
iajs-2860	74	26	𝟒(𝑥,𝑟𝑖	𝟒(𝑥,𝑟𝑖	PROPN
iajs-2860	74	27	)	)	PUNCT
iajs-2860	74	28	𝑘4	𝑘4	PROPN
iajs-2860	74	29	(	(	PUNCT
iajs-2860	74	30	16	16	NUM
iajs-2860	74	31	)	)	PUNCT
iajs-2860	74	32	𝑏𝑖	𝑏𝑖	ADP
iajs-2860	74	33	=	=	PUNCT
iajs-2860	75	1	−	−	PROPN
iajs-2860	75	2	𝑄𝑖	𝑄𝑖	PROPN
iajs-2860	75	3	′′′(𝑥,𝑟𝑖	′′′(𝑥,𝑟𝑖	PROPN
iajs-2860	75	4	)	)	PUNCT
iajs-2860	75	5	𝑘3	𝑘3	PROPN
iajs-2860	75	6	(	(	PUNCT
iajs-2860	75	7	17	17	NUM
iajs-2860	75	8	)	)	PUNCT
iajs-2860	75	9	𝑐𝑖	𝑐𝑖	NOUN
iajs-2860	75	10	=	=	SYM
iajs-2860	75	11	𝑄′𝑖(𝑥	𝑄′𝑖(𝑥	PROPN
iajs-2860	75	12	,	,	PUNCT
iajs-2860	75	13	𝑟𝑖	𝑟𝑖	PUNCT
iajs-2860	75	14	)	)	PUNCT
iajs-2860	75	15	′	′	NUM
iajs-2860	76	1	−	−	PROPN
iajs-2860	76	2	𝑘𝑏𝑖	𝑘𝑏𝑖	INTJ
iajs-2860	76	3	(	(	PUNCT
iajs-2860	76	4	18	18	NUM
iajs-2860	76	5	)	)	PUNCT
iajs-2860	76	6	𝑑𝑖	𝑑𝑖	PART
iajs-2860	77	1	=	=	SYM
iajs-2860	77	2	𝑄′′𝑖(𝑥,𝑟𝑖)′+𝑘2𝑎𝑖	𝑄′′𝑖(𝑥,𝑟𝑖)′+𝑘2𝑎𝑖	X
iajs-2860	77	3	𝟐	𝟐	NUM
iajs-2860	77	4	(	(	PUNCT
iajs-2860	77	5	19	19	NUM
iajs-2860	77	6	)	)	PUNCT
iajs-2860	77	7	𝑒𝑖	𝑒𝑖	NOUN
iajs-2860	77	8	=	=	PUNCT
iajs-2860	77	9	𝑄𝑖(𝑥	𝑄𝑖(𝑥	PROPN
iajs-2860	77	10	,	,	PUNCT
iajs-2860	77	11	𝑟𝑖	𝑟𝑖	X
iajs-2860	77	12	)	)	PUNCT
iajs-2860	77	13	′	′	NUM
iajs-2860	78	1	−	−	NOUN
iajs-2860	78	2	𝑎𝑖	𝑎𝑖	INTJ
iajs-2860	78	3	(	(	PUNCT
iajs-2860	78	4	20	20	NUM
iajs-2860	78	5	)	)	PUNCT
iajs-2860	78	6	4	4	NUM
iajs-2860	78	7	.	.	X
iajs-2860	79	1	lnps	lnps	PROPN
iajs-2860	79	2	and	and	CCONJ
iajs-2860	79	3	qnps	qnps	ADJ
iajs-2860	79	4	methods	method	NOUN
iajs-2860	79	5	of	of	ADP
iajs-2860	79	6	fviewsk	fviewsk	NOUN
iajs-2860	79	7	:	:	PUNCT
iajs-2860	79	8	in	in	ADP
iajs-2860	79	9	this	this	DET
iajs-2860	79	10	section	section	NOUN
iajs-2860	79	11	,	,	PUNCT
iajs-2860	79	12	we	we	PRON
iajs-2860	79	13	use	use	VERB
iajs-2860	79	14	lnps	lnps	NOUN
iajs-2860	79	15	and	and	CCONJ
iajs-2860	79	16	qnps	qnps	ADJ
iajs-2860	79	17	methods	method	NOUN
iajs-2860	79	18	to	to	PART
iajs-2860	79	19	compute	compute	VERB
iajs-2860	79	20	numerical	numerical	ADJ
iajs-2860	79	21	solution	solution	NOUN
iajs-2860	79	22	of	of	ADP
iajs-2860	79	23	fviewsk	fviewsk	NOUN
iajs-2860	79	24	which	which	PRON
iajs-2860	79	25	is	be	AUX
iajs-2860	79	26	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	79	27	,	,	PUNCT
iajs-2860	79	28	𝑟	𝑟	NOUN
iajs-2860	79	29	)	)	PUNCT
iajs-2860	79	30	−	−	NUM
iajs-2860	79	31	∫	∫	PROPN
iajs-2860	79	32	𝑡𝜇−1	𝑡𝜇−1	PROPN
iajs-2860	79	33	𝑥𝜇	𝑥𝜇	CCONJ
iajs-2860	79	34	𝑥	𝑥	NOUN
iajs-2860	79	35	0	0	NUM
iajs-2860	79	36	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	79	37	,	,	PUNCT
iajs-2860	79	38	𝑡)𝑑𝑡	𝑡)𝑑𝑡	PROPN
iajs-2860	79	39	=	=	SYM
iajs-2860	79	40	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	79	41	,	,	PUNCT
iajs-2860	79	42	𝑟	𝑟	NOUN
iajs-2860	79	43	)	)	PUNCT
iajs-2860	79	44	,	,	PUNCT
iajs-2860	80	1	𝑥	𝑥	PRON
iajs-2860	80	2	∈	∈	PROPN
iajs-2860	81	1	[	[	X
iajs-2860	81	2	0	0	NUM
iajs-2860	81	3	,	,	PUNCT
iajs-2860	81	4	𝑇	𝑇	PROPN
iajs-2860	81	5	]	]	X
iajs-2860	81	6	(	(	PUNCT
iajs-2860	81	7	21	21	NUM
iajs-2860	81	8	)	)	PUNCT
iajs-2860	81	9	where	where	SCONJ
iajs-2860	81	10	0	0	X
iajs-2860	81	11	<	<	X
iajs-2860	81	12	𝜇	𝜇	X
iajs-2860	81	13	<	<	X
iajs-2860	81	14	1	1	NUM
iajs-2860	81	15	and	and	CCONJ
iajs-2860	81	16	f	f	PROPN
iajs-2860	81	17	is	be	AUX
iajs-2860	81	18	a	a	DET
iajs-2860	81	19	known	know	VERB
iajs-2860	81	20	function	function	NOUN
iajs-2860	81	21	[	[	X
iajs-2860	81	22	11	11	NUM
iajs-2860	81	23	]	]	PUNCT
iajs-2860	81	24	.	.	PUNCT
iajs-2860	82	1	there	there	PRON
iajs-2860	82	2	is	be	VERB
iajs-2860	82	3	singularity	singularity	NOUN
iajs-2860	82	4	at	at	ADP
iajs-2860	82	5	𝑥	𝑥	NOUN
iajs-2860	82	6	=	=	SYM
iajs-2860	82	7	0	0	NUM
iajs-2860	82	8	and	and	CCONJ
iajs-2860	82	9	𝑡	𝑡	X
iajs-2860	82	10	=	=	NOUN
iajs-2860	82	11	0	0	NUM
iajs-2860	82	12	for	for	ADP
iajs-2860	82	13	any	any	DET
iajs-2860	82	14	positive	positive	ADJ
iajs-2860	82	15	value	value	NOUN
iajs-2860	82	16	of	of	ADP
iajs-2860	82	17	t	t	PROPN
iajs-2860	82	18	.	.	PUNCT
iajs-2860	83	1	to	to	PART
iajs-2860	83	2	solve	solve	VERB
iajs-2860	83	3	equation	equation	NOUN
iajs-2860	83	4	(	(	PUNCT
iajs-2860	83	5	21	21	NUM
iajs-2860	83	6	)	)	PUNCT
iajs-2860	83	7	,	,	PUNCT
iajs-2860	83	8	we	we	PRON
iajs-2860	83	9	multiply	multiply	VERB
iajs-2860	83	10	both	both	DET
iajs-2860	83	11	sides	side	NOUN
iajs-2860	83	12	by	by	ADP
iajs-2860	83	13	𝑥𝜇	𝑥𝜇	ADP
iajs-2860	83	14	to	to	PART
iajs-2860	83	15	get	get	VERB
iajs-2860	83	16	𝑥𝜇	𝑥𝜇	ADP
iajs-2860	83	17	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	83	18	,	,	PUNCT
iajs-2860	83	19	𝑟	𝑟	NOUN
iajs-2860	83	20	)	)	PUNCT
iajs-2860	84	1	−	−	ADP
iajs-2860	84	2	∫	∫	PROPN
iajs-2860	85	1	𝑡𝜇−1𝑥	𝑡𝜇−1𝑥	PROPN
iajs-2860	85	2	0	0	NUM
iajs-2860	85	3	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	85	4	,	,	PUNCT
iajs-2860	85	5	𝑡)𝑑𝑡	𝑡)𝑑𝑡	PROPN
iajs-2860	85	6	=	=	SYM
iajs-2860	85	7	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	85	8	,	,	PUNCT
iajs-2860	85	9	𝑟)𝑥𝜇	𝑟)𝑥𝜇	PROPN
iajs-2860	85	10	(	(	PUNCT
iajs-2860	85	11	22	22	NUM
iajs-2860	85	12	)	)	PUNCT
iajs-2860	85	13	hence	hence	ADV
iajs-2860	85	14	,	,	PUNCT
iajs-2860	85	15	we	we	PRON
iajs-2860	85	16	differentiate	differentiate	VERB
iajs-2860	85	17	equation	equation	NOUN
iajs-2860	85	18	(	(	PUNCT
iajs-2860	85	19	22	22	NUM
iajs-2860	85	20	)	)	PUNCT
iajs-2860	85	21	with	with	ADP
iajs-2860	85	22	respect	respect	NOUN
iajs-2860	85	23	to	to	ADP
iajs-2860	85	24	x	x	PRON
iajs-2860	85	25	,	,	PUNCT
iajs-2860	85	26	then	then	ADV
iajs-2860	85	27	we	we	PRON
iajs-2860	85	28	get	get	VERB
iajs-2860	85	29	𝑥𝜇	𝑥𝜇	ADP
iajs-2860	85	30	𝑢′(𝑥	𝑢′(𝑥	PRON
iajs-2860	85	31	,	,	PUNCT
iajs-2860	85	32	𝑟	𝑟	X
iajs-2860	85	33	)	)	PUNCT
iajs-2860	86	1	+	+	CCONJ
iajs-2860	86	2	𝜇𝑥𝜇−1𝑢(𝑥	𝜇𝑥𝜇−1𝑢(𝑥	NOUN
iajs-2860	86	3	,	,	PUNCT
iajs-2860	86	4	𝑟	𝑟	NOUN
iajs-2860	86	5	)	)	PUNCT
iajs-2860	86	6	−	−	PROPN
iajs-2860	86	7	1	1	NUM
iajs-2860	86	8	𝑥1−𝜇	𝑥1−𝜇	PROPN
iajs-2860	86	9	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	86	10	,	,	PUNCT
iajs-2860	86	11	𝑟	𝑟	NOUN
iajs-2860	86	12	)	)	PUNCT
iajs-2860	86	13	=	=	SYM
iajs-2860	86	14	𝜇𝑥𝜇−1𝑓(𝑥	𝜇𝑥𝜇−1𝑓(𝑥	NOUN
iajs-2860	86	15	,	,	PUNCT
iajs-2860	86	16	𝑟	𝑟	NOUN
iajs-2860	86	17	)	)	PUNCT
iajs-2860	86	18	+	+	CCONJ
iajs-2860	86	19	𝑥𝑓′(𝑥	𝑥𝑓′(𝑥	ADJ
iajs-2860	86	20	,	,	PUNCT
iajs-2860	86	21	𝑟)𝑥𝜇	𝑟)𝑥𝜇	PROPN
iajs-2860	86	22	(	(	PUNCT
iajs-2860	86	23	23	23	NUM
iajs-2860	86	24	)	)	PUNCT
iajs-2860	86	25	and	and	CCONJ
iajs-2860	86	26	multiplying	multiply	VERB
iajs-2860	86	27	both	both	DET
iajs-2860	86	28	sides	side	NOUN
iajs-2860	86	29	by	by	ADP
iajs-2860	86	30	𝑥1−𝜇	𝑥1−𝜇	NOUN
iajs-2860	86	31	to	to	PART
iajs-2860	86	32	get	get	VERB
iajs-2860	86	33	𝑥𝑢′(𝑥	𝑥𝑢′(𝑥	PROPN
iajs-2860	86	34	,	,	PUNCT
iajs-2860	86	35	𝑟	𝑟	NOUN
iajs-2860	86	36	)	)	PUNCT
iajs-2860	86	37	+	+	CCONJ
iajs-2860	86	38	(	(	PUNCT
iajs-2860	86	39	1	1	NUM
iajs-2860	86	40	−	−	PROPN
iajs-2860	86	41	𝜇)𝑢(𝑥	𝜇)𝑢(𝑥	NOUN
iajs-2860	86	42	,	,	PUNCT
iajs-2860	86	43	𝑟	𝑟	NOUN
iajs-2860	86	44	)	)	PUNCT
iajs-2860	86	45	=	=	SYM
iajs-2860	86	46	𝜇𝑓(𝑥	𝜇𝑓(𝑥	NOUN
iajs-2860	86	47	,	,	PUNCT
iajs-2860	86	48	𝑟	𝑟	NOUN
iajs-2860	86	49	)	)	PUNCT
iajs-2860	86	50	+	+	CCONJ
iajs-2860	86	51	𝑥𝑓′(𝑥	𝑥𝑓′(𝑥	ADJ
iajs-2860	86	52	,	,	PUNCT
iajs-2860	86	53	𝑟	𝑟	NOUN
iajs-2860	86	54	)	)	PUNCT
iajs-2860	86	55	(	(	PUNCT
iajs-2860	86	56	24	24	NUM
iajs-2860	86	57	)	)	PUNCT
iajs-2860	86	58	hence	hence	ADV
iajs-2860	86	59	,	,	PUNCT
iajs-2860	86	60	we	we	PRON
iajs-2860	86	61	differentiate	differentiate	VERB
iajs-2860	86	62	equation	equation	NOUN
iajs-2860	86	63	(	(	PUNCT
iajs-2860	86	64	24	24	NUM
iajs-2860	86	65	)	)	PUNCT
iajs-2860	86	66	four	four	NUM
iajs-2860	86	67	times	time	NOUN
iajs-2860	86	68	with	with	ADP
iajs-2860	86	69	respect	respect	NOUN
iajs-2860	86	70	to	to	ADP
iajs-2860	86	71	𝑥	𝑥	PROPN
iajs-2860	86	72	and	and	CCONJ
iajs-2860	86	73	replace	replace	VERB
iajs-2860	86	74	𝑥	𝑥	NOUN
iajs-2860	86	75	with	with	ADP
iajs-2860	86	76	the	the	DET
iajs-2860	86	77	relation	relation	NOUN
iajs-2860	86	78	to	to	PART
iajs-2860	86	79	yield	yield	VERB
iajs-2860	86	80	:	:	PUNCT
iajs-2860	86	81	𝑢0	𝑢0	PROPN
iajs-2860	86	82	=	=	SYM
iajs-2860	86	83	𝜇	𝜇	ADP
iajs-2860	86	84	𝜇−1	𝜇−1	PROPN
iajs-2860	86	85	𝑓(𝑎	𝑓(𝑎	NOUN
iajs-2860	86	86	,	,	PUNCT
iajs-2860	86	87	𝑎	𝑎	NOUN
iajs-2860	86	88	)	)	PUNCT
iajs-2860	86	89	𝑢′	𝑢′	ADJ
iajs-2860	86	90	0	0	NUM
iajs-2860	86	91	=	=	SYM
iajs-2860	86	92	𝜇+1	𝜇+1	NUM
iajs-2860	86	93	𝜇	𝜇	ADP
iajs-2860	86	94	𝑓′(𝑎	𝑓′(𝑎	NOUN
iajs-2860	86	95	,	,	PUNCT
iajs-2860	86	96	𝑎	𝑎	NOUN
iajs-2860	86	97	)	)	PUNCT
iajs-2860	86	98	ihjpas	ihjpa	NOUN
iajs-2860	86	99	.	.	PUNCT
iajs-2860	87	1	36(1)2023	36(1)2023	NUM
iajs-2860	87	2	411	411	NUM
iajs-2860	87	3	𝑢′′0	𝑢′′0	NOUN
iajs-2860	87	4	=	=	SYM
iajs-2860	87	5	𝜇+2	𝜇+2	PROPN
iajs-2860	87	6	𝜇+1	𝜇+1	PROPN
iajs-2860	87	7	𝑓′′(𝑎	𝑓′′(𝑎	NOUN
iajs-2860	87	8	,	,	PUNCT
iajs-2860	87	9	𝑎	𝑎	NOUN
iajs-2860	87	10	)	)	PUNCT
iajs-2860	87	11	(	(	PUNCT
iajs-2860	87	12	25	25	NUM
iajs-2860	87	13	)	)	PUNCT
iajs-2860	87	14	𝑢0	𝑢0	NOUN
iajs-2860	87	15	′′′	′′′	NOUN
iajs-2860	88	1	=	=	SYM
iajs-2860	89	1	𝜇+3	𝜇+3	PROPN
iajs-2860	89	2	𝜇+2	𝜇+2	ADP
iajs-2860	89	3	𝑓′′′(𝑎	𝑓′′′(𝑎	PROPN
iajs-2860	89	4	,	,	PUNCT
iajs-2860	89	5	𝑎	𝑎	NOUN
iajs-2860	89	6	)	)	PUNCT
iajs-2860	89	7	𝑢0	𝑢0	NOUN
iajs-2860	89	8	4	4	NUM
iajs-2860	89	9	=	=	SYM
iajs-2860	89	10	𝜇+4	𝜇+4	NUM
iajs-2860	89	11	𝜇+3	𝜇+3	PROPN
iajs-2860	89	12	𝑓4(𝑎	𝑓4(𝑎	PROPN
iajs-2860	89	13	,	,	PUNCT
iajs-2860	89	14	𝑎	𝑎	NOUN
iajs-2860	89	15	)	)	PUNCT
iajs-2860	89	16	to	to	PART
iajs-2860	89	17	approximate	approximate	VERB
iajs-2860	89	18	the	the	DET
iajs-2860	89	19	solution	solution	NOUN
iajs-2860	89	20	of	of	ADP
iajs-2860	89	21	linear	linear	PROPN
iajs-2860	89	22	fviewsk	fviewsk	PROPN
iajs-2860	89	23	in	in	ADP
iajs-2860	89	24	(	(	PUNCT
iajs-2860	89	25	25	25	NUM
iajs-2860	89	26	)	)	PUNCT
iajs-2860	89	27	using	use	VERB
iajs-2860	89	28	linear	linear	PROPN
iajs-2860	89	29	nps	nps	PROPN
iajs-2860	89	30	function	function	NOUN
iajs-2860	89	31	,	,	PUNCT
iajs-2860	89	32	we	we	PRON
iajs-2860	89	33	present	present	VERB
iajs-2860	89	34	a	a	DET
iajs-2860	89	35	method	method	NOUN
iajs-2860	89	36	of	of	ADP
iajs-2860	89	37	solution	solution	NOUN
iajs-2860	89	38	following	follow	VERB
iajs-2860	89	39	the	the	DET
iajs-2860	89	40	algorithm	algorithm	NOUN
iajs-2860	89	41	.	.	PUNCT
iajs-2860	90	1	algorithm	algorithm	NOUN
iajs-2860	90	2	of	of	ADP
iajs-2860	90	3	(	(	PUNCT
iajs-2860	90	4	lnps	lnps	PROPN
iajs-2860	90	5	)	)	PUNCT
iajs-2860	91	1	step1	step1	PROPN
iajs-2860	91	2	:	:	PUNCT
iajs-2860	91	3	input	input	NOUN
iajs-2860	91	4	ℎ	ℎ	X
iajs-2860	91	5	=	=	PUNCT
iajs-2860	91	6	𝑏−𝑎	𝑏−𝑎	PROPN
iajs-2860	91	7	𝑛	𝑛	PRON
iajs-2860	91	8	,	,	PUNCT
iajs-2860	91	9	𝑟𝑖	𝑟𝑖	X
iajs-2860	91	10	=	=	SYM
iajs-2860	91	11	𝑟0	𝑟0	PROPN
iajs-2860	91	12	+	+	CCONJ
iajs-2860	91	13	𝑖ℎ	𝑖ℎ	PRON
iajs-2860	91	14	,	,	PUNCT
iajs-2860	91	15	𝑖	𝑖	NOUN
iajs-2860	91	16	=	=	SYM
iajs-2860	91	17	0,1	0,1	NUM
iajs-2860	91	18	,	,	PUNCT
iajs-2860	91	19	…	…	PUNCT
iajs-2860	91	20	,	,	PUNCT
iajs-2860	91	21	𝑛	𝑛	PROPN
iajs-2860	91	22	and	and	CCONJ
iajs-2860	91	23	𝑢0	𝑢0	PROPN
iajs-2860	91	24	=	=	SYM
iajs-2860	91	25	𝜇	𝜇	ADP
iajs-2860	91	26	𝜇−1	𝜇−1	PROPN
iajs-2860	91	27	𝑓(𝑎	𝑓(𝑎	PROPN
iajs-2860	91	28	,	,	PUNCT
iajs-2860	91	29	𝑎	𝑎	NOUN
iajs-2860	91	30	)	)	PUNCT
iajs-2860	91	31	step2	step2	NOUN
iajs-2860	91	32	:	:	PUNCT
iajs-2860	91	33	compute	compute	PROPN
iajs-2860	91	34	𝑎0	𝑎0	PROPN
iajs-2860	91	35	,	,	PUNCT
iajs-2860	91	36	𝑏0	𝑏0	NOUN
iajs-2860	91	37	,	,	PUNCT
iajs-2860	91	38	𝑐0	𝑐0	VERB
iajs-2860	91	39	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2860	91	40	𝑑0	𝑑0	NOUN
iajs-2860	91	41	by	by	ADP
iajs-2860	91	42	''	''	PUNCT
iajs-2860	91	43	substituting	substitute	VERB
iajs-2860	91	44	'	'	PUNCT
iajs-2860	91	45	the	the	DET
iajs-2860	91	46	equations	equation	NOUN
iajs-2860	91	47	(	(	PUNCT
iajs-2860	91	48	11	11	NUM
iajs-2860	91	49	–	–	SYM
iajs-2860	91	50	14	14	NUM
iajs-2860	91	51	)	)	PUNCT
iajs-2860	91	52	into	into	ADP
iajs-2860	91	53	equations	equation	NOUN
iajs-2860	91	54	'	'	PART
iajs-2860	91	55	(	(	PUNCT
iajs-2860	91	56	7	7	NUM
iajs-2860	91	57	–	–	SYM
iajs-2860	91	58	10	10	NUM
iajs-2860	91	59	)	)	PUNCT
iajs-2860	91	60	step3	step3	PROPN
iajs-2860	91	61	:	:	PUNCT
iajs-2860	91	62	evaluates	evaluate	VERB
iajs-2860	91	63	s0(r	s0(r	PRON
iajs-2860	91	64	)	)	PUNCT
iajs-2860	91	65	using	use	VERB
iajs-2860	91	66	step2	step2	PROPN
iajs-2860	91	67	and	and	CCONJ
iajs-2860	91	68	equation	equation	NOUN
iajs-2860	91	69	(	(	PUNCT
iajs-2860	91	70	6	6	NUM
iajs-2860	91	71	)	)	PUNCT
iajs-2860	91	72	for	for	ADP
iajs-2860	91	73	i	i	PRON
iajs-2860	91	74	=	=	SYM
iajs-2860	91	75	0	0	NUM
iajs-2860	91	76	"	"	PUNCT
iajs-2860	91	77	step4	step4	NOUN
iajs-2860	91	78	:	:	PUNCT
iajs-2860	91	79	approximates"𝑢1	approximates"𝑢1	PROPN
iajs-2860	91	80	=	=	SYM
iajs-2860	91	81	𝑢(𝑟1	𝑢(𝑟1	PROPN
iajs-2860	91	82	)	)	PUNCT
iajs-2860	92	1	≈	≈	PROPN
iajs-2860	92	2	𝑆0(𝑟1)𝑠	𝑆0(𝑟1)𝑠	PROPN
iajs-2860	92	3	step5	step5	PROPN
iajs-2860	92	4	:	:	PUNCT
iajs-2860	92	5	do	do	VERB
iajs-2860	92	6	the	the	DET
iajs-2860	92	7	following	follow	VERB
iajs-2860	92	8	steps	step	NOUN
iajs-2860	92	9	for	for	ADP
iajs-2860	92	10	𝑖	𝑖	NOUN
iajs-2860	92	11	=	=	SYM
iajs-2860	92	12	1	1	NUM
iajs-2860	92	13	to	to	ADP
iajs-2860	92	14	𝑛	𝑛	PROPN
iajs-2860	92	15	−	−	NUM
iajs-2860	92	16	1	1	NUM
iajs-2860	92	17	step6	step6	NOUN
iajs-2860	92	18	:	:	PUNCT
iajs-2860	92	19	compute	compute	NOUN
iajs-2860	92	20	𝑎𝑖	𝑎𝑖	PROPN
iajs-2860	92	21	,	,	PUNCT
iajs-2860	92	22	𝑏𝑖	𝑏𝑖	PROPN
iajs-2860	92	23	,	,	PUNCT
iajs-2860	92	24	𝑐𝑖,𝑎𝑛𝑑	𝑐𝑖,𝑎𝑛𝑑	NOUN
iajs-2860	92	25	𝑑𝑖	𝑑𝑖	X
iajs-2860	92	26	by	by	ADP
iajs-2860	92	27	using	use	VERB
iajs-2860	92	28	equations	equation	NOUN
iajs-2860	92	29	(	(	PUNCT
iajs-2860	92	30	7	7	NUM
iajs-2860	92	31	–	–	SYM
iajs-2860	92	32	10	10	NUM
iajs-2860	92	33	)	)	PUNCT
iajs-2860	92	34	and	and	CCONJ
iajs-2860	92	35	replacing	replace	VERB
iajs-2860	92	36	𝑢0(𝑟𝑖	𝑢0(𝑟𝑖	NOUN
iajs-2860	92	37	)	)	PUNCT
iajs-2860	92	38	,	,	PUNCT
iajs-2860	92	39	𝑢′0(𝑟𝑖	𝑢′0(𝑟𝑖	NOUN
iajs-2860	92	40	)	)	PUNCT
iajs-2860	92	41	,	,	PUNCT
iajs-2860	92	42	𝑢0	𝑢0	PROPN
iajs-2860	92	43	′′(𝑟𝑖	′′(𝑟𝑖	PROPN
iajs-2860	92	44	)	)	PUNCT
iajs-2860	92	45	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2860	92	46	𝑢0	𝑢0	PROPN
iajs-2860	92	47	′′′(𝑟𝑖	′′′(𝑟𝑖	PROPN
iajs-2860	92	48	)	)	PUNCT
iajs-2860	92	49	𝑖𝑛	𝑖𝑛	NOUN
iajs-2860	92	50	𝑆(𝑟𝑖	𝑆(𝑟𝑖	NOUN
iajs-2860	92	51	)	)	PUNCT
iajs-2860	92	52	,	,	PUNCT
iajs-2860	92	53	s′(𝑟𝑖	s′(𝑟𝑖	PROPN
iajs-2860	92	54	)	)	PUNCT
iajs-2860	92	55	,	,	PUNCT
iajs-2860	92	56	𝑆′′(𝑟𝑖)𝑎𝑛𝑑	𝑆′′(𝑟𝑖)𝑎𝑛𝑑	PROPN
iajs-2860	92	57	𝑆′′′(𝑟𝑖	𝑆′′′(𝑟𝑖	PROPN
iajs-2860	92	58	)	)	PUNCT
iajs-2860	92	59	step7	step7	NOUN
iajs-2860	92	60	:	:	PUNCT
iajs-2860	92	61	calculate	calculate	NOUN
iajs-2860	92	62	𝑆𝑖(𝑟	𝑆𝑖(𝑟	PROPN
iajs-2860	92	63	)	)	PUNCT
iajs-2860	92	64	using	use	VERB
iajs-2860	92	65	step	step	NOUN
iajs-2860	92	66	6	6	NUM
iajs-2860	92	67	and	and	CCONJ
iajs-2860	92	68	equations	equation	NOUN
iajs-2860	92	69	(	(	PUNCT
iajs-2860	92	70	6	6	NUM
iajs-2860	92	71	)	)	PUNCT
iajs-2860	92	72	step8	step8	ADJ
iajs-2860	92	73	:	:	PUNCT
iajs-2860	92	74	approximate	approximate	ADJ
iajs-2860	92	75	𝑢𝑖+1	𝑢𝑖+1	PROPN
iajs-2860	92	76	=	=	SYM
iajs-2860	92	77	𝑆𝑖(𝑟𝑖+1	𝑆𝑖(𝑟𝑖+1	PROPN
iajs-2860	92	78	)	)	PUNCT
iajs-2860	92	79	remark	remark	NOUN
iajs-2860	92	80	:	:	PUNCT
iajs-2860	92	81	to	to	PART
iajs-2860	92	82	approximate	approximate	VERB
iajs-2860	92	83	solution	solution	NOUN
iajs-2860	92	84	of	of	ADP
iajs-2860	92	85	qnps	qnps	NOUN
iajs-2860	92	86	by	by	ADP
iajs-2860	92	87	the	the	DET
iajs-2860	92	88	above	above	ADJ
iajs-2860	92	89	.	.	PUNCT
iajs-2860	93	1	algorithm	algorithm	NOUN
iajs-2860	93	2	and	and	CCONJ
iajs-2860	93	3	replace	replace	VERB
iajs-2860	93	4	step	step	NOUN
iajs-2860	93	5	6	6	NUM
iajs-2860	93	6	by	by	ADP
iajs-2860	93	7	(	(	PUNCT
iajs-2860	93	8	compute	compute	NOUN
iajs-2860	93	9	𝑎𝑖	𝑎𝑖	PROPN
iajs-2860	93	10	,	,	PUNCT
iajs-2860	93	11	𝑏𝑖,𝑐𝑖	𝑏𝑖,𝑐𝑖	NOUN
iajs-2860	93	12	,	,	PUNCT
iajs-2860	93	13	𝑑𝑖	𝑑𝑖	PROPN
iajs-2860	93	14	and	and	CCONJ
iajs-2860	93	15	𝑒𝑖	𝑒𝑖	X
iajs-2860	93	16	using	use	VERB
iajs-2860	93	17	equations	equation	NOUN
iajs-2860	93	18	(	(	PUNCT
iajs-2860	93	19	16	16	NUM
iajs-2860	93	20	–	–	SYM
iajs-2860	93	21	.20	.20	NUM
iajs-2860	93	22	)	)	PUNCT
iajs-2860	93	23	and	and	CCONJ
iajs-2860	93	24	replacing	replace	VERB
iajs-2860	93	25	𝑄𝑖(𝑟𝑖	𝑄𝑖(𝑟𝑖	NOUN
iajs-2860	93	26	)	)	PUNCT
iajs-2860	93	27	,	,	PUNCT
iajs-2860	93	28	q′𝑖(𝑟𝑖	q′𝑖(𝑟𝑖	NOUN
iajs-2860	93	29	)	)	PUNCT
iajs-2860	93	30	,	,	PUNCT
iajs-2860	93	31	q′′𝑖(𝑟𝑖	q′′𝑖(𝑟𝑖	PROPN
iajs-2860	93	32	)	)	PUNCT
iajs-2860	93	33	,	,	PUNCT
iajs-2860	93	34	𝑄0	𝑄0	PROPN
iajs-2860	93	35	′′′(𝑟𝑖	′′′(𝑟𝑖	PROPN
iajs-2860	93	36	)	)	PUNCT
iajs-2860	93	37	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2860	93	38	𝑄𝑖	𝑄𝑖	PROPN
iajs-2860	93	39	(	(	PUNCT
iajs-2860	93	40	𝟒)(𝑟𝑖	𝟒)(𝑟𝑖	PROPN
iajs-2860	93	41	)	)	PUNCT
iajs-2860	93	42	𝑖𝑛	𝑖𝑛	NOUN
iajs-2860	93	43	𝑆(𝑟𝑖	𝑆(𝑟𝑖	NOUN
iajs-2860	93	44	)	)	PUNCT
iajs-2860	93	45	,	,	PUNCT
iajs-2860	93	46	s′(𝑟𝑖	s′(𝑟𝑖	PROPN
iajs-2860	93	47	)	)	PUNCT
iajs-2860	93	48	,	,	PUNCT
iajs-2860	93	49	𝑆′′(𝑟𝑖	𝑆′′(𝑟𝑖	PROPN
iajs-2860	93	50	)	)	PUNCT
iajs-2860	93	51	,	,	PUNCT
iajs-2860	93	52	𝑆′′′(𝑟𝑖)𝑎𝑛𝑑	𝑆′′′(𝑟𝑖)𝑎𝑛𝑑	PROPN
iajs-2860	93	53	𝑆𝑖	𝑆𝑖	PROPN
iajs-2860	93	54	(	(	PUNCT
iajs-2860	93	55	𝟒)(𝑟𝑖	𝟒)(𝑟𝑖	NUM
iajs-2860	93	56	)	)	PUNCT
iajs-2860	93	57	and	and	CCONJ
iajs-2860	93	58	replace	replace	VERB
iajs-2860	93	59	step7	step7	VERB
iajs-2860	93	60	by	by	ADP
iajs-2860	93	61	(	(	PUNCT
iajs-2860	93	62	calculate	calculate	NOUN
iajs-2860	93	63	𝑆𝑖(𝑟	𝑆𝑖(𝑟	PROPN
iajs-2860	93	64	)	)	PUNCT
iajs-2860	93	65	using	use	VERB
iajs-2860	93	66	step	step	NOUN
iajs-2860	93	67	6	6	NUM
iajs-2860	93	68	and	and	CCONJ
iajs-2860	93	69	equation	equation	NOUN
iajs-2860	93	70	(	(	PUNCT
iajs-2860	93	71	15	15	NUM
iajs-2860	93	72	)	)	PUNCT
iajs-2860	93	73	)	)	PUNCT
iajs-2860	93	74	.	.	PUNCT
iajs-2860	94	1	5	5	X
iajs-2860	94	2	.	.	X
iajs-2860	94	3	nps	nps	PROPN
iajs-2860	94	4	of	of	ADP
iajs-2860	94	5	fvie	fvie	PROPN
iajs-2860	94	6	with	with	ADP
iajs-2860	94	7	abel	abel	PROPN
iajs-2860	94	8	's	's	PART
iajs-2860	94	9	type	type	NOUN
iajs-2860	94	10	kernel	kernel	NOUN
iajs-2860	94	11	:	:	PUNCT
iajs-2860	94	12	singular	singular	PROPN
iajs-2860	94	13	fvie	fvie	ADJ
iajs-2860	94	14	with	with	ADP
iajs-2860	94	15	abel	abel	PROPN
iajs-2860	94	16	's	's	PART
iajs-2860	94	17	type	type	NOUN
iajs-2860	94	18	kernel	kernel	NOUN
iajs-2860	95	1	[	[	X
iajs-2860	95	2	3	3	X
iajs-2860	95	3	]	]	PUNCT
iajs-2860	95	4	can	can	AUX
iajs-2860	95	5	be	be	AUX
iajs-2860	95	6	written	write	VERB
iajs-2860	95	7	in	in	ADP
iajs-2860	95	8	a	a	DET
iajs-2860	95	9	general	general	ADJ
iajs-2860	95	10	form	form	NOUN
iajs-2860	95	11	as	as	ADP
iajs-2860	95	12	�	�	PROPN
iajs-2860	95	13	̃	̃	PROPN
iajs-2860	95	14	�	�	PROPN
iajs-2860	95	15	(𝑥	(𝑥	PROPN
iajs-2860	95	16	,	,	PUNCT
iajs-2860	95	17	𝑟	𝑟	NOUN
iajs-2860	95	18	)	)	PUNCT
iajs-2860	95	19	=	=	SYM
iajs-2860	95	20	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	95	21	,	,	PUNCT
iajs-2860	95	22	𝑟	𝑟	NOUN
iajs-2860	95	23	)	)	PUNCT
iajs-2860	96	1	+	+	NUM
iajs-2860	96	2	∫	∫	PROPN
iajs-2860	96	3	𝑢(𝑡,𝑟	𝑢(𝑡,𝑟	X
iajs-2860	96	4	)	)	PUNCT
iajs-2860	96	5	√𝑥−𝑡	√𝑥−𝑡	PROPN
iajs-2860	96	6	𝑥	𝑥	NOUN
iajs-2860	96	7	0	0	NUM
iajs-2860	96	8	𝑑𝑡.	𝑑𝑡.	X
iajs-2860	96	9	(	(	PUNCT
iajs-2860	96	10	26	26	NUM
iajs-2860	96	11	)	)	PUNCT
iajs-2860	96	12	where	where	SCONJ
iajs-2860	96	13	𝑟	𝑟	X
iajs-2860	96	14	,	,	PUNCT
iajs-2860	96	15	𝑥	𝑥	DET
iajs-2860	96	16	∈	∈	NOUN
iajs-2860	97	1	[	[	X
iajs-2860	97	2	0,1	0,1	NUM
iajs-2860	97	3	]	]	PUNCT
iajs-2860	97	4	to	to	PART
iajs-2860	97	5	solve	solve	VERB
iajs-2860	97	6	equation	equation	NOUN
iajs-2860	97	7	(	(	PUNCT
iajs-2860	97	8	26	26	NUM
iajs-2860	97	9	)	)	PUNCT
iajs-2860	97	10	,	,	PUNCT
iajs-2860	97	11	applying	apply	VERB
iajs-2860	97	12	laplace	laplace	NOUN
iajs-2860	97	13	transformation	transformation	NOUN
iajs-2860	97	14	to	to	ADP
iajs-2860	97	15	both	both	DET
iajs-2860	97	16	sides	side	NOUN
iajs-2860	97	17	,	,	PUNCT
iajs-2860	97	18	we	we	PRON
iajs-2860	97	19	have	have	VERB
iajs-2860	97	20	ℒ	ℒ	PROPN
iajs-2860	97	21	�	�	PROPN
iajs-2860	97	22	̃	̃	PROPN
iajs-2860	97	23	�	�	PROPN
iajs-2860	97	24	(𝑥	(𝑥	PROPN
iajs-2860	97	25	,	,	PUNCT
iajs-2860	97	26	𝑟	𝑟	NOUN
iajs-2860	97	27	)	)	PUNCT
iajs-2860	97	28	=	=	SYM
iajs-2860	97	29	ℒ𝑓(𝑥	ℒ𝑓(𝑥	NOUN
iajs-2860	97	30	,	,	PUNCT
iajs-2860	97	31	𝑟	𝑟	NOUN
iajs-2860	97	32	)	)	PUNCT
iajs-2860	98	1	+	+	CCONJ
iajs-2860	98	2	ℒ	ℒ	PROPN
iajs-2860	98	3	[	[	X
iajs-2860	98	4	∫	∫	PROPN
iajs-2860	98	5	𝑢(𝑡,𝑟	𝑢(𝑡,𝑟	X
iajs-2860	98	6	)	)	PUNCT
iajs-2860	98	7	√𝑥−𝑡	√𝑥−𝑡	VERB
iajs-2860	98	8	𝑥	𝑥	NOUN
iajs-2860	98	9	0	0	NUM
iajs-2860	98	10	𝑑𝑡	𝑑𝑡	ADP
iajs-2860	98	11	]	]	X
iajs-2860	98	12	(	(	PUNCT
iajs-2860	98	13	27	27	NUM
iajs-2860	98	14	)	)	PUNCT
iajs-2860	98	15	by	by	ADP
iajs-2860	98	16	convolution	convolution	NOUN
iajs-2860	98	17	theorem	theorem	VERB
iajs-2860	98	18	,	,	PUNCT
iajs-2860	98	19	equation	equation	NOUN
iajs-2860	98	20	(	(	PUNCT
iajs-2860	98	21	27	27	NUM
iajs-2860	98	22	)	)	PUNCT
iajs-2860	98	23	yields	yield	VERB
iajs-2860	98	24	ℒ	ℒ	PROPN
iajs-2860	98	25	�	�	PROPN
iajs-2860	98	26	̃	̃	PROPN
iajs-2860	98	27	�	�	PROPN
iajs-2860	98	28	(𝑥	(𝑥	PROPN
iajs-2860	98	29	,	,	PUNCT
iajs-2860	98	30	𝑟	𝑟	NOUN
iajs-2860	98	31	)	)	PUNCT
iajs-2860	98	32	=	=	SYM
iajs-2860	98	33	ℒ	ℒ	ADJ
iajs-2860	98	34	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	98	35	,	,	PUNCT
iajs-2860	98	36	𝑟	𝑟	NOUN
iajs-2860	98	37	)	)	PUNCT
iajs-2860	99	1	+	+	CCONJ
iajs-2860	99	2	ℒ[𝑥−1	ℒ[𝑥−1	NOUN
iajs-2860	99	3	2⁄	2⁄	NUM
iajs-2860	99	4	]	]	PUNCT
iajs-2860	99	5	ℒ	ℒ	PROPN
iajs-2860	99	6	[	[	X
iajs-2860	99	7	�	�	PROPN
iajs-2860	99	8	̃	̃	PROPN
iajs-2860	99	9	�	�	PROPN
iajs-2860	99	10	(𝑥	(𝑥	PROPN
iajs-2860	99	11	,	,	PUNCT
iajs-2860	99	12	𝑟	𝑟	NOUN
iajs-2860	99	13	)	)	PUNCT
iajs-2860	99	14	(	(	PUNCT
iajs-2860	99	15	28	28	NUM
iajs-2860	99	16	)	)	PUNCT
iajs-2860	99	17	then	then	ADV
iajs-2860	99	18	,	,	PUNCT
iajs-2860	99	19	we	we	PRON
iajs-2860	99	20	have	have	VERB
iajs-2860	99	21	ℒ	ℒ	PROPN
iajs-2860	99	22	�	�	PROPN
iajs-2860	99	23	̃	̃	PROPN
iajs-2860	99	24	�	�	PROPN
iajs-2860	99	25	(𝑥	(𝑥	PROPN
iajs-2860	99	26	,	,	PUNCT
iajs-2860	99	27	𝑟	𝑟	NOUN
iajs-2860	99	28	)	)	PUNCT
iajs-2860	99	29	=	=	SYM
iajs-2860	99	30	ℒ	ℒ	ADJ
iajs-2860	99	31	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	99	32	,	,	PUNCT
iajs-2860	99	33	𝑟	𝑟	NOUN
iajs-2860	99	34	)	)	PUNCT
iajs-2860	100	1	+	+	CCONJ
iajs-2860	100	2	√	√	VERB
iajs-2860	100	3	𝜋	𝜋	VERB
iajs-2860	100	4	𝑠	𝑠	PRON
iajs-2860	100	5	ℒ	ℒ	PROPN
iajs-2860	100	6	[	[	X
iajs-2860	100	7	�	�	PROPN
iajs-2860	100	8	̃	̃	PROPN
iajs-2860	100	9	�	�	PROPN
iajs-2860	100	10	(𝑥	(𝑥	PROPN
iajs-2860	100	11	,	,	PUNCT
iajs-2860	100	12	𝑟	𝑟	NOUN
iajs-2860	100	13	)	)	PUNCT
iajs-2860	100	14	(	(	PUNCT
iajs-2860	100	15	29	29	NUM
iajs-2860	100	16	)	)	PUNCT
iajs-2860	100	17	upon	upon	SCONJ
iajs-2860	100	18	using	use	VERB
iajs-2860	100	19	the	the	DET
iajs-2860	100	20	inverse	inverse	NOUN
iajs-2860	100	21	of	of	ADP
iajs-2860	100	22	the	the	DET
iajs-2860	100	23	laplace	laplace	NOUN
iajs-2860	100	24	transformation	transformation	NOUN
iajs-2860	100	25	to	to	ADP
iajs-2860	100	26	both	both	DET
iajs-2860	100	27	sides	side	NOUN
iajs-2860	100	28	of	of	ADP
iajs-2860	100	29	equation	equation	NOUN
iajs-2860	100	30	(	(	PUNCT
iajs-2860	100	31	29	29	NUM
iajs-2860	100	32	)	)	PUNCT
iajs-2860	100	33	,	,	PUNCT
iajs-2860	100	34	we	we	PRON
iajs-2860	100	35	obtain	obtain	VERB
iajs-2860	100	36	�	�	PROPN
iajs-2860	100	37	̃	̃	PROPN
iajs-2860	100	38	�	�	PROPN
iajs-2860	100	39	(𝑥	(𝑥	PROPN
iajs-2860	100	40	,	,	PUNCT
iajs-2860	100	41	𝑟	𝑟	NOUN
iajs-2860	100	42	)	)	PUNCT
iajs-2860	100	43	=	=	SYM
iajs-2860	100	44	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	100	45	,	,	PUNCT
iajs-2860	100	46	𝑟	𝑟	NOUN
iajs-2860	100	47	)	)	PUNCT
iajs-2860	101	1	+	+	CCONJ
iajs-2860	101	2	ℒ−1	ℒ−1	PROPN
iajs-2860	102	1	[	[	X
iajs-2860	102	2	√	√	X
iajs-2860	102	3	𝜋	𝜋	VERB
iajs-2860	102	4	𝑠	𝑠	PRON
iajs-2860	102	5	ℒ	ℒ	PROPN
iajs-2860	102	6	[	[	X
iajs-2860	102	7	�	�	PROPN
iajs-2860	102	8	̃	̃	PROPN
iajs-2860	102	9	�	�	PROPN
iajs-2860	102	10	(𝑥	(𝑥	PROPN
iajs-2860	102	11	,	,	PUNCT
iajs-2860	102	12	𝑟	𝑟	NOUN
iajs-2860	102	13	)	)	PUNCT
iajs-2860	102	14	]	]	PUNCT
iajs-2860	102	15	(	(	PUNCT
iajs-2860	102	16	30	30	X
iajs-2860	102	17	)	)	PUNCT
iajs-2860	102	18	let	let	VERB
iajs-2860	102	19	the	the	DET
iajs-2860	102	20	solution	solution	NOUN
iajs-2860	102	21	of	of	ADP
iajs-2860	102	22	equation	equation	NOUN
iajs-2860	102	23	(	(	PUNCT
iajs-2860	102	24	30	30	NUM
iajs-2860	102	25	)	)	PUNCT
iajs-2860	102	26	be	be	AUX
iajs-2860	102	27	in	in	ADP
iajs-2860	102	28	the	the	DET
iajs-2860	102	29	form	form	NOUN
iajs-2860	102	30	of	of	ADP
iajs-2860	102	31	the	the	DET
iajs-2860	102	32	spline	spline	NOUN
iajs-2860	102	33	where	where	SCONJ
iajs-2860	102	34	ihjpas	ihjpa	VERB
iajs-2860	102	35	.	.	PUNCT
iajs-2860	103	1	36(1)2023	36(1)2023	NUM
iajs-2860	103	2	412	412	NUM
iajs-2860	103	3	�	�	PROPN
iajs-2860	103	4	̃	̃	PROPN
iajs-2860	103	5	�	�	PROPN
iajs-2860	103	6	(𝑥	(𝑥	PROPN
iajs-2860	103	7	,	,	PUNCT
iajs-2860	103	8	𝑟	𝑟	NOUN
iajs-2860	103	9	)	)	PUNCT
iajs-2860	103	10	=	=	NOUN
iajs-2860	104	1	[	[	X
iajs-2860	104	2	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	104	3	,	,	PUNCT
iajs-2860	104	4	𝑟	𝑟	NOUN
iajs-2860	104	5	)	)	PUNCT
iajs-2860	104	6	,	,	PUNCT
iajs-2860	104	7	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	104	8	,	,	PUNCT
iajs-2860	104	9	𝑟	𝑟	NOUN
iajs-2860	104	10	)	)	PUNCT
iajs-2860	104	11	]	]	PUNCT
iajs-2860	105	1	=	=	PUNCT
iajs-2860	105	2	𝑎	𝑎	X
iajs-2860	105	3	cos(𝑥	cos(𝑥	NOUN
iajs-2860	105	4	)	)	PUNCT
iajs-2860	105	5	+	+	CCONJ
iajs-2860	105	6	𝑏	𝑏	DET
iajs-2860	105	7	sin(𝑥	sin(𝑥	NOUN
iajs-2860	105	8	)	)	PUNCT
iajs-2860	105	9	+	+	X
iajs-2860	105	10	𝑐(𝑥	𝑐(𝑥	ADJ
iajs-2860	105	11	)	)	PUNCT
iajs-2860	106	1	+	+	CCONJ
iajs-2860	106	2	𝑑	𝑑	PROPN
iajs-2860	106	3	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	106	4	,	,	PUNCT
iajs-2860	106	5	𝑟	𝑟	NOUN
iajs-2860	106	6	)	)	PUNCT
iajs-2860	107	1	=	=	NOUN
iajs-2860	108	1	[	[	X
iajs-2860	108	2	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	108	3	,	,	PUNCT
iajs-2860	108	4	𝑟	𝑟	NOUN
iajs-2860	108	5	)	)	PUNCT
iajs-2860	108	6	,	,	PUNCT
iajs-2860	108	7	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	108	8	,	,	PUNCT
iajs-2860	108	9	𝑟	𝑟	NOUN
iajs-2860	108	10	)	)	PUNCT
iajs-2860	108	11	]	]	PUNCT
iajs-2860	108	12	by	by	ADP
iajs-2860	108	13	substituting	substitute	VERB
iajs-2860	108	14	the	the	DET
iajs-2860	108	15	above	above	ADJ
iajs-2860	108	16	two	two	NUM
iajs-2860	108	17	equations	equation	NOUN
iajs-2860	108	18	into	into	ADP
iajs-2860	108	19	equation	equation	NOUN
iajs-2860	108	20	(	(	PUNCT
iajs-2860	108	21	30	30	NUM
iajs-2860	108	22	)	)	PUNCT
iajs-2860	108	23	to	to	PART
iajs-2860	108	24	get	get	VERB
iajs-2860	108	25	:	:	PUNCT
iajs-2860	108	26	𝑎	𝑎	PROPN
iajs-2860	108	27	cos(𝑥	cos(𝑥	NOUN
iajs-2860	108	28	)	)	PUNCT
iajs-2860	108	29	+	+	CCONJ
iajs-2860	108	30	𝑏	𝑏	DET
iajs-2860	108	31	sin(𝑥	sin(𝑥	NOUN
iajs-2860	108	32	)	)	PUNCT
iajs-2860	108	33	+	+	X
iajs-2860	108	34	𝑐(𝑥	𝑐(𝑥	ADJ
iajs-2860	108	35	)	)	PUNCT
iajs-2860	109	1	+	+	CCONJ
iajs-2860	109	2	𝑑	𝑑	NOUN
iajs-2860	109	3	=	=	SYM
iajs-2860	109	4	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	109	5	,	,	PUNCT
iajs-2860	109	6	𝑟	𝑟	NOUN
iajs-2860	109	7	)	)	PUNCT
iajs-2860	110	1	+	+	CCONJ
iajs-2860	110	2	ℒ−1	ℒ−1	PROPN
iajs-2860	111	1	[	[	X
iajs-2860	111	2	√	√	X
iajs-2860	111	3	𝜋	𝜋	VERB
iajs-2860	111	4	𝑠	𝑠	PROPN
iajs-2860	111	5	ℒ[𝑎	ℒ[𝑎	X
iajs-2860	111	6	cos(𝑥	cos(𝑥	PROPN
iajs-2860	111	7	)	)	PUNCT
iajs-2860	111	8	+	+	CCONJ
iajs-2860	111	9	𝑏	𝑏	DET
iajs-2860	111	10	sin(𝑥	sin(𝑥	NOUN
iajs-2860	111	11	)	)	PUNCT
iajs-2860	111	12	+	+	X
iajs-2860	112	1	𝑐(𝑥	𝑐(𝑥	ADJ
iajs-2860	112	2	)	)	PUNCT
iajs-2860	113	1	+	+	CCONJ
iajs-2860	113	2	𝑑	𝑑	X
iajs-2860	113	3	]	]	X
iajs-2860	113	4	]	]	X
iajs-2860	113	5	(	(	PUNCT
iajs-2860	113	6	31	31	NUM
iajs-2860	113	7	)	)	PUNCT
iajs-2860	113	8	now	now	ADV
iajs-2860	113	9	,	,	PUNCT
iajs-2860	113	10	by	by	ADP
iajs-2860	113	11	simplifying	simplify	VERB
iajs-2860	113	12	equation	equation	NOUN
iajs-2860	113	13	(	(	PUNCT
iajs-2860	113	14	31	31	NUM
iajs-2860	113	15	)	)	PUNCT
iajs-2860	113	16	to	to	PART
iajs-2860	113	17	obtain	obtain	VERB
iajs-2860	113	18	:	:	PUNCT
iajs-2860	113	19	[	[	X
iajs-2860	113	20	𝑎(cos(𝑥	𝑎(cos(𝑥	NOUN
iajs-2860	113	21	)	)	PUNCT
iajs-2860	113	22	−	−	NUM
iajs-2860	113	23	√𝜋	√𝜋	X
iajs-2860	113	24	cos(𝑥	cos(𝑥	NOUN
iajs-2860	113	25	)	)	PUNCT
iajs-2860	113	26	(	(	PUNCT
iajs-2860	113	27	√2	√2	NOUN
iajs-2860	113	28	−	−	NOUN
iajs-2860	113	29	1	1	NUM
iajs-2860	113	30	)	)	PUNCT
iajs-2860	113	31	−	−	NUM
iajs-2860	113	32	√𝜋	√𝜋	SYM
iajs-2860	113	33	sin(𝑥	sin(𝑥	NOUN
iajs-2860	113	34	)	)	PUNCT
iajs-2860	113	35	(	(	PUNCT
iajs-2860	113	36	2	2	NUM
iajs-2860	113	37	−	−	NOUN
iajs-2860	113	38	√2	√2	NOUN
iajs-2860	113	39	)	)	PUNCT
iajs-2860	113	40	+	+	NOUN
iajs-2860	113	41	𝑏	𝑏	PROPN
iajs-2860	113	42	(	(	PUNCT
iajs-2860	113	43	sin(𝑥	sin(𝑥	NOUN
iajs-2860	113	44	)	)	PUNCT
iajs-2860	113	45	−	−	NOUN
iajs-2860	113	46	√𝜋	√𝜋	X
iajs-2860	113	47	cos(𝑥	cos(𝑥	NOUN
iajs-2860	113	48	)	)	PUNCT
iajs-2860	113	49	(	(	PUNCT
iajs-2860	113	50	−1	−1	NOUN
iajs-2860	113	51	+	+	CCONJ
iajs-2860	113	52	1	1	NUM
iajs-2860	113	53	√2	√2	PROPN
iajs-2860	113	54	)	)	PUNCT
iajs-2860	113	55	−	−	PROPN
iajs-2860	113	56	√𝜋	√𝜋	SYM
iajs-2860	113	57	sin(𝑥	sin(𝑥	NOUN
iajs-2860	113	58	)	)	PUNCT
iajs-2860	113	59	(	(	PUNCT
iajs-2860	113	60	2	2	NUM
iajs-2860	113	61	−	−	NUM
iajs-2860	113	62	1	1	NUM
iajs-2860	113	63	√2	√2	PROPN
iajs-2860	113	64	)	)	PUNCT
iajs-2860	114	1	+	+	CCONJ
iajs-2860	114	2	𝑐	𝑐	PROPN
iajs-2860	114	3	(	(	PUNCT
iajs-2860	114	4	𝑥	𝑥	NOUN
iajs-2860	114	5	−	−	NUM
iajs-2860	114	6	4𝑥3	4𝑥3	NUM
iajs-2860	114	7	2⁄	2⁄	NUM
iajs-2860	114	8	3	3	NUM
iajs-2860	114	9	)	)	PUNCT
iajs-2860	115	1	+	+	PUNCT
iajs-2860	115	2	𝑑(1	𝑑(1	ADJ
iajs-2860	115	3	−	−	NOUN
iajs-2860	115	4	2√𝑥	2√𝑥	NUM
iajs-2860	115	5	)	)	PUNCT
iajs-2860	115	6	]	]	PUNCT
iajs-2860	116	1	=	=	SYM
iajs-2860	116	2	𝑓	𝑓	PROPN
iajs-2860	116	3	(	(	PUNCT
iajs-2860	116	4	𝑥	𝑥	PROPN
iajs-2860	116	5	,	,	PUNCT
iajs-2860	116	6	𝑟	𝑟	NOUN
iajs-2860	116	7	)	)	PUNCT
iajs-2860	116	8	(	(	PUNCT
iajs-2860	116	9	32	32	NUM
iajs-2860	116	10	)	)	PUNCT
iajs-2860	116	11	let	let	VERB
iajs-2860	116	12	∆	∆	PROPN
iajs-2860	116	13	be	be	AUX
iajs-2860	116	14	a	a	DET
iajs-2860	116	15	partition	partition	NOUN
iajs-2860	116	16	for	for	ADP
iajs-2860	116	17	the	the	DET
iajs-2860	116	18	x	x	PROPN
iajs-2860	116	19	,	,	PUNCT
iajs-2860	116	20	s.t	s.t	PROPN
iajs-2860	116	21	∆	∆	PROPN
iajs-2860	116	22	:	:	PUNCT
iajs-2860	116	23	0	0	NUM
iajs-2860	116	24	=	=	SYM
iajs-2860	116	25	𝑥0	𝑥0	PROPN
iajs-2860	116	26	<	<	X
iajs-2860	116	27	𝑥1	𝑥1	NOUN
iajs-2860	116	28	<	<	X
iajs-2860	116	29	𝑥2	𝑥2	PROPN
iajs-2860	116	30	<	<	X
iajs-2860	116	31	𝑥3	𝑥3	NOUN
iajs-2860	116	32	=	=	ADJ
iajs-2860	116	33	1	1	X
iajs-2860	116	34	.	.	PUNCT
iajs-2860	116	35	where	where	SCONJ
iajs-2860	116	36	ℎ	ℎ	X
iajs-2860	116	37	=	=	SYM
iajs-2860	116	38	1	1	NUM
iajs-2860	116	39	3⁄	3⁄	NUM
iajs-2860	116	40	then	then	ADV
iajs-2860	116	41	𝑥0	𝑥0	VERB
iajs-2860	116	42	=	=	SYM
iajs-2860	116	43	0	0	NUM
iajs-2860	116	44	,	,	PUNCT
iajs-2860	116	45	𝑥1	𝑥1	NOUN
iajs-2860	116	46	=	=	NOUN
iajs-2860	116	47	1	1	NUM
iajs-2860	116	48	3⁄	3⁄	NUM
iajs-2860	116	49	,	,	PUNCT
iajs-2860	116	50	𝑥2	𝑥2	NOUN
iajs-2860	116	51	=	=	SYM
iajs-2860	116	52	1	1	NUM
iajs-2860	116	53	3⁄	3⁄	NUM
iajs-2860	116	54	and	and	CCONJ
iajs-2860	116	55	𝑥3	𝑥3	NOUN
iajs-2860	116	56	=	=	SYM
iajs-2860	116	57	1	1	X
iajs-2860	116	58	.	.	PUNCT
iajs-2860	117	1	the	the	DET
iajs-2860	117	2	approximate	approximate	ADJ
iajs-2860	117	3	equation	equation	NOUN
iajs-2860	117	4	�	�	PROPN
iajs-2860	117	5	̃	̃	PROPN
iajs-2860	117	6	�	�	PROPN
iajs-2860	117	7	(𝑥	(𝑥	PROPN
iajs-2860	117	8	,	,	PUNCT
iajs-2860	117	9	𝑟	𝑟	NOUN
iajs-2860	117	10	)	)	PUNCT
iajs-2860	117	11	=	=	SYM
iajs-2860	117	12	𝑓	𝑓	PROPN
iajs-2860	117	13	(	(	PUNCT
iajs-2860	117	14	𝑥	𝑥	NOUN
iajs-2860	117	15	,	,	PUNCT
iajs-2860	117	16	𝑟)[𝑎	𝑟)[𝑎	PROPN
iajs-2860	117	17	cos(𝑥	cos(𝑥	PROPN
iajs-2860	117	18	)	)	PUNCT
iajs-2860	117	19	+	+	CCONJ
iajs-2860	117	20	𝑏	𝑏	DET
iajs-2860	117	21	sin(𝑥	sin(𝑥	NOUN
iajs-2860	117	22	)	)	PUNCT
iajs-2860	117	23	+	+	X
iajs-2860	117	24	𝑐(𝑥	𝑐(𝑥	ADJ
iajs-2860	117	25	)	)	PUNCT
iajs-2860	118	1	+	+	CCONJ
iajs-2860	118	2	𝑑	𝑑	X
iajs-2860	118	3	]	]	X
iajs-2860	118	4	(	(	PUNCT
iajs-2860	118	5	33	33	NUM
iajs-2860	118	6	)	)	PUNCT
iajs-2860	118	7	now	now	ADV
iajs-2860	118	8	,	,	PUNCT
iajs-2860	118	9	we	we	PRON
iajs-2860	118	10	need	need	VERB
iajs-2860	118	11	to	to	PART
iajs-2860	118	12	solve	solve	VERB
iajs-2860	118	13	equation	equation	NOUN
iajs-2860	118	14	(	(	PUNCT
iajs-2860	118	15	33	33	NUM
iajs-2860	118	16	)	)	PUNCT
iajs-2860	118	17	to	to	PART
iajs-2860	118	18	find	find	VERB
iajs-2860	118	19	the	the	DET
iajs-2860	118	20	constants	constant	NOUN
iajs-2860	118	21	(	(	PUNCT
iajs-2860	118	22	𝑎	𝑎	X
iajs-2860	118	23	,	,	PUNCT
iajs-2860	118	24	𝑏	𝑏	NOUN
iajs-2860	118	25	,	,	PUNCT
iajs-2860	118	26	𝑐	𝑐	PROPN
iajs-2860	118	27	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
iajs-2860	118	28	𝑑	𝑑	NOUN
iajs-2860	118	29	)	)	PUNCT
iajs-2860	118	30	using	use	VERB
iajs-2860	118	31	the	the	DET
iajs-2860	118	32	system	system	NOUN
iajs-2860	118	33	:	:	PUNCT
iajs-2860	118	34	𝑀𝐶	𝑀𝐶	PROPN
iajs-2860	118	35	=	=	SYM
iajs-2860	118	36	𝐹	𝐹	PROPN
iajs-2860	118	37	to	to	PART
iajs-2860	118	38	find	find	VERB
iajs-2860	118	39	𝐶	𝐶	PROPN
iajs-2860	118	40	calculate	calculate	VERB
iajs-2860	118	41	𝐶	𝐶	PROPN
iajs-2860	118	42	=	=	SYM
iajs-2860	118	43	𝑀−1𝐹	𝑀−1𝐹	PROPN
iajs-2860	118	44	and	and	CCONJ
iajs-2860	118	45	substitute	substitute	VERB
iajs-2860	118	46	this	this	DET
iajs-2860	118	47	solution	solution	NOUN
iajs-2860	118	48	in	in	ADP
iajs-2860	118	49	equation	equation	NOUN
iajs-2860	118	50	(	(	PUNCT
iajs-2860	118	51	33	33	NUM
iajs-2860	118	52	)	)	PUNCT
iajs-2860	118	53	to	to	PART
iajs-2860	118	54	find	find	VERB
iajs-2860	118	55	approximation	approximation	NOUN
iajs-2860	118	56	solution	solution	NOUN
iajs-2860	118	57	.	.	PUNCT
iajs-2860	119	1	6	6	X
iajs-2860	119	2	.	.	X
iajs-2860	119	3	illustrative	illustrative	ADJ
iajs-2860	119	4	examples	example	NOUN
iajs-2860	119	5	in	in	ADP
iajs-2860	119	6	this	this	DET
iajs-2860	119	7	section	section	NOUN
iajs-2860	119	8	,	,	PUNCT
iajs-2860	119	9	two	two	NUM
iajs-2860	119	10	test	test	NOUN
iajs-2860	119	11	examples	example	NOUN
iajs-2860	119	12	are	be	AUX
iajs-2860	119	13	illustrated	illustrate	VERB
iajs-2860	119	14	below	below	ADV
iajs-2860	119	15	to	to	PART
iajs-2860	119	16	solve	solve	VERB
iajs-2860	119	17	(	(	PUNCT
iajs-2860	119	18	fviewsk	fviewsk	PROPN
iajs-2860	119	19	)	)	PUNCT
iajs-2860	119	20	and	and	CCONJ
iajs-2860	119	21	fvie	fvie	ADJ
iajs-2860	119	22	with	with	ADP
iajs-2860	119	23	abel	abel	PROPN
iajs-2860	119	24	's	's	PART
iajs-2860	119	25	types	type	NOUN
iajs-2860	119	26	kernel	kernel	VERB
iajs-2860	119	27	in	in	ADP
iajs-2860	119	28	upper	upper	ADJ
iajs-2860	119	29	and	and	CCONJ
iajs-2860	119	30	lower	low	ADJ
iajs-2860	119	31	solution	solution	NOUN
iajs-2860	119	32	,	,	PUNCT
iajs-2860	119	33	𝑠(𝑥	𝑠(𝑥	PROPN
iajs-2860	119	34	,	,	PUNCT
iajs-2860	119	35	𝑟	𝑟	NOUN
iajs-2860	119	36	)	)	PUNCT
iajs-2860	119	37	the	the	DET
iajs-2860	119	38	approximate	approximate	ADJ
iajs-2860	119	39	solution	solution	NOUN
iajs-2860	119	40	by	by	ADP
iajs-2860	119	41	the	the	DET
iajs-2860	119	42	proposed	propose	VERB
iajs-2860	119	43	method	method	NOUN
iajs-2860	119	44	and	and	CCONJ
iajs-2860	119	45	error	error	NOUN
iajs-2860	119	46	=	=	SYM
iajs-2860	119	47	|𝑢(𝑥	|𝑢(𝑥	PROPN
iajs-2860	119	48	,	,	PUNCT
iajs-2860	119	49	𝑟	𝑟	NOUN
iajs-2860	119	50	)	)	PUNCT
iajs-2860	120	1	−	−	PROPN
iajs-2860	121	1	𝑠(𝑥	𝑠(𝑥	PROPN
iajs-2860	121	2	,	,	PUNCT
iajs-2860	121	3	𝑟)|	𝑟)|	ADJ
iajs-2860	121	4	where	where	SCONJ
iajs-2860	121	5	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	121	6	,	,	PUNCT
iajs-2860	121	7	𝑟	𝑟	NOUN
iajs-2860	121	8	)	)	PUNCT
iajs-2860	121	9	the	the	DET
iajs-2860	121	10	exact	exact	ADJ
iajs-2860	121	11	solution	solution	NOUN
iajs-2860	121	12	.	.	PUNCT
iajs-2860	122	1	example	example	NOUN
iajs-2860	122	2	(	(	PUNCT
iajs-2860	122	3	1	1	NUM
iajs-2860	122	4	):	):	PUNCT
iajs-2860	122	5	consider	consider	VERB
iajs-2860	122	6	lfvies	lfvie	NOUN
iajs-2860	122	7	with	with	ADP
iajs-2860	122	8	weakly	weakly	ADJ
iajs-2860	122	9	singular	singular	ADJ
iajs-2860	122	10	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	122	11	,	,	PUNCT
iajs-2860	122	12	𝑟	𝑟	NOUN
iajs-2860	122	13	)	)	PUNCT
iajs-2860	123	1	−	−	NUM
iajs-2860	123	2	∫	∫	PROPN
iajs-2860	123	3	𝑡𝛽−1	𝑡𝛽−1	PROPN
iajs-2860	123	4	𝑥𝛽	𝑥𝛽	VERB
iajs-2860	123	5	𝑥	𝑥	NOUN
iajs-2860	123	6	0	0	NUM
iajs-2860	123	7	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	123	8	,	,	PUNCT
iajs-2860	123	9	𝑡)𝑑𝑡	𝑡)𝑑𝑡	PROPN
iajs-2860	123	10	=	=	SYM
iajs-2860	123	11	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	123	12	,	,	PUNCT
iajs-2860	123	13	𝑟	𝑟	NOUN
iajs-2860	123	14	)	)	PUNCT
iajs-2860	123	15	,	,	PUNCT
iajs-2860	123	16	where	where	SCONJ
iajs-2860	123	17	𝑓(𝑥	𝑓(𝑥	NOUN
iajs-2860	123	18	,	,	PUNCT
iajs-2860	123	19	𝑟	𝑟	NOUN
iajs-2860	123	20	)	)	PUNCT
iajs-2860	123	21	=	=	SYM
iajs-2860	124	1	[	[	X
iajs-2860	124	2	(	(	PUNCT
iajs-2860	124	3	0.71428571𝑥3	0.71428571𝑥3	NUM
iajs-2860	124	4	−	−	NOUN
iajs-2860	124	5	0.6𝑥2)(𝑟	0.6𝑥2)(𝑟	NOUN
iajs-2860	124	6	−	−	PROPN
iajs-2860	124	7	1	1	NUM
iajs-2860	124	8	)	)	PUNCT
iajs-2860	124	9	;	;	PUNCT
iajs-2860	124	10	(	(	PUNCT
iajs-2860	124	11	0.71428571𝑥3	0.71428571𝑥3	NUM
iajs-2860	124	12	−	−	NOUN
iajs-2860	124	13	0.6𝑥2)(1	0.6𝑥2)(1	NUM
iajs-2860	124	14	−	−	NUM
iajs-2860	124	15	𝑟	𝑟	NOUN
iajs-2860	124	16	)	)	PUNCT
iajs-2860	124	17	]	]	PUNCT
iajs-2860	124	18	the	the	DET
iajs-2860	124	19	exact	exact	ADJ
iajs-2860	124	20	solution	solution	NOUN
iajs-2860	124	21	is	be	AUX
iajs-2860	124	22	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	124	23	,	,	PUNCT
iajs-2860	124	24	𝑟	𝑟	NOUN
iajs-2860	124	25	)	)	PUNCT
iajs-2860	124	26	=	=	SYM
iajs-2860	125	1	[	[	X
iajs-2860	125	2	(	(	PUNCT
iajs-2860	125	3	𝑥3	𝑥3	NOUN
iajs-2860	125	4	−	−	PROPN
iajs-2860	125	5	𝑥2)(𝑟	𝑥2)(𝑟	CCONJ
iajs-2860	125	6	−	−	PROPN
iajs-2860	125	7	1	1	NUM
iajs-2860	125	8	)	)	PUNCT
iajs-2860	125	9	;	;	PUNCT
iajs-2860	125	10	(	(	PUNCT
iajs-2860	125	11	𝑥3	𝑥3	NOUN
iajs-2860	125	12	−	−	PROPN
iajs-2860	125	13	𝑥2)(1	𝑥2)(1	NOUN
iajs-2860	125	14	−	−	PROPN
iajs-2860	125	15	𝑟	𝑟	NOUN
iajs-2860	125	16	)	)	PUNCT
iajs-2860	125	17	]	]	PUNCT
iajs-2860	126	1	[	[	X
iajs-2860	126	2	11	11	NUM
iajs-2860	126	3	]	]	SYM
iajs-2860	126	4	table	table	NOUN
iajs-2860	126	5	1	1	NUM
iajs-2860	126	6	.	.	PUNCT
iajs-2860	126	7	shows	show	VERB
iajs-2860	126	8	the	the	DET
iajs-2860	126	9	results	result	NOUN
iajs-2860	126	10	of	of	ADP
iajs-2860	126	11	lfvies	lfvie	NOUN
iajs-2860	126	12	with	with	ADP
iajs-2860	126	13	weakly	weakly	ADJ
iajs-2860	126	14	singular	singular	NOUN
iajs-2860	126	15	in	in	ADP
iajs-2860	126	16	lower	low	ADJ
iajs-2860	126	17	solution	solution	NOUN
iajs-2860	126	18	𝒙	𝒙	PROPN
iajs-2860	126	19	𝒖(𝒙	𝒖(𝒙	PROPN
iajs-2860	126	20	,	,	PUNCT
iajs-2860	126	21	𝒓	𝒓	NOUN
iajs-2860	126	22	)	)	PUNCT
iajs-2860	126	23	𝒔	𝒔	X
iajs-2860	126	24	(	(	PUNCT
iajs-2860	126	25	𝒙	𝒙	PROPN
iajs-2860	126	26	,	,	PUNCT
iajs-2860	126	27	𝒓	𝒓	NOUN
iajs-2860	126	28	)	)	PUNCT
iajs-2860	126	29	in	in	ADP
iajs-2860	126	30	error	error	NOUN
iajs-2860	126	31	in	in	ADP
iajs-2860	126	32	linear	linear	PROPN
iajs-2860	126	33	quadratic	quadratic	ADJ
iajs-2860	126	34	linear	linear	ADJ
iajs-2860	126	35	quadratic	quadratic	ADJ
iajs-2860	126	36	0	0	NUM
iajs-2860	126	37	0.000000000	0.000000000	NUM
iajs-2860	126	38	0.000000000	0.000000000	NUM
iajs-2860	126	39	-0.000000000	-0.000000000	NOUN
iajs-2860	127	1	0.000000000	0.000000000	NUM
iajs-2860	127	2	0.000000000	0.000000000	NUM
iajs-2860	127	3	0.1	0.1	NUM
iajs-2860	127	4	−8.1	−8.1	PROPN
iajs-2860	127	5	×	×	PROPN
iajs-2860	127	6	10−3	10−3	NUM
iajs-2860	127	7	-0.008092924	-0.008092924	NOUN
iajs-2860	128	1	−8.1	−8.1	PRON
iajs-2860	128	2	×	×	NOUN
iajs-2860	128	3	10−3	10−3	NUM
iajs-2860	128	4	7.004	7.004	NUM
iajs-2860	128	5	×	×	NOUN
iajs-2860	128	6	10−6	10−6	NUM
iajs-2860	128	7	4.499	4.499	NUM
iajs-2860	128	8	×	×	NOUN
iajs-2860	128	9	10−7	10−7	NUM
iajs-2860	128	10	0.2	0.2	NUM
iajs-2860	128	11	-0.02900000	-0.02900000	NOUN
iajs-2860	128	12	-0.028694546	-0.028694546	PROPN
iajs-2860	128	13	-0.028814338	-0.028814338	PUNCT
iajs-2860	129	1	1.055	1.055	NUM
iajs-2860	129	2	×	×	PROPN
iajs-2860	129	3	10−4	10−4	NUM
iajs-2860	129	4	1.439	1.439	NUM
iajs-2860	129	5	×	×	NOUN
iajs-2860	129	6	10−5	10−5	NUM
iajs-2860	129	7	0.3	0.3	NUM
iajs-2860	129	8	-0.05700000	-0.05700000	NOUN
iajs-2860	129	9	-0.056203437	-0.056203437	NOUN
iajs-2860	129	10	-0.056809116	-0.056809116	VERB
iajs-2860	130	1	4.966	4.966	NUM
iajs-2860	130	2	×	×	NOUN
iajs-2860	130	3	10−4	10−4	NUM
iajs-2860	130	4	1.091	1.091	NUM
iajs-2860	130	5	×	×	NOUN
iajs-2860	130	6	10−4	10−4	NUM
iajs-2860	130	7	0.4	0.4	NUM
iajs-2860	130	8	-0.08600000	-0.08600000	NOUN
iajs-2860	130	9	-0.849492630	-0.849492630	PROPN
iajs-2860	130	10	-0.086859048	-0.086859048	PROPN
iajs-2860	131	1	1.451	1.451	NUM
iajs-2860	131	2	×	×	NOUN
iajs-2860	131	3	10−3	10−3	NUM
iajs-2860	131	4	4.590	4.590	NUM
iajs-2860	131	5	×	×	NOUN
iajs-2860	131	6	10−4	10−4	NUM
iajs-2860	131	7	0.5	0.5	NUM
iajs-2860	131	8	-0.11300000	-0.11300000	X
iajs-2860	131	9	-0.109249304	-0.109249304	PROPN
iajs-2860	131	10	-0.113897909	-0.113897909	PROPN
iajs-2860	132	1	3.251	3.251	NUM
iajs-2860	132	2	×	×	NOUN
iajs-2860	132	3	10−3	10−3	NUM
iajs-2860	132	4	1.398	1.398	NUM
iajs-2860	132	5	×	×	PROPN
iajs-2860	132	6	10−3	10−3	NUM
iajs-2860	132	7	0.6	0.6	NUM
iajs-2860	132	8	-0.13200000	-0.13200000	NOUN
iajs-2860	132	9	-0.123465262	-0.123465262	PROPN
iajs-2860	132	10	-0.133069357	-0.133069357	VERB
iajs-2860	132	11	6.135	6.135	NUM
iajs-2860	132	12	×	×	PROPN
iajs-2860	132	13	10−3	10−3	NUM
iajs-2860	132	14	3.469	3.469	NUM
iajs-2860	132	15	×	×	NOUN
iajs-2860	132	16	10−3	10−3	NUM
iajs-2860	132	17	0.7	0.7	NUM
iajs-2860	132	18	-0.13200000	-0.13200000	NOUN
iajs-2860	132	19	-0.122059594	-0.122059594	NOUN
iajs-2860	132	20	-0.139775512	-0.139775512	NOUN
iajs-2860	132	21	0.010000000	0.010000000	NUM
iajs-2860	132	22	7.476	7.476	NUM
iajs-2860	132	23	×	×	NOUN
iajs-2860	132	24	10−3	10−3	NUM
iajs-2860	132	25	0.8	0.8	NUM
iajs-2860	132	26	-0.11500000	-0.11500000	NOUN
iajs-2860	132	27	-0.099650843	-0.099650843	NOUN
iajs-2860	132	28	-0.129722893	-0.129722893	VERB
iajs-2860	133	1	0.016000000	0.016000000	NUM
iajs-2860	134	1	0.015000000	0.015000000	NUM
iajs-2860	134	2	0.9	0.9	NUM
iajs-2860	134	3	-0.07300000	-0.07300000	NOUN
iajs-2860	134	4	-0.051067411	-0.051067411	PROPN
iajs-2860	134	5	-0.098965315	-0.098965315	PROPN
iajs-2860	134	6	0.022000000	0.022000000	NUM
iajs-2860	134	7	0.026000000	0.026000000	NUM
iajs-2860	134	8	1	1	NUM
iajs-2860	134	9	0.000000000	0.000000000	NUM
iajs-2860	134	10	0.02860077	0.02860077	NUM
iajs-2860	134	11	-0.043943323	-0.043943323	NOUN
iajs-2860	135	1	0.029000000	0.029000000	NUM
iajs-2860	135	2	0.044000000	0.044000000	NUM
iajs-2860	135	3	table	table	NOUN
iajs-2860	135	4	2	2	NUM
iajs-2860	135	5	.	.	PUNCT
iajs-2860	135	6	shows	show	VERB
iajs-2860	135	7	the	the	DET
iajs-2860	135	8	results	result	NOUN
iajs-2860	135	9	of	of	ADP
iajs-2860	135	10	lfvies	lfvie	NOUN
iajs-2860	135	11	with	with	ADP
iajs-2860	135	12	weakly	weakly	ADJ
iajs-2860	135	13	singular	singular	NOUN
iajs-2860	135	14	in	in	ADP
iajs-2860	135	15	upper	upper	ADJ
iajs-2860	135	16	solution	solution	NOUN
iajs-2860	135	17	𝒙	𝒙	PROPN
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iajs-2860	135	19	,	,	PUNCT
iajs-2860	135	20	𝒓	𝒓	NOUN
iajs-2860	135	21	)	)	PUNCT
iajs-2860	135	22	𝒔(𝒙	𝒔(𝒙	PROPN
iajs-2860	135	23	,	,	PUNCT
iajs-2860	135	24	𝒓	𝒓	NOUN
iajs-2860	135	25	)	)	PUNCT
iajs-2860	135	26	in	in	ADP
iajs-2860	135	27	error	error	NOUN
iajs-2860	135	28	in	in	ADP
iajs-2860	135	29	linear	linear	PROPN
iajs-2860	135	30	quadratic	quadratic	ADJ
iajs-2860	135	31	linear	linear	ADJ
iajs-2860	135	32	quadratic	quadratic	ADJ
iajs-2860	135	33	0	0	NUM
iajs-2860	135	34	0.000000000	0.000000000	NUM
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iajs-2860	135	36	0.000000000	0.000000000	NUM
iajs-2860	135	37	0.000000000	0.000000000	NUM
iajs-2860	135	38	0.000000000	0.000000000	NUM
iajs-2860	135	39	0.1	0.1	NUM
iajs-2860	135	40	8.1	8.1	NUM
iajs-2860	135	41	×	×	PROPN
iajs-2860	135	42	10−3	10−3	NUM
iajs-2860	135	43	0.008092924	0.008092924	NUM
iajs-2860	135	44	8.10	8.10	NUM
iajs-2860	135	45	×	×	NOUN
iajs-2860	135	46	10−3	10−3	NUM
iajs-2860	135	47	7.004	7.004	NUM
iajs-2860	135	48	×	×	NOUN
iajs-2860	135	49	10−6	10−6	NUM
iajs-2860	135	50	4.499	4.499	NUM
iajs-2860	135	51	×	×	NOUN
iajs-2860	135	52	10−7	10−7	NUM
iajs-2860	135	53	0.2	0.2	NUM
iajs-2860	135	54	0.029000000	0.029000000	NUM
iajs-2860	135	55	0.028694546	0.028694546	NUM
iajs-2860	135	56	0.028814338	0.028814338	NUM
iajs-2860	135	57	1.055	1.055	NUM
iajs-2860	135	58	×	×	NOUN
iajs-2860	135	59	10−4	10−4	NUM
iajs-2860	135	60	1.439	1.439	NUM
iajs-2860	135	61	×	×	NOUN
iajs-2860	135	62	10−5	10−5	NUM
iajs-2860	135	63	ihjpas	ihjpa	NOUN
iajs-2860	135	64	.	.	PUNCT
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iajs-2860	136	2	413	413	NUM
iajs-2860	136	3	0.3	0.3	NUM
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iajs-2860	136	5	0.056203437	0.056203437	NUM
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iajs-2860	136	8	×	×	NOUN
iajs-2860	136	9	10−4	10−4	NUM
iajs-2860	136	10	1.091	1.091	NUM
iajs-2860	136	11	×	×	NOUN
iajs-2860	136	12	10−4	10−4	NUM
iajs-2860	136	13	0.4	0.4	NUM
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iajs-2860	136	15	0.084949263	0.084949263	NUM
iajs-2860	136	16	0.086859048	0.086859048	NUM
iajs-2860	136	17	1.451	1.451	NUM
iajs-2860	136	18	×	×	NOUN
iajs-2860	136	19	10−3	10−3	NUM
iajs-2860	136	20	4.590	4.590	NUM
iajs-2860	136	21	×	×	NOUN
iajs-2860	136	22	10−4	10−4	NUM
iajs-2860	136	23	0.5	0.5	NUM
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iajs-2860	136	25	0.109249304	0.109249304	NUM
iajs-2860	136	26	0.113897909	0.113897909	NUM
iajs-2860	136	27	3.251	3.251	NUM
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iajs-2860	136	29	10−3	10−3	NUM
iajs-2860	136	30	1.398	1.398	NUM
iajs-2860	136	31	×	×	NOUN
iajs-2860	136	32	10−3	10−3	NUM
iajs-2860	136	33	0.6	0.6	NUM
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iajs-2860	136	35	0.123465262	0.123465262	NUM
iajs-2860	136	36	0.133069357	0.133069357	NUM
iajs-2860	136	37	6.135	6.135	NUM
iajs-2860	136	38	×	×	NOUN
iajs-2860	136	39	10−3	10−3	NUM
iajs-2860	136	40	3.469	3.469	NUM
iajs-2860	136	41	×	×	NOUN
iajs-2860	136	42	10−3	10−3	NUM
iajs-2860	136	43	0.7	0.7	NUM
iajs-2860	136	44	0.132000000	0.132000000	NUM
iajs-2860	136	45	0.122059594	0.122059594	NUM
iajs-2860	136	46	0.139775512	0.139775512	NUM
iajs-2860	136	47	0.010000000	0.010000000	NUM
iajs-2860	136	48	7.476	7.476	NUM
iajs-2860	136	49	×	×	NOUN
iajs-2860	136	50	10−3	10−3	NUM
iajs-2860	136	51	0.8	0.8	NUM
iajs-2860	136	52	0.115000000	0.115000000	NUM
iajs-2860	136	53	0.099650843	0.099650843	NUM
iajs-2860	136	54	0.129722893	0.129722893	NUM
iajs-2860	136	55	0.016000000	0.016000000	NUM
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iajs-2860	136	57	0.9	0.9	NUM
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iajs-2860	136	59	0.051067411	0.051067411	NUM
iajs-2860	136	60	0.098965315	0.098965315	NUM
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iajs-2860	136	62	0.026000000	0.026000000	NUM
iajs-2860	136	63	1	1	NUM
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iajs-2860	136	68	0.044000000	0.044000000	NUM
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iajs-2860	136	71	2	2	NUM
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iajs-2860	136	73	consider	consider	VERB
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iajs-2860	136	75	fvie	fvie	NOUN
iajs-2860	136	76	with	with	ADP
iajs-2860	136	77	abel	abel	PROPN
iajs-2860	136	78	's	's	PART
iajs-2860	136	79	type	type	NOUN
iajs-2860	136	80	kernel	kernel	NOUN
iajs-2860	136	81	{	{	PUNCT
iajs-2860	136	82	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	136	83	,	,	PUNCT
iajs-2860	136	84	𝑟	𝑟	NOUN
iajs-2860	136	85	)	)	PUNCT
iajs-2860	136	86	=	=	SYM
iajs-2860	136	87	(	(	PUNCT
iajs-2860	137	1	𝑥	𝑥	X
iajs-2860	137	2	+	+	NUM
iajs-2860	137	3	4	4	NUM
iajs-2860	137	4	3	3	NUM
iajs-2860	137	5	𝑥3	𝑥3	NOUN
iajs-2860	137	6	2⁄	2⁄	NUM
iajs-2860	137	7	)	)	PUNCT
iajs-2860	137	8	(	(	PUNCT
iajs-2860	137	9	4	4	NUM
iajs-2860	137	10	+	+	NUM
iajs-2860	137	11	𝑟	𝑟	NOUN
iajs-2860	137	12	)	)	PUNCT
iajs-2860	137	13	−	−	NUM
iajs-2860	137	14	∫	∫	PROPN
iajs-2860	137	15	𝑢(𝑡,𝑟	𝑢(𝑡,𝑟	X
iajs-2860	137	16	)	)	PUNCT
iajs-2860	137	17	√𝑥−𝑡	√𝑥−𝑡	VERB
iajs-2860	137	18	𝑥	𝑥	NOUN
iajs-2860	137	19	0	0	NUM
iajs-2860	137	20	𝑑𝑡	𝑑𝑡	ADP
iajs-2860	137	21	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	137	22	,	,	PUNCT
iajs-2860	137	23	𝑟	𝑟	NOUN
iajs-2860	137	24	)	)	PUNCT
iajs-2860	137	25	=	=	SYM
iajs-2860	138	1	(	(	PUNCT
iajs-2860	138	2	𝑥	𝑥	X
iajs-2860	138	3	+	+	NUM
iajs-2860	138	4	4	4	NUM
iajs-2860	138	5	3	3	NUM
iajs-2860	138	6	𝑥3	𝑥3	NOUN
iajs-2860	138	7	2⁄	2⁄	NUM
iajs-2860	138	8	)	)	PUNCT
iajs-2860	138	9	(	(	PUNCT
iajs-2860	138	10	𝑟	𝑟	X
iajs-2860	138	11	−	−	PROPN
iajs-2860	138	12	6	6	NUM
iajs-2860	138	13	)	)	PUNCT
iajs-2860	138	14	−	−	PROPN
iajs-2860	138	15	∫	∫	PROPN
iajs-2860	138	16	𝑢(𝑡,𝑟	𝑢(𝑡,𝑟	X
iajs-2860	138	17	)	)	PUNCT
iajs-2860	138	18	√𝑥−𝑡	√𝑥−𝑡	VERB
iajs-2860	138	19	𝑥	𝑥	NOUN
iajs-2860	138	20	0	0	NUM
iajs-2860	138	21	𝑑𝑡	𝑑𝑡	ADP
iajs-2860	138	22	where	where	SCONJ
iajs-2860	138	23	the	the	DET
iajs-2860	138	24	exact	exact	ADJ
iajs-2860	138	25	solution	solution	NOUN
iajs-2860	138	26	is	be	AUX
iajs-2860	138	27	𝑢(𝑥	𝑢(𝑥	PROPN
iajs-2860	138	28	,	,	PUNCT
iajs-2860	138	29	𝑟	𝑟	NOUN
iajs-2860	138	30	)	)	PUNCT
iajs-2860	138	31	=	=	SYM
iajs-2860	139	1	[	[	X
iajs-2860	139	2	(	(	PUNCT
iajs-2860	139	3	4	4	NUM
iajs-2860	139	4	+	+	CCONJ
iajs-2860	139	5	𝑟)𝑥	𝑟)𝑥	NOUN
iajs-2860	139	6	,	,	PUNCT
iajs-2860	139	7	(	(	PUNCT
iajs-2860	139	8	6	6	NUM
iajs-2860	139	9	−	−	NOUN
iajs-2860	139	10	𝑟)𝑥][3	𝑟)𝑥][3	NOUN
iajs-2860	139	11	]	]	PUNCT
iajs-2860	139	12	table	table	NOUN
iajs-2860	139	13	3	3	X
iajs-2860	139	14	.	.	PUNCT
iajs-2860	139	15	fvie	fvie	ADJ
iajs-2860	139	16	with	with	ADP
iajs-2860	139	17	abel	abel	PROPN
iajs-2860	139	18	's	's	PART
iajs-2860	139	19	type	type	NOUN
iajs-2860	139	20	kernel	kernel	NOUN
iajs-2860	139	21	in	in	ADP
iajs-2860	139	22	the	the	DET
iajs-2860	139	23	parametric	parametric	ADJ
iajs-2860	139	24	form	form	NOUN
iajs-2860	139	25	in	in	ADP
iajs-2860	139	26	upper	upper	ADJ
iajs-2860	139	27	and	and	CCONJ
iajs-2860	139	28	lower	low	ADJ
iajs-2860	139	29	solution	solution	NOUN
iajs-2860	139	30	.	.	PUNCT
iajs-2860	140	1	𝒙	𝒙	PROPN
iajs-2860	140	2	𝒖(𝒙	𝒖(𝒙	NOUN
iajs-2860	140	3	,	,	PUNCT
iajs-2860	140	4	𝒓	𝒓	NOUN
iajs-2860	140	5	)	)	PUNCT
iajs-2860	140	6	𝒔(𝒙	𝒔(𝒙	PROPN
iajs-2860	140	7	,	,	PUNCT
iajs-2860	140	8	𝒓	𝒓	NOUN
iajs-2860	140	9	)	)	PUNCT
iajs-2860	140	10	error	error	NOUN
iajs-2860	140	11	𝒖(𝒙	𝒖(𝒙	NOUN
iajs-2860	140	12	,	,	PUNCT
iajs-2860	140	13	𝒓	𝒓	NOUN
iajs-2860	140	14	)	)	PUNCT
iajs-2860	140	15	𝒔(𝒙	𝒔(𝒙	PROPN
iajs-2860	140	16	,	,	PUNCT
iajs-2860	140	17	𝒓	𝒓	NOUN
iajs-2860	140	18	)	)	PUNCT
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iajs-2860	140	20	0	0	NUM
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iajs-2860	140	22	0.000000000	0.000000000	NUM
iajs-2860	140	23	0.000000000	0.000000000	NUM
iajs-2860	140	24	0.0000000000	0.0000000000	NUM
iajs-2860	140	25	0.0000000000	0.0000000000	NUM
iajs-2860	140	26	0.000000000	0.000000000	NUM
iajs-2860	140	27	0.1	0.1	NUM
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iajs-2860	141	1	0.000000000	0.000000000	NUM
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iajs-2860	141	4	0.000000000	0.000000000	NUM
iajs-2860	141	5	0.2	0.2	NUM
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iajs-2860	141	7	-1.180000000	-1.180000000	NOUN
iajs-2860	142	1	0.000000000	0.000000000	NUM
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iajs-2860	142	3	0.8200000000	0.8200000000	NUM
iajs-2860	142	4	0.000000000	0.000000000	NUM
iajs-2860	142	5	0.3	0.3	NUM
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iajs-2860	142	7	-1.770000000	-1.770000000	PUNCT
iajs-2860	143	1	0.000000000	0.000000000	NUM
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iajs-2860	144	4	0.4	0.4	NUM
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iajs-2860	145	1	0.000000000	0.000000000	NUM
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iajs-2860	145	3	1.6400000000	1.6400000000	NUM
iajs-2860	145	4	0.000000000	0.000000000	NUM
iajs-2860	145	5	0.5	0.5	NUM
iajs-2860	145	6	-2.95000000	-2.95000000	NOUN
iajs-2860	145	7	-2.950000000	-2.950000000	PROPN
iajs-2860	146	1	0.000000000	0.000000000	NUM
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iajs-2860	146	3	2.0500000000	2.0500000000	NUM
iajs-2860	146	4	0.000000000	0.000000000	NUM
iajs-2860	146	5	0.6	0.6	NUM
iajs-2860	146	6	-3.54000000	-3.54000000	NOUN
iajs-2860	146	7	-3.540000000	-3.540000000	NOUN
iajs-2860	147	1	0.000000000	0.000000000	NUM
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iajs-2860	147	3	2.4600000000	2.4600000000	NUM
iajs-2860	147	4	0.000000000	0.000000000	NUM
iajs-2860	147	5	0.7	0.7	NUM
iajs-2860	147	6	-4.13000000	-4.13000000	ADJ
iajs-2860	147	7	-4.130000000	-4.130000000	NOUN
iajs-2860	147	8	0.000000000	0.000000000	NUM
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iajs-2860	147	10	2.8700000000	2.8700000000	NUM
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iajs-2860	147	14	-4.720000000	-4.720000000	NOUN
iajs-2860	148	1	0.000000000	0.000000000	NUM
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iajs-2860	148	4	0.000000000	0.000000000	NUM
iajs-2860	148	5	0.9	0.9	NUM
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iajs-2860	148	7	-5.310000000	-5.310000000	PUNCT
iajs-2860	149	1	0.000000000	0.000000000	NUM
iajs-2860	149	2	3.6900000000	3.6900000000	NUM
iajs-2860	149	3	3.6900000000	3.6900000000	NUM
iajs-2860	149	4	0.000000000	0.000000000	NUM
iajs-2860	149	5	1	1	NUM
iajs-2860	149	6	-5.90000000	-5.90000000	NOUN
iajs-2860	149	7	-5.900000000	-5.900000000	NOUN
iajs-2860	150	1	0.000000000	0.000000000	NUM
iajs-2860	150	2	4.1000000000	4.1000000000	NUM
iajs-2860	150	3	4.1000000000	4.1000000000	NUM
iajs-2860	150	4	0.000000000	0.000000000	NUM
iajs-2860	150	5	figure	figure	NOUN
iajs-2860	150	6	1	1	NUM
iajs-2860	150	7	.	.	PUNCT
iajs-2860	150	8	results	result	NOUN
iajs-2860	150	9	of	of	ADP
iajs-2860	150	10	example	example	NOUN
iajs-2860	150	11	(	(	PUNCT
iajs-2860	150	12	1	1	X
iajs-2860	150	13	)	)	PUNCT
iajs-2860	150	14	figure	figure	NOUN
iajs-2860	150	15	2.results	2.results	NUM
iajs-2860	150	16	of	of	ADP
iajs-2860	150	17	example	example	NOUN
iajs-2860	150	18	(	(	PUNCT
iajs-2860	150	19	2	2	NUM
iajs-2860	150	20	)	)	PUNCT
iajs-2860	150	21	in	in	ADP
iajs-2860	150	22	figures	figure	NOUN
iajs-2860	150	23	1	1	NUM
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iajs-2860	150	25	2	2	NUM
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iajs-2860	150	28	plot	plot	VERB
iajs-2860	150	29	the	the	DET
iajs-2860	150	30	graphs	graph	NOUN
iajs-2860	150	31	of	of	ADP
iajs-2860	150	32	the	the	DET
iajs-2860	150	33	approximate	approximate	ADJ
iajs-2860	150	34	(	(	PUNCT
iajs-2860	150	35	(	(	PUNCT
iajs-2860	150	36	+	+	NOUN
iajs-2860	150	37	)	)	PUNCT
iajs-2860	150	38	for	for	ADP
iajs-2860	150	39	upper	upper	ADJ
iajs-2860	150	40	and	and	CCONJ
iajs-2860	150	41	(	(	PUNCT
iajs-2860	150	42	+	+	NOUN
iajs-2860	150	43	)	)	PUNCT
iajs-2860	150	44	for	for	ADP
iajs-2860	150	45	lower	low	ADJ
iajs-2860	150	46	)	)	PUNCT
iajs-2860	150	47	solution	solution	NOUN
iajs-2860	150	48	and	and	CCONJ
iajs-2860	150	49	exact	exact	ADJ
iajs-2860	150	50	(	(	PUNCT
iajs-2860	150	51	(	(	PUNCT
iajs-2860	150	52	×	×	NOUN
iajs-2860	150	53	)	)	PUNCT
iajs-2860	150	54	for	for	ADP
iajs-2860	150	55	upper	upper	ADJ
iajs-2860	150	56	and	and	CCONJ
iajs-2860	150	57	(	(	PUNCT
iajs-2860	150	58	×	×	NOUN
iajs-2860	150	59	)	)	PUNCT
iajs-2860	150	60	for	for	ADP
iajs-2860	150	61	lower	low	ADJ
iajs-2860	150	62	)	)	PUNCT
iajs-2860	150	63	solution	solution	NOUN
iajs-2860	150	64	among	among	ADP
iajs-2860	150	65	different	different	ADJ
iajs-2860	150	66	values	value	NOUN
iajs-2860	150	67	of	of	ADP
iajs-2860	150	68	x	x	PUNCT
iajs-2860	150	69	with	with	ADP
iajs-2860	150	70	fuzzy	fuzzy	ADJ
iajs-2860	150	71	parameter	parameter	NOUN
iajs-2860	150	72	𝑟	𝑟	NOUN
iajs-2860	150	73	=	=	SYM
iajs-2860	150	74	0.1	0.1	NUM
iajs-2860	150	75	,	,	PUNCT
iajs-2860	150	76	0	0	NUM
iajs-2860	150	77	≤	≤	NUM
iajs-2860	150	78	𝑥	𝑥	PUNCT
iajs-2860	150	79	≤	≤	NUM
iajs-2860	150	80	1	1	NUM
iajs-2860	150	81	,	,	PUNCT
iajs-2860	150	82	where	where	SCONJ
iajs-2860	150	83	s(x	s(x	NOUN
iajs-2860	150	84	)	)	PUNCT
iajs-2860	150	85	.is	.is	PUNCT
iajs-2860	150	86	approximate	approximate	ADJ
iajs-2860	150	87	,	,	PUNCT
iajs-2860	150	88	and	and	CCONJ
iajs-2860	150	89	e(x	e(x	NUM
iajs-2860	150	90	)	)	PUNCT
iajs-2860	150	91	is	be	AUX
iajs-2860	150	92	exact	exact	ADJ
iajs-2860	150	93	for	for	ADP
iajs-2860	150	94	the	the	DET
iajs-2860	150	95	upper	upper	ADJ
iajs-2860	150	96	parametric	parametric	ADJ
iajs-2860	150	97	form	form	NOUN
iajs-2860	150	98	and	and	CCONJ
iajs-2860	150	99	s1(x	s1(x	NOUN
iajs-2860	150	100	)	)	PUNCT
iajs-2860	150	101	.is	.is	PUNCT
iajs-2860	150	102	approximate	approximate	ADJ
iajs-2860	150	103	,	,	PUNCT
iajs-2860	150	104	and	and	CCONJ
iajs-2860	150	105	e1(x	e1(x	NUM
iajs-2860	150	106	)	)	PUNCT
iajs-2860	150	107	is	be	AUX
iajs-2860	150	108	exact	exact	ADJ
iajs-2860	150	109	for	for	ADP
iajs-2860	150	110	the	the	DET
iajs-2860	150	111	lower	low	ADJ
iajs-2860	150	112	parametric	parametric	ADJ
iajs-2860	150	113	form	form	NOUN
iajs-2860	150	114	.	.	PUNCT
iajs-2860	151	1	7	7	X
iajs-2860	151	2	.	.	X
iajs-2860	151	3	conclusions	conclusion	NOUN
iajs-2860	151	4	ihjpas	ihjpa	VERB
iajs-2860	151	5	.	.	PUNCT
iajs-2860	152	1	36(1)2023	36(1)2023	NUM
iajs-2860	152	2	414	414	NUM
iajs-2860	152	3	the	the	DET
iajs-2860	152	4	examples	example	NOUN
iajs-2860	152	5	show	show	VERB
iajs-2860	152	6	that	that	SCONJ
iajs-2860	152	7	the	the	DET
iajs-2860	152	8	results	result	NOUN
iajs-2860	152	9	of	of	ADP
iajs-2860	152	10	the	the	DET
iajs-2860	152	11	method	method	NOUN
iajs-2860	152	12	are	be	AUX
iajs-2860	152	13	convergent	convergent	ADJ
iajs-2860	152	14	to	to	ADP
iajs-2860	152	15	the	the	DET
iajs-2860	152	16	exact	exact	ADJ
iajs-2860	152	17	solution	solution	NOUN
iajs-2860	152	18	.	.	PUNCT
iajs-2860	153	1	the	the	DET
iajs-2860	153	2	non	non	ADJ
iajs-2860	153	3	-	-	ADJ
iajs-2860	153	4	polynomial	polynomial	ADJ
iajs-2860	153	5	spline	spline	NOUN
iajs-2860	153	6	has	have	AUX
iajs-2860	153	7	been	be	AUX
iajs-2860	153	8	successfully	successfully	ADV
iajs-2860	153	9	used	use	VERB
iajs-2860	153	10	to	to	PART
iajs-2860	153	11	obtain	obtain	VERB
iajs-2860	153	12	the	the	DET
iajs-2860	153	13	.approximate	.approximate	ADJ
iajs-2860	153	14	solutions	solution	NOUN
iajs-2860	153	15	;	;	PUNCT
iajs-2860	153	16	the	the	DET
iajs-2860	153	17	trigonometric	trigonometric	ADJ
iajs-2860	153	18	term	term	NOUN
iajs-2860	153	19	of	of	ADP
iajs-2860	153	20	this	this	DET
iajs-2860	153	21	spline	spline	NOUN
iajs-2860	153	22	has	have	AUX
iajs-2860	153	23	infinite	infinite	ADJ
iajs-2860	153	24	derivatives	derivative	NOUN
iajs-2860	153	25	,	,	PUNCT
iajs-2860	153	26	which	which	PRON
iajs-2860	153	27	agree	agree	VERB
iajs-2860	153	28	with	with	ADP
iajs-2860	153	29	the	the	DET
iajs-2860	153	30	exact	exact	ADJ
iajs-2860	153	31	solution	solution	NOUN
iajs-2860	153	32	.	.	PUNCT
iajs-2860	154	1	given	give	VERB
iajs-2860	154	2	the	the	DET
iajs-2860	154	3	results	result	NOUN
iajs-2860	154	4	,	,	PUNCT
iajs-2860	154	5	tables	table	NOUN
iajs-2860	154	6	and	and	CCONJ
iajs-2860	154	7	figures	figure	NOUN
iajs-2860	154	8	show	show	VERB
iajs-2860	154	9	that	that	SCONJ
iajs-2860	154	10	the	the	DET
iajs-2860	154	11	proposed	propose	VERB
iajs-2860	154	12	technique	technique	NOUN
iajs-2860	154	13	is	be	AUX
iajs-2860	154	14	a	a	DET
iajs-2860	154	15	powerful	powerful	ADJ
iajs-2860	154	16	mathematical	mathematical	ADJ
iajs-2860	154	17	tool	tool	NOUN
iajs-2860	154	18	for	for	ADP
iajs-2860	154	19	solving	solve	VERB
iajs-2860	154	20	fviewsk	fviewsk	NOUN
iajs-2860	154	21	with	with	ADP
iajs-2860	154	22	mathcad	mathcad	PROPN
iajs-2860	154	23	programming	programming	NOUN
iajs-2860	154	24	implementation	implementation	NOUN
iajs-2860	154	25	.	.	PUNCT
iajs-2860	155	1	we	we	PRON
iajs-2860	155	2	will	will	AUX
iajs-2860	155	3	consider	consider	VERB
iajs-2860	155	4	fviewsk	fviewsk	NOUN
iajs-2860	155	5	,	,	PUNCT
iajs-2860	155	6	algorithm	algorithm	NOUN
iajs-2860	155	7	,	,	PUNCT
iajs-2860	155	8	and	and	CCONJ
iajs-2860	155	9	applied	apply	VERB
iajs-2860	155	10	fractional	fractional	ADJ
iajs-2860	155	11	order	order	NOUN
iajs-2860	155	12	problems	problem	NOUN
iajs-2860	155	13	for	for	ADP
iajs-2860	155	14	future	future	ADJ
iajs-2860	155	15	work	work	NOUN
iajs-2860	155	16	.	.	PUNCT
iajs-2860	156	1	references	reference	NOUN
iajs-2860	156	2	1	1	NUM
iajs-2860	156	3	.	.	PUNCT
iajs-2860	157	1	zadeh	zadeh	PROPN
iajs-2860	157	2	,	,	PUNCT
iajs-2860	157	3	l.	l.	PROPN
iajs-2860	157	4	fuzzy	fuzzy	PROPN
iajs-2860	157	5	sets	set	VERB
iajs-2860	157	6	information	information	NOUN
iajs-2860	157	7	and	and	CCONJ
iajs-2860	157	8	control	control	NOUN
iajs-2860	157	9	,	,	PUNCT
iajs-2860	157	10	1965	1965	NUM
iajs-2860	157	11	,	,	PUNCT
iajs-2860	157	12	8	8	NUM
iajs-2860	157	13	,	,	PUNCT
iajs-2860	157	14	338	338	NUM
iajs-2860	157	15	-	-	SYM
iajs-2860	157	16	353	353	NUM
iajs-2860	157	17	.	.	NOUN
iajs-2860	158	1	2	2	NUM
iajs-2860	158	2	.	.	X
iajs-2860	158	3	mahmoud	mahmoud	PROPN
iajs-2860	158	4	p.	p.	PROPN
iajs-2860	158	5	;	;	PUNCT
iajs-2860	158	6	farshid	farshid	VERB
iajs-2860	158	7	m.and	m.and	PROPN
iajs-2860	158	8	mohammad	mohammad	PROPN
iajs-2860	158	9	k.	k.	PROPN
iajs-2860	158	10	solving	solving	PROPN
iajs-2860	158	11	linear	linear	PROPN
iajs-2860	158	12	and	and	CCONJ
iajs-2860	158	13	nonlinear	nonlinear	ADJ
iajs-2860	158	14	abel	abel	PROPN
iajs-2860	158	15	fuzzy	fuzzy	ADJ
iajs-2860	158	16	integral	integral	ADJ
iajs-2860	158	17	equation	equation	NOUN
iajs-2860	158	18	by	by	ADP
iajs-2860	158	19	fuzzy	fuzzy	ADJ
iajs-2860	158	20	laplace	laplace	NOUN
iajs-2860	158	21	transforms	transform	VERB
iajs-2860	158	22	.	.	PUNCT
iajs-2860	159	1	mathematical	mathematical	ADJ
iajs-2860	159	2	inverse	inverse	NOUN
iajs-2860	159	3	problems	problem	NOUN
iajs-2860	159	4	,	,	PUNCT
iajs-2860	159	5	2014	2014	NUM
iajs-2860	159	6	,	,	PUNCT
iajs-2860	159	7	1(2	1(2	NUM
iajs-2860	159	8	)	)	PUNCT
iajs-2860	159	9	.	.	PUNCT
iajs-2860	160	1	3	3	X
iajs-2860	160	2	.	.	X
iajs-2860	160	3	bushnaq	bushnaq	PROPN
iajs-2860	160	4	,	,	PUNCT
iajs-2860	160	5	s.	s.	PROPN
iajs-2860	160	6	;	;	PUNCT
iajs-2860	160	7	ullah	ullah	PROPN
iajs-2860	160	8	,	,	PUNCT
iajs-2860	160	9	z.	z.	PROPN
iajs-2860	160	10	;	;	PUNCT
iajs-2860	160	11	ullah	ullah	PROPN
iajs-2860	160	12	,	,	PUNCT
iajs-2860	160	13	a.	a.	NOUN
iajs-2860	160	14	and	and	CCONJ
iajs-2860	160	15	shah	shah	PROPN
iajs-2860	160	16	,	,	PUNCT
iajs-2860	160	17	k.	k.	PROPN
iajs-2860	160	18	solution	solution	NOUN
iajs-2860	160	19	of	of	ADP
iajs-2860	160	20	integral	integral	ADJ
iajs-2860	160	21	equation	equation	NOUN
iajs-2860	160	22	with	with	ADP
iajs-2860	160	23	abel	abel	PROPN
iajs-2860	160	24	's	's	PART
iajs-2860	160	25	type	type	NOUN
iajs-2860	160	26	kernel	kernel	NOUN
iajs-2860	160	27	using	use	VERB
iajs-2860	160	28	a	a	DET
iajs-2860	160	29	novel	novel	ADJ
iajs-2860	160	30	hybrid	hybrid	NOUN
iajs-2860	160	31	method	method	NOUN
iajs-2860	160	32	.	.	PUNCT
iajs-2860	161	1	advances	advance	NOUN
iajs-2860	161	2	in	in	ADP
iajs-2860	161	3	difference	difference	NOUN
iajs-2860	161	4	equation	equation	NOUN
iajs-2860	161	5	,	,	PUNCT
iajs-2860	161	6	springer	springer	NOUN
iajs-2860	161	7	open	open	PROPN
iajs-2860	161	8	journal	journal	PROPN
iajs-2860	161	9	,	,	PUNCT
iajs-2860	161	10	2020	2020	NUM
iajs-2860	161	11	,	,	PUNCT
iajs-2860	161	12	165	165	NUM
iajs-2860	161	13	.	.	NOUN
iajs-2860	161	14	4	4	NUM
iajs-2860	161	15	.	.	X
iajs-2860	162	1	zahra	zahra	NOUN
iajs-2860	162	2	,	,	PUNCT
iajs-2860	162	3	a.	a.	NOUN
iajs-2860	162	4	collocation	collocation	NOUN
iajs-2860	162	5	method	method	NOUN
iajs-2860	162	6	for	for	ADP
iajs-2860	162	7	fuzzy	fuzzy	ADJ
iajs-2860	162	8	volterra	volterra	NOUN
iajs-2860	162	9	integral	integral	ADJ
iajs-2860	162	10	equation	equation	NOUN
iajs-2860	162	11	of	of	ADP
iajs-2860	162	12	the	the	DET
iajs-2860	162	13	second	second	ADJ
iajs-2860	162	14	kind	kind	NOUN
iajs-2860	162	15	with	with	ADP
iajs-2860	162	16	weakly	weakly	ADJ
iajs-2860	162	17	singular	singular	ADJ
iajs-2860	162	18	kernels	kernel	NOUN
iajs-2860	162	19	,	,	PUNCT
iajs-2860	162	20	research	research	NOUN
iajs-2860	162	21	square	square	NOUN
iajs-2860	162	22	,	,	PUNCT
iajs-2860	162	23	2021	2021	NUM
iajs-2860	162	24	.	.	PUNCT
iajs-2860	163	1	5	5	NUM
iajs-2860	163	2	.	.	X
iajs-2860	163	3	elgaili	elgaili	NOUN
iajs-2860	163	4	,	,	PUNCT
iajs-2860	163	5	a.i	a.i	PROPN
iajs-2860	163	6	.	.	PROPN
iajs-2860	163	7	;	;	PUNCT
iajs-2860	163	8	abdel	abdel	PROPN
iajs-2860	163	9	radi	radi	PROPN
iajs-2860	163	10	,	,	PUNCT
iajs-2860	163	11	a.a	a.a	PROPN
iajs-2860	163	12	.	.	PROPN
iajs-2860	163	13	;	;	PUNCT
iajs-2860	163	14	zakieldeen	zakieldeen	NUM
iajs-2860	163	15	a.m.	a.m.	NOUN
iajs-2860	163	16	;	;	PUNCT
iajs-2860	163	17	reem	reem	PROPN
iajs-2860	163	18	,	,	PUNCT
iajs-2860	163	19	h.i	h.i	PROPN
iajs-2860	163	20	.	.	PROPN
iajs-2860	163	21	analytical	analytical	ADJ
iajs-2860	163	22	and	and	CCONJ
iajs-2860	163	23	numerical	numerical	ADJ
iajs-2860	163	24	solution	solution	NOUN
iajs-2860	163	25	of	of	ADP
iajs-2860	163	26	linear	linear	PROPN
iajs-2860	163	27	volterra	volterra	PROPN
iajs-2860	163	28	integral	integral	ADJ
iajs-2860	163	29	equations	equation	NOUN
iajs-2860	163	30	of	of	ADP
iajs-2860	163	31	the	the	DET
iajs-2860	163	32	second	second	ADJ
iajs-2860	163	33	kind	kind	NOUN
iajs-2860	163	34	with	with	ADP
iajs-2860	163	35	weakly	weakly	ADJ
iajs-2860	163	36	singular	singular	ADJ
iajs-2860	163	37	kernel	kernel	NOUN
iajs-2860	163	38	by	by	ADP
iajs-2860	163	39	using	use	VERB
iajs-2860	163	40	the	the	DET
iajs-2860	163	41	sixth	sixth	ADJ
iajs-2860	163	42	order	order	NOUN
iajs-2860	163	43	of	of	ADP
iajs-2860	163	44	non	non	ADJ
iajs-2860	163	45	-	-	ADJ
iajs-2860	163	46	polynomial	polynomial	ADJ
iajs-2860	163	47	spline	spline	NOUN
iajs-2860	163	48	functions	function	NOUN
iajs-2860	163	49	by	by	ADP
iajs-2860	163	50	matlab.turkish	matlab.turkish	ADJ
iajs-2860	163	51	journal	journal	NOUN
iajs-2860	163	52	of	of	ADP
iajs-2860	163	53	computer	computer	NOUN
iajs-2860	163	54	and	and	CCONJ
iajs-2860	163	55	mathematics	mathematic	NOUN
iajs-2860	163	56	education	education	NOUN
iajs-2860	163	57	,	,	PUNCT
iajs-2860	163	58	2021	2021	NUM
iajs-2860	163	59	,	,	PUNCT
iajs-2860	163	60	12(14	12(14	NUM
iajs-2860	163	61	)	)	PUNCT
iajs-2860	163	62	.	.	PUNCT
iajs-2860	164	1	6	6	X
iajs-2860	164	2	.	.	X
iajs-2860	164	3	biswas	biswas	PROPN
iajs-2860	164	4	,	,	PUNCT
iajs-2860	164	5	s.	s.	PROPN
iajs-2860	164	6	;	;	PUNCT
iajs-2860	164	7	roy	roy	PROPN
iajs-2860	164	8	,	,	PUNCT
iajs-2860	164	9	t.k	t.k	PROPN
iajs-2860	164	10	.	.	PROPN
iajs-2860	164	11	fuzzy	fuzzy	ADJ
iajs-2860	164	12	linear	linear	ADJ
iajs-2860	164	13	integral	integral	ADJ
iajs-2860	164	14	equation	equation	NOUN
iajs-2860	164	15	and	and	CCONJ
iajs-2860	164	16	its	its	PRON
iajs-2860	164	17	application	application	NOUN
iajs-2860	164	18	inbiomathematical	inbiomathematical	ADJ
iajs-2860	164	19	model	model	NOUN
iajs-2860	164	20	.	.	PUNCT
iajs-2860	165	1	advances	advance	NOUN
iajs-2860	165	2	in	in	ADP
iajs-2860	165	3	fuzzy	fuzzy	ADJ
iajs-2860	165	4	mathematics	mathematic	NOUN
iajs-2860	165	5	,	,	PUNCT
iajs-2860	165	6	2017	2017	NUM
iajs-2860	165	7	,	,	PUNCT
iajs-2860	165	8	12(5	12(5	NUM
iajs-2860	165	9	)	)	PUNCT
iajs-2860	165	10	,	,	PUNCT
iajs-2860	165	11	1137–1157	1137–1157	NUM
iajs-2860	165	12	.	.	PUNCT
iajs-2860	166	1	7	7	X
iajs-2860	166	2	.	.	X
iajs-2860	166	3	waffa	waffa	PROPN
iajs-2860	166	4	,	,	PUNCT
iajs-2860	166	5	a.i	a.i	PROPN
iajs-2860	166	6	.	.	PROPN
iajs-2860	166	7	numerical	numerical	PROPN
iajs-2860	166	8	method	method	NOUN
iajs-2860	166	9	for	for	ADP
iajs-2860	166	10	solving	solve	VERB
iajs-2860	166	11	integral	integral	ADJ
iajs-2860	166	12	equations	equation	NOUN
iajs-2860	166	13	,	,	PUNCT
iajs-2860	166	14	2016	2016	NUM
iajs-2860	166	15	,	,	PUNCT
iajs-2860	166	16	m.sc	m.sc	PROPN
iajs-2860	166	17	.	.	PUNCT
iajs-2860	167	1	thesis	thesis	PROPN
iajs-2860	167	2	mustansiriayah	mustansiriayah	PROPN
iajs-2860	167	3	university	university	NOUN
iajs-2860	167	4	.	.	PUNCT
iajs-2860	168	1	8	8	NUM
iajs-2860	168	2	.	.	X
iajs-2860	168	3	ali	ali	PROPN
iajs-2860	168	4	,	,	PUNCT
iajs-2860	168	5	f.j	f.j	PROPN
iajs-2860	168	6	;	;	PUNCT
iajs-2860	168	7	anakira	anakira	PROPN
iajs-2860	168	8	,	,	PUNCT
iajs-2860	168	9	r.	r.	PROPN
iajs-2860	168	10	n.	n.	PROPN
iajs-2860	168	11	;	;	PUNCT
iajs-2860	168	12	alomari	alomari	PROPN
iajs-2860	168	13	,	,	PUNCT
iajs-2860	168	14	a.k	a.k	PROPN
iajs-2860	168	15	.	.	PUNCT
iajs-2860	168	16	;	;	PUNCT
iajs-2860	168	17	noraziah	noraziah	PROPN
iajs-2860	168	18	,	,	PUNCT
iajs-2860	168	19	m.	m.	NOUN
iajs-2860	168	20	solution	solution	NOUN
iajs-2860	168	21	and	and	CCONJ
iajs-2860	168	22	analysis	analysis	NOUN
iajs-2860	168	23	of	of	ADP
iajs-2860	168	24	the	the	DET
iajs-2860	168	25	fuzzy	fuzzy	ADJ
iajs-2860	168	26	volterra	volterra	NOUN
iajs-2860	168	27	integral	integral	ADJ
iajs-2860	168	28	equation	equation	NOUN
iajs-2860	168	29	via	via	ADP
iajs-2860	168	30	homotopy	homotopy	NOUN
iajs-2860	168	31	analysis	analysis	NOUN
iajs-2860	168	32	method	method	NOUN
iajs-2860	168	33	.	.	PUNCT
iajs-2860	169	1	computer	computer	NOUN
iajs-2860	169	2	modeling	modeling	NOUN
iajs-2860	169	3	in	in	ADP
iajs-2860	169	4	engineering	engineering	NOUN
iajs-2860	169	5	and	and	CCONJ
iajs-2860	169	6	sciences	science	NOUN
iajs-2860	169	7	,	,	PUNCT
iajs-2860	169	8	2021,127(3	2021,127(3	NUM
iajs-2860	169	9	)	)	PUNCT
iajs-2860	169	10	.	.	PUNCT
iajs-2860	170	1	9	9	X
iajs-2860	170	2	.	.	X
iajs-2860	170	3	muna	muna	PROPN
iajs-2860	170	4	,	,	PUNCT
iajs-2860	170	5	m.	m.	NOUN
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iajs-2860	170	12	second	second	ADJ
iajs-2860	170	13	kind	kind	NOUN
iajs-2860	170	14	volterra	volterra	NOUN
iajs-2860	170	15	integrals	integral	VERB
iajs-2860	170	16	equations	equation	NOUN
iajs-2860	170	17	using	use	VERB
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iajs-2860	170	19	polynomial	polynomial	ADJ
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iajs-2860	170	21	functions	function	NOUN
iajs-2860	170	22	.	.	PUNCT
iajs-2860	171	1	baghdad	baghdad	PROPN
iajs-2860	171	2	science	science	PROPN
iajs-2860	171	3	journal	journal	PROPN
iajs-2860	171	4	,	,	PUNCT
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iajs-2860	171	6	,	,	PUNCT
iajs-2860	171	7	11(2	11(2	NUM
iajs-2860	171	8	)	)	PUNCT
iajs-2860	171	9	.	.	PUNCT
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iajs-2860	172	2	.	.	PUNCT
iajs-2860	172	3	rawaa	rawaa	PROPN
iajs-2860	172	4	,	,	PUNCT
iajs-2860	172	5	i.e.	i.e.	X
iajs-2860	172	6	;	;	PUNCT
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iajs-2860	172	8	,	,	PUNCT
iajs-2860	172	9	j.s	j.s	PROPN
iajs-2860	172	10	.	.	PROPN
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iajs-2860	173	2	treatment	treatment	NOUN
iajs-2860	173	3	of	of	ADP
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iajs-2860	173	5	order	order	NOUN
iajs-2860	173	6	volterra	volterra	PROPN
iajs-2860	173	7	integro	integro	PROPN
iajs-2860	173	8	-	-	PUNCT
iajs-2860	173	9	differential	differential	NOUN
iajs-2860	173	10	equation	equation	NOUN
iajs-2860	173	11	using	use	VERB
iajs-2860	173	12	non	non	ADJ
iajs-2860	173	13	-	-	ADJ
iajs-2860	173	14	polynomial	polynomial	ADJ
iajs-2860	173	15	spline	spline	NOUN
iajs-2860	173	16	function	function	NOUN
iajs-2860	173	17	.	.	PUNCT
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iajs-2860	174	7	,	,	PUNCT
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iajs-2860	176	4	.	.	PROPN
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iajs-2860	176	7	equation	equation	NOUN
iajs-2860	176	8	using	use	VERB
iajs-2860	176	9	non	non	ADJ
iajs-2860	176	10	-	-	ADJ
iajs-2860	176	11	polynomial	polynomial	ADJ
iajs-2860	176	12	spline	spline	NOUN
iajs-2860	176	13	functions	function	NOUN
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iajs-2860	177	6	university	university	PROPN
iajs-2860	177	7	.	.	PUNCT