id	sid	tid	token	lemma	pos
ap-9438	1	1	acta	acta	PROPN
ap-9438	1	2	polytechnica	polytechnica	PROPN
ap-9438	1	3	https://doi.org/10.14311/ap.2024.64.0414	https://doi.org/10.14311/ap.2024.64.0414	PROPN
ap-9438	1	4	acta	acta	PROPN
ap-9438	1	5	polytechnica	polytechnica	PROPN
ap-9438	1	6	64(5):414–419	64(5):414–419	PROPN
ap-9438	1	7	,	,	PUNCT
ap-9438	1	8	2024	2024	NUM
ap-9438	1	9	©	©	ADP
ap-9438	1	10	2024	2024	NUM
ap-9438	1	11	the	the	DET
ap-9438	1	12	author(s	author(s	NOUN
ap-9438	1	13	)	)	PUNCT
ap-9438	1	14	.	.	PUNCT
ap-9438	2	1	licensed	license	VERB
ap-9438	2	2	under	under	ADP
ap-9438	2	3	a	a	DET
ap-9438	2	4	cc	cc	NOUN
ap-9438	2	5	-	-	PUNCT
ap-9438	2	6	by	by	ADP
ap-9438	2	7	4.0	4.0	NUM
ap-9438	2	8	licence	licence	NOUN
ap-9438	2	9	published	publish	VERB
ap-9438	2	10	by	by	ADP
ap-9438	2	11	the	the	DET
ap-9438	2	12	czech	czech	PROPN
ap-9438	2	13	technical	technical	PROPN
ap-9438	2	14	university	university	PROPN
ap-9438	2	15	in	in	ADP
ap-9438	2	16	prague	prague	PROPN
ap-9438	2	17	a	a	DET
ap-9438	2	18	novel	novel	ADJ
ap-9438	2	19	approach	approach	NOUN
ap-9438	2	20	to	to	ADP
ap-9438	2	21	nonlinear	nonlinear	ADJ
ap-9438	2	22	fractional	fractional	PROPN
ap-9438	2	23	volterra	volterra	PROPN
ap-9438	2	24	integral	integral	ADJ
ap-9438	2	25	equations	equation	NOUN
ap-9438	2	26	mohammed	mohammed	PROPN
ap-9438	2	27	abdulshareef	abdulshareef	PROPN
ap-9438	2	28	husseina	husseina	PROPN
ap-9438	2	29	,	,	PUNCT
ap-9438	2	30	b,∗	b,∗	PROPN
ap-9438	2	31	,	,	PUNCT
ap-9438	2	32	hassan	hassan	PROPN
ap-9438	2	33	kamil	kamil	PROPN
ap-9438	2	34	jassimc	jassimc	PROPN
ap-9438	2	35	a	a	DET
ap-9438	2	36	al	al	PROPN
ap-9438	2	37	-	-	PUNCT
ap-9438	2	38	ayen	ayen	PROPN
ap-9438	2	39	iraqi	iraqi	ADJ
ap-9438	2	40	university	university	NOUN
ap-9438	2	41	,	,	PUNCT
ap-9438	2	42	scientific	scientific	ADJ
ap-9438	2	43	research	research	NOUN
ap-9438	2	44	center	center	NOUN
ap-9438	2	45	,	,	PUNCT
ap-9438	2	46	64001	64001	NUM
ap-9438	2	47	nasiriyah	nasiriyah	NOUN
ap-9438	2	48	,	,	PUNCT
ap-9438	2	49	iraq	iraq	PROPN
ap-9438	2	50	b	b	PROPN
ap-9438	2	51	ministry	ministry	PROPN
ap-9438	2	52	of	of	ADP
ap-9438	2	53	education	education	PROPN
ap-9438	2	54	,	,	PUNCT
ap-9438	2	55	education	education	NOUN
ap-9438	2	56	directorate	directorate	NOUN
ap-9438	2	57	of	of	ADP
ap-9438	2	58	thi	thi	PROPN
ap-9438	2	59	-	-	PUNCT
ap-9438	2	60	qar	qar	PROPN
ap-9438	2	61	,	,	PUNCT
ap-9438	2	62	64001	64001	NUM
ap-9438	2	63	nasiriyah	nasiriyah	NOUN
ap-9438	2	64	,	,	PUNCT
ap-9438	2	65	iraq	iraq	PROPN
ap-9438	2	66	c	c	PROPN
ap-9438	2	67	university	university	PROPN
ap-9438	2	68	of	of	ADP
ap-9438	2	69	thi	thi	PROPN
ap-9438	2	70	-	-	PUNCT
ap-9438	2	71	qar	qar	PROPN
ap-9438	2	72	,	,	PUNCT
ap-9438	2	73	college	college	NOUN
ap-9438	2	74	of	of	ADP
ap-9438	2	75	education	education	NOUN
ap-9438	2	76	for	for	ADP
ap-9438	2	77	pure	pure	ADJ
ap-9438	2	78	science	science	NOUN
ap-9438	2	79	,	,	PUNCT
ap-9438	2	80	department	department	NOUN
ap-9438	2	81	of	of	ADP
ap-9438	2	82	mathematics	mathematic	NOUN
ap-9438	2	83	,	,	PUNCT
ap-9438	2	84	64001	64001	NUM
ap-9438	2	85	thi	thi	PROPN
ap-9438	2	86	-	-	PUNCT
ap-9438	2	87	qar	qar	PROPN
ap-9438	2	88	,	,	PUNCT
ap-9438	2	89	iraq	iraq	PROPN
ap-9438	2	90	∗	∗	VERB
ap-9438	2	91	corresponding	correspond	VERB
ap-9438	2	92	author	author	NOUN
ap-9438	2	93	:	:	PUNCT
ap-9438	2	94	mshirq@utq.edu.iq	mshirq@utq.edu.iq	ADJ
ap-9438	2	95	abstract	abstract	NOUN
ap-9438	2	96	.	.	PUNCT
ap-9438	3	1	nonlinear	nonlinear	ADJ
ap-9438	3	2	fractional	fractional	PROPN
ap-9438	3	3	volterra	volterra	PROPN
ap-9438	3	4	integral	integral	ADJ
ap-9438	3	5	equations	equation	NOUN
ap-9438	3	6	(	(	PUNCT
ap-9438	3	7	fvies	fvie	NOUN
ap-9438	3	8	)	)	PUNCT
ap-9438	3	9	of	of	ADP
ap-9438	3	10	the	the	DET
ap-9438	3	11	first	first	ADJ
ap-9438	3	12	kind	kind	NOUN
ap-9438	3	13	present	present	ADJ
ap-9438	3	14	challenges	challenge	NOUN
ap-9438	3	15	due	due	ADP
ap-9438	3	16	to	to	ADP
ap-9438	3	17	their	their	PRON
ap-9438	3	18	intricate	intricate	ADJ
ap-9438	3	19	nature	nature	NOUN
ap-9438	3	20	,	,	PUNCT
ap-9438	3	21	combining	combine	VERB
ap-9438	3	22	fractional	fractional	ADJ
ap-9438	3	23	calculus	calculus	NOUN
ap-9438	3	24	and	and	CCONJ
ap-9438	3	25	integral	integral	ADJ
ap-9438	3	26	equations	equation	NOUN
ap-9438	3	27	.	.	PUNCT
ap-9438	4	1	in	in	ADP
ap-9438	4	2	this	this	DET
ap-9438	4	3	research	research	NOUN
ap-9438	4	4	paper	paper	NOUN
ap-9438	4	5	,	,	PUNCT
ap-9438	4	6	we	we	PRON
ap-9438	4	7	introduce	introduce	VERB
ap-9438	4	8	a	a	DET
ap-9438	4	9	novel	novel	ADJ
ap-9438	4	10	method	method	NOUN
ap-9438	4	11	for	for	ADP
ap-9438	4	12	solving	solve	VERB
ap-9438	4	13	such	such	ADJ
ap-9438	4	14	equations	equation	NOUN
ap-9438	4	15	using	use	VERB
ap-9438	4	16	leibniz	leibniz	PROPN
ap-9438	4	17	integral	integral	ADJ
ap-9438	4	18	rules	rule	NOUN
ap-9438	4	19	.	.	PUNCT
ap-9438	5	1	our	our	PRON
ap-9438	5	2	study	study	NOUN
ap-9438	5	3	focuses	focus	VERB
ap-9438	5	4	on	on	ADP
ap-9438	5	5	a	a	DET
ap-9438	5	6	thorough	thorough	ADJ
ap-9438	5	7	analysis	analysis	NOUN
ap-9438	5	8	and	and	CCONJ
ap-9438	5	9	application	application	NOUN
ap-9438	5	10	of	of	ADP
ap-9438	5	11	the	the	DET
ap-9438	5	12	proposed	propose	VERB
ap-9438	5	13	algorithm	algorithm	NOUN
ap-9438	5	14	to	to	PART
ap-9438	5	15	solve	solve	VERB
ap-9438	5	16	fractional	fractional	ADJ
ap-9438	5	17	volterra	volterra	PROPN
ap-9438	5	18	integral	integral	ADJ
ap-9438	5	19	equations	equation	NOUN
ap-9438	5	20	.	.	PUNCT
ap-9438	6	1	by	by	ADP
ap-9438	6	2	using	use	VERB
ap-9438	6	3	leibniz	leibniz	PROPN
ap-9438	6	4	integral	integral	ADJ
ap-9438	6	5	rules	rule	NOUN
ap-9438	6	6	,	,	PUNCT
ap-9438	6	7	we	we	PRON
ap-9438	6	8	offer	offer	VERB
ap-9438	6	9	a	a	DET
ap-9438	6	10	fresh	fresh	ADJ
ap-9438	6	11	perspective	perspective	NOUN
ap-9438	6	12	on	on	ADP
ap-9438	6	13	handling	handle	VERB
ap-9438	6	14	these	these	DET
ap-9438	6	15	equations	equation	NOUN
ap-9438	6	16	,	,	PUNCT
ap-9438	6	17	shedding	shed	VERB
ap-9438	6	18	light	light	NOUN
ap-9438	6	19	on	on	ADP
ap-9438	6	20	their	their	PRON
ap-9438	6	21	fundamental	fundamental	ADJ
ap-9438	6	22	properties	property	NOUN
ap-9438	6	23	and	and	CCONJ
ap-9438	6	24	behaviours	behaviour	NOUN
ap-9438	6	25	.	.	PUNCT
ap-9438	7	1	as	as	ADP
ap-9438	7	2	a	a	DET
ap-9438	7	3	result	result	NOUN
ap-9438	7	4	of	of	ADP
ap-9438	7	5	this	this	DET
ap-9438	7	6	study	study	NOUN
ap-9438	7	7	,	,	PUNCT
ap-9438	7	8	we	we	PRON
ap-9438	7	9	anticipate	anticipate	VERB
ap-9438	7	10	contributing	contribute	VERB
ap-9438	7	11	distinctively	distinctively	ADV
ap-9438	7	12	to	to	ADP
ap-9438	7	13	the	the	DET
ap-9438	7	14	broader	broad	ADJ
ap-9438	7	15	development	development	NOUN
ap-9438	7	16	of	of	ADP
ap-9438	7	17	analytical	analytical	ADJ
ap-9438	7	18	tools	tool	NOUN
ap-9438	7	19	and	and	CCONJ
ap-9438	7	20	techniques	technique	NOUN
ap-9438	7	21	.	.	PUNCT
ap-9438	8	1	by	by	ADP
ap-9438	8	2	bridging	bridge	VERB
ap-9438	8	3	the	the	DET
ap-9438	8	4	gap	gap	NOUN
ap-9438	8	5	between	between	ADP
ap-9438	8	6	fractional	fractional	ADJ
ap-9438	8	7	calculus	calculus	NOUN
ap-9438	8	8	and	and	CCONJ
ap-9438	8	9	integral	integral	ADJ
ap-9438	8	10	equations	equation	NOUN
ap-9438	8	11	,	,	PUNCT
ap-9438	8	12	our	our	PRON
ap-9438	8	13	approach	approach	NOUN
ap-9438	8	14	not	not	PART
ap-9438	8	15	only	only	ADV
ap-9438	8	16	offers	offer	VERB
ap-9438	8	17	a	a	DET
ap-9438	8	18	valuable	valuable	ADJ
ap-9438	8	19	computational	computational	ADJ
ap-9438	8	20	methodology	methodology	NOUN
ap-9438	8	21	but	but	CCONJ
ap-9438	8	22	also	also	ADV
ap-9438	8	23	paves	pave	VERB
ap-9438	8	24	the	the	DET
ap-9438	8	25	way	way	NOUN
ap-9438	8	26	for	for	ADP
ap-9438	8	27	new	new	ADJ
ap-9438	8	28	insights	insight	NOUN
ap-9438	8	29	into	into	ADP
ap-9438	8	30	the	the	DET
ap-9438	8	31	application	application	NOUN
ap-9438	8	32	domains	domain	NOUN
ap-9438	8	33	in	in	ADP
ap-9438	8	34	which	which	PRON
ap-9438	8	35	such	such	ADJ
ap-9438	8	36	equations	equation	NOUN
ap-9438	8	37	arise	arise	VERB
ap-9438	8	38	.	.	PUNCT
ap-9438	9	1	keywords	keyword	NOUN
ap-9438	9	2	:	:	PUNCT
ap-9438	9	3	integral	integral	ADJ
ap-9438	9	4	equations	equation	NOUN
ap-9438	9	5	,	,	PUNCT
ap-9438	9	6	fractional	fractional	ADJ
ap-9438	9	7	calculus	calculus	NOUN
ap-9438	9	8	,	,	PUNCT
ap-9438	9	9	leibniz	leibniz	PROPN
ap-9438	9	10	integral	integral	ADJ
ap-9438	9	11	rule	rule	NOUN
ap-9438	9	12	,	,	PUNCT
ap-9438	9	13	mitteg	mitteg	NOUN
ap-9438	9	14	-	-	PUNCT
ap-9438	9	15	leffler	leffler	NOUN
ap-9438	9	16	function	function	NOUN
ap-9438	9	17	,	,	PUNCT
ap-9438	9	18	caputo	caputo	PROPN
ap-9438	9	19	fractional	fractional	PROPN
ap-9438	9	20	operator	operator	NOUN
ap-9438	9	21	,	,	PUNCT
ap-9438	9	22	riemann	riemann	PROPN
ap-9438	9	23	-	-	PUNCT
ap-9438	9	24	liouville	liouville	VERB
ap-9438	9	25	fractional	fractional	ADJ
ap-9438	9	26	operator	operator	NOUN
ap-9438	9	27	.	.	PUNCT
ap-9438	10	1	1	1	X
ap-9438	10	2	.	.	X
ap-9438	10	3	introduction	introduction	NOUN
ap-9438	10	4	fractional	fractional	ADJ
ap-9438	10	5	calculus	calculus	NOUN
ap-9438	10	6	is	be	AUX
ap-9438	10	7	a	a	DET
ap-9438	10	8	branch	branch	NOUN
ap-9438	10	9	of	of	ADP
ap-9438	10	10	mathematical	mathematical	ADJ
ap-9438	10	11	analysis	analysis	NOUN
ap-9438	10	12	that	that	PRON
ap-9438	10	13	generalises	generalise	VERB
ap-9438	10	14	the	the	DET
ap-9438	10	15	concept	concept	NOUN
ap-9438	10	16	of	of	ADP
ap-9438	10	17	differentiation	differentiation	NOUN
ap-9438	10	18	and	and	CCONJ
ap-9438	10	19	integration	integration	NOUN
ap-9438	10	20	to	to	ADP
ap-9438	10	21	non	non	ADJ
ap-9438	10	22	-	-	ADJ
ap-9438	10	23	integer	integer	ADJ
ap-9438	10	24	orders	order	NOUN
ap-9438	10	25	.	.	PUNCT
ap-9438	11	1	unlike	unlike	ADP
ap-9438	11	2	traditional	traditional	ADJ
ap-9438	11	3	calculus	calculus	NOUN
ap-9438	11	4	,	,	PUNCT
ap-9438	11	5	which	which	PRON
ap-9438	11	6	deals	deal	VERB
ap-9438	11	7	exclusively	exclusively	ADV
ap-9438	11	8	with	with	ADP
ap-9438	11	9	integer	integer	NOUN
ap-9438	11	10	-	-	PUNCT
ap-9438	11	11	order	order	NOUN
ap-9438	11	12	derivatives	derivative	NOUN
ap-9438	11	13	and	and	CCONJ
ap-9438	11	14	integrals	integral	NOUN
ap-9438	11	15	,	,	PUNCT
ap-9438	11	16	fractional	fractional	ADJ
ap-9438	11	17	calculus	calculus	NOUN
ap-9438	11	18	allows	allow	VERB
ap-9438	11	19	for	for	ADP
ap-9438	11	20	operations	operation	NOUN
ap-9438	11	21	involving	involve	VERB
ap-9438	11	22	fractional	fractional	ADJ
ap-9438	11	23	powers	power	NOUN
ap-9438	11	24	of	of	ADP
ap-9438	11	25	the	the	DET
ap-9438	11	26	differentiation	differentiation	NOUN
ap-9438	11	27	operator	operator	NOUN
ap-9438	11	28	.	.	PUNCT
ap-9438	12	1	it	it	PRON
ap-9438	12	2	finds	find	VERB
ap-9438	12	3	applications	application	NOUN
ap-9438	12	4	in	in	ADP
ap-9438	12	5	various	various	ADJ
ap-9438	12	6	scientific	scientific	ADJ
ap-9438	12	7	fields	field	NOUN
ap-9438	12	8	,	,	PUNCT
ap-9438	12	9	such	such	ADJ
ap-9438	12	10	as	as	ADP
ap-9438	12	11	physics	physics	NOUN
ap-9438	12	12	,	,	PUNCT
ap-9438	12	13	engineering	engineering	NOUN
ap-9438	12	14	,	,	PUNCT
ap-9438	12	15	and	and	CCONJ
ap-9438	12	16	biology	biology	NOUN
ap-9438	12	17	,	,	PUNCT
ap-9438	12	18	where	where	SCONJ
ap-9438	12	19	phenomena	phenomenon	NOUN
ap-9438	12	20	exhibit	exhibit	VERB
ap-9438	12	21	fractal	fractal	ADJ
ap-9438	12	22	or	or	CCONJ
ap-9438	12	23	non	non	ADJ
ap-9438	12	24	-	-	ADJ
ap-9438	12	25	local	local	ADJ
ap-9438	12	26	behaviors	behavior	NOUN
ap-9438	12	27	that	that	PRON
ap-9438	12	28	can	can	AUX
ap-9438	12	29	not	not	PART
ap-9438	12	30	be	be	AUX
ap-9438	12	31	adequately	adequately	ADV
ap-9438	12	32	described	describe	VERB
ap-9438	12	33	by	by	ADP
ap-9438	12	34	classical	classical	ADJ
ap-9438	12	35	methods	method	NOUN
ap-9438	12	36	.	.	PUNCT
ap-9438	13	1	the	the	DET
ap-9438	13	2	study	study	NOUN
ap-9438	13	3	of	of	ADP
ap-9438	13	4	fractional	fractional	ADJ
ap-9438	13	5	calculus	calculus	NOUN
ap-9438	13	6	enables	enable	VERB
ap-9438	13	7	a	a	DET
ap-9438	13	8	deeper	deep	ADJ
ap-9438	13	9	understanding	understanding	NOUN
ap-9438	13	10	of	of	ADP
ap-9438	13	11	complex	complex	ADJ
ap-9438	13	12	systems	system	NOUN
ap-9438	13	13	and	and	CCONJ
ap-9438	13	14	provides	provide	VERB
ap-9438	13	15	powerful	powerful	ADJ
ap-9438	13	16	tools	tool	NOUN
ap-9438	13	17	for	for	ADP
ap-9438	13	18	modeling	modeling	NOUN
ap-9438	13	19	and	and	CCONJ
ap-9438	13	20	analyzing	analyze	VERB
ap-9438	13	21	intricate	intricate	ADJ
ap-9438	13	22	dynamical	dynamical	ADJ
ap-9438	13	23	processes	process	NOUN
ap-9438	13	24	with	with	ADP
ap-9438	13	25	fractional	fractional	ADJ
ap-9438	13	26	dimensions	dimension	NOUN
ap-9438	13	27	or	or	CCONJ
ap-9438	13	28	memory	memory	NOUN
ap-9438	13	29	effects	effect	NOUN
ap-9438	13	30	.	.	PUNCT
ap-9438	14	1	see	see	VERB
ap-9438	14	2	for	for	ADP
ap-9438	14	3	instance	instance	NOUN
ap-9438	14	4	[	[	X
ap-9438	14	5	1–4	1–4	NOUN
ap-9438	14	6	]	]	X
ap-9438	14	7	.	.	PUNCT
ap-9438	15	1	many	many	ADJ
ap-9438	15	2	researchers	researcher	NOUN
ap-9438	15	3	have	have	AUX
ap-9438	15	4	studied	study	VERB
ap-9438	15	5	different	different	ADJ
ap-9438	15	6	methods	method	NOUN
ap-9438	15	7	for	for	ADP
ap-9438	15	8	solving	solve	VERB
ap-9438	15	9	integral	integral	ADJ
ap-9438	15	10	equations	equation	NOUN
ap-9438	15	11	of	of	ADP
ap-9438	15	12	various	various	ADJ
ap-9438	15	13	types	type	NOUN
ap-9438	15	14	:	:	PUNCT
ap-9438	15	15	linear	linear	ADJ
ap-9438	15	16	,	,	PUNCT
ap-9438	15	17	nonlinear	nonlinear	ADJ
ap-9438	15	18	,	,	PUNCT
ap-9438	15	19	homogeneous	homogeneous	ADJ
ap-9438	15	20	,	,	PUNCT
ap-9438	15	21	heterogeneous	heterogeneous	ADJ
ap-9438	15	22	,	,	PUNCT
ap-9438	15	23	volterra	volterra	PROPN
ap-9438	15	24	,	,	PUNCT
ap-9438	15	25	friedholm	friedholm	NOUN
ap-9438	15	26	,	,	PUNCT
ap-9438	15	27	first	first	ADJ
ap-9438	15	28	kind	kind	NOUN
ap-9438	15	29	and	and	CCONJ
ap-9438	15	30	second	second	ADJ
ap-9438	15	31	kind	kind	NOUN
ap-9438	15	32	.	.	PUNCT
ap-9438	16	1	here	here	ADV
ap-9438	16	2	are	be	AUX
ap-9438	16	3	some	some	PRON
ap-9438	16	4	of	of	ADP
ap-9438	16	5	these	these	DET
ap-9438	16	6	studies	study	NOUN
ap-9438	16	7	,	,	PUNCT
ap-9438	16	8	in	in	ADP
ap-9438	16	9	2010	2010	NUM
ap-9438	16	10	mohammed	mohammed	PROPN
ap-9438	16	11	belmekki	belmekki	PROPN
ap-9438	16	12	and	and	CCONJ
ap-9438	16	13	mouffak	mouffak	PROPN
ap-9438	16	14	benchohra	benchohra	NOUN
ap-9438	16	15	established	establish	VERB
ap-9438	16	16	sufficient	sufficient	ADJ
ap-9438	16	17	conditions	condition	NOUN
ap-9438	16	18	for	for	ADP
ap-9438	16	19	the	the	DET
ap-9438	16	20	existence	existence	NOUN
ap-9438	16	21	and	and	CCONJ
ap-9438	16	22	uniqueness	uniqueness	NOUN
ap-9438	16	23	of	of	ADP
ap-9438	16	24	solutions	solution	NOUN
ap-9438	16	25	for	for	ADP
ap-9438	16	26	some	some	DET
ap-9438	16	27	nondensely	nondensely	ADV
ap-9438	16	28	defined	define	VERB
ap-9438	16	29	semilinear	semilinear	ADJ
ap-9438	16	30	functional	functional	ADJ
ap-9438	16	31	differential	differential	ADJ
ap-9438	16	32	equations	equation	NOUN
ap-9438	16	33	involving	involve	VERB
ap-9438	16	34	the	the	DET
ap-9438	16	35	riemann	riemann	PROPN
ap-9438	16	36	–	–	PUNCT
ap-9438	17	1	liouville	liouville	VERB
ap-9438	17	2	derivative	derivative	NOUN
ap-9438	17	3	[	[	X
ap-9438	17	4	1	1	NUM
ap-9438	17	5	]	]	PUNCT
ap-9438	17	6	,	,	PUNCT
ap-9438	17	7	in	in	ADP
ap-9438	17	8	2015	2015	NUM
ap-9438	17	9	salman	salman	PROPN
ap-9438	17	10	jahanshahi	jahanshahi	PROPN
ap-9438	17	11	,	,	PUNCT
ap-9438	17	12	esmail	esmail	ADJ
ap-9438	17	13	babolian	babolian	NOUN
ap-9438	17	14	,	,	PUNCT
ap-9438	17	15	delfim	delfim	NOUN
ap-9438	17	16	torres	torre	NOUN
ap-9438	17	17	,	,	PUNCT
ap-9438	17	18	and	and	CCONJ
ap-9438	17	19	alireza	alireza	PROPN
ap-9438	17	20	vahidi	vahidi	PROPN
ap-9438	17	21	introduced	introduce	VERB
ap-9438	17	22	a	a	DET
ap-9438	17	23	new	new	ADJ
ap-9438	17	24	method	method	NOUN
ap-9438	17	25	for	for	ADP
ap-9438	17	26	numerically	numerically	ADV
ap-9438	17	27	solving	solve	VERB
ap-9438	17	28	abel	abel	PROPN
ap-9438	17	29	integral	integral	ADJ
ap-9438	17	30	equations	equation	NOUN
ap-9438	17	31	of	of	ADP
ap-9438	17	32	the	the	DET
ap-9438	17	33	first	first	ADJ
ap-9438	17	34	kind	kind	NOUN
ap-9438	17	35	[	[	X
ap-9438	17	36	5	5	NUM
ap-9438	17	37	]	]	PUNCT
ap-9438	17	38	,	,	PUNCT
ap-9438	17	39	in	in	ADP
ap-9438	17	40	2020	2020	NUM
ap-9438	17	41	giuseppe	giuseppe	PROPN
ap-9438	17	42	pellicane	pellicane	PROPN
ap-9438	17	43	,	,	PUNCT
ap-9438	17	44	lloyd	lloyd	PROPN
ap-9438	17	45	lee	lee	PROPN
ap-9438	17	46	,	,	PUNCT
ap-9438	17	47	and	and	CCONJ
ap-9438	17	48	carlo	carlo	PROPN
ap-9438	17	49	caccamo	caccamo	PROPN
ap-9438	17	50	briefly	briefly	ADV
ap-9438	17	51	reviewed	review	VERB
ap-9438	17	52	the	the	DET
ap-9438	17	53	application	application	NOUN
ap-9438	17	54	of	of	ADP
ap-9438	17	55	integral	integral	ADJ
ap-9438	17	56	equation	equation	NOUN
ap-9438	17	57	theories	theory	NOUN
ap-9438	17	58	(	(	PUNCT
ap-9438	17	59	iets	iet	NOUN
ap-9438	17	60	)	)	PUNCT
ap-9438	17	61	of	of	ADP
ap-9438	17	62	the	the	DET
ap-9438	17	63	fluid	fluid	ADJ
ap-9438	17	64	state	state	NOUN
ap-9438	17	65	to	to	PART
ap-9438	17	66	predict	predict	VERB
ap-9438	17	67	fluid	fluid	ADJ
ap-9438	17	68	phase	phase	NOUN
ap-9438	17	69	equilibria	equilibrium	NOUN
ap-9438	17	70	of	of	ADP
ap-9438	17	71	simple	simple	ADJ
ap-9438	17	72	fluids	fluid	NOUN
ap-9438	17	73	[	[	X
ap-9438	17	74	6	6	NUM
ap-9438	17	75	]	]	PUNCT
ap-9438	17	76	,	,	PUNCT
ap-9438	17	77	in	in	ADP
ap-9438	17	78	2024	2024	NUM
ap-9438	17	79	mohammed	mohammed	PROPN
ap-9438	17	80	abdulshareef	abdulshareef	PROPN
ap-9438	17	81	hussein	hussein	PROPN
ap-9438	17	82	,	,	PUNCT
ap-9438	17	83	hassan	hassan	PROPN
ap-9438	17	84	kamil	kamil	PROPN
ap-9438	17	85	jassim	jassim	PROPN
ap-9438	17	86	,	,	PUNCT
ap-9438	17	87	and	and	CCONJ
ap-9438	17	88	ali	ali	PROPN
ap-9438	17	89	kareem	kareem	PROPN
ap-9438	17	90	jassim	jassim	PROPN
ap-9438	17	91	introduced	introduce	VERB
ap-9438	17	92	the	the	DET
ap-9438	17	93	hussein	hussein	NOUN
ap-9438	17	94	-	-	PUNCT
ap-9438	17	95	jassim	jassim	NOUN
ap-9438	17	96	method	method	NOUN
ap-9438	17	97	(	(	PUNCT
ap-9438	17	98	hj	hj	NOUN
ap-9438	17	99	-	-	PUNCT
ap-9438	17	100	method	method	NOUN
ap-9438	17	101	)	)	PUNCT
ap-9438	17	102	for	for	ADP
ap-9438	17	103	solving	solve	VERB
ap-9438	17	104	volterra	volterra	NOUN
ap-9438	17	105	integral	integral	ADJ
ap-9438	17	106	equations	equation	NOUN
ap-9438	17	107	of	of	ADP
ap-9438	17	108	the	the	DET
ap-9438	17	109	second	second	ADJ
ap-9438	17	110	kind	kind	NOUN
ap-9438	17	111	(	(	PUNCT
ap-9438	17	112	viesks	viesk	NOUN
ap-9438	17	113	)	)	PUNCT
ap-9438	18	1	[	[	X
ap-9438	18	2	7	7	NUM
ap-9438	18	3	]	]	PUNCT
ap-9438	18	4	,	,	PUNCT
ap-9438	18	5	in	in	ADP
ap-9438	18	6	2024	2024	NUM
ap-9438	18	7	hamid	hamid	PROPN
ap-9438	18	8	mottaghi	mottaghi	PROPN
ap-9438	18	9	golshan	golshan	PROPN
ap-9438	18	10	proposed	propose	VERB
ap-9438	18	11	a	a	DET
ap-9438	18	12	numerical	numerical	ADJ
ap-9438	18	13	iterative	iterative	NOUN
ap-9438	18	14	method	method	NOUN
ap-9438	18	15	to	to	PART
ap-9438	18	16	solve	solve	VERB
ap-9438	18	17	multidimensional	multidimensional	ADJ
ap-9438	18	18	integral	integral	ADJ
ap-9438	18	19	equations	equation	NOUN
ap-9438	18	20	based	base	VERB
ap-9438	18	21	on	on	ADP
ap-9438	18	22	picard	picard	PROPN
ap-9438	18	23	iteration	iteration	PROPN
ap-9438	18	24	and	and	CCONJ
ap-9438	18	25	newton	newton	PROPN
ap-9438	18	26	–	–	PUNCT
ap-9438	18	27	cotes	cote	NOUN
ap-9438	18	28	rules	rule	NOUN
ap-9438	18	29	in	in	ADP
ap-9438	18	30	a	a	DET
ap-9438	18	31	cubic	cubic	ADJ
ap-9438	18	32	domain	domain	NOUN
ap-9438	18	33	[	[	X
ap-9438	18	34	8	8	NUM
ap-9438	18	35	]	]	PUNCT
ap-9438	18	36	.	.	PUNCT
ap-9438	19	1	in	in	ADP
ap-9438	19	2	addition	addition	NOUN
ap-9438	19	3	to	to	ADP
ap-9438	19	4	other	other	ADJ
ap-9438	19	5	studies	study	NOUN
ap-9438	19	6	[	[	X
ap-9438	19	7	9–16	9–16	NOUN
ap-9438	19	8	]	]	PUNCT
ap-9438	19	9	.	.	PUNCT
ap-9438	20	1	the	the	DET
ap-9438	20	2	fractional	fractional	PROPN
ap-9438	20	3	leibniz	leibniz	PROPN
ap-9438	20	4	integral	integral	ADJ
ap-9438	20	5	rules	rule	NOUN
ap-9438	20	6	form	form	VERB
ap-9438	20	7	a	a	DET
ap-9438	20	8	key	key	ADJ
ap-9438	20	9	framework	framework	NOUN
ap-9438	20	10	in	in	ADP
ap-9438	20	11	the	the	DET
ap-9438	20	12	field	field	NOUN
ap-9438	20	13	of	of	ADP
ap-9438	20	14	fractional	fractional	ADJ
ap-9438	20	15	calculus	calculus	NOUN
ap-9438	20	16	,	,	PUNCT
ap-9438	20	17	notably	notably	ADV
ap-9438	20	18	for	for	ADP
ap-9438	20	19	the	the	DET
ap-9438	20	20	riemann	riemann	PROPN
ap-9438	20	21	-	-	PUNCT
ap-9438	20	22	liouville	liouville	PROPN
ap-9438	20	23	and	and	CCONJ
ap-9438	20	24	caputo	caputo	PROPN
ap-9438	20	25	fractional	fractional	ADJ
ap-9438	20	26	derivatives	derivative	NOUN
ap-9438	20	27	,	,	PUNCT
ap-9438	20	28	demonstrating	demonstrate	VERB
ap-9438	20	29	their	their	PRON
ap-9438	20	30	importance	importance	NOUN
ap-9438	20	31	in	in	ADP
ap-9438	20	32	a	a	DET
ap-9438	20	33	variety	variety	NOUN
ap-9438	20	34	of	of	ADP
ap-9438	20	35	applications	application	NOUN
ap-9438	20	36	.	.	PUNCT
ap-9438	21	1	these	these	DET
ap-9438	21	2	rules	rule	NOUN
ap-9438	21	3	are	be	AUX
ap-9438	21	4	an	an	DET
ap-9438	21	5	extension	extension	NOUN
ap-9438	21	6	of	of	ADP
ap-9438	21	7	the	the	DET
ap-9438	21	8	traditional	traditional	ADJ
ap-9438	21	9	leibniz	leibniz	PROPN
ap-9438	21	10	differentiation	differentiation	NOUN
ap-9438	21	11	rule	rule	NOUN
ap-9438	21	12	,	,	PUNCT
ap-9438	21	13	modified	modify	VERB
ap-9438	21	14	to	to	PART
ap-9438	21	15	include	include	VERB
ap-9438	21	16	fractional	fractional	ADJ
ap-9438	21	17	orders	order	NOUN
ap-9438	21	18	.	.	PUNCT
ap-9438	22	1	specifically	specifically	ADV
ap-9438	22	2	,	,	PUNCT
ap-9438	22	3	they	they	PRON
ap-9438	22	4	allow	allow	VERB
ap-9438	22	5	for	for	ADP
ap-9438	22	6	the	the	DET
ap-9438	22	7	differentiation	differentiation	NOUN
ap-9438	22	8	of	of	ADP
ap-9438	22	9	intricate	intricate	ADJ
ap-9438	22	10	compositions	composition	NOUN
ap-9438	22	11	and	and	CCONJ
ap-9438	22	12	products	product	NOUN
ap-9438	22	13	using	use	VERB
ap-9438	22	14	fractional	fractional	ADJ
ap-9438	22	15	derivatives	derivative	NOUN
ap-9438	22	16	,	,	PUNCT
ap-9438	22	17	giving	give	VERB
ap-9438	22	18	a	a	DET
ap-9438	22	19	methodical	methodical	ADJ
ap-9438	22	20	way	way	NOUN
ap-9438	22	21	to	to	ADP
ap-9438	22	22	unraveling	unravel	VERB
ap-9438	22	23	the	the	DET
ap-9438	22	24	complexity	complexity	NOUN
ap-9438	22	25	of	of	ADP
ap-9438	22	26	these	these	DET
ap-9438	22	27	derivatives	derivative	NOUN
ap-9438	22	28	.	.	PUNCT
ap-9438	23	1	in	in	ADP
ap-9438	23	2	the	the	DET
ap-9438	23	3	context	context	NOUN
ap-9438	23	4	of	of	ADP
ap-9438	23	5	the	the	DET
ap-9438	23	6	riemann	riemann	PROPN
ap-9438	23	7	-	-	PUNCT
ap-9438	23	8	liouville	liouville	VERB
ap-9438	23	9	fractional	fractional	ADJ
ap-9438	23	10	derivative	derivative	NOUN
ap-9438	23	11	,	,	PUNCT
ap-9438	23	12	the	the	DET
ap-9438	23	13	fractional	fractional	PROPN
ap-9438	23	14	leibniz	leibniz	PROPN
ap-9438	23	15	rules	rule	NOUN
ap-9438	23	16	are	be	AUX
ap-9438	23	17	critical	critical	ADJ
ap-9438	23	18	in	in	ADP
ap-9438	23	19	assessing	assess	VERB
ap-9438	23	20	mixed	mixed	ADJ
ap-9438	23	21	-	-	PUNCT
ap-9438	23	22	order	order	NOUN
ap-9438	23	23	derivatives	derivative	NOUN
ap-9438	23	24	of	of	ADP
ap-9438	23	25	complex	complex	ADJ
ap-9438	23	26	functions	function	NOUN
ap-9438	23	27	,	,	PUNCT
ap-9438	23	28	shedding	shed	VERB
ap-9438	23	29	light	light	NOUN
ap-9438	23	30	on	on	ADP
ap-9438	23	31	their	their	PRON
ap-9438	23	32	complicated	complicated	ADJ
ap-9438	23	33	properties	property	NOUN
ap-9438	23	34	[	[	X
ap-9438	23	35	17–19	17–19	NUM
ap-9438	23	36	]	]	PUNCT
ap-9438	23	37	.	.	PUNCT
ap-9438	24	1	this	this	DET
ap-9438	24	2	paper	paper	NOUN
ap-9438	24	3	aims	aim	VERB
ap-9438	24	4	to	to	PART
ap-9438	24	5	introduce	introduce	VERB
ap-9438	24	6	a	a	DET
ap-9438	24	7	new	new	ADJ
ap-9438	24	8	analytic	analytic	ADJ
ap-9438	24	9	method	method	NOUN
ap-9438	24	10	for	for	ADP
ap-9438	24	11	obtaining	obtain	VERB
ap-9438	24	12	the	the	DET
ap-9438	24	13	exact	exact	ADJ
ap-9438	24	14	solution	solution	NOUN
ap-9438	24	15	of	of	ADP
ap-9438	24	16	the	the	DET
ap-9438	24	17	following	follow	VERB
ap-9438	24	18	nonlinear	nonlinear	PROPN
ap-9438	24	19	fractional	fractional	PROPN
ap-9438	24	20	volterra	volterra	PROPN
ap-9438	24	21	integral	integral	ADJ
ap-9438	24	22	equation	equation	NOUN
ap-9438	24	23	of	of	ADP
ap-9438	24	24	the	the	DET
ap-9438	24	25	first	first	ADJ
ap-9438	24	26	kind	kind	NOUN
ap-9438	24	27	using	use	VERB
ap-9438	24	28	fractional	fractional	ADJ
ap-9438	24	29	leibniz	leibniz	PROPN
ap-9438	24	30	integral	integral	ADJ
ap-9438	24	31	rules	rule	NOUN
ap-9438	24	32	:	:	PUNCT
ap-9438	24	33	f(t	f(t	NOUN
ap-9438	24	34	)	)	PUNCT
ap-9438	25	1	=	=	SYM
ap-9438	25	2	∫	∫	PROPN
ap-9438	25	3	t	t	PROPN
ap-9438	25	4	a	a	DET
ap-9438	25	5	(	(	PUNCT
ap-9438	25	6	t−	t−	PROPN
ap-9438	25	7	s)γg(s)n(ψ(s))ds	s)γg(s)n(ψ(s))ds	PROPN
ap-9438	25	8	,	,	PUNCT
ap-9438	25	9	t	t	X
ap-9438	25	10	>	>	X
ap-9438	25	11	a	a	PRON
ap-9438	25	12	,	,	PUNCT
ap-9438	25	13	(	(	PUNCT
ap-9438	25	14	1	1	X
ap-9438	25	15	)	)	PUNCT
ap-9438	25	16	where	where	SCONJ
ap-9438	25	17	f(t	f(t	NOUN
ap-9438	25	18	)	)	PUNCT
ap-9438	25	19	and	and	CCONJ
ap-9438	25	20	g(s	g(	NOUN
ap-9438	25	21	)	)	PUNCT
ap-9438	25	22	are	be	AUX
ap-9438	25	23	given	give	VERB
ap-9438	25	24	,	,	PUNCT
ap-9438	25	25	ψ(s	ψ(s	PROPN
ap-9438	25	26	)	)	PUNCT
ap-9438	25	27	is	be	AUX
ap-9438	25	28	an	an	DET
ap-9438	25	29	unknown	unknown	ADJ
ap-9438	25	30	function	function	NOUN
ap-9438	25	31	,	,	PUNCT
ap-9438	25	32	n	n	X
ap-9438	25	33	is	be	AUX
ap-9438	25	34	a	a	DET
ap-9438	25	35	nonlinear	nonlinear	ADJ
ap-9438	25	36	term	term	NOUN
ap-9438	25	37	,	,	PUNCT
ap-9438	25	38	and	and	CCONJ
ap-9438	25	39	γ	γ	X
ap-9438	25	40	≥	≥	X
ap-9438	25	41	0	0	NUM
ap-9438	25	42	is	be	AUX
ap-9438	25	43	the	the	DET
ap-9438	25	44	414	414	NUM
ap-9438	25	45	https://doi.org/10.14311/ap.2024.64.0414	https://doi.org/10.14311/ap.2024.64.0414	ADJ
ap-9438	25	46	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
ap-9438	25	47	https://www.cvut.cz/en	https://www.cvut.cz/en	PROPN
ap-9438	25	48	vol	vol	NOUN
ap-9438	25	49	.	.	PROPN
ap-9438	26	1	64	64	NUM
ap-9438	26	2	no	no	NOUN
ap-9438	26	3	.	.	PUNCT
ap-9438	27	1	5/2024	5/2024	NUM
ap-9438	27	2	a	a	DET
ap-9438	27	3	novel	novel	ADJ
ap-9438	27	4	approach	approach	NOUN
ap-9438	27	5	to	to	ADP
ap-9438	27	6	nonlinear	nonlinear	ADJ
ap-9438	27	7	fractional	fractional	PROPN
ap-9438	27	8	volterra	volterra	PROPN
ap-9438	27	9	integral	integral	ADJ
ap-9438	27	10	equations	equation	NOUN
ap-9438	27	11	order	order	NOUN
ap-9438	27	12	of	of	ADP
ap-9438	27	13	the	the	DET
ap-9438	27	14	integral	integral	ADJ
ap-9438	27	15	equation	equation	NOUN
ap-9438	27	16	.	.	PUNCT
ap-9438	28	1	the	the	DET
ap-9438	28	2	main	main	ADJ
ap-9438	28	3	idea	idea	NOUN
ap-9438	28	4	here	here	ADV
ap-9438	28	5	is	be	AUX
ap-9438	28	6	to	to	PART
ap-9438	28	7	develop	develop	VERB
ap-9438	28	8	an	an	DET
ap-9438	28	9	analytical	analytical	ADJ
ap-9438	28	10	approach	approach	NOUN
ap-9438	28	11	to	to	PART
ap-9438	28	12	solve	solve	VERB
ap-9438	28	13	this	this	DET
ap-9438	28	14	type	type	NOUN
ap-9438	28	15	of	of	ADP
ap-9438	28	16	equation	equation	NOUN
ap-9438	28	17	using	use	VERB
ap-9438	28	18	concepts	concept	NOUN
ap-9438	28	19	of	of	ADP
ap-9438	28	20	fractional	fractional	ADJ
ap-9438	28	21	calculus	calculus	NOUN
ap-9438	28	22	and	and	CCONJ
ap-9438	28	23	to	to	PART
ap-9438	28	24	apply	apply	VERB
ap-9438	28	25	fractional	fractional	ADJ
ap-9438	28	26	leibniz	leibniz	PROPN
ap-9438	28	27	integral	integral	ADJ
ap-9438	28	28	rules	rule	NOUN
ap-9438	28	29	to	to	PART
ap-9438	28	30	arrive	arrive	VERB
ap-9438	28	31	at	at	ADP
ap-9438	28	32	exact	exact	ADJ
ap-9438	28	33	solutions	solution	NOUN
ap-9438	28	34	.	.	PUNCT
ap-9438	29	1	this	this	DET
ap-9438	29	2	research	research	NOUN
ap-9438	29	3	paper	paper	NOUN
ap-9438	29	4	may	may	AUX
ap-9438	29	5	contribute	contribute	VERB
ap-9438	29	6	to	to	ADP
ap-9438	29	7	a	a	DET
ap-9438	29	8	better	well	ADJ
ap-9438	29	9	understanding	understanding	NOUN
ap-9438	29	10	of	of	ADP
ap-9438	29	11	how	how	SCONJ
ap-9438	29	12	to	to	PART
ap-9438	29	13	solve	solve	VERB
ap-9438	29	14	volterra	volterra	NOUN
ap-9438	29	15	integral	integral	ADJ
ap-9438	29	16	equations	equation	NOUN
ap-9438	29	17	of	of	ADP
ap-9438	29	18	the	the	DET
ap-9438	29	19	first	first	ADJ
ap-9438	29	20	kind	kind	NOUN
ap-9438	29	21	and	and	CCONJ
ap-9438	29	22	may	may	AUX
ap-9438	29	23	have	have	VERB
ap-9438	29	24	practical	practical	ADJ
ap-9438	29	25	applications	application	NOUN
ap-9438	29	26	in	in	ADP
ap-9438	29	27	various	various	ADJ
ap-9438	29	28	fields	field	NOUN
ap-9438	29	29	,	,	PUNCT
ap-9438	29	30	such	such	ADJ
ap-9438	29	31	as	as	ADP
ap-9438	29	32	mthematics	mthematic	NOUN
ap-9438	29	33	,	,	PUNCT
ap-9438	29	34	science	science	NOUN
ap-9438	29	35	,	,	PUNCT
ap-9438	29	36	engineering	engineering	NOUN
ap-9438	29	37	,	,	PUNCT
ap-9438	29	38	and	and	CCONJ
ap-9438	29	39	others	other	NOUN
ap-9438	29	40	.	.	PUNCT
ap-9438	30	1	following	follow	VERB
ap-9438	30	2	is	be	AUX
ap-9438	30	3	an	an	DET
ap-9438	30	4	outline	outline	NOUN
ap-9438	30	5	of	of	ADP
ap-9438	30	6	the	the	DET
ap-9438	30	7	paper	paper	NOUN
ap-9438	30	8	.	.	PUNCT
ap-9438	31	1	in	in	ADP
ap-9438	31	2	the	the	DET
ap-9438	31	3	second	second	ADJ
ap-9438	31	4	section	section	NOUN
ap-9438	31	5	of	of	ADP
ap-9438	31	6	this	this	DET
ap-9438	31	7	paper	paper	NOUN
ap-9438	31	8	,	,	PUNCT
ap-9438	31	9	we	we	PRON
ap-9438	31	10	present	present	VERB
ap-9438	31	11	several	several	ADJ
ap-9438	31	12	mathematical	mathematical	ADJ
ap-9438	31	13	concepts	concept	NOUN
ap-9438	31	14	related	relate	VERB
ap-9438	31	15	to	to	ADP
ap-9438	31	16	our	our	PRON
ap-9438	31	17	study	study	NOUN
ap-9438	31	18	,	,	PUNCT
ap-9438	31	19	in	in	ADP
ap-9438	31	20	the	the	DET
ap-9438	31	21	third	third	ADJ
ap-9438	31	22	section	section	NOUN
ap-9438	31	23	,	,	PUNCT
ap-9438	31	24	we	we	PRON
ap-9438	31	25	discuss	discuss	VERB
ap-9438	31	26	the	the	DET
ap-9438	31	27	algorithm	algorithm	NOUN
ap-9438	31	28	and	and	CCONJ
ap-9438	31	29	analysis	analysis	NOUN
ap-9438	31	30	of	of	ADP
ap-9438	31	31	the	the	DET
ap-9438	31	32	new	new	ADJ
ap-9438	31	33	method	method	NOUN
ap-9438	31	34	,	,	PUNCT
ap-9438	31	35	in	in	ADP
ap-9438	31	36	the	the	DET
ap-9438	31	37	fourth	fourth	ADJ
ap-9438	31	38	section	section	NOUN
ap-9438	31	39	we	we	PRON
ap-9438	31	40	apply	apply	VERB
ap-9438	31	41	this	this	DET
ap-9438	31	42	method	method	NOUN
ap-9438	31	43	to	to	ADP
ap-9438	31	44	several	several	ADJ
ap-9438	31	45	illustrative	illustrative	ADJ
ap-9438	31	46	examples	example	NOUN
ap-9438	31	47	to	to	PART
ap-9438	31	48	ensure	ensure	VERB
ap-9438	31	49	the	the	DET
ap-9438	31	50	effectiveness	effectiveness	NOUN
ap-9438	31	51	of	of	ADP
ap-9438	31	52	the	the	DET
ap-9438	31	53	method	method	NOUN
ap-9438	31	54	,	,	PUNCT
ap-9438	31	55	and	and	CCONJ
ap-9438	31	56	in	in	ADP
ap-9438	31	57	the	the	DET
ap-9438	31	58	final	final	ADJ
ap-9438	31	59	section	section	NOUN
ap-9438	31	60	,	,	PUNCT
ap-9438	31	61	we	we	PRON
ap-9438	31	62	review	review	VERB
ap-9438	31	63	the	the	DET
ap-9438	31	64	results	result	NOUN
ap-9438	31	65	and	and	CCONJ
ap-9438	31	66	recommendations	recommendation	NOUN
ap-9438	31	67	.	.	PUNCT
ap-9438	32	1	2	2	X
ap-9438	32	2	.	.	X
ap-9438	32	3	mathematical	mathematical	ADJ
ap-9438	32	4	preliminaries	preliminary	NOUN
ap-9438	32	5	definition	definition	NOUN
ap-9438	32	6	1	1	NUM
ap-9438	32	7	(	(	PUNCT
ap-9438	32	8	the	the	DET
ap-9438	32	9	gamma	gamma	PROPN
ap-9438	32	10	function	function	NOUN
ap-9438	32	11	[	[	X
ap-9438	32	12	20	20	NUM
ap-9438	32	13	]	]	NUM
ap-9438	32	14	)	)	PUNCT
ap-9438	32	15	.	.	PUNCT
ap-9438	33	1	the	the	DET
ap-9438	33	2	gamma	gamma	PROPN
ap-9438	33	3	function	function	PROPN
ap-9438	33	4	,	,	PUNCT
ap-9438	33	5	denoted	denote	VERB
ap-9438	33	6	by	by	ADP
ap-9438	33	7	γ(γ	γ(γ	NOUN
ap-9438	33	8	)	)	PUNCT
ap-9438	33	9	is	be	AUX
ap-9438	33	10	a	a	DET
ap-9438	33	11	mathematical	mathematical	ADJ
ap-9438	33	12	function	function	NOUN
ap-9438	33	13	that	that	PRON
ap-9438	33	14	generalises	generalise	VERB
ap-9438	33	15	the	the	DET
ap-9438	33	16	concept	concept	NOUN
ap-9438	33	17	of	of	ADP
ap-9438	33	18	factorial	factorial	NOUN
ap-9438	33	19	to	to	ADP
ap-9438	33	20	non	non	ADJ
ap-9438	33	21	-	-	ADJ
ap-9438	33	22	integer	integer	ADJ
ap-9438	33	23	values	value	NOUN
ap-9438	33	24	.	.	PUNCT
ap-9438	34	1	it	it	PRON
ap-9438	34	2	is	be	AUX
ap-9438	34	3	defined	define	VERB
ap-9438	34	4	as	as	ADP
ap-9438	34	5	follows	follow	VERB
ap-9438	34	6	:	:	PUNCT
ap-9438	34	7	γ(γ	γ(γ	NOUN
ap-9438	34	8	)	)	PUNCT
ap-9438	35	1	=	=	SYM
ap-9438	35	2	∫	∫	PROPN
ap-9438	36	1	∞	∞	PROPN
ap-9438	36	2	0	0	NUM
ap-9438	36	3	tγ−1e−tdt	tγ−1e−tdt	NOUN
ap-9438	36	4	,	,	PUNCT
ap-9438	36	5	re(γ	re(γ	NOUN
ap-9438	36	6	)	)	PUNCT
ap-9438	36	7	>	>	X
ap-9438	36	8	0	0	X
ap-9438	36	9	.	.	PUNCT
ap-9438	37	1	(	(	PUNCT
ap-9438	37	2	2	2	X
ap-9438	37	3	)	)	PUNCT
ap-9438	37	4	these	these	PRON
ap-9438	37	5	are	be	AUX
ap-9438	37	6	the	the	DET
ap-9438	37	7	two	two	NUM
ap-9438	37	8	important	important	ADJ
ap-9438	37	9	properties	property	NOUN
ap-9438	37	10	of	of	ADP
ap-9438	37	11	the	the	DET
ap-9438	37	12	gamma	gamma	NOUN
ap-9438	37	13	function	function	NOUN
ap-9438	37	14	[	[	X
ap-9438	37	15	20	20	NUM
ap-9438	37	16	]	]	SYM
ap-9438	37	17	:	:	PUNCT
ap-9438	37	18	1	1	X
ap-9438	37	19	.	.	X
ap-9438	37	20	γ(γ	γ(γ	PROPN
ap-9438	38	1	+	+	CCONJ
ap-9438	38	2	1	1	X
ap-9438	38	3	)	)	PUNCT
ap-9438	38	4	=	=	NOUN
ap-9438	38	5	γγ(γ	γγ(γ	PRON
ap-9438	38	6	)	)	PUNCT
ap-9438	38	7	,	,	PUNCT
ap-9438	38	8	γ	γ	X
ap-9438	38	9	>	>	X
ap-9438	38	10	0	0	NUM
ap-9438	38	11	,	,	PUNCT
ap-9438	38	12	(	(	PUNCT
ap-9438	38	13	3	3	NUM
ap-9438	38	14	)	)	PUNCT
ap-9438	38	15	2	2	NUM
ap-9438	38	16	.	.	X
ap-9438	38	17	γ(γ	γ(γ	PROPN
ap-9438	38	18	+	+	CCONJ
ap-9438	38	19	1	1	X
ap-9438	38	20	)	)	PUNCT
ap-9438	38	21	=	=	SYM
ap-9438	38	22	γ	γ	X
ap-9438	38	23	!	!	PROPN
ap-9438	38	24	,	,	PUNCT
ap-9438	38	25	γ	γ	PROPN
ap-9438	38	26	is	be	AUX
ap-9438	38	27	a	a	DET
ap-9438	38	28	non	non	ADJ
ap-9438	38	29	negative	negative	ADJ
ap-9438	38	30	integer	integer	NOUN
ap-9438	38	31	number	number	NOUN
ap-9438	38	32	.	.	PUNCT
ap-9438	39	1	(	(	PUNCT
ap-9438	39	2	4	4	X
ap-9438	39	3	)	)	PUNCT
ap-9438	39	4	definition	definition	NOUN
ap-9438	39	5	2	2	NUM
ap-9438	39	6	(	(	PUNCT
ap-9438	39	7	the	the	DET
ap-9438	39	8	mittag	mittag	ADJ
ap-9438	39	9	-	-	PUNCT
ap-9438	39	10	leffler	leffler	NOUN
ap-9438	39	11	function	function	NOUN
ap-9438	39	12	[	[	X
ap-9438	39	13	21	21	NUM
ap-9438	39	14	]	]	PUNCT
ap-9438	39	15	)	)	PUNCT
ap-9438	39	16	.	.	PUNCT
ap-9438	40	1	the	the	DET
ap-9438	40	2	mittag	mittag	ADJ
ap-9438	40	3	-	-	PUNCT
ap-9438	40	4	leffler	leffler	NOUN
ap-9438	40	5	function	function	NOUN
ap-9438	40	6	of	of	ADP
ap-9438	40	7	one	one	NUM
ap-9438	40	8	parameter	parameter	NOUN
ap-9438	40	9	is	be	AUX
ap-9438	40	10	defined	define	VERB
ap-9438	40	11	by	by	ADP
ap-9438	40	12	the	the	DET
ap-9438	40	13	following	follow	VERB
ap-9438	40	14	power	power	NOUN
ap-9438	40	15	series	series	NOUN
ap-9438	40	16	:	:	PUNCT
ap-9438	40	17	eγ(z	eγ(z	NUM
ap-9438	40	18	)	)	PUNCT
ap-9438	40	19	=	=	PUNCT
ap-9438	41	1	∞∑	∞∑	NUM
ap-9438	41	2	n=0	n=0	NUM
ap-9438	41	3	zn	zn	NOUN
ap-9438	41	4	γ(nγ	γ(nγ	NOUN
ap-9438	41	5	+	+	CCONJ
ap-9438	41	6	1	1	NUM
ap-9438	41	7	)	)	PUNCT
ap-9438	41	8	,	,	PUNCT
ap-9438	41	9	re(γ	re(γ	NOUN
ap-9438	41	10	)	)	PUNCT
ap-9438	41	11	>	>	X
ap-9438	41	12	0	0	NUM
ap-9438	41	13	,	,	PUNCT
ap-9438	41	14	(	(	PUNCT
ap-9438	41	15	5	5	NUM
ap-9438	41	16	)	)	PUNCT
ap-9438	41	17	and	and	CCONJ
ap-9438	41	18	of	of	ADP
ap-9438	41	19	two	two	NUM
ap-9438	41	20	parameters	parameter	NOUN
ap-9438	41	21	is	be	AUX
ap-9438	41	22	defined	define	VERB
ap-9438	41	23	by	by	ADP
ap-9438	41	24	:	:	PUNCT
ap-9438	41	25	eγ	eγ	PROPN
ap-9438	41	26	,	,	PUNCT
ap-9438	41	27	β(z	β(z	PROPN
ap-9438	41	28	)	)	PUNCT
ap-9438	41	29	=	=	PUNCT
ap-9438	42	1	∞∑	∞∑	NUM
ap-9438	42	2	n=0	n=0	NUM
ap-9438	42	3	zn	zn	NUM
ap-9438	42	4	γ(γn+	γ(γn+	X
ap-9438	42	5	β	β	NOUN
ap-9438	42	6	)	)	PUNCT
ap-9438	42	7	,	,	PUNCT
ap-9438	42	8	re(γ	re(γ	NOUN
ap-9438	42	9	)	)	PUNCT
ap-9438	42	10	>	>	X
ap-9438	42	11	0	0	NUM
ap-9438	42	12	,	,	PUNCT
ap-9438	42	13	re(β	re(β	X
ap-9438	42	14	)	)	PUNCT
ap-9438	42	15	>	>	X
ap-9438	42	16	0	0	X
ap-9438	42	17	.	.	PUNCT
ap-9438	43	1	(	(	PUNCT
ap-9438	43	2	6	6	NUM
ap-9438	43	3	)	)	PUNCT
ap-9438	43	4	theorem	theorem	NOUN
ap-9438	43	5	1	1	NUM
ap-9438	44	1	[	[	X
ap-9438	44	2	22	22	NUM
ap-9438	44	3	]	]	PUNCT
ap-9438	44	4	.	.	PUNCT
ap-9438	45	1	let	let	VERB
ap-9438	45	2	eγ(z	eγ(z	NUM
ap-9438	45	3	)	)	PUNCT
ap-9438	45	4	and	and	CCONJ
ap-9438	45	5	eγ	eγ	PROPN
ap-9438	45	6	,	,	PUNCT
ap-9438	45	7	β(z	β(z	PROPN
ap-9438	45	8	)	)	PUNCT
ap-9438	45	9	are	be	AUX
ap-9438	45	10	the	the	DET
ap-9438	45	11	mittag	mittag	ADJ
ap-9438	45	12	-	-	PUNCT
ap-9438	45	13	leffler	leffler	NOUN
ap-9438	45	14	functions	function	NOUN
ap-9438	45	15	of	of	ADP
ap-9438	45	16	one	one	NUM
ap-9438	45	17	and	and	CCONJ
ap-9438	45	18	two	two	NUM
ap-9438	45	19	parameters	parameter	NOUN
ap-9438	45	20	,	,	PUNCT
ap-9438	45	21	respectively	respectively	ADV
ap-9438	45	22	.	.	PUNCT
ap-9438	46	1	then	then	ADV
ap-9438	46	2	:	:	PUNCT
ap-9438	46	3	1	1	X
ap-9438	46	4	.	.	X
ap-9438	46	5	eγ,1(z	eγ,1(z	PROPN
ap-9438	46	6	)	)	PUNCT
ap-9438	46	7	=	=	SYM
ap-9438	46	8	eγ(z	eγ(z	NUM
ap-9438	46	9	)	)	PUNCT
ap-9438	46	10	,	,	PUNCT
ap-9438	46	11	(	(	PUNCT
ap-9438	46	12	7	7	X
ap-9438	46	13	)	)	SYM
ap-9438	46	14	2	2	NUM
ap-9438	46	15	.	.	PUNCT
ap-9438	47	1	e1,1(z	e1,1(z	PROPN
ap-9438	47	2	)	)	PUNCT
ap-9438	47	3	=	=	SYM
ap-9438	47	4	ez	ez	PROPN
ap-9438	47	5	,	,	PUNCT
ap-9438	47	6	(	(	PUNCT
ap-9438	47	7	8)	8)	NUM
ap-9438	47	8	3	3	NUM
ap-9438	47	9	.	.	PUNCT
ap-9438	47	10	e2,2(−z2	e2,2(−z2	NOUN
ap-9438	47	11	)	)	PUNCT
ap-9438	48	1	=	=	PUNCT
ap-9438	48	2	sin	sin	NOUN
ap-9438	48	3	z	z	PROPN
ap-9438	48	4	z	z	NOUN
ap-9438	48	5	,	,	PUNCT
ap-9438	48	6	(	(	PUNCT
ap-9438	48	7	9	9	NUM
ap-9438	48	8	)	)	SYM
ap-9438	48	9	4	4	NUM
ap-9438	48	10	.	.	X
ap-9438	48	11	e2,1(−z2	e2,1(−z2	NOUN
ap-9438	48	12	)	)	PUNCT
ap-9438	49	1	=	=	PUNCT
ap-9438	49	2	cos	cos	PROPN
ap-9438	49	3	z.	z.	PROPN
ap-9438	49	4	(	(	PUNCT
ap-9438	49	5	10	10	NUM
ap-9438	49	6	)	)	PUNCT
ap-9438	49	7	definition	definition	NOUN
ap-9438	49	8	3	3	NUM
ap-9438	49	9	(	(	PUNCT
ap-9438	49	10	the	the	DET
ap-9438	49	11	riemann	riemann	PROPN
ap-9438	49	12	-	-	PUNCT
ap-9438	49	13	liouville	liouville	VERB
ap-9438	49	14	integral	integral	ADJ
ap-9438	49	15	[	[	X
ap-9438	49	16	22	22	NUM
ap-9438	49	17	,	,	PUNCT
ap-9438	49	18	23	23	NUM
ap-9438	49	19	]	]	PUNCT
ap-9438	49	20	)	)	PUNCT
ap-9438	49	21	.	.	PUNCT
ap-9438	50	1	the	the	DET
ap-9438	50	2	riemann	riemann	PROPN
ap-9438	50	3	-	-	PUNCT
ap-9438	50	4	liouville	liouville	VERB
ap-9438	50	5	integral	integral	ADJ
ap-9438	50	6	of	of	ADP
ap-9438	50	7	fractional	fractional	ADJ
ap-9438	50	8	order	order	NOUN
ap-9438	50	9	γ	γ	X
ap-9438	50	10	>	>	X
ap-9438	50	11	0	0	NUM
ap-9438	50	12	of	of	ADP
ap-9438	50	13	a	a	DET
ap-9438	50	14	f	f	PROPN
ap-9438	50	15	∈	∈	PROPN
ap-9438	50	16	c[a	c[a	PROPN
ap-9438	50	17	,	,	PUNCT
ap-9438	50	18	b	b	AUX
ap-9438	50	19	]	]	PUNCT
ap-9438	50	20	is	be	AUX
ap-9438	50	21	defined	define	VERB
ap-9438	50	22	as	as	SCONJ
ap-9438	50	23	follows	follow	VERB
ap-9438	50	24	:	:	PUNCT
ap-9438	51	1	ai	ai	VERB
ap-9438	51	2	γ	γ	PROPN
ap-9438	51	3	t	t	PROPN
ap-9438	51	4	f(t	f(t	PROPN
ap-9438	51	5	)	)	PUNCT
ap-9438	51	6	=	=	SYM
ap-9438	51	7	1	1	NUM
ap-9438	51	8	γ(γ	γ(γ	NOUN
ap-9438	51	9	)	)	PUNCT
ap-9438	51	10	∫	∫	PROPN
ap-9438	51	11	t	t	PROPN
ap-9438	51	12	a	a	X
ap-9438	51	13	(	(	PUNCT
ap-9438	51	14	t−	t−	PROPN
ap-9438	51	15	τ)γ−1f(τ)dτ	τ)γ−1f(τ)dτ	NOUN
ap-9438	51	16	.	.	PUNCT
ap-9438	52	1	(	(	PUNCT
ap-9438	52	2	11	11	NUM
ap-9438	52	3	)	)	PUNCT
ap-9438	52	4	definition	definition	NOUN
ap-9438	52	5	4	4	NUM
ap-9438	52	6	(	(	PUNCT
ap-9438	52	7	the	the	DET
ap-9438	52	8	riemann	riemann	PROPN
ap-9438	52	9	-	-	PUNCT
ap-9438	52	10	liouville	liouville	VERB
ap-9438	52	11	derivative	derivative	NOUN
ap-9438	52	12	[	[	X
ap-9438	52	13	22	22	NUM
ap-9438	52	14	,	,	PUNCT
ap-9438	52	15	23	23	NUM
ap-9438	52	16	]	]	PUNCT
ap-9438	52	17	)	)	PUNCT
ap-9438	52	18	.	.	PUNCT
ap-9438	53	1	the	the	DET
ap-9438	53	2	riemann	riemann	PROPN
ap-9438	53	3	-	-	PUNCT
ap-9438	53	4	liouville	liouville	VERB
ap-9438	53	5	derivative	derivative	NOUN
ap-9438	53	6	of	of	ADP
ap-9438	53	7	fractional	fractional	ADJ
ap-9438	53	8	order	order	NOUN
ap-9438	53	9	n−	n−	NOUN
ap-9438	53	10	1	1	NUM
ap-9438	53	11	<	<	X
ap-9438	53	12	γ	γ	X
ap-9438	53	13	≤	≤	NOUN
ap-9438	53	14	n	n	CCONJ
ap-9438	53	15	,	,	PUNCT
ap-9438	53	16	n	n	CCONJ
ap-9438	53	17	∈	∈	PROPN
ap-9438	53	18	n	n	NOUN
ap-9438	53	19	of	of	ADP
ap-9438	53	20	a	a	DET
ap-9438	53	21	f	f	PROPN
ap-9438	53	22	∈	∈	PROPN
ap-9438	53	23	c[a	c[a	PROPN
ap-9438	53	24	,	,	PUNCT
ap-9438	53	25	b	b	AUX
ap-9438	53	26	]	]	PUNCT
ap-9438	53	27	is	be	AUX
ap-9438	53	28	defined	define	VERB
ap-9438	53	29	as	as	SCONJ
ap-9438	53	30	follows	follow	VERB
ap-9438	53	31	:	:	PUNCT
ap-9438	53	32	rl	rl	VERB
ap-9438	53	33	a	a	DET
ap-9438	53	34	dγ	dγ	PROPN
ap-9438	53	35	t	t	NOUN
ap-9438	53	36	f(t	f(t	PROPN
ap-9438	53	37	)	)	PUNCT
ap-9438	54	1	=	=	SYM
ap-9438	54	2	dn	dn	PROPN
ap-9438	54	3	t	t	PROPN
ap-9438	54	4	(	(	PUNCT
ap-9438	54	5	ai	ai	VERB
ap-9438	54	6	n−γ	n−γ	PROPN
ap-9438	54	7	t	t	PROPN
ap-9438	54	8	f(t	f(t	PROPN
ap-9438	54	9	)	)	PUNCT
ap-9438	54	10	)	)	PUNCT
ap-9438	54	11	=	=	SYM
ap-9438	55	1	1	1	NUM
ap-9438	55	2	γ(n−	γ(n−	PROPN
ap-9438	55	3	γ)d	γ)d	PROPN
ap-9438	55	4	n	n	NUM
ap-9438	55	5	t	t	PROPN
ap-9438	55	6	∫	∫	PROPN
ap-9438	55	7	t	t	PROPN
ap-9438	55	8	a	a	X
ap-9438	55	9	(	(	PUNCT
ap-9438	55	10	t−	t−	PROPN
ap-9438	55	11	τ)n−γ−1f(τ)dτ	τ)n−γ−1f(τ)dτ	NOUN
ap-9438	55	12	.	.	PUNCT
ap-9438	56	1	(	(	PUNCT
ap-9438	56	2	12	12	NUM
ap-9438	56	3	)	)	PUNCT
ap-9438	56	4	theorem	theorem	NOUN
ap-9438	56	5	2	2	NUM
ap-9438	56	6	[	[	X
ap-9438	56	7	23	23	NUM
ap-9438	56	8	]	]	PUNCT
ap-9438	56	9	.	.	PUNCT
ap-9438	57	1	let	let	VERB
ap-9438	57	2	γ	γ	X
ap-9438	57	3	,	,	PUNCT
ap-9438	57	4	β	β	X
ap-9438	57	5	,	,	PUNCT
ap-9438	57	6	µ	µ	NOUN
ap-9438	57	7	,	,	PUNCT
ap-9438	57	8	λ	λ	PROPN
ap-9438	57	9	∈	∈	PROPN
ap-9438	57	10	c	c	NOUN
ap-9438	57	11	,	,	PUNCT
ap-9438	57	12	re(γ	re(γ	X
ap-9438	57	13	)	)	PUNCT
ap-9438	57	14	>	>	X
ap-9438	57	15	0	0	NUM
ap-9438	57	16	,	,	PUNCT
ap-9438	57	17	re(β	re(β	X
ap-9438	57	18	)	)	PUNCT
ap-9438	57	19	>	>	X
ap-9438	58	1	−1	−1	NOUN
ap-9438	58	2	,	,	PUNCT
ap-9438	58	3	n−	n−	NOUN
ap-9438	58	4	1	1	NUM
ap-9438	58	5	<	<	X
ap-9438	58	6	γ	γ	X
ap-9438	58	7	≤	≤	NOUN
ap-9438	58	8	n	n	CCONJ
ap-9438	58	9	and	and	CCONJ
ap-9438	58	10	n	n	PRON
ap-9438	58	11	∈	∈	PROPN
ap-9438	58	12	n.	n.	NOUN
ap-9438	58	13	then	then	ADV
ap-9438	58	14	:	:	PUNCT
ap-9438	58	15	1	1	X
ap-9438	58	16	.	.	X
ap-9438	58	17	rldn	rldn	ADJ
ap-9438	58	18	t	t	PROPN
ap-9438	58	19	f(t	f(t	PROPN
ap-9438	58	20	)	)	PUNCT
ap-9438	58	21	=	=	SYM
ap-9438	58	22	dn	dn	PROPN
ap-9438	58	23	t	t	PROPN
ap-9438	58	24	f(t	f(t	PROPN
ap-9438	58	25	)	)	PUNCT
ap-9438	58	26	,	,	PUNCT
ap-9438	58	27	(	(	PUNCT
ap-9438	58	28	13	13	NUM
ap-9438	58	29	)	)	SYM
ap-9438	58	30	2	2	NUM
ap-9438	58	31	.	.	PUNCT
ap-9438	58	32	rldγ	rldγ	PROPN
ap-9438	58	33	t	t	PROPN
ap-9438	58	34	t	t	PROPN
ap-9438	58	35	β	β	X
ap-9438	58	36	=	=	PUNCT
ap-9438	58	37	γ(β	γ(β	PROPN
ap-9438	58	38	+	+	CCONJ
ap-9438	58	39	1	1	X
ap-9438	58	40	)	)	PUNCT
ap-9438	58	41	γ(β	γ(β	PROPN
ap-9438	58	42	−	−	PROPN
ap-9438	58	43	γ	γ	X
ap-9438	58	44	+	+	NOUN
ap-9438	58	45	1	1	NUM
ap-9438	58	46	)	)	PUNCT
ap-9438	58	47	t	t	PROPN
ap-9438	58	48	β−γ	β−γ	PROPN
ap-9438	58	49	,	,	PUNCT
ap-9438	58	50	(	(	PUNCT
ap-9438	58	51	14	14	NUM
ap-9438	58	52	)	)	SYM
ap-9438	58	53	3	3	NUM
ap-9438	58	54	.	.	X
ap-9438	58	55	rldγ	rldγ	PROPN
ap-9438	58	56	t	t	PROPN
ap-9438	58	57	t	t	PROPN
ap-9438	58	58	βeµ,β+1	βeµ,β+1	PROPN
ap-9438	58	59	(	(	PUNCT
ap-9438	58	60	λtµ	λtµ	NOUN
ap-9438	58	61	)	)	PUNCT
ap-9438	58	62	=	=	SYM
ap-9438	59	1	tβ−γeµ,β−γ+1	tβ−γeµ,β−γ+1	PROPN
ap-9438	59	2	(	(	PUNCT
ap-9438	59	3	λtµ	λtµ	NOUN
ap-9438	59	4	)	)	PUNCT
ap-9438	59	5	.	.	PUNCT
ap-9438	60	1	(	(	PUNCT
ap-9438	60	2	15	15	NUM
ap-9438	60	3	)	)	PUNCT
ap-9438	60	4	proposition	proposition	NOUN
ap-9438	60	5	(	(	PUNCT
ap-9438	60	6	the	the	DET
ap-9438	60	7	leibniz	leibniz	PROPN
ap-9438	60	8	integral	integral	ADJ
ap-9438	60	9	rule	rule	NOUN
ap-9438	60	10	[	[	X
ap-9438	60	11	24	24	NUM
ap-9438	60	12	]	]	PUNCT
ap-9438	60	13	)	)	PUNCT
ap-9438	60	14	.	.	PUNCT
ap-9438	61	1	let	let	VERB
ap-9438	61	2	f	f	PROPN
ap-9438	61	3	(	(	PUNCT
ap-9438	61	4	x	x	PROPN
ap-9438	61	5	,	,	PUNCT
ap-9438	61	6	t	t	PROPN
ap-9438	61	7	)	)	PUNCT
ap-9438	61	8	be	be	AUX
ap-9438	61	9	a	a	DET
ap-9438	61	10	function	function	NOUN
ap-9438	61	11	of	of	ADP
ap-9438	61	12	two	two	NUM
ap-9438	61	13	variables	variable	NOUN
ap-9438	61	14	,	,	PUNCT
ap-9438	61	15	x	x	PUNCT
ap-9438	61	16	and	and	CCONJ
ap-9438	61	17	t.	t.	NOUN
ap-9438	61	18	if	if	SCONJ
ap-9438	61	19	the	the	DET
ap-9438	61	20	function	function	NOUN
ap-9438	61	21	f	f	X
ap-9438	61	22	(	(	PUNCT
ap-9438	61	23	x	x	PROPN
ap-9438	61	24	,	,	PUNCT
ap-9438	61	25	t	t	PROPN
ap-9438	61	26	)	)	PUNCT
ap-9438	61	27	and	and	CCONJ
ap-9438	61	28	its	its	PRON
ap-9438	61	29	partial	partial	ADJ
ap-9438	61	30	derivative	derivative	ADJ
ap-9438	61	31	∂f	∂f	PROPN
ap-9438	61	32	∂x	∂x	NOUN
ap-9438	61	33	are	be	AUX
ap-9438	61	34	continuous	continuous	ADJ
ap-9438	61	35	in	in	ADP
ap-9438	61	36	a	a	DET
ap-9438	61	37	region	region	NOUN
ap-9438	61	38	that	that	PRON
ap-9438	61	39	includes	include	VERB
ap-9438	61	40	the	the	DET
ap-9438	61	41	interval	interval	NOUN
ap-9438	61	42	[	[	X
ap-9438	61	43	a	a	X
ap-9438	61	44	,	,	PUNCT
ap-9438	61	45	b	b	NOUN
ap-9438	61	46	]	]	X
ap-9438	61	47	for	for	ADP
ap-9438	61	48	x	x	NOUN
ap-9438	61	49	,	,	PUNCT
ap-9438	61	50	and	and	CCONJ
ap-9438	61	51	a(t	a(t	NOUN
ap-9438	61	52	)	)	PUNCT
ap-9438	61	53	and	and	CCONJ
ap-9438	61	54	b(t	b(t	NOUN
ap-9438	61	55	)	)	PUNCT
ap-9438	61	56	are	be	AUX
ap-9438	61	57	continuously	continuously	ADV
ap-9438	61	58	differentiable	differentiable	ADJ
ap-9438	61	59	functions	function	NOUN
ap-9438	61	60	of	of	ADP
ap-9438	61	61	t	t	PROPN
ap-9438	61	62	,	,	PUNCT
ap-9438	61	63	then	then	ADV
ap-9438	61	64	the	the	DET
ap-9438	61	65	derivative	derivative	NOUN
ap-9438	61	66	of	of	ADP
ap-9438	61	67	the	the	DET
ap-9438	61	68	integral	integral	ADJ
ap-9438	61	69	is	be	AUX
ap-9438	61	70	given	give	VERB
ap-9438	61	71	by	by	ADP
ap-9438	61	72	:	:	PUNCT
ap-9438	61	73	d	d	X
ap-9438	61	74	dt	dt	X
ap-9438	61	75	∫	∫	PROPN
ap-9438	61	76	b(t	b(t	PROPN
ap-9438	61	77	)	)	PUNCT
ap-9438	61	78	a(t	a(t	NOUN
ap-9438	61	79	)	)	PUNCT
ap-9438	61	80	f	f	NOUN
ap-9438	61	81	(	(	PUNCT
ap-9438	61	82	x	x	PROPN
ap-9438	61	83	,	,	PUNCT
ap-9438	61	84	t	t	PROPN
ap-9438	61	85	)	)	PUNCT
ap-9438	61	86	dx	dx	PROPN
ap-9438	62	1	=	=	SYM
ap-9438	62	2	f	f	PROPN
ap-9438	62	3	(	(	PUNCT
ap-9438	62	4	b(t	b(t	PROPN
ap-9438	62	5	)	)	PUNCT
ap-9438	62	6	,	,	PUNCT
ap-9438	62	7	t	t	PROPN
ap-9438	62	8	)	)	PUNCT
ap-9438	62	9	·	·	PUNCT
ap-9438	63	1	db	db	PROPN
ap-9438	64	1	dt	dt	X
ap-9438	65	1	−	−	PROPN
ap-9438	65	2	f	f	X
ap-9438	65	3	(	(	PUNCT
ap-9438	65	4	a(t	a(t	PROPN
ap-9438	65	5	)	)	PUNCT
ap-9438	65	6	,	,	PUNCT
ap-9438	65	7	t	t	PROPN
ap-9438	65	8	)	)	PUNCT
ap-9438	65	9	·	·	PUNCT
ap-9438	65	10	da	da	NOUN
ap-9438	65	11	dt	dt	X
ap-9438	65	12	+	+	NUM
ap-9438	65	13	∫	∫	PROPN
ap-9438	65	14	b(t	b(t	PROPN
ap-9438	65	15	)	)	PUNCT
ap-9438	65	16	a(t	a(t	NOUN
ap-9438	65	17	)	)	PUNCT
ap-9438	66	1	∂f	∂f	PROPN
ap-9438	66	2	∂t	∂t	PROPN
ap-9438	66	3	dx	dx	PROPN
ap-9438	66	4	.	.	PUNCT
ap-9438	67	1	(	(	PUNCT
ap-9438	67	2	16	16	NUM
ap-9438	67	3	)	)	PUNCT
ap-9438	67	4	theorem	theorem	VERB
ap-9438	67	5	3	3	NUM
ap-9438	68	1	[	[	X
ap-9438	68	2	25	25	NUM
ap-9438	68	3	]	]	PUNCT
ap-9438	68	4	.	.	PUNCT
ap-9438	69	1	let	let	VERB
ap-9438	69	2	γ	γ	X
ap-9438	69	3	>	>	X
ap-9438	69	4	0	0	PROPN
ap-9438	69	5	,	,	PUNCT
ap-9438	69	6	t	t	X
ap-9438	69	7	>	>	X
ap-9438	69	8	a	a	PROPN
ap-9438	69	9	and	and	CCONJ
ap-9438	69	10	k	k	PROPN
ap-9438	69	11	∈	∈	PROPN
ap-9438	69	12	c2[a	c2[a	NOUN
ap-9438	69	13	,	,	PUNCT
ap-9438	69	14	b	b	NOUN
ap-9438	69	15	]	]	X
ap-9438	69	16	.	.	PUNCT
ap-9438	70	1	then	then	ADV
ap-9438	70	2	leibniz	leibniz	PROPN
ap-9438	70	3	integral	integral	ADJ
ap-9438	70	4	rule	rule	NOUN
ap-9438	70	5	of	of	ADP
ap-9438	70	6	higher	high	ADJ
ap-9438	70	7	order	order	NOUN
ap-9438	70	8	derivative	derivative	ADJ
ap-9438	70	9	and	and	CCONJ
ap-9438	70	10	fractional	fractional	ADJ
ap-9438	70	11	leibniz	leibniz	PROPN
ap-9438	70	12	integral	integral	ADJ
ap-9438	70	13	rules	rule	NOUN
ap-9438	70	14	in	in	ADP
ap-9438	70	15	the	the	DET
ap-9438	70	16	sense	sense	NOUN
ap-9438	70	17	of	of	ADP
ap-9438	70	18	riemann	riemann	PROPN
ap-9438	70	19	-	-	PUNCT
ap-9438	70	20	liouville	liouville	PROPN
ap-9438	70	21	and	and	CCONJ
ap-9438	70	22	caputo	caputo	PROPN
ap-9438	70	23	derivatives	derivative	NOUN
ap-9438	70	24	respectively	respectively	ADV
ap-9438	70	25	,	,	PUNCT
ap-9438	70	26	are	be	AUX
ap-9438	70	27	given	give	VERB
ap-9438	70	28	by	by	ADP
ap-9438	70	29	:	:	PUNCT
ap-9438	70	30	1	1	X
ap-9438	70	31	.	.	X
ap-9438	70	32	dn	dn	PROPN
ap-9438	70	33	t	t	PROPN
ap-9438	70	34	∫	∫	PROPN
ap-9438	70	35	t	t	PROPN
ap-9438	70	36	a	a	DET
ap-9438	70	37	k(t	k(t	PROPN
ap-9438	70	38	,	,	PUNCT
ap-9438	71	1	s)ds	s)ds	PROPN
ap-9438	71	2	=	=	PUNCT
ap-9438	71	3	n∑	n∑	PROPN
ap-9438	71	4	i=1	i=1	PROPN
ap-9438	72	1	di−1	di−1	PROPN
ap-9438	72	2	t	t	PROPN
ap-9438	72	3	lim	lim	PROPN
ap-9438	72	4	s→t	s→t	PROPN
ap-9438	72	5	dn−i	dn−i	PROPN
ap-9438	72	6	t	t	PROPN
ap-9438	72	7	k(t	k(t	PROPN
ap-9438	72	8	,	,	PUNCT
ap-9438	72	9	s	s	PART
ap-9438	72	10	)	)	PUNCT
ap-9438	73	1	+	+	CCONJ
ap-9438	73	2	∫	∫	PROPN
ap-9438	73	3	t	t	PROPN
ap-9438	73	4	a	a	DET
ap-9438	73	5	dn	dn	PROPN
ap-9438	73	6	t	t	PROPN
ap-9438	73	7	k(t	k(t	PROPN
ap-9438	73	8	,	,	PUNCT
ap-9438	73	9	s)ds	s)ds	PROPN
ap-9438	73	10	,	,	PUNCT
ap-9438	73	11	(	(	PUNCT
ap-9438	73	12	17	17	NUM
ap-9438	73	13	)	)	PUNCT
ap-9438	73	14	2	2	NUM
ap-9438	73	15	.	.	X
ap-9438	73	16	rl	rl	VERB
ap-9438	73	17	a	a	DET
ap-9438	73	18	dγ	dγ	PROPN
ap-9438	73	19	t	t	PROPN
ap-9438	73	20	∫	∫	PROPN
ap-9438	73	21	t	t	PROPN
ap-9438	73	22	a	a	DET
ap-9438	73	23	k(t	k(t	PROPN
ap-9438	73	24	,	,	PUNCT
ap-9438	74	1	s)ds	s)ds	PROPN
ap-9438	74	2	=	=	PUNCT
ap-9438	74	3	n∑	n∑	PROPN
ap-9438	74	4	i=1	i=1	PROPN
ap-9438	75	1	di−1	di−1	PROPN
ap-9438	75	2	t	t	PROPN
ap-9438	75	3	lim	lim	PROPN
ap-9438	75	4	s→t	s→t	NUM
ap-9438	75	5	rl	rl	PROPN
ap-9438	75	6	s	s	PROPN
ap-9438	75	7	dγ−i	dγ−i	PROPN
ap-9438	75	8	t	t	PROPN
ap-9438	75	9	k(t	k(t	PROPN
ap-9438	75	10	,	,	PUNCT
ap-9438	75	11	s	s	PART
ap-9438	75	12	)	)	PUNCT
ap-9438	76	1	+	+	CCONJ
ap-9438	76	2	∫	∫	PROPN
ap-9438	76	3	t	t	PROPN
ap-9438	76	4	a	a	DET
ap-9438	76	5	rl	rl	PROPN
ap-9438	76	6	s	s	X
ap-9438	76	7	dγ	dγ	NOUN
ap-9438	76	8	tk(t	tk(t	PRON
ap-9438	76	9	,	,	PUNCT
ap-9438	76	10	s)ds	s)ds	PROPN
ap-9438	76	11	.	.	PUNCT
ap-9438	77	1	(	(	PUNCT
ap-9438	77	2	18	18	NUM
ap-9438	77	3	)	)	PUNCT
ap-9438	77	4	3	3	NUM
ap-9438	77	5	.	.	PUNCT
ap-9438	78	1	the	the	DET
ap-9438	78	2	approach	approach	NOUN
ap-9438	78	3	analysis	analysis	NOUN
ap-9438	78	4	in	in	ADP
ap-9438	78	5	this	this	DET
ap-9438	78	6	section	section	NOUN
ap-9438	78	7	of	of	ADP
ap-9438	78	8	the	the	DET
ap-9438	78	9	research	research	NOUN
ap-9438	78	10	paper	paper	NOUN
ap-9438	78	11	,	,	PUNCT
ap-9438	78	12	we	we	PRON
ap-9438	78	13	will	will	AUX
ap-9438	78	14	provide	provide	VERB
ap-9438	78	15	a	a	DET
ap-9438	78	16	detailed	detailed	ADJ
ap-9438	78	17	analysis	analysis	NOUN
ap-9438	78	18	of	of	ADP
ap-9438	78	19	the	the	DET
ap-9438	78	20	algorithm	algorithm	NOUN
ap-9438	78	21	of	of	ADP
ap-9438	78	22	the	the	DET
ap-9438	78	23	new	new	ADJ
ap-9438	78	24	method	method	NOUN
ap-9438	78	25	in	in	ADP
ap-9438	78	26	the	the	DET
ap-9438	78	27	context	context	NOUN
ap-9438	78	28	of	of	ADP
ap-9438	78	29	fractional	fractional	PROPN
ap-9438	78	30	volterra	volterra	PROPN
ap-9438	78	31	integral	integral	ADJ
ap-9438	78	32	equations	equation	NOUN
ap-9438	78	33	.	.	PUNCT
ap-9438	79	1	this	this	DET
ap-9438	79	2	method	method	NOUN
ap-9438	79	3	relies	rely	VERB
ap-9438	79	4	on	on	ADP
ap-9438	79	5	leibniz	leibniz	PROPN
ap-9438	79	6	integral	integral	ADJ
ap-9438	79	7	rules	rule	NOUN
ap-9438	79	8	and	and	CCONJ
ap-9438	79	9	the	the	DET
ap-9438	79	10	algorithm	algorithm	NOUN
ap-9438	79	11	employed	employ	VERB
ap-9438	79	12	in	in	ADP
ap-9438	79	13	this	this	DET
ap-9438	79	14	study	study	NOUN
ap-9438	79	15	constitutes	constitute	VERB
ap-9438	79	16	an	an	DET
ap-9438	79	17	innovative	innovative	ADJ
ap-9438	79	18	approach	approach	NOUN
ap-9438	79	19	to	to	ADP
ap-9438	79	20	addressing	address	VERB
ap-9438	79	21	the	the	DET
ap-9438	79	22	challenges	challenge	NOUN
ap-9438	79	23	posed	pose	VERB
ap-9438	79	24	by	by	ADP
ap-9438	79	25	fractional	fractional	ADJ
ap-9438	79	26	volterra	volterra	PROPN
ap-9438	79	27	integral	integral	ADJ
ap-9438	79	28	equations	equation	NOUN
ap-9438	79	29	.	.	PUNCT
ap-9438	80	1	this	this	DET
ap-9438	80	2	section	section	NOUN
ap-9438	80	3	focuses	focus	VERB
ap-9438	80	4	on	on	ADP
ap-9438	80	5	the	the	DET
ap-9438	80	6	analysis	analysis	NOUN
ap-9438	80	7	and	and	CCONJ
ap-9438	80	8	use	use	NOUN
ap-9438	80	9	of	of	ADP
ap-9438	80	10	this	this	DET
ap-9438	80	11	algorithm	algorithm	NOUN
ap-9438	80	12	for	for	ADP
ap-9438	80	13	solving	solve	VERB
ap-9438	80	14	fractional	fractional	ADJ
ap-9438	80	15	volterra	volterra	NOUN
ap-9438	80	16	integral	integral	ADJ
ap-9438	80	17	equations	equation	NOUN
ap-9438	80	18	.	.	PUNCT
ap-9438	81	1	through	through	ADP
ap-9438	81	2	this	this	DET
ap-9438	81	3	approach	approach	NOUN
ap-9438	81	4	,	,	PUNCT
ap-9438	81	5	we	we	PRON
ap-9438	81	6	strive	strive	VERB
ap-9438	81	7	to	to	PART
ap-9438	81	8	deepen	deepen	VERB
ap-9438	81	9	our	our	PRON
ap-9438	81	10	understanding	understanding	NOUN
ap-9438	81	11	of	of	ADP
ap-9438	81	12	the	the	DET
ap-9438	81	13	fundamental	fundamental	ADJ
ap-9438	81	14	properties	property	NOUN
ap-9438	81	15	of	of	ADP
ap-9438	81	16	fractional	fractional	ADJ
ap-9438	81	17	volterra	volterra	PROPN
ap-9438	81	18	integral	integral	ADJ
ap-9438	81	19	equations	equation	NOUN
ap-9438	81	20	.	.	PUNCT
ap-9438	82	1	furthermore	furthermore	ADV
ap-9438	82	2	,	,	PUNCT
ap-9438	82	3	we	we	PRON
ap-9438	82	4	will	will	AUX
ap-9438	82	5	415	415	NUM
ap-9438	82	6	mohammed	mohammed	PROPN
ap-9438	82	7	abdulshareef	abdulshareef	PROPN
ap-9438	82	8	hussein	hussein	PROPN
ap-9438	82	9	,	,	PUNCT
ap-9438	82	10	hassan	hassan	PROPN
ap-9438	82	11	kamil	kamil	PROPN
ap-9438	82	12	jassim	jassim	PROPN
ap-9438	82	13	acta	acta	PROPN
ap-9438	82	14	polytechnica	polytechnica	PROPN
ap-9438	82	15	contribute	contribute	VERB
ap-9438	82	16	distinctly	distinctly	ADV
ap-9438	82	17	to	to	ADP
ap-9438	82	18	the	the	DET
ap-9438	82	19	development	development	NOUN
ap-9438	82	20	of	of	ADP
ap-9438	82	21	analytical	analytical	ADJ
ap-9438	82	22	tools	tool	NOUN
ap-9438	82	23	and	and	CCONJ
ap-9438	82	24	techniques	technique	NOUN
ap-9438	82	25	available	available	ADJ
ap-9438	82	26	to	to	ADP
ap-9438	82	27	the	the	DET
ap-9438	82	28	scientific	scientific	ADJ
ap-9438	82	29	research	research	NOUN
ap-9438	82	30	community	community	NOUN
ap-9438	82	31	.	.	PUNCT
ap-9438	83	1	consider	consider	VERB
ap-9438	83	2	the	the	DET
ap-9438	83	3	following	follow	VERB
ap-9438	83	4	fractional	fractional	PROPN
ap-9438	83	5	volterra	volterra	PROPN
ap-9438	83	6	integral	integral	ADJ
ap-9438	83	7	equation	equation	NOUN
ap-9438	83	8	of	of	ADP
ap-9438	83	9	first	first	ADJ
ap-9438	83	10	kind	kind	NOUN
ap-9438	83	11	such	such	ADJ
ap-9438	83	12	as	as	ADP
ap-9438	83	13	g(t	g(t	PROPN
ap-9438	83	14	)	)	PUNCT
ap-9438	83	15	̸=	̸=	PROPN
ap-9438	83	16	0	0	NUM
ap-9438	83	17	:	:	PUNCT
ap-9438	83	18	f(t	f(t	NOUN
ap-9438	83	19	)	)	PUNCT
ap-9438	84	1	=	=	SYM
ap-9438	84	2	∫	∫	PROPN
ap-9438	84	3	t	t	PROPN
ap-9438	84	4	a	a	DET
ap-9438	84	5	(	(	PUNCT
ap-9438	84	6	t−	t−	PROPN
ap-9438	84	7	s)γg(s)n(ψ(s))ds	s)γg(s)n(ψ(s))ds	PROPN
ap-9438	84	8	,	,	PUNCT
ap-9438	84	9	(	(	PUNCT
ap-9438	84	10	19	19	NUM
ap-9438	84	11	)	)	PUNCT
ap-9438	84	12	where	where	SCONJ
ap-9438	84	13	n	n	X
ap-9438	84	14	−	−	PROPN
ap-9438	84	15	1	1	NUM
ap-9438	84	16	≤	≤	NUM
ap-9438	84	17	γ	γ	X
ap-9438	84	18	≤	≤	NOUN
ap-9438	84	19	n	n	CCONJ
ap-9438	84	20	and	and	CCONJ
ap-9438	84	21	n	n	PRON
ap-9438	84	22	∈	∈	PROPN
ap-9438	84	23	z+	z+	NUM
ap-9438	84	24	.	.	PUNCT
ap-9438	85	1	now	now	ADV
ap-9438	85	2	,	,	PUNCT
ap-9438	85	3	we	we	PRON
ap-9438	85	4	will	will	AUX
ap-9438	85	5	find	find	VERB
ap-9438	85	6	the	the	DET
ap-9438	85	7	exact	exact	ADJ
ap-9438	85	8	solution	solution	NOUN
ap-9438	85	9	to	to	ADP
ap-9438	85	10	equation	equation	NOUN
ap-9438	85	11	(	(	PUNCT
ap-9438	85	12	19	19	NUM
ap-9438	85	13	)	)	PUNCT
ap-9438	85	14	using	use	VERB
ap-9438	85	15	the	the	DET
ap-9438	85	16	fractional	fractional	ADJ
ap-9438	85	17	derivative	derivative	NOUN
ap-9438	85	18	of	of	ADP
ap-9438	85	19	riemann	riemann	PROPN
ap-9438	85	20	-	-	PUNCT
ap-9438	85	21	liouville	liouville	NOUN
ap-9438	85	22	.	.	PUNCT
ap-9438	86	1	by	by	ADP
ap-9438	86	2	taking	take	VERB
ap-9438	86	3	the	the	DET
ap-9438	86	4	fractional	fractional	ADJ
ap-9438	86	5	derivative	derivative	NOUN
ap-9438	86	6	rl	rl	ADP
ap-9438	86	7	a	a	DET
ap-9438	86	8	dγ+1	dγ+1	PROPN
ap-9438	86	9	t	t	NOUN
ap-9438	86	10	to	to	ADP
ap-9438	86	11	both	both	DET
ap-9438	86	12	sides	side	NOUN
ap-9438	86	13	of	of	ADP
ap-9438	86	14	equation	equation	NOUN
ap-9438	86	15	(	(	PUNCT
ap-9438	86	16	19	19	NUM
ap-9438	86	17	)	)	PUNCT
ap-9438	86	18	,	,	PUNCT
ap-9438	86	19	we	we	PRON
ap-9438	86	20	obtain	obtain	VERB
ap-9438	86	21	:	:	PUNCT
ap-9438	86	22	rl	rl	VERB
ap-9438	86	23	a	a	DET
ap-9438	86	24	dγ+1	dγ+1	PROPN
ap-9438	86	25	t	t	NOUN
ap-9438	86	26	f(t	f(t	PROPN
ap-9438	86	27	)	)	PUNCT
ap-9438	87	1	=	=	SYM
ap-9438	87	2	rl	rl	VERB
ap-9438	87	3	a	a	DET
ap-9438	87	4	dγ+1	dγ+1	PROPN
ap-9438	87	5	t	t	PROPN
ap-9438	87	6	∫	∫	PROPN
ap-9438	87	7	t	t	PROPN
ap-9438	87	8	a	a	DET
ap-9438	87	9	(	(	PUNCT
ap-9438	87	10	t−	t−	PROPN
ap-9438	87	11	s)γg(s)n(ψ(s))ds	s)γg(s)n(ψ(s))ds	PROPN
ap-9438	87	12	,	,	PUNCT
ap-9438	87	13	by	by	ADP
ap-9438	87	14	using	use	VERB
ap-9438	87	15	theorem	theorem	NOUN
ap-9438	87	16	3	3	NUM
ap-9438	87	17	,	,	PUNCT
ap-9438	87	18	we	we	PRON
ap-9438	87	19	get	get	VERB
ap-9438	87	20	:	:	PUNCT
ap-9438	87	21	rl	rl	VERB
ap-9438	87	22	a	a	DET
ap-9438	87	23	dγ+1	dγ+1	PROPN
ap-9438	87	24	t	t	NOUN
ap-9438	87	25	f(t	f(t	PROPN
ap-9438	87	26	)	)	PUNCT
ap-9438	87	27	=	=	PUNCT
ap-9438	88	1	=	=	PUNCT
ap-9438	88	2	dt	dt	X
ap-9438	88	3	(	(	PUNCT
ap-9438	88	4	n∑	n∑	NOUN
ap-9438	88	5	i=1	i=1	PROPN
ap-9438	89	1	di−1	di−1	PROPN
ap-9438	89	2	t	t	PROPN
ap-9438	89	3	lim	lim	PROPN
ap-9438	89	4	s→t	s→t	NUM
ap-9438	89	5	rl	rl	PROPN
ap-9438	89	6	s	s	PROPN
ap-9438	89	7	dγ−i	dγ−i	PROPN
ap-9438	89	8	t	t	NOUN
ap-9438	89	9	(	(	PUNCT
ap-9438	89	10	t−	t−	PROPN
ap-9438	89	11	s)γg(s)n(ψ(s	s)γg(s)n(ψ(s	PROPN
ap-9438	89	12	)	)	PUNCT
ap-9438	89	13	)	)	PUNCT
ap-9438	90	1	+	+	CCONJ
ap-9438	90	2	∫	∫	PROPN
ap-9438	90	3	t	t	PROPN
ap-9438	90	4	a	a	DET
ap-9438	90	5	rl	rl	PROPN
ap-9438	90	6	s	s	PROPN
ap-9438	90	7	dγ	dγ	ADP
ap-9438	90	8	t	t	PROPN
ap-9438	90	9	(	(	PUNCT
ap-9438	90	10	t−	t−	PROPN
ap-9438	90	11	s)γg(s)n(ψ(s))ds	s)γg(s)n(ψ(s))ds	PROPN
ap-9438	90	12	)	)	PUNCT
ap-9438	90	13	,	,	PUNCT
ap-9438	90	14	=	=	PRON
ap-9438	91	1	dt	dt	X
ap-9438	91	2	(	(	PUNCT
ap-9438	91	3	n∑	n∑	NOUN
ap-9438	91	4	i=1	i=1	PROPN
ap-9438	91	5	di−1	di−1	PROPN
ap-9438	91	6	t	t	PROPN
ap-9438	91	7	lim	lim	PROPN
ap-9438	91	8	s→t	s→t	NUM
ap-9438	91	9	γ(γ	γ(γ	PROPN
ap-9438	91	10	+	+	CCONJ
ap-9438	91	11	1	1	X
ap-9438	91	12	)	)	PUNCT
ap-9438	91	13	γ(i+	γ(i+	NUM
ap-9438	91	14	1	1	NUM
ap-9438	91	15	)	)	PUNCT
ap-9438	91	16	(	(	PUNCT
ap-9438	91	17	t−	t−	PROPN
ap-9438	91	18	s)ig(s)n(ψ(s	s)ig(s)n(ψ(s	PROPN
ap-9438	91	19	)	)	PUNCT
ap-9438	91	20	)	)	PUNCT
ap-9438	92	1	+	+	CCONJ
ap-9438	93	1	∫	∫	PROPN
ap-9438	93	2	t	t	PROPN
ap-9438	93	3	a	a	DET
ap-9438	93	4	γ(γ	γ(γ	PROPN
ap-9438	93	5	+	+	CCONJ
ap-9438	93	6	1)g(s)n(ψ(s))ds	1)g(s)n(ψ(s))ds	NUM
ap-9438	93	7	)	)	PUNCT
ap-9438	93	8	,	,	PUNCT
ap-9438	93	9	=	=	PRON
ap-9438	93	10	dt	dt	X
ap-9438	93	11	(	(	PUNCT
ap-9438	93	12	∫	∫	PROPN
ap-9438	93	13	t	t	PROPN
ap-9438	93	14	a	a	DET
ap-9438	93	15	γ(γ	γ(γ	PROPN
ap-9438	93	16	+	+	CCONJ
ap-9438	93	17	1)g(s)n(ψ(s))ds	1)g(s)n(ψ(s))ds	NUM
ap-9438	93	18	)	)	PUNCT
ap-9438	93	19	.	.	PUNCT
ap-9438	94	1	by	by	ADP
ap-9438	94	2	definition	definition	NOUN
ap-9438	94	3	of	of	ADP
ap-9438	94	4	the	the	DET
ap-9438	94	5	leibniz	leibniz	PROPN
ap-9438	94	6	integral	integral	ADJ
ap-9438	94	7	rule	rule	NOUN
ap-9438	94	8	:	:	PUNCT
ap-9438	94	9	rl	rl	VERB
ap-9438	94	10	a	a	DET
ap-9438	94	11	dγ+1	dγ+1	PROPN
ap-9438	94	12	t	t	NOUN
ap-9438	94	13	f(t	f(t	PROPN
ap-9438	94	14	)	)	PUNCT
ap-9438	94	15	=	=	SYM
ap-9438	94	16	γ(γ	γ(γ	PROPN
ap-9438	94	17	+	+	CCONJ
ap-9438	94	18	1)g(t)n(ψ(t	1)g(t)n(ψ(t	NUM
ap-9438	94	19	)	)	PUNCT
ap-9438	94	20	)	)	PUNCT
ap-9438	94	21	.	.	PUNCT
ap-9438	95	1	assuming	assume	VERB
ap-9438	95	2	that	that	SCONJ
ap-9438	95	3	g(t	g(t	PROPN
ap-9438	95	4	)	)	PUNCT
ap-9438	95	5	̸=	̸=	PROPN
ap-9438	95	6	0	0	NUM
ap-9438	95	7	,	,	PUNCT
ap-9438	95	8	we	we	PRON
ap-9438	95	9	get	get	VERB
ap-9438	95	10	:	:	PUNCT
ap-9438	95	11	n(ψ(t	n(ψ(t	PROPN
ap-9438	95	12	)	)	PUNCT
ap-9438	95	13	)	)	PUNCT
ap-9438	96	1	=	=	PRON
ap-9438	96	2	rl	rl	VERB
ap-9438	96	3	a	a	DET
ap-9438	96	4	dγ+1	dγ+1	PROPN
ap-9438	96	5	t	t	NOUN
ap-9438	96	6	f(t	f(t	PROPN
ap-9438	96	7	)	)	PUNCT
ap-9438	96	8	γ(γ	γ(γ	PROPN
ap-9438	97	1	+	+	CCONJ
ap-9438	97	2	1)g(t	1)g(t	NUM
ap-9438	97	3	)	)	PUNCT
ap-9438	97	4	.	.	PUNCT
ap-9438	98	1	by	by	ADP
ap-9438	98	2	the	the	DET
ap-9438	98	3	concept	concept	NOUN
ap-9438	98	4	of	of	ADP
ap-9438	98	5	the	the	DET
ap-9438	98	6	inverse	inverse	NOUN
ap-9438	98	7	of	of	ADP
ap-9438	98	8	the	the	DET
ap-9438	98	9	function	function	NOUN
ap-9438	98	10	:	:	PUNCT
ap-9438	98	11	ψ(t	ψ(t	PROPN
ap-9438	98	12	)	)	PUNCT
ap-9438	99	1	=	=	SYM
ap-9438	99	2	n−1	n−1	PROPN
ap-9438	99	3	(	(	PUNCT
ap-9438	99	4	rl	rl	ADP
ap-9438	99	5	a	a	DET
ap-9438	99	6	dγ+1	dγ+1	PROPN
ap-9438	99	7	t	t	NOUN
ap-9438	99	8	f(t	f(t	PROPN
ap-9438	99	9	)	)	PUNCT
ap-9438	99	10	γ(γ	γ(γ	PROPN
ap-9438	99	11	+	+	CCONJ
ap-9438	99	12	1)g(t	1)g(t	NUM
ap-9438	99	13	)	)	PUNCT
ap-9438	99	14	)	)	PUNCT
ap-9438	99	15	.	.	PUNCT
ap-9438	100	1	(	(	PUNCT
ap-9438	100	2	20	20	NUM
ap-9438	100	3	)	)	PUNCT
ap-9438	100	4	thus	thus	ADV
ap-9438	100	5	,	,	PUNCT
ap-9438	100	6	the	the	DET
ap-9438	100	7	exact	exact	ADJ
ap-9438	100	8	solution	solution	NOUN
ap-9438	100	9	of	of	ADP
ap-9438	100	10	equation	equation	NOUN
ap-9438	100	11	(	(	PUNCT
ap-9438	100	12	19	19	NUM
ap-9438	100	13	)	)	PUNCT
ap-9438	100	14	when	when	SCONJ
ap-9438	100	15	γ	γ	X
ap-9438	100	16	=	=	SYM
ap-9438	100	17	n	n	X
ap-9438	100	18	is	be	AUX
ap-9438	100	19	a	a	DET
ap-9438	100	20	non	non	ADJ
ap-9438	100	21	-	-	ADJ
ap-9438	100	22	negative	negative	ADJ
ap-9438	100	23	integer	integer	NOUN
ap-9438	100	24	number	number	NOUN
ap-9438	100	25	is	be	AUX
ap-9438	100	26	:	:	PUNCT
ap-9438	100	27	ψ(t	ψ(t	PROPN
ap-9438	100	28	)	)	PUNCT
ap-9438	100	29	=	=	SYM
ap-9438	100	30	n−1	n−1	PROPN
ap-9438	100	31	(	(	PUNCT
ap-9438	100	32	dn+1	dn+1	NOUN
ap-9438	100	33	t	t	PRON
ap-9438	100	34	f(t	f(t	NOUN
ap-9438	100	35	)	)	PUNCT
ap-9438	100	36	n!g(t	n!g(t	NOUN
ap-9438	100	37	)	)	PUNCT
ap-9438	100	38	)	)	PUNCT
ap-9438	100	39	.	.	PUNCT
ap-9438	101	1	(	(	PUNCT
ap-9438	101	2	21	21	NUM
ap-9438	101	3	)	)	PUNCT
ap-9438	101	4	4	4	NUM
ap-9438	101	5	.	.	X
ap-9438	101	6	illustrative	illustrative	ADJ
ap-9438	101	7	examples	example	NOUN
ap-9438	101	8	in	in	ADP
ap-9438	101	9	this	this	DET
ap-9438	101	10	section	section	NOUN
ap-9438	101	11	,	,	PUNCT
ap-9438	101	12	we	we	PRON
ap-9438	101	13	will	will	AUX
ap-9438	101	14	use	use	VERB
ap-9438	101	15	the	the	DET
ap-9438	101	16	new	new	ADJ
ap-9438	101	17	method	method	NOUN
ap-9438	101	18	we	we	PRON
ap-9438	101	19	have	have	AUX
ap-9438	101	20	developed	develop	VERB
ap-9438	101	21	to	to	PART
ap-9438	101	22	solve	solve	VERB
ap-9438	101	23	various	various	ADJ
ap-9438	101	24	volterra	volterra	NOUN
ap-9438	101	25	integral	integral	ADJ
ap-9438	101	26	equations	equation	NOUN
ap-9438	101	27	.	.	PUNCT
ap-9438	102	1	our	our	PRON
ap-9438	102	2	goal	goal	NOUN
ap-9438	102	3	is	be	AUX
ap-9438	102	4	to	to	PART
ap-9438	102	5	obtain	obtain	VERB
ap-9438	102	6	precise	precise	ADJ
ap-9438	102	7	solutions	solution	NOUN
ap-9438	102	8	for	for	ADP
ap-9438	102	9	these	these	DET
ap-9438	102	10	equations	equation	NOUN
ap-9438	102	11	and	and	CCONJ
ap-9438	102	12	assess	assess	VERB
ap-9438	102	13	the	the	DET
ap-9438	102	14	accuracy	accuracy	NOUN
ap-9438	102	15	and	and	CCONJ
ap-9438	102	16	efficacy	efficacy	NOUN
ap-9438	102	17	of	of	ADP
ap-9438	102	18	our	our	PRON
ap-9438	102	19	method	method	NOUN
ap-9438	102	20	in	in	ADP
ap-9438	102	21	achieving	achieve	VERB
ap-9438	102	22	these	these	DET
ap-9438	102	23	solutions	solution	NOUN
ap-9438	102	24	.	.	PUNCT
ap-9438	103	1	throughout	throughout	ADP
ap-9438	103	2	this	this	DET
ap-9438	103	3	section	section	NOUN
ap-9438	103	4	,	,	PUNCT
ap-9438	103	5	we	we	PRON
ap-9438	103	6	will	will	AUX
ap-9438	103	7	provide	provide	VERB
ap-9438	103	8	detailed	detailed	ADJ
ap-9438	103	9	explanations	explanation	NOUN
ap-9438	103	10	of	of	ADP
ap-9438	103	11	the	the	DET
ap-9438	103	12	steps	step	NOUN
ap-9438	103	13	we	we	PRON
ap-9438	103	14	take	take	VERB
ap-9438	103	15	,	,	PUNCT
ap-9438	103	16	the	the	DET
ap-9438	103	17	equations	equation	NOUN
ap-9438	103	18	we	we	PRON
ap-9438	103	19	solve	solve	VERB
ap-9438	103	20	,	,	PUNCT
ap-9438	103	21	and	and	CCONJ
ap-9438	103	22	the	the	DET
ap-9438	103	23	results	result	NOUN
ap-9438	103	24	we	we	PRON
ap-9438	103	25	achieve	achieve	VERB
ap-9438	103	26	.	.	PUNCT
ap-9438	104	1	this	this	DET
ap-9438	104	2	section	section	NOUN
ap-9438	104	3	will	will	AUX
ap-9438	104	4	showcase	showcase	VERB
ap-9438	104	5	our	our	PRON
ap-9438	104	6	novel	novel	ADJ
ap-9438	104	7	method	method	NOUN
ap-9438	104	8	’s	’s	PART
ap-9438	104	9	application	application	NOUN
ap-9438	104	10	to	to	ADP
ap-9438	104	11	various	various	ADJ
ap-9438	104	12	volterra	volterra	NOUN
ap-9438	104	13	integral	integral	ADJ
ap-9438	104	14	equations	equation	NOUN
ap-9438	104	15	.	.	PUNCT
ap-9438	105	1	by	by	ADP
ap-9438	105	2	obtaining	obtain	VERB
ap-9438	105	3	exact	exact	ADJ
ap-9438	105	4	solutions	solution	NOUN
ap-9438	105	5	and	and	CCONJ
ap-9438	105	6	evaluating	evaluate	VERB
ap-9438	105	7	their	their	PRON
ap-9438	105	8	accuracy	accuracy	NOUN
ap-9438	105	9	and	and	CCONJ
ap-9438	105	10	computational	computational	ADJ
ap-9438	105	11	efficiency	efficiency	NOUN
ap-9438	105	12	,	,	PUNCT
ap-9438	105	13	we	we	PRON
ap-9438	105	14	aim	aim	VERB
ap-9438	105	15	to	to	PART
ap-9438	105	16	establish	establish	VERB
ap-9438	105	17	the	the	DET
ap-9438	105	18	credibility	credibility	NOUN
ap-9438	105	19	and	and	CCONJ
ap-9438	105	20	utility	utility	NOUN
ap-9438	105	21	of	of	ADP
ap-9438	105	22	our	our	PRON
ap-9438	105	23	method	method	NOUN
ap-9438	105	24	as	as	ADP
ap-9438	105	25	a	a	DET
ap-9438	105	26	valuable	valuable	ADJ
ap-9438	105	27	tool	tool	NOUN
ap-9438	105	28	for	for	ADP
ap-9438	105	29	solving	solve	VERB
ap-9438	105	30	these	these	DET
ap-9438	105	31	types	type	NOUN
ap-9438	105	32	of	of	ADP
ap-9438	105	33	equations	equation	NOUN
ap-9438	105	34	in	in	ADP
ap-9438	105	35	diverse	diverse	ADJ
ap-9438	105	36	scenarios	scenario	NOUN
ap-9438	105	37	.	.	PUNCT
ap-9438	106	1	example	example	NOUN
ap-9438	106	2	1	1	NUM
ap-9438	106	3	.	.	X
ap-9438	107	1	consider	consider	VERB
ap-9438	107	2	the	the	DET
ap-9438	107	3	following	follow	VERB
ap-9438	107	4	abel	abel	PROPN
ap-9438	107	5	’s	’s	PART
ap-9438	107	6	integral	integral	ADJ
ap-9438	107	7	equation	equation	NOUN
ap-9438	107	8	[	[	X
ap-9438	107	9	26	26	NUM
ap-9438	107	10	]	]	X
ap-9438	107	11	:	:	PUNCT
ap-9438	108	1	π	π	X
ap-9438	108	2	+	+	CCONJ
ap-9438	108	3	t	t	X
ap-9438	108	4	=	=	SYM
ap-9438	108	5	∫	∫	PROPN
ap-9438	108	6	t	t	PROPN
ap-9438	108	7	0	0	NUM
ap-9438	108	8	eψ(s	eψ(s	PRON
ap-9438	108	9	)	)	PUNCT
ap-9438	108	10	√	√	ADP
ap-9438	108	11	t−	t−	PROPN
ap-9438	108	12	s	s	PART
ap-9438	108	13	ds	ds	NOUN
ap-9438	108	14	,	,	PUNCT
ap-9438	108	15	(	(	PUNCT
ap-9438	108	16	22	22	NUM
ap-9438	108	17	)	)	PUNCT
ap-9438	108	18	with	with	ADP
ap-9438	108	19	the	the	DET
ap-9438	108	20	algorithm	algorithm	NOUN
ap-9438	108	21	of	of	ADP
ap-9438	108	22	the	the	DET
ap-9438	108	23	new	new	ADJ
ap-9438	108	24	method	method	NOUN
ap-9438	108	25	,	,	PUNCT
ap-9438	108	26	we	we	PRON
ap-9438	108	27	get	get	VERB
ap-9438	108	28	:	:	PUNCT
ap-9438	108	29	f(t	f(t	NOUN
ap-9438	108	30	)	)	PUNCT
ap-9438	108	31	=	=	PUNCT
ap-9438	109	1	π	π	PROPN
ap-9438	109	2	+	+	CCONJ
ap-9438	109	3	t	t	PROPN
ap-9438	109	4	,	,	PUNCT
ap-9438	109	5	g(t	g(t	PROPN
ap-9438	109	6	)	)	PUNCT
ap-9438	109	7	=	=	SYM
ap-9438	110	1	1	1	NUM
ap-9438	110	2	,	,	PUNCT
ap-9438	110	3	γ	γ	X
ap-9438	110	4	=	=	SYM
ap-9438	110	5	−1	−1	NOUN
ap-9438	110	6	2	2	NUM
ap-9438	110	7	.	.	PUNCT
ap-9438	111	1	now	now	ADV
ap-9438	111	2	:	:	PUNCT
ap-9438	111	3	eψ(s	eψ(s	X
ap-9438	111	4	)	)	PUNCT
ap-9438	112	1	=	=	SYM
ap-9438	112	2	rld	rld	NOUN
ap-9438	112	3	1	1	NUM
ap-9438	112	4	2	2	NUM
ap-9438	112	5	t	t	NOUN
ap-9438	112	6	(	(	PUNCT
ap-9438	112	7	π	π	PROPN
ap-9438	112	8	+	+	PROPN
ap-9438	112	9	t	t	PROPN
ap-9438	112	10	)	)	PUNCT
ap-9438	112	11	γ(1/2	γ(1/2	NOUN
ap-9438	112	12	)	)	PUNCT
ap-9438	113	1	=	=	SYM
ap-9438	113	2	2	2	NUM
ap-9438	113	3	(	(	PUNCT
ap-9438	113	4	2	2	NUM
ap-9438	113	5	t+	t+	PUNCT
ap-9438	113	6	π	π	PROPN
ap-9438	113	7	π	π	NOUN
ap-9438	113	8	√	√	PROPN
ap-9438	113	9	t	t	PROPN
ap-9438	113	10	)	)	PUNCT
ap-9438	113	11	.	.	PUNCT
ap-9438	114	1	thus	thus	ADV
ap-9438	114	2	,	,	PUNCT
ap-9438	114	3	the	the	DET
ap-9438	114	4	exact	exact	ADJ
ap-9438	114	5	solution	solution	NOUN
ap-9438	114	6	of	of	ADP
ap-9438	114	7	equation	equation	NOUN
ap-9438	114	8	(	(	PUNCT
ap-9438	114	9	22	22	NUM
ap-9438	114	10	)	)	PUNCT
ap-9438	114	11	is	be	AUX
ap-9438	114	12	given	give	VERB
ap-9438	114	13	by	by	ADP
ap-9438	114	14	:	:	PUNCT
ap-9438	114	15	ψ(t	ψ(t	PROPN
ap-9438	114	16	)	)	PUNCT
ap-9438	114	17	=	=	SYM
ap-9438	115	1	ln	ln	NOUN
ap-9438	115	2	(	(	PUNCT
ap-9438	115	3	2	2	NUM
ap-9438	115	4	2	2	NUM
ap-9438	115	5	t+	t+	PUNCT
ap-9438	115	6	π	π	PROPN
ap-9438	115	7	π	π	NOUN
ap-9438	115	8	√	√	PROPN
ap-9438	115	9	t	t	PROPN
ap-9438	115	10	)	)	PUNCT
ap-9438	115	11	.	.	PUNCT
ap-9438	116	1	example	example	NOUN
ap-9438	117	1	2	2	NUM
ap-9438	117	2	.	.	X
ap-9438	117	3	assume	assume	VERB
ap-9438	117	4	the	the	DET
ap-9438	117	5	following	follow	VERB
ap-9438	117	6	volterra	volterra	NOUN
ap-9438	117	7	integral	integral	ADJ
ap-9438	117	8	equation	equation	NOUN
ap-9438	117	9	[	[	X
ap-9438	117	10	27	27	NUM
ap-9438	117	11	]	]	SYM
ap-9438	117	12	:	:	PUNCT
ap-9438	117	13	9	9	NUM
ap-9438	117	14	7	7	NUM
ap-9438	117	15	γ	γ	X
ap-9438	117	16	(	(	PUNCT
ap-9438	117	17	2	2	NUM
ap-9438	117	18	3	3	NUM
ap-9438	117	19	)	)	PUNCT
ap-9438	117	20	γ	γ	X
ap-9438	117	21	(	(	PUNCT
ap-9438	117	22	5	5	NUM
ap-9438	117	23	6	6	NUM
ap-9438	117	24	)	)	PUNCT
ap-9438	118	1	t7/6	t7/6	NOUN
ap-9438	118	2	√	√	NOUN
ap-9438	119	1	π	π	X
ap-9438	119	2	=	=	SYM
ap-9438	119	3	∫	∫	PROPN
ap-9438	119	4	t	t	PROPN
ap-9438	119	5	0	0	NUM
ap-9438	120	1	ln(ψ(s	ln(ψ(s	NUM
ap-9438	120	2	)	)	PUNCT
ap-9438	120	3	)	)	PUNCT
ap-9438	121	1	3	3	NUM
ap-9438	121	2	√	√	NUM
ap-9438	121	3	t−	t−	PROPN
ap-9438	121	4	s	s	PART
ap-9438	121	5	ds	ds	NOUN
ap-9438	121	6	,	,	PUNCT
ap-9438	121	7	(	(	PUNCT
ap-9438	121	8	23	23	NUM
ap-9438	121	9	)	)	PUNCT
ap-9438	121	10	from	from	ADP
ap-9438	121	11	equation	equation	NOUN
ap-9438	121	12	(	(	PUNCT
ap-9438	121	13	20	20	NUM
ap-9438	121	14	)	)	PUNCT
ap-9438	121	15	,	,	PUNCT
ap-9438	121	16	we	we	PRON
ap-9438	121	17	obtain	obtain	VERB
ap-9438	121	18	:	:	PUNCT
ap-9438	121	19	f(t	f(t	PROPN
ap-9438	121	20	)	)	PUNCT
ap-9438	121	21	=	=	SYM
ap-9438	121	22	9	9	NUM
ap-9438	121	23	7	7	NUM
ap-9438	121	24	γ	γ	X
ap-9438	121	25	(	(	PUNCT
ap-9438	121	26	2	2	NUM
ap-9438	121	27	3	3	NUM
ap-9438	121	28	)	)	PUNCT
ap-9438	121	29	γ	γ	X
ap-9438	121	30	(	(	PUNCT
ap-9438	121	31	5	5	NUM
ap-9438	121	32	6	6	NUM
ap-9438	121	33	)	)	PUNCT
ap-9438	121	34	t7/6	t7/6	NOUN
ap-9438	121	35	√	√	PROPN
ap-9438	121	36	π	π	PROPN
ap-9438	121	37	,	,	PUNCT
ap-9438	121	38	g(t	g(t	PROPN
ap-9438	121	39	)	)	PUNCT
ap-9438	121	40	=	=	SYM
ap-9438	121	41	1	1	NUM
ap-9438	121	42	,	,	PUNCT
ap-9438	121	43	γ	γ	X
ap-9438	121	44	=	=	SYM
ap-9438	121	45	−1	−1	NOUN
ap-9438	121	46	3	3	NUM
ap-9438	121	47	.	.	PUNCT
ap-9438	122	1	now	now	ADV
ap-9438	122	2	:	:	PUNCT
ap-9438	122	3	ln(ψ(t	ln(ψ(t	PROPN
ap-9438	122	4	)	)	PUNCT
ap-9438	122	5	)	)	PUNCT
ap-9438	123	1	=	=	SYM
ap-9438	123	2	1	1	NUM
ap-9438	123	3	γ(2/3	γ(2/3	ADJ
ap-9438	123	4	)	)	PUNCT
ap-9438	123	5	rld	rld	NOUN
ap-9438	123	6	2	2	NUM
ap-9438	123	7	3	3	NUM
ap-9438	123	8	t	t	NOUN
ap-9438	123	9	(	(	PUNCT
ap-9438	123	10	9	9	NUM
ap-9438	123	11	7	7	NUM
ap-9438	123	12	γ	γ	X
ap-9438	123	13	(	(	PUNCT
ap-9438	123	14	2	2	NUM
ap-9438	123	15	3	3	NUM
ap-9438	123	16	)	)	PUNCT
ap-9438	123	17	γ	γ	X
ap-9438	123	18	(	(	PUNCT
ap-9438	123	19	5	5	NUM
ap-9438	123	20	6	6	NUM
ap-9438	123	21	)	)	PUNCT
ap-9438	123	22	t7/6	t7/6	NOUN
ap-9438	123	23	√	√	PROPN
ap-9438	123	24	π	π	NOUN
ap-9438	123	25	)	)	PUNCT
ap-9438	124	1	=	=	PUNCT
ap-9438	124	2	√	√	PROPN
ap-9438	124	3	t.	t.	NOUN
ap-9438	125	1	thus	thus	ADV
ap-9438	125	2	,	,	PUNCT
ap-9438	125	3	we	we	PRON
ap-9438	125	4	have	have	VERB
ap-9438	125	5	the	the	DET
ap-9438	125	6	exact	exact	ADJ
ap-9438	125	7	solution	solution	NOUN
ap-9438	125	8	of	of	ADP
ap-9438	125	9	equation	equation	NOUN
ap-9438	125	10	(	(	PUNCT
ap-9438	125	11	23	23	NUM
ap-9438	125	12	):	):	PUNCT
ap-9438	125	13	ψ(t	ψ(t	PROPN
ap-9438	125	14	)	)	PUNCT
ap-9438	126	1	=	=	SYM
ap-9438	126	2	e	e	NOUN
ap-9438	126	3	√	√	ADJ
ap-9438	126	4	t.	t.	NOUN
ap-9438	126	5	example	example	NOUN
ap-9438	126	6	3	3	X
ap-9438	126	7	.	.	PUNCT
ap-9438	126	8	suppose	suppose	VERB
ap-9438	126	9	that	that	SCONJ
ap-9438	126	10	the	the	DET
ap-9438	126	11	following	follow	VERB
ap-9438	126	12	fvie	fvie	NOUN
ap-9438	126	13	[	[	X
ap-9438	126	14	27	27	NUM
ap-9438	126	15	]	]	SYM
ap-9438	126	16	:	:	PUNCT
ap-9438	126	17	6	6	NUM
ap-9438	126	18	t5/6	t5/6	NOUN
ap-9438	126	19	(	(	PUNCT
ap-9438	126	20	11	11	NUM
ap-9438	126	21	+	+	SYM
ap-9438	126	22	6	6	NUM
ap-9438	126	23	t	t	NOUN
ap-9438	126	24	)	)	PUNCT
ap-9438	126	25	55	55	NUM
ap-9438	126	26	=	=	SYM
ap-9438	126	27	∫	∫	PROPN
ap-9438	126	28	t	t	PROPN
ap-9438	126	29	0	0	NUM
ap-9438	126	30	arcsin(ψ(s	arcsin(ψ(s	PROPN
ap-9438	126	31	)	)	PUNCT
ap-9438	126	32	)	)	PUNCT
ap-9438	126	33	6	6	NUM
ap-9438	126	34	√	√	NUM
ap-9438	126	35	t−	t−	PROPN
ap-9438	126	36	s	s	PART
ap-9438	126	37	ds	ds	NOUN
ap-9438	126	38	,	,	PUNCT
ap-9438	126	39	(	(	PUNCT
ap-9438	126	40	24	24	NUM
ap-9438	126	41	)	)	PUNCT
ap-9438	126	42	by	by	ADP
ap-9438	126	43	using	use	VERB
ap-9438	126	44	our	our	PRON
ap-9438	126	45	algorithm	algorithm	NOUN
ap-9438	126	46	,	,	PUNCT
ap-9438	126	47	we	we	PRON
ap-9438	126	48	obtain	obtain	VERB
ap-9438	126	49	:	:	PUNCT
ap-9438	126	50	f	f	X
ap-9438	126	51	=	=	SYM
ap-9438	126	52	6	6	NUM
ap-9438	126	53	t5/6	t5/6	NOUN
ap-9438	126	54	(	(	PUNCT
ap-9438	126	55	11	11	NUM
ap-9438	126	56	+	+	SYM
ap-9438	126	57	6	6	NUM
ap-9438	126	58	t	t	NOUN
ap-9438	126	59	)	)	PUNCT
ap-9438	126	60	55	55	NUM
ap-9438	126	61	,	,	PUNCT
ap-9438	126	62	g(t	g(t	PROPN
ap-9438	126	63	)	)	PUNCT
ap-9438	126	64	=	=	SYM
ap-9438	126	65	1	1	NUM
ap-9438	126	66	,	,	PUNCT
ap-9438	126	67	γ	γ	X
ap-9438	126	68	=	=	SYM
ap-9438	126	69	−1	−1	NOUN
ap-9438	126	70	6	6	NUM
ap-9438	126	71	.	.	PUNCT
ap-9438	127	1	416	416	NUM
ap-9438	127	2	vol	vol	NOUN
ap-9438	127	3	.	.	PUNCT
ap-9438	128	1	64	64	NUM
ap-9438	128	2	no	no	NOUN
ap-9438	128	3	.	.	PUNCT
ap-9438	129	1	5/2024	5/2024	NUM
ap-9438	129	2	a	a	DET
ap-9438	129	3	novel	novel	ADJ
ap-9438	129	4	approach	approach	NOUN
ap-9438	129	5	to	to	ADP
ap-9438	129	6	nonlinear	nonlinear	ADJ
ap-9438	129	7	fractional	fractional	PROPN
ap-9438	129	8	volterra	volterra	PROPN
ap-9438	129	9	integral	integral	ADJ
ap-9438	129	10	equations	equation	NOUN
ap-9438	129	11	now	now	ADV
ap-9438	129	12	:	:	PUNCT
ap-9438	129	13	arcsin	arcsin	PROPN
ap-9438	129	14	(	(	PUNCT
ap-9438	129	15	ψ(t	ψ(t	PROPN
ap-9438	129	16	)	)	PUNCT
ap-9438	129	17	)	)	PUNCT
ap-9438	130	1	=	=	X
ap-9438	130	2	rl	rl	NOUN
ap-9438	130	3	d	d	PROPN
ap-9438	130	4	3/2	3/2	NUM
ap-9438	130	5	t	t	PROPN
ap-9438	130	6	6	6	NUM
ap-9438	130	7	t5/6	t5/6	NOUN
ap-9438	130	8	(	(	PUNCT
ap-9438	130	9	11	11	NUM
ap-9438	130	10	+	+	SYM
ap-9438	130	11	6	6	NUM
ap-9438	130	12	t	t	NOUN
ap-9438	130	13	)	)	PUNCT
ap-9438	130	14	55	55	NUM
ap-9438	130	15	=	=	SYM
ap-9438	130	16	t+	t+	PUNCT
ap-9438	130	17	1	1	NUM
ap-9438	130	18	.	.	PUNCT
ap-9438	131	1	hence	hence	ADV
ap-9438	131	2	,	,	PUNCT
ap-9438	131	3	we	we	PRON
ap-9438	131	4	obtain	obtain	VERB
ap-9438	131	5	the	the	DET
ap-9438	131	6	exact	exact	ADJ
ap-9438	131	7	solution	solution	NOUN
ap-9438	131	8	of	of	ADP
ap-9438	131	9	equation	equation	NOUN
ap-9438	131	10	(	(	PUNCT
ap-9438	131	11	24	24	NUM
ap-9438	131	12	):	):	PUNCT
ap-9438	131	13	ψ(t	ψ(t	PROPN
ap-9438	131	14	)	)	PUNCT
ap-9438	131	15	=	=	SYM
ap-9438	131	16	sin	sin	NOUN
ap-9438	131	17	(	(	PUNCT
ap-9438	131	18	t+	t+	NOUN
ap-9438	131	19	1	1	NUM
ap-9438	131	20	)	)	PUNCT
ap-9438	131	21	.	.	PUNCT
ap-9438	132	1	example	example	NOUN
ap-9438	133	1	4	4	X
ap-9438	133	2	.	.	PUNCT
ap-9438	133	3	consider	consider	VERB
ap-9438	133	4	the	the	DET
ap-9438	133	5	following	follow	VERB
ap-9438	133	6	fvie	fvie	NOUN
ap-9438	133	7	of	of	ADP
ap-9438	133	8	first	first	ADJ
ap-9438	133	9	kind	kind	NOUN
ap-9438	133	10	[	[	X
ap-9438	133	11	27	27	NUM
ap-9438	133	12	]	]	PUNCT
ap-9438	133	13	:	:	PUNCT
ap-9438	133	14	√	√	PROPN
ap-9438	133	15	π	π	PROPN
ap-9438	133	16	γ	γ	X
ap-9438	133	17	(	(	PUNCT
ap-9438	133	18	5	5	NUM
ap-9438	133	19	3	3	NUM
ap-9438	133	20	)	)	PUNCT
ap-9438	133	21	38	38	NUM
ap-9438	133	22	γ	γ	X
ap-9438	133	23	(	(	PUNCT
ap-9438	133	24	19	19	NUM
ap-9438	133	25	6	6	NUM
ap-9438	133	26	)	)	PUNCT
ap-9438	133	27	(	(	PUNCT
ap-9438	133	28	10	10	NUM
ap-9438	133	29	t	t	PROPN
ap-9438	133	30	19	19	NUM
ap-9438	133	31	6	6	NUM
ap-9438	133	32	+	+	NUM
ap-9438	133	33	19	19	NUM
ap-9438	133	34	t	t	PROPN
ap-9438	133	35	13	13	NUM
ap-9438	133	36	6	6	NUM
ap-9438	133	37	)	)	PUNCT
ap-9438	134	1	=	=	SYM
ap-9438	134	2	∫	∫	PROPN
ap-9438	135	1	t	t	PROPN
ap-9438	135	2	0	0	NUM
ap-9438	135	3	√	√	PROPN
ap-9438	135	4	t−	t−	PROPN
ap-9438	135	5	s(1	s(1	PROPN
ap-9438	135	6	+	+	CCONJ
ap-9438	135	7	s)u2(s)ds	s)u2(s)ds	NOUN
ap-9438	135	8	,	,	PUNCT
ap-9438	135	9	(	(	PUNCT
ap-9438	135	10	25	25	NUM
ap-9438	135	11	)	)	PUNCT
ap-9438	135	12	by	by	ADP
ap-9438	135	13	equation	equation	NOUN
ap-9438	135	14	(	(	PUNCT
ap-9438	135	15	20	20	NUM
ap-9438	135	16	)	)	PUNCT
ap-9438	135	17	,	,	PUNCT
ap-9438	135	18	we	we	PRON
ap-9438	135	19	get	get	VERB
ap-9438	135	20	:	:	PUNCT
ap-9438	135	21	f(t	f(t	NOUN
ap-9438	135	22	)	)	PUNCT
ap-9438	135	23	=	=	PUNCT
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ap-9438	136	2	π	π	PROPN
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ap-9438	136	4	(	(	PUNCT
ap-9438	136	5	5	5	NUM
ap-9438	136	6	3	3	NUM
ap-9438	136	7	)	)	PUNCT
ap-9438	136	8	38	38	NUM
ap-9438	136	9	γ	γ	X
ap-9438	136	10	(	(	PUNCT
ap-9438	136	11	19	19	NUM
ap-9438	136	12	6	6	NUM
ap-9438	136	13	)	)	PUNCT
ap-9438	136	14	(	(	PUNCT
ap-9438	136	15	10t19/6	10t19/6	NUM
ap-9438	136	16	+	+	NUM
ap-9438	136	17	19t13/6	19t13/6	NUM
ap-9438	136	18	)	)	PUNCT
ap-9438	136	19	,	,	PUNCT
ap-9438	136	20	g(t	g(t	PROPN
ap-9438	136	21	)	)	PUNCT
ap-9438	136	22	=	=	SYM
ap-9438	136	23	1	1	NUM
ap-9438	136	24	+	+	NUM
ap-9438	136	25	t	t	PROPN
ap-9438	136	26	,	,	PUNCT
ap-9438	136	27	γ	γ	NOUN
ap-9438	136	28	=	=	SYM
ap-9438	136	29	1	1	NUM
ap-9438	136	30	2	2	NUM
ap-9438	136	31	.	.	PUNCT
ap-9438	137	1	now	now	ADV
ap-9438	137	2	:	:	PUNCT
ap-9438	137	3	u2(t)=	u2(t)=	ADJ
ap-9438	137	4	√	√	NUM
ap-9438	137	5	π	π	PROPN
ap-9438	137	6	γ	γ	X
ap-9438	137	7	(	(	PUNCT
ap-9438	137	8	5	5	NUM
ap-9438	137	9	3	3	NUM
ap-9438	137	10	)	)	PUNCT
ap-9438	137	11	38	38	NUM
ap-9438	137	12	γ	γ	X
ap-9438	137	13	(	(	PUNCT
ap-9438	137	14	19	19	NUM
ap-9438	137	15	6	6	NUM
ap-9438	137	16	)	)	PUNCT
ap-9438	137	17	γ	γ	X
ap-9438	137	18	(	(	PUNCT
ap-9438	137	19	3	3	NUM
ap-9438	137	20	2	2	NUM
ap-9438	137	21	)	)	PUNCT
ap-9438	137	22	(	(	PUNCT
ap-9438	137	23	1	1	NUM
ap-9438	137	24	+	+	NUM
ap-9438	137	25	t	t	PROPN
ap-9438	137	26	)	)	PUNCT
ap-9438	137	27	rld	rld	NOUN
ap-9438	137	28	3/2	3/2	NUM
ap-9438	137	29	t	t	NOUN
ap-9438	137	30	(	(	PUNCT
ap-9438	137	31	10	10	NUM
ap-9438	137	32	t	t	PROPN
ap-9438	137	33	19	19	NUM
ap-9438	137	34	6	6	NUM
ap-9438	137	35	+	+	NUM
ap-9438	137	36	19	19	NUM
ap-9438	137	37	t	t	PROPN
ap-9438	137	38	13	13	NUM
ap-9438	137	39	6	6	NUM
ap-9438	137	40	)	)	PUNCT
ap-9438	137	41	=	=	PUNCT
ap-9438	138	1	t5/3	t5/3	X
ap-9438	139	1	+	+	X
ap-9438	139	2	t2/3	t2/3	NUM
ap-9438	139	3	1	1	NUM
ap-9438	139	4	+	+	NUM
ap-9438	139	5	t	t	NOUN
ap-9438	139	6	=	=	SYM
ap-9438	139	7	t2/3	t2/3	PROPN
ap-9438	139	8	.	.	PUNCT
ap-9438	140	1	therefore	therefore	ADV
ap-9438	140	2	,	,	PUNCT
ap-9438	140	3	the	the	DET
ap-9438	140	4	following	follow	VERB
ap-9438	140	5	solution	solution	NOUN
ap-9438	140	6	results	result	VERB
ap-9438	140	7	from	from	ADP
ap-9438	140	8	equation	equation	NOUN
ap-9438	140	9	(	(	PUNCT
ap-9438	140	10	25	25	NUM
ap-9438	140	11	):	):	PUNCT
ap-9438	140	12	ψ(t	ψ(t	PROPN
ap-9438	140	13	)	)	PUNCT
ap-9438	140	14	=	=	SYM
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ap-9438	140	16	3	3	NUM
ap-9438	140	17	√	√	PROPN
ap-9438	140	18	t.	t.	NOUN
ap-9438	140	19	example	example	NOUN
ap-9438	141	1	5	5	X
ap-9438	141	2	.	.	PUNCT
ap-9438	141	3	let	let	VERB
ap-9438	141	4	the	the	DET
ap-9438	141	5	following	follow	VERB
ap-9438	141	6	fvie	fvie	NOUN
ap-9438	141	7	[	[	X
ap-9438	141	8	26	26	NUM
ap-9438	141	9	]	]	SYM
ap-9438	141	10	:	:	PUNCT
ap-9438	141	11	1	1	NUM
ap-9438	141	12	2	2	NUM
ap-9438	141	13	√	√	NUM
ap-9438	141	14	πx3e1	πx3e1	PROPN
ap-9438	141	15	,	,	PUNCT
ap-9438	141	16	3	3	NUM
ap-9438	141	17	2	2	NUM
ap-9438	141	18	(	(	PUNCT
ap-9438	141	19	t2	t2	NOUN
ap-9438	141	20	)	)	PUNCT
ap-9438	141	21	=	=	SYM
ap-9438	142	1	∫	∫	PROPN
ap-9438	142	2	t	t	PROPN
ap-9438	142	3	0	0	NUM
ap-9438	142	4	√	√	PROPN
ap-9438	142	5	(	(	PUNCT
ap-9438	142	6	t−	t−	PROPN
ap-9438	142	7	s)ψ(s)ds	s)ψ(s)ds	NOUN
ap-9438	142	8	,	,	PUNCT
ap-9438	142	9	(	(	PUNCT
ap-9438	142	10	26	26	NUM
ap-9438	142	11	)	)	PUNCT
ap-9438	142	12	with	with	ADP
ap-9438	142	13	equation	equation	NOUN
ap-9438	142	14	(	(	PUNCT
ap-9438	142	15	20	20	NUM
ap-9438	142	16	)	)	PUNCT
ap-9438	142	17	,	,	PUNCT
ap-9438	142	18	we	we	PRON
ap-9438	142	19	get	get	VERB
ap-9438	142	20	:	:	PUNCT
ap-9438	142	21	f(t	f(t	NOUN
ap-9438	142	22	)	)	PUNCT
ap-9438	142	23	=	=	SYM
ap-9438	142	24	1	1	NUM
ap-9438	142	25	2	2	NUM
ap-9438	142	26	√	√	NUM
ap-9438	142	27	πx3e1	πx3e1	PROPN
ap-9438	142	28	,	,	PUNCT
ap-9438	142	29	3	3	NUM
ap-9438	142	30	2	2	NUM
ap-9438	142	31	(	(	PUNCT
ap-9438	142	32	t2	t2	NOUN
ap-9438	142	33	)	)	PUNCT
ap-9438	142	34	,	,	PUNCT
ap-9438	142	35	g(t	g(t	PROPN
ap-9438	142	36	)	)	PUNCT
ap-9438	142	37	=	=	SYM
ap-9438	143	1	1	1	NUM
ap-9438	143	2	,	,	PUNCT
ap-9438	143	3	γ	γ	NOUN
ap-9438	143	4	=	=	SYM
ap-9438	143	5	1	1	NUM
ap-9438	143	6	2	2	NUM
ap-9438	143	7	.	.	PUNCT
ap-9438	144	1	now	now	ADV
ap-9438	144	2	:	:	PUNCT
ap-9438	144	3	√	√	PUNCT
ap-9438	144	4	ψ(t	ψ(t	PROPN
ap-9438	144	5	)	)	PUNCT
ap-9438	145	1	=	=	NOUN
ap-9438	145	2	1	1	NUM
ap-9438	145	3	2	2	NUM
ap-9438	145	4	rld	rld	NOUN
ap-9438	145	5	3/2	3/2	NUM
ap-9438	145	6	t	t	NOUN
ap-9438	145	7	(	(	PUNCT
ap-9438	145	8	√	√	PROPN
ap-9438	145	9	πx3e1,3/2(1	πx3e1,3/2(1	SYM
ap-9438	145	10	2	2	NUM
ap-9438	145	11	t	t	NOUN
ap-9438	145	12	)	)	PUNCT
ap-9438	145	13	)	)	PUNCT
ap-9438	146	1	=	=	SYM
ap-9438	146	2	e1,1(1	e1,1(1	X
ap-9438	146	3	2	2	NUM
ap-9438	146	4	t	t	NOUN
ap-9438	146	5	)	)	PUNCT
ap-9438	146	6	=	=	PUNCT
ap-9438	146	7	e	e	X
ap-9438	146	8	t	t	PROPN
ap-9438	146	9	2	2	NUM
ap-9438	146	10	.	.	PUNCT
ap-9438	147	1	thus	thus	ADV
ap-9438	147	2	,	,	PUNCT
ap-9438	147	3	we	we	PRON
ap-9438	147	4	get	get	VERB
ap-9438	147	5	an	an	DET
ap-9438	147	6	accurate	accurate	ADJ
ap-9438	147	7	solution	solution	NOUN
ap-9438	147	8	of	of	ADP
ap-9438	147	9	equation	equation	NOUN
ap-9438	147	10	(	(	PUNCT
ap-9438	147	11	26	26	NUM
ap-9438	147	12	):	):	PUNCT
ap-9438	147	13	ψ(t	ψ(t	PROPN
ap-9438	147	14	)	)	PUNCT
ap-9438	147	15	=	=	SYM
ap-9438	147	16	et	et	PROPN
ap-9438	147	17	.	.	PUNCT
ap-9438	147	18	example	example	NOUN
ap-9438	148	1	6	6	NUM
ap-9438	148	2	.	.	PUNCT
ap-9438	148	3	assume	assume	VERB
ap-9438	148	4	the	the	DET
ap-9438	148	5	following	follow	VERB
ap-9438	148	6	singular	singular	NOUN
ap-9438	148	7	fvie	fvie	NOUN
ap-9438	149	1	[	[	X
ap-9438	149	2	28	28	NUM
ap-9438	149	3	]	]	SYM
ap-9438	149	4	4x3/2	4x3/2	NUM
ap-9438	149	5	3	3	NUM
ap-9438	149	6	−	−	PROPN
ap-9438	149	7	32x7/2	32x7/2	NUM
ap-9438	149	8	35	35	NUM
ap-9438	149	9	=	=	SYM
ap-9438	149	10	∫	∫	PROPN
ap-9438	149	11	t	t	PROPN
ap-9438	149	12	0	0	NUM
ap-9438	149	13	ψ(s)√	ψ(s)√	PROPN
ap-9438	149	14	t−	t−	PROPN
ap-9438	149	15	s	s	PART
ap-9438	149	16	ds	ds	X
ap-9438	149	17	(	(	PUNCT
ap-9438	149	18	27	27	NUM
ap-9438	149	19	)	)	PUNCT
ap-9438	149	20	by	by	ADP
ap-9438	149	21	the	the	DET
ap-9438	149	22	previous	previous	ADJ
ap-9438	149	23	algorithm	algorithm	NOUN
ap-9438	149	24	,	,	PUNCT
ap-9438	149	25	we	we	PRON
ap-9438	149	26	get	get	VERB
ap-9438	149	27	:	:	PUNCT
ap-9438	149	28	f(t	f(t	NOUN
ap-9438	149	29	)	)	PUNCT
ap-9438	150	1	=	=	SYM
ap-9438	150	2	4x3/2	4x3/2	NUM
ap-9438	150	3	3	3	NUM
ap-9438	150	4	−	−	PROPN
ap-9438	150	5	32x7/2	32x7/2	NUM
ap-9438	150	6	35	35	NUM
ap-9438	150	7	,	,	PUNCT
ap-9438	150	8	g(t	g(t	PROPN
ap-9438	150	9	)	)	PUNCT
ap-9438	150	10	=	=	SYM
ap-9438	151	1	1	1	NUM
ap-9438	151	2	,	,	PUNCT
ap-9438	151	3	γ	γ	X
ap-9438	151	4	=	=	SYM
ap-9438	151	5	−1	−1	NOUN
ap-9438	151	6	2	2	NUM
ap-9438	151	7	.	.	PUNCT
ap-9438	152	1	now	now	ADV
ap-9438	152	2	:	:	PUNCT
ap-9438	152	3	ψ(t	ψ(t	PROPN
ap-9438	152	4	)	)	PUNCT
ap-9438	152	5	=	=	SYM
ap-9438	152	6	1	1	NUM
ap-9438	152	7	γ(1/2	γ(1/2	NOUN
ap-9438	152	8	)	)	PUNCT
ap-9438	152	9	rld	rld	NOUN
ap-9438	153	1	3/2	3/2	NUM
ap-9438	153	2	t	t	NOUN
ap-9438	153	3	(	(	PUNCT
ap-9438	153	4	4x3/2	4x3/2	PROPN
ap-9438	153	5	3	3	NUM
ap-9438	153	6	−	−	PROPN
ap-9438	153	7	32x7/2	32x7/2	NUM
ap-9438	153	8	35	35	NUM
ap-9438	153	9	)	)	PUNCT
ap-9438	153	10	.	.	PUNCT
ap-9438	154	1	thus	thus	ADV
ap-9438	154	2	,	,	PUNCT
ap-9438	154	3	the	the	DET
ap-9438	154	4	solution	solution	NOUN
ap-9438	154	5	of	of	ADP
ap-9438	154	6	equation	equation	NOUN
ap-9438	154	7	(	(	PUNCT
ap-9438	154	8	27	27	NUM
ap-9438	154	9	)	)	PUNCT
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ap-9438	154	11	:	:	PUNCT
ap-9438	154	12	ψ(t	ψ(t	PROPN
ap-9438	154	13	)	)	PUNCT
ap-9438	154	14	=	=	SYM
ap-9438	154	15	t−	t−	PROPN
ap-9438	154	16	t3	t3	PROPN
ap-9438	154	17	.	.	PUNCT
ap-9438	154	18	example	example	NOUN
ap-9438	155	1	7	7	NUM
ap-9438	155	2	.	.	PUNCT
ap-9438	155	3	let	let	VERB
ap-9438	155	4	the	the	DET
ap-9438	155	5	following	follow	VERB
ap-9438	155	6	abel	abel	PROPN
ap-9438	155	7	integral	integral	ADJ
ap-9438	155	8	equation	equation	NOUN
ap-9438	155	9	[	[	X
ap-9438	155	10	28	28	NUM
ap-9438	155	11	]	]	X
ap-9438	155	12	:	:	PUNCT
ap-9438	155	13	2	2	NUM
ap-9438	155	14	√	√	NOUN
ap-9438	155	15	t	t	PROPN
ap-9438	155	16	(	(	PUNCT
ap-9438	155	17	−48	−48	PROPN
ap-9438	155	18	t3	t3	NOUN
ap-9438	155	19	−	−	PROPN
ap-9438	155	20	56	56	NUM
ap-9438	155	21	t2	t2	NOUN
ap-9438	155	22	+	+	CCONJ
ap-9438	155	23	105	105	NUM
ap-9438	155	24	)	)	PUNCT
ap-9438	155	25	105	105	NUM
ap-9438	155	26	=	=	SYM
ap-9438	155	27	∫	∫	PROPN
ap-9438	155	28	t	t	PROPN
ap-9438	155	29	0	0	NUM
ap-9438	155	30	ψ(s)√	ψ(s)√	PROPN
ap-9438	155	31	t−	t−	PROPN
ap-9438	155	32	s	s	PART
ap-9438	155	33	ds	ds	PROPN
ap-9438	155	34	.	.	PUNCT
ap-9438	156	1	(	(	PUNCT
ap-9438	156	2	28	28	NUM
ap-9438	156	3	)	)	PUNCT
ap-9438	156	4	by	by	ADP
ap-9438	156	5	equation	equation	NOUN
ap-9438	156	6	(	(	PUNCT
ap-9438	156	7	20	20	NUM
ap-9438	156	8	)	)	PUNCT
ap-9438	156	9	,	,	PUNCT
ap-9438	156	10	we	we	PRON
ap-9438	156	11	get	get	VERB
ap-9438	156	12	:	:	PUNCT
ap-9438	156	13	f(t	f(t	NOUN
ap-9438	156	14	)	)	PUNCT
ap-9438	157	1	=	=	SYM
ap-9438	157	2	2	2	NUM
ap-9438	157	3	√	√	NUM
ap-9438	157	4	t	t	PROPN
ap-9438	157	5	(	(	PUNCT
ap-9438	157	6	−48	−48	PROPN
ap-9438	157	7	t3	t3	NOUN
ap-9438	157	8	−	−	PROPN
ap-9438	157	9	56	56	NUM
ap-9438	157	10	t2	t2	NOUN
ap-9438	157	11	+	+	CCONJ
ap-9438	157	12	105	105	NUM
ap-9438	157	13	)	)	PUNCT
ap-9438	157	14	105	105	NUM
ap-9438	157	15	,	,	PUNCT
ap-9438	157	16	g(t	g(t	PROPN
ap-9438	157	17	)	)	PUNCT
ap-9438	157	18	=	=	SYM
ap-9438	157	19	1	1	NUM
ap-9438	157	20	,	,	PUNCT
ap-9438	157	21	γ	γ	X
ap-9438	157	22	=	=	SYM
ap-9438	157	23	−1	−1	NOUN
ap-9438	157	24	2	2	NUM
ap-9438	157	25	.	.	PUNCT
ap-9438	158	1	now	now	ADV
ap-9438	158	2	:	:	PUNCT
ap-9438	158	3	ψ(t	ψ(t	PROPN
ap-9438	158	4	)	)	PUNCT
ap-9438	158	5	=	=	SYM
ap-9438	158	6	1	1	NUM
ap-9438	158	7	γ	γ	X
ap-9438	158	8	(	(	PUNCT
ap-9438	158	9	1	1	NUM
ap-9438	158	10	2	2	NUM
ap-9438	158	11	)	)	PUNCT
ap-9438	158	12	rld	rld	NOUN
ap-9438	158	13	1/2	1/2	NUM
ap-9438	158	14	t	t	NOUN
ap-9438	158	15	(	(	PUNCT
ap-9438	158	16	2	2	NUM
ap-9438	158	17	√	√	PROPN
ap-9438	158	18	t	t	PROPN
ap-9438	158	19	(	(	PUNCT
ap-9438	158	20	−48	−48	PROPN
ap-9438	158	21	t3	t3	NOUN
ap-9438	158	22	−	−	PROPN
ap-9438	158	23	56	56	NUM
ap-9438	158	24	t2	t2	NOUN
ap-9438	158	25	+	+	CCONJ
ap-9438	158	26	105	105	NUM
ap-9438	158	27	)	)	PUNCT
ap-9438	158	28	105	105	NUM
ap-9438	158	29	)	)	PUNCT
ap-9438	158	30	.	.	PUNCT
ap-9438	159	1	thus	thus	ADV
ap-9438	159	2	,	,	PUNCT
ap-9438	159	3	we	we	PRON
ap-9438	159	4	have	have	AUX
ap-9438	159	5	ψ(t	ψ(t	PROPN
ap-9438	159	6	)	)	PUNCT
ap-9438	159	7	=	=	SYM
ap-9438	160	1	1	1	NUM
ap-9438	160	2	−	−	PROPN
ap-9438	160	3	t2	t2	PROPN
ap-9438	160	4	−	−	PROPN
ap-9438	160	5	t3	t3	PROPN
ap-9438	160	6	.	.	PUNCT
ap-9438	160	7	remark	remark	PROPN
ap-9438	160	8	.	.	PUNCT
ap-9438	161	1	the	the	DET
ap-9438	161	2	exact	exact	ADJ
ap-9438	161	3	solutions	solution	NOUN
ap-9438	161	4	in	in	ADP
ap-9438	161	5	examples	example	NOUN
ap-9438	161	6	1	1	NUM
ap-9438	161	7	and	and	CCONJ
ap-9438	161	8	5	5	NUM
ap-9438	161	9	obtained	obtain	VERB
ap-9438	161	10	by	by	ADP
ap-9438	161	11	the	the	DET
ap-9438	161	12	new	new	ADJ
ap-9438	161	13	method	method	NOUN
ap-9438	161	14	are	be	AUX
ap-9438	161	15	completely	completely	ADV
ap-9438	161	16	identical	identical	ADJ
ap-9438	161	17	to	to	ADP
ap-9438	161	18	the	the	DET
ap-9438	161	19	precise	precise	ADJ
ap-9438	161	20	solutions	solution	NOUN
ap-9438	161	21	in	in	ADP
ap-9438	161	22	the	the	DET
ap-9438	161	23	reference	reference	NOUN
ap-9438	161	24	[	[	X
ap-9438	161	25	26	26	NUM
ap-9438	161	26	]	]	PUNCT
ap-9438	161	27	.	.	PUNCT
ap-9438	162	1	in	in	ADP
ap-9438	162	2	examples	example	NOUN
ap-9438	162	3	1	1	NUM
ap-9438	162	4	,	,	PUNCT
ap-9438	162	5	2	2	NUM
ap-9438	162	6	,	,	PUNCT
ap-9438	162	7	and	and	CCONJ
ap-9438	162	8	3	3	NUM
ap-9438	162	9	,	,	PUNCT
ap-9438	162	10	we	we	PRON
ap-9438	162	11	obtained	obtain	VERB
ap-9438	162	12	precise	precise	ADJ
ap-9438	162	13	solutions	solution	NOUN
ap-9438	162	14	to	to	ADP
ap-9438	162	15	the	the	DET
ap-9438	162	16	integral	integral	ADJ
ap-9438	162	17	equations	equation	NOUN
ap-9438	162	18	using	use	VERB
ap-9438	162	19	the	the	DET
ap-9438	162	20	new	new	ADJ
ap-9438	162	21	method	method	NOUN
ap-9438	162	22	,	,	PUNCT
ap-9438	162	23	identical	identical	ADJ
ap-9438	162	24	to	to	ADP
ap-9438	162	25	the	the	DET
ap-9438	162	26	exact	exact	ADJ
ap-9438	162	27	solutions	solution	NOUN
ap-9438	162	28	in	in	ADP
ap-9438	162	29	the	the	DET
ap-9438	162	30	reference	reference	NOUN
ap-9438	162	31	[	[	X
ap-9438	162	32	27	27	NUM
ap-9438	162	33	]	]	PUNCT
ap-9438	162	34	.	.	PUNCT
ap-9438	163	1	examples	example	NOUN
ap-9438	163	2	6	6	NUM
ap-9438	163	3	and	and	CCONJ
ap-9438	163	4	7	7	NUM
ap-9438	163	5	have	have	VERB
ap-9438	163	6	numerical	numerical	ADJ
ap-9438	163	7	solutions	solution	NOUN
ap-9438	163	8	that	that	PRON
ap-9438	163	9	converge	converge	VERB
ap-9438	163	10	to	to	ADP
ap-9438	163	11	the	the	DET
ap-9438	163	12	exact	exact	ADJ
ap-9438	163	13	solutions	solution	NOUN
ap-9438	163	14	we	we	PRON
ap-9438	163	15	obtained	obtain	VERB
ap-9438	163	16	using	use	VERB
ap-9438	163	17	the	the	DET
ap-9438	163	18	new	new	ADJ
ap-9438	163	19	method	method	NOUN
ap-9438	163	20	;	;	PUNCT
ap-9438	163	21	see	see	VERB
ap-9438	163	22	the	the	DET
ap-9438	163	23	reference	reference	NOUN
ap-9438	163	24	[	[	X
ap-9438	163	25	28	28	NUM
ap-9438	163	26	]	]	PUNCT
ap-9438	163	27	.	.	PUNCT
ap-9438	164	1	from	from	ADP
ap-9438	164	2	the	the	DET
ap-9438	164	3	above	above	NOUN
ap-9438	164	4	,	,	PUNCT
ap-9438	164	5	it	it	PRON
ap-9438	164	6	becomes	become	VERB
ap-9438	164	7	clear	clear	ADJ
ap-9438	164	8	that	that	SCONJ
ap-9438	164	9	the	the	DET
ap-9438	164	10	method	method	NOUN
ap-9438	164	11	that	that	PRON
ap-9438	164	12	was	be	AUX
ap-9438	164	13	used	use	VERB
ap-9438	164	14	in	in	ADP
ap-9438	164	15	this	this	DET
ap-9438	164	16	study	study	NOUN
ap-9438	164	17	is	be	AUX
ap-9438	164	18	an	an	DET
ap-9438	164	19	effective	effective	ADJ
ap-9438	164	20	and	and	CCONJ
ap-9438	164	21	efficient	efficient	ADJ
ap-9438	164	22	method	method	NOUN
ap-9438	164	23	for	for	ADP
ap-9438	164	24	solving	solve	VERB
ap-9438	164	25	fractional	fractional	ADJ
ap-9438	164	26	integral	integral	ADJ
ap-9438	164	27	equations	equation	NOUN
ap-9438	164	28	of	of	ADP
ap-9438	164	29	the	the	DET
ap-9438	164	30	first	first	ADJ
ap-9438	164	31	kind	kind	NOUN
ap-9438	164	32	.	.	PUNCT
ap-9438	165	1	this	this	DET
ap-9438	165	2	method	method	NOUN
ap-9438	165	3	is	be	AUX
ap-9438	165	4	characterised	characterise	VERB
ap-9438	165	5	by	by	ADP
ap-9438	165	6	its	its	PRON
ap-9438	165	7	simple	simple	ADJ
ap-9438	165	8	and	and	CCONJ
ap-9438	165	9	concise	concise	ADJ
ap-9438	165	10	algorithm	algorithm	NOUN
ap-9438	165	11	.	.	PUNCT
ap-9438	166	1	5	5	X
ap-9438	166	2	.	.	X
ap-9438	166	3	results	result	NOUN
ap-9438	166	4	and	and	CCONJ
ap-9438	166	5	conclusion	conclusion	NOUN
ap-9438	166	6	in	in	ADP
ap-9438	166	7	conclusion	conclusion	NOUN
ap-9438	166	8	,	,	PUNCT
ap-9438	166	9	this	this	DET
ap-9438	166	10	study	study	NOUN
ap-9438	166	11	introduces	introduce	VERB
ap-9438	166	12	a	a	DET
ap-9438	166	13	new	new	ADJ
ap-9438	166	14	method	method	NOUN
ap-9438	166	15	using	use	VERB
ap-9438	166	16	the	the	DET
ap-9438	166	17	leibniz	leibniz	NOUN
ap-9438	166	18	integration	integration	NOUN
ap-9438	166	19	rule	rule	NOUN
ap-9438	166	20	to	to	PART
ap-9438	166	21	solve	solve	VERB
ap-9438	166	22	exactly	exactly	ADV
ap-9438	166	23	nonlinear	nonlinear	ADJ
ap-9438	166	24	volterra	volterra	ADJ
ap-9438	166	25	integral	integral	ADJ
ap-9438	166	26	equations	equation	NOUN
ap-9438	166	27	of	of	ADP
ap-9438	166	28	the	the	DET
ap-9438	166	29	first	first	ADJ
ap-9438	166	30	kind	kind	NOUN
ap-9438	166	31	with	with	ADP
ap-9438	166	32	fractional	fractional	ADJ
ap-9438	166	33	orders	order	NOUN
ap-9438	166	34	.	.	PUNCT
ap-9438	167	1	the	the	DET
ap-9438	167	2	method	method	NOUN
ap-9438	167	3	’s	’s	PART
ap-9438	167	4	effectiveness	effectiveness	NOUN
ap-9438	167	5	is	be	AUX
ap-9438	167	6	demonstrated	demonstrate	VERB
ap-9438	167	7	through	through	ADP
ap-9438	167	8	illustrative	illustrative	ADJ
ap-9438	167	9	examples	example	NOUN
ap-9438	167	10	provided	provide	VERB
ap-9438	167	11	in	in	ADP
ap-9438	167	12	the	the	DET
ap-9438	167	13	study	study	NOUN
ap-9438	167	14	.	.	PUNCT
ap-9438	168	1	applying	apply	VERB
ap-9438	168	2	this	this	DET
ap-9438	168	3	approach	approach	NOUN
ap-9438	168	4	to	to	ADP
ap-9438	168	5	several	several	ADJ
ap-9438	168	6	equations	equation	NOUN
ap-9438	168	7	of	of	ADP
ap-9438	168	8	fractional	fractional	ADJ
ap-9438	168	9	orders	order	NOUN
ap-9438	168	10	consistently	consistently	ADV
ap-9438	168	11	yielded	yield	VERB
ap-9438	168	12	satisfactory	satisfactory	ADJ
ap-9438	168	13	results	result	NOUN
ap-9438	168	14	.	.	PUNCT
ap-9438	169	1	researchers	researcher	NOUN
ap-9438	169	2	are	be	AUX
ap-9438	169	3	encouraged	encourage	VERB
ap-9438	169	4	to	to	PART
ap-9438	169	5	further	far	ADV
ap-9438	169	6	develop	develop	VERB
ap-9438	169	7	and	and	CCONJ
ap-9438	169	8	refine	refine	VERB
ap-9438	169	9	this	this	DET
ap-9438	169	10	method	method	NOUN
ap-9438	169	11	for	for	ADP
ap-9438	169	12	solving	solve	VERB
ap-9438	169	13	a	a	DET
ap-9438	169	14	wide	wide	ADJ
ap-9438	169	15	range	range	NOUN
ap-9438	169	16	of	of	ADP
ap-9438	169	17	integral	integral	ADJ
ap-9438	169	18	and	and	CCONJ
ap-9438	169	19	differential	differential	ADJ
ap-9438	169	20	equations	equation	NOUN
ap-9438	169	21	.	.	PUNCT
ap-9438	170	1	data	datum	NOUN
ap-9438	170	2	availability	availability	NOUN
ap-9438	170	3	no	no	DET
ap-9438	170	4	underlying	underlie	VERB
ap-9438	170	5	data	datum	NOUN
ap-9438	170	6	were	be	AUX
ap-9438	170	7	collected	collect	VERB
ap-9438	170	8	or	or	CCONJ
ap-9438	170	9	produced	produce	VERB
ap-9438	170	10	in	in	ADP
ap-9438	170	11	this	this	DET
ap-9438	170	12	study	study	NOUN
ap-9438	170	13	.	.	PUNCT
ap-9438	171	1	conflicts	conflict	NOUN
ap-9438	171	2	of	of	ADP
ap-9438	171	3	interest	interest	NOUN
ap-9438	171	4	we	we	PRON
ap-9438	171	5	,	,	PUNCT
ap-9438	171	6	the	the	DET
ap-9438	171	7	authors	author	NOUN
ap-9438	171	8	,	,	PUNCT
ap-9438	171	9	declare	declare	VERB
ap-9438	171	10	no	no	DET
ap-9438	171	11	conflict	conflict	NOUN
ap-9438	171	12	of	of	ADP
ap-9438	171	13	interest	interest	NOUN
ap-9438	171	14	.	.	PUNCT
ap-9438	172	1	funding	fund	VERB
ap-9438	172	2	the	the	DET
ap-9438	172	3	authors	author	NOUN
ap-9438	172	4	did	do	AUX
ap-9438	172	5	not	not	PART
ap-9438	172	6	receive	receive	VERB
ap-9438	172	7	any	any	DET
ap-9438	172	8	financial	financial	ADJ
ap-9438	172	9	support	support	NOUN
ap-9438	172	10	.	.	PUNCT
ap-9438	173	1	acknowledgements	acknowledgement	NOUN
ap-9438	173	2	the	the	DET
ap-9438	173	3	authors	author	NOUN
ap-9438	173	4	would	would	AUX
ap-9438	173	5	like	like	VERB
ap-9438	173	6	to	to	PART
ap-9438	173	7	express	express	VERB
ap-9438	173	8	their	their	PRON
ap-9438	173	9	sincere	sincere	ADJ
ap-9438	173	10	gratitude	gratitude	NOUN
ap-9438	173	11	to	to	ADP
ap-9438	173	12	all	all	DET
ap-9438	173	13	those	those	PRON
ap-9438	173	14	who	who	PRON
ap-9438	173	15	contributed	contribute	VERB
ap-9438	173	16	to	to	ADP
ap-9438	173	17	this	this	DET
ap-9438	173	18	work	work	NOUN
ap-9438	173	19	.	.	PUNCT
ap-9438	174	1	417	417	NUM
ap-9438	174	2	mohammed	mohammed	PROPN
ap-9438	174	3	abdulshareef	abdulshareef	PROPN
ap-9438	174	4	hussein	hussein	PROPN
ap-9438	174	5	,	,	PUNCT
ap-9438	174	6	hassan	hassan	PROPN
ap-9438	174	7	kamil	kamil	PROPN
ap-9438	174	8	jassim	jassim	PROPN
ap-9438	174	9	acta	acta	PROPN
ap-9438	174	10	polytechnica	polytechnica	PROPN
ap-9438	174	11	references	reference	NOUN
ap-9438	174	12	[	[	X
ap-9438	174	13	1	1	NUM
ap-9438	174	14	]	]	PUNCT
ap-9438	174	15	m.	m.	NOUN
ap-9438	174	16	belmekki	belmekki	PROPN
ap-9438	174	17	,	,	PUNCT
ap-9438	174	18	m.	m.	NOUN
ap-9438	174	19	benchohra	benchohra	NOUN
ap-9438	174	20	.	.	PUNCT
ap-9438	175	1	existence	existence	NOUN
ap-9438	175	2	results	result	VERB
ap-9438	175	3	for	for	ADP
ap-9438	175	4	fractional	fractional	ADJ
ap-9438	175	5	order	order	NOUN
ap-9438	175	6	semilinear	semilinear	PROPN
ap-9438	175	7	functional	functional	ADJ
ap-9438	175	8	differential	differential	ADJ
ap-9438	175	9	equations	equation	NOUN
ap-9438	175	10	with	with	ADP
ap-9438	175	11	nondense	nondense	NOUN
ap-9438	175	12	domain	domain	NOUN
ap-9438	175	13	.	.	PUNCT
ap-9438	176	1	nonlinear	nonlinear	ADJ
ap-9438	176	2	analysis	analysis	NOUN
ap-9438	176	3	:	:	PUNCT
ap-9438	176	4	theory	theory	NOUN
ap-9438	176	5	,	,	PUNCT
ap-9438	176	6	methods	method	NOUN
ap-9438	176	7	&	&	CCONJ
ap-9438	176	8	applications	application	NOUN
ap-9438	176	9	72(2):925–932	72(2):925–932	PROPN
ap-9438	176	10	,	,	PUNCT
ap-9438	176	11	2010	2010	NUM
ap-9438	176	12	.	.	PUNCT
ap-9438	177	1	https://doi.org/10.1016/j.na.2009.07.034	https://doi.org/10.1016/j.na.2009.07.034	PUNCT
ap-9438	177	2	[	[	X
ap-9438	177	3	2	2	NUM
ap-9438	177	4	]	]	PUNCT
ap-9438	177	5	h.	h.	PROPN
ap-9438	177	6	k.	k.	PROPN
ap-9438	177	7	jassim	jassim	PROPN
ap-9438	177	8	,	,	PUNCT
ap-9438	177	9	m.	m.	NOUN
ap-9438	177	10	a.	a.	PROPN
ap-9438	177	11	hussein	hussein	PROPN
ap-9438	177	12	.	.	PUNCT
ap-9438	178	1	a	a	DET
ap-9438	178	2	new	new	ADJ
ap-9438	178	3	approach	approach	NOUN
ap-9438	178	4	for	for	ADP
ap-9438	178	5	solving	solve	VERB
ap-9438	178	6	nonlinear	nonlinear	ADJ
ap-9438	178	7	fractional	fractional	ADJ
ap-9438	178	8	ordinary	ordinary	ADJ
ap-9438	178	9	differential	differential	ADJ
ap-9438	178	10	equations	equation	NOUN
ap-9438	178	11	.	.	PUNCT
ap-9438	179	1	mathematics	mathematic	NOUN
ap-9438	179	2	11(7):1565	11(7):1565	NUM
ap-9438	179	3	,	,	PUNCT
ap-9438	179	4	2023	2023	NUM
ap-9438	179	5	.	.	PUNCT
ap-9438	180	1	https://doi.org/10.3390/math11071565	https://doi.org/10.3390/math11071565	PROPN
ap-9438	180	2	[	[	X
ap-9438	180	3	3	3	NUM
ap-9438	180	4	]	]	PUNCT
ap-9438	180	5	m.	m.	NOUN
ap-9438	180	6	a.	a.	PROPN
ap-9438	180	7	hussein	hussein	PROPN
ap-9438	180	8	,	,	PUNCT
ap-9438	180	9	h.	h.	PROPN
ap-9438	180	10	k.	k.	PROPN
ap-9438	180	11	jassim	jassim	PROPN
ap-9438	180	12	.	.	PUNCT
ap-9438	181	1	analysis	analysis	NOUN
ap-9438	181	2	of	of	ADP
ap-9438	181	3	fractional	fractional	ADJ
ap-9438	181	4	differential	differential	ADJ
ap-9438	181	5	equations	equation	NOUN
ap-9438	181	6	with	with	ADP
ap-9438	181	7	antagana	antagana	PROPN
ap-9438	181	8	-	-	PUNCT
ap-9438	181	9	baleanu	baleanu	ADJ
ap-9438	181	10	fractional	fractional	ADJ
ap-9438	181	11	operator	operator	NOUN
ap-9438	181	12	.	.	PUNCT
ap-9438	182	1	progress	progress	NOUN
ap-9438	182	2	in	in	ADP
ap-9438	182	3	fractional	fractional	ADJ
ap-9438	182	4	differentiation	differentiation	NOUN
ap-9438	182	5	and	and	CCONJ
ap-9438	182	6	applications	application	NOUN
ap-9438	182	7	9(4):681–686	9(4):681–686	NUM
ap-9438	182	8	,	,	PUNCT
ap-9438	182	9	2023	2023	NUM
ap-9438	182	10	.	.	PUNCT
ap-9438	183	1	https://doi.org/10.18576/pfda/090411	https://doi.org/10.18576/pfda/090411	PROPN
ap-9438	184	1	[	[	X
ap-9438	184	2	4	4	NUM
ap-9438	184	3	]	]	PUNCT
ap-9438	184	4	m.	m.	NOUN
ap-9438	184	5	higazy	higazy	NOUN
ap-9438	184	6	,	,	PUNCT
ap-9438	184	7	s.	s.	PROPN
ap-9438	184	8	aggarwal	aggarwal	PROPN
ap-9438	184	9	,	,	PUNCT
ap-9438	184	10	t.	t.	PROPN
ap-9438	184	11	a.	a.	NOUN
ap-9438	184	12	nofal	nofal	PROPN
ap-9438	184	13	.	.	PUNCT
ap-9438	185	1	sawi	sawi	ADJ
ap-9438	185	2	decomposition	decomposition	NOUN
ap-9438	185	3	method	method	NOUN
ap-9438	185	4	for	for	ADP
ap-9438	185	5	volterra	volterra	NOUN
ap-9438	185	6	integral	integral	ADJ
ap-9438	185	7	equation	equation	NOUN
ap-9438	185	8	with	with	ADP
ap-9438	185	9	application	application	NOUN
ap-9438	185	10	.	.	PUNCT
ap-9438	186	1	journal	journal	NOUN
ap-9438	186	2	of	of	ADP
ap-9438	186	3	mathematics	mathematics	PROPN
ap-9438	186	4	2020(1):6687134	2020(1):6687134	NUM
ap-9438	186	5	,	,	PUNCT
ap-9438	186	6	2020	2020	NUM
ap-9438	186	7	.	.	PUNCT
ap-9438	187	1	https://doi.org/10.1155/2020/6687134	https://doi.org/10.1155/2020/6687134	PROPN
ap-9438	188	1	[	[	X
ap-9438	188	2	5	5	X
ap-9438	188	3	]	]	PUNCT
ap-9438	188	4	s.	s.	PROPN
ap-9438	188	5	jahanshahi	jahanshahi	PROPN
ap-9438	188	6	,	,	PUNCT
ap-9438	188	7	e.	e.	PROPN
ap-9438	188	8	babolian	babolian	PROPN
ap-9438	188	9	,	,	PUNCT
ap-9438	188	10	d.	d.	PROPN
ap-9438	188	11	f.	f.	PROPN
ap-9438	188	12	m.	m.	PROPN
ap-9438	188	13	torres	torres	PROPN
ap-9438	188	14	,	,	PUNCT
ap-9438	188	15	a.	a.	NOUN
ap-9438	188	16	vahidi	vahidi	NOUN
ap-9438	188	17	.	.	PUNCT
ap-9438	189	1	solving	solve	VERB
ap-9438	189	2	abel	abel	PROPN
ap-9438	189	3	integral	integral	ADJ
ap-9438	189	4	equations	equation	NOUN
ap-9438	189	5	of	of	ADP
ap-9438	189	6	first	first	ADJ
ap-9438	189	7	kind	kind	NOUN
ap-9438	189	8	via	via	ADP
ap-9438	189	9	fractional	fractional	ADJ
ap-9438	189	10	calculus	calculus	NOUN
ap-9438	189	11	.	.	PUNCT
ap-9438	190	1	journal	journal	PROPN
ap-9438	190	2	of	of	ADP
ap-9438	190	3	king	king	PROPN
ap-9438	190	4	saud	saud	PROPN
ap-9438	190	5	university	university	PROPN
ap-9438	190	6	–	–	PUNCT
ap-9438	190	7	science	science	NOUN
ap-9438	190	8	27(2):161–167	27(2):161–167	PROPN
ap-9438	190	9	,	,	PUNCT
ap-9438	190	10	2015	2015	NUM
ap-9438	190	11	.	.	PUNCT
ap-9438	191	1	https://doi.org/10.1016/j.jksus.2014.09.004	https://doi.org/10.1016/j.jksus.2014.09.004	NOUN
ap-9438	191	2	[	[	X
ap-9438	191	3	6	6	NUM
ap-9438	191	4	]	]	X
ap-9438	191	5	g.	g.	PROPN
ap-9438	191	6	pellicane	pellicane	PROPN
ap-9438	191	7	,	,	PUNCT
ap-9438	191	8	l.	l.	PROPN
ap-9438	191	9	l.	l.	PROPN
ap-9438	191	10	lee	lee	PROPN
ap-9438	191	11	,	,	PUNCT
ap-9438	191	12	c.	c.	PROPN
ap-9438	191	13	caccamo	caccamo	PROPN
ap-9438	191	14	.	.	PUNCT
ap-9438	192	1	integral	integral	ADJ
ap-9438	192	2	-	-	PUNCT
ap-9438	192	3	equation	equation	NOUN
ap-9438	192	4	theories	theory	NOUN
ap-9438	192	5	of	of	ADP
ap-9438	192	6	fluid	fluid	ADJ
ap-9438	192	7	phase	phase	NOUN
ap-9438	192	8	equilibria	equilibrium	NOUN
ap-9438	192	9	in	in	ADP
ap-9438	192	10	simple	simple	ADJ
ap-9438	192	11	fluids	fluid	NOUN
ap-9438	192	12	.	.	PUNCT
ap-9438	193	1	fluid	fluid	ADJ
ap-9438	193	2	phase	phase	NOUN
ap-9438	193	3	equilibria	equilibrium	NOUN
ap-9438	193	4	521:112665	521:112665	NUM
ap-9438	193	5	,	,	PUNCT
ap-9438	193	6	2020	2020	NUM
ap-9438	193	7	.	.	PUNCT
ap-9438	194	1	https://doi.org/10.1016/j.fluid.2020.112665	https://doi.org/10.1016/j.fluid.2020.112665	NOUN
ap-9438	194	2	[	[	X
ap-9438	194	3	7	7	NUM
ap-9438	194	4	]	]	PUNCT
ap-9438	194	5	m.	m.	NOUN
ap-9438	194	6	a.	a.	PROPN
ap-9438	194	7	hussein	hussein	PROPN
ap-9438	194	8	,	,	PUNCT
ap-9438	194	9	h.	h.	PROPN
ap-9438	194	10	k.	k.	PROPN
ap-9438	194	11	jassim	jassim	PROPN
ap-9438	194	12	,	,	PUNCT
ap-9438	194	13	a.	a.	PROPN
ap-9438	194	14	k.	k.	PROPN
ap-9438	194	15	jassim	jassim	PROPN
ap-9438	194	16	.	.	PUNCT
ap-9438	195	1	an	an	DET
ap-9438	195	2	innovative	innovative	ADJ
ap-9438	195	3	iterative	iterative	NOUN
ap-9438	195	4	approach	approach	NOUN
ap-9438	195	5	to	to	ADP
ap-9438	195	6	solving	solve	VERB
ap-9438	195	7	volterra	volterra	NOUN
ap-9438	195	8	integral	integral	ADJ
ap-9438	195	9	equations	equation	NOUN
ap-9438	195	10	of	of	ADP
ap-9438	195	11	second	second	ADJ
ap-9438	195	12	kind	kind	NOUN
ap-9438	195	13	.	.	PUNCT
ap-9438	196	1	acta	acta	PROPN
ap-9438	196	2	polytechnica	polytechnica	PROPN
ap-9438	196	3	64(2):87–102	64(2):87–102	NUM
ap-9438	196	4	,	,	PUNCT
ap-9438	196	5	2024	2024	NUM
ap-9438	196	6	.	.	PUNCT
ap-9438	197	1	https://doi.org/10.14311/ap.2024.64.0087	https://doi.org/10.14311/ap.2024.64.0087	NOUN
ap-9438	198	1	[	[	X
ap-9438	198	2	8	8	X
ap-9438	198	3	]	]	X
ap-9438	198	4	h.	h.	NOUN
ap-9438	198	5	mottaghi	mottaghi	PROPN
ap-9438	198	6	golshan	golshan	PROPN
ap-9438	198	7	.	.	PUNCT
ap-9438	199	1	numerical	numerical	ADJ
ap-9438	199	2	solution	solution	NOUN
ap-9438	199	3	of	of	ADP
ap-9438	199	4	nonlinear	nonlinear	ADJ
ap-9438	199	5	m	m	ADJ
ap-9438	199	6	-	-	ADJ
ap-9438	199	7	dimensional	dimensional	ADJ
ap-9438	199	8	fredholm	fredholm	ADJ
ap-9438	199	9	integral	integral	ADJ
ap-9438	199	10	equations	equation	NOUN
ap-9438	199	11	using	use	VERB
ap-9438	199	12	iterative	iterative	NOUN
ap-9438	199	13	newton	newton	PROPN
ap-9438	199	14	-	-	PUNCT
ap-9438	199	15	cotes	cote	NOUN
ap-9438	199	16	rules	rule	NOUN
ap-9438	199	17	.	.	PUNCT
ap-9438	200	1	journal	journal	NOUN
ap-9438	200	2	of	of	ADP
ap-9438	200	3	computational	computational	ADJ
ap-9438	200	4	and	and	CCONJ
ap-9438	200	5	applied	apply	VERB
ap-9438	200	6	mathematics	mathematics	PROPN
ap-9438	200	7	448:115917	448:115917	NUM
ap-9438	200	8	,	,	PUNCT
ap-9438	200	9	2024	2024	NUM
ap-9438	200	10	.	.	PUNCT
ap-9438	200	11	https://doi.org/10.1016/j.cam.2024.115917	https://doi.org/10.1016/j.cam.2024.115917	PUNCT
ap-9438	201	1	[	[	X
ap-9438	201	2	9	9	NUM
ap-9438	201	3	]	]	PUNCT
ap-9438	201	4	m.	m.	NOUN
ap-9438	201	5	a.	a.	PROPN
ap-9438	201	6	hussein	hussein	PROPN
ap-9438	201	7	.	.	PUNCT
ap-9438	202	1	the	the	DET
ap-9438	202	2	approximate	approximate	ADJ
ap-9438	202	3	solutions	solution	NOUN
ap-9438	202	4	of	of	ADP
ap-9438	202	5	fractional	fractional	ADJ
ap-9438	202	6	differential	differential	ADJ
ap-9438	202	7	equations	equation	NOUN
ap-9438	202	8	with	with	ADP
ap-9438	202	9	antagana	antagana	PROPN
ap-9438	202	10	-	-	PUNCT
ap-9438	202	11	baleanu	baleanu	ADJ
ap-9438	202	12	fractional	fractional	ADJ
ap-9438	202	13	operator	operator	NOUN
ap-9438	202	14	.	.	PUNCT
ap-9438	203	1	mathematics	mathematic	NOUN
ap-9438	203	2	and	and	CCONJ
ap-9438	203	3	computational	computational	ADJ
ap-9438	203	4	sciences	science	NOUN
ap-9438	203	5	3(3):29–39	3(3):29–39	NUM
ap-9438	203	6	,	,	PUNCT
ap-9438	203	7	2022	2022	NUM
ap-9438	203	8	.	.	PUNCT
ap-9438	204	1	https://doi.org/10.30511/mcs.2022.560414.1077	https://doi.org/10.30511/mcs.2022.560414.1077	X
ap-9438	205	1	[	[	X
ap-9438	205	2	10	10	NUM
ap-9438	205	3	]	]	X
ap-9438	205	4	h.	h.	PROPN
ap-9438	205	5	k.	k.	PROPN
ap-9438	205	6	jassim	jassim	PROPN
ap-9438	205	7	,	,	PUNCT
ap-9438	205	8	m.	m.	NOUN
ap-9438	205	9	a.	a.	PROPN
ap-9438	205	10	s.	s.	PROPN
ap-9438	205	11	hussain	hussain	PROPN
ap-9438	205	12	.	.	PUNCT
ap-9438	206	1	on	on	ADP
ap-9438	206	2	approximate	approximate	ADJ
ap-9438	206	3	solutions	solution	NOUN
ap-9438	206	4	for	for	ADP
ap-9438	206	5	fractional	fractional	ADJ
ap-9438	206	6	system	system	NOUN
ap-9438	206	7	of	of	ADP
ap-9438	206	8	differential	differential	ADJ
ap-9438	206	9	equations	equation	NOUN
ap-9438	206	10	with	with	ADP
ap-9438	206	11	caputo	caputo	PROPN
ap-9438	206	12	-	-	PUNCT
ap-9438	206	13	fabrizio	fabrizio	PROPN
ap-9438	206	14	fractional	fractional	ADJ
ap-9438	206	15	operator	operator	NOUN
ap-9438	206	16	.	.	PUNCT
ap-9438	207	1	journal	journal	PROPN
ap-9438	207	2	of	of	ADP
ap-9438	207	3	mathematics	mathematic	NOUN
ap-9438	207	4	and	and	CCONJ
ap-9438	207	5	computer	computer	NOUN
ap-9438	207	6	science	science	NOUN
ap-9438	207	7	23(1):58–66	23(1):58–66	NUM
ap-9438	207	8	,	,	PUNCT
ap-9438	207	9	2021	2021	NUM
ap-9438	207	10	.	.	PUNCT
ap-9438	208	1	https://doi.org/10.22436/jmcs.023.01.06	https://doi.org/10.22436/jmcs.023.01.06	NUM
ap-9438	209	1	[	[	X
ap-9438	209	2	11	11	NUM
ap-9438	209	3	]	]	X
ap-9438	209	4	r.	r.	PROPN
ap-9438	209	5	katani	katani	PROPN
ap-9438	209	6	,	,	PUNCT
ap-9438	209	7	s.	s.	PROPN
ap-9438	209	8	mckee	mckee	PROPN
ap-9438	209	9	.	.	PUNCT
ap-9438	210	1	a	a	DET
ap-9438	210	2	hybrid	hybrid	NOUN
ap-9438	210	3	legendre	legendre	NOUN
ap-9438	210	4	block	block	PROPN
ap-9438	210	5	-	-	PUNCT
ap-9438	210	6	pulse	pulse	NOUN
ap-9438	210	7	method	method	NOUN
ap-9438	210	8	for	for	ADP
ap-9438	210	9	mixed	mixed	ADJ
ap-9438	210	10	volterra	volterra	NOUN
ap-9438	210	11	-	-	PUNCT
ap-9438	210	12	fredholm	fredholm	NOUN
ap-9438	210	13	integral	integral	ADJ
ap-9438	210	14	equations	equation	NOUN
ap-9438	210	15	.	.	PUNCT
ap-9438	211	1	journal	journal	NOUN
ap-9438	211	2	of	of	ADP
ap-9438	211	3	computational	computational	ADJ
ap-9438	211	4	and	and	CCONJ
ap-9438	211	5	applied	apply	VERB
ap-9438	211	6	mathematics	mathematics	NOUN
ap-9438	211	7	376:112867	376:112867	NUM
ap-9438	211	8	,	,	PUNCT
ap-9438	211	9	2020	2020	NUM
ap-9438	211	10	.	.	PUNCT
ap-9438	212	1	https://doi.org/10.1016/j.cam.2020.112867	https://doi.org/10.1016/j.cam.2020.112867	PUNCT
ap-9438	213	1	[	[	X
ap-9438	213	2	12	12	NUM
ap-9438	213	3	]	]	PUNCT
ap-9438	213	4	h.	h.	PROPN
ap-9438	213	5	k.	k.	PROPN
ap-9438	213	6	jassim	jassim	PROPN
ap-9438	213	7	,	,	PUNCT
ap-9438	213	8	m.	m.	NOUN
ap-9438	213	9	a.	a.	PROPN
ap-9438	213	10	hussein	hussein	PROPN
ap-9438	213	11	.	.	PUNCT
ap-9438	214	1	a	a	DET
ap-9438	214	2	novel	novel	ADJ
ap-9438	214	3	formulation	formulation	NOUN
ap-9438	214	4	of	of	ADP
ap-9438	214	5	the	the	DET
ap-9438	214	6	fractional	fractional	ADJ
ap-9438	214	7	derivative	derivative	NOUN
ap-9438	214	8	with	with	ADP
ap-9438	214	9	the	the	DET
ap-9438	214	10	order	order	NOUN
ap-9438	214	11	α	α	PRON
ap-9438	214	12	≥	≥	NOUN
ap-9438	214	13	0	0	NUM
ap-9438	214	14	and	and	CCONJ
ap-9438	214	15	without	without	ADP
ap-9438	214	16	the	the	DET
ap-9438	214	17	singular	singular	ADJ
ap-9438	214	18	kernel	kernel	NOUN
ap-9438	214	19	.	.	PUNCT
ap-9438	215	1	mathematics	mathematics	PROPN
ap-9438	215	2	10(21):4123	10(21):4123	NUM
ap-9438	215	3	,	,	PUNCT
ap-9438	215	4	2022	2022	NUM
ap-9438	215	5	.	.	PUNCT
ap-9438	216	1	https://doi.org/10.3390/math10214123	https://doi.org/10.3390/math10214123	NOUN
ap-9438	216	2	[	[	X
ap-9438	216	3	13	13	NUM
ap-9438	216	4	]	]	PUNCT
ap-9438	216	5	j.	j.	PROPN
ap-9438	216	6	wu	wu	PROPN
ap-9438	216	7	,	,	PUNCT
ap-9438	216	8	y.	y.	PROPN
ap-9438	216	9	zhou	zhou	PROPN
ap-9438	216	10	,	,	PUNCT
ap-9438	216	11	c.	c.	PROPN
ap-9438	216	12	hang	hang	PROPN
ap-9438	216	13	.	.	PUNCT
ap-9438	217	1	a	a	DET
ap-9438	217	2	singularity	singularity	NOUN
ap-9438	217	3	free	free	ADJ
ap-9438	217	4	and	and	CCONJ
ap-9438	217	5	derivative	derivative	ADJ
ap-9438	217	6	free	free	ADJ
ap-9438	217	7	approach	approach	NOUN
ap-9438	217	8	for	for	ADP
ap-9438	217	9	abel	abel	PROPN
ap-9438	217	10	integral	integral	ADJ
ap-9438	217	11	equation	equation	NOUN
ap-9438	217	12	in	in	ADP
ap-9438	217	13	analyzing	analyze	VERB
ap-9438	217	14	the	the	DET
ap-9438	217	15	laser	laser	NOUN
ap-9438	217	16	-	-	PUNCT
ap-9438	217	17	induced	induce	VERB
ap-9438	217	18	breakdown	breakdown	NOUN
ap-9438	217	19	spectroscopy	spectroscopy	NOUN
ap-9438	217	20	.	.	PUNCT
ap-9438	218	1	spectrochimica	spectrochimica	PROPN
ap-9438	218	2	acta	acta	PROPN
ap-9438	218	3	part	part	PROPN
ap-9438	218	4	b	b	PROPN
ap-9438	218	5	:	:	PUNCT
ap-9438	218	6	atomic	atomic	ADJ
ap-9438	218	7	spectroscopy	spectroscopy	NOUN
ap-9438	218	8	167:105791	167:105791	NUM
ap-9438	218	9	,	,	PUNCT
ap-9438	218	10	2020	2020	NUM
ap-9438	218	11	.	.	PUNCT
ap-9438	218	12	https://doi.org/10.1016/j.sab.2020.105791	https://doi.org/10.1016/j.sab.2020.105791	PUNCT
ap-9438	219	1	[	[	X
ap-9438	219	2	14	14	NUM
ap-9438	219	3	]	]	PUNCT
ap-9438	219	4	m.	m.	NOUN
ap-9438	219	5	a.	a.	PROPN
ap-9438	219	6	hussein	hussein	PROPN
ap-9438	219	7	.	.	PUNCT
ap-9438	220	1	using	use	VERB
ap-9438	220	2	the	the	DET
ap-9438	220	3	elzaki	elzaki	NOUN
ap-9438	220	4	decomposition	decomposition	NOUN
ap-9438	220	5	method	method	NOUN
ap-9438	220	6	to	to	PART
ap-9438	220	7	solve	solve	VERB
ap-9438	220	8	nonlinear	nonlinear	ADJ
ap-9438	220	9	fractional	fractional	ADJ
ap-9438	220	10	differential	differential	ADJ
ap-9438	220	11	equations	equation	NOUN
ap-9438	220	12	with	with	ADP
ap-9438	220	13	the	the	DET
ap-9438	220	14	caputo	caputo	PROPN
ap-9438	220	15	-	-	PUNCT
ap-9438	220	16	fabrizio	fabrizio	PROPN
ap-9438	220	17	fractional	fractional	ADJ
ap-9438	220	18	operator	operator	NOUN
ap-9438	220	19	.	.	PUNCT
ap-9438	221	1	baghdad	baghdad	PROPN
ap-9438	221	2	science	science	PROPN
ap-9438	221	3	journal	journal	PROPN
ap-9438	221	4	21(3):1044–1054	21(3):1044–1054	NUM
ap-9438	221	5	,	,	PUNCT
ap-9438	221	6	2024	2024	NUM
ap-9438	221	7	.	.	PUNCT
ap-9438	222	1	https://doi.org/10.21123/bsj.2023.7310	https://doi.org/10.21123/bsj.2023.7310	PRON
ap-9438	222	2	[	[	X
ap-9438	222	3	15	15	NUM
ap-9438	222	4	]	]	X
ap-9438	222	5	s.	s.	PROPN
ap-9438	222	6	aggarwal	aggarwal	PROPN
ap-9438	222	7	,	,	PUNCT
ap-9438	222	8	n.	n.	PROPN
ap-9438	222	9	sharma	sharma	PROPN
ap-9438	222	10	.	.	PUNCT
ap-9438	223	1	laplace	laplace	PROPN
ap-9438	223	2	transform	transform	VERB
ap-9438	223	3	for	for	ADP
ap-9438	223	4	the	the	DET
ap-9438	223	5	solution	solution	NOUN
ap-9438	223	6	of	of	ADP
ap-9438	223	7	first	first	ADJ
ap-9438	223	8	kind	kind	ADJ
ap-9438	223	9	linear	linear	PROPN
ap-9438	223	10	volterra	volterra	PROPN
ap-9438	223	11	integral	integral	ADJ
ap-9438	223	12	equation	equation	NOUN
ap-9438	223	13	.	.	PUNCT
ap-9438	224	1	journal	journal	NOUN
ap-9438	224	2	of	of	ADP
ap-9438	224	3	advanced	advanced	ADJ
ap-9438	224	4	research	research	NOUN
ap-9438	224	5	in	in	ADP
ap-9438	224	6	applied	apply	VERB
ap-9438	224	7	mathematics	mathematic	NOUN
ap-9438	224	8	and	and	CCONJ
ap-9438	224	9	statistics	statistic	NOUN
ap-9438	224	10	4(3&4):16–23	4(3&4):16–23	PROPN
ap-9438	224	11	,	,	PUNCT
ap-9438	224	12	2019	2019	NUM
ap-9438	224	13	.	.	PUNCT
ap-9438	225	1	[	[	X
ap-9438	225	2	16	16	NUM
ap-9438	225	3	]	]	X
ap-9438	225	4	s.	s.	PROPN
ap-9438	225	5	aggarwal	aggarwal	PROPN
ap-9438	225	6	,	,	PUNCT
ap-9438	225	7	r.	r.	PROPN
ap-9438	225	8	kumar	kumar	PROPN
ap-9438	225	9	,	,	PUNCT
ap-9438	225	10	j.	j.	PROPN
ap-9438	225	11	chandel	chandel	PROPN
ap-9438	225	12	.	.	PUNCT
ap-9438	225	13	solution	solution	NOUN
ap-9438	225	14	of	of	ADP
ap-9438	225	15	linear	linear	PROPN
ap-9438	225	16	volterra	volterra	PROPN
ap-9438	225	17	integral	integral	ADJ
ap-9438	225	18	equation	equation	NOUN
ap-9438	225	19	of	of	ADP
ap-9438	225	20	second	second	ADJ
ap-9438	225	21	kind	kind	NOUN
ap-9438	225	22	via	via	ADP
ap-9438	225	23	rishi	rishi	PROPN
ap-9438	225	24	transform	transform	PROPN
ap-9438	225	25	.	.	PUNCT
ap-9438	226	1	journal	journal	PROPN
ap-9438	226	2	of	of	ADP
ap-9438	226	3	scientific	scientific	ADJ
ap-9438	226	4	research	research	NOUN
ap-9438	226	5	15(1):111–119	15(1):111–119	PROPN
ap-9438	226	6	,	,	PUNCT
ap-9438	226	7	2023	2023	NUM
ap-9438	226	8	.	.	PUNCT
ap-9438	227	1	https://doi.org/10.3329/jsr.v15i1.60337	https://doi.org/10.3329/jsr.v15i1.60337	NOUN
ap-9438	228	1	[	[	X
ap-9438	228	2	17	17	NUM
ap-9438	228	3	]	]	PUNCT
ap-9438	228	4	k.	k.	PROPN
ap-9438	228	5	sayevand	sayevand	PROPN
ap-9438	228	6	,	,	PUNCT
ap-9438	228	7	j.	j.	PROPN
ap-9438	228	8	tenreiro	tenreiro	PROPN
ap-9438	228	9	machado	machado	PROPN
ap-9438	228	10	,	,	PUNCT
ap-9438	228	11	d.	d.	PROPN
ap-9438	228	12	baleanu	baleanu	PROPN
ap-9438	228	13	.	.	PUNCT
ap-9438	229	1	a	a	DET
ap-9438	229	2	new	new	ADJ
ap-9438	229	3	glance	glance	NOUN
ap-9438	229	4	on	on	ADP
ap-9438	229	5	the	the	DET
ap-9438	229	6	leibniz	leibniz	NOUN
ap-9438	229	7	rule	rule	NOUN
ap-9438	229	8	for	for	ADP
ap-9438	229	9	fractional	fractional	ADJ
ap-9438	229	10	derivatives	derivative	NOUN
ap-9438	229	11	.	.	PUNCT
ap-9438	230	1	communications	communication	NOUN
ap-9438	230	2	in	in	ADP
ap-9438	230	3	nonlinear	nonlinear	ADJ
ap-9438	230	4	science	science	NOUN
ap-9438	230	5	and	and	CCONJ
ap-9438	230	6	numerical	numerical	PROPN
ap-9438	230	7	simulation	simulation	PROPN
ap-9438	230	8	62:244–249	62:244–249	PROPN
ap-9438	230	9	,	,	PUNCT
ap-9438	230	10	2018	2018	NUM
ap-9438	230	11	.	.	PUNCT
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ap-9438	231	3	18	18	NUM
ap-9438	231	4	]	]	PUNCT
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ap-9438	231	7	,	,	PUNCT
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ap-9438	231	10	.	.	PUNCT
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ap-9438	232	8	multiple	multiple	ADJ
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ap-9438	232	10	.	.	PUNCT
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ap-9438	233	8	,	,	PUNCT
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ap-9438	233	10	.	.	PUNCT
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ap-9438	240	8	.	.	PUNCT
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ap-9438	241	3	-	-	SYM
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ap-9438	241	5	-	-	PUNCT
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ap-9438	241	7	-	-	PUNCT
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ap-9438	241	9	-	-	SYM
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ap-9438	243	12	.	.	PUNCT
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ap-9438	244	8	,	,	PUNCT
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ap-9438	244	10	.	.	PUNCT
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ap-9438	246	10	.	.	PUNCT
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ap-9438	247	6	,	,	PUNCT
ap-9438	247	7	2018	2018	NUM
ap-9438	247	8	.	.	PUNCT
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ap-9438	249	2	23	23	NUM
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ap-9438	250	8	.	.	PUNCT
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ap-9438	251	8	,	,	PUNCT
ap-9438	251	9	2011	2011	NUM
ap-9438	251	10	.	.	PUNCT
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ap-9438	252	4	]	]	PUNCT
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ap-9438	252	7	.	.	PUNCT
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ap-9438	254	8	.	.	PUNCT
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ap-9438	258	8	.	.	PUNCT
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ap-9438	273	11	.	.	PUNCT
ap-9438	274	1	https://doi.org/10.22075/ijnaa.2021.5290	https://doi.org/10.22075/ijnaa.2021.5290	PROPN
ap-9438	274	2	419	419	NUM
ap-9438	274	3	https://doi.org/10.1007/978-3-642-21449-3	https://doi.org/10.1007/978-3-642-21449-3	PROPN
ap-9438	274	4	https://doi.org/10.22075/ijnaa.2021.5290	https://doi.org/10.22075/ijnaa.2021.5290	PROPN
ap-9438	274	5	acta	acta	PROPN
ap-9438	274	6	polytechnica	polytechnica	PROPN
ap-9438	274	7	64(5):414–419	64(5):414–419	NUM
ap-9438	274	8	,	,	PUNCT
ap-9438	274	9	2024	2024	NUM
ap-9438	274	10	1	1	NUM
ap-9438	274	11	introduction	introduction	NOUN
ap-9438	274	12	2	2	NUM
ap-9438	274	13	mathematical	mathematical	ADJ
ap-9438	274	14	preliminaries	preliminary	NOUN
ap-9438	274	15	3	3	NUM
ap-9438	274	16	the	the	DET
ap-9438	274	17	approach	approach	NOUN
ap-9438	274	18	analysis	analysis	NOUN
ap-9438	274	19	4	4	NUM
ap-9438	274	20	illustrative	illustrative	ADJ
ap-9438	274	21	examples	example	NOUN
ap-9438	274	22	5	5	NUM
ap-9438	274	23	results	result	NOUN
ap-9438	274	24	and	and	CCONJ
ap-9438	274	25	conclusion	conclusion	NOUN
ap-9438	274	26	acknowledgements	acknowledgement	NOUN
ap-9438	274	27	references	reference	NOUN