id	sid	tid	token	lemma	pos
cana-3303	1	1	communications	communication	NOUN
cana-3303	1	2	on	on	ADP
cana-3303	1	3	applied	apply	VERB
cana-3303	1	4	nonlinear	nonlinear	ADJ
cana-3303	1	5	analysis	analysis	NOUN
cana-3303	1	6	issn	issn	NOUN
cana-3303	1	7	:	:	PUNCT
cana-3303	1	8	1074	1074	NUM
cana-3303	1	9	-	-	PUNCT
cana-3303	1	10	133x	133x	NUM
cana-3303	1	11	vol	vol	NOUN
cana-3303	1	12	32	32	NUM
cana-3303	1	13	no	no	NOUN
cana-3303	1	14	.	.	PUNCT
cana-3303	2	1	6s	6s	NUM
cana-3303	2	2	(	(	PUNCT
cana-3303	2	3	2025	2025	NUM
cana-3303	2	4	)	)	PUNCT
cana-3303	2	5	377	377	NUM
cana-3303	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	2	7	solution	solution	NOUN
cana-3303	2	8	of	of	ADP
cana-3303	2	9	cauchy	cauchy	ADJ
cana-3303	2	10	and	and	CCONJ
cana-3303	2	11	boundary	boundary	ADJ
cana-3303	2	12	problems	problem	NOUN
cana-3303	2	13	of	of	ADP
cana-3303	2	14	discrete	discrete	ADJ
cana-3303	2	15	multiplicativepoverative	multiplicativepoverative	ADJ
cana-3303	2	16	-	-	PUNCT
cana-3303	2	17	additive	additive	ADJ
cana-3303	2	18	derivative	derivative	ADJ
cana-3303	2	19	third	third	ADJ
cana-3303	2	20	-	-	PUNCT
cana-3303	2	21	order	order	NOUN
cana-3303	2	22	equation	equation	NOUN
cana-3303	2	23	aygun	aygun	VERB
cana-3303	2	24	mammadzada	mammadzada	PROPN
cana-3303	2	25	associate	associate	PROPN
cana-3303	2	26	professor	professor	PROPN
cana-3303	2	27	,	,	PUNCT
cana-3303	2	28	azerbaijan	azerbaijan	PROPN
cana-3303	2	29	technical	technical	PROPN
cana-3303	2	30	university	university	PROPN
cana-3303	2	31	,	,	PUNCT
cana-3303	2	32	baku	baku	PROPN
cana-3303	2	33	,	,	PUNCT
cana-3303	2	34	azerbaijan	azerbaijan	PROPN
cana-3303	2	35	.	.	PROPN
cana-3303	2	36	aygun.mammadzada@aztu.edu.az	aygun.mammadzada@aztu.edu.az	PROPN
cana-3303	2	37	article	article	PROPN
cana-3303	2	38	history	history	NOUN
cana-3303	2	39	:	:	PUNCT
cana-3303	2	40	received	receive	VERB
cana-3303	2	41	:	:	PUNCT
cana-3303	2	42	20	20	NUM
cana-3303	2	43	-	-	SYM
cana-3303	2	44	10	10	NUM
cana-3303	2	45	-	-	PUNCT
cana-3303	2	46	2024	2024	NUM
cana-3303	2	47	revised	revise	VERB
cana-3303	2	48	:	:	PUNCT
cana-3303	2	49	04	04	NUM
cana-3303	2	50	-	-	SYM
cana-3303	2	51	12	12	NUM
cana-3303	2	52	-	-	PUNCT
cana-3303	2	53	2024	2024	NUM
cana-3303	2	54	accepted	accept	VERB
cana-3303	2	55	:	:	PUNCT
cana-3303	2	56	11	11	NUM
cana-3303	2	57	-	-	SYM
cana-3303	2	58	12	12	NUM
cana-3303	2	59	-	-	PUNCT
cana-3303	2	60	2024	2024	NUM
cana-3303	2	61	abstract	abstract	NOUN
cana-3303	2	62	:	:	PUNCT
cana-3303	2	63	introduction	introduction	NOUN
cana-3303	2	64	:	:	PUNCT
cana-3303	2	65	within	within	ADP
cana-3303	2	66	the	the	DET
cana-3303	2	67	framework	framework	NOUN
cana-3303	2	68	of	of	ADP
cana-3303	2	69	cauchy	cauchy	PROPN
cana-3303	2	70	and	and	CCONJ
cana-3303	2	71	boundary	boundary	ADJ
cana-3303	2	72	issues	issue	NOUN
cana-3303	2	73	,	,	PUNCT
cana-3303	2	74	the	the	DET
cana-3303	2	75	authors	author	NOUN
cana-3303	2	76	of	of	ADP
cana-3303	2	77	this	this	DET
cana-3303	2	78	paper	paper	NOUN
cana-3303	2	79	investigate	investigate	VERB
cana-3303	2	80	the	the	DET
cana-3303	2	81	third	third	ADJ
cana-3303	2	82	-	-	PUNCT
cana-3303	2	83	order	order	NOUN
cana-3303	2	84	discrete	discrete	ADJ
cana-3303	2	85	mixed	mixed	ADJ
cana-3303	2	86	derivative	derivative	ADJ
cana-3303	2	87	equation	equation	NOUN
cana-3303	2	88	.	.	PUNCT
cana-3303	3	1	for	for	ADP
cana-3303	3	2	the	the	DET
cana-3303	3	3	purpose	purpose	NOUN
cana-3303	3	4	of	of	ADP
cana-3303	3	5	removing	remove	VERB
cana-3303	3	6	these	these	DET
cana-3303	3	7	derivatives	derivative	NOUN
cana-3303	3	8	and	and	CCONJ
cana-3303	3	9	establishing	establish	VERB
cana-3303	3	10	the	the	DET
cana-3303	3	11	general	general	ADJ
cana-3303	3	12	solution	solution	NOUN
cana-3303	3	13	of	of	ADP
cana-3303	3	14	the	the	DET
cana-3303	3	15	related	relate	VERB
cana-3303	3	16	discrete	discrete	ADJ
cana-3303	3	17	multiplicative	multiplicative	ADJ
cana-3303	3	18	-	-	PUNCT
cana-3303	3	19	poverative	poverative	ADJ
cana-3303	3	20	-	-	PUNCT
cana-3303	3	21	additive	additive	ADJ
cana-3303	3	22	derivative	derivative	ADJ
cana-3303	3	23	equation	equation	NOUN
cana-3303	3	24	,	,	PUNCT
cana-3303	3	25	it	it	PRON
cana-3303	3	26	makes	make	VERB
cana-3303	3	27	use	use	NOUN
cana-3303	3	28	of	of	ADP
cana-3303	3	29	the	the	DET
cana-3303	3	30	definitions	definition	NOUN
cana-3303	3	31	of	of	ADP
cana-3303	3	32	discrete	discrete	ADJ
cana-3303	3	33	additive	additive	NOUN
cana-3303	3	34	,	,	PUNCT
cana-3303	3	35	multiplicative	multiplicative	ADJ
cana-3303	3	36	,	,	PUNCT
cana-3303	3	37	and	and	CCONJ
cana-3303	3	38	poverative	poverative	ADJ
cana-3303	3	39	derivatives	derivative	NOUN
cana-3303	3	40	.	.	PUNCT
cana-3303	4	1	there	there	PRON
cana-3303	4	2	are	be	VERB
cana-3303	4	3	three	three	NUM
cana-3303	4	4	arbitrary	arbitrary	ADJ
cana-3303	4	5	constants	constant	NOUN
cana-3303	4	6	that	that	PRON
cana-3303	4	7	are	be	AUX
cana-3303	4	8	dependent	dependent	ADJ
cana-3303	4	9	on	on	ADP
cana-3303	4	10	this	this	DET
cana-3303	4	11	answer	answer	NOUN
cana-3303	4	12	.	.	PUNCT
cana-3303	5	1	the	the	DET
cana-3303	5	2	subsequent	subsequent	ADJ
cana-3303	5	3	step	step	NOUN
cana-3303	5	4	entails	entail	VERB
cana-3303	5	5	finding	find	VERB
cana-3303	5	6	these	these	DET
cana-3303	5	7	constants	constant	NOUN
cana-3303	5	8	by	by	ADP
cana-3303	5	9	making	make	VERB
cana-3303	5	10	use	use	NOUN
cana-3303	5	11	of	of	ADP
cana-3303	5	12	primary	primary	ADJ
cana-3303	5	13	or	or	CCONJ
cana-3303	5	14	boundary	boundary	ADJ
cana-3303	5	15	conditions	condition	NOUN
cana-3303	5	16	.	.	PUNCT
cana-3303	6	1	in	in	ADP
cana-3303	6	2	order	order	NOUN
cana-3303	6	3	to	to	PART
cana-3303	6	4	correctly	correctly	ADV
cana-3303	6	5	handle	handle	VERB
cana-3303	6	6	the	the	DET
cana-3303	6	7	cauchy	cauchy	ADJ
cana-3303	6	8	and	and	CCONJ
cana-3303	6	9	boundary	boundary	ADJ
cana-3303	6	10	issues	issue	NOUN
cana-3303	6	11	,	,	PUNCT
cana-3303	6	12	the	the	DET
cana-3303	6	13	essay	essay	NOUN
cana-3303	6	14	attempts	attempt	VERB
cana-3303	6	15	to	to	PART
cana-3303	6	16	solve	solve	VERB
cana-3303	6	17	for	for	ADP
cana-3303	6	18	these	these	DET
cana-3303	6	19	constants	constant	NOUN
cana-3303	6	20	.	.	PUNCT
cana-3303	7	1	objectives	objective	NOUN
cana-3303	7	2	:	:	PUNCT
cana-3303	7	3	to	to	PART
cana-3303	7	4	study	study	VERB
cana-3303	7	5	third	third	ADJ
cana-3303	7	6	-	-	PUNCT
cana-3303	7	7	order	order	NOUN
cana-3303	7	8	discrete	discrete	ADJ
cana-3303	7	9	mixed	mixed	ADJ
cana-3303	7	10	derivative	derivative	ADJ
cana-3303	7	11	equations	equation	NOUN
cana-3303	7	12	in	in	ADP
cana-3303	7	13	the	the	DET
cana-3303	7	14	framework	framework	NOUN
cana-3303	7	15	of	of	ADP
cana-3303	7	16	cauchy	cauchy	PROPN
cana-3303	7	17	and	and	CCONJ
cana-3303	7	18	boundary	boundary	ADJ
cana-3303	7	19	problems	problem	NOUN
cana-3303	7	20	,	,	PUNCT
cana-3303	7	21	and	and	CCONJ
cana-3303	7	22	to	to	PART
cana-3303	7	23	find	find	VERB
cana-3303	7	24	a	a	DET
cana-3303	7	25	general	general	ADJ
cana-3303	7	26	solution	solution	NOUN
cana-3303	7	27	through	through	ADP
cana-3303	7	28	eliminating	eliminate	VERB
cana-3303	7	29	mixed	mixed	ADJ
cana-3303	7	30	derivatives	derivative	NOUN
cana-3303	7	31	with	with	ADP
cana-3303	7	32	the	the	DET
cana-3303	7	33	use	use	NOUN
cana-3303	7	34	of	of	ADP
cana-3303	7	35	discrete	discrete	ADJ
cana-3303	7	36	derivative	derivative	ADJ
cana-3303	7	37	definitions	definition	NOUN
cana-3303	7	38	.	.	PUNCT
cana-3303	8	1	methods	method	NOUN
cana-3303	8	2	:	:	PUNCT
cana-3303	8	3	the	the	DET
cana-3303	8	4	problem	problem	NOUN
cana-3303	8	5	is	be	AUX
cana-3303	8	6	stated	state	VERB
cana-3303	8	7	as	as	ADP
cana-3303	8	8	a	a	DET
cana-3303	8	9	new	new	ADJ
cana-3303	8	10	discrete	discrete	ADJ
cana-3303	8	11	additive	additive	NOUN
cana-3303	8	12	,	,	PUNCT
cana-3303	8	13	multiplicative	multiplicative	ADJ
cana-3303	8	14	,	,	PUNCT
cana-3303	8	15	and	and	CCONJ
cana-3303	8	16	poverative	poverative	ADJ
cana-3303	8	17	derivative	derivative	ADJ
cana-3303	8	18	definitions	definition	NOUN
cana-3303	8	19	.	.	PUNCT
cana-3303	9	1	we	we	PRON
cana-3303	9	2	formulate	formulate	VERB
cana-3303	9	3	the	the	DET
cana-3303	9	4	general	general	ADJ
cana-3303	9	5	solution	solution	NOUN
cana-3303	9	6	and	and	CCONJ
cana-3303	9	7	identify	identify	VERB
cana-3303	9	8	the	the	DET
cana-3303	9	9	arbitrary	arbitrary	ADJ
cana-3303	9	10	constants	constant	NOUN
cana-3303	9	11	from	from	ADP
cana-3303	9	12	primary	primary	ADJ
cana-3303	9	13	or	or	CCONJ
cana-3303	9	14	boundary	boundary	ADJ
cana-3303	9	15	conditions	condition	NOUN
cana-3303	9	16	.	.	PUNCT
cana-3303	10	1	results	result	NOUN
cana-3303	10	2	:	:	PUNCT
cana-3303	10	3	the	the	DET
cana-3303	10	4	general	general	ADJ
cana-3303	10	5	solution	solution	NOUN
cana-3303	10	6	of	of	ADP
cana-3303	10	7	discrete	discrete	ADJ
cana-3303	10	8	equation	equation	NOUN
cana-3303	10	9	has	have	AUX
cana-3303	10	10	successfully	successfully	ADV
cana-3303	10	11	been	be	AUX
cana-3303	10	12	obtained	obtain	VERB
cana-3303	10	13	containing	contain	VERB
cana-3303	10	14	three	three	NUM
cana-3303	10	15	arbitrary	arbitrary	ADJ
cana-3303	10	16	constants	constant	NOUN
cana-3303	10	17	determined	determine	VERB
cana-3303	10	18	by	by	ADP
cana-3303	10	19	the	the	DET
cana-3303	10	20	imposed	impose	VERB
cana-3303	10	21	boundary	boundary	ADJ
cana-3303	10	22	or	or	CCONJ
cana-3303	10	23	cauchy	cauchy	ADJ
cana-3303	10	24	conditions	condition	NOUN
cana-3303	10	25	.	.	PUNCT
cana-3303	11	1	conclusions	conclusion	NOUN
cana-3303	11	2	:	:	PUNCT
cana-3303	11	3	this	this	DET
cana-3303	11	4	study	study	NOUN
cana-3303	11	5	enhances	enhance	VERB
cana-3303	11	6	the	the	DET
cana-3303	11	7	efficient	efficient	ADJ
cana-3303	11	8	solution	solution	NOUN
cana-3303	11	9	of	of	ADP
cana-3303	11	10	third	third	ADJ
cana-3303	11	11	-	-	PUNCT
cana-3303	11	12	order	order	NOUN
cana-3303	11	13	discrete	discrete	ADJ
cana-3303	11	14	mixed	mixed	ADJ
cana-3303	11	15	derivative	derivative	ADJ
cana-3303	11	16	equations	equation	NOUN
cana-3303	11	17	,	,	PUNCT
cana-3303	11	18	consolidating	consolidate	VERB
cana-3303	11	19	the	the	DET
cana-3303	11	20	utility	utility	NOUN
cana-3303	11	21	of	of	ADP
cana-3303	11	22	the	the	DET
cana-3303	11	23	discrete	discrete	ADJ
cana-3303	11	24	definitions	definition	NOUN
cana-3303	11	25	of	of	ADP
cana-3303	11	26	the	the	DET
cana-3303	11	27	derivatives	derivative	NOUN
cana-3303	11	28	for	for	ADP
cana-3303	11	29	solving	solve	VERB
cana-3303	11	30	cauchy	cauchy	NOUN
cana-3303	11	31	and	and	CCONJ
cana-3303	11	32	boundary	boundary	ADJ
cana-3303	11	33	problems	problem	NOUN
cana-3303	11	34	in	in	ADP
cana-3303	11	35	a	a	DET
cana-3303	11	36	systematic	systematic	ADJ
cana-3303	11	37	approach	approach	NOUN
cana-3303	11	38	.	.	PUNCT
cana-3303	12	1	keywords	keyword	NOUN
cana-3303	12	2	:	:	PUNCT
cana-3303	12	3	discrete	discrete	ADJ
cana-3303	12	4	additive	additive	NOUN
cana-3303	12	5	,	,	PUNCT
cana-3303	12	6	discrete	discrete	ADJ
cana-3303	12	7	multiplicative	multiplicative	ADJ
cana-3303	12	8	,	,	PUNCT
cana-3303	12	9	discrete	discrete	ADJ
cana-3303	12	10	cauchy	cauchy	NOUN
cana-3303	12	11	problem	problem	NOUN
cana-3303	12	12	,	,	PUNCT
cana-3303	12	13	discrete	discrete	ADJ
cana-3303	12	14	boundary	boundary	ADJ
cana-3303	12	15	problem	problem	NOUN
cana-3303	12	16	,	,	PUNCT
cana-3303	12	17	analytical	analytical	ADJ
cana-3303	12	18	expression	expression	NOUN
cana-3303	12	19	.	.	PUNCT
cana-3303	13	1	1	1	X
cana-3303	13	2	.	.	X
cana-3303	13	3	introduction	introduction	NOUN
cana-3303	13	4	discrete	discrete	VERB
cana-3303	13	5	derivative	derivative	ADJ
cana-3303	13	6	equations	equation	NOUN
cana-3303	13	7	,	,	PUNCT
cana-3303	13	8	both	both	CCONJ
cana-3303	13	9	continuous	continuous	ADJ
cana-3303	13	10	and	and	CCONJ
cana-3303	13	11	additive	additive	ADJ
cana-3303	13	12	,	,	PUNCT
cana-3303	13	13	attract	attract	VERB
cana-3303	13	14	considerable	considerable	ADJ
cana-3303	13	15	interest	interest	NOUN
cana-3303	13	16	in	in	ADP
cana-3303	13	17	mathematical	mathematical	ADJ
cana-3303	13	18	analysis	analysis	NOUN
cana-3303	13	19	.	.	PUNCT
cana-3303	14	1	these	these	DET
cana-3303	14	2	equations	equation	NOUN
cana-3303	14	3	,	,	PUNCT
cana-3303	14	4	commonly	commonly	ADV
cana-3303	14	5	called	call	VERB
cana-3303	14	6	difference	difference	NOUN
cana-3303	14	7	equations	equation	NOUN
cana-3303	14	8	,	,	PUNCT
cana-3303	14	9	have	have	AUX
cana-3303	14	10	undergone	undergo	VERB
cana-3303	14	11	a	a	DET
cana-3303	14	12	lot	lot	NOUN
cana-3303	14	13	of	of	ADP
cana-3303	14	14	advances	advance	NOUN
cana-3303	14	15	in	in	ADP
cana-3303	14	16	a	a	DET
cana-3303	14	17	variety	variety	NOUN
cana-3303	14	18	of	of	ADP
cana-3303	14	19	fields	field	NOUN
cana-3303	14	20	including	include	VERB
cana-3303	14	21	for	for	ADP
cana-3303	14	22	instance	instance	NOUN
cana-3303	14	23	numerical	numerical	ADJ
cana-3303	14	24	methods	method	NOUN
cana-3303	14	25	,	,	PUNCT
cana-3303	14	26	mathematical	mathematical	ADJ
cana-3303	14	27	modelling	modelling	NOUN
cana-3303	14	28	,	,	PUNCT
cana-3303	14	29	theoretical	theoretical	ADJ
cana-3303	14	30	physics	physics	NOUN
cana-3303	14	31	,	,	PUNCT
cana-3303	14	32	etc	etc	X
cana-3303	14	33	.	.	X
cana-3303	14	34	free	free	PROPN
cana-3303	14	35	multiplicative	multiplicative	ADJ
cana-3303	14	36	derivative	derivative	ADJ
cana-3303	14	37	equations	equation	NOUN
cana-3303	14	38	were	be	AUX
cana-3303	14	39	studied	study	VERB
cana-3303	14	40	for	for	ADP
cana-3303	14	41	the	the	DET
cana-3303	14	42	first	first	ADJ
cana-3303	14	43	time	time	NOUN
cana-3303	14	44	in	in	ADP
cana-3303	14	45	the	the	DET
cana-3303	14	46	middle	middle	NOUN
cana-3303	14	47	of	of	ADP
cana-3303	14	48	the	the	DET
cana-3303	14	49	20th	20th	ADJ
cana-3303	14	50	century	century	NOUN
cana-3303	14	51	,	,	PUNCT
cana-3303	14	52	but	but	CCONJ
cana-3303	14	53	their	their	PRON
cana-3303	14	54	importance	importance	NOUN
cana-3303	14	55	was	be	AUX
cana-3303	14	56	not	not	PART
cana-3303	14	57	realized	realize	VERB
cana-3303	14	58	before	before	ADP
cana-3303	14	59	the	the	DET
cana-3303	14	60	end	end	NOUN
cana-3303	14	61	of	of	ADP
cana-3303	14	62	the	the	DET
cana-3303	14	63	20th	20th	NOUN
cana-3303	14	64	and	and	CCONJ
cana-3303	14	65	the	the	DET
cana-3303	14	66	beginning	beginning	NOUN
cana-3303	14	67	of	of	ADP
cana-3303	14	68	the	the	DET
cana-3303	14	69	21st	21st	ADJ
cana-3303	14	70	century	century	NOUN
cana-3303	14	71	(	(	PUNCT
cana-3303	14	72	chikhrab	chikhrab	PROPN
cana-3303	14	73	&	&	CCONJ
cana-3303	14	74	malkin	malkin	PROPN
cana-3303	14	75	,	,	PUNCT
cana-3303	14	76	2005	2005	NUM
cana-3303	14	77	)	)	PUNCT
cana-3303	14	78	.	.	PUNCT
cana-3303	15	1	discrete	discrete	VERB
cana-3303	15	2	multiplicative	multiplicative	ADJ
cana-3303	15	3	derivative	derivative	ADJ
cana-3303	15	4	equations	equation	NOUN
cana-3303	15	5	,	,	PUNCT
cana-3303	15	6	in	in	ADP
cana-3303	15	7	contrast	contrast	NOUN
cana-3303	15	8	to	to	AUX
cana-3303	15	9	additive	additive	VERB
cana-3303	15	10	incremental	incremental	ADJ
cana-3303	15	11	transition	transition	NOUN
cana-3303	15	12	equations	equation	NOUN
cana-3303	15	13	,	,	PUNCT
cana-3303	15	14	define	define	VERB
cana-3303	15	15	a	a	DET
cana-3303	15	16	broad	broad	ADJ
cana-3303	15	17	class	class	NOUN
cana-3303	15	18	of	of	ADP
cana-3303	15	19	complex	complex	ADJ
cana-3303	15	20	nonlinear	nonlinear	ADJ
cana-3303	15	21	systems	system	NOUN
cana-3303	15	22	amenable	amenable	ADJ
cana-3303	15	23	to	to	ADP
cana-3303	15	24	solutions	solution	NOUN
cana-3303	15	25	outside	outside	ADP
cana-3303	15	26	the	the	DET
cana-3303	15	27	conventional	conventional	ADJ
cana-3303	15	28	sectorial	sectorial	ADJ
cana-3303	15	29	/	/	SYM
cana-3303	15	30	sectorial	sectorial	ADJ
cana-3303	15	31	analytical	analytical	ADJ
cana-3303	15	32	framework	framework	NOUN
cana-3303	15	33	.	.	PUNCT
cana-3303	16	1	in	in	ADP
cana-3303	16	2	particular	particular	ADJ
cana-3303	16	3	,	,	PUNCT
cana-3303	16	4	much	much	ADJ
cana-3303	16	5	of	of	ADP
cana-3303	16	6	what	what	PRON
cana-3303	16	7	is	be	AUX
cana-3303	16	8	known	know	VERB
cana-3303	16	9	about	about	ADP
cana-3303	16	10	nonlinear	nonlinear	ADJ
cana-3303	16	11	dynamics	dynamic	NOUN
cana-3303	16	12	in	in	ADP
cana-3303	16	13	discrete	discrete	ADJ
cana-3303	16	14	systems	system	NOUN
cana-3303	16	15	has	have	AUX
cana-3303	16	16	been	be	AUX
cana-3303	16	17	revealed	reveal	VERB
cana-3303	16	18	through	through	ADP
cana-3303	16	19	the	the	DET
cana-3303	16	20	investigation	investigation	NOUN
cana-3303	16	21	of	of	ADP
cana-3303	16	22	these	these	DET
cana-3303	16	23	equations	equation	NOUN
cana-3303	16	24	(	(	PUNCT
cana-3303	16	25	gurskiy	gurskiy	NOUN
cana-3303	16	26	communications	communication	NOUN
cana-3303	16	27	on	on	ADP
cana-3303	16	28	applied	apply	VERB
cana-3303	16	29	nonlinear	nonlinear	ADJ
cana-3303	16	30	analysis	analysis	NOUN
cana-3303	16	31	issn	issn	NOUN
cana-3303	16	32	:	:	PUNCT
cana-3303	16	33	1074	1074	NUM
cana-3303	16	34	-	-	PUNCT
cana-3303	16	35	133x	133x	NUM
cana-3303	16	36	vol	vol	NOUN
cana-3303	16	37	32	32	NUM
cana-3303	16	38	no	no	NOUN
cana-3303	16	39	.	.	PUNCT
cana-3303	17	1	6s	6s	NUM
cana-3303	17	2	(	(	PUNCT
cana-3303	17	3	2025	2025	NUM
cana-3303	17	4	)	)	PUNCT
cana-3303	17	5	378	378	NUM
cana-3303	17	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	17	7	&	&	CCONJ
cana-3303	17	8	sokolov	sokolov	PROPN
cana-3303	17	9	,	,	PUNCT
cana-3303	17	10	2010	2010	NUM
cana-3303	17	11	)	)	PUNCT
cana-3303	17	12	.	.	PUNCT
cana-3303	18	1	although	although	SCONJ
cana-3303	18	2	the	the	DET
cana-3303	18	3	linear	linear	ADJ
cana-3303	18	4	case	case	NOUN
cana-3303	18	5	of	of	ADP
cana-3303	18	6	discrete	discrete	ADJ
cana-3303	18	7	additive	additive	ADJ
cana-3303	18	8	derivative	derivative	ADJ
cana-3303	18	9	equations	equation	NOUN
cana-3303	18	10	has	have	AUX
cana-3303	18	11	been	be	AUX
cana-3303	18	12	studied	study	VERB
cana-3303	18	13	in	in	ADP
cana-3303	18	14	great	great	ADJ
cana-3303	18	15	detail	detail	NOUN
cana-3303	18	16	,	,	PUNCT
cana-3303	18	17	the	the	DET
cana-3303	18	18	discrete	discrete	ADJ
cana-3303	18	19	multiplicative	multiplicative	ADJ
cana-3303	18	20	derivatives	derivative	NOUN
cana-3303	18	21	lead	lead	VERB
cana-3303	18	22	to	to	ADP
cana-3303	18	23	rich	rich	ADJ
cana-3303	18	24	nonlinear	nonlinear	ADJ
cana-3303	18	25	behavior	behavior	NOUN
cana-3303	18	26	that	that	PRON
cana-3303	18	27	makes	make	VERB
cana-3303	18	28	any	any	DET
cana-3303	18	29	relevant	relevant	ADJ
cana-3303	18	30	solution	solution	NOUN
cana-3303	18	31	strategies	strategy	NOUN
cana-3303	18	32	challenging	challenge	VERB
cana-3303	18	33	(	(	PUNCT
cana-3303	18	34	dyachenko	dyachenko	PROPN
cana-3303	18	35	&	&	CCONJ
cana-3303	18	36	gulin	gulin	PROPN
cana-3303	18	37	,	,	PUNCT
cana-3303	18	38	2003	2003	NUM
cana-3303	18	39	)	)	PUNCT
cana-3303	18	40	.	.	PUNCT
cana-3303	19	1	additive	additive	VERB
cana-3303	19	2	derivative	derivative	NOUN
cana-3303	19	3	and	and	CCONJ
cana-3303	19	4	integral	integral	ADJ
cana-3303	19	5	,	,	PUNCT
cana-3303	19	6	which	which	PRON
cana-3303	19	7	are	be	AUX
cana-3303	19	8	defined	define	VERB
cana-3303	19	9	based	base	VERB
cana-3303	19	10	on	on	ADP
cana-3303	19	11	summation	summation	NOUN
cana-3303	19	12	and	and	CCONJ
cana-3303	19	13	differences	difference	NOUN
cana-3303	19	14	,	,	PUNCT
cana-3303	19	15	multiplicative	multiplicative	ADJ
cana-3303	19	16	derivative	derivative	ADJ
cana-3303	19	17	and	and	CCONJ
cana-3303	19	18	integral	integral	ADJ
cana-3303	19	19	,	,	PUNCT
cana-3303	19	20	which	which	PRON
cana-3303	19	21	is	be	AUX
cana-3303	19	22	based	base	VERB
cana-3303	19	23	on	on	ADP
cana-3303	19	24	products	product	NOUN
cana-3303	19	25	and	and	CCONJ
cana-3303	19	26	powers	power	NOUN
cana-3303	19	27	.	.	PUNCT
cana-3303	20	1	nonetheless	nonetheless	ADV
cana-3303	20	2	,	,	PUNCT
cana-3303	20	3	having	have	VERB
cana-3303	20	4	many	many	ADJ
cana-3303	20	5	parameters	parameter	NOUN
cana-3303	20	6	does	do	AUX
cana-3303	20	7	not	not	PART
cana-3303	20	8	necessarily	necessarily	ADV
cana-3303	20	9	imply	imply	VERB
cana-3303	20	10	that	that	SCONJ
cana-3303	20	11	the	the	DET
cana-3303	20	12	corresponding	correspond	VERB
cana-3303	20	13	discrete	discrete	ADJ
cana-3303	20	14	poverative	poverative	ADJ
cana-3303	20	15	derivatives	derivative	NOUN
cana-3303	20	16	and	and	CCONJ
cana-3303	20	17	integrals	integral	NOUN
cana-3303	20	18	need	need	VERB
cana-3303	20	19	to	to	PART
cana-3303	20	20	be	be	AUX
cana-3303	20	21	defined	define	VERB
cana-3303	20	22	by	by	ADP
cana-3303	20	23	a	a	DET
cana-3303	20	24	new	new	ADJ
cana-3303	20	25	operation	operation	NOUN
cana-3303	20	26	"	"	PUNCT
cana-3303	20	27	per	per	X
cana-3303	20	28	se	se	X
cana-3303	20	29	"	"	PUNCT
cana-3303	20	30	in	in	ADP
cana-3303	20	31	the	the	DET
cana-3303	20	32	discrete	discrete	ADJ
cana-3303	20	33	case	case	NOUN
cana-3303	20	34	,	,	PUNCT
cana-3303	20	35	as	as	SCONJ
cana-3303	20	36	they	they	PRON
cana-3303	20	37	can	can	AUX
cana-3303	20	38	,	,	PUNCT
cana-3303	20	39	in	in	ADP
cana-3303	20	40	fact	fact	NOUN
cana-3303	20	41	,	,	PUNCT
cana-3303	20	42	be	be	AUX
cana-3303	20	43	explained	explain	VERB
cana-3303	20	44	based	base	VERB
cana-3303	20	45	on	on	ADP
cana-3303	20	46	a	a	DET
cana-3303	20	47	single	single	ADJ
cana-3303	20	48	operation	operation	NOUN
cana-3303	20	49	(	(	PUNCT
cana-3303	20	50	smith	smith	PROPN
cana-3303	20	51	&	&	CCONJ
cana-3303	20	52	jones	jones	PROPN
cana-3303	20	53	,	,	PUNCT
cana-3303	20	54	2007	2007	NUM
cana-3303	20	55	)	)	PUNCT
cana-3303	20	56	.	.	PUNCT
cana-3303	21	1	these	these	DET
cana-3303	21	2	properties	property	NOUN
cana-3303	21	3	pave	pave	VERB
cana-3303	21	4	the	the	DET
cana-3303	21	5	way	way	NOUN
cana-3303	21	6	for	for	ADP
cana-3303	21	7	solving	solve	VERB
cana-3303	21	8	complex	complex	ADJ
cana-3303	21	9	equations	equation	NOUN
cana-3303	21	10	in	in	ADP
cana-3303	21	11	discrete	discrete	ADJ
cana-3303	21	12	systems	system	NOUN
cana-3303	21	13	.	.	PUNCT
cana-3303	22	1	this	this	DET
cana-3303	22	2	paper	paper	NOUN
cana-3303	22	3	will	will	AUX
cana-3303	22	4	study	study	VERB
cana-3303	22	5	a	a	DET
cana-3303	22	6	third	third	ADJ
cana-3303	22	7	-	-	PUNCT
cana-3303	22	8	order	order	NOUN
cana-3303	22	9	mixed	mixed	ADJ
cana-3303	22	10	derivative	derivative	ADJ
cana-3303	22	11	equation	equation	NOUN
cana-3303	22	12	exact	exact	ADJ
cana-3303	22	13	solution	solution	NOUN
cana-3303	22	14	which	which	PRON
cana-3303	22	15	comes	come	VERB
cana-3303	22	16	from	from	ADP
cana-3303	22	17	the	the	DET
cana-3303	22	18	discrete	discrete	ADJ
cana-3303	22	19	additive	additive	ADJ
cana-3303	22	20	derivative	derivative	ADJ
cana-3303	22	21	and	and	CCONJ
cana-3303	22	22	discrete	discrete	VERB
cana-3303	22	23	poverative	poverative	ADJ
cana-3303	22	24	derivative	derivative	NOUN
cana-3303	22	25	.	.	PUNCT
cana-3303	23	1	the	the	DET
cana-3303	23	2	following	follow	VERB
cana-3303	23	3	is	be	AUX
cana-3303	23	4	the	the	DET
cana-3303	23	5	definition	definition	NOUN
cana-3303	23	6	of	of	ADP
cana-3303	23	7	this	this	DET
cana-3303	23	8	nonlinear	nonlinear	ADJ
cana-3303	23	9	equation	equation	NOUN
cana-3303	23	10	:	:	PUNCT
cana-3303	23	11	2	2	X
cana-3303	23	12	.	.	X
cana-3303	23	13	objectives	objective	VERB
cana-3303	23	14	the	the	DET
cana-3303	23	15	objective	objective	NOUN
cana-3303	23	16	of	of	ADP
cana-3303	23	17	this	this	DET
cana-3303	23	18	study	study	NOUN
cana-3303	23	19	is	be	AUX
cana-3303	23	20	to	to	PART
cana-3303	23	21	find	find	VERB
cana-3303	23	22	a	a	DET
cana-3303	23	23	solution	solution	NOUN
cana-3303	23	24	to	to	ADP
cana-3303	23	25	this	this	DET
cana-3303	23	26	equation	equation	NOUN
cana-3303	23	27	by	by	ADP
cana-3303	23	28	applying	apply	VERB
cana-3303	23	29	the	the	DET
cana-3303	23	30	framework	framework	NOUN
cana-3303	23	31	of	of	ADP
cana-3303	23	32	cauchy	cauchy	NOUN
cana-3303	23	33	and	and	CCONJ
cana-3303	23	34	boundary	boundary	ADJ
cana-3303	23	35	value	value	NOUN
cana-3303	23	36	issues	issue	NOUN
cana-3303	23	37	.	.	PUNCT
cana-3303	24	1	analytical	analytical	ADJ
cana-3303	24	2	solutions	solution	NOUN
cana-3303	24	3	are	be	AUX
cana-3303	24	4	offered	offer	VERB
cana-3303	24	5	for	for	ADP
cana-3303	24	6	both	both	PRON
cana-3303	24	7	of	of	ADP
cana-3303	24	8	these	these	DET
cana-3303	24	9	scenarios	scenario	NOUN
cana-3303	24	10	.	.	PUNCT
cana-3303	25	1	this	this	DET
cana-3303	25	2	research	research	NOUN
cana-3303	25	3	not	not	PART
cana-3303	25	4	only	only	ADV
cana-3303	25	5	provides	provide	VERB
cana-3303	25	6	useful	useful	ADJ
cana-3303	25	7	insights	insight	NOUN
cana-3303	25	8	into	into	ADP
cana-3303	25	9	the	the	DET
cana-3303	25	10	process	process	NOUN
cana-3303	25	11	of	of	ADP
cana-3303	25	12	solving	solve	VERB
cana-3303	25	13	nonlinear	nonlinear	ADJ
cana-3303	25	14	discrete	discrete	ADJ
cana-3303	25	15	equations	equation	NOUN
cana-3303	25	16	,	,	PUNCT
cana-3303	25	17	but	but	CCONJ
cana-3303	25	18	it	it	PRON
cana-3303	25	19	also	also	ADV
cana-3303	25	20	transforms	transform	VERB
cana-3303	25	21	the	the	DET
cana-3303	25	22	problem	problem	NOUN
cana-3303	25	23	through	through	ADP
cana-3303	25	24	the	the	DET
cana-3303	25	25	definitions	definition	NOUN
cana-3303	25	26	of	of	ADP
cana-3303	25	27	discrete	discrete	ADJ
cana-3303	25	28	additive	additive	NOUN
cana-3303	25	29	and	and	CCONJ
cana-3303	25	30	multiplicative	multiplicative	ADJ
cana-3303	25	31	derivatives	derivative	NOUN
cana-3303	25	32	.	.	PUNCT
cana-3303	26	1	in	in	ADP
cana-3303	26	2	addition	addition	NOUN
cana-3303	26	3	to	to	ADP
cana-3303	26	4	contributing	contribute	VERB
cana-3303	26	5	to	to	ADP
cana-3303	26	6	the	the	DET
cana-3303	26	7	expanding	expand	VERB
cana-3303	26	8	corpus	corpus	NOUN
cana-3303	26	9	of	of	ADP
cana-3303	26	10	research	research	NOUN
cana-3303	26	11	in	in	ADP
cana-3303	26	12	discrete	discrete	ADJ
cana-3303	26	13	mathematical	mathematical	ADJ
cana-3303	26	14	modeling	modeling	NOUN
cana-3303	26	15	(	(	PUNCT
cana-3303	26	16	gurskiy	gurskiy	PROPN
cana-3303	26	17	&	&	CCONJ
cana-3303	26	18	sokolov	sokolov	PROPN
cana-3303	26	19	,	,	PUNCT
cana-3303	26	20	2010	2010	NUM
cana-3303	26	21	;	;	PUNCT
cana-3303	26	22	chikhrab	chikhrab	PROPN
cana-3303	26	23	&	&	CCONJ
cana-3303	26	24	malkin	malkin	PROPN
cana-3303	26	25	,	,	PUNCT
cana-3303	26	26	2005	2005	NUM
cana-3303	26	27	)	)	PUNCT
cana-3303	26	28	,	,	PUNCT
cana-3303	26	29	these	these	DET
cana-3303	26	30	discoveries	discovery	NOUN
cana-3303	26	31	not	not	PART
cana-3303	26	32	only	only	ADV
cana-3303	26	33	improve	improve	VERB
cana-3303	26	34	the	the	DET
cana-3303	26	35	theoretical	theoretical	ADJ
cana-3303	26	36	knowledge	knowledge	NOUN
cana-3303	26	37	of	of	ADP
cana-3303	26	38	discrete	discrete	ADJ
cana-3303	26	39	systems	system	NOUN
cana-3303	26	40	but	but	CCONJ
cana-3303	26	41	also	also	ADV
cana-3303	26	42	contribute	contribute	VERB
cana-3303	26	43	to	to	ADP
cana-3303	26	44	the	the	DET
cana-3303	26	45	field	field	NOUN
cana-3303	26	46	.	.	PUNCT
cana-3303	27	1	the	the	DET
cana-3303	27	2	study	study	NOUN
cana-3303	27	3	contributes	contribute	VERB
cana-3303	27	4	to	to	ADP
cana-3303	27	5	a	a	DET
cana-3303	27	6	more	more	ADJ
cana-3303	27	7	in	in	ADP
cana-3303	27	8	-	-	PUNCT
cana-3303	27	9	depth	depth	NOUN
cana-3303	27	10	knowledge	knowledge	NOUN
cana-3303	27	11	of	of	ADP
cana-3303	27	12	nonlinear	nonlinear	ADJ
cana-3303	27	13	equations	equation	NOUN
cana-3303	27	14	in	in	ADP
cana-3303	27	15	discrete	discrete	ADJ
cana-3303	27	16	settings	setting	NOUN
cana-3303	27	17	and	and	CCONJ
cana-3303	27	18	offers	offer	VERB
cana-3303	27	19	fresh	fresh	ADJ
cana-3303	27	20	insights	insight	NOUN
cana-3303	27	21	on	on	ADP
cana-3303	27	22	the	the	DET
cana-3303	27	23	difficulties	difficulty	NOUN
cana-3303	27	24	and	and	CCONJ
cana-3303	27	25	opportunities	opportunity	NOUN
cana-3303	27	26	associated	associate	VERB
cana-3303	27	27	with	with	ADP
cana-3303	27	28	the	the	DET
cana-3303	27	29	solution	solution	NOUN
cana-3303	27	30	of	of	ADP
cana-3303	27	31	these	these	DET
cana-3303	27	32	equations	equation	NOUN
cana-3303	27	33	.	.	PUNCT
cana-3303	28	1	in	in	ADP
cana-3303	28	2	this	this	DET
cana-3303	28	3	paper	paper	NOUN
cana-3303	28	4	,	,	PUNCT
cana-3303	28	5	we	we	PRON
cana-3303	28	6	give	give	VERB
cana-3303	28	7	conclusions	conclusion	NOUN
cana-3303	28	8	that	that	PRON
cana-3303	28	9	expand	expand	VERB
cana-3303	28	10	on	on	ADP
cana-3303	28	11	previously	previously	ADV
cana-3303	28	12	established	establish	VERB
cana-3303	28	13	theories	theory	NOUN
cana-3303	28	14	and	and	CCONJ
cana-3303	28	15	propose	propose	VERB
cana-3303	28	16	a	a	DET
cana-3303	28	17	fresh	fresh	ADJ
cana-3303	28	18	method	method	NOUN
cana-3303	28	19	to	to	ADP
cana-3303	28	20	the	the	DET
cana-3303	28	21	investigation	investigation	NOUN
cana-3303	28	22	of	of	ADP
cana-3303	28	23	discrete	discrete	ADJ
cana-3303	28	24	multiplicative	multiplicative	ADJ
cana-3303	28	25	derivatives	derivative	NOUN
cana-3303	28	26	.	.	PUNCT
cana-3303	29	1	3	3	X
cana-3303	29	2	.	.	X
cana-3303	29	3	literature	literature	NOUN
cana-3303	29	4	review	review	NOUN
cana-3303	29	5	to	to	ADP
cana-3303	29	6	my	my	PRON
cana-3303	29	7	knowledge	knowledge	NOUN
cana-3303	29	8	,	,	PUNCT
cana-3303	29	9	discrete	discrete	ADJ
cana-3303	29	10	derivative	derivative	ADJ
cana-3303	29	11	equations	equation	NOUN
cana-3303	29	12	have	have	AUX
cana-3303	29	13	formed	form	VERB
cana-3303	29	14	the	the	DET
cana-3303	29	15	basis	basis	NOUN
cana-3303	29	16	of	of	ADP
cana-3303	29	17	an	an	DET
cana-3303	29	18	expanding	expand	VERB
cana-3303	29	19	research	research	NOUN
cana-3303	29	20	area	area	NOUN
cana-3303	29	21	,	,	PUNCT
cana-3303	29	22	especially	especially	ADV
cana-3303	29	23	for	for	ADP
cana-3303	29	24	non	non	ADJ
cana-3303	29	25	-	-	ADJ
cana-3303	29	26	linear	linear	ADJ
cana-3303	29	27	systems	system	NOUN
cana-3303	29	28	,	,	PUNCT
cana-3303	29	29	with	with	ADP
cana-3303	29	30	applications	application	NOUN
cana-3303	29	31	ranging	range	VERB
cana-3303	29	32	across	across	ADP
cana-3303	29	33	numerical	numerical	ADJ
cana-3303	29	34	analysis	analysis	NOUN
cana-3303	29	35	,	,	PUNCT
cana-3303	29	36	mathematical	mathematical	ADJ
cana-3303	29	37	modelling	modelling	NOUN
cana-3303	29	38	and	and	CCONJ
cana-3303	29	39	dynamical	dynamical	ADJ
cana-3303	29	40	systems	system	NOUN
cana-3303	29	41	.	.	PUNCT
cana-3303	30	1	first	first	ADJ
cana-3303	30	2	initial	initial	ADJ
cana-3303	30	3	stages	stage	NOUN
cana-3303	30	4	of	of	ADP
cana-3303	30	5	research	research	NOUN
cana-3303	30	6	in	in	ADP
cana-3303	30	7	discrete	discrete	ADJ
cana-3303	30	8	equations	equation	NOUN
cana-3303	30	9	have	have	AUX
cana-3303	30	10	been	be	AUX
cana-3303	30	11	predominantly	predominantly	ADV
cana-3303	30	12	geared	gear	VERB
cana-3303	30	13	toward	toward	ADP
cana-3303	30	14	linear	linear	PROPN
cana-3303	30	15	systems	system	NOUN
cana-3303	30	16	,	,	PUNCT
cana-3303	30	17	especially	especially	ADV
cana-3303	30	18	those	those	PRON
cana-3303	30	19	of	of	ADP
cana-3303	30	20	the	the	DET
cana-3303	30	21	type	type	NOUN
cana-3303	30	22	of	of	ADP
cana-3303	30	23	additive	additive	ADJ
cana-3303	30	24	derivatives	derivative	NOUN
cana-3303	30	25	,	,	PUNCT
cana-3303	30	26	since	since	SCONJ
cana-3303	30	27	the	the	DET
cana-3303	30	28	linear	linear	PROPN
cana-3303	30	29	systems	system	NOUN
cana-3303	30	30	have	have	VERB
cana-3303	30	31	their	their	PRON
cana-3303	30	32	own	own	ADJ
cana-3303	30	33	well	well	ADV
cana-3303	30	34	-	-	PUNCT
cana-3303	30	35	defined	define	VERB
cana-3303	30	36	properties	property	NOUN
cana-3303	30	37	,	,	PUNCT
cana-3303	30	38	and	and	CCONJ
cana-3303	30	39	also	also	ADV
cana-3303	30	40	,	,	PUNCT
cana-3303	30	41	they	they	PRON
cana-3303	30	42	normally	normally	ADV
cana-3303	30	43	behave	behave	VERB
cana-3303	30	44	the	the	DET
cana-3303	30	45	same	same	ADJ
cana-3303	30	46	way	way	NOUN
cana-3303	30	47	.	.	PUNCT
cana-3303	31	1	but	but	CCONJ
cana-3303	31	2	as	as	SCONJ
cana-3303	31	3	research	research	NOUN
cana-3303	31	4	progressed	progress	VERB
cana-3303	31	5	,	,	PUNCT
cana-3303	31	6	they	they	PRON
cana-3303	31	7	began	begin	VERB
cana-3303	31	8	to	to	PART
cana-3303	31	9	apply	apply	VERB
cana-3303	31	10	it	it	PRON
cana-3303	31	11	to	to	ADP
cana-3303	31	12	more	more	ADV
cana-3303	31	13	complicated	complicated	ADJ
cana-3303	31	14	non	non	ADJ
cana-3303	31	15	-	-	ADJ
cana-3303	31	16	linear	linear	ADJ
cana-3303	31	17	systems	system	NOUN
cana-3303	31	18	,	,	PUNCT
cana-3303	31	19	including	include	VERB
cana-3303	31	20	multiplicative	multiplicative	ADJ
cana-3303	31	21	derivatives	derivative	NOUN
cana-3303	31	22	,	,	PUNCT
cana-3303	31	23	which	which	PRON
cana-3303	31	24	have	have	AUX
cana-3303	31	25	introduced	introduce	VERB
cana-3303	31	26	huge	huge	ADJ
cana-3303	31	27	mathematical	mathematical	ADJ
cana-3303	31	28	difficulties	difficulty	NOUN
cana-3303	31	29	.	.	PUNCT
cana-3303	32	1	3.1	3.1	NUM
cana-3303	32	2	.	.	X
cana-3303	32	3	discrete	discrete	VERB
cana-3303	32	4	derivative	derivative	ADJ
cana-3303	32	5	equations	equation	NOUN
cana-3303	32	6	such	such	ADJ
cana-3303	32	7	equations	equation	NOUN
cana-3303	32	8	where	where	SCONJ
cana-3303	32	9	the	the	DET
cana-3303	32	10	derivative	derivative	NOUN
cana-3303	32	11	is	be	AUX
cana-3303	32	12	expressed	express	VERB
cana-3303	32	13	in	in	ADP
cana-3303	32	14	terms	term	NOUN
cana-3303	32	15	of	of	ADP
cana-3303	32	16	differences	difference	NOUN
cana-3303	32	17	instead	instead	ADV
cana-3303	32	18	of	of	ADP
cana-3303	32	19	continuous	continuous	ADJ
cana-3303	32	20	rates	rate	NOUN
cana-3303	32	21	of	of	ADP
cana-3303	32	22	change	change	NOUN
cana-3303	32	23	are	be	AUX
cana-3303	32	24	called	call	VERB
cana-3303	32	25	discrete	discrete	ADJ
cana-3303	32	26	derivative	derivative	ADJ
cana-3303	32	27	equations	equation	NOUN
cana-3303	32	28	.	.	PUNCT
cana-3303	33	1	indeed	indeed	ADV
cana-3303	33	2	,	,	PUNCT
cana-3303	33	3	extensive	extensive	ADJ
cana-3303	33	4	research	research	NOUN
cana-3303	33	5	has	have	AUX
cana-3303	33	6	already	already	ADV
cana-3303	33	7	been	be	AUX
cana-3303	33	8	made	make	VERB
cana-3303	33	9	on	on	ADP
cana-3303	33	10	such	such	ADJ
cana-3303	33	11	equations	equation	NOUN
cana-3303	33	12	within	within	ADP
cana-3303	33	13	the	the	DET
cana-3303	33	14	study	study	NOUN
cana-3303	33	15	of	of	ADP
cana-3303	33	16	numerical	numerical	ADJ
cana-3303	33	17	methods	method	NOUN
cana-3303	33	18	since	since	SCONJ
cana-3303	33	19	they	they	PRON
cana-3303	33	20	are	be	AUX
cana-3303	33	21	relatively	relatively	ADV
cana-3303	33	22	common	common	ADJ
cana-3303	33	23	in	in	ADP
cana-3303	33	24	practical	practical	ADJ
cana-3303	33	25	problems	problem	NOUN
cana-3303	33	26	of	of	ADP
cana-3303	33	27	the	the	DET
cana-3303	33	28	discretization	discretization	NOUN
cana-3303	33	29	of	of	ADP
cana-3303	33	30	continuous	continuous	ADJ
cana-3303	33	31	problems	problem	NOUN
cana-3303	33	32	.	.	PUNCT
cana-3303	34	1	linear	linear	ADJ
cana-3303	34	2	additive	additive	ADJ
cana-3303	34	3	derivative	derivative	ADJ
cana-3303	34	4	equations	equation	NOUN
cana-3303	34	5	have	have	AUX
cana-3303	34	6	been	be	AUX
cana-3303	34	7	well	well	ADV
cana-3303	34	8	defined	define	VERB
cana-3303	34	9	in	in	ADP
cana-3303	34	10	the	the	DET
cana-3303	34	11	literature	literature	NOUN
cana-3303	34	12	(	(	PUNCT
cana-3303	34	13	chikhrab	chikhrab	PROPN
cana-3303	34	14	&	&	CCONJ
cana-3303	34	15	malkin	malkin	PROPN
cana-3303	34	16	,	,	PUNCT
cana-3303	34	17	2005	2005	NUM
cana-3303	34	18	)	)	PUNCT
cana-3303	34	19	.	.	PUNCT
cana-3303	35	1	chikhrab	chikhrab	PROPN
cana-3303	35	2	and	and	CCONJ
cana-3303	35	3	malkin	malkin	PROPN
cana-3303	35	4	(	(	PUNCT
cana-3303	35	5	2005	2005	NUM
cana-3303	35	6	)	)	PUNCT
cana-3303	35	7	applied	apply	VERB
cana-3303	35	8	communications	communication	NOUN
cana-3303	35	9	on	on	ADP
cana-3303	35	10	applied	apply	VERB
cana-3303	35	11	nonlinear	nonlinear	ADJ
cana-3303	35	12	analysis	analysis	NOUN
cana-3303	35	13	issn	issn	NOUN
cana-3303	35	14	:	:	PUNCT
cana-3303	35	15	1074	1074	NUM
cana-3303	35	16	-	-	PUNCT
cana-3303	35	17	133x	133x	NUM
cana-3303	35	18	vol	vol	NOUN
cana-3303	35	19	32	32	NUM
cana-3303	36	1	no	no	NOUN
cana-3303	36	2	.	.	PUNCT
cana-3303	37	1	6s	6s	NUM
cana-3303	37	2	(	(	PUNCT
cana-3303	37	3	2025	2025	NUM
cana-3303	37	4	)	)	PUNCT
cana-3303	37	5	379	379	NUM
cana-3303	37	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	37	7	difference	difference	NOUN
cana-3303	37	8	equations	equation	NOUN
cana-3303	37	9	to	to	ADP
cana-3303	37	10	the	the	DET
cana-3303	37	11	scales	scale	NOUN
cana-3303	37	12	of	of	ADP
cana-3303	37	13	problem	problem	NOUN
cana-3303	37	14	solving	solve	VERB
cana-3303	37	15	and	and	CCONJ
cana-3303	37	16	offered	offer	VERB
cana-3303	37	17	a	a	DET
cana-3303	37	18	complete	complete	ADJ
cana-3303	37	19	introduction	introduction	NOUN
cana-3303	37	20	to	to	AUX
cana-3303	37	21	additive	additive	ADJ
cana-3303	37	22	derivatives	derivative	NOUN
cana-3303	37	23	.	.	PUNCT
cana-3303	38	1	more	more	ADV
cana-3303	38	2	recently	recently	ADV
cana-3303	38	3	there	there	PRON
cana-3303	38	4	has	have	AUX
cana-3303	38	5	been	be	AUX
cana-3303	38	6	an	an	DET
cana-3303	38	7	increased	increase	VERB
cana-3303	38	8	interest	interest	NOUN
cana-3303	38	9	in	in	ADP
cana-3303	38	10	non	non	ADJ
cana-3303	38	11	-	-	ADJ
cana-3303	38	12	linear	linear	ADJ
cana-3303	38	13	systems	system	NOUN
cana-3303	38	14	,	,	PUNCT
cana-3303	38	15	specifically	specifically	ADV
cana-3303	38	16	those	those	PRON
cana-3303	38	17	with	with	ADP
cana-3303	38	18	multiplicative	multiplicative	ADJ
cana-3303	38	19	derivatives	derivative	NOUN
cana-3303	38	20	.	.	PUNCT
cana-3303	39	1	as	as	SCONJ
cana-3303	39	2	highlighted	highlight	VERB
cana-3303	39	3	by	by	ADP
cana-3303	39	4	gurskiy	gurskiy	NOUN
cana-3303	39	5	and	and	CCONJ
cana-3303	39	6	sokolov	sokolov	PROPN
cana-3303	39	7	(	(	PUNCT
cana-3303	39	8	2010	2010	NUM
cana-3303	39	9	)	)	PUNCT
cana-3303	39	10	,	,	PUNCT
cana-3303	39	11	these	these	DET
cana-3303	39	12	interactions	interaction	NOUN
cana-3303	39	13	are	be	AUX
cana-3303	39	14	multiplicative	multiplicative	ADJ
cana-3303	39	15	and	and	CCONJ
cana-3303	39	16	so	so	ADV
cana-3303	39	17	lead	lead	VERB
cana-3303	39	18	to	to	ADP
cana-3303	39	19	more	more	ADV
cana-3303	39	20	complex	complex	ADJ
cana-3303	39	21	behavior	behavior	NOUN
cana-3303	39	22	(	(	PUNCT
cana-3303	39	23	non	non	ADJ
cana-3303	39	24	-	-	ADJ
cana-3303	39	25	linear	linear	ADJ
cana-3303	39	26	)	)	PUNCT
cana-3303	39	27	in	in	ADP
cana-3303	39	28	both	both	DET
cana-3303	39	29	systems	system	NOUN
cana-3303	39	30	.	.	PUNCT
cana-3303	40	1	multiplicative	multiplicative	ADJ
cana-3303	40	2	derivatives	derivative	NOUN
cana-3303	40	3	are	be	AUX
cana-3303	40	4	more	more	ADV
cana-3303	40	5	difficult	difficult	ADJ
cana-3303	40	6	to	to	PART
cana-3303	40	7	analyze	analyze	VERB
cana-3303	40	8	and	and	CCONJ
cana-3303	40	9	need	need	VERB
cana-3303	40	10	more	more	ADJ
cana-3303	40	11	complex	complex	ADJ
cana-3303	40	12	mathematical	mathematical	ADJ
cana-3303	40	13	methods	method	NOUN
cana-3303	40	14	for	for	ADP
cana-3303	40	15	their	their	PRON
cana-3303	40	16	solution	solution	NOUN
cana-3303	40	17	compared	compare	VERB
cana-3303	40	18	to	to	ADP
cana-3303	40	19	the	the	DET
cana-3303	40	20	additive	additive	ADJ
cana-3303	40	21	derivatives	derivative	NOUN
cana-3303	40	22	.	.	PUNCT
cana-3303	41	1	this	this	DET
cana-3303	41	2	approach	approach	NOUN
cana-3303	41	3	has	have	AUX
cana-3303	41	4	been	be	AUX
cana-3303	41	5	useful	useful	ADJ
cana-3303	41	6	for	for	ADP
cana-3303	41	7	physical	physical	ADJ
cana-3303	41	8	systems	system	NOUN
cana-3303	41	9	,	,	PUNCT
cana-3303	41	10	since	since	SCONJ
cana-3303	41	11	many	many	ADJ
cana-3303	41	12	,	,	PUNCT
cana-3303	41	13	if	if	SCONJ
cana-3303	41	14	not	not	PART
cana-3303	41	15	all	all	PRON
cana-3303	41	16	,	,	PUNCT
cana-3303	41	17	systems	system	NOUN
cana-3303	41	18	of	of	ADP
cana-3303	41	19	interest	interest	NOUN
cana-3303	41	20	may	may	AUX
cana-3303	41	21	be	be	AUX
cana-3303	41	22	modeled	model	VERB
cana-3303	41	23	with	with	ADP
cana-3303	41	24	multiplicative	multiplicative	ADJ
cana-3303	41	25	derivatives	derivative	NOUN
cana-3303	41	26	,	,	PUNCT
cana-3303	41	27	as	as	SCONJ
cana-3303	41	28	shown	show	VERB
cana-3303	41	29	by	by	ADP
cana-3303	41	30	gurskiy	gurskiy	NOUN
cana-3303	41	31	and	and	CCONJ
cana-3303	41	32	sokolov	sokolov	PROPN
cana-3303	41	33	(	(	PUNCT
cana-3303	41	34	2010	2010	NUM
cana-3303	41	35	)	)	PUNCT
cana-3303	41	36	.	.	PUNCT
cana-3303	42	1	3.2	3.2	NUM
cana-3303	42	2	.	.	PUNCT
cana-3303	42	3	discrete	discrete	ADJ
cana-3303	42	4	multiplicative	multiplicative	ADJ
cana-3303	42	5	derivatives	derivative	NOUN
cana-3303	42	6	discrete	discrete	VERB
cana-3303	42	7	multiplicative	multiplicative	ADJ
cana-3303	42	8	derivatives	derivative	NOUN
cana-3303	42	9	are	be	AUX
cana-3303	42	10	of	of	ADP
cana-3303	42	11	particular	particular	ADJ
cana-3303	42	12	interest	interest	NOUN
cana-3303	42	13	given	give	VERB
cana-3303	42	14	their	their	PRON
cana-3303	42	15	non	non	ADJ
cana-3303	42	16	-	-	NOUN
cana-3303	42	17	linearity	linearity	NOUN
cana-3303	42	18	,	,	PUNCT
cana-3303	42	19	which	which	PRON
cana-3303	42	20	presents	present	VERB
cana-3303	42	21	both	both	DET
cana-3303	42	22	theoretical	theoretical	ADJ
cana-3303	42	23	intractability	intractability	NOUN
cana-3303	42	24	and	and	CCONJ
cana-3303	42	25	computational	computational	ADJ
cana-3303	42	26	challenges	challenge	NOUN
cana-3303	42	27	.	.	PUNCT
cana-3303	43	1	one	one	NUM
cana-3303	43	2	reason	reason	NOUN
cana-3303	43	3	these	these	DET
cana-3303	43	4	equations	equation	NOUN
cana-3303	43	5	appear	appear	VERB
cana-3303	43	6	often	often	ADV
cana-3303	43	7	is	be	AUX
cana-3303	43	8	because	because	SCONJ
cana-3303	43	9	they	they	PRON
cana-3303	43	10	frequently	frequently	ADV
cana-3303	43	11	result	result	VERB
cana-3303	43	12	from	from	ADP
cana-3303	43	13	examining	examine	VERB
cana-3303	43	14	growth	growth	NOUN
cana-3303	43	15	models	model	NOUN
cana-3303	43	16	,	,	PUNCT
cana-3303	43	17	systems	system	NOUN
cana-3303	43	18	where	where	SCONJ
cana-3303	43	19	interactions	interaction	NOUN
cana-3303	43	20	are	be	AUX
cana-3303	43	21	exponential	exponential	ADJ
cana-3303	43	22	,	,	PUNCT
cana-3303	43	23	and	and	CCONJ
cana-3303	43	24	certain	certain	ADJ
cana-3303	43	25	kinds	kind	NOUN
cana-3303	43	26	of	of	ADP
cana-3303	43	27	feedback	feedback	NOUN
cana-3303	43	28	.	.	PUNCT
cana-3303	44	1	the	the	DET
cana-3303	44	2	work	work	NOUN
cana-3303	44	3	of	of	ADP
cana-3303	44	4	dyachenko	dyachenko	NOUN
cana-3303	44	5	and	and	CCONJ
cana-3303	44	6	gulin	gulin	PROPN
cana-3303	44	7	(	(	PUNCT
cana-3303	44	8	2003	2003	NUM
cana-3303	44	9	)	)	PUNCT
cana-3303	44	10	systematically	systematically	ADV
cana-3303	44	11	analyzed	analyze	VERB
cana-3303	44	12	discrete	discrete	ADJ
cana-3303	44	13	nonlinear	nonlinear	ADJ
cana-3303	44	14	equations	equation	NOUN
cana-3303	44	15	and	and	CCONJ
cana-3303	44	16	their	their	PRON
cana-3303	44	17	multiplicative	multiplicative	ADJ
cana-3303	44	18	derivatives	derivative	NOUN
cana-3303	44	19	,	,	PUNCT
cana-3303	44	20	forming	form	VERB
cana-3303	44	21	a	a	DET
cana-3303	44	22	theoretical	theoretical	ADJ
cana-3303	44	23	basis	basis	NOUN
cana-3303	44	24	to	to	PART
cana-3303	44	25	provide	provide	VERB
cana-3303	44	26	insight	insight	NOUN
cana-3303	44	27	into	into	ADP
cana-3303	44	28	such	such	ADJ
cana-3303	44	29	complex	complex	ADJ
cana-3303	44	30	systems	system	NOUN
cana-3303	44	31	.	.	PUNCT
cana-3303	45	1	their	their	PRON
cana-3303	45	2	emphasized	emphasize	VERB
cana-3303	45	3	the	the	DET
cana-3303	45	4	mathematical	mathematical	ADJ
cana-3303	45	5	methods	method	NOUN
cana-3303	45	6	to	to	PART
cana-3303	45	7	solve	solve	VERB
cana-3303	45	8	these	these	DET
cana-3303	45	9	types	type	NOUN
cana-3303	45	10	of	of	ADP
cana-3303	45	11	equations	equation	NOUN
cana-3303	45	12	such	such	ADJ
cana-3303	45	13	as	as	ADP
cana-3303	45	14	transformation	transformation	NOUN
cana-3303	45	15	techniques	technique	NOUN
cana-3303	45	16	and	and	CCONJ
cana-3303	45	17	solution	solution	NOUN
cana-3303	45	18	strategies	strategy	NOUN
cana-3303	45	19	of	of	ADP
cana-3303	45	20	nonlinear	nonlinear	ADJ
cana-3303	45	21	difference	difference	NOUN
cana-3303	45	22	equations	equation	NOUN
cana-3303	45	23	.	.	PUNCT
cana-3303	46	1	in	in	ADP
cana-3303	46	2	addition	addition	NOUN
cana-3303	46	3	,	,	PUNCT
cana-3303	46	4	the	the	DET
cana-3303	46	5	application	application	NOUN
cana-3303	46	6	of	of	ADP
cana-3303	46	7	poverative	poverative	ADJ
cana-3303	46	8	derivatives	derivative	NOUN
cana-3303	46	9	and	and	CCONJ
cana-3303	46	10	integrals	integral	NOUN
cana-3303	46	11	in	in	ADP
cana-3303	46	12	the	the	DET
cana-3303	46	13	discrete	discrete	ADJ
cana-3303	46	14	context	context	NOUN
cana-3303	46	15	was	be	AUX
cana-3303	46	16	also	also	ADV
cana-3303	46	17	been	be	AUX
cana-3303	46	18	acknowledged	acknowledge	VERB
cana-3303	46	19	.	.	PUNCT
cana-3303	47	1	in	in	ADP
cana-3303	47	2	the	the	DET
cana-3303	47	3	discrete	discrete	ADJ
cana-3303	47	4	case	case	NOUN
cana-3303	47	5	,	,	PUNCT
cana-3303	47	6	such	such	ADJ
cana-3303	47	7	derivatives	derivative	NOUN
cana-3303	47	8	are	be	AUX
cana-3303	47	9	defined	define	VERB
cana-3303	47	10	with	with	ADP
cana-3303	47	11	a	a	DET
cana-3303	47	12	single	single	ADJ
cana-3303	47	13	operator	operator	NOUN
cana-3303	47	14	rather	rather	ADV
cana-3303	47	15	than	than	ADP
cana-3303	47	16	multiple	multiple	ADJ
cana-3303	47	17	operators	operator	NOUN
cana-3303	47	18	,	,	PUNCT
cana-3303	47	19	thus	thus	ADV
cana-3303	47	20	making	make	VERB
cana-3303	47	21	them	they	PRON
cana-3303	47	22	tractable	tractable	ADJ
cana-3303	47	23	.	.	PUNCT
cana-3303	48	1	are	be	AUX
cana-3303	48	2	smith	smith	PROPN
cana-3303	48	3	and	and	CCONJ
cana-3303	48	4	jones	jones	PROPN
cana-3303	48	5	(	(	PUNCT
cana-3303	48	6	2007	2007	NUM
cana-3303	48	7	)	)	PUNCT
cana-3303	48	8	proved	prove	VERB
cana-3303	48	9	efficacy	efficacy	NOUN
cana-3303	48	10	of	of	ADP
cana-3303	48	11	using	use	VERB
cana-3303	48	12	poverative	poverative	ADJ
cana-3303	48	13	derivatives	derivative	NOUN
cana-3303	48	14	in	in	ADP
cana-3303	48	15	boundary	boundary	ADJ
cana-3303	48	16	value	value	NOUN
cana-3303	48	17	problems	problem	NOUN
cana-3303	48	18	,	,	PUNCT
cana-3303	48	19	thus	thus	ADV
cana-3303	48	20	enabling	enable	VERB
cana-3303	48	21	solution	solution	NOUN
cana-3303	48	22	of	of	ADP
cana-3303	48	23	complex	complex	ADJ
cana-3303	48	24	systems	system	NOUN
cana-3303	48	25	with	with	ADP
cana-3303	48	26	non	non	ADJ
cana-3303	48	27	-	-	ADJ
cana-3303	48	28	linear	linear	ADJ
cana-3303	48	29	dynamics	dynamic	NOUN
cana-3303	48	30	.	.	PUNCT
cana-3303	49	1	subsequent	subsequent	ADJ
cana-3303	49	2	work	work	NOUN
cana-3303	49	3	prepared	prepare	VERB
cana-3303	49	4	them	they	PRON
cana-3303	49	5	to	to	PART
cana-3303	49	6	apply	apply	VERB
cana-3303	49	7	these	these	DET
cana-3303	49	8	methods	method	NOUN
cana-3303	49	9	to	to	PART
cana-3303	49	10	solve	solve	VERB
cana-3303	49	11	equations	equation	NOUN
cana-3303	49	12	involving	involve	VERB
cana-3303	49	13	discrete	discrete	ADJ
cana-3303	49	14	multiplicative	multiplicative	ADJ
cana-3303	49	15	derivatives	derivative	NOUN
cana-3303	49	16	.	.	PUNCT
cana-3303	50	1	3.3	3.3	NUM
cana-3303	50	2	.	.	PUNCT
cana-3303	51	1	exercise	exercise	NOUN
cana-3303	51	2	of	of	ADP
cana-3303	51	3	discrete	discrete	ADJ
cana-3303	51	4	derivatives	derivative	NOUN
cana-3303	51	5	in	in	ADP
cana-3303	51	6	boundary	boundary	ADJ
cana-3303	51	7	problems	problem	NOUN
cana-3303	51	8	the	the	DET
cana-3303	51	9	boundary	boundary	ADJ
cana-3303	51	10	conditions	condition	NOUN
cana-3303	51	11	are	be	AUX
cana-3303	51	12	one	one	NUM
cana-3303	51	13	of	of	ADP
cana-3303	51	14	the	the	DET
cana-3303	51	15	major	major	ADJ
cana-3303	51	16	problems	problem	NOUN
cana-3303	51	17	in	in	ADP
cana-3303	51	18	the	the	DET
cana-3303	51	19	theory	theory	NOUN
cana-3303	51	20	of	of	ADP
cana-3303	51	21	discrete	discrete	ADJ
cana-3303	51	22	derivative	derivative	ADJ
cana-3303	51	23	equations	equation	NOUN
cana-3303	51	24	.	.	PUNCT
cana-3303	52	1	boundary	boundary	ADJ
cana-3303	52	2	value	value	NOUN
cana-3303	52	3	problems	problem	NOUN
cana-3303	52	4	are	be	AUX
cana-3303	52	5	important	important	ADJ
cana-3303	52	6	to	to	ADP
cana-3303	52	7	both	both	CCONJ
cana-3303	52	8	theoretical	theoretical	ADJ
cana-3303	52	9	and	and	CCONJ
cana-3303	52	10	applied	apply	VERB
cana-3303	52	11	mathematics	mathematic	NOUN
cana-3303	52	12	,	,	PUNCT
cana-3303	52	13	as	as	SCONJ
cana-3303	52	14	they	they	PRON
cana-3303	52	15	describe	describe	VERB
cana-3303	52	16	numerous	numerous	ADJ
cana-3303	52	17	physical	physical	ADJ
cana-3303	52	18	phenomena	phenomenon	NOUN
cana-3303	52	19	where	where	SCONJ
cana-3303	52	20	the	the	DET
cana-3303	52	21	values	value	NOUN
cana-3303	52	22	at	at	ADP
cana-3303	52	23	the	the	DET
cana-3303	52	24	edges	edge	NOUN
cana-3303	52	25	of	of	ADP
cana-3303	52	26	the	the	DET
cana-3303	52	27	system	system	NOUN
cana-3303	52	28	are	be	AUX
cana-3303	52	29	known	know	VERB
cana-3303	52	30	.	.	PUNCT
cana-3303	53	1	in	in	ADP
cana-3303	53	2	this	this	DET
cana-3303	53	3	section	section	NOUN
cana-3303	53	4	we	we	PRON
cana-3303	53	5	highlight	highlight	VERB
cana-3303	53	6	the	the	DET
cana-3303	53	7	implications	implication	NOUN
cana-3303	53	8	of	of	ADP
cana-3303	53	9	boundary	boundary	ADJ
cana-3303	53	10	conditions	condition	NOUN
cana-3303	53	11	in	in	ADP
cana-3303	53	12	discrete	discrete	ADJ
cana-3303	53	13	derivative	derivative	ADJ
cana-3303	53	14	equations	equation	NOUN
cana-3303	53	15	and	and	CCONJ
cana-3303	53	16	emphasis	emphasis	VERB
cana-3303	53	17	the	the	DET
cana-3303	53	18	intuition	intuition	NOUN
cana-3303	53	19	behind	behind	ADP
cana-3303	53	20	specific	specific	ADJ
cana-3303	53	21	net	net	NOUN
cana-3303	53	22	.	.	PUNCT
cana-3303	54	1	smith	smith	PROPN
cana-3303	54	2	and	and	CCONJ
cana-3303	54	3	jones	jones	PROPN
cana-3303	54	4	(	(	PUNCT
cana-3303	54	5	2007	2007	NUM
cana-3303	54	6	)	)	PUNCT
cana-3303	54	7	examined	examine	VERB
cana-3303	54	8	the	the	DET
cana-3303	54	9	methods	method	NOUN
cana-3303	54	10	for	for	ADP
cana-3303	54	11	treating	treat	VERB
cana-3303	54	12	boundary	boundary	ADJ
cana-3303	54	13	value	value	NOUN
cana-3303	54	14	problems	problem	NOUN
cana-3303	54	15	with	with	ADP
cana-3303	54	16	discrete	discrete	ADJ
cana-3303	54	17	derivatives	derivative	NOUN
cana-3303	54	18	,	,	PUNCT
cana-3303	54	19	demonstrating	demonstrate	VERB
cana-3303	54	20	that	that	SCONJ
cana-3303	54	21	solutions	solution	NOUN
cana-3303	54	22	can	can	AUX
cana-3303	54	23	be	be	AUX
cana-3303	54	24	reached	reach	VERB
cana-3303	54	25	under	under	ADP
cana-3303	54	26	certain	certain	ADJ
cana-3303	54	27	conditions	condition	NOUN
cana-3303	54	28	.	.	PUNCT
cana-3303	55	1	here	here	ADV
cana-3303	55	2	we	we	PRON
cana-3303	55	3	consider	consider	VERB
cana-3303	55	4	a	a	DET
cana-3303	55	5	third	third	ADJ
cana-3303	55	6	-	-	PUNCT
cana-3303	55	7	order	order	NOUN
cana-3303	55	8	mixed	mixed	ADJ
cana-3303	55	9	derivative	derivative	ADJ
cana-3303	55	10	equation	equation	NOUN
cana-3303	55	11	that	that	PRON
cana-3303	55	12	is	be	AUX
cana-3303	55	13	obtained	obtain	VERB
cana-3303	55	14	using	use	VERB
cana-3303	55	15	discrete	discrete	ADJ
cana-3303	55	16	additive	additive	ADJ
cana-3303	55	17	and	and	CCONJ
cana-3303	55	18	poverative	poverative	ADJ
cana-3303	55	19	derivatives	derivative	NOUN
cana-3303	55	20	.	.	PUNCT
cana-3303	56	1	for	for	ADP
cana-3303	56	2	its	its	PRON
cana-3303	56	3	non	non	ADJ
cana-3303	56	4	-	-	ADJ
cana-3303	56	5	linear	linear	ADJ
cana-3303	56	6	nature	nature	NOUN
cana-3303	56	7	and	and	CCONJ
cana-3303	56	8	the	the	DET
cana-3303	56	9	multiple	multiple	ADJ
cana-3303	56	10	derivatives	derivative	NOUN
cana-3303	56	11	each	each	PRON
cana-3303	56	12	requiring	require	VERB
cana-3303	56	13	specific	specific	ADJ
cana-3303	56	14	attention	attention	NOUN
cana-3303	56	15	make	make	VERB
cana-3303	56	16	this	this	DET
cana-3303	56	17	type	type	NOUN
cana-3303	56	18	of	of	ADP
cana-3303	56	19	equation	equation	NOUN
cana-3303	56	20	a	a	DET
cana-3303	56	21	special	special	ADJ
cana-3303	56	22	case	case	NOUN
cana-3303	56	23	.	.	PUNCT
cana-3303	57	1	the	the	DET
cana-3303	57	2	issue	issue	NOUN
cana-3303	57	3	of	of	ADP
cana-3303	57	4	boundary	boundary	ADJ
cana-3303	57	5	conditions	condition	NOUN
cana-3303	57	6	for	for	ADP
cana-3303	57	7	this	this	DET
cana-3303	57	8	equation	equation	NOUN
cana-3303	57	9	,	,	PUNCT
cana-3303	57	10	which	which	PRON
cana-3303	57	11	we	we	PRON
cana-3303	57	12	shall	shall	AUX
cana-3303	57	13	not	not	PART
cana-3303	57	14	be	be	AUX
cana-3303	57	15	addressed	address	VERB
cana-3303	57	16	to	to	ADP
cana-3303	57	17	the	the	DET
cana-3303	57	18	same	same	ADJ
cana-3303	57	19	extent	extent	NOUN
cana-3303	57	20	in	in	ADP
cana-3303	57	21	previous	previous	ADJ
cana-3303	57	22	papers	paper	NOUN
cana-3303	57	23	(	(	PUNCT
cana-3303	57	24	gurskiy	gurskiy	PROPN
cana-3303	57	25	&	&	CCONJ
cana-3303	57	26	sokolov	sokolov	PROPN
cana-3303	57	27	,	,	PUNCT
cana-3303	57	28	2010	2010	NUM
cana-3303	57	29	)	)	PUNCT
cana-3303	57	30	,	,	PUNCT
cana-3303	57	31	is	be	AUX
cana-3303	57	32	obviously	obviously	ADV
cana-3303	57	33	critical	critical	ADJ
cana-3303	57	34	for	for	ADP
cana-3303	57	35	the	the	DET
cana-3303	57	36	construction	construction	NOUN
cana-3303	57	37	of	of	ADP
cana-3303	57	38	a	a	DET
cana-3303	57	39	complete	complete	ADJ
cana-3303	57	40	solution	solution	NOUN
cana-3303	57	41	.	.	PUNCT
cana-3303	58	1	communications	communication	NOUN
cana-3303	58	2	on	on	ADP
cana-3303	58	3	applied	apply	VERB
cana-3303	58	4	nonlinear	nonlinear	ADJ
cana-3303	58	5	analysis	analysis	NOUN
cana-3303	58	6	issn	issn	NOUN
cana-3303	58	7	:	:	PUNCT
cana-3303	58	8	1074	1074	NUM
cana-3303	58	9	-	-	PUNCT
cana-3303	58	10	133x	133x	NUM
cana-3303	58	11	vol	vol	NOUN
cana-3303	58	12	32	32	NUM
cana-3303	58	13	no	no	NOUN
cana-3303	58	14	.	.	PUNCT
cana-3303	59	1	6s	6s	NUM
cana-3303	59	2	(	(	PUNCT
cana-3303	59	3	2025	2025	NUM
cana-3303	59	4	)	)	PUNCT
cana-3303	59	5	380	380	NUM
cana-3303	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	59	7	3.4	3.4	NUM
cana-3303	59	8	.	.	PUNCT
cana-3303	60	1	recent	recent	ADJ
cana-3303	60	2	developments	development	NOUN
cana-3303	60	3	and	and	CCONJ
cana-3303	60	4	outstanding	outstanding	ADJ
cana-3303	60	5	questions	question	NOUN
cana-3303	60	6	discrete	discrete	VERB
cana-3303	60	7	multiplicative	multiplicative	ADJ
cana-3303	60	8	derivatives	derivative	NOUN
cana-3303	60	9	emerged	emerge	VERB
cana-3303	60	10	as	as	ADP
cana-3303	60	11	a	a	DET
cana-3303	60	12	new	new	ADJ
cana-3303	60	13	path	path	NOUN
cana-3303	60	14	toward	toward	ADP
cana-3303	60	15	more	more	ADJ
cana-3303	60	16	complex	complex	ADJ
cana-3303	60	17	systems	system	NOUN
cana-3303	60	18	.	.	PUNCT
cana-3303	61	1	we	we	PRON
cana-3303	61	2	have	have	AUX
cana-3303	61	3	made	make	VERB
cana-3303	61	4	great	great	ADJ
cana-3303	61	5	strides	stride	NOUN
cana-3303	61	6	towards	towards	ADP
cana-3303	61	7	understanding	understanding	NOUN
cana-3303	61	8	and	and	CCONJ
cana-3303	61	9	solving	solve	VERB
cana-3303	61	10	discrete	discrete	ADJ
cana-3303	61	11	derivative	derivative	ADJ
cana-3303	61	12	equations	equation	NOUN
cana-3303	61	13	,	,	PUNCT
cana-3303	61	14	but	but	CCONJ
cana-3303	61	15	have	have	VERB
cana-3303	61	16	a	a	DET
cana-3303	61	17	long	long	ADJ
cana-3303	61	18	way	way	NOUN
cana-3303	61	19	to	to	PART
cana-3303	61	20	go	go	VERB
cana-3303	61	21	,	,	PUNCT
cana-3303	61	22	especially	especially	ADV
cana-3303	61	23	in	in	ADP
cana-3303	61	24	the	the	DET
cana-3303	61	25	non	non	ADJ
cana-3303	61	26	-	-	ADJ
cana-3303	61	27	linear	linear	ADJ
cana-3303	61	28	regime	regime	NOUN
cana-3303	61	29	.	.	PUNCT
cana-3303	62	1	one	one	NUM
cana-3303	62	2	major	major	ADJ
cana-3303	62	3	open	open	ADJ
cana-3303	62	4	problem	problem	NOUN
cana-3303	62	5	is	be	AUX
cana-3303	62	6	that	that	SCONJ
cana-3303	62	7	,	,	PUNCT
cana-3303	62	8	even	even	ADV
cana-3303	62	9	about	about	ADP
cana-3303	62	10	equations	equation	NOUN
cana-3303	62	11	like	like	ADP
cana-3303	62	12	this	this	PRON
cana-3303	62	13	,	,	PUNCT
cana-3303	62	14	boundary	boundary	ADJ
cana-3303	62	15	value	value	NOUN
cana-3303	62	16	problems	problem	NOUN
cana-3303	62	17	are	be	AUX
cana-3303	62	18	still	still	ADV
cana-3303	62	19	poorly	poorly	ADV
cana-3303	62	20	understood	understand	VERB
cana-3303	62	21	and	and	CCONJ
cana-3303	62	22	the	the	DET
cana-3303	62	23	question	question	NOUN
cana-3303	62	24	of	of	ADP
cana-3303	62	25	what	what	DET
cana-3303	62	26	years	year	NOUN
cana-3303	62	27	dependent	dependent	ADJ
cana-3303	62	28	conditions	condition	NOUN
cana-3303	62	29	on	on	ADP
cana-3303	62	30	the	the	DET
cana-3303	62	31	boundary	boundary	NOUN
cana-3303	62	32	may	may	AUX
cana-3303	62	33	leads	lead	VERB
cana-3303	62	34	to	to	ADP
cana-3303	62	35	such	such	ADJ
cana-3303	62	36	solutions	solution	NOUN
cana-3303	62	37	is	be	AUX
cana-3303	62	38	wide	wide	ADV
cana-3303	62	39	open	open	ADJ
cana-3303	62	40	.	.	PUNCT
cana-3303	63	1	they	they	PRON
cana-3303	63	2	are	be	AUX
cana-3303	63	3	fundamental	fundamental	ADJ
cana-3303	63	4	for	for	ADP
cana-3303	63	5	the	the	DET
cana-3303	63	6	theoretical	theoretical	ADJ
cana-3303	63	7	aspects	aspect	NOUN
cana-3303	63	8	of	of	ADP
cana-3303	63	9	the	the	DET
cana-3303	63	10	field	field	NOUN
cana-3303	63	11	and	and	CCONJ
cana-3303	63	12	are	be	AUX
cana-3303	63	13	also	also	ADV
cana-3303	63	14	fundamental	fundamental	ADJ
cana-3303	63	15	since	since	SCONJ
cana-3303	63	16	most	most	ADJ
cana-3303	63	17	physical	physical	ADJ
cana-3303	63	18	systems	system	NOUN
cana-3303	63	19	are	be	AUX
cana-3303	63	20	implemented	implement	VERB
cana-3303	63	21	and	and	CCONJ
cana-3303	63	22	controlled	control	VERB
cana-3303	63	23	as	as	ADP
cana-3303	63	24	discrete	discrete	ADJ
cana-3303	63	25	systems	system	NOUN
cana-3303	63	26	.	.	PUNCT
cana-3303	64	1	also	also	ADV
cana-3303	64	2	,	,	PUNCT
cana-3303	64	3	the	the	DET
cana-3303	64	4	literature	literature	NOUN
cana-3303	64	5	has	have	AUX
cana-3303	64	6	recently	recently	ADV
cana-3303	64	7	started	start	VERB
cana-3303	64	8	to	to	PART
cana-3303	64	9	tackle	tackle	VERB
cana-3303	64	10	some	some	PRON
cana-3303	64	11	of	of	ADP
cana-3303	64	12	the	the	DET
cana-3303	64	13	open	open	ADJ
cana-3303	64	14	questions	question	NOUN
cana-3303	64	15	in	in	ADP
cana-3303	64	16	the	the	DET
cana-3303	64	17	area	area	NOUN
cana-3303	64	18	.	.	PUNCT
cana-3303	65	1	the	the	DET
cana-3303	65	2	use	use	NOUN
cana-3303	65	3	and	and	CCONJ
cana-3303	65	4	applications	application	NOUN
cana-3303	65	5	of	of	ADP
cana-3303	65	6	discrete	discrete	ADJ
cana-3303	65	7	multiplicative	multiplicative	ADJ
cana-3303	65	8	derivatives	derivative	NOUN
cana-3303	65	9	,	,	PUNCT
cana-3303	65	10	for	for	ADP
cana-3303	65	11	example	example	NOUN
cana-3303	65	12	,	,	PUNCT
cana-3303	65	13	are	be	AUX
cana-3303	65	14	still	still	ADV
cana-3303	65	15	an	an	DET
cana-3303	65	16	open	open	ADJ
cana-3303	65	17	area	area	NOUN
cana-3303	65	18	of	of	ADP
cana-3303	65	19	study	study	NOUN
cana-3303	65	20	in	in	ADP
cana-3303	65	21	fields	field	NOUN
cana-3303	65	22	such	such	ADJ
cana-3303	65	23	as	as	ADP
cana-3303	65	24	modeling	model	VERB
cana-3303	65	25	non	non	ADJ
cana-3303	65	26	-	-	ADJ
cana-3303	65	27	linear	linear	ADJ
cana-3303	65	28	systems	system	NOUN
cana-3303	65	29	,	,	PUNCT
cana-3303	65	30	including	include	VERB
cana-3303	65	31	population	population	NOUN
cana-3303	65	32	dynamics	dynamic	NOUN
cana-3303	65	33	,	,	PUNCT
cana-3303	65	34	heat	heat	NOUN
cana-3303	65	35	conduction	conduction	NOUN
cana-3303	65	36	,	,	PUNCT
cana-3303	65	37	and	and	CCONJ
cana-3303	65	38	other	other	ADJ
cana-3303	65	39	physical	physical	ADJ
cana-3303	65	40	processes	process	NOUN
cana-3303	65	41	.	.	PUNCT
cana-3303	66	1	in	in	ADP
cana-3303	66	2	this	this	DET
cana-3303	66	3	regard	regard	NOUN
cana-3303	66	4	,	,	PUNCT
cana-3303	66	5	we	we	PRON
cana-3303	66	6	can	can	AUX
cana-3303	66	7	highlight	highlight	VERB
cana-3303	66	8	the	the	DET
cana-3303	66	9	requirement	requirement	NOUN
cana-3303	66	10	of	of	ADP
cana-3303	66	11	new	new	ADJ
cana-3303	66	12	solution	solution	NOUN
cana-3303	66	13	methods	method	NOUN
cana-3303	66	14	in	in	ADP
cana-3303	66	15	particular	particular	ADJ
cana-3303	66	16	for	for	ADP
cana-3303	66	17	equations	equation	NOUN
cana-3303	66	18	having	have	VERB
cana-3303	66	19	non	non	ADJ
cana-3303	66	20	-	-	ADJ
cana-3303	66	21	local	local	ADJ
cana-3303	66	22	boundary	boundary	ADJ
cana-3303	66	23	conditions	condition	NOUN
cana-3303	66	24	as	as	SCONJ
cana-3303	66	25	some	some	PRON
cana-3303	66	26	of	of	ADP
cana-3303	66	27	the	the	DET
cana-3303	66	28	future	future	ADJ
cana-3303	66	29	research	research	NOUN
cana-3303	66	30	will	will	AUX
cana-3303	66	31	be	be	AUX
cana-3303	66	32	vital	vital	ADJ
cana-3303	66	33	in	in	ADP
cana-3303	66	34	this	this	DET
cana-3303	66	35	area	area	NOUN
cana-3303	66	36	.	.	PUNCT
cana-3303	67	1	4	4	X
cana-3303	67	2	.	.	X
cana-3303	67	3	methods	method	NOUN
cana-3303	67	4	the	the	DET
cana-3303	67	5	method	method	NOUN
cana-3303	67	6	of	of	ADP
cana-3303	67	7	this	this	DET
cana-3303	67	8	study	study	NOUN
cana-3303	67	9	is	be	AUX
cana-3303	67	10	based	base	VERB
cana-3303	67	11	on	on	ADP
cana-3303	67	12	a	a	DET
cana-3303	67	13	third	third	ADJ
cana-3303	67	14	-	-	PUNCT
cana-3303	67	15	order	order	NOUN
cana-3303	67	16	nonlinear	nonlinear	ADJ
cana-3303	67	17	mixed	mix	VERB
cana-3303	67	18	derivative	derivative	ADJ
cana-3303	67	19	equation	equation	NOUN
cana-3303	67	20	obtained	obtain	VERB
cana-3303	67	21	through	through	ADP
cana-3303	67	22	discrete	discrete	ADJ
cana-3303	67	23	additive	additive	NOUN
cana-3303	67	24	and	and	CCONJ
cana-3303	67	25	multiplicative	multiplicative	ADJ
cana-3303	67	26	derivative	derivative	ADJ
cana-3303	67	27	operations	operation	NOUN
cana-3303	67	28	.	.	PUNCT
cana-3303	68	1	this	this	PRON
cana-3303	68	2	is	be	AUX
cana-3303	68	3	an	an	DET
cana-3303	68	4	analytical	analytical	ADJ
cana-3303	68	5	research	research	NOUN
cana-3303	68	6	utilizing	utilize	VERB
cana-3303	68	7	mathematical	mathematical	ADJ
cana-3303	68	8	metamorphoses	metamorphosis	NOUN
cana-3303	68	9	based	base	VERB
cana-3303	68	10	on	on	ADP
cana-3303	68	11	existing	exist	VERB
cana-3303	68	12	causal	causal	ADJ
cana-3303	68	13	theorems	theorem	NOUN
cana-3303	68	14	of	of	ADP
cana-3303	68	15	discrete	discrete	ADJ
cana-3303	68	16	derivatives	derivative	NOUN
cana-3303	68	17	.	.	PUNCT
cana-3303	69	1	absorption	absorption	NOUN
cana-3303	69	2	of	of	ADP
cana-3303	69	3	a	a	DET
cana-3303	69	4	nonlinear	nonlinear	ADJ
cana-3303	69	5	equation	equation	NOUN
cana-3303	69	6	here	here	ADV
cana-3303	69	7	we	we	PRON
cana-3303	69	8	will	will	AUX
cana-3303	69	9	be	be	AUX
cana-3303	69	10	solving	solve	VERB
cana-3303	69	11	a	a	DET
cana-3303	69	12	third	third	ADJ
cana-3303	69	13	-	-	PUNCT
cana-3303	69	14	order	order	NOUN
cana-3303	69	15	nonlinear	nonlinear	ADJ
cana-3303	69	16	equation	equation	NOUN
cana-3303	69	17	involving	involve	VERB
cana-3303	69	18	additive	additive	ADJ
cana-3303	69	19	and	and	CCONJ
cana-3303	69	20	multiplicative	multiplicative	ADJ
cana-3303	69	21	derivatives	derivative	NOUN
cana-3303	69	22	.	.	PUNCT
cana-3303	70	1	using	use	VERB
cana-3303	70	2	specialized	specialized	ADJ
cana-3303	70	3	methods	method	NOUN
cana-3303	70	4	,	,	PUNCT
cana-3303	70	5	this	this	DET
cana-3303	70	6	complex	complex	ADJ
cana-3303	70	7	equation	equation	NOUN
cana-3303	70	8	can	can	AUX
cana-3303	70	9	be	be	AUX
cana-3303	70	10	simplified	simplify	VERB
cana-3303	70	11	and	and	CCONJ
cana-3303	70	12	solved	solve	VERB
cana-3303	70	13	.	.	PUNCT
cana-3303	71	1	discrete	discrete	ADJ
cana-3303	71	2	derivatives	derivative	NOUN
cana-3303	71	3	the	the	DET
cana-3303	71	4	methodology	methodology	NOUN
cana-3303	71	5	relies	rely	VERB
cana-3303	71	6	heavily	heavily	ADV
cana-3303	71	7	on	on	ADP
cana-3303	71	8	discrete	discrete	ADJ
cana-3303	71	9	derivatives	derivative	NOUN
cana-3303	71	10	.	.	PUNCT
cana-3303	72	1	discrete	discrete	ADJ
cana-3303	72	2	additive	additive	ADJ
cana-3303	72	3	derivatives	derivative	NOUN
cana-3303	72	4	(	(	PUNCT
cana-3303	72	5	which	which	PRON
cana-3303	72	6	work	work	VERB
cana-3303	72	7	on	on	ADP
cana-3303	72	8	differences	difference	NOUN
cana-3303	72	9	between	between	ADP
cana-3303	72	10	terms	term	NOUN
cana-3303	72	11	)	)	PUNCT
cana-3303	72	12	and	and	CCONJ
cana-3303	72	13	discrete	discrete	VERB
cana-3303	72	14	multiplicative	multiplicative	ADJ
cana-3303	72	15	derivatives	derivative	NOUN
cana-3303	72	16	(	(	PUNCT
cana-3303	72	17	which	which	PRON
cana-3303	72	18	work	work	VERB
cana-3303	72	19	on	on	ADP
cana-3303	72	20	the	the	DET
cana-3303	72	21	ratio	ratio	NOUN
cana-3303	72	22	of	of	ADP
cana-3303	72	23	terms	term	NOUN
cana-3303	72	24	)	)	PUNCT
cana-3303	72	25	,	,	PUNCT
cana-3303	72	26	are	be	AUX
cana-3303	72	27	first	first	ADV
cana-3303	72	28	used	use	VERB
cana-3303	72	29	to	to	PART
cana-3303	72	30	transform	transform	VERB
cana-3303	72	31	the	the	DET
cana-3303	72	32	equation	equation	NOUN
cana-3303	72	33	.	.	PUNCT
cana-3303	73	1	these	these	DET
cana-3303	73	2	transformations	transformation	NOUN
cana-3303	73	3	also	also	ADV
cana-3303	73	4	enable	enable	VERB
cana-3303	73	5	the	the	DET
cana-3303	73	6	nonlinear	nonlinear	ADJ
cana-3303	73	7	equation	equation	NOUN
cana-3303	73	8	to	to	PART
cana-3303	73	9	be	be	AUX
cana-3303	73	10	written	write	VERB
cana-3303	73	11	in	in	ADP
cana-3303	73	12	a	a	DET
cana-3303	73	13	simpler	simple	ADJ
cana-3303	73	14	form	form	NOUN
cana-3303	73	15	.	.	PUNCT
cana-3303	74	1	step	step	NOUN
cana-3303	74	2	-	-	PUNCT
cana-3303	74	3	by	by	ADP
cana-3303	74	4	-	-	PUNCT
cana-3303	74	5	step	step	NOUN
cana-3303	74	6	derivation	derivation	NOUN
cana-3303	74	7	of	of	ADP
cana-3303	74	8	expressions	expression	NOUN
cana-3303	74	9	the	the	DET
cana-3303	74	10	process	process	NOUN
cana-3303	74	11	of	of	ADP
cana-3303	74	12	finding	find	VERB
cana-3303	74	13	solution	solution	NOUN
cana-3303	74	14	is	be	AUX
cana-3303	74	15	iterative	iterative	ADJ
cana-3303	74	16	and	and	CCONJ
cana-3303	74	17	derives	derive	VERB
cana-3303	74	18	expressions	expression	NOUN
cana-3303	74	19	one	one	NUM
cana-3303	74	20	after	after	ADP
cana-3303	74	21	another	another	PRON
cana-3303	74	22	.	.	PUNCT
cana-3303	75	1	this	this	DET
cana-3303	75	2	iterative	iterative	NOUN
cana-3303	75	3	approach	approach	NOUN
cana-3303	75	4	has	have	AUX
cana-3303	75	5	been	be	AUX
cana-3303	75	6	applied	apply	VERB
cana-3303	75	7	repeatedly	repeatedly	ADV
cana-3303	75	8	to	to	PART
cana-3303	75	9	progressively	progressively	ADV
cana-3303	75	10	solve	solve	VERB
cana-3303	75	11	the	the	DET
cana-3303	75	12	equation	equation	NOUN
cana-3303	75	13	for	for	ADP
cana-3303	75	14	varying	vary	VERB
cana-3303	75	15	parameter	parameter	NOUN
cana-3303	75	16	values	value	NOUN
cana-3303	75	17	from	from	ADP
cana-3303	75	18	which	which	PRON
cana-3303	75	19	an	an	DET
cana-3303	75	20	overall	overall	ADJ
cana-3303	75	21	solution	solution	NOUN
cana-3303	75	22	can	can	AUX
cana-3303	75	23	be	be	AUX
cana-3303	75	24	arrived	arrive	VERB
cana-3303	75	25	at	at	ADP
cana-3303	75	26	.	.	PUNCT
cana-3303	76	1	problems	problem	NOUN
cana-3303	76	2	of	of	ADP
cana-3303	76	3	cauchy	cauchy	PROPN
cana-3303	76	4	and	and	CCONJ
cana-3303	76	5	boundary	boundary	ADJ
cana-3303	76	6	value	value	NOUN
cana-3303	76	7	two	two	NUM
cana-3303	76	8	classes	class	NOUN
cana-3303	76	9	(	(	PUNCT
cana-3303	76	10	problems	problem	NOUN
cana-3303	76	11	are	be	AUX
cana-3303	76	12	studied	study	VERB
cana-3303	76	13	)	)	PUNCT
cana-3303	76	14	,	,	PUNCT
cana-3303	76	15	cauchy	cauchy	PROPN
cana-3303	76	16	(	(	PUNCT
cana-3303	76	17	problems	problem	NOUN
cana-3303	76	18	(	(	PUNCT
cana-3303	76	19	problem	problem	NOUN
cana-3303	76	20	with	with	ADP
cana-3303	76	21	known	know	VERB
cana-3303	76	22	initial	initial	ADJ
cana-3303	76	23	conditions	condition	NOUN
cana-3303	76	24	)	)	PUNCT
cana-3303	76	25	and	and	CCONJ
cana-3303	76	26	boundary	boundary	ADJ
cana-3303	76	27	value	value	NOUN
cana-3303	76	28	problem	problem	NOUN
cana-3303	76	29	(	(	PUNCT
cana-3303	76	30	problem	problem	NOUN
cana-3303	76	31	with	with	ADP
cana-3303	76	32	known	know	VERB
cana-3303	76	33	boundary	boundary	ADJ
cana-3303	76	34	values	value	NOUN
cana-3303	76	35	)	)	PUNCT
cana-3303	76	36	.	.	PUNCT
cana-3303	77	1	applying	apply	VERB
cana-3303	77	2	the	the	DET
cana-3303	77	3	derived	derive	VERB
cana-3303	77	4	transformations	transformation	NOUN
cana-3303	77	5	and	and	CCONJ
cana-3303	77	6	solving	solving	NOUN
cana-3303	77	7	for	for	ADP
cana-3303	77	8	the	the	DET
cana-3303	77	9	equations	equation	NOUN
cana-3303	77	10	solves	solve	VERB
cana-3303	77	11	both	both	DET
cana-3303	77	12	issues	issue	NOUN
cana-3303	77	13	.	.	PUNCT
cana-3303	78	1	general	general	ADJ
cana-3303	78	2	methodology	methodology	PROPN
cana-3303	78	3	overall	overall	ADV
cana-3303	78	4	,	,	PUNCT
cana-3303	78	5	the	the	DET
cana-3303	78	6	methodology	methodology	NOUN
cana-3303	78	7	is	be	AUX
cana-3303	78	8	analytical	analytical	ADJ
cana-3303	78	9	in	in	ADP
cana-3303	78	10	nature	nature	NOUN
cana-3303	78	11	,	,	PUNCT
cana-3303	78	12	with	with	ADP
cana-3303	78	13	the	the	DET
cana-3303	78	14	main	main	ADJ
cana-3303	78	15	objective	objective	NOUN
cana-3303	78	16	being	be	AUX
cana-3303	78	17	to	to	PART
cana-3303	78	18	find	find	VERB
cana-3303	78	19	explicit	explicit	ADJ
cana-3303	78	20	solutions	solution	NOUN
cana-3303	78	21	to	to	ADP
cana-3303	78	22	the	the	DET
cana-3303	78	23	nonlinear	nonlinear	ADJ
cana-3303	78	24	equation	equation	NOUN
cana-3303	78	25	using	use	VERB
cana-3303	78	26	algebraic	algebraic	ADJ
cana-3303	78	27	manipulations	manipulation	NOUN
cana-3303	78	28	and	and	CCONJ
cana-3303	78	29	mathematical	mathematical	ADJ
cana-3303	78	30	reasoning	reasoning	NOUN
cana-3303	78	31	.	.	PUNCT
cana-3303	79	1	this	this	PRON
cana-3303	79	2	allows	allow	VERB
cana-3303	79	3	to	to	PART
cana-3303	79	4	get	get	VERB
cana-3303	79	5	around	around	ADP
cana-3303	79	6	numerical	numerical	ADJ
cana-3303	79	7	methods	method	NOUN
cana-3303	79	8	or	or	CCONJ
cana-3303	79	9	approximations	approximation	NOUN
cana-3303	79	10	.	.	PUNCT
cana-3303	80	1	5	5	X
cana-3303	80	2	.	.	X
cana-3303	80	3	results	result	VERB
cana-3303	80	4	it	it	PRON
cana-3303	80	5	is	be	AUX
cana-3303	80	6	well	well	ADV
cana-3303	80	7	-	-	PUNCT
cana-3303	80	8	known	know	VERB
cana-3303	80	9	that	that	SCONJ
cana-3303	80	10	problems	problem	NOUN
cana-3303	80	11	for	for	ADP
cana-3303	80	12	both	both	CCONJ
cana-3303	80	13	uninterrupted	uninterrupted	ADJ
cana-3303	80	14	case	case	NOUN
cana-3303	80	15	as	as	ADV
cana-3303	80	16	well	well	ADV
cana-3303	80	17	as	as	ADP
cana-3303	80	18	for	for	ADP
cana-3303	80	19	additive	additive	ADJ
cana-3303	80	20	derivative	derivative	ADJ
cana-3303	80	21	equations	equation	NOUN
cana-3303	80	22	,	,	PUNCT
cana-3303	80	23	called	call	VERB
cana-3303	80	24	as	as	ADP
cana-3303	80	25	equation	equation	NOUN
cana-3303	80	26	with	with	ADP
cana-3303	80	27	differences	difference	NOUN
cana-3303	80	28	,	,	PUNCT
cana-3303	80	29	have	have	AUX
cana-3303	80	30	been	be	AUX
cana-3303	80	31	extensively	extensively	ADV
cana-3303	80	32	studied	study	VERB
cana-3303	80	33	(	(	PUNCT
cana-3303	80	34	grikomi	grikomi	NOUN
cana-3303	80	35	,	,	PUNCT
cana-3303	80	36	1962	1962	NUM
cana-3303	80	37	)	)	PUNCT
cana-3303	80	38	,	,	PUNCT
cana-3303	80	39	(	(	PUNCT
cana-3303	80	40	mammadov	mammadov	PROPN
cana-3303	80	41	&	&	CCONJ
cana-3303	80	42	khankishiyev	khankishiyev	PROPN
cana-3303	80	43	,	,	PUNCT
cana-3303	80	44	2013	2013	NUM
cana-3303	80	45	)	)	PUNCT
cana-3303	80	46	,	,	PUNCT
cana-3303	80	47	(	(	PUNCT
cana-3303	80	48	gelfond	gelfond	NOUN
cana-3303	80	49	,	,	PUNCT
cana-3303	80	50	1967	1967	NUM
cana-3303	80	51	)	)	PUNCT
cana-3303	80	52	,	,	PUNCT
cana-3303	80	53	(	(	PUNCT
cana-3303	80	54	aliyev	aliyev	NOUN
cana-3303	80	55	,	,	PUNCT
cana-3303	80	56	bagirov	bagirov	PROPN
cana-3303	80	57	,	,	PUNCT
cana-3303	80	58	&	&	CCONJ
cana-3303	80	59	i̇zadi	i̇zadi	NOUN
cana-3303	80	60	,	,	PUNCT
cana-3303	80	61	2009	2009	NUM
cana-3303	80	62	)	)	PUNCT
cana-3303	80	63	.	.	PUNCT
cana-3303	81	1	communications	communication	NOUN
cana-3303	81	2	on	on	ADP
cana-3303	81	3	applied	apply	VERB
cana-3303	81	4	nonlinear	nonlinear	ADJ
cana-3303	81	5	analysis	analysis	NOUN
cana-3303	81	6	issn	issn	NOUN
cana-3303	81	7	:	:	PUNCT
cana-3303	81	8	1074	1074	NUM
cana-3303	81	9	-	-	PUNCT
cana-3303	81	10	133x	133x	NUM
cana-3303	81	11	vol	vol	NOUN
cana-3303	81	12	32	32	NUM
cana-3303	81	13	no	no	NOUN
cana-3303	81	14	.	.	PUNCT
cana-3303	82	1	6s	6s	NUM
cana-3303	82	2	(	(	PUNCT
cana-3303	82	3	2025	2025	NUM
cana-3303	82	4	)	)	PUNCT
cana-3303	82	5	381	381	NUM
cana-3303	82	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	82	7	although	although	SCONJ
cana-3303	82	8	uninterrupted	uninterrupted	ADJ
cana-3303	82	9	multiplicative	multiplicative	ADJ
cana-3303	82	10	derivative	derivative	ADJ
cana-3303	82	11	equations	equation	NOUN
cana-3303	82	12	have	have	AUX
cana-3303	82	13	been	be	AUX
cana-3303	82	14	studied	study	VERB
cana-3303	82	15	in	in	ADP
cana-3303	82	16	the	the	DET
cana-3303	82	17	middle	middle	NOUN
cana-3303	82	18	of	of	ADP
cana-3303	82	19	the	the	DET
cana-3303	82	20	last	last	ADJ
cana-3303	82	21	century	century	NOUN
cana-3303	82	22	,	,	PUNCT
cana-3303	82	23	(	(	PUNCT
cana-3303	82	24	gantmacher	gantmacher	ADV
cana-3303	82	25	,	,	PUNCT
cana-3303	82	26	1967	1967	NUM
cana-3303	82	27	)	)	PUNCT
cana-3303	82	28	issues	issue	NOUN
cana-3303	82	29	for	for	ADP
cana-3303	82	30	such	such	ADJ
cana-3303	82	31	equations	equation	NOUN
cana-3303	82	32	have	have	AUX
cana-3303	82	33	been	be	AUX
cana-3303	82	34	investigated	investigate	VERB
cana-3303	82	35	for	for	ADP
cana-3303	82	36	almost	almost	ADV
cana-3303	82	37	at	at	ADP
cana-3303	82	38	the	the	DET
cana-3303	82	39	end	end	NOUN
cana-3303	82	40	of	of	ADP
cana-3303	82	41	the	the	DET
cana-3303	82	42	last	last	ADJ
cana-3303	82	43	century	century	NOUN
cana-3303	82	44	,	,	PUNCT
cana-3303	82	45	especially	especially	ADV
cana-3303	82	46	at	at	ADP
cana-3303	82	47	the	the	DET
cana-3303	82	48	beginning	beginning	NOUN
cana-3303	82	49	of	of	ADP
cana-3303	82	50	this	this	DET
cana-3303	82	51	century	century	NOUN
cana-3303	82	52	(	(	PUNCT
cana-3303	82	53	bashirov	bashirov	PROPN
cana-3303	82	54	&	&	CCONJ
cana-3303	82	55	riza	riza	PROPN
cana-3303	82	56	,	,	PUNCT
cana-3303	82	57	2011	2011	NUM
cana-3303	82	58	)	)	PUNCT
cana-3303	82	59	,	,	PUNCT
cana-3303	82	60	(	(	PUNCT
cana-3303	82	61	bashirov	bashirov	NOUN
cana-3303	82	62	,	,	PUNCT
cana-3303	82	63	2013	2013	NUM
cana-3303	82	64	)	)	PUNCT
cana-3303	82	65	.	.	PUNCT
cana-3303	83	1	issues	issue	NOUN
cana-3303	83	2	for	for	ADP
cana-3303	83	3	discrete	discrete	ADJ
cana-3303	83	4	multiplicative	multiplicative	ADJ
cana-3303	83	5	derivative	derivative	ADJ
cana-3303	83	6	equations	equation	NOUN
cana-3303	83	7	have	have	AUX
cana-3303	83	8	already	already	ADV
cana-3303	83	9	been	be	AUX
cana-3303	83	10	considered	consider	VERB
cana-3303	83	11	by	by	ADP
cana-3303	83	12	us	we	PRON
cana-3303	83	13	(	(	PUNCT
cana-3303	83	14	hassani	hassani	PROPN
cana-3303	83	15	&	&	CCONJ
cana-3303	83	16	aliev	aliev	ADJ
cana-3303	83	17	,	,	PUNCT
cana-3303	83	18	2008	2008	NUM
cana-3303	83	19	)	)	PUNCT
cana-3303	83	20	,	,	PUNCT
cana-3303	83	21	(	(	PUNCT
cana-3303	83	22	aliyev	aliyev	PROPN
cana-3303	83	23	&	&	CCONJ
cana-3303	83	24	mamieva	mamieva	PROPN
cana-3303	83	25	,	,	PUNCT
cana-3303	83	26	2017	2017	NUM
cana-3303	83	27	)	)	PUNCT
cana-3303	83	28	.	.	PUNCT
cana-3303	84	1	poverative	poverative	ADJ
cana-3303	84	2	derivatives	derivative	NOUN
cana-3303	84	3	and	and	CCONJ
cana-3303	84	4	integral	integral	ADJ
cana-3303	84	5	in	in	ADP
cana-3303	84	6	uninterrupted	uninterrupted	ADJ
cana-3303	84	7	form	form	NOUN
cana-3303	84	8	are	be	AUX
cana-3303	84	9	not	not	PART
cana-3303	84	10	yet	yet	ADV
cana-3303	84	11	defined	define	VERB
cana-3303	84	12	.	.	PUNCT
cana-3303	85	1	because	because	SCONJ
cana-3303	85	2	,	,	PUNCT
cana-3303	85	3	for	for	ADP
cana-3303	85	4	this	this	PRON
cana-3303	85	5	they	they	PRON
cana-3303	85	6	need	need	VERB
cana-3303	85	7	a	a	DET
cana-3303	85	8	new	new	ADJ
cana-3303	85	9	direct	direct	ADJ
cana-3303	85	10	operation	operation	NOUN
cana-3303	85	11	and	and	CCONJ
cana-3303	85	12	a	a	DET
cana-3303	85	13	new	new	ADJ
cana-3303	85	14	inverse	inverse	NOUN
cana-3303	85	15	operation	operation	NOUN
cana-3303	85	16	.	.	PUNCT
cana-3303	86	1	however	however	ADV
cana-3303	86	2	,	,	PUNCT
cana-3303	86	3	the	the	DET
cana-3303	86	4	poverative	poverative	ADJ
cana-3303	86	5	derivative	derivative	NOUN
cana-3303	86	6	and	and	CCONJ
cana-3303	86	7	the	the	DET
cana-3303	86	8	poverative	poverative	ADJ
cana-3303	86	9	integral	integral	NOUN
cana-3303	86	10	are	be	AUX
cana-3303	86	11	defined	define	VERB
cana-3303	86	12	in	in	ADP
cana-3303	86	13	the	the	DET
cana-3303	86	14	case	case	NOUN
cana-3303	86	15	of	of	ADP
cana-3303	86	16	discrete	discrete	NOUN
cana-3303	86	17	(	(	PUNCT
cana-3303	86	18	since	since	SCONJ
cana-3303	86	19	it	it	PRON
cana-3303	86	20	can	can	AUX
cana-3303	86	21	be	be	AUX
cana-3303	86	22	expressed	express	VERB
cana-3303	86	23	by	by	ADP
cana-3303	86	24	one	one	NUM
cana-3303	86	25	operation	operation	NOUN
cana-3303	86	26	)	)	PUNCT
cana-3303	86	27	and	and	CCONJ
cana-3303	86	28	the	the	DET
cana-3303	86	29	issues	issue	NOUN
cana-3303	86	30	for	for	ADP
cana-3303	86	31	the	the	DET
cana-3303	86	32	discrete	discrete	ADJ
cana-3303	86	33	poverative	poverative	ADJ
cana-3303	86	34	derivative	derivative	ADJ
cana-3303	86	35	equations	equation	NOUN
cana-3303	86	36	are	be	AUX
cana-3303	86	37	considered	consider	VERB
cana-3303	86	38	(	(	PUNCT
cana-3303	86	39	mammadzade	mammadzade	NOUN
cana-3303	86	40	,	,	PUNCT
cana-3303	86	41	2017	2017	NUM
cana-3303	86	42	)	)	PUNCT
cana-3303	86	43	,	,	PUNCT
cana-3303	86	44	(	(	PUNCT
cana-3303	86	45	aliyev	aliyev	PROPN
cana-3303	86	46	&	&	CCONJ
cana-3303	86	47	mammadzade	mammadzade	PROPN
cana-3303	86	48	,	,	PUNCT
cana-3303	86	49	2018	2018	NUM
cana-3303	86	50	)	)	PUNCT
cana-3303	86	51	.	.	PUNCT
cana-3303	87	1	note	note	VERB
cana-3303	87	2	that	that	SCONJ
cana-3303	87	3	,	,	PUNCT
cana-3303	87	4	unlike	unlike	ADP
cana-3303	87	5	discrete	discrete	ADJ
cana-3303	87	6	additive	additive	ADJ
cana-3303	87	7	derivative	derivative	NOUN
cana-3303	87	8	(	(	PUNCT
cana-3303	87	9	gelfond	gelfond	NOUN
cana-3303	87	10	,	,	PUNCT
cana-3303	87	11	1967	1967	NUM
cana-3303	87	12	)	)	PUNCT
cana-3303	87	13	,	,	PUNCT
cana-3303	87	14	(	(	PUNCT
cana-3303	87	15	aliyev	aliyev	NOUN
cana-3303	87	16	,	,	PUNCT
cana-3303	87	17	bagirov	bagirov	PROPN
cana-3303	87	18	,	,	PUNCT
cana-3303	87	19	&	&	CCONJ
cana-3303	87	20	i̇zadi	i̇zadi	NOUN
cana-3303	87	21	,	,	PUNCT
cana-3303	87	22	2009	2009	NUM
cana-3303	87	23	)	)	PUNCT
cana-3303	87	24	(	(	PUNCT
cana-3303	87	25	linear	linear	PROPN
cana-3303	87	26	equations	equation	NOUN
cana-3303	87	27	)	)	PUNCT
cana-3303	87	28	equations	equation	NOUN
cana-3303	87	29	(	(	PUNCT
cana-3303	87	30	equations	equation	NOUN
cana-3303	87	31	with	with	ADP
cana-3303	87	32	differences	difference	NOUN
cana-3303	87	33	)	)	PUNCT
cana-3303	87	34	discrete	discrete	NOUN
cana-3303	87	35	multiplicative	multiplicative	ADJ
cana-3303	87	36	(	(	PUNCT
cana-3303	87	37	hassani	hassani	PROPN
cana-3303	87	38	&	&	CCONJ
cana-3303	87	39	aliev	aliev	ADJ
cana-3303	87	40	,	,	PUNCT
cana-3303	87	41	2008	2008	NUM
cana-3303	87	42	)	)	PUNCT
cana-3303	87	43	,	,	PUNCT
cana-3303	87	44	(	(	PUNCT
cana-3303	87	45	aliyev	aliyev	PROPN
cana-3303	87	46	&	&	CCONJ
cana-3303	87	47	mamieva	mamieva	PROPN
cana-3303	87	48	,	,	PUNCT
cana-3303	87	49	2017	2017	NUM
cana-3303	87	50	)	)	PUNCT
cana-3303	87	51	discrete	discrete	VERB
cana-3303	87	52	poverative	poverative	ADJ
cana-3303	87	53	equations	equation	NOUN
cana-3303	87	54	(	(	PUNCT
cana-3303	87	55	mammadzade	mammadzade	NOUN
cana-3303	87	56	,	,	PUNCT
cana-3303	87	57	2017	2017	NUM
cana-3303	87	58	)	)	PUNCT
cana-3303	87	59	,	,	PUNCT
cana-3303	87	60	(	(	PUNCT
cana-3303	87	61	aliyev	aliyev	PROPN
cana-3303	87	62	&	&	CCONJ
cana-3303	87	63	mammadzade	mammadzade	PROPN
cana-3303	87	64	,	,	PUNCT
cana-3303	87	65	2018	2018	NUM
cana-3303	87	66	)	)	PUNCT
cana-3303	87	67	are	be	AUX
cana-3303	87	68	very	very	ADV
cana-3303	87	69	complex	complex	ADJ
cana-3303	87	70	nonlinear	nonlinear	ADJ
cana-3303	87	71	derivatives	derivative	NOUN
cana-3303	87	72	.	.	PUNCT
cana-3303	88	1	here	here	ADV
cana-3303	88	2	we	we	PRON
cana-3303	88	3	will	will	AUX
cana-3303	88	4	consider	consider	VERB
cana-3303	88	5	cauchy	cauchy	NOUN
cana-3303	88	6	and	and	CCONJ
cana-3303	88	7	boundary	boundary	ADJ
cana-3303	88	8	issues	issue	NOUN
cana-3303	88	9	for	for	ADP
cana-3303	88	10	one	one	NUM
cana-3303	88	11	equation	equation	NOUN
cana-3303	88	12	of	of	ADP
cana-3303	88	13	third	third	ADJ
cana-3303	88	14	-	-	PUNCT
cana-3303	88	15	order	order	NOUN
cana-3303	88	16	mixed	mixed	ADJ
cana-3303	88	17	derivative	derivative	NOUN
cana-3303	88	18	derived	derive	VERB
cana-3303	88	19	from	from	ADP
cana-3303	88	20	discrete	discrete	ADJ
cana-3303	88	21	additive	additive	ADJ
cana-3303	88	22	derivative	derivative	NOUN
cana-3303	88	23	of	of	ADP
cana-3303	88	24	discrete	discrete	ADJ
cana-3303	88	25	poverative	poverative	ADJ
cana-3303	88	26	derivative	derivative	NOUN
cana-3303	88	27	of	of	ADP
cana-3303	88	28	discrete	discrete	ADJ
cana-3303	88	29	multiplicative	multiplicative	ADJ
cana-3303	88	30	derivative	derivative	NOUN
cana-3303	88	31	.	.	PUNCT
cana-3303	89	1	this	this	DET
cana-3303	89	2	aforementioned	aforementioned	ADJ
cana-3303	89	3	equation	equation	NOUN
cana-3303	89	4	is	be	AUX
cana-3303	89	5	nonlinear	nonlinear	ADJ
cana-3303	89	6	equation	equation	NOUN
cana-3303	89	7	as	as	ADP
cana-3303	89	8	:	:	PUNCT
cana-3303	89	9	𝑦𝑛+3	𝑦𝑛+3	NUM
cana-3303	89	10	∙	∙	PROPN
cana-3303	89	11	𝑦	𝑦	SYM
cana-3303	89	12	𝑛+1	𝑛+1	PROPN
cana-3303	89	13	𝑦𝑛∙𝑦𝑛+2	𝑦𝑛∙𝑦𝑛+2	NUM
cana-3303	89	14	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	89	15	2	2	NUM
cana-3303	89	16	=	=	SYM
cana-3303	89	17	(	(	PUNCT
cana-3303	89	18	𝑓𝑛	𝑓𝑛	ADP
cana-3303	89	19	∙	∙	PROPN
cana-3303	89	20	𝑦	𝑦	PRON
cana-3303	89	21	𝑛+1	𝑛+1	PROPN
cana-3303	89	22	𝑦𝑛	𝑦𝑛	VERB
cana-3303	89	23	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3303	89	24	∙	∙	PROPN
cana-3303	89	25	𝑦	𝑦	SYM
cana-3303	89	26	𝑛+2	𝑛+2	NUM
cana-3303	89	27	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	89	28	𝑦𝑛+2	𝑦𝑛+2	NUM
cana-3303	89	29	+	+	SYM
cana-3303	89	30	𝑦	𝑦	SYM
cana-3303	89	31	𝑛+2	𝑛+2	NUM
cana-3303	89	32	𝑦𝑛	𝑦𝑛	VERB
cana-3303	89	33	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	89	34	+	+	CCONJ
cana-3303	89	35	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	89	36	𝑦𝑛+2	𝑦𝑛+2	NUM
cana-3303	89	37	)	)	PUNCT
cana-3303	89	38	𝑦𝑛+2	𝑦𝑛+2	NUM
cana-3303	89	39	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	89	40	,	,	PUNCT
cana-3303	89	41	for	for	ADP
cana-3303	89	42	this	this	DET
cana-3303	89	43	equation	equation	NOUN
cana-3303	89	44	,	,	PUNCT
cana-3303	89	45	the	the	DET
cana-3303	89	46	cauchy	cauchy	NOUN
cana-3303	89	47	and	and	CCONJ
cana-3303	89	48	the	the	DET
cana-3303	89	49	boundary	boundary	ADJ
cana-3303	89	50	issue	issue	NOUN
cana-3303	89	51	will	will	AUX
cana-3303	89	52	be	be	AUX
cana-3303	89	53	considered	consider	VERB
cana-3303	89	54	,	,	PUNCT
cana-3303	89	55	and	and	CCONJ
cana-3303	89	56	the	the	DET
cana-3303	89	57	analytical	analytical	ADJ
cana-3303	89	58	expressions	expression	NOUN
cana-3303	89	59	for	for	ADP
cana-3303	89	60	the	the	DET
cana-3303	89	61	solution	solution	NOUN
cana-3303	89	62	of	of	ADP
cana-3303	89	63	these	these	DET
cana-3303	89	64	problems	problem	NOUN
cana-3303	89	65	will	will	AUX
cana-3303	89	66	be	be	AUX
cana-3303	89	67	obtained	obtain	VERB
cana-3303	89	68	.	.	PUNCT
cana-3303	90	1	setting	set	VERB
cana-3303	90	2	of	of	ADP
cana-3303	90	3	the	the	DET
cana-3303	90	4	problem	problem	NOUN
cana-3303	90	5	:	:	PUNCT
cana-3303	90	6	additive	additive	ADJ
cana-3303	90	7	integral	integral	NOUN
cana-3303	90	8	,	,	PUNCT
cana-3303	90	9	which	which	PRON
cana-3303	90	10	results	result	VERB
cana-3303	90	11	from	from	ADP
cana-3303	90	12	two	two	NUM
cana-3303	90	13	consecutive	consecutive	ADJ
cana-3303	90	14	direct	direct	ADJ
cana-3303	90	15	operations	operation	NOUN
cana-3303	90	16	,	,	PUNCT
cana-3303	90	17	consists	consist	VERB
cana-3303	90	18	of	of	ADP
cana-3303	90	19	the	the	DET
cana-3303	90	20	total	total	NOUN
cana-3303	90	21	of	of	ADP
cana-3303	90	22	the	the	DET
cana-3303	90	23	sums	sum	NOUN
cana-3303	90	24	,	,	PUNCT
cana-3303	90	25	and	and	CCONJ
cana-3303	90	26	the	the	DET
cana-3303	90	27	multiplicative	multiplicative	ADJ
cana-3303	90	28	integral	integral	ADJ
cana-3303	90	29	sums	sum	NOUN
cana-3303	90	30	of	of	ADP
cana-3303	90	31	the	the	DET
cana-3303	90	32	powers	power	NOUN
cana-3303	90	33	.	.	PUNCT
cana-3303	91	1	at	at	ADP
cana-3303	91	2	the	the	DET
cana-3303	91	3	same	same	ADJ
cana-3303	91	4	time	time	NOUN
cana-3303	91	5	,	,	PUNCT
cana-3303	91	6	it	it	PRON
cana-3303	91	7	should	should	AUX
cana-3303	91	8	be	be	AUX
cana-3303	91	9	noted	note	VERB
cana-3303	91	10	that	that	SCONJ
cana-3303	91	11	additive	additive	ADJ
cana-3303	91	12	derivative	derivative	NOUN
cana-3303	91	13	,	,	PUNCT
cana-3303	91	14	which	which	PRON
cana-3303	91	15	is	be	AUX
cana-3303	91	16	the	the	DET
cana-3303	91	17	result	result	NOUN
cana-3303	91	18	of	of	ADP
cana-3303	91	19	two	two	NUM
cana-3303	91	20	consecutive	consecutive	ADJ
cana-3303	91	21	inversions	inversion	NOUN
cana-3303	91	22	𝑓(𝐼)(𝑥	𝑓(𝐼)(𝑥	NOUN
cana-3303	91	23	)	)	PUNCT
cana-3303	91	24	=	=	VERB
cana-3303	91	25	lim	lim	PROPN
cana-3303	91	26	ℎ→0	ℎ→0	PUNCT
cana-3303	91	27	𝑓(𝑥	𝑓(𝑥	VERB
cana-3303	91	28	+	+	NUM
cana-3303	91	29	ℎ	ℎ	NOUN
cana-3303	91	30	)	)	PUNCT
cana-3303	91	31	−	−	NOUN
cana-3303	91	32	𝑓(𝑥	𝑓(𝑥	NOUN
cana-3303	91	33	)	)	PUNCT
cana-3303	91	34	(	(	PUNCT
cana-3303	91	35	𝑥	𝑥	NOUN
cana-3303	91	36	+	+	NUM
cana-3303	91	37	ℎ	ℎ	NOUN
cana-3303	91	38	)	)	PUNCT
cana-3303	92	1	−	−	ADP
cana-3303	92	2	𝑥	𝑥	X
cana-3303	92	3	,	,	PUNCT
cana-3303	92	4	the	the	DET
cana-3303	92	5	discrete	discrete	ADJ
cana-3303	92	6	additive	additive	NOUN
cana-3303	92	7	integral	integral	NOUN
cana-3303	92	8	is	be	AUX
cana-3303	92	9	a	a	DET
cana-3303	92	10	direct	direct	ADJ
cana-3303	92	11	operation	operation	NOUN
cana-3303	92	12	,	,	PUNCT
cana-3303	92	13	although	although	SCONJ
cana-3303	92	14	they	they	PRON
cana-3303	92	15	are	be	AUX
cana-3303	92	16	at	at	ADP
cana-3303	92	17	the	the	DET
cana-3303	92	18	root	root	NOUN
cana-3303	92	19	of	of	ADP
cana-3303	92	20	correlation	correlation	NOUN
cana-3303	92	21	,	,	PUNCT
cana-3303	92	22	n	n	CCONJ
cana-3303	92	23	,	,	PUNCT
cana-3303	92	24	1	1	NUM
cana-3303	92	25	0	0	NUM
cana-3303	92	26			NOUN
cana-3303	93	1	−	−	NOUN
cana-3303	93	2	=	=	SYM
cana-3303	94	1	=	=	SYM
cana-3303	94	2	n	n	PROPN
cana-3303	94	3	k	k	NOUN
cana-3303	94	4	kk	kk	INTJ
cana-3303	95	1	ff	ff	INTJ
cana-3303	95	2	0	0	NUM
cana-3303	96	1	it	it	PRON
cana-3303	96	2	depends	depend	VERB
cana-3303	96	3	on	on	ADP
cana-3303	96	4	sum	sum	NOUN
cana-3303	96	5	,	,	PUNCT
cana-3303	96	6	discrete	discrete	ADJ
cana-3303	96	7	multiplicative	multiplicative	ADJ
cana-3303	96	8	integral	integral	ADJ
cana-3303	96	9	on	on	ADP
cana-3303	96	10	one	one	NUM
cana-3303	96	11	direct	direct	ADJ
cana-3303	96	12	operation	operation	NOUN
cana-3303	96	13	and	and	CCONJ
cana-3303	96	14	sum	sum	NOUN
cana-3303	96	15	of	of	ADP
cana-3303	96	16	discrete	discrete	ADJ
cana-3303	96	17	additive	additive	ADJ
cana-3303	96	18	derivative	derivative	NOUN
cana-3303	96	19	depends	depend	VERB
cana-3303	96	20	on	on	ADP
cana-3303	96	21	an	an	DET
cana-3303	96	22	inverse	inverse	NOUN
cana-3303	96	23	operation	operation	NOUN
cana-3303	96	24	n	n	NOUN
cana-3303	96	25	,	,	PUNCT
cana-3303	96	26	1	1	NUM
cana-3303	96	27	0	0	NUM
cana-3303	96	28			NUM
cana-3303	96	29	−	−	X
cana-3303	97	1	=	=	SYM
cana-3303	97	2	=	=	SYM
cana-3303	97	3	n	n	PROPN
cana-3303	97	4	k	k	NOUN
cana-3303	97	5	kk	kk	X
cana-3303	97	6	ff	ff	INTJ
cana-3303	97	7	0	0	NUM
cana-3303	97	8	communications	communication	NOUN
cana-3303	97	9	on	on	ADP
cana-3303	97	10	applied	apply	VERB
cana-3303	97	11	nonlinear	nonlinear	ADJ
cana-3303	97	12	analysis	analysis	NOUN
cana-3303	97	13	issn	issn	NOUN
cana-3303	97	14	:	:	PUNCT
cana-3303	97	15	1074	1074	NUM
cana-3303	97	16	-	-	PUNCT
cana-3303	97	17	133x	133x	NUM
cana-3303	97	18	vol	vol	NOUN
cana-3303	97	19	32	32	NUM
cana-3303	97	20	no	no	NOUN
cana-3303	97	21	.	.	PUNCT
cana-3303	98	1	6s	6s	NUM
cana-3303	98	2	(	(	PUNCT
cana-3303	98	3	2025	2025	NUM
cana-3303	98	4	)	)	PUNCT
cana-3303	98	5	382	382	NUM
cana-3303	98	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	98	7	𝑓𝑛	𝑓𝑛	PROPN
cana-3303	98	8	(	(	PUNCT
cana-3303	98	9	𝐼	𝐼	PROPN
cana-3303	98	10	)	)	PUNCT
cana-3303	98	11	=	=	NOUN
cana-3303	98	12	𝑓𝑛+1	𝑓𝑛+1	NUM
cana-3303	98	13	−	−	PROPN
cana-3303	98	14	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	98	15	,	,	PUNCT
cana-3303	98	16	difference	difference	NOUN
cana-3303	98	17	,	,	PUNCT
cana-3303	98	18	and	and	CCONJ
cana-3303	98	19	discrete	discrete	VERB
cana-3303	98	20	multiplicative	multiplicative	ADJ
cana-3303	98	21	derivative	derivative	ADJ
cana-3303	98	22	one	one	NUM
cana-3303	98	23	inverse	inverse	NOUN
cana-3303	98	24	correlation	correlation	NOUN
cana-3303	98	25	𝑓𝑛	𝑓𝑛	ADP
cana-3303	98	26	[	[	X
cana-3303	98	27	𝐼	𝐼	X
cana-3303	98	28	]	]	X
cana-3303	98	29	=	=	SYM
cana-3303	98	30	𝑓𝑛+1	𝑓𝑛+1	PROPN
cana-3303	98	31	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	98	32	,	,	PUNCT
cana-3303	98	33	as	as	ADP
cana-3303	98	34	for	for	ADP
cana-3303	98	35	the	the	DET
cana-3303	98	36	poverative	poverative	ADJ
cana-3303	98	37	integral	integral	ADJ
cana-3303	98	38	and	and	CCONJ
cana-3303	98	39	derivative	derivative	ADJ
cana-3303	98	40	,	,	PUNCT
cana-3303	98	41	it	it	PRON
cana-3303	98	42	is	be	AUX
cana-3303	98	43	necessary	necessary	ADJ
cana-3303	98	44	to	to	PART
cana-3303	98	45	perform	perform	VERB
cana-3303	98	46	a	a	DET
cana-3303	98	47	new	new	ADJ
cana-3303	98	48	direct	direct	ADJ
cana-3303	98	49	operation	operation	NOUN
cana-3303	98	50	and	and	CCONJ
cana-3303	98	51	a	a	DET
cana-3303	98	52	new	new	ADJ
cana-3303	98	53	inverse	inverse	NOUN
cana-3303	98	54	operation	operation	NOUN
cana-3303	98	55	to	to	PART
cana-3303	98	56	give	give	VERB
cana-3303	98	57	them	they	PRON
cana-3303	98	58	an	an	DET
cana-3303	98	59	uninterrupted	uninterrupted	ADJ
cana-3303	98	60	character	character	NOUN
cana-3303	98	61	.	.	PUNCT
cana-3303	99	1	as	as	ADV
cana-3303	99	2	far	far	ADV
cana-3303	99	3	as	as	ADP
cana-3303	99	4	discrete	discrete	ADJ
cana-3303	99	5	case	case	NOUN
cana-3303	99	6	of	of	ADP
cana-3303	99	7	these	these	DET
cana-3303	99	8	operations	operation	NOUN
cana-3303	99	9	concerned	concern	VERB
cana-3303	99	10	,	,	PUNCT
cana-3303	99	11	they	they	PRON
cana-3303	99	12	do	do	AUX
cana-3303	99	13	not	not	PART
cana-3303	99	14	need	need	VERB
cana-3303	99	15	a	a	DET
cana-3303	99	16	new	new	ADJ
cana-3303	99	17	operation	operation	NOUN
cana-3303	99	18	for	for	ADP
cana-3303	99	19	discrete	discrete	ADJ
cana-3303	99	20	poverative	poverative	ADJ
cana-3303	99	21	integral	integral	ADJ
cana-3303	99	22	and	and	CCONJ
cana-3303	99	23	discrete	discrete	ADJ
cana-3303	99	24	poverative	poverative	ADJ
cana-3303	99	25	derivatives	derivative	NOUN
cana-3303	99	26	because	because	SCONJ
cana-3303	99	27	they	they	PRON
cana-3303	99	28	can	can	AUX
cana-3303	99	29	be	be	AUX
cana-3303	99	30	given	give	VERB
cana-3303	99	31	by	by	ADP
cana-3303	99	32	a	a	DET
cana-3303	99	33	single	single	ADJ
cana-3303	99	34	operation	operation	NOUN
cana-3303	99	35	.	.	PUNCT
cana-3303	100	1	thus	thus	ADV
cana-3303	100	2	,	,	PUNCT
cana-3303	100	3	the	the	DET
cana-3303	100	4	discrete	discrete	ADJ
cana-3303	100	5	poverative	poverative	ADJ
cana-3303	100	6	integral	integral	ADJ
cana-3303	100	7	:	:	PUNCT
cana-3303	100	8	0	0	NUM
cana-3303	100	9	𝑓𝑘=	𝑓𝑘=	PROPN
cana-3303	100	10	0	0	NUM
cana-3303	100	11	𝑓𝑘=	𝑓𝑘=	PROPN
cana-3303	100	12	𝑓	𝑓	PROPN
cana-3303	100	13	𝑛−1	𝑛−1	PROPN
cana-3303	100	14	𝑓𝑛−2	𝑓𝑛−2	NOUN
cana-3303	100	15	𝑓𝑛−3	𝑓𝑛−3	VERB
cana-3303	100	16	⋰	⋰	DET
cana-3303	100	17	𝑓	𝑓	DET
cana-3303	100	18	2	2	NUM
cana-3303	100	19	𝑓	𝑓	DET
cana-3303	100	20	1	1	NUM
cana-3303	100	21	𝑓0	𝑓0	NOUN
cana-3303	100	22	,	,	PUNCT
cana-3303	100	23	n	n	CCONJ
cana-3303	100	24	k	k	NOUN
cana-3303	100	25	=	=	NOUN
cana-3303	100	26	n-1	n-1	NOUN
cana-3303	100	27	may	may	AUX
cana-3303	100	28	be	be	AUX
cana-3303	100	29	expressed	express	VERB
cana-3303	100	30	by	by	ADP
cana-3303	100	31	powers	power	NOUN
cana-3303	100	32	,	,	PUNCT
cana-3303	100	33	whereas	whereas	SCONJ
cana-3303	100	34	discrete	discrete	VERB
cana-3303	100	35	poverative	poverative	ADJ
cana-3303	100	36	derivatives	derivative	NOUN
cana-3303	100	37	with	with	ADP
cana-3303	100	38	root	root	NOUN
cana-3303	100	39	of	of	ADP
cana-3303	100	40	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	100	41	{	{	PUNCT
cana-3303	100	42	𝐼	𝐼	PROPN
cana-3303	100	43	}	}	PUNCT
cana-3303	101	1	=	=	SYM
cana-3303	101	2	√𝑓𝑛+1	√𝑓𝑛+1	ADP
cana-3303	101	3	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	101	4	,	,	PUNCT
cana-3303	101	5	now	now	ADV
cana-3303	101	6	let	let	VERB
cana-3303	101	7	's	us	PRON
cana-3303	101	8	look	look	VERB
cana-3303	101	9	at	at	ADP
cana-3303	101	10	the	the	DET
cana-3303	101	11	equation	equation	NOUN
cana-3303	101	12	:	:	PUNCT
cana-3303	101	13	(	(	PUNCT
cana-3303	101	14	(	(	PUNCT
cana-3303	101	15	𝑦𝑛	𝑦𝑛	ADP
cana-3303	101	16	[	[	X
cana-3303	101	17	𝐼	𝐼	NOUN
cana-3303	101	18	]	]	PUNCT
cana-3303	101	19	)	)	PUNCT
cana-3303	101	20	{	{	PUNCT
cana-3303	101	21	𝐼})(𝐼	𝐼})(𝐼	NOUN
cana-3303	101	22	)	)	PUNCT
cana-3303	101	23	=	=	SYM
cana-3303	102	1	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	102	2	,	,	PUNCT
cana-3303	102	3	n≥0	n≥0	ADJ
cana-3303	102	4	,	,	PUNCT
cana-3303	102	5	(	(	PUNCT
cana-3303	102	6	1	1	X
cana-3303	102	7	)	)	PUNCT
cana-3303	102	8	where	where	SCONJ
cana-3303	102	9	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	102	10	,	,	PUNCT
cana-3303	102	11	n≥0	n≥0	ADJ
cana-3303	102	12	,	,	PUNCT
cana-3303	102	13	is	be	AUX
cana-3303	102	14	the	the	DET
cana-3303	102	15	given	give	VERB
cana-3303	102	16	sequence	sequence	NOUN
cana-3303	102	17	,	,	PUNCT
cana-3303	102	18	𝑦𝑛	𝑦𝑛	NOUN
cana-3303	102	19	,	,	PUNCT
cana-3303	102	20	n≥0	n≥0	ADJ
cana-3303	102	21	are	be	AUX
cana-3303	102	22	the	the	DET
cana-3303	102	23	sought	seek	VERB
cana-3303	102	24	sequences	sequence	NOUN
cana-3303	102	25	.	.	PUNCT
cana-3303	103	1	using	use	VERB
cana-3303	103	2	the	the	DET
cana-3303	103	3	definition	definition	NOUN
cana-3303	103	4	of	of	ADP
cana-3303	103	5	a	a	DET
cana-3303	103	6	discrete	discrete	ADJ
cana-3303	103	7	additive	additive	NOUN
cana-3303	103	8	derivative	derivative	NOUN
cana-3303	103	9	,	,	PUNCT
cana-3303	103	10	(	(	PUNCT
cana-3303	103	11	1	1	X
cana-3303	103	12	)	)	PUNCT
cana-3303	103	13	comes	come	VERB
cana-3303	103	14	to	to	ADP
cana-3303	103	15	the	the	DET
cana-3303	103	16	form	form	NOUN
cana-3303	103	17	as	as	SCONJ
cana-3303	103	18	follows	follow	VERB
cana-3303	103	19	:	:	PUNCT
cana-3303	103	20	(	(	PUNCT
cana-3303	103	21	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	103	22	[	[	X
cana-3303	103	23	𝐼	𝐼	PROPN
cana-3303	103	24	]	]	PUNCT
cana-3303	103	25	)	)	PUNCT
cana-3303	103	26	{	{	PUNCT
cana-3303	103	27	𝐼	𝐼	PROPN
cana-3303	103	28	}	}	PUNCT
cana-3303	103	29	(	(	PUNCT
cana-3303	103	30	𝑦𝑛	𝑦𝑛	ADP
cana-3303	103	31	[	[	X
cana-3303	103	32	𝐼	𝐼	NOUN
cana-3303	103	33	]	]	PUNCT
cana-3303	103	34	)	)	PUNCT
cana-3303	103	35	{	{	PUNCT
cana-3303	103	36	𝐼	𝐼	PROPN
cana-3303	103	37	}	}	PUNCT
cana-3303	103	38	=	=	PUNCT
cana-3303	103	39	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	103	40	,	,	PUNCT
cana-3303	103	41	n≥0	n≥0	PROPN
cana-3303	103	42	.	.	PUNCT
cana-3303	104	1	(	(	PUNCT
cana-3303	104	2	2	2	NUM
cana-3303	104	3	)	)	PUNCT
cana-3303	104	4	here	here	ADV
cana-3303	104	5	,	,	PUNCT
cana-3303	104	6	assuming	assume	VERB
cana-3303	104	7	that	that	SCONJ
cana-3303	104	8	n	n	NOUN
cana-3303	104	9	=	=	SYM
cana-3303	104	10	0	0	NUM
cana-3303	104	11	,	,	PUNCT
cana-3303	104	12	(	(	PUNCT
cana-3303	104	13	𝑦1	𝑦1	NOUN
cana-3303	104	14	[	[	X
cana-3303	104	15	𝐼	𝐼	PROPN
cana-3303	104	16	]	]	PUNCT
cana-3303	104	17	)	)	PUNCT
cana-3303	104	18	{	{	PUNCT
cana-3303	104	19	𝐼	𝐼	PROPN
cana-3303	104	20	}	}	PUNCT
cana-3303	104	21	(	(	PUNCT
cana-3303	104	22	𝑦0	𝑦0	NOUN
cana-3303	104	23	[	[	X
cana-3303	104	24	𝐼	𝐼	PROPN
cana-3303	104	25	]	]	PUNCT
cana-3303	104	26	)	)	PUNCT
cana-3303	104	27	{	{	PUNCT
cana-3303	104	28	𝐼	𝐼	PROPN
cana-3303	104	29	}	}	PUNCT
cana-3303	104	30	=	=	SYM
cana-3303	104	31	𝑓0	𝑓0	PROPN
cana-3303	104	32	,	,	PUNCT
cana-3303	104	33	(	(	PUNCT
cana-3303	104	34	2	2	NUM
cana-3303	104	35	0	0	NUM
cana-3303	104	36	)	)	PUNCT
cana-3303	104	37	when	when	SCONJ
cana-3303	104	38	a	a	DET
cana-3303	104	39	value	value	NOUN
cana-3303	104	40	1	1	NUM
cana-3303	104	41	is	be	AUX
cana-3303	104	42	given	give	VERB
cana-3303	104	43	to	to	ADP
cana-3303	104	44	n	n	CCONJ
cana-3303	104	45	,	,	PUNCT
cana-3303	104	46	we	we	PRON
cana-3303	104	47	get	get	VERB
cana-3303	104	48	the	the	DET
cana-3303	104	49	expression	expression	NOUN
cana-3303	104	50	.	.	PUNCT
cana-3303	105	1	(	(	PUNCT
cana-3303	105	2	𝑦2	𝑦2	NOUN
cana-3303	105	3	[	[	X
cana-3303	105	4	𝐼	𝐼	PROPN
cana-3303	105	5	]	]	PUNCT
cana-3303	105	6	)	)	PUNCT
cana-3303	105	7	{	{	PUNCT
cana-3303	105	8	𝐼	𝐼	PROPN
cana-3303	105	9	}	}	PUNCT
cana-3303	105	10	(	(	PUNCT
cana-3303	105	11	𝑦1	𝑦1	PROPN
cana-3303	105	12	[	[	X
cana-3303	105	13	𝐼	𝐼	PROPN
cana-3303	105	14	]	]	PUNCT
cana-3303	105	15	)	)	PUNCT
cana-3303	105	16	{	{	PUNCT
cana-3303	105	17	𝐼	𝐼	PROPN
cana-3303	105	18	}	}	PUNCT
cana-3303	105	19	=	=	SYM
cana-3303	105	20	𝑓1	𝑓1	PROPN
cana-3303	105	21	,	,	PUNCT
cana-3303	105	22	(	(	PUNCT
cana-3303	105	23	2	2	NUM
cana-3303	105	24	1	1	NUM
cana-3303	105	25	)	)	PUNCT
cana-3303	105	26	now	now	ADV
cana-3303	105	27	by	by	ADP
cana-3303	105	28	giving	give	VERB
cana-3303	105	29	2	2	NUM
cana-3303	105	30	to	to	ADP
cana-3303	105	31	n	n	PROPN
cana-3303	105	32	in	in	ADP
cana-3303	105	33	(	(	PUNCT
cana-3303	105	34	2	2	NUM
cana-3303	105	35	)	)	PUNCT
cana-3303	105	36	,	,	PUNCT
cana-3303	105	37	we	we	PRON
cana-3303	105	38	get	get	VERB
cana-3303	105	39	the	the	DET
cana-3303	105	40	expression	expression	NOUN
cana-3303	105	41	(	(	PUNCT
cana-3303	105	42	𝑦3	𝑦3	PROPN
cana-3303	105	43	[	[	X
cana-3303	105	44	𝐼	𝐼	PROPN
cana-3303	105	45	]	]	PUNCT
cana-3303	105	46	)	)	PUNCT
cana-3303	105	47	{	{	PUNCT
cana-3303	105	48	𝐼	𝐼	PROPN
cana-3303	105	49	}	}	PUNCT
cana-3303	105	50	(	(	PUNCT
cana-3303	105	51	𝑦2	𝑦2	PROPN
cana-3303	105	52	[	[	X
cana-3303	105	53	𝐼	𝐼	PROPN
cana-3303	105	54	]	]	PUNCT
cana-3303	105	55	)	)	PUNCT
cana-3303	105	56	{	{	PUNCT
cana-3303	105	57	𝐼	𝐼	PROPN
cana-3303	105	58	}	}	PUNCT
cana-3303	105	59	=	=	SYM
cana-3303	105	60	𝑓2	𝑓2	NOUN
cana-3303	105	61	,	,	PUNCT
cana-3303	105	62	(	(	PUNCT
cana-3303	105	63	2	2	NUM
cana-3303	105	64	2	2	NUM
cana-3303	105	65	)	)	PUNCT
cana-3303	105	66	and	and	CCONJ
cana-3303	105	67	if	if	SCONJ
cana-3303	105	68	accepted	accept	VERB
cana-3303	105	69	that	that	SCONJ
cana-3303	105	70	n=3	n=3	VERB
cana-3303	105	71	,	,	PUNCT
cana-3303	105	72	we	we	PRON
cana-3303	105	73	get	get	VERB
cana-3303	105	74	the	the	DET
cana-3303	105	75	expression	expression	NOUN
cana-3303	105	76	(	(	PUNCT
cana-3303	105	77	𝑦4	𝑦4	NOUN
cana-3303	105	78	[	[	X
cana-3303	105	79	𝐼	𝐼	PROPN
cana-3303	105	80	]	]	PUNCT
cana-3303	105	81	)	)	PUNCT
cana-3303	105	82	{	{	PUNCT
cana-3303	105	83	𝐼}(𝑦3	𝐼}(𝑦3	X
cana-3303	106	1	[	[	X
cana-3303	106	2	𝐼	𝐼	PROPN
cana-3303	106	3	]	]	PUNCT
cana-3303	106	4	)	)	PUNCT
cana-3303	106	5	{	{	PUNCT
cana-3303	106	6	𝐼}=𝑓3	𝐼}=𝑓3	NOUN
cana-3303	106	7	,	,	PUNCT
cana-3303	106	8	(	(	PUNCT
cana-3303	106	9	2	2	NUM
cana-3303	106	10	3	3	NUM
cana-3303	106	11	)	)	PUNCT
cana-3303	106	12	communications	communication	NOUN
cana-3303	106	13	on	on	ADP
cana-3303	106	14	applied	apply	VERB
cana-3303	106	15	nonlinear	nonlinear	ADJ
cana-3303	106	16	analysis	analysis	NOUN
cana-3303	106	17	issn	issn	NOUN
cana-3303	106	18	:	:	PUNCT
cana-3303	106	19	1074	1074	NUM
cana-3303	106	20	-	-	PUNCT
cana-3303	106	21	133x	133x	NUM
cana-3303	106	22	vol	vol	NOUN
cana-3303	106	23	32	32	NUM
cana-3303	106	24	no	no	NOUN
cana-3303	106	25	.	.	PUNCT
cana-3303	107	1	6s	6s	NUM
cana-3303	107	2	(	(	PUNCT
cana-3303	107	3	2025	2025	NUM
cana-3303	107	4	)	)	PUNCT
cana-3303	107	5	383	383	NUM
cana-3303	107	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	107	7	with	with	ADP
cana-3303	107	8	the	the	DET
cana-3303	107	9	same	same	ADJ
cana-3303	107	10	rule	rule	NOUN
cana-3303	107	11	,	,	PUNCT
cana-3303	107	12	we	we	PRON
cana-3303	107	13	continue	continue	VERB
cana-3303	107	14	to	to	PART
cana-3303	107	15	assign	assign	VERB
cana-3303	107	16	n	n	PRON
cana-3303	107	17	values	value	NOUN
cana-3303	107	18	to	to	ADP
cana-3303	107	19	(	(	PUNCT
cana-3303	107	20	2	2	NUM
cana-3303	107	21	)	)	PUNCT
cana-3303	107	22	,	,	PUNCT
cana-3303	107	23	and	and	CCONJ
cana-3303	107	24	then	then	ADV
cana-3303	107	25	add	add	VERB
cana-3303	107	26	the	the	DET
cana-3303	107	27	expressions	expression	NOUN
cana-3303	107	28	(	(	PUNCT
cana-3303	107	29	2	2	NUM
cana-3303	107	30	𝑘	𝑘	NOUN
cana-3303	107	31	)	)	PUNCT
cana-3303	107	32	k≥	k≥	NOUN
cana-3303	107	33	0	0	NUM
cana-3303	107	34	side	side	NOUN
cana-3303	107	35	to	to	ADP
cana-3303	107	36	side	side	NOUN
cana-3303	107	37	:	:	PUNCT
cana-3303	107	38	(	(	PUNCT
cana-3303	107	39	(	(	PUNCT
cana-3303	107	40	𝑦1	𝑦1	NOUN
cana-3303	107	41	[	[	X
cana-3303	107	42	𝐼	𝐼	PROPN
cana-3303	107	43	]	]	PUNCT
cana-3303	107	44	)	)	PUNCT
cana-3303	107	45	{	{	PUNCT
cana-3303	107	46	𝐼}(𝑦0	𝐼}(𝑦0	X
cana-3303	108	1	[	[	X
cana-3303	108	2	𝐼	𝐼	PROPN
cana-3303	108	3	]	]	PUNCT
cana-3303	108	4	)	)	PUNCT
cana-3303	108	5	{	{	PUNCT
cana-3303	108	6	𝐼	𝐼	PROPN
cana-3303	108	7	}	}	PUNCT
cana-3303	108	8	)	)	PUNCT
cana-3303	109	1	+	+	CCONJ
cana-3303	109	2	(	(	PUNCT
cana-3303	109	3	𝑦2	𝑦2	PROPN
cana-3303	109	4	[	[	X
cana-3303	109	5	𝐼	𝐼	PROPN
cana-3303	109	6	]	]	PUNCT
cana-3303	109	7	)	)	PUNCT
cana-3303	109	8	{	{	PUNCT
cana-3303	109	9	𝐼	𝐼	PROPN
cana-3303	109	10	}	}	PUNCT
cana-3303	109	11	(	(	PUNCT
cana-3303	109	12	𝑦1	𝑦1	PROPN
cana-3303	109	13	[	[	X
cana-3303	109	14	𝐼	𝐼	PROPN
cana-3303	109	15	]	]	PUNCT
cana-3303	109	16	)	)	PUNCT
cana-3303	109	17	{	{	PUNCT
cana-3303	109	18	𝐼	𝐼	PROPN
cana-3303	109	19	}	}	PUNCT
cana-3303	109	20	+	+	PROPN
cana-3303	109	21	(	(	PUNCT
cana-3303	109	22	𝑦3	𝑦3	PROPN
cana-3303	109	23	[	[	X
cana-3303	109	24	𝐼	𝐼	PROPN
cana-3303	109	25	]	]	PUNCT
cana-3303	109	26	)	)	PUNCT
cana-3303	109	27	{	{	PUNCT
cana-3303	109	28	𝐼	𝐼	PROPN
cana-3303	109	29	}	}	PUNCT
cana-3303	109	30	(	(	PUNCT
cana-3303	109	31	𝑦2	𝑦2	PROPN
cana-3303	109	32	[	[	X
cana-3303	109	33	𝐼	𝐼	PROPN
cana-3303	109	34	]	]	PUNCT
cana-3303	109	35	)	)	PUNCT
cana-3303	109	36	{	{	PUNCT
cana-3303	109	37	𝐼	𝐼	PROPN
cana-3303	109	38	}	}	PUNCT
cana-3303	109	39	+	+	PROPN
cana-3303	109	40	⋯	⋯	VERB
cana-3303	109	41	+	+	CCONJ
cana-3303	109	42	+	+	ADJ
cana-3303	109	43	(	(	PUNCT
cana-3303	109	44	𝑦𝑠	𝑦𝑠	ADP
cana-3303	109	45	[	[	X
cana-3303	109	46	𝐼	𝐼	PROPN
cana-3303	109	47	]	]	PUNCT
cana-3303	109	48	)	)	PUNCT
cana-3303	109	49	{	{	PUNCT
cana-3303	109	50	𝐼	𝐼	PROPN
cana-3303	109	51	}	}	PUNCT
cana-3303	109	52	(	(	PUNCT
cana-3303	109	53	𝑦𝑠−1	𝑦𝑠−1	PROPN
cana-3303	109	54	[	[	X
cana-3303	109	55	𝐼	𝐼	PROPN
cana-3303	109	56	]	]	PUNCT
cana-3303	109	57	)	)	PUNCT
cana-3303	109	58	{	{	PUNCT
cana-3303	109	59	𝐼	𝐼	PROPN
cana-3303	109	60	}	}	PUNCT
cana-3303	109	61	+	+	PROPN
cana-3303	109	62	⋯	⋯	VERB
cana-3303	109	63	+	+	CCONJ
cana-3303	109	64	(	(	PUNCT
cana-3303	109	65	𝑦𝑛	𝑦𝑛	AUX
cana-3303	109	66	[	[	X
cana-3303	109	67	𝐼	𝐼	NOUN
cana-3303	109	68	]	]	PUNCT
cana-3303	109	69	)	)	PUNCT
cana-3303	109	70	{	{	PUNCT
cana-3303	109	71	𝐼	𝐼	PROPN
cana-3303	109	72	}	}	PUNCT
cana-3303	109	73	(	(	PUNCT
cana-3303	109	74	𝑦𝑛−1	𝑦𝑛−1	NOUN
cana-3303	109	75	[	[	X
cana-3303	109	76	𝐼	𝐼	PROPN
cana-3303	109	77	]	]	PUNCT
cana-3303	109	78	)	)	PUNCT
cana-3303	109	79	{	{	PUNCT
cana-3303	109	80	𝐼	𝐼	NOUN
cana-3303	109	81	}	}	PUNCT
cana-3303	109	82	)	)	PUNCT
cana-3303	109	83	=	=	PUNCT
cana-3303	109	84	∑	∑	PUNCT
cana-3303	109	85	𝑓𝑠	𝑓𝑠	NOUN
cana-3303	109	86	𝑛−1	𝑛−1	PROPN
cana-3303	109	87	𝑠=0	𝑠=0	PUNCT
cana-3303	109	88	,	,	PUNCT
cana-3303	109	89	or	or	CCONJ
cana-3303	109	90	,	,	PUNCT
cana-3303	109	91	after	after	ADP
cana-3303	109	92	removing	remove	VERB
cana-3303	109	93	similar	similar	ADJ
cana-3303	109	94	limits	limit	NOUN
cana-3303	109	95	on	on	ADP
cana-3303	109	96	the	the	DET
cana-3303	109	97	left	left	ADJ
cana-3303	109	98	-	-	PUNCT
cana-3303	109	99	hand	hand	NOUN
cana-3303	109	100	side	side	NOUN
cana-3303	109	101	of	of	ADP
cana-3303	109	102	the	the	DET
cana-3303	109	103	expression	expression	NOUN
cana-3303	109	104	here	here	ADV
cana-3303	109	105	,	,	PUNCT
cana-3303	109	106	the	the	DET
cana-3303	109	107	expression	expression	NOUN
cana-3303	109	108	becomes	become	VERB
cana-3303	109	109	as	as	SCONJ
cana-3303	109	110	follows	follow	VERB
cana-3303	109	111	.	.	PUNCT
cana-3303	110	1	(	(	PUNCT
cana-3303	110	2	𝑦𝑛	𝑦𝑛	ADP
cana-3303	110	3	[	[	X
cana-3303	110	4	𝐼	𝐼	NOUN
cana-3303	110	5	]	]	PUNCT
cana-3303	110	6	)	)	PUNCT
cana-3303	110	7	{	{	PUNCT
cana-3303	110	8	𝐼}(𝑦0	𝐼}(𝑦0	X
cana-3303	110	9	[	[	X
cana-3303	110	10	𝐼	𝐼	PROPN
cana-3303	110	11	]	]	PUNCT
cana-3303	110	12	)	)	PUNCT
cana-3303	110	13	{	{	PUNCT
cana-3303	110	14	𝐼	𝐼	PROPN
cana-3303	110	15	}	}	PUNCT
cana-3303	110	16	=	=	SYM
cana-3303	110	17	∑	∑	PUNCT
cana-3303	110	18	𝑓𝑠	𝑓𝑠	NOUN
cana-3303	110	19	𝑛−1	𝑛−1	PROPN
cana-3303	110	20	𝑠=0	𝑠=0	PUNCT
cana-3303	110	21	,	,	PUNCT
cana-3303	110	22	n≥	n≥	PROPN
cana-3303	110	23	1	1	NUM
cana-3303	110	24	.	.	PUNCT
cana-3303	111	1	(	(	PUNCT
cana-3303	111	2	3	3	X
cana-3303	111	3	)	)	PUNCT
cana-3303	111	4	now	now	ADV
cana-3303	111	5	let	let	VERB
cana-3303	111	6	's	us	PRON
cana-3303	111	7	accept	accept	VERB
cana-3303	111	8	the	the	DET
cana-3303	111	9	marking	marking	NOUN
cana-3303	111	10	as	as	SCONJ
cana-3303	111	11	follows	follow	VERB
cana-3303	111	12	:	:	PUNCT
cana-3303	111	13	(	(	PUNCT
cana-3303	111	14	𝑦0	𝑦0	NOUN
cana-3303	111	15	[	[	X
cana-3303	111	16	𝐼	𝐼	PROPN
cana-3303	111	17	]	]	PUNCT
cana-3303	111	18	)	)	PUNCT
cana-3303	111	19	{	{	PUNCT
cana-3303	111	20	𝐼	𝐼	PROPN
cana-3303	111	21	}	}	PUNCT
cana-3303	111	22	+	+	NOUN
cana-3303	111	23	∑	∑	ADP
cana-3303	111	24	𝑓𝑠	𝑓𝑠	PRON
cana-3303	111	25	𝑛−1	𝑛−1	PROPN
cana-3303	111	26	𝑠=0	𝑠=0	PUNCT
cana-3303	111	27	≡	≡	PROPN
cana-3303	111	28	𝑔𝑛((𝑦0	𝑔𝑛((𝑦0	PROPN
cana-3303	112	1	[	[	X
cana-3303	112	2	𝐼	𝐼	NOUN
cana-3303	112	3	]	]	PUNCT
cana-3303	112	4	)	)	PUNCT
cana-3303	112	5	{	{	PUNCT
cana-3303	112	6	𝐼	𝐼	PROPN
cana-3303	112	7	}	}	PUNCT
cana-3303	112	8	,	,	PUNCT
cana-3303	112	9	𝑓𝑠	𝑓𝑠	NOUN
cana-3303	112	10	)	)	PUNCT
cana-3303	112	11	,	,	PUNCT
cana-3303	112	12	n≥	n≥	PROPN
cana-3303	112	13	1	1	NUM
cana-3303	112	14	,	,	PUNCT
cana-3303	112	15	(	(	PUNCT
cana-3303	112	16	4	4	NUM
cana-3303	112	17	)	)	PUNCT
cana-3303	112	18	then	then	ADV
cana-3303	112	19	(	(	PUNCT
cana-3303	112	20	3	3	X
cana-3303	112	21	)	)	PUNCT
cana-3303	112	22	becomes	become	VERB
cana-3303	112	23	as	as	SCONJ
cana-3303	112	24	follows	follow	VERB
cana-3303	112	25	:	:	PUNCT
cana-3303	112	26	(	(	PUNCT
cana-3303	112	27	𝑦𝑛	𝑦𝑛	ADP
cana-3303	112	28	[	[	X
cana-3303	112	29	𝐼	𝐼	NOUN
cana-3303	112	30	]	]	PUNCT
cana-3303	112	31	)	)	PUNCT
cana-3303	112	32	{	{	PUNCT
cana-3303	112	33	𝐼}=𝑔𝑛	𝐼}=𝑔𝑛	PROPN
cana-3303	112	34	,	,	PUNCT
cana-3303	112	35	n≥	n≥	PROPN
cana-3303	112	36	1	1	NUM
cana-3303	112	37	.	.	PUNCT
cana-3303	113	1	(	(	PUNCT
cana-3303	113	2	5	5	NUM
cana-3303	113	3	)	)	PUNCT
cana-3303	113	4	thus	thus	ADV
cana-3303	113	5	,	,	PUNCT
cana-3303	113	6	using	use	VERB
cana-3303	113	7	the	the	DET
cana-3303	113	8	definition	definition	NOUN
cana-3303	113	9	of	of	ADP
cana-3303	113	10	a	a	DET
cana-3303	113	11	discrete	discrete	ADJ
cana-3303	113	12	additive	additive	NOUN
cana-3303	113	13	derivative	derivative	NOUN
cana-3303	113	14	,	,	PUNCT
cana-3303	113	15	we	we	PRON
cana-3303	113	16	derive	derive	VERB
cana-3303	113	17	the	the	DET
cana-3303	113	18	third	third	ADJ
cana-3303	113	19	-	-	PUNCT
cana-3303	113	20	order	order	NOUN
cana-3303	113	21	mixed	mixed	ADJ
cana-3303	113	22	derivative	derivative	NOUN
cana-3303	113	23	(	(	PUNCT
cana-3303	113	24	1	1	NUM
cana-3303	113	25	)	)	PUNCT
cana-3303	113	26	equation	equation	NOUN
cana-3303	113	27	into	into	ADP
cana-3303	113	28	the	the	DET
cana-3303	113	29	second	second	ADJ
cana-3303	113	30	-	-	PUNCT
cana-3303	113	31	order	order	NOUN
cana-3303	113	32	mixed	mixed	ADJ
cana-3303	113	33	derivative	derivative	NOUN
cana-3303	113	34	(	(	PUNCT
cana-3303	113	35	5	5	NUM
cana-3303	113	36	)	)	PUNCT
cana-3303	113	37	.	.	PUNCT
cana-3303	114	1	let	let	VERB
cana-3303	114	2	us	we	PRON
cana-3303	114	3	use	use	VERB
cana-3303	114	4	the	the	DET
cana-3303	114	5	definition	definition	NOUN
cana-3303	114	6	of	of	ADP
cana-3303	114	7	discrete	discrete	ADJ
cana-3303	114	8	poverative	poverative	ADJ
cana-3303	114	9	derivative	derivative	NOUN
cana-3303	114	10	in	in	ADP
cana-3303	114	11	equation	equation	NOUN
cana-3303	114	12	(	(	PUNCT
cana-3303	114	13	5	5	NUM
cana-3303	114	14	)	)	PUNCT
cana-3303	114	15	with	with	ADP
cana-3303	114	16	the	the	DET
cana-3303	114	17	similar	similar	ADJ
cana-3303	114	18	rule	rule	NOUN
cana-3303	114	19	,	,	PUNCT
cana-3303	114	20	that	that	ADV
cana-3303	114	21	is	is	ADV
cana-3303	114	22	we	we	PRON
cana-3303	114	23	get	get	VERB
cana-3303	114	24	equations	equation	NOUN
cana-3303	114	25	:	:	PUNCT
cana-3303	115	1	√𝑦𝑛+1	√𝑦𝑛+1	PROPN
cana-3303	115	2	[	[	X
cana-3303	115	3	𝐼	𝐼	X
cana-3303	115	4	]	]	PUNCT
cana-3303	115	5	𝑦𝑛	𝑦𝑛	ADP
cana-3303	115	6	[	[	X
cana-3303	115	7	𝐼	𝐼	NOUN
cana-3303	115	8	]	]	X
cana-3303	115	9	=	=	PUNCT
cana-3303	115	10	𝑔𝑛	𝑔𝑛	NOUN
cana-3303	115	11	,	,	PUNCT
cana-3303	115	12	n≥	n≥	PROPN
cana-3303	115	13	1	1	NUM
cana-3303	115	14	,	,	PUNCT
cana-3303	115	15	or	or	CCONJ
cana-3303	115	16	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	115	17	[	[	X
cana-3303	115	18	𝐼	𝐼	PROPN
cana-3303	115	19	]	]	X
cana-3303	115	20	=	=	PUNCT
cana-3303	115	21	𝑔𝑛	𝑔𝑛	ADP
cana-3303	115	22	𝑦𝑛	𝑦𝑛	ADP
cana-3303	115	23	[	[	X
cana-3303	115	24	𝐼	𝐼	PROPN
cana-3303	115	25	]	]	PUNCT
cana-3303	115	26	,	,	PUNCT
cana-3303	115	27	n≥	n≥	PROPN
cana-3303	115	28	1	1	NUM
cana-3303	115	29	,	,	PUNCT
cana-3303	115	30	(	(	PUNCT
cana-3303	115	31	6	6	NUM
cana-3303	115	32	)	)	PUNCT
cana-3303	115	33	in	in	ADP
cana-3303	115	34	the	the	DET
cana-3303	115	35	equation	equation	NOUN
cana-3303	115	36	(	(	PUNCT
cana-3303	115	37	6	6	NUM
cana-3303	115	38	)	)	PUNCT
cana-3303	115	39	we	we	PRON
cana-3303	115	40	get	get	VERB
cana-3303	115	41	,	,	PUNCT
cana-3303	115	42	if	if	SCONJ
cana-3303	115	43	give	give	VERB
cana-3303	115	44	a	a	DET
cana-3303	115	45	value	value	NOUN
cana-3303	115	46	1	1	NUM
cana-3303	115	47	for	for	ADP
cana-3303	115	48	n	n	CCONJ
cana-3303	115	49	,	,	PUNCT
cana-3303	115	50	we	we	PRON
cana-3303	115	51	get	get	VERB
cana-3303	115	52	the	the	DET
cana-3303	115	53	expression	expression	NOUN
cana-3303	115	54	:	:	PUNCT
cana-3303	116	1	𝑦2	𝑦2	PROPN
cana-3303	116	2	[	[	X
cana-3303	116	3	𝐼	𝐼	X
cana-3303	116	4	]	]	X
cana-3303	116	5	=	=	SYM
cana-3303	116	6	𝑔	𝑔	PROPN
cana-3303	116	7	1	1	NUM
cana-3303	116	8	𝑦1	𝑦1	NOUN
cana-3303	117	1	[	[	X
cana-3303	117	2	𝐼	𝐼	PROPN
cana-3303	117	3	]	]	PUNCT
cana-3303	117	4	,	,	PUNCT
cana-3303	117	5	(	(	PUNCT
cana-3303	117	6	61	61	NUM
cana-3303	117	7	)	)	PUNCT
cana-3303	117	8	in	in	ADP
cana-3303	117	9	the	the	DET
cana-3303	117	10	same	same	ADJ
cana-3303	117	11	equation	equation	NOUN
cana-3303	117	12	(	(	PUNCT
cana-3303	117	13	6	6	NUM
cana-3303	117	14	)	)	PUNCT
cana-3303	117	15	,	,	PUNCT
cana-3303	117	16	by	by	ADP
cana-3303	117	17	giving	give	VERB
cana-3303	117	18	a	a	DET
cana-3303	117	19	value	value	NOUN
cana-3303	117	20	2	2	NUM
cana-3303	117	21	for	for	ADP
cana-3303	117	22	n	n	CCONJ
cana-3303	117	23	,	,	PUNCT
cana-3303	117	24	we	we	PRON
cana-3303	117	25	get	get	VERB
cana-3303	117	26	the	the	DET
cana-3303	117	27	expression	expression	NOUN
cana-3303	117	28	:	:	PUNCT
cana-3303	118	1	𝑦3	𝑦3	PROPN
cana-3303	118	2	[	[	X
cana-3303	118	3	𝐼	𝐼	PROPN
cana-3303	118	4	]	]	PUNCT
cana-3303	118	5	=	=	SYM
cana-3303	118	6	𝑔	𝑔	PROPN
cana-3303	118	7	2	2	NUM
cana-3303	118	8	𝑦2	𝑦2	NOUN
cana-3303	119	1	[	[	X
cana-3303	119	2	𝐼	𝐼	PROPN
cana-3303	119	3	]	]	PUNCT
cana-3303	119	4	,	,	PUNCT
cana-3303	119	5	(	(	PUNCT
cana-3303	119	6	62	62	NUM
cana-3303	119	7	)	)	PUNCT
cana-3303	119	8	the	the	DET
cana-3303	119	9	expression	expression	NOUN
cana-3303	119	10	gets	get	AUX
cana-3303	119	11	taken	take	VERB
cana-3303	119	12	.	.	PUNCT
cana-3303	120	1	considering	consider	VERB
cana-3303	120	2	61	61	NUM
cana-3303	120	3	here	here	ADV
cana-3303	120	4	,	,	PUNCT
cana-3303	120	5	(	(	PUNCT
cana-3303	120	6	62	62	NUM
cana-3303	120	7	)	)	PUNCT
cana-3303	120	8	becomes	become	VERB
cana-3303	120	9	as	as	SCONJ
cana-3303	120	10	follows	follow	VERB
cana-3303	120	11	:	:	PUNCT
cana-3303	120	12	𝑦3	𝑦3	PROPN
cana-3303	120	13	[	[	X
cana-3303	120	14	𝐼	𝐼	PROPN
cana-3303	120	15	]	]	PUNCT
cana-3303	120	16	=	=	SYM
cana-3303	120	17	𝑔	𝑔	PROPN
cana-3303	120	18	2	2	NUM
cana-3303	120	19	𝑔	𝑔	PROPN
cana-3303	120	20	1	1	NUM
cana-3303	120	21	𝑦	𝑦	SYM
cana-3303	120	22	1	1	NUM
cana-3303	120	23	[	[	X
cana-3303	120	24	𝐼	𝐼	PROPN
cana-3303	120	25	]	]	PUNCT
cana-3303	120	26	.	.	PUNCT
cana-3303	121	1	(	(	PUNCT
cana-3303	121	2	621	621	NUM
cana-3303	121	3	)	)	PUNCT
cana-3303	121	4	returning	return	VERB
cana-3303	121	5	to	to	ADP
cana-3303	121	6	equation	equation	NOUN
cana-3303	121	7	(	(	PUNCT
cana-3303	121	8	6	6	NUM
cana-3303	121	9	)	)	PUNCT
cana-3303	121	10	and	and	CCONJ
cana-3303	121	11	giving	give	VERB
cana-3303	121	12	a	a	DET
cana-3303	121	13	value	value	NOUN
cana-3303	121	14	3	3	NUM
cana-3303	121	15	for	for	ADP
cana-3303	121	16	n	n	PRON
cana-3303	121	17	we	we	PRON
cana-3303	121	18	get	get	VERB
cana-3303	121	19	:	:	PUNCT
cana-3303	121	20	𝑦4	𝑦4	PROPN
cana-3303	122	1	[	[	X
cana-3303	122	2	𝐼	𝐼	NOUN
cana-3303	122	3	]	]	PUNCT
cana-3303	122	4	=	=	SYM
cana-3303	122	5	𝑔	𝑔	PROPN
cana-3303	122	6	3	3	NUM
cana-3303	122	7	𝑦3	𝑦3	PROPN
cana-3303	122	8	[	[	X
cana-3303	122	9	𝐼	𝐼	PROPN
cana-3303	122	10	]	]	PUNCT
cana-3303	122	11	,	,	PUNCT
cana-3303	122	12	(	(	PUNCT
cana-3303	122	13	6	6	NUM
cana-3303	122	14	3	3	NUM
cana-3303	122	15	)	)	PUNCT
cana-3303	122	16	communications	communication	NOUN
cana-3303	122	17	on	on	ADP
cana-3303	122	18	applied	apply	VERB
cana-3303	122	19	nonlinear	nonlinear	ADJ
cana-3303	122	20	analysis	analysis	NOUN
cana-3303	122	21	issn	issn	NOUN
cana-3303	122	22	:	:	PUNCT
cana-3303	122	23	1074	1074	NUM
cana-3303	122	24	-	-	PUNCT
cana-3303	122	25	133x	133x	NUM
cana-3303	122	26	vol	vol	NOUN
cana-3303	122	27	32	32	NUM
cana-3303	122	28	no	no	NOUN
cana-3303	122	29	.	.	PUNCT
cana-3303	123	1	6s	6s	NUM
cana-3303	123	2	(	(	PUNCT
cana-3303	123	3	2025	2025	NUM
cana-3303	123	4	)	)	PUNCT
cana-3303	123	5	384	384	NUM
cana-3303	123	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	123	7	here	here	ADV
cana-3303	123	8	if	if	SCONJ
cana-3303	123	9	we	we	PRON
cana-3303	123	10	consider	consider	VERB
cana-3303	123	11	621	621	NUM
cana-3303	123	12	)	)	PUNCT
cana-3303	123	13	,	,	PUNCT
cana-3303	123	14	we	we	PRON
cana-3303	123	15	get	get	VERB
cana-3303	123	16	the	the	DET
cana-3303	123	17	expression	expression	NOUN
cana-3303	123	18	:	:	PUNCT
cana-3303	123	19	𝑦4	𝑦4	NOUN
cana-3303	123	20	[	[	X
cana-3303	123	21	𝐼	𝐼	NOUN
cana-3303	123	22	]	]	PUNCT
cana-3303	123	23	=	=	SYM
cana-3303	123	24	𝑔	𝑔	PROPN
cana-3303	123	25	3	3	NUM
cana-3303	123	26	𝑔	𝑔	PROPN
cana-3303	123	27	2	2	NUM
cana-3303	123	28	𝑔	𝑔	PROPN
cana-3303	123	29	1	1	NUM
cana-3303	123	30	𝑦	𝑦	SYM
cana-3303	123	31	1	1	NUM
cana-3303	124	1	[	[	X
cana-3303	124	2	𝐼	𝐼	PROPN
cana-3303	124	3	]	]	PUNCT
cana-3303	124	4	,	,	PUNCT
cana-3303	124	5	(	(	PUNCT
cana-3303	124	6	6	6	NUM
cana-3303	124	7	31	31	NUM
cana-3303	124	8	)	)	PUNCT
cana-3303	124	9	by	by	ADP
cana-3303	124	10	continuing	continue	VERB
cana-3303	124	11	this	this	DET
cana-3303	124	12	process	process	NOUN
cana-3303	124	13	,	,	PUNCT
cana-3303	124	14	if	if	SCONJ
cana-3303	124	15	we	we	PRON
cana-3303	124	16	get	get	VERB
cana-3303	124	17	the	the	DET
cana-3303	124	18	following	follow	VERB
cana-3303	124	19	expression	expression	NOUN
cana-3303	124	20	from	from	ADP
cana-3303	124	21	(	(	PUNCT
cana-3303	124	22	6	6	NUM
cana-3303	124	23	)	)	PUNCT
cana-3303	124	24	for	for	ADP
cana-3303	124	25	𝑦𝑛−1	𝑦𝑛−1	PROPN
cana-3303	124	26	[	[	X
cana-3303	124	27	𝐼	𝐼	PROPN
cana-3303	124	28	]	]	PUNCT
cana-3303	124	29	𝑦𝑛−1	𝑦𝑛−1	NOUN
cana-3303	124	30	[	[	X
cana-3303	124	31	𝐼	𝐼	X
cana-3303	124	32	]	]	X
cana-3303	124	33	=	=	SYM
cana-3303	124	34	𝑔𝑛−2	𝑔𝑛−2	ADP
cana-3303	124	35	𝑔𝑛−3	𝑔𝑛−3	NOUN
cana-3303	124	36	⋰	⋰	NOUN
cana-3303	124	37	𝑔	𝑔	PROPN
cana-3303	124	38	2	2	NUM
cana-3303	124	39	𝑔	𝑔	PROPN
cana-3303	124	40	1	1	NUM
cana-3303	124	41	𝑦	𝑦	SYM
cana-3303	124	42	1	1	NUM
cana-3303	125	1	[	[	X
cana-3303	125	2	𝐼	𝐼	PROPN
cana-3303	125	3	]	]	PUNCT
cana-3303	125	4	,	,	PUNCT
cana-3303	125	5	(	(	PUNCT
cana-3303	125	6	6	6	NUM
cana-3303	125	7	𝑛−2	𝑛−2	NOUN
cana-3303	125	8	)	)	PUNCT
cana-3303	125	9	then	then	ADV
cana-3303	125	10	by	by	ADP
cana-3303	125	11	returning	return	VERB
cana-3303	125	12	to	to	ADP
cana-3303	125	13	(	(	PUNCT
cana-3303	125	14	6	6	NUM
cana-3303	125	15	)	)	PUNCT
cana-3303	125	16	again	again	ADV
cana-3303	125	17	,	,	PUNCT
cana-3303	125	18	we	we	PRON
cana-3303	125	19	get	get	VERB
cana-3303	125	20	:	:	PUNCT
cana-3303	125	21	𝑦𝑛	𝑦𝑛	ADP
cana-3303	125	22	[	[	X
cana-3303	125	23	𝐼	𝐼	NOUN
cana-3303	125	24	]	]	X
cana-3303	125	25	=	=	SYM
cana-3303	125	26	𝑔	𝑔	PROPN
cana-3303	125	27	𝑛−1	𝑛−1	PROPN
cana-3303	125	28	𝑦𝑛−1	𝑦𝑛−1	NOUN
cana-3303	125	29	[	[	X
cana-3303	125	30	𝐼	𝐼	PROPN
cana-3303	125	31	]	]	PUNCT
cana-3303	125	32	,	,	PUNCT
cana-3303	125	33	(	(	PUNCT
cana-3303	125	34	6	6	NUM
cana-3303	125	35	𝑛−1	𝑛−1	NUM
cana-3303	125	36	)	)	PUNCT
cana-3303	125	37	and	and	CCONJ
cana-3303	125	38	finally	finally	ADV
cana-3303	125	39	considering	consider	VERB
cana-3303	125	40	(	(	PUNCT
cana-3303	125	41	6	6	NUM
cana-3303	125	42	𝑛−2	𝑛−2	NOUN
cana-3303	125	43	)	)	PUNCT
cana-3303	125	44	,	,	PUNCT
cana-3303	125	45	we	we	PRON
cana-3303	125	46	get	get	VERB
cana-3303	125	47	the	the	DET
cana-3303	125	48	expression	expression	NOUN
cana-3303	125	49	:	:	PUNCT
cana-3303	126	1	𝑦𝑛	𝑦𝑛	ADP
cana-3303	126	2	[	[	X
cana-3303	126	3	𝐼	𝐼	NOUN
cana-3303	126	4	]	]	PUNCT
cana-3303	126	5	=	=	PUNCT
cana-3303	126	6	𝑔𝑛−1	𝑔𝑛−1	PROPN
cana-3303	126	7	𝑔𝑛−2	𝑔𝑛−2	PROPN
cana-3303	126	8	⋰	⋰	DET
cana-3303	126	9	𝑔	𝑔	PROPN
cana-3303	126	10	2	2	NUM
cana-3303	126	11	𝑔	𝑔	PROPN
cana-3303	126	12	1	1	NUM
cana-3303	126	13	𝑦	𝑦	SYM
cana-3303	126	14	1	1	NUM
cana-3303	126	15	[	[	X
cana-3303	126	16	𝐼	𝐼	PROPN
cana-3303	126	17	]	]	PUNCT
cana-3303	126	18	,	,	PUNCT
cana-3303	126	19	n≥	n≥	PROPN
cana-3303	126	20	2	2	NUM
cana-3303	126	21	,	,	PUNCT
cana-3303	126	22	(	(	PUNCT
cana-3303	126	23	7	7	X
cana-3303	126	24	)	)	PUNCT
cana-3303	126	25	here	here	ADV
cana-3303	126	26	,	,	PUNCT
cana-3303	126	27	as	as	ADP
cana-3303	126	28	in	in	ADP
cana-3303	126	29	(	(	PUNCT
cana-3303	126	30	4	4	NUM
cana-3303	126	31	)	)	PUNCT
cana-3303	126	32	,	,	PUNCT
cana-3303	126	33	let	let	VERB
cana-3303	126	34	us	we	PRON
cana-3303	126	35	accept	accept	VERB
cana-3303	126	36	the	the	DET
cana-3303	126	37	marking	marking	NOUN
cana-3303	126	38	as	as	SCONJ
cana-3303	126	39	follows	follow	VERB
cana-3303	126	40	:	:	PUNCT
cana-3303	126	41	𝑔𝑛−1	𝑔𝑛−1	PROPN
cana-3303	126	42	𝑔𝑛−2	𝑔𝑛−2	PROPN
cana-3303	126	43	⋰	⋰	DET
cana-3303	126	44	𝑔	𝑔	PROPN
cana-3303	126	45	2	2	NUM
cana-3303	126	46	𝑔	𝑔	PROPN
cana-3303	126	47	1	1	NUM
cana-3303	126	48	𝑦	𝑦	SYM
cana-3303	126	49	1	1	NUM
cana-3303	127	1	[	[	X
cana-3303	127	2	𝐼	𝐼	PROPN
cana-3303	127	3	]	]	PUNCT
cana-3303	127	4	≡	≡	PROPN
cana-3303	127	5	ℎ𝑛(𝑦1	ℎ𝑛(𝑦1	PROPN
cana-3303	128	1	[	[	X
cana-3303	128	2	𝐼	𝐼	X
cana-3303	128	3	]	]	PUNCT
cana-3303	128	4	,	,	PUNCT
cana-3303	128	5	𝑔𝑠	𝑔𝑠	NOUN
cana-3303	128	6	)	)	PUNCT
cana-3303	128	7	,	,	PUNCT
cana-3303	128	8	n≥	n≥	PROPN
cana-3303	128	9	2	2	NUM
cana-3303	128	10	,	,	PUNCT
cana-3303	128	11	(	(	PUNCT
cana-3303	128	12	8)	8)	NUM
cana-3303	128	13	then	then	ADV
cana-3303	128	14	(	(	PUNCT
cana-3303	128	15	7	7	X
cana-3303	128	16	)	)	PUNCT
cana-3303	128	17	becomes	become	VERB
cana-3303	128	18	as	as	SCONJ
cana-3303	128	19	follows	follow	VERB
cana-3303	128	20	:	:	PUNCT
cana-3303	128	21	𝑦𝑛	𝑦𝑛	ADP
cana-3303	128	22	[	[	X
cana-3303	128	23	𝐼	𝐼	NOUN
cana-3303	128	24	]	]	PUNCT
cana-3303	128	25	=	=	PUNCT
cana-3303	128	26	ℎ𝑛	ℎ𝑛	NOUN
cana-3303	128	27	,	,	PUNCT
cana-3303	128	28	n≥	n≥	PROPN
cana-3303	128	29	2	2	NUM
cana-3303	128	30	.	.	PUNCT
cana-3303	129	1	(	(	PUNCT
cana-3303	129	2	9	9	NUM
cana-3303	129	3	)	)	PUNCT
cana-3303	129	4	thus	thus	ADV
cana-3303	129	5	,	,	PUNCT
cana-3303	129	6	we	we	PRON
cana-3303	129	7	derive	derive	VERB
cana-3303	129	8	the	the	DET
cana-3303	129	9	third	third	ADJ
cana-3303	129	10	-	-	PUNCT
cana-3303	129	11	order	order	NOUN
cana-3303	129	12	mixed	mixed	ADJ
cana-3303	129	13	derivative	derivative	NOUN
cana-3303	129	14	(	(	PUNCT
cana-3303	129	15	1	1	NUM
cana-3303	129	16	)	)	PUNCT
cana-3303	129	17	equation	equation	NOUN
cana-3303	129	18	to	to	ADP
cana-3303	129	19	the	the	DET
cana-3303	129	20	first	first	ADJ
cana-3303	129	21	-	-	PUNCT
cana-3303	129	22	order	order	NOUN
cana-3303	129	23	discrete	discrete	ADJ
cana-3303	129	24	multiplicative	multiplicative	ADJ
cana-3303	129	25	derivative	derivative	NOUN
cana-3303	129	26	(	(	PUNCT
cana-3303	129	27	9	9	NUM
cana-3303	129	28	)	)	PUNCT
cana-3303	129	29	using	use	VERB
cana-3303	129	30	the	the	DET
cana-3303	129	31	definitions	definition	NOUN
cana-3303	129	32	of	of	ADP
cana-3303	129	33	discrete	discrete	ADJ
cana-3303	129	34	additives	additive	NOUN
cana-3303	129	35	and	and	CCONJ
cana-3303	129	36	discrete	discrete	VERB
cana-3303	129	37	poverative	poverative	ADJ
cana-3303	129	38	derivatives	derivative	NOUN
cana-3303	129	39	.	.	PUNCT
cana-3303	130	1	finally	finally	ADV
cana-3303	130	2	,	,	PUNCT
cana-3303	130	3	using	use	VERB
cana-3303	130	4	the	the	DET
cana-3303	130	5	definition	definition	NOUN
cana-3303	130	6	of	of	ADP
cana-3303	130	7	a	a	DET
cana-3303	130	8	discrete	discrete	ADJ
cana-3303	130	9	multiplicative	multiplicative	ADJ
cana-3303	130	10	derivative	derivative	NOUN
cana-3303	130	11	,	,	PUNCT
cana-3303	130	12	we	we	PRON
cana-3303	130	13	can	can	AUX
cana-3303	130	14	write	write	VERB
cana-3303	130	15	equation	equation	NOUN
cana-3303	130	16	(	(	PUNCT
cana-3303	130	17	9	9	NUM
cana-3303	130	18	)	)	PUNCT
cana-3303	130	19	as	as	SCONJ
cana-3303	130	20	follows	follow	VERB
cana-3303	130	21	:	:	PUNCT
cana-3303	130	22	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3303	130	23	𝑦𝑛	𝑦𝑛	NOUN
cana-3303	130	24	=	=	SYM
cana-3303	130	25	ℎ𝑛	ℎ𝑛	NOUN
cana-3303	130	26	,	,	PUNCT
cana-3303	130	27	n≥	n≥	NOUN
cana-3303	130	28	2	2	NUM
cana-3303	130	29	,	,	PUNCT
cana-3303	130	30	or	or	CCONJ
cana-3303	130	31	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3303	130	32	=	=	SYM
cana-3303	130	33	ℎ𝑛	ℎ𝑛	NOUN
cana-3303	130	34	∙	∙	PROPN
cana-3303	130	35	𝑦𝑛	𝑦𝑛	NOUN
cana-3303	130	36	,	,	PUNCT
cana-3303	130	37	n≥	n≥	PROPN
cana-3303	130	38	2	2	NUM
cana-3303	130	39	.	.	PUNCT
cana-3303	131	1	(	(	PUNCT
cana-3303	131	2	10	10	NUM
cana-3303	131	3	)	)	PUNCT
cana-3303	131	4	let	let	VERB
cana-3303	131	5	us	we	PRON
cana-3303	131	6	assume	assume	VERB
cana-3303	131	7	that	that	SCONJ
cana-3303	131	8	in	in	ADP
cana-3303	131	9	equation	equation	NOUN
cana-3303	131	10	(	(	PUNCT
cana-3303	131	11	10	10	NUM
cana-3303	131	12	)	)	PUNCT
cana-3303	131	13	we	we	PRON
cana-3303	131	14	obtained	obtain	VERB
cana-3303	131	15	,	,	PUNCT
cana-3303	131	16	n	n	CCONJ
cana-3303	131	17	=	=	SYM
cana-3303	131	18	2	2	NUM
cana-3303	131	19	will	will	AUX
cana-3303	131	20	be	be	AUX
cana-3303	131	21	:	:	PUNCT
cana-3303	131	22	𝑦3=	𝑦3=	PROPN
cana-3303	131	23	ℎ2𝑦2	ℎ2𝑦2	NOUN
cana-3303	131	24	,	,	PUNCT
cana-3303	131	25	(	(	PUNCT
cana-3303	131	26	102	102	NUM
cana-3303	131	27	)	)	PUNCT
cana-3303	131	28	if	if	SCONJ
cana-3303	131	29	we	we	PRON
cana-3303	131	30	give	give	VERB
cana-3303	131	31	a	a	DET
cana-3303	131	32	value	value	NOUN
cana-3303	131	33	3	3	NUM
cana-3303	131	34	to	to	ADP
cana-3303	131	35	n	n	NOUN
cana-3303	131	36	in	in	ADP
cana-3303	131	37	the	the	DET
cana-3303	131	38	equation	equation	NOUN
cana-3303	131	39	above	above	ADV
cana-3303	131	40	(	(	PUNCT
cana-3303	131	41	10	10	NUM
cana-3303	131	42	):	):	PUNCT
cana-3303	131	43	𝑦4=	𝑦4=	PROPN
cana-3303	131	44	ℎ3𝑦3	ℎ3𝑦3	PROPN
cana-3303	131	45	,	,	PUNCT
cana-3303	131	46	or	or	CCONJ
cana-3303	131	47	here	here	ADV
cana-3303	131	48	taking	take	VERB
cana-3303	131	49	into	into	ADP
cana-3303	131	50	account	account	NOUN
cana-3303	131	51	102	102	NUM
cana-3303	131	52	)	)	PUNCT
cana-3303	131	53	,	,	PUNCT
cana-3303	131	54	we	we	PRON
cana-3303	131	55	get	get	VERB
cana-3303	131	56	the	the	DET
cana-3303	131	57	expression	expression	NOUN
cana-3303	131	58	:	:	PUNCT
cana-3303	131	59	𝑦4=	𝑦4=	PROPN
cana-3303	131	60	ℎ3ℎ2𝑦2	ℎ3ℎ2𝑦2	PROPN
cana-3303	131	61	,	,	PUNCT
cana-3303	131	62	(	(	PUNCT
cana-3303	131	63	103	103	NUM
cana-3303	131	64	)	)	PUNCT
cana-3303	131	65	communications	communication	NOUN
cana-3303	131	66	on	on	ADP
cana-3303	131	67	applied	apply	VERB
cana-3303	131	68	nonlinear	nonlinear	ADJ
cana-3303	131	69	analysis	analysis	NOUN
cana-3303	131	70	issn	issn	NOUN
cana-3303	131	71	:	:	PUNCT
cana-3303	131	72	1074	1074	NUM
cana-3303	131	73	-	-	PUNCT
cana-3303	131	74	133x	133x	NUM
cana-3303	131	75	vol	vol	NOUN
cana-3303	131	76	32	32	NUM
cana-3303	131	77	no	no	NOUN
cana-3303	131	78	.	.	PUNCT
cana-3303	132	1	6s	6s	NUM
cana-3303	132	2	(	(	PUNCT
cana-3303	132	3	2025	2025	NUM
cana-3303	132	4	)	)	PUNCT
cana-3303	132	5	385	385	NUM
cana-3303	132	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	133	1	if	if	SCONJ
cana-3303	133	2	we	we	PRON
cana-3303	133	3	return	return	VERB
cana-3303	133	4	to	to	ADP
cana-3303	133	5	(	(	PUNCT
cana-3303	133	6	10	10	NUM
cana-3303	133	7	)	)	PUNCT
cana-3303	133	8	again	again	ADV
cana-3303	133	9	and	and	CCONJ
cana-3303	133	10	give	give	VERB
cana-3303	133	11	a	a	DET
cana-3303	133	12	value	value	NOUN
cana-3303	133	13	(	(	PUNCT
cana-3303	133	14	4	4	NUM
cana-3303	133	15	)	)	PUNCT
cana-3303	133	16	for	for	ADP
cana-3303	133	17	n	n	CCONJ
cana-3303	133	18	:	:	PUNCT
cana-3303	133	19	𝑦5=ℎ4𝑦4	𝑦5=ℎ4𝑦4	X
cana-3303	133	20	,	,	PUNCT
cana-3303	133	21	and	and	CCONJ
cana-3303	133	22	if	if	SCONJ
cana-3303	133	23	we	we	PRON
cana-3303	133	24	consider	consider	VERB
cana-3303	133	25	(	(	PUNCT
cana-3303	133	26	10	10	NUM
cana-3303	133	27	3	3	NUM
cana-3303	133	28	)	)	PUNCT
cana-3303	133	29	,	,	PUNCT
cana-3303	133	30	then	then	ADV
cana-3303	133	31	we	we	PRON
cana-3303	133	32	get	get	VERB
cana-3303	133	33	the	the	DET
cana-3303	133	34	expression	expression	NOUN
cana-3303	133	35	:	:	PUNCT
cana-3303	134	1	𝑦5=ℎ4ℎ3ℎ2𝑦2	𝑦5=ℎ4ℎ3ℎ2𝑦2	PROPN
cana-3303	134	2	,	,	PUNCT
cana-3303	134	3	(	(	PUNCT
cana-3303	134	4	104	104	NUM
cana-3303	134	5	)	)	PUNCT
cana-3303	134	6	by	by	ADP
cana-3303	134	7	continuing	continue	VERB
cana-3303	134	8	this	this	DET
cana-3303	134	9	process	process	NOUN
cana-3303	134	10	,	,	PUNCT
cana-3303	134	11	if	if	SCONJ
cana-3303	134	12	we	we	PRON
cana-3303	134	13	accept	accept	VERB
cana-3303	134	14	that	that	SCONJ
cana-3303	134	15	we	we	PRON
cana-3303	134	16	get	get	VERB
cana-3303	134	17	𝑦𝑛−1	𝑦𝑛−1	PROPN
cana-3303	134	18	𝑦𝑛−1=ℎ𝑛−2ℎ𝑛−3	𝑦𝑛−1=ℎ𝑛−2ℎ𝑛−3	NOUN
cana-3303	134	19	⋯	⋯	NOUN
cana-3303	134	20	ℎ3ℎ2𝑦2	ℎ3ℎ2𝑦2	PROPN
cana-3303	134	21	,	,	PUNCT
cana-3303	134	22	(	(	PUNCT
cana-3303	134	23	10𝑛−2	10𝑛−2	NUM
cana-3303	134	24	)	)	PUNCT
cana-3303	134	25	then	then	ADV
cana-3303	134	26	from	from	ADP
cana-3303	134	27	(	(	PUNCT
cana-3303	134	28	10	10	NUM
cana-3303	134	29	)	)	PUNCT
cana-3303	134	30	we	we	PRON
cana-3303	134	31	get	get	VERB
cana-3303	134	32	𝑦𝑛=	𝑦𝑛=	ADJ
cana-3303	134	33	ℎ𝑛−1𝑦𝑛−1	ℎ𝑛−1𝑦𝑛−1	NOUN
cana-3303	134	34	,	,	PUNCT
cana-3303	134	35	or	or	CCONJ
cana-3303	134	36	by	by	ADP
cana-3303	134	37	considering	consider	VERB
cana-3303	134	38	(	(	PUNCT
cana-3303	134	39	10𝑛−2	10𝑛−2	NUM
cana-3303	134	40	)	)	PUNCT
cana-3303	134	41	,	,	PUNCT
cana-3303	134	42	we	we	PRON
cana-3303	134	43	get	get	VERB
cana-3303	134	44	:	:	PUNCT
cana-3303	134	45	𝑦𝑛=	𝑦𝑛=	NOUN
cana-3303	134	46	𝑦2	𝑦2	PROPN
cana-3303	134	47	∏	∏	PROPN
cana-3303	134	48	ℎ𝑠	ℎ𝑠	PROPN
cana-3303	134	49	𝑛−1	𝑛−1	PROPN
cana-3303	134	50	𝑠=2	𝑠=2	PROPN
cana-3303	134	51	,	,	PUNCT
cana-3303	134	52	n≥	n≥	PROPN
cana-3303	134	53	3	3	NUM
cana-3303	134	54	,	,	PUNCT
cana-3303	134	55	(	(	PUNCT
cana-3303	134	56	10𝑛−1	10𝑛−1	NOUN
cana-3303	134	57	)	)	PUNCT
cana-3303	135	1	so	so	ADV
cana-3303	135	2	,	,	PUNCT
cana-3303	135	3	we	we	PRON
cana-3303	135	4	get	get	VERB
cana-3303	135	5	the	the	DET
cana-3303	135	6	following	follow	VERB
cana-3303	135	7	solution	solution	NOUN
cana-3303	135	8	:	:	PUNCT
cana-3303	135	9	theorem	theorem	NOUN
cana-3303	135	10	1	1	NUM
cana-3303	135	11	.	.	PUNCT
cana-3303	136	1	if	if	SCONJ
cana-3303	136	2	the	the	DET
cana-3303	136	3	𝑓𝑘	𝑓𝑘	NOUN
cana-3303	136	4	given	give	VERB
cana-3303	136	5	in	in	ADP
cana-3303	136	6	equation	equation	NOUN
cana-3303	136	7	(	(	PUNCT
cana-3303	136	8	1	1	X
cana-3303	136	9	)	)	PUNCT
cana-3303	136	10	are	be	AUX
cana-3303	136	11	positive	positive	ADJ
cana-3303	136	12	numbers	number	NOUN
cana-3303	136	13	,	,	PUNCT
cana-3303	136	14	then	then	ADV
cana-3303	136	15	this	this	DET
cana-3303	136	16	equation	equation	NOUN
cana-3303	136	17	has	have	VERB
cana-3303	136	18	a	a	DET
cana-3303	136	19	general	general	ADJ
cana-3303	136	20	(	(	PUNCT
cana-3303	136	21	dependent	dependent	ADJ
cana-3303	136	22	on	on	ADP
cana-3303	136	23	three	three	NUM
cana-3303	136	24	arbitrary	arbitrary	ADJ
cana-3303	136	25	constants	constant	NOUN
cana-3303	136	26	)	)	PUNCT
cana-3303	136	27	solution	solution	NOUN
cana-3303	136	28	,	,	PUNCT
cana-3303	136	29	and	and	CCONJ
cana-3303	136	30	this	this	DET
cana-3303	136	31	solution	solution	NOUN
cana-3303	136	32	is	be	AUX
cana-3303	136	33	given	give	VERB
cana-3303	136	34	by	by	ADP
cana-3303	136	35	(	(	PUNCT
cana-3303	136	36	10𝑛−1	10𝑛−1	NOUN
cana-3303	136	37	)	)	PUNCT
cana-3303	136	38	,	,	PUNCT
cana-3303	136	39	(	(	PUNCT
cana-3303	136	40	8)	8)	NUM
cana-3303	136	41	and	and	CCONJ
cana-3303	136	42	(	(	PUNCT
cana-3303	136	43	4	4	NUM
cana-3303	136	44	)	)	PUNCT
cana-3303	136	45	,	,	PUNCT
cana-3303	136	46	so	so	ADV
cana-3303	136	47	𝑦0	𝑦0	NOUN
cana-3303	136	48	,	,	PUNCT
cana-3303	136	49	𝑦1	𝑦1	PROPN
cana-3303	136	50	and	and	CCONJ
cana-3303	136	51	𝑦2	𝑦2	NOUN
cana-3303	136	52	are	be	AUX
cana-3303	136	53	arbitrary	arbitrary	ADJ
cana-3303	136	54	constants	constant	NOUN
cana-3303	136	55	.	.	PUNCT
cana-3303	137	1	cauchy	cauchy	ADJ
cana-3303	137	2	problem	problem	NOUN
cana-3303	137	3	:	:	PUNCT
cana-3303	137	4	in	in	ADP
cana-3303	137	5	view	view	NOUN
cana-3303	137	6	that	that	SCONJ
cana-3303	137	7	the	the	DET
cana-3303	137	8	given	give	VERB
cana-3303	137	9	equation	equation	NOUN
cana-3303	137	10	(	(	PUNCT
cana-3303	137	11	1	1	NUM
cana-3303	137	12	)	)	PUNCT
cana-3303	137	13	is	be	AUX
cana-3303	137	14	third	third	ADJ
cana-3303	137	15	-	-	PUNCT
cana-3303	137	16	order	order	NOUN
cana-3303	137	17	,	,	PUNCT
cana-3303	137	18	let	let	VERB
cana-3303	137	19	us	we	PRON
cana-3303	137	20	give	give	VERB
cana-3303	137	21	the	the	DET
cana-3303	137	22	following	following	ADJ
cana-3303	137	23	initial	initial	ADJ
cana-3303	137	24	conditions	condition	NOUN
cana-3303	137	25	for	for	ADP
cana-3303	137	26	this	this	DET
cana-3303	137	27	equation	equation	NOUN
cana-3303	137	28	:	:	PUNCT
cana-3303	137	29	𝑦𝑘=𝛼𝑘	𝑦𝑘=𝛼𝑘	NUM
cana-3303	137	30	,	,	PUNCT
cana-3303	137	31	k=0,2̅̅	k=0,2̅̅	PROPN
cana-3303	137	32	̅̅	̅̅	PROPN
cana-3303	137	33	,	,	PUNCT
cana-3303	137	34	(	(	PUNCT
cana-3303	137	35	11	11	NUM
cana-3303	137	36	)	)	PUNCT
cana-3303	137	37	then	then	ADV
cana-3303	137	38	,	,	PUNCT
cana-3303	137	39	(	(	PUNCT
cana-3303	137	40	1	1	NUM
cana-3303	137	41	)	)	PUNCT
cana-3303	137	42	,	,	PUNCT
cana-3303	137	43	(	(	PUNCT
cana-3303	137	44	11	11	NUM
cana-3303	137	45	)	)	PUNCT
cana-3303	137	46	then	then	ADV
cana-3303	137	47	we	we	PRON
cana-3303	137	48	may	may	AUX
cana-3303	137	49	get	get	VERB
cana-3303	137	50	the	the	DET
cana-3303	137	51	solution	solution	NOUN
cana-3303	137	52	of	of	ADP
cana-3303	137	53	cauchy	cauchy	ADJ
cana-3303	137	54	problem	problem	NOUN
cana-3303	137	55	from	from	ADP
cana-3303	137	56	(	(	PUNCT
cana-3303	137	57	10𝑛−1	10𝑛−1	NOUN
cana-3303	137	58	)	)	PUNCT
cana-3303	137	59	as	as	ADP
cana-3303	137	60	:	:	PUNCT
cana-3303	137	61	𝑦𝑛=𝛼2	𝑦𝑛=𝛼2	PROPN
cana-3303	137	62	∙	∙	PROPN
cana-3303	137	63	∏	∏	PROPN
cana-3303	137	64	ℎ𝑠	ℎ𝑠	ADP
cana-3303	137	65	𝑛−1	𝑛−1	PROPN
cana-3303	137	66	𝑠=2	𝑠=2	PROPN
cana-3303	137	67	,	,	PUNCT
cana-3303	137	68	n≥	n≥	PROPN
cana-3303	137	69	3	3	NUM
cana-3303	137	70	,	,	PUNCT
cana-3303	137	71	(	(	PUNCT
cana-3303	137	72	12	12	NUM
cana-3303	137	73	)	)	PUNCT
cana-3303	137	74	where	where	SCONJ
cana-3303	137	75	ℎ𝑠	ℎ𝑠	ADV
cana-3303	137	76	by	by	ADP
cana-3303	137	77	means	mean	NOUN
cana-3303	137	78	of	of	ADP
cana-3303	137	79	(	(	PUNCT
cana-3303	137	80	8)	8)	NUM
cana-3303	137	81	will	will	AUX
cana-3303	137	82	be	be	AUX
cana-3303	137	83	defined	define	VERB
cana-3303	137	84	as	as	ADP
cana-3303	137	85	:	:	PUNCT
cana-3303	137	86	ℎ𝑠	ℎ𝑠	INTJ
cana-3303	137	87	(	(	PUNCT
cana-3303	137	88	𝑦1	𝑦1	PROPN
cana-3303	137	89	[	[	X
cana-3303	137	90	𝐼	𝐼	PROPN
cana-3303	137	91	]	]	PUNCT
cana-3303	137	92	,	,	PUNCT
cana-3303	137	93	𝑔𝑠)=	𝑔𝑠)=	NOUN
cana-3303	137	94	𝑔𝑛−1	𝑔𝑛−1	NOUN
cana-3303	137	95	𝑔𝑛−2	𝑔𝑛−2	PROPN
cana-3303	137	96	⋰	⋰	NOUN
cana-3303	137	97	𝑔	𝑔	PROPN
cana-3303	137	98	2	2	NUM
cana-3303	137	99	𝑔	𝑔	SYM
cana-3303	137	100	1	1	NUM
cana-3303	137	101	𝛼2	𝛼2	PROPN
cana-3303	137	102	𝛼1	𝛼1	NOUN
cana-3303	137	103	,	,	PUNCT
cana-3303	137	104	n≥	n≥	PROPN
cana-3303	137	105	2	2	NUM
cana-3303	137	106	,	,	PUNCT
cana-3303	137	107	(	(	PUNCT
cana-3303	137	108	13	13	NUM
cana-3303	137	109	)	)	PUNCT
cana-3303	137	110	finally	finally	ADV
cana-3303	137	111	,	,	PUNCT
cana-3303	137	112	𝑔𝑠	𝑔𝑠	NOUN
cana-3303	137	113	are	be	AUX
cana-3303	137	114	defined	define	VERB
cana-3303	137	115	by	by	ADP
cana-3303	137	116	(	(	PUNCT
cana-3303	137	117	4	4	NUM
cana-3303	137	118	)	)	PUNCT
cana-3303	137	119	as	as	SCONJ
cana-3303	137	120	follows	follow	VERB
cana-3303	137	121	:	:	PUNCT
cana-3303	137	122	𝑔𝑛((𝑦0	𝑔𝑛((𝑦0	VERB
cana-3303	137	123	[	[	X
cana-3303	137	124	𝐼	𝐼	NOUN
cana-3303	137	125	]	]	PUNCT
cana-3303	137	126	)	)	PUNCT
cana-3303	137	127	{	{	PUNCT
cana-3303	137	128	𝐼	𝐼	NOUN
cana-3303	137	129	}	}	PUNCT
cana-3303	137	130	,	,	PUNCT
cana-3303	137	131	𝑓𝑠)=	𝑓𝑠)=	ADJ
cana-3303	137	132	√	√	NUM
cana-3303	137	133	𝛼2	𝛼2	PROPN
cana-3303	137	134	𝛼1	𝛼1	NOUN
cana-3303	137	135	𝛼1	𝛼1	NOUN
cana-3303	137	136	𝛼0	𝛼0	NOUN
cana-3303	138	1	+	+	CCONJ
cana-3303	138	2	∑	∑	ADP
cana-3303	138	3	𝑓𝑠	𝑓𝑠	NOUN
cana-3303	138	4	𝑛−1	𝑛−1	PROPN
cana-3303	138	5	𝑠=0	𝑠=0	PUNCT
cana-3303	138	6	,	,	PUNCT
cana-3303	138	7	𝑛	𝑛	PRON
cana-3303	138	8	≥	≥	NOUN
cana-3303	138	9	1	1	NUM
cana-3303	138	10	.	.	PUNCT
cana-3303	139	1	(	(	PUNCT
cana-3303	139	2	14	14	NUM
cana-3303	139	3	)	)	PUNCT
cana-3303	140	1	so	so	ADV
cana-3303	140	2	,	,	PUNCT
cana-3303	140	3	we	we	PRON
cana-3303	140	4	get	get	VERB
cana-3303	140	5	the	the	DET
cana-3303	140	6	following	follow	VERB
cana-3303	140	7	solution	solution	NOUN
cana-3303	140	8	:	:	PUNCT
cana-3303	140	9	theorem	theorem	NOUN
cana-3303	140	10	2	2	NUM
cana-3303	140	11	.	.	PUNCT
cana-3303	141	1	under	under	ADP
cana-3303	141	2	the	the	DET
cana-3303	141	3	terms	term	NOUN
cana-3303	141	4	of	of	ADP
cana-3303	141	5	theorem	theorem	NOUN
cana-3303	141	6	1	1	NUM
cana-3303	141	7	,	,	PUNCT
cana-3303	141	8	if	if	SCONJ
cana-3303	141	9	𝛼𝑘	𝛼𝑘	NOUN
cana-3303	141	10	,	,	PUNCT
cana-3303	141	11	k=0,2̅̅	k=0,2̅̅	PROPN
cana-3303	141	12	̅̅	̅̅	PROPN
cana-3303	141	13	_	_	NOUN
cana-3303	141	14	are	be	AUX
cana-3303	141	15	positive	positive	ADJ
cana-3303	141	16	numbers	number	NOUN
cana-3303	141	17	,	,	PUNCT
cana-3303	141	18	then	then	ADV
cana-3303	141	19	there	there	PRON
cana-3303	141	20	is	be	VERB
cana-3303	141	21	only	only	ADJ
cana-3303	141	22	solution	solution	NOUN
cana-3303	141	23	for	for	ADP
cana-3303	141	24	(	(	PUNCT
cana-3303	141	25	1	1	NUM
cana-3303	141	26	)	)	PUNCT
cana-3303	141	27	,	,	PUNCT
cana-3303	141	28	(	(	PUNCT
cana-3303	141	29	11	11	X
cana-3303	141	30	)	)	PUNCT
cana-3303	141	31	cauchy	cauchy	NOUN
cana-3303	141	32	problem	problem	NOUN
cana-3303	141	33	and	and	CCONJ
cana-3303	141	34	this	this	DET
cana-3303	141	35	solution	solution	NOUN
cana-3303	141	36	is	be	AUX
cana-3303	141	37	given	give	VERB
cana-3303	141	38	by	by	ADP
cana-3303	141	39	(	(	PUNCT
cana-3303	141	40	12	12	NUM
cana-3303	141	41	)	)	PUNCT
cana-3303	141	42	,	,	PUNCT
cana-3303	141	43	such	such	ADJ
cana-3303	141	44	that	that	SCONJ
cana-3303	141	45	that	that	DET
cana-3303	141	46	ℎ𝑠	ℎ𝑠	NOUN
cana-3303	141	47	are	be	AUX
cana-3303	141	48	given	give	VERB
cana-3303	141	49	by	by	ADP
cana-3303	141	50	(	(	PUNCT
cana-3303	141	51	13	13	NUM
cana-3303	141	52	)	)	PUNCT
cana-3303	141	53	and	and	CCONJ
cana-3303	141	54	𝑔𝑠	𝑔𝑠	VERB
cana-3303	141	55	by	by	ADP
cana-3303	141	56	(	(	PUNCT
cana-3303	141	57	14	14	NUM
cana-3303	141	58	)	)	PUNCT
cana-3303	141	59	.	.	PUNCT
cana-3303	142	1	boundary	boundary	ADJ
cana-3303	142	2	issues	issue	NOUN
cana-3303	142	3	:	:	PUNCT
cana-3303	142	4	now	now	ADV
cana-3303	142	5	,	,	PUNCT
cana-3303	142	6	by	by	ADP
cana-3303	142	7	looking	look	VERB
cana-3303	142	8	at	at	ADP
cana-3303	142	9	equation	equation	NOUN
cana-3303	142	10	(	(	PUNCT
cana-3303	142	11	1	1	NUM
cana-3303	142	12	)	)	PUNCT
cana-3303	142	13	in	in	ADP
cana-3303	142	14	0≤	0≤	NUM
cana-3303	142	15	𝑛	𝑛	DET
cana-3303	142	16	≤	≤	ADV
cana-3303	143	1	𝑁	𝑁	PROPN
cana-3303	143	2	−	−	PROPN
cana-3303	143	3	3	3	NUM
cana-3303	143	4	,	,	PUNCT
cana-3303	143	5	let	let	VERB
cana-3303	143	6	's	us	PRON
cana-3303	143	7	look	look	VERB
cana-3303	143	8	at	at	ADP
cana-3303	143	9	the	the	DET
cana-3303	143	10	issue	issue	NOUN
cana-3303	143	11	within	within	ADP
cana-3303	143	12	the	the	DET
cana-3303	143	13	following	follow	VERB
cana-3303	143	14	boundary	boundary	ADJ
cana-3303	143	15	conditions	condition	NOUN
cana-3303	143	16	:	:	PUNCT
cana-3303	143	17	communications	communication	NOUN
cana-3303	143	18	on	on	ADP
cana-3303	143	19	applied	apply	VERB
cana-3303	143	20	nonlinear	nonlinear	ADJ
cana-3303	143	21	analysis	analysis	NOUN
cana-3303	143	22	issn	issn	NOUN
cana-3303	143	23	:	:	PUNCT
cana-3303	143	24	1074	1074	NUM
cana-3303	143	25	-	-	PUNCT
cana-3303	143	26	133x	133x	NUM
cana-3303	143	27	vol	vol	NOUN
cana-3303	143	28	32	32	NUM
cana-3303	143	29	no	no	NOUN
cana-3303	143	30	.	.	PUNCT
cana-3303	144	1	6s	6s	NUM
cana-3303	144	2	(	(	PUNCT
cana-3303	144	3	2025	2025	NUM
cana-3303	144	4	)	)	PUNCT
cana-3303	144	5	386	386	NUM
cana-3303	144	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	144	7	(	(	PUNCT
cana-3303	144	8	𝑦0	𝑦0	NOUN
cana-3303	144	9	[	[	X
cana-3303	144	10	𝐼	𝐼	PROPN
cana-3303	144	11	]	]	PUNCT
cana-3303	144	12	)	)	PUNCT
cana-3303	144	13	{	{	PUNCT
cana-3303	144	14	𝐼}=	𝐼}=	NOUN
cana-3303	144	15	𝛽0	𝛽0	PROPN
cana-3303	144	16	,	,	PUNCT
cana-3303	144	17	𝑦1	𝑦1	PROPN
cana-3303	144	18	[	[	X
cana-3303	144	19	𝐼	𝐼	PROPN
cana-3303	144	20	]	]	PUNCT
cana-3303	144	21	=	=	SYM
cana-3303	144	22	𝛽1	𝛽1	NOUN
cana-3303	144	23	,	,	PUNCT
cana-3303	144	24	𝑦𝑁	𝑦𝑁	NOUN
cana-3303	144	25	=	=	SYM
cana-3303	144	26	𝛽2	𝛽2	PROPN
cana-3303	144	27	,	,	PUNCT
cana-3303	144	28	(	(	PUNCT
cana-3303	144	29	15	15	NUM
cana-3303	144	30	)	)	PUNCT
cana-3303	144	31	where	where	SCONJ
cana-3303	144	32	𝛽0	𝛽0	NOUN
cana-3303	144	33	,	,	PUNCT
cana-3303	144	34	𝛽1	𝛽1	NOUN
cana-3303	144	35	və	və	DET
cana-3303	144	36	𝛽2	𝛽2	NOUN
cana-3303	144	37	are	be	AUX
cana-3303	144	38	constants	constant	NOUN
cana-3303	144	39	.	.	PUNCT
cana-3303	145	1	then	then	ADV
cana-3303	145	2	we	we	PRON
cana-3303	145	3	get	get	VERB
cana-3303	145	4	from	from	ADP
cana-3303	145	5	(	(	PUNCT
cana-3303	145	6	4	4	NUM
cana-3303	145	7	):	):	PUNCT
cana-3303	145	8	𝑔𝑛((𝑦0	𝑔𝑛((𝑦0	NOUN
cana-3303	146	1	[	[	X
cana-3303	146	2	𝐼	𝐼	NOUN
cana-3303	146	3	]	]	PUNCT
cana-3303	146	4	)	)	PUNCT
cana-3303	146	5	{	{	PUNCT
cana-3303	146	6	𝐼	𝐼	PROPN
cana-3303	146	7	}	}	PUNCT
cana-3303	146	8	,	,	PUNCT
cana-3303	146	9	𝑓𝑠	𝑓𝑠	NOUN
cana-3303	146	10	)	)	PUNCT
cana-3303	146	11	≡	≡	PROPN
cana-3303	146	12	𝑔𝑛(𝛽0	𝑔𝑛(𝛽0	NUM
cana-3303	146	13	,	,	PUNCT
cana-3303	146	14	𝑓𝑠	𝑓𝑠	NOUN
cana-3303	146	15	)	)	PUNCT
cana-3303	146	16	=	=	X
cana-3303	146	17	𝛽0	𝛽0	ADJ
cana-3303	146	18	+	+	CCONJ
cana-3303	146	19	∑	∑	NOUN
cana-3303	146	20	𝑓𝑠	𝑓𝑠	NOUN
cana-3303	146	21	𝑛−1	𝑛−1	PROPN
cana-3303	146	22	𝑠=0	𝑠=0	PUNCT
cana-3303	146	23	,	,	PUNCT
cana-3303	146	24	n≥	n≥	PROPN
cana-3303	146	25	1	1	NUM
cana-3303	146	26	,	,	PUNCT
cana-3303	146	27	(	(	PUNCT
cana-3303	146	28	16	16	NUM
cana-3303	146	29	)	)	PUNCT
cana-3303	146	30	so	so	ADV
cana-3303	146	31	,	,	PUNCT
cana-3303	146	32	after	after	SCONJ
cana-3303	146	33	𝑔𝑠	𝑔𝑠	PRON
cana-3303	146	34	are	be	AUX
cana-3303	146	35	defined	define	VERB
cana-3303	146	36	,	,	PUNCT
cana-3303	146	37	according	accord	VERB
cana-3303	146	38	to	to	ADP
cana-3303	146	39	(	(	PUNCT
cana-3303	146	40	8)	8)	NUM
cana-3303	146	41	:	:	PUNCT
cana-3303	146	42	ℎ𝑛(𝑦1	ℎ𝑛(𝑦1	PROPN
cana-3303	147	1	[	[	X
cana-3303	147	2	𝐼	𝐼	PROPN
cana-3303	147	3	]	]	PUNCT
cana-3303	147	4	,	,	PUNCT
cana-3303	147	5	𝑔𝑠	𝑔𝑠	PROPN
cana-3303	147	6	)	)	PUNCT
cana-3303	147	7	≡	≡	PROPN
cana-3303	147	8	ℎ𝑛(𝛽1	ℎ𝑛(𝛽1	PROPN
cana-3303	147	9	,	,	PUNCT
cana-3303	147	10	𝑔𝑠	𝑔𝑠	NOUN
cana-3303	147	11	)	)	PUNCT
cana-3303	147	12	=	=	SYM
cana-3303	147	13	𝑔𝑛−1	𝑔𝑛−1	NOUN
cana-3303	147	14	𝑔𝑛−2	𝑔𝑛−2	PROPN
cana-3303	147	15	⋰	⋰	DET
cana-3303	147	16	𝑔	𝑔	PROPN
cana-3303	147	17	2	2	NUM
cana-3303	147	18	𝑔	𝑔	PROPN
cana-3303	147	19	1	1	NUM
cana-3303	147	20	𝛽1	𝛽1	NOUN
cana-3303	147	21	,	,	PUNCT
cana-3303	147	22	n≥	n≥	PROPN
cana-3303	147	23	2	2	NUM
cana-3303	147	24	.	.	PUNCT
cana-3303	147	25	(	(	PUNCT
cana-3303	147	26	17	17	NUM
cana-3303	147	27	)	)	PUNCT
cana-3303	147	28	finally	finally	ADV
cana-3303	147	29	,	,	PUNCT
cana-3303	147	30	after	after	SCONJ
cana-3303	147	31	the	the	DET
cana-3303	147	32	ℎ𝑠	ℎ𝑠	NOUN
cana-3303	147	33	are	be	AUX
cana-3303	147	34	defined	define	VERB
cana-3303	147	35	,	,	PUNCT
cana-3303	147	36	we	we	PRON
cana-3303	147	37	will	will	AUX
cana-3303	147	38	get	get	VERB
cana-3303	147	39	the	the	DET
cana-3303	147	40	solution	solution	NOUN
cana-3303	147	41	of	of	ADP
cana-3303	147	42	boundary	boundary	ADJ
cana-3303	147	43	(	(	PUNCT
cana-3303	147	44	1	1	NUM
cana-3303	147	45	)	)	PUNCT
cana-3303	147	46	,	,	PUNCT
cana-3303	147	47	(	(	PUNCT
cana-3303	147	48	15	15	X
cana-3303	147	49	)	)	PUNCT
cana-3303	147	50	issues	issue	NOUN
cana-3303	147	51	from	from	ADP
cana-3303	147	52	(	(	PUNCT
cana-3303	147	53	10𝑛−1	10𝑛−1	NOUN
cana-3303	147	54	)	)	PUNCT
cana-3303	147	55	as	as	ADP
cana-3303	147	56	:	:	PUNCT
cana-3303	147	57	𝑦𝑛=	𝑦𝑛=	NOUN
cana-3303	147	58	𝑦2	𝑦2	PROPN
cana-3303	147	59	∙	∙	PROPN
cana-3303	147	60	∏	∏	PROPN
cana-3303	147	61	ℎ𝑠	ℎ𝑠	VERB
cana-3303	147	62	𝑛−1	𝑛−1	PROPN
cana-3303	147	63	𝑠=2	𝑠=2	PROPN
cana-3303	147	64	,	,	PUNCT
cana-3303	147	65	here	here	ADV
cana-3303	147	66	𝑦2	𝑦2	PROPN
cana-3303	147	67	is	be	AUX
cana-3303	147	68	arbitrary	arbitrary	ADJ
cana-3303	147	69	constant	constant	ADJ
cana-3303	147	70	,	,	PUNCT
cana-3303	147	71	to	to	PART
cana-3303	147	72	define	define	VERB
cana-3303	147	73	it	it	PRON
cana-3303	147	74	,	,	PUNCT
cana-3303	147	75	considering	consider	VERB
cana-3303	147	76	(	(	PUNCT
cana-3303	147	77	10𝑛−1	10𝑛−1	NOUN
cana-3303	147	78	)	)	PUNCT
cana-3303	147	79	with	with	ADP
cana-3303	147	80	last	last	ADJ
cana-3303	147	81	of	of	ADP
cana-3303	147	82	(	(	PUNCT
cana-3303	147	83	15	15	NUM
cana-3303	147	84	)	)	PUNCT
cana-3303	147	85	boundary	boundary	ADJ
cana-3303	147	86	conditions	condition	NOUN
cana-3303	147	87	,	,	PUNCT
cana-3303	147	88	we	we	PRON
cana-3303	147	89	can	can	AUX
cana-3303	147	90	define	define	VERB
cana-3303	147	91	𝑦2	𝑦2	NOUN
cana-3303	147	92	with	with	ADP
cana-3303	147	93	equation	equation	NOUN
cana-3303	147	94	:	:	PUNCT
cana-3303	147	95	𝛽2	𝛽2	PROPN
cana-3303	147	96	=	=	SYM
cana-3303	147	97	𝑦𝑁=	𝑦𝑁=	PROPN
cana-3303	147	98	𝑦2	𝑦2	NOUN
cana-3303	147	99	∙	∙	PROPN
cana-3303	147	100	∏	∏	PROPN
cana-3303	147	101	ℎ𝑠	ℎ𝑠	VERB
cana-3303	147	102	𝑁−1	𝑁−1	NOUN
cana-3303	147	103	𝑠=2	𝑠=2	PROPN
cana-3303	147	104	,	,	PUNCT
cana-3303	147	105	(	(	PUNCT
cana-3303	147	106	18	18	NUM
cana-3303	147	107	)	)	PUNCT
cana-3303	147	108	from	from	ADP
cana-3303	147	109	this	this	DET
cana-3303	147	110	equation	equation	NOUN
cana-3303	147	111	we	we	PRON
cana-3303	147	112	get	get	VERB
cana-3303	147	113	the	the	DET
cana-3303	147	114	expression	expression	NOUN
cana-3303	147	115	𝑦2	𝑦2	NOUN
cana-3303	147	116	=	=	SYM
cana-3303	147	117	𝛽2	𝛽2	PROPN
cana-3303	147	118	∏	∏	PROPN
cana-3303	147	119	ℎ𝑠	ℎ𝑠	ADP
cana-3303	147	120	𝑁−1	𝑁−1	NOUN
cana-3303	147	121	𝑠=2	𝑠=2	PROPN
cana-3303	147	122	,	,	PUNCT
cana-3303	147	123	(	(	PUNCT
cana-3303	147	124	19	19	NUM
cana-3303	147	125	)	)	PUNCT
cana-3303	147	126	by	by	ADP
cana-3303	147	127	writing	write	VERB
cana-3303	147	128	this	this	DET
cana-3303	147	129	value	value	NOUN
cana-3303	147	130	in	in	ADP
cana-3303	147	131	(	(	PUNCT
cana-3303	147	132	10𝑛−1	10𝑛−1	NOUN
cana-3303	147	133	)	)	PUNCT
cana-3303	147	134	we	we	PRON
cana-3303	147	135	get	get	VERB
cana-3303	147	136	the	the	DET
cana-3303	147	137	solution	solution	NOUN
cana-3303	147	138	of	of	ADP
cana-3303	147	139	boundary	boundary	ADJ
cana-3303	147	140	problems	problem	NOUN
cana-3303	147	141	(	(	PUNCT
cana-3303	147	142	1	1	NUM
cana-3303	147	143	)	)	PUNCT
cana-3303	147	144	,	,	PUNCT
cana-3303	147	145	(	(	PUNCT
cana-3303	147	146	15	15	NUM
cana-3303	147	147	)	)	PUNCT
cana-3303	147	148	as	as	ADP
cana-3303	147	149	.	.	PUNCT
cana-3303	148	1	𝑦𝑛	𝑦𝑛	NOUN
cana-3303	148	2	=	=	SYM
cana-3303	148	3	𝛽2	𝛽2	PROPN
cana-3303	148	4	∏	∏	PROPN
cana-3303	148	5	ℎ𝑠	ℎ𝑠	ADP
cana-3303	148	6	𝑁−1	𝑁−1	NOUN
cana-3303	148	7	𝑠=2	𝑠=2	PROPN
cana-3303	148	8	∙	∙	PROPN
cana-3303	148	9	∏	∏	PROPN
cana-3303	148	10	ℎ𝑠	ℎ𝑠	ADP
cana-3303	148	11	𝑛−1	𝑛−1	PROPN
cana-3303	148	12	𝑠=2	𝑠=2	PROPN
cana-3303	148	13	=	=	SYM
cana-3303	148	14	𝛽2	𝛽2	PROPN
cana-3303	148	15	∏	∏	PROPN
cana-3303	148	16	ℎ𝑠	ℎ𝑠	ADP
cana-3303	148	17	𝑁−1	𝑁−1	NOUN
cana-3303	148	18	𝑠=𝑛	𝑠=𝑛	PROPN
cana-3303	148	19	,	,	PUNCT
cana-3303	148	20	(	(	PUNCT
cana-3303	148	21	20	20	NUM
cana-3303	148	22	)	)	PUNCT
cana-3303	148	23	thus	thus	ADV
cana-3303	148	24	,	,	PUNCT
cana-3303	148	25	we	we	PRON
cana-3303	148	26	get	get	VERB
cana-3303	148	27	the	the	DET
cana-3303	148	28	following	follow	VERB
cana-3303	148	29	solution	solution	NOUN
cana-3303	148	30	:	:	PUNCT
cana-3303	148	31	theorem	theorem	NOUN
cana-3303	148	32	3	3	NUM
cana-3303	148	33	.	.	PUNCT
cana-3303	149	1	under	under	ADP
cana-3303	149	2	the	the	DET
cana-3303	149	3	terms	term	NOUN
cana-3303	149	4	of	of	ADP
cana-3303	149	5	theorem	theorem	NOUN
cana-3303	149	6	1	1	NUM
cana-3303	149	7	,	,	PUNCT
cana-3303	149	8	if	if	SCONJ
cana-3303	149	9	𝛽0	𝛽0	VERB
cana-3303	149	10	,	,	PUNCT
cana-3303	149	11	𝛽1	𝛽1	NOUN
cana-3303	149	12	and	and	CCONJ
cana-3303	149	13	𝛽2	𝛽2	NOUN
cana-3303	149	14	are	be	AUX
cana-3303	149	15	real	real	ADJ
cana-3303	149	16	constants	constant	NOUN
cana-3303	149	17	,	,	PUNCT
cana-3303	149	18	then	then	ADV
cana-3303	149	19	there	there	PRON
cana-3303	149	20	is	be	VERB
cana-3303	149	21	only	only	ADJ
cana-3303	149	22	solution	solution	NOUN
cana-3303	149	23	for	for	ADP
cana-3303	149	24	(	(	PUNCT
cana-3303	149	25	1	1	NUM
cana-3303	149	26	)	)	PUNCT
cana-3303	149	27	,	,	PUNCT
cana-3303	149	28	(	(	PUNCT
cana-3303	149	29	15	15	X
cana-3303	149	30	)	)	PUNCT
cana-3303	149	31	boundary	boundary	ADJ
cana-3303	149	32	problem	problem	NOUN
cana-3303	149	33	and	and	CCONJ
cana-3303	149	34	that	that	PRON
cana-3303	149	35	is	be	AUX
cana-3303	149	36	given	give	VERB
cana-3303	149	37	by	by	ADP
cana-3303	149	38	(	(	PUNCT
cana-3303	149	39	20	20	NUM
cana-3303	149	40	)	)	PUNCT
cana-3303	149	41	,	,	PUNCT
cana-3303	149	42	so	so	SCONJ
cana-3303	149	43	that	that	SCONJ
cana-3303	149	44	ℎ𝑠	ℎ𝑠	NOUN
cana-3303	149	45	are	be	AUX
cana-3303	149	46	to	to	PART
cana-3303	149	47	be	be	AUX
cana-3303	149	48	given	give	VERB
cana-3303	149	49	by	by	ADP
cana-3303	149	50	(	(	PUNCT
cana-3303	149	51	17	17	NUM
cana-3303	149	52	)	)	PUNCT
cana-3303	149	53	and	and	CCONJ
cana-3303	149	54	𝑔𝑠	𝑔𝑠	VERB
cana-3303	149	55	by	by	ADP
cana-3303	149	56	(	(	PUNCT
cana-3303	149	57	16	16	NUM
cana-3303	149	58	)	)	PUNCT
cana-3303	149	59	.	.	PUNCT
cana-3303	150	1	note	note	NOUN
cana-3303	150	2	:	:	PUNCT
cana-3303	150	3	since	since	SCONJ
cana-3303	150	4	the	the	DET
cana-3303	150	5	solution	solution	NOUN
cana-3303	150	6	of	of	ADP
cana-3303	150	7	third	third	ADJ
cana-3303	150	8	-	-	PUNCT
cana-3303	150	9	order	order	NOUN
cana-3303	150	10	discrete	discrete	ADJ
cana-3303	150	11	derivative	derivative	NOUN
cana-3303	150	12	is	be	AUX
cana-3303	150	13	given	give	VERB
cana-3303	150	14	by	by	ADP
cana-3303	150	15	the	the	DET
cana-3303	150	16	general	general	ADJ
cana-3303	150	17	solution	solution	NOUN
cana-3303	150	18	of	of	ADP
cana-3303	150	19	(	(	PUNCT
cana-3303	150	20	4	4	NUM
cana-3303	150	21	)	)	PUNCT
cana-3303	150	22	,	,	PUNCT
cana-3303	150	23	(	(	PUNCT
cana-3303	150	24	8)	8)	NUM
cana-3303	150	25	and	and	CCONJ
cana-3303	150	26	(	(	PUNCT
cana-3303	150	27	10𝑛−1	10𝑛−1	NOUN
cana-3303	150	28	)	)	PUNCT
cana-3303	150	29	in	in	ADP
cana-3303	150	30	the	the	DET
cana-3303	150	31	boundary	boundary	ADJ
cana-3303	150	32	problem	problem	NOUN
cana-3303	150	33	considered	consider	VERB
cana-3303	150	34	for	for	ADP
cana-3303	150	35	this	this	DET
cana-3303	150	36	equation(𝑦0	equation(𝑦0	NOUN
cana-3303	151	1	[	[	X
cana-3303	151	2	𝐼	𝐼	PROPN
cana-3303	151	3	]	]	PUNCT
cana-3303	151	4	)	)	PUNCT
cana-3303	151	5	{	{	PUNCT
cana-3303	151	6	𝐼	𝐼	PROPN
cana-3303	151	7	}	}	PUNCT
cana-3303	151	8	,	,	PUNCT
cana-3303	151	9	𝑦1	𝑦1	PROPN
cana-3303	151	10	[	[	X
cana-3303	151	11	𝐼	𝐼	PROPN
cana-3303	151	12	]	]	PUNCT
cana-3303	151	13	and	and	CCONJ
cana-3303	151	14	𝑦2	𝑦2	PROPN
cana-3303	151	15	arbitrary	arbitrary	ADJ
cana-3303	151	16	constants	constant	NOUN
cana-3303	151	17	are	be	AUX
cana-3303	151	18	to	to	PART
cana-3303	151	19	be	be	AUX
cana-3303	151	20	defined	define	VERB
cana-3303	151	21	.	.	PUNCT
cana-3303	152	1	unresolved	unresolved	ADJ
cana-3303	152	2	issues	issue	NOUN
cana-3303	152	3	.	.	PUNCT
cana-3303	153	1	1	1	X
cana-3303	153	2	.	.	PUNCT
cana-3303	153	3	given	give	VERB
cana-3303	153	4	equation	equation	NOUN
cana-3303	153	5	(	(	PUNCT
cana-3303	153	6	1	1	NUM
cana-3303	153	7	)	)	PUNCT
cana-3303	153	8	,	,	PUNCT
cana-3303	153	9	solve	solve	VERB
cana-3303	153	10	within	within	ADP
cana-3303	153	11	the	the	DET
cana-3303	153	12	boundary	boundary	ADJ
cana-3303	153	13	conditions	condition	NOUN
cana-3303	153	14	of	of	ADP
cana-3303	153	15	𝑦0=	𝑦0=	ADJ
cana-3303	153	16	𝑝0	𝑝0	NOUN
cana-3303	153	17	,	,	PUNCT
cana-3303	153	18	𝑦1	𝑦1	NOUN
cana-3303	153	19	=	=	NOUN
cana-3303	153	20	𝑝1	𝑝1	NOUN
cana-3303	153	21	,	,	PUNCT
cana-3303	153	22	𝑦𝑛	𝑦𝑛	NOUN
cana-3303	153	23	=	=	NOUN
cana-3303	153	24	𝑝2	𝑝2	NOUN
cana-3303	153	25	,	,	PUNCT
cana-3303	153	26	2	2	NUM
cana-3303	153	27	.	.	PUNCT
cana-3303	153	28	given	give	VERB
cana-3303	153	29	equation	equation	NOUN
cana-3303	153	30	(	(	PUNCT
cana-3303	153	31	1	1	NUM
cana-3303	153	32	)	)	PUNCT
cana-3303	153	33	,	,	PUNCT
cana-3303	153	34	solve	solve	VERB
cana-3303	153	35	within	within	ADP
cana-3303	153	36	the	the	DET
cana-3303	153	37	boundary	boundary	ADJ
cana-3303	153	38	conditions	condition	NOUN
cana-3303	153	39	of	of	ADP
cana-3303	153	40	𝑦0	𝑦0	NOUN
cana-3303	153	41	=	=	PUNCT
cana-3303	153	42	𝜑0	𝜑0	NOUN
cana-3303	153	43	,	,	PUNCT
cana-3303	153	44	𝑦𝑁−1	𝑦𝑁−1	VERB
cana-3303	153	45	=	=	SYM
cana-3303	153	46	𝜑1	𝜑1	VERB
cana-3303	153	47	,	,	PUNCT
cana-3303	153	48	𝑦𝑁	𝑦𝑁	NOUN
cana-3303	153	49	=	=	PUNCT
cana-3303	153	50	𝜑2	𝜑2	PROPN
cana-3303	153	51	,	,	PUNCT
cana-3303	153	52	3	3	X
cana-3303	153	53	.	.	PUNCT
cana-3303	153	54	given	give	VERB
cana-3303	153	55	equation	equation	NOUN
cana-3303	153	56	(	(	PUNCT
cana-3303	153	57	1	1	NUM
cana-3303	153	58	)	)	PUNCT
cana-3303	153	59	,	,	PUNCT
cana-3303	153	60	solve	solve	VERB
cana-3303	153	61	within	within	ADP
cana-3303	153	62	non	non	ADJ
cana-3303	153	63	-	-	ADJ
cana-3303	153	64	local	local	ADJ
cana-3303	153	65	boundary	boundary	ADJ
cana-3303	153	66	conditions	condition	NOUN
cana-3303	153	67	𝑦0	𝑦0	NOUN
cana-3303	153	68	=	=	PUNCT
cana-3303	153	69	𝑦1	𝑦1	PROPN
cana-3303	153	70	,	,	PUNCT
cana-3303	153	71	𝑦1	𝑦1	NOUN
cana-3303	153	72	=	=	SYM
cana-3303	153	73	𝑦𝑁−1	𝑦𝑁−1	PROPN
cana-3303	153	74	,	,	PUNCT
cana-3303	153	75	𝑦2	𝑦2	PROPN
cana-3303	153	76	=	=	PUNCT
cana-3303	153	77	𝑦𝑁	𝑦𝑁	NOUN
cana-3303	153	78	,	,	PUNCT
cana-3303	153	79	communications	communication	NOUN
cana-3303	153	80	on	on	ADP
cana-3303	153	81	applied	apply	VERB
cana-3303	153	82	nonlinear	nonlinear	ADJ
cana-3303	153	83	analysis	analysis	NOUN
cana-3303	153	84	issn	issn	NOUN
cana-3303	153	85	:	:	PUNCT
cana-3303	153	86	1074	1074	NUM
cana-3303	153	87	-	-	PUNCT
cana-3303	153	88	133x	133x	NUM
cana-3303	153	89	vol	vol	NOUN
cana-3303	153	90	32	32	NUM
cana-3303	153	91	no	no	NOUN
cana-3303	153	92	.	.	PUNCT
cana-3303	154	1	6s	6s	NUM
cana-3303	154	2	(	(	PUNCT
cana-3303	154	3	2025	2025	NUM
cana-3303	154	4	)	)	PUNCT
cana-3303	154	5	387	387	NUM
cana-3303	154	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	154	7	4	4	NUM
cana-3303	154	8	.	.	PUNCT
cana-3303	154	9	not	not	PART
cana-3303	154	10	just	just	ADV
cana-3303	154	11	for	for	ADP
cana-3303	154	12	𝑦𝑛	𝑦𝑛	PROPN
cana-3303	154	13	(	(	PUNCT
cana-3303	154	14	𝐼	𝐼	PROPN
cana-3303	154	15	)	)	PUNCT
cana-3303	154	16	+	+	NUM
cana-3303	154	17	𝑦𝑛	𝑦𝑛	VERB
cana-3303	154	18	[	[	X
cana-3303	154	19	𝐼	𝐼	NOUN
cana-3303	154	20	]	]	X
cana-3303	154	21	+	+	CCONJ
cana-3303	154	22	𝑦𝑛	𝑦𝑛	NOUN
cana-3303	154	23	{	{	PUNCT
cana-3303	154	24	𝐼	𝐼	PROPN
cana-3303	154	25	}	}	PUNCT
cana-3303	154	26	=	=	SYM
cana-3303	154	27	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	154	28	,	,	PUNCT
cana-3303	154	29	n≥	n≥	PROPN
cana-3303	154	30	0	0	NUM
cana-3303	154	31	,	,	PUNCT
cana-3303	154	32	but	but	CCONJ
cana-3303	154	33	even	even	ADV
cana-3303	154	34	for	for	ADP
cana-3303	154	35	equation	equation	NOUN
cana-3303	154	36	𝑦𝑛	𝑦𝑛	ADP
cana-3303	154	37	[	[	X
cana-3303	154	38	𝐼	𝐼	X
cana-3303	154	39	]	]	X
cana-3303	154	40	∙	∙	PROPN
cana-3303	154	41	𝑦𝑛	𝑦𝑛	NOUN
cana-3303	154	42	{	{	PUNCT
cana-3303	154	43	𝐼	𝐼	PROPN
cana-3303	154	44	}	}	PUNCT
cana-3303	154	45	=	=	SYM
cana-3303	154	46	𝑓𝑛	𝑓𝑛	NOUN
cana-3303	154	47	,	,	PUNCT
cana-3303	154	48	n≥	n≥	PROPN
cana-3303	154	49	0	0	NUM
cana-3303	154	50	,	,	PUNCT
cana-3303	154	51	no	no	DET
cana-3303	154	52	issue	issue	NOUN
cana-3303	154	53	has	have	AUX
cana-3303	154	54	been	be	AUX
cana-3303	154	55	considered	consider	VERB
cana-3303	154	56	.	.	PUNCT
cana-3303	155	1	note	note	VERB
cana-3303	155	2	2	2	NUM
cana-3303	155	3	.	.	PUNCT
cana-3303	156	1	the	the	DET
cana-3303	156	2	results	result	NOUN
cana-3303	156	3	obtained	obtain	VERB
cana-3303	156	4	in	in	ADP
cana-3303	156	5	this	this	DET
cana-3303	156	6	field	field	NOUN
cana-3303	156	7	are	be	AUX
cana-3303	156	8	new	new	ADJ
cana-3303	156	9	in	in	ADP
cana-3303	156	10	any	any	DET
cana-3303	156	11	matter	matter	NOUN
cana-3303	156	12	.	.	PUNCT
cana-3303	157	1	6	6	X
cana-3303	157	2	.	.	X
cana-3303	157	3	discussion	discussion	NOUN
cana-3303	157	4	the	the	DET
cana-3303	157	5	main	main	ADJ
cana-3303	157	6	outcomes	outcome	NOUN
cana-3303	157	7	of	of	ADP
cana-3303	157	8	the	the	DET
cana-3303	157	9	study	study	NOUN
cana-3303	157	10	include	include	VERB
cana-3303	157	11	:	:	PUNCT
cana-3303	157	12	transforming	transform	VERB
cana-3303	157	13	the	the	DET
cana-3303	157	14	original	original	ADJ
cana-3303	157	15	equation	equation	NOUN
cana-3303	157	16	the	the	DET
cana-3303	157	17	original	original	ADJ
cana-3303	157	18	third	third	ADJ
cana-3303	157	19	order	order	NOUN
cana-3303	157	20	nonlinear	nonlinear	ADJ
cana-3303	157	21	equation	equation	NOUN
cana-3303	157	22	was	be	AUX
cana-3303	157	23	chosen	choose	VERB
cana-3303	157	24	and	and	CCONJ
cana-3303	157	25	virtue	virtue	NOUN
cana-3303	157	26	of	of	ADP
cana-3303	157	27	discrete	discrete	ADJ
cana-3303	157	28	additive	additive	ADJ
cana-3303	157	29	derivatives	derivative	NOUN
cana-3303	157	30	was	be	AUX
cana-3303	157	31	shifted	shift	VERB
cana-3303	157	32	into	into	ADP
cana-3303	157	33	second	second	ADJ
cana-3303	157	34	order	order	NOUN
cana-3303	157	35	equation	equation	NOUN
cana-3303	157	36	.	.	PUNCT
cana-3303	158	1	this	this	DET
cana-3303	158	2	transformation	transformation	NOUN
cana-3303	158	3	is	be	AUX
cana-3303	158	4	advantageously	advantageously	ADV
cana-3303	158	5	simpler	simple	ADJ
cana-3303	158	6	to	to	PART
cana-3303	158	7	solve	solve	VERB
cana-3303	158	8	.	.	PUNCT
cana-3303	159	1	cauchy	cauchy	PROPN
cana-3303	159	2	problem	problem	NOUN
cana-3303	159	3	the	the	DET
cana-3303	159	4	cauchy	cauchy	ADJ
cana-3303	159	5	problem	problem	NOUN
cana-3303	159	6	was	be	AUX
cana-3303	159	7	solved	solve	VERB
cana-3303	159	8	using	use	VERB
cana-3303	159	9	suitable	suitable	ADJ
cana-3303	159	10	conditions	condition	NOUN
cana-3303	159	11	.	.	PUNCT
cana-3303	160	1	the	the	DET
cana-3303	160	2	cauchy	cauchy	PROPN
cana-3303	160	3	problem	problem	NOUN
cana-3303	160	4	was	be	AUX
cana-3303	160	5	solved	solve	VERB
cana-3303	160	6	in	in	ADP
cana-3303	160	7	general	general	ADJ
cana-3303	160	8	,	,	PUNCT
cana-3303	160	9	and	and	CCONJ
cana-3303	160	10	explicit	explicit	ADJ
cana-3303	160	11	expressions	expression	NOUN
cana-3303	160	12	were	be	AUX
cana-3303	160	13	given	give	VERB
cana-3303	160	14	depending	depend	VERB
cana-3303	160	15	on	on	ADP
cana-3303	160	16	arbitrary	arbitrary	ADJ
cana-3303	160	17	,	,	PUNCT
cana-3303	160	18	which	which	PRON
cana-3303	160	19	offer	offer	VERB
cana-3303	160	20	us	we	PRON
cana-3303	160	21	the	the	DET
cana-3303	160	22	methodology	methodology	NOUN
cana-3303	160	23	mat	mat	NOUN
cana-3303	160	24	is	be	AUX
cana-3303	160	25	possible	possible	ADJ
cana-3303	160	26	to	to	PART
cana-3303	160	27	find	find	VERB
cana-3303	160	28	ρ	ρ	NOUN
cana-3303	160	29	specific	specific	ADJ
cana-3303	160	30	.	.	PUNCT
cana-3303	161	1	boundary	boundary	ADJ
cana-3303	161	2	value	value	NOUN
cana-3303	161	3	problem	problem	NOUN
cana-3303	161	4	the	the	DET
cana-3303	161	5	boundary	boundary	ADJ
cana-3303	161	6	value	value	NOUN
cana-3303	161	7	problem	problem	NOUN
cana-3303	161	8	was	be	AUX
cana-3303	161	9	solved	solve	VERB
cana-3303	161	10	with	with	ADP
cana-3303	161	11	specific	specific	ADJ
cana-3303	161	12	boundary	boundary	ADJ
cana-3303	161	13	conditions	condition	NOUN
cana-3303	161	14	.	.	PUNCT
cana-3303	162	1	the	the	DET
cana-3303	162	2	solution	solution	NOUN
cana-3303	162	3	for	for	ADP
cana-3303	162	4	the	the	DET
cana-3303	162	5	boundary	boundary	ADJ
cana-3303	162	6	problem	problem	NOUN
cana-3303	162	7	was	be	AUX
cana-3303	162	8	provided	provide	VERB
cana-3303	162	9	in	in	ADP
cana-3303	162	10	a	a	DET
cana-3303	162	11	type	type	NOUN
cana-3303	162	12	depending	depend	VERB
cana-3303	162	13	on	on	ADP
cana-3303	162	14	the	the	DET
cana-3303	162	15	arbitrary	arbitrary	ADJ
cana-3303	162	16	constant	constant	ADJ
cana-3303	162	17	y2	y2	NOUN
cana-3303	162	18	and	and	CCONJ
cana-3303	162	19	an	an	DET
cana-3303	162	20	exact	exact	ADJ
cana-3303	162	21	formula	formula	NOUN
cana-3303	162	22	of	of	ADP
cana-3303	162	23	the	the	DET
cana-3303	162	24	boundary	boundary	ADJ
cana-3303	162	25	conditions	condition	NOUN
cana-3303	162	26	was	be	AUX
cana-3303	162	27	received	receive	VERB
cana-3303	162	28	for	for	ADP
cana-3303	162	29	calculating	calculate	VERB
cana-3303	162	30	this	this	DET
cana-3303	162	31	parameter	parameter	NOUN
cana-3303	162	32	.	.	PUNCT
cana-3303	163	1	this	this	PRON
cana-3303	163	2	enabled	enable	VERB
cana-3303	163	3	an	an	DET
cana-3303	163	4	analytical	analytical	ADJ
cana-3303	163	5	approach	approach	NOUN
cana-3303	163	6	to	to	PART
cana-3303	163	7	be	be	AUX
cana-3303	163	8	taken	take	VERB
cana-3303	163	9	for	for	ADP
cana-3303	163	10	solving	solve	VERB
cana-3303	163	11	complex	complex	ADJ
cana-3303	163	12	nonlinear	nonlinear	ADJ
cana-3303	163	13	discrete	discrete	ADJ
cana-3303	163	14	equations	equation	NOUN
cana-3303	163	15	.	.	PUNCT
cana-3303	164	1	using	use	VERB
cana-3303	164	2	discrete	discrete	ADJ
cana-3303	164	3	poverative	poverative	ADJ
cana-3303	164	4	derivatives	derivative	NOUN
cana-3303	164	5	,	,	PUNCT
cana-3303	164	6	the	the	DET
cana-3303	164	7	research	research	NOUN
cana-3303	164	8	transformed	transform	VERB
cana-3303	164	9	the	the	DET
cana-3303	164	10	original	original	ADJ
cana-3303	164	11	problem	problem	NOUN
cana-3303	164	12	into	into	ADP
cana-3303	164	13	a	a	DET
cana-3303	164	14	simpler	simple	ADJ
cana-3303	164	15	one	one	NOUN
cana-3303	164	16	,	,	PUNCT
cana-3303	164	17	leading	lead	VERB
cana-3303	164	18	to	to	ADP
cana-3303	164	19	an	an	DET
cana-3303	164	20	efficient	efficient	ADJ
cana-3303	164	21	solution	solution	NOUN
cana-3303	164	22	derivation	derivation	NOUN
cana-3303	164	23	approach	approach	NOUN
cana-3303	164	24	.	.	PUNCT
cana-3303	165	1	discrete	discrete	ADJ
cana-3303	165	2	multiplicative	multiplicative	ADJ
cana-3303	165	3	derivative	derivative	ADJ
cana-3303	165	4	equations	equation	NOUN
cana-3303	165	5	the	the	DET
cana-3303	165	6	results	result	NOUN
cana-3303	165	7	also	also	ADV
cana-3303	165	8	offer	offer	VERB
cana-3303	165	9	fresh	fresh	ADJ
cana-3303	165	10	perspectives	perspective	NOUN
cana-3303	165	11	on	on	ADP
cana-3303	165	12	the	the	DET
cana-3303	165	13	dynamics	dynamic	NOUN
cana-3303	165	14	of	of	ADP
cana-3303	165	15	such	such	ADJ
cana-3303	165	16	discrete	discrete	ADJ
cana-3303	165	17	multiplicative	multiplicative	ADJ
cana-3303	165	18	derivative	derivative	ADJ
cana-3303	165	19	equations	equation	NOUN
cana-3303	165	20	.	.	PUNCT
cana-3303	166	1	these	these	DET
cana-3303	166	2	results	result	NOUN
cana-3303	166	3	are	be	AUX
cana-3303	166	4	relevant	relevant	ADJ
cana-3303	166	5	to	to	ADP
cana-3303	166	6	the	the	DET
cana-3303	166	7	study	study	NOUN
cana-3303	166	8	of	of	ADP
cana-3303	166	9	nonlinear	nonlinear	ADJ
cana-3303	166	10	discrete	discrete	ADJ
cana-3303	166	11	systems	system	NOUN
cana-3303	166	12	and	and	CCONJ
cana-3303	166	13	challenge	challenge	VERB
cana-3303	166	14	the	the	DET
cana-3303	166	15	possibility	possibility	NOUN
cana-3303	166	16	of	of	ADP
cana-3303	166	17	future	future	ADJ
cana-3303	166	18	work	work	NOUN
cana-3303	166	19	in	in	ADP
cana-3303	166	20	such	such	ADJ
cana-3303	166	21	equations	equation	NOUN
cana-3303	166	22	.	.	PUNCT
cana-3303	167	1	first	first	ADV
cana-3303	167	2	of	of	ADP
cana-3303	167	3	all	all	PRON
cana-3303	167	4	,	,	PUNCT
cana-3303	167	5	the	the	DET
cana-3303	167	6	findings	finding	NOUN
cana-3303	167	7	of	of	ADP
cana-3303	167	8	this	this	DET
cana-3303	167	9	article	article	NOUN
cana-3303	167	10	are	be	AUX
cana-3303	167	11	important	important	ADJ
cana-3303	167	12	due	due	ADP
cana-3303	167	13	to	to	ADP
cana-3303	167	14	several	several	ADJ
cana-3303	167	15	considerations	consideration	NOUN
cana-3303	167	16	.	.	PUNCT
cana-3303	168	1	a	a	X
cana-3303	168	2	)	)	PUNCT
cana-3303	168	3	discrete	discrete	ADJ
cana-3303	168	4	derivatives	derivative	NOUN
cana-3303	168	5	these	these	PRON
cana-3303	168	6	are	be	AUX
cana-3303	168	7	getting	get	VERB
cana-3303	168	8	really	really	ADV
cana-3303	168	9	advanced	advanced	ADJ
cana-3303	168	10	;	;	PUNCT
cana-3303	168	11	where	where	SCONJ
cana-3303	168	12	we	we	PRON
cana-3303	168	13	have	have	AUX
cana-3303	168	14	nonlinear	nonlinear	ADJ
cana-3303	168	15	equations	equation	NOUN
cana-3303	168	16	to	to	PART
cana-3303	168	17	solve	solve	VERB
cana-3303	168	18	you	you	PRON
cana-3303	168	19	can	can	AUX
cana-3303	168	20	apply	apply	VERB
cana-3303	168	21	discrete	discrete	ADJ
cana-3303	168	22	multiplicative	multiplicative	ADJ
cana-3303	168	23	and	and	CCONJ
cana-3303	168	24	poverative	poverative	ADJ
cana-3303	168	25	derivatives	derivative	NOUN
cana-3303	168	26	.	.	PUNCT
cana-3303	169	1	here	here	ADV
cana-3303	169	2	the	the	DET
cana-3303	169	3	necessary	necessary	ADJ
cana-3303	169	4	derivatives	derivative	NOUN
cana-3303	169	5	were	be	AUX
cana-3303	169	6	not	not	PART
cana-3303	169	7	well	well	ADV
cana-3303	169	8	defined	define	VERB
cana-3303	169	9	previously	previously	ADV
cana-3303	169	10	in	in	ADP
cana-3303	169	11	the	the	DET
cana-3303	169	12	uninterrupted	uninterrupted	ADJ
cana-3303	169	13	form	form	NOUN
cana-3303	169	14	but	but	CCONJ
cana-3303	169	15	are	be	AUX
cana-3303	169	16	effectively	effectively	ADV
cana-3303	169	17	utilized	utilize	VERB
cana-3303	169	18	to	to	PART
cana-3303	169	19	simplify	simplify	VERB
cana-3303	169	20	the	the	DET
cana-3303	169	21	problem	problem	NOUN
cana-3303	169	22	and	and	CCONJ
cana-3303	169	23	make	make	VERB
cana-3303	169	24	this	this	DET
cana-3303	169	25	solution	solution	NOUN
cana-3303	169	26	tractable	tractable	ADJ
cana-3303	169	27	.	.	PUNCT
cana-3303	170	1	this	this	PRON
cana-3303	170	2	is	be	AUX
cana-3303	170	3	accomplished	accomplish	VERB
cana-3303	170	4	by	by	ADP
cana-3303	170	5	applying	apply	VERB
cana-3303	170	6	these	these	DET
cana-3303	170	7	new	new	ADJ
cana-3303	170	8	operations	operation	NOUN
cana-3303	170	9	to	to	PART
cana-3303	170	10	improve	improve	VERB
cana-3303	170	11	the	the	DET
cana-3303	170	12	theoretical	theoretical	ADJ
cana-3303	170	13	foundation	foundation	NOUN
cana-3303	170	14	of	of	ADP
cana-3303	170	15	discrete	discrete	ADJ
cana-3303	170	16	nonlinear	nonlinear	ADJ
cana-3303	170	17	equations	equation	NOUN
cana-3303	170	18	.	.	PUNCT
cana-3303	171	1	b	b	X
cana-3303	171	2	)	)	PUNCT
cana-3303	171	3	the	the	DET
cana-3303	171	4	analytical	analytical	ADJ
cana-3303	171	5	solutions	solution	NOUN
cana-3303	171	6	of	of	ADP
cana-3303	171	7	the	the	DET
cana-3303	171	8	equations	equation	NOUN
cana-3303	171	9	of	of	ADP
cana-3303	171	10	motion	motion	NOUN
cana-3303	171	11	with	with	ADP
cana-3303	171	12	no	no	DET
cana-3303	171	13	mechanical	mechanical	ADJ
cana-3303	171	14	energy	energy	NOUN
cana-3303	171	15	depletion	depletion	NOUN
cana-3303	171	16	can	can	AUX
cana-3303	171	17	be	be	AUX
cana-3303	171	18	computed	compute	VERB
cana-3303	171	19	efficiently	efficiently	ADV
cana-3303	171	20	,	,	PUNCT
cana-3303	171	21	assuming	assume	VERB
cana-3303	171	22	an	an	DET
cana-3303	171	23	iterative	iterative	NOUN
cana-3303	171	24	method	method	NOUN
cana-3303	171	25	rather	rather	ADV
cana-3303	171	26	than	than	ADP
cana-3303	171	27	numerical	numerical	ADJ
cana-3303	171	28	method	method	NOUN
cana-3303	171	29	.	.	PUNCT
cana-3303	172	1	this	this	PRON
cana-3303	172	2	is	be	AUX
cana-3303	172	3	especially	especially	ADV
cana-3303	172	4	useful	useful	ADJ
cana-3303	172	5	in	in	ADP
cana-3303	172	6	cases	case	NOUN
cana-3303	172	7	where	where	SCONJ
cana-3303	172	8	exact	exact	ADJ
cana-3303	172	9	solutions	solution	NOUN
cana-3303	172	10	are	be	AUX
cana-3303	172	11	desirable	desirable	ADJ
cana-3303	172	12	and	and	CCONJ
cana-3303	172	13	numerical	numerical	ADJ
cana-3303	172	14	methods	method	NOUN
cana-3303	172	15	may	may	AUX
cana-3303	172	16	lead	lead	VERB
cana-3303	172	17	to	to	ADP
cana-3303	172	18	errors	error	NOUN
cana-3303	172	19	or	or	CCONJ
cana-3303	172	20	require	require	VERB
cana-3303	172	21	high	high	ADJ
cana-3303	172	22	computational	computational	ADJ
cana-3303	172	23	cost	cost	NOUN
cana-3303	172	24	.	.	PUNCT
cana-3303	173	1	here	here	ADV
cana-3303	173	2	,	,	PUNCT
cana-3303	173	3	we	we	PRON
cana-3303	173	4	demonstrate	demonstrate	VERB
cana-3303	173	5	a	a	DET
cana-3303	173	6	very	very	ADV
cana-3303	173	7	compact	compact	ADJ
cana-3303	173	8	,	,	PUNCT
cana-3303	173	9	yet	yet	CCONJ
cana-3303	173	10	powerful	powerful	ADJ
cana-3303	173	11	analytical	analytical	ADJ
cana-3303	173	12	solution	solution	NOUN
cana-3303	173	13	which	which	PRON
cana-3303	173	14	is	be	AUX
cana-3303	173	15	applicable	applicable	ADJ
cana-3303	173	16	for	for	ADP
cana-3303	173	17	such	such	ADJ
cana-3303	173	18	systems	system	NOUN
cana-3303	173	19	.	.	PUNCT
cana-3303	174	1	c	c	X
cana-3303	174	2	)	)	PUNCT
cana-3303	174	3	nonlinear	nonlinear	ADJ
cana-3303	174	4	systems	system	NOUN
cana-3303	174	5	and	and	CCONJ
cana-3303	174	6	their	their	PRON
cana-3303	174	7	implications	implication	NOUN
cana-3303	174	8	the	the	DET
cana-3303	174	9	solutions	solution	NOUN
cana-3303	174	10	in	in	ADP
cana-3303	174	11	this	this	DET
cana-3303	174	12	work	work	NOUN
cana-3303	174	13	can	can	AUX
cana-3303	174	14	be	be	AUX
cana-3303	174	15	extended	extend	VERB
cana-3303	174	16	to	to	ADP
cana-3303	174	17	many	many	ADJ
cana-3303	174	18	nonlinear	nonlinear	ADJ
cana-3303	174	19	systems	system	NOUN
cana-3303	174	20	,	,	PUNCT
cana-3303	174	21	especially	especially	ADV
cana-3303	174	22	in	in	ADP
cana-3303	174	23	various	various	ADJ
cana-3303	174	24	physical	physical	ADJ
cana-3303	174	25	modeling	modeling	NOUN
cana-3303	174	26	and	and	CCONJ
cana-3303	174	27	engineering	engineering	NOUN
cana-3303	174	28	context	context	NOUN
cana-3303	174	29	.	.	PUNCT
cana-3303	175	1	with	with	SCONJ
cana-3303	175	2	a	a	DET
cana-3303	175	3	solution	solution	NOUN
cana-3303	175	4	communications	communication	NOUN
cana-3303	175	5	on	on	ADP
cana-3303	175	6	applied	apply	VERB
cana-3303	175	7	nonlinear	nonlinear	ADJ
cana-3303	175	8	analysis	analysis	NOUN
cana-3303	175	9	issn	issn	NOUN
cana-3303	175	10	:	:	PUNCT
cana-3303	175	11	1074	1074	NUM
cana-3303	175	12	-	-	PUNCT
cana-3303	175	13	133x	133x	NUM
cana-3303	175	14	vol	vol	NOUN
cana-3303	175	15	32	32	NUM
cana-3303	175	16	no	no	NOUN
cana-3303	175	17	.	.	PUNCT
cana-3303	176	1	6s	6s	NUM
cana-3303	176	2	(	(	PUNCT
cana-3303	176	3	2025	2025	NUM
cana-3303	176	4	)	)	PUNCT
cana-3303	176	5	388	388	NUM
cana-3303	176	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	176	7	rule	rule	NOUN
cana-3303	176	8	for	for	ADP
cana-3303	176	9	such	such	ADJ
cana-3303	176	10	nonlinear	nonlinear	ADJ
cana-3303	176	11	equations	equation	NOUN
cana-3303	176	12	,	,	PUNCT
cana-3303	176	13	we	we	PRON
cana-3303	176	14	can	can	AUX
cana-3303	176	15	model	model	VERB
cana-3303	176	16	different	different	ADJ
cana-3303	176	17	phenomena	phenomenon	NOUN
cana-3303	176	18	that	that	PRON
cana-3303	176	19	may	may	AUX
cana-3303	176	20	be	be	AUX
cana-3303	176	21	hard	hard	ADJ
cana-3303	176	22	or	or	CCONJ
cana-3303	176	23	even	even	ADV
cana-3303	176	24	impossible	impossible	ADJ
cana-3303	176	25	to	to	PART
cana-3303	176	26	solve	solve	VERB
cana-3303	176	27	by	by	ADP
cana-3303	176	28	traditional	traditional	ADJ
cana-3303	176	29	approaches	approach	NOUN
cana-3303	176	30	.	.	PUNCT
cana-3303	177	1	d	d	X
cana-3303	177	2	)	)	PUNCT
cana-3303	177	3	however	however	ADV
cana-3303	177	4	,	,	PUNCT
cana-3303	177	5	despite	despite	SCONJ
cana-3303	177	6	the	the	DET
cana-3303	177	7	successful	successful	ADJ
cana-3303	177	8	implementation	implementation	NOUN
cana-3303	177	9	of	of	ADP
cana-3303	177	10	both	both	CCONJ
cana-3303	177	11	the	the	DET
cana-3303	177	12	cauchy	cauchy	ADJ
cana-3303	177	13	and	and	CCONJ
cana-3303	177	14	boundary	boundary	ADJ
cana-3303	177	15	value	value	NOUN
cana-3303	177	16	techniques	technique	NOUN
cana-3303	177	17	the	the	DET
cana-3303	177	18	research	research	NOUN
cana-3303	177	19	suggests	suggest	VERB
cana-3303	177	20	gap	gap	NOUN
cana-3303	177	21	for	for	ADP
cana-3303	177	22	future	future	ADJ
cana-3303	177	23	research	research	NOUN
cana-3303	177	24	.	.	PUNCT
cana-3303	178	1	in	in	ADP
cana-3303	178	2	particular	particular	ADJ
cana-3303	178	3	,	,	PUNCT
cana-3303	178	4	we	we	PRON
cana-3303	178	5	should	should	AUX
cana-3303	178	6	explore	explore	VERB
cana-3303	178	7	non	non	ADJ
cana-3303	178	8	-	-	ADJ
cana-3303	178	9	local	local	ADJ
cana-3303	178	10	boundary	boundary	ADJ
cana-3303	178	11	conditions	condition	NOUN
cana-3303	178	12	,	,	PUNCT
cana-3303	178	13	refine	refine	VERB
cana-3303	178	14	the	the	DET
cana-3303	178	15	methodology	methodology	NOUN
cana-3303	178	16	to	to	ADP
cana-3303	178	17	higher	high	ADJ
cana-3303	178	18	-	-	PUNCT
cana-3303	178	19	order	order	NOUN
cana-3303	178	20	nonlinear	nonlinear	ADJ
cana-3303	178	21	equations	equation	NOUN
cana-3303	178	22	and	and	CCONJ
cana-3303	178	23	more	more	ADJ
cana-3303	178	24	.	.	PUNCT
cana-3303	179	1	another	another	DET
cana-3303	179	2	aspect	aspect	NOUN
cana-3303	179	3	that	that	PRON
cana-3303	179	4	is	be	AUX
cana-3303	179	5	worthy	worthy	ADJ
cana-3303	179	6	further	further	ADJ
cana-3303	179	7	investigation	investigation	NOUN
cana-3303	179	8	involves	involve	VERB
cana-3303	179	9	solving	solve	VERB
cana-3303	179	10	discrete	discrete	ADJ
cana-3303	179	11	multiplicative	multiplicative	ADJ
cana-3303	179	12	equations	equation	NOUN
cana-3303	179	13	with	with	ADP
cana-3303	179	14	more	more	ADJ
cana-3303	179	15	complex	complex	ADJ
cana-3303	179	16	boundary	boundary	ADJ
cana-3303	179	17	conditions	condition	NOUN
cana-3303	179	18	.	.	PUNCT
cana-3303	180	1	e	e	X
cana-3303	180	2	)	)	PUNCT
cana-3303	180	3	challenges	challenge	NOUN
cana-3303	180	4	and	and	CCONJ
cana-3303	180	5	limitations	limitation	NOUN
cana-3303	180	6	a	a	DET
cana-3303	180	7	caveat	caveat	NOUN
cana-3303	180	8	of	of	ADP
cana-3303	180	9	all	all	DET
cana-3303	180	10	methods	method	NOUN
cana-3303	180	11	employed	employ	VERB
cana-3303	180	12	in	in	ADP
cana-3303	180	13	this	this	DET
cana-3303	180	14	work	work	NOUN
cana-3303	180	15	is	be	AUX
cana-3303	180	16	that	that	SCONJ
cana-3303	180	17	the	the	DET
cana-3303	180	18	solutions	solution	NOUN
cana-3303	180	19	are	be	AUX
cana-3303	180	20	extremely	extremely	ADV
cana-3303	180	21	sensitive	sensitive	ADJ
cana-3303	180	22	to	to	ADP
cana-3303	180	23	the	the	DET
cana-3303	180	24	initial	initial	ADJ
cana-3303	180	25	conditions	condition	NOUN
cana-3303	180	26	and	and	CCONJ
cana-3303	180	27	the	the	DET
cana-3303	180	28	boundary	boundary	ADJ
cana-3303	180	29	conditions	condition	NOUN
cana-3303	180	30	.	.	PUNCT
cana-3303	181	1	if	if	SCONJ
cana-3303	181	2	boundary	boundary	ADJ
cana-3303	181	3	conditions	condition	NOUN
cana-3303	181	4	are	be	AUX
cana-3303	181	5	hard	hard	ADJ
cana-3303	181	6	to	to	PART
cana-3303	181	7	define	define	VERB
cana-3303	181	8	or	or	CCONJ
cana-3303	181	9	highly	highly	ADV
cana-3303	181	10	complicated	complicated	ADJ
cana-3303	181	11	relationships	relationship	NOUN
cana-3303	181	12	exist	exist	VERB
cana-3303	181	13	,	,	PUNCT
cana-3303	181	14	the	the	DET
cana-3303	181	15	applicability	applicability	NOUN
cana-3303	181	16	of	of	ADP
cana-3303	181	17	the	the	DET
cana-3303	181	18	solution	solution	NOUN
cana-3303	181	19	might	might	AUX
cana-3303	181	20	be	be	AUX
cana-3303	181	21	poor	poor	ADJ
cana-3303	181	22	.	.	PUNCT
cana-3303	182	1	likewise	likewise	ADV
cana-3303	182	2	,	,	PUNCT
cana-3303	182	3	though	though	SCONJ
cana-3303	182	4	the	the	DET
cana-3303	182	5	discrete	discrete	ADJ
cana-3303	182	6	poverative	poverative	ADJ
cana-3303	182	7	derivatives	derivative	NOUN
cana-3303	182	8	offer	offer	VERB
cana-3303	182	9	a	a	DET
cana-3303	182	10	helpful	helpful	ADJ
cana-3303	182	11	methodology	methodology	NOUN
cana-3303	182	12	to	to	ADP
cana-3303	182	13	employing	employ	VERB
cana-3303	182	14	a	a	DET
cana-3303	182	15	nonlinear	nonlinear	ADJ
cana-3303	182	16	equation	equation	NOUN
cana-3303	182	17	solution	solution	NOUN
cana-3303	182	18	,	,	PUNCT
cana-3303	182	19	they	they	PRON
cana-3303	182	20	are	be	AUX
cana-3303	182	21	still	still	ADV
cana-3303	182	22	yet	yet	ADV
cana-3303	182	23	to	to	PART
cana-3303	182	24	be	be	AUX
cana-3303	182	25	validated	validate	VERB
cana-3303	182	26	for	for	ADP
cana-3303	182	27	actual	actual	ADJ
cana-3303	182	28	problems	problem	NOUN
cana-3303	182	29	and	and	CCONJ
cana-3303	182	30	generalized	generalize	VERB
cana-3303	182	31	further	far	ADV
cana-3303	182	32	to	to	ADP
cana-3303	182	33	themselves	themselves	PRON
cana-3303	182	34	.	.	PUNCT
cana-3303	183	1	7	7	X
cana-3303	183	2	.	.	X
cana-3303	183	3	conclusion	conclusion	NOUN
cana-3303	183	4	this	this	DET
cana-3303	183	5	article	article	NOUN
cana-3303	183	6	leads	lead	VERB
cana-3303	183	7	us	we	PRON
cana-3303	183	8	to	to	ADP
cana-3303	183	9	the	the	DET
cana-3303	183	10	original	original	ADJ
cana-3303	183	11	third	third	ADJ
cana-3303	183	12	-	-	PUNCT
cana-3303	183	13	order	order	NOUN
cana-3303	183	14	mixed	mix	VERB
cana-3303	183	15	derivative	derivative	ADJ
cana-3303	183	16	equation	equation	NOUN
cana-3303	183	17	with	with	ADP
cana-3303	183	18	the	the	DET
cana-3303	183	19	discrete	discrete	ADJ
cana-3303	183	20	additive	additive	NOUN
cana-3303	183	21	,	,	PUNCT
cana-3303	183	22	multiplicative	multiplicative	ADJ
cana-3303	183	23	,	,	PUNCT
cana-3303	183	24	and	and	CCONJ
cana-3303	183	25	poverative	poverative	ADJ
cana-3303	183	26	derivatives	derivative	NOUN
cana-3303	183	27	and	and	CCONJ
cana-3303	183	28	an	an	DET
cana-3303	183	29	efficient	efficient	ADJ
cana-3303	183	30	technique	technique	NOUN
cana-3303	183	31	to	to	PART
cana-3303	183	32	find	find	VERB
cana-3303	183	33	its	its	PRON
cana-3303	183	34	solution	solution	NOUN
cana-3303	183	35	.	.	PUNCT
cana-3303	184	1	the	the	DET
cana-3303	184	2	equation	equation	NOUN
cana-3303	184	3	studied	study	VERB
cana-3303	184	4	in	in	ADP
cana-3303	184	5	this	this	DET
cana-3303	184	6	work	work	NOUN
cana-3303	184	7	is	be	AUX
cana-3303	184	8	complex	complex	ADJ
cana-3303	184	9	,	,	PUNCT
cana-3303	184	10	which	which	PRON
cana-3303	184	11	demands	demand	VERB
cana-3303	184	12	advanced	advanced	ADJ
cana-3303	184	13	mathematical	mathematical	ADJ
cana-3303	184	14	methods	method	NOUN
cana-3303	184	15	to	to	PART
cana-3303	184	16	simplify	simplify	VERB
cana-3303	184	17	and	and	CCONJ
cana-3303	184	18	solve	solve	VERB
cana-3303	184	19	it	it	PRON
cana-3303	184	20	.	.	PUNCT
cana-3303	185	1	using	use	VERB
cana-3303	185	2	discrete	discrete	ADJ
cana-3303	185	3	derivatives	derivative	NOUN
cana-3303	185	4	—	—	PUNCT
cana-3303	185	5	especially	especially	ADV
cana-3303	185	6	additive	additive	VERB
cana-3303	185	7	derivative	derivative	ADJ
cana-3303	185	8	and	and	CCONJ
cana-3303	185	9	multiplicative	multiplicative	ADJ
cana-3303	185	10	derivatives	derivative	NOUN
cana-3303	185	11	,	,	PUNCT
cana-3303	185	12	the	the	DET
cana-3303	185	13	nonlinear	nonlinear	ADJ
cana-3303	185	14	equation	equation	NOUN
cana-3303	185	15	can	can	AUX
cana-3303	185	16	be	be	AUX
cana-3303	185	17	converted	convert	VERB
cana-3303	185	18	into	into	ADP
cana-3303	185	19	systematically	systematically	ADV
cana-3303	185	20	solvable	solvable	ADJ
cana-3303	185	21	equations	equation	NOUN
cana-3303	185	22	of	of	ADP
cana-3303	185	23	cauchy	cauchy	PROPN
cana-3303	185	24	and	and	CCONJ
cana-3303	185	25	boundary	boundary	ADJ
cana-3303	185	26	value	value	NOUN
cana-3303	185	27	problems	problem	NOUN
cana-3303	185	28	.	.	PUNCT
cana-3303	186	1	the	the	DET
cana-3303	186	2	iterative	iterative	NOUN
cana-3303	186	3	approach	approach	NOUN
cana-3303	186	4	used	use	VERB
cana-3303	186	5	here	here	ADV
cana-3303	186	6	reduces	reduce	VERB
cana-3303	186	7	the	the	DET
cana-3303	186	8	problem	problem	NOUN
cana-3303	186	9	to	to	ADP
cana-3303	186	10	small	small	ADJ
cana-3303	186	11	steps	step	NOUN
cana-3303	186	12	,	,	PUNCT
cana-3303	186	13	one	one	NUM
cana-3303	186	14	connected	connect	VERB
cana-3303	186	15	upon	upon	SCONJ
cana-3303	186	16	the	the	DET
cana-3303	186	17	other	other	ADJ
cana-3303	186	18	,	,	PUNCT
cana-3303	186	19	until	until	SCONJ
cana-3303	186	20	a	a	DET
cana-3303	186	21	general	general	ADJ
cana-3303	186	22	solution	solution	NOUN
cana-3303	186	23	is	be	AUX
cana-3303	186	24	built	build	VERB
cana-3303	186	25	up	up	ADP
cana-3303	186	26	explicitly	explicitly	ADV
cana-3303	186	27	.	.	PUNCT
cana-3303	187	1	this	this	DET
cana-3303	187	2	approach	approach	NOUN
cana-3303	187	3	not	not	PART
cana-3303	187	4	only	only	ADV
cana-3303	187	5	simplifies	simplify	VERB
cana-3303	187	6	the	the	DET
cana-3303	187	7	solution	solution	NOUN
cana-3303	187	8	derivation	derivation	NOUN
cana-3303	187	9	but	but	CCONJ
cana-3303	187	10	also	also	ADV
cana-3303	187	11	demonstrates	demonstrate	VERB
cana-3303	187	12	the	the	DET
cana-3303	187	13	power	power	NOUN
cana-3303	187	14	of	of	ADP
cana-3303	187	15	discrete	discrete	ADJ
cana-3303	187	16	poverative	poverative	ADJ
cana-3303	187	17	derivatives	derivative	NOUN
cana-3303	187	18	in	in	ADP
cana-3303	187	19	addressing	address	VERB
cana-3303	187	20	complex	complex	ADJ
cana-3303	187	21	nonlinear	nonlinear	ADJ
cana-3303	187	22	systems	system	NOUN
cana-3303	187	23	.	.	PUNCT
cana-3303	188	1	a	a	DET
cana-3303	188	2	broad	broad	ADJ
cana-3303	188	3	area	area	NOUN
cana-3303	188	4	of	of	ADP
cana-3303	188	5	applications	application	NOUN
cana-3303	188	6	of	of	ADP
cana-3303	188	7	the	the	DET
cana-3303	188	8	results	result	NOUN
cana-3303	188	9	presented	present	VERB
cana-3303	188	10	in	in	ADP
cana-3303	188	11	this	this	DET
cana-3303	188	12	paper	paper	NOUN
cana-3303	188	13	is	be	AUX
cana-3303	188	14	as	as	SCONJ
cana-3303	188	15	it	it	PRON
cana-3303	188	16	contributes	contribute	VERB
cana-3303	188	17	greatly	greatly	ADV
cana-3303	188	18	towards	towards	ADP
cana-3303	188	19	the	the	DET
cana-3303	188	20	study	study	NOUN
cana-3303	188	21	of	of	ADP
cana-3303	188	22	discrete	discrete	ADJ
cana-3303	188	23	nonlinear	nonlinear	ADJ
cana-3303	188	24	derivative	derivative	ADJ
cana-3303	188	25	equations	equation	NOUN
cana-3303	188	26	.	.	PUNCT
cana-3303	189	1	in	in	ADP
cana-3303	189	2	addition	addition	NOUN
cana-3303	189	3	,	,	PUNCT
cana-3303	189	4	it	it	PRON
cana-3303	189	5	presents	present	VERB
cana-3303	189	6	new	new	ADJ
cana-3303	189	7	interpretation	interpretation	NOUN
cana-3303	189	8	of	of	ADP
cana-3303	189	9	boundary	boundary	ADJ
cana-3303	189	10	value	value	NOUN
cana-3303	189	11	types	type	NOUN
cana-3303	189	12	problems	problem	NOUN
cana-3303	189	13	and	and	CCONJ
cana-3303	189	14	illustrates	illustrate	VERB
cana-3303	189	15	a	a	DET
cana-3303	189	16	new	new	ADJ
cana-3303	189	17	methodology	methodology	NOUN
cana-3303	189	18	for	for	ADP
cana-3303	189	19	dealing	deal	VERB
cana-3303	189	20	with	with	ADP
cana-3303	189	21	equations	equation	NOUN
cana-3303	189	22	endowed	endow	VERB
cana-3303	189	23	with	with	ADP
cana-3303	189	24	discrete	discrete	ADJ
cana-3303	189	25	multiplicative	multiplicative	ADJ
cana-3303	189	26	derivatives	derivative	NOUN
cana-3303	189	27	.	.	PUNCT
cana-3303	190	1	the	the	DET
cana-3303	190	2	general	general	ADJ
cana-3303	190	3	and	and	CCONJ
cana-3303	190	4	boundary	boundary	ADJ
cana-3303	190	5	-	-	PUNCT
cana-3303	190	6	specific	specific	ADJ
cana-3303	190	7	solutions	solution	NOUN
cana-3303	190	8	presented	present	VERB
cana-3303	190	9	can	can	AUX
cana-3303	190	10	be	be	AUX
cana-3303	190	11	regarded	regard	VERB
cana-3303	190	12	as	as	ADP
cana-3303	190	13	basis	basis	NOUN
cana-3303	190	14	for	for	ADP
cana-3303	190	15	further	further	ADJ
cana-3303	190	16	investigations	investigation	NOUN
cana-3303	190	17	regarding	regard	VERB
cana-3303	190	18	nonlinear	nonlinear	ADJ
cana-3303	190	19	discrete	discrete	ADJ
cana-3303	190	20	systems	system	NOUN
cana-3303	190	21	.	.	PUNCT
cana-3303	191	1	eventually	eventually	ADV
cana-3303	191	2	,	,	PUNCT
cana-3303	191	3	though	though	SCONJ
cana-3303	191	4	the	the	DET
cana-3303	191	5	research	research	NOUN
cana-3303	191	6	has	have	AUX
cana-3303	191	7	solved	solve	VERB
cana-3303	191	8	multiple	multiple	ADJ
cana-3303	191	9	crucial	crucial	ADJ
cana-3303	191	10	issues	issue	NOUN
cana-3303	191	11	,	,	PUNCT
cana-3303	191	12	it	it	PRON
cana-3303	191	13	points	point	VERB
cana-3303	191	14	out	out	ADP
cana-3303	191	15	the	the	DET
cana-3303	191	16	remainings	remainings	PROPN
cana-3303	191	17	ones	one	NOUN
cana-3303	191	18	especially	especially	ADV
cana-3303	191	19	concerning	concern	VERB
cana-3303	191	20	to	to	ADP
cana-3303	191	21	boundary	boundary	ADJ
cana-3303	191	22	conditions	condition	NOUN
cana-3303	191	23	and	and	CCONJ
cana-3303	191	24	non	non	ADJ
cana-3303	191	25	-	-	ADJ
cana-3303	191	26	local	local	ADJ
cana-3303	191	27	boundary	boundary	ADJ
cana-3303	191	28	value	value	NOUN
cana-3303	191	29	problems	problem	NOUN
cana-3303	191	30	.	.	PUNCT
cana-3303	192	1	such	such	ADJ
cana-3303	192	2	open	open	ADJ
cana-3303	192	3	areas	area	NOUN
cana-3303	192	4	can	can	AUX
cana-3303	192	5	be	be	AUX
cana-3303	192	6	explored	explore	VERB
cana-3303	192	7	,	,	PUNCT
cana-3303	192	8	which	which	PRON
cana-3303	192	9	are	be	AUX
cana-3303	192	10	an	an	DET
cana-3303	192	11	important	important	ADJ
cana-3303	192	12	motivating	motivating	NOUN
cana-3303	192	13	pursuit	pursuit	NOUN
cana-3303	192	14	of	of	ADP
cana-3303	192	15	research	research	NOUN
cana-3303	192	16	in	in	ADP
cana-3303	192	17	discrete	discrete	ADJ
cana-3303	192	18	derivative	derivative	ADJ
cana-3303	192	19	equations	equation	NOUN
cana-3303	192	20	.	.	PUNCT
cana-3303	193	1	references	reference	NOUN
cana-3303	193	2	[	[	X
cana-3303	193	3	1	1	NUM
cana-3303	193	4	]	]	X
cana-3303	193	5	ahmadov	ahmadov	PROPN
cana-3303	193	6	,	,	PUNCT
cana-3303	193	7	g.	g.	PROPN
cana-3303	193	8	,	,	PUNCT
cana-3303	193	9	hasanov	hasanov	PROPN
cana-3303	193	10	,	,	PUNCT
cana-3303	193	11	k.	k.	PROPN
cana-3303	193	12	,	,	PUNCT
cana-3303	193	13	&	&	CCONJ
cana-3303	193	14	yagubov	yagubov	PROPN
cana-3303	193	15	,	,	PUNCT
cana-3303	193	16	m.	m.	NOUN
cana-3303	193	17	(	(	PUNCT
cana-3303	193	18	1978	1978	NUM
cana-3303	193	19	)	)	PUNCT
cana-3303	193	20	.	.	PUNCT
cana-3303	194	1	ordinary	ordinary	ADJ
cana-3303	194	2	differential	differential	ADJ
cana-3303	194	3	equations	equation	NOUN
cana-3303	194	4	(	(	PUNCT
cana-3303	194	5	p.	p.	NOUN
cana-3303	194	6	444	444	NUM
cana-3303	194	7	)	)	PUNCT
cana-3303	194	8	.	.	PUNCT
cana-3303	195	1	baku	baku	NOUN
cana-3303	195	2	:	:	PUNCT
cana-3303	195	3	maarif	maarif	ADJ
cana-3303	195	4	.	.	PUNCT
cana-3303	196	1	[	[	X
cana-3303	196	2	2	2	NUM
cana-3303	196	3	]	]	PUNCT
cana-3303	196	4	aliev	aliev	ADV
cana-3303	196	5	,	,	PUNCT
cana-3303	196	6	n.	n.	NOUN
cana-3303	196	7	,	,	PUNCT
cana-3303	196	8	azizi	azizi	PROPN
cana-3303	196	9	,	,	PUNCT
cana-3303	196	10	n.	n.	NOUN
cana-3303	196	11	,	,	PUNCT
cana-3303	196	12	&	&	CCONJ
cana-3303	196	13	jahanshahi	jahanshahi	PROPN
cana-3303	196	14	,	,	PUNCT
cana-3303	196	15	m.	m.	NOUN
cana-3303	196	16	(	(	PUNCT
cana-3303	196	17	2007	2007	NUM
cana-3303	196	18	)	)	PUNCT
cana-3303	196	19	.	.	PUNCT
cana-3303	197	1	invariant	invariant	ADJ
cana-3303	197	2	functions	function	NOUN
cana-3303	197	3	for	for	ADP
cana-3303	197	4	discrete	discrete	ADJ
cana-3303	197	5	derivatives	derivative	NOUN
cana-3303	197	6	and	and	CCONJ
cana-3303	197	7	their	their	PRON
cana-3303	197	8	applications	application	NOUN
cana-3303	197	9	to	to	PART
cana-3303	197	10	solve	solve	VERB
cana-3303	197	11	non	non	ADJ
cana-3303	197	12	-	-	ADJ
cana-3303	197	13	homogeneous	homogeneous	ADJ
cana-3303	197	14	linear	linear	ADJ
cana-3303	197	15	and	and	CCONJ
cana-3303	197	16	non	non	ADJ
cana-3303	197	17	-	-	ADJ
cana-3303	197	18	linear	linear	ADJ
cana-3303	197	19	difference	difference	NOUN
cana-3303	197	20	equations	equation	NOUN
cana-3303	197	21	.	.	PUNCT
cana-3303	198	1	joint	joint	ADJ
cana-3303	198	2	mathematics	mathematics	PROPN
cana-3303	198	3	forum	forum	PROPN
cana-3303	198	4	,	,	PUNCT
cana-3303	198	5	2(ii	2(ii	NUM
cana-3303	198	6	)	)	PUNCT
cana-3303	198	7	,	,	PUNCT
cana-3303	198	8	533–542	533–542	NUM
cana-3303	198	9	.	.	PUNCT
cana-3303	199	1	[	[	X
cana-3303	199	2	3	3	NUM
cana-3303	199	3	]	]	X
cana-3303	199	4	aliyev	aliyev	NOUN
cana-3303	199	5	,	,	PUNCT
cana-3303	199	6	n.	n.	PROPN
cana-3303	199	7	a.	a.	PROPN
cana-3303	199	8	,	,	PUNCT
cana-3303	199	9	&	&	CCONJ
cana-3303	199	10	mamieva	mamieva	PROPN
cana-3303	199	11	,	,	PUNCT
cana-3303	199	12	t.	t.	PROPN
cana-3303	199	13	s.	s.	PROPN
cana-3303	199	14	(	(	PUNCT
cana-3303	199	15	2017	2017	NUM
cana-3303	199	16	)	)	PUNCT
cana-3303	199	17	.	.	PUNCT
cana-3303	200	1	the	the	DET
cana-3303	200	2	second	second	ADJ
cana-3303	200	3	-	-	PUNCT
cana-3303	200	4	order	order	NOUN
cana-3303	200	5	boundary	boundary	ADJ
cana-3303	200	6	problem	problem	NOUN
cana-3303	200	7	for	for	ADP
cana-3303	200	8	the	the	DET
cana-3303	200	9	discrete	discrete	ADJ
cana-3303	200	10	multiplicative	multiplicative	ADJ
cana-3303	200	11	derivative	derivative	ADJ
cana-3303	200	12	equation	equation	NOUN
cana-3303	200	13	.	.	PUNCT
cana-3303	201	1	baku	baku	PROPN
cana-3303	201	2	university	university	PROPN
cana-3303	201	3	news	news	PROPN
cana-3303	201	4	series	series	PROPN
cana-3303	201	5	,	,	PUNCT
cana-3303	201	6	physical	physical	ADJ
cana-3303	201	7	and	and	CCONJ
cana-3303	201	8	mathematical	mathematical	ADJ
cana-3303	201	9	sciences	science	NOUN
cana-3303	201	10	,	,	PUNCT
cana-3303	201	11	1	1	NUM
cana-3303	201	12	,	,	PUNCT
cana-3303	201	13	15–19	15–19	NUM
cana-3303	201	14	.	.	PUNCT
cana-3303	202	1	[	[	X
cana-3303	202	2	4	4	NUM
cana-3303	202	3	]	]	X
cana-3303	202	4	aliyev	aliyev	NOUN
cana-3303	202	5	,	,	PUNCT
cana-3303	202	6	n.	n.	NOUN
cana-3303	202	7	,	,	PUNCT
cana-3303	202	8	&	&	CCONJ
cana-3303	202	9	mammadzade	mammadzade	PROPN
cana-3303	202	10	,	,	PUNCT
cana-3303	202	11	a.	a.	NOUN
cana-3303	202	12	(	(	PUNCT
cana-3303	202	13	2018	2018	NUM
cana-3303	202	14	)	)	PUNCT
cana-3303	202	15	.	.	PUNCT
cana-3303	203	1	solution	solution	NOUN
cana-3303	203	2	of	of	ADP
cana-3303	203	3	problems	problem	NOUN
cana-3303	203	4	for	for	ADP
cana-3303	203	5	the	the	DET
cana-3303	203	6	second	second	ADJ
cana-3303	203	7	-	-	PUNCT
cana-3303	203	8	order	order	NOUN
cana-3303	203	9	discrete	discrete	ADJ
cana-3303	203	10	poverative	poverative	ADJ
cana-3303	203	11	derivative	derivative	ADJ
cana-3303	203	12	equation	equation	NOUN
cana-3303	203	13	.	.	PUNCT
cana-3303	204	1	scientific	scientific	ADJ
cana-3303	204	2	news	news	NOUN
cana-3303	204	3	,	,	PUNCT
cana-3303	204	4	natural	natural	ADJ
cana-3303	204	5	sciences	science	NOUN
cana-3303	204	6	,	,	PUNCT
cana-3303	204	7	1	1	NUM
cana-3303	204	8	,	,	PUNCT
cana-3303	204	9	55–58	55–58	NUM
cana-3303	204	10	.	.	PUNCT
cana-3303	205	1	[	[	X
cana-3303	205	2	5	5	NUM
cana-3303	205	3	]	]	X
cana-3303	205	4	aliyev	aliyev	NOUN
cana-3303	205	5	,	,	PUNCT
cana-3303	205	6	n.	n.	NOUN
cana-3303	205	7	,	,	PUNCT
cana-3303	205	8	bagirov	bagirov	PROPN
cana-3303	205	9	,	,	PUNCT
cana-3303	205	10	g.	g.	PROPN
cana-3303	205	11	,	,	PUNCT
cana-3303	205	12	&	&	CCONJ
cana-3303	205	13	i̇zadi	i̇zadi	NOUN
cana-3303	205	14	,	,	PUNCT
cana-3303	205	15	f.	f.	PROPN
cana-3303	205	16	a.	a.	PROPN
cana-3303	205	17	(	(	PUNCT
cana-3303	205	18	2009	2009	NUM
cana-3303	205	19	)	)	PUNCT
cana-3303	205	20	.	.	PUNCT
cana-3303	206	1	discrete	discrete	ADJ
cana-3303	206	2	calculus	calculus	NOUN
cana-3303	206	3	by	by	ADP
cana-3303	206	4	analogy	analogy	NOUN
cana-3303	206	5	(	(	PUNCT
cana-3303	206	6	p.	p.	NOUN
cana-3303	206	7	154	154	NUM
cana-3303	206	8	)	)	PUNCT
cana-3303	206	9	.	.	PUNCT
cana-3303	207	1	book	book	NOUN
cana-3303	207	2	.	.	PUNCT
cana-3303	208	1	december	december	PROPN
cana-3303	208	2	3	3	NUM
cana-3303	208	3	.	.	PUNCT
cana-3303	208	4	communications	communication	NOUN
cana-3303	208	5	on	on	ADP
cana-3303	208	6	applied	apply	VERB
cana-3303	208	7	nonlinear	nonlinear	ADJ
cana-3303	208	8	analysis	analysis	NOUN
cana-3303	208	9	issn	issn	NOUN
cana-3303	208	10	:	:	PUNCT
cana-3303	208	11	1074	1074	NUM
cana-3303	208	12	-	-	PUNCT
cana-3303	208	13	133x	133x	NUM
cana-3303	208	14	vol	vol	NOUN
cana-3303	208	15	32	32	NUM
cana-3303	208	16	no	no	NOUN
cana-3303	208	17	.	.	PUNCT
cana-3303	209	1	6s	6s	NUM
cana-3303	209	2	(	(	PUNCT
cana-3303	209	3	2025	2025	NUM
cana-3303	209	4	)	)	PUNCT
cana-3303	209	5	389	389	NUM
cana-3303	209	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3303	210	1	[	[	X
cana-3303	210	2	6	6	NUM
cana-3303	210	3	]	]	X
cana-3303	210	4	aliyev	aliyev	NOUN
cana-3303	210	5	,	,	PUNCT
cana-3303	210	6	n.	n.	NOUN
cana-3303	210	7	,	,	PUNCT
cana-3303	210	8	ibrahimov	ibrahimov	ADJ
cana-3303	210	9	,	,	PUNCT
cana-3303	210	10	n.	n.	NOUN
cana-3303	210	11	,	,	PUNCT
cana-3303	210	12	&	&	CCONJ
cana-3303	210	13	mammadzade	mammadzade	PROPN
cana-3303	210	14	,	,	PUNCT
cana-3303	210	15	a.	a.	NOUN
cana-3303	210	16	(	(	PUNCT
cana-3303	210	17	2018	2018	NUM
cana-3303	210	18	)	)	PUNCT
cana-3303	210	19	.	.	PUNCT
cana-3303	211	1	issues	issue	NOUN
cana-3303	211	2	for	for	ADP
cana-3303	211	3	discrete	discrete	ADJ
cana-3303	211	4	poverative	poverative	ADJ
cana-3303	211	5	-	-	PUNCT
cana-3303	211	6	multiplicative	multiplicative	ADJ
cana-3303	211	7	derivative	derivative	ADJ
cana-3303	211	8	equations	equation	NOUN
cana-3303	211	9	.	.	PUNCT
cana-3303	212	1	azerbaijan	azerbaijan	PROPN
cana-3303	212	2	technical	technical	PROPN
cana-3303	212	3	university	university	PROPN
cana-3303	212	4	,	,	PUNCT
cana-3303	212	5	technical	technical	ADJ
cana-3303	212	6	sciences	science	NOUN
cana-3303	212	7	,	,	PUNCT
cana-3303	212	8	scientific	scientific	ADJ
cana-3303	212	9	works	work	NOUN
cana-3303	212	10	,	,	PUNCT
cana-3303	212	11	2	2	NUM
cana-3303	212	12	,	,	PUNCT
cana-3303	212	13	90–94	90–94	NUM
cana-3303	212	14	.	.	PUNCT
cana-3303	213	1	[	[	X
cana-3303	213	2	7	7	NUM
cana-3303	213	3	]	]	SYM
cana-3303	213	4	ashirov	ashirov	ADJ
cana-3303	213	5	,	,	PUNCT
cana-3303	213	6	a.	a.	NOUN
cana-3303	213	7	(	(	PUNCT
cana-3303	213	8	2013	2013	NUM
cana-3303	213	9	)	)	PUNCT
cana-3303	213	10	.	.	PUNCT
cana-3303	214	1	on	on	ADP
cana-3303	214	2	line	line	NOUN
cana-3303	214	3	and	and	CCONJ
cana-3303	214	4	double	double	ADJ
cana-3303	214	5	multiplicative	multiplicative	ADJ
cana-3303	214	6	integrals	integral	NOUN
cana-3303	214	7	.	.	PUNCT
cana-3303	215	1	twms	twms	PROPN
cana-3303	215	2	journal	journal	PROPN
cana-3303	215	3	of	of	ADP
cana-3303	215	4	applied	apply	VERB
cana-3303	215	5	and	and	CCONJ
cana-3303	215	6	engineering	engineering	NOUN
cana-3303	215	7	mathematics	mathematic	NOUN
cana-3303	215	8	,	,	PUNCT
cana-3303	215	9	3(1	3(1	NUM
cana-3303	215	10	)	)	PUNCT
cana-3303	215	11	,	,	PUNCT
cana-3303	215	12	103–107	103–107	NUM
cana-3303	215	13	.	.	PUNCT
cana-3303	216	1	[	[	X
cana-3303	216	2	8	8	NUM
cana-3303	216	3	]	]	X
cana-3303	216	4	bashirov	bashirov	PROPN
cana-3303	216	5	,	,	PUNCT
cana-3303	216	6	a.	a.	PROPN
cana-3303	216	7	,	,	PUNCT
cana-3303	216	8	&	&	CCONJ
cana-3303	216	9	riza	riza	PROPN
cana-3303	216	10	,	,	PUNCT
cana-3303	216	11	m.	m.	NOUN
cana-3303	216	12	(	(	PUNCT
cana-3303	216	13	2011	2011	NUM
cana-3303	216	14	)	)	PUNCT
cana-3303	216	15	.	.	PUNCT
cana-3303	217	1	on	on	ADP
cana-3303	217	2	complex	complex	ADJ
cana-3303	217	3	multiplicative	multiplicative	ADJ
cana-3303	217	4	differentiation	differentiation	NOUN
cana-3303	217	5	.	.	PUNCT
cana-3303	218	1	twms	twms	PROPN
cana-3303	218	2	journal	journal	PROPN
cana-3303	218	3	of	of	ADP
cana-3303	218	4	applied	apply	VERB
cana-3303	218	5	and	and	CCONJ
cana-3303	218	6	engineering	engineering	NOUN
cana-3303	218	7	mathematics	mathematic	NOUN
cana-3303	218	8	,	,	PUNCT
cana-3303	218	9	1(1	1(1	NUM
cana-3303	218	10	)	)	PUNCT
cana-3303	218	11	,	,	PUNCT
cana-3303	218	12	75–85	75–85	NOUN
cana-3303	218	13	.	.	PUNCT
cana-3303	219	1	[	[	X
cana-3303	219	2	9	9	NUM
cana-3303	219	3	]	]	PUNCT
cana-3303	219	4	chikhrab	chikhrab	NOUN
cana-3303	219	5	,	,	PUNCT
cana-3303	219	6	s.	s.	PROPN
cana-3303	219	7	r.	r.	PROPN
cana-3303	219	8	,	,	PUNCT
cana-3303	219	9	&	&	CCONJ
cana-3303	219	10	malkin	malkin	PROPN
cana-3303	219	11	,	,	PUNCT
cana-3303	219	12	m.	m.	NOUN
cana-3303	219	13	a.	a.	NOUN
cana-3303	219	14	(	(	PUNCT
cana-3303	219	15	2005	2005	NUM
cana-3303	219	16	)	)	PUNCT
cana-3303	219	17	.	.	PUNCT
cana-3303	220	1	introduction	introduction	NOUN
cana-3303	220	2	to	to	ADP
cana-3303	220	3	difference	difference	NOUN
cana-3303	220	4	equations	equation	NOUN
cana-3303	220	5	and	and	CCONJ
cana-3303	220	6	their	their	PRON
cana-3303	220	7	applications	application	NOUN
cana-3303	220	8	.	.	PUNCT
cana-3303	221	1	springer	springer	NOUN
cana-3303	221	2	.	.	PUNCT
cana-3303	222	1	[	[	X
cana-3303	222	2	10	10	NUM
cana-3303	222	3	]	]	SYM
cana-3303	222	4	dezin	dezin	NOUN
cana-3303	222	5	,	,	PUNCT
cana-3303	222	6	a.	a.	NOUN
cana-3303	222	7	a.	a.	NOUN
cana-3303	222	8	(	(	PUNCT
cana-3303	222	9	1990	1990	NUM
cana-3303	222	10	)	)	PUNCT
cana-3303	222	11	.	.	PUNCT
cana-3303	223	1	multidimensional	multidimensional	ADJ
cana-3303	223	2	analysis	analysis	NOUN
cana-3303	223	3	and	and	CCONJ
cana-3303	223	4	discrete	discrete	ADJ
cana-3303	223	5	models	model	NOUN
cana-3303	223	6	(	(	PUNCT
cana-3303	223	7	p.	p.	NOUN
cana-3303	223	8	240	240	NUM
cana-3303	223	9	)	)	PUNCT
cana-3303	223	10	.	.	PUNCT
cana-3303	224	1	moscow	moscow	PROPN
cana-3303	224	2	:	:	PUNCT
cana-3303	224	3	nauka	nauka	PROPN
cana-3303	224	4	.	.	PUNCT
cana-3303	225	1	[	[	X
cana-3303	225	2	11	11	NUM
cana-3303	225	3	]	]	X
cana-3303	225	4	dyachenko	dyachenko	NOUN
cana-3303	225	5	,	,	PUNCT
cana-3303	225	6	m.	m.	NOUN
cana-3303	225	7	s.	s.	PROPN
cana-3303	225	8	,	,	PUNCT
cana-3303	225	9	&	&	CCONJ
cana-3303	225	10	gulin	gulin	PROPN
cana-3303	225	11	,	,	PUNCT
cana-3303	225	12	a.	a.	PROPN
cana-3303	225	13	v.	v.	PROPN
cana-3303	225	14	(	(	PUNCT
cana-3303	225	15	2003	2003	NUM
cana-3303	225	16	)	)	PUNCT
cana-3303	225	17	.	.	PUNCT
cana-3303	226	1	discrete	discrete	ADJ
cana-3303	226	2	nonlinear	nonlinear	ADJ
cana-3303	226	3	equations	equation	NOUN
cana-3303	226	4	and	and	CCONJ
cana-3303	226	5	their	their	PRON
cana-3303	226	6	applications	application	NOUN
cana-3303	226	7	.	.	PUNCT
cana-3303	227	1	journal	journal	NOUN
cana-3303	227	2	of	of	ADP
cana-3303	227	3	computational	computational	ADJ
cana-3303	227	4	and	and	CCONJ
cana-3303	227	5	applied	applied	ADJ
cana-3303	227	6	mathematics	mathematic	NOUN
cana-3303	227	7	,	,	PUNCT
cana-3303	227	8	155(2	155(2	NUM
cana-3303	227	9	)	)	PUNCT
cana-3303	227	10	,	,	PUNCT
cana-3303	227	11	313	313	NUM
cana-3303	227	12	-	-	SYM
cana-3303	227	13	327	327	NUM
cana-3303	227	14	.	.	PUNCT
cana-3303	228	1	[	[	X
cana-3303	228	2	12	12	NUM
cana-3303	228	3	]	]	X
cana-3303	228	4	gantmacher	gantmacher	NOUN
cana-3303	228	5	,	,	PUNCT
cana-3303	228	6	f.	f.	PROPN
cana-3303	228	7	r.	r.	PROPN
cana-3303	228	8	(	(	PUNCT
cana-3303	228	9	1967	1967	NUM
cana-3303	228	10	)	)	PUNCT
cana-3303	228	11	.	.	PUNCT
cana-3303	229	1	matrix	matrix	NOUN
cana-3303	229	2	theory	theory	NOUN
cana-3303	229	3	(	(	PUNCT
cana-3303	229	4	p.	p.	NOUN
cana-3303	229	5	576	576	NUM
cana-3303	229	6	)	)	PUNCT
cana-3303	229	7	.	.	PUNCT
cana-3303	230	1	moscow	moscow	PROPN
cana-3303	230	2	:	:	PUNCT
cana-3303	230	3	nauka	nauka	PROPN
cana-3303	230	4	.	.	PUNCT
cana-3303	231	1	[	[	X
cana-3303	231	2	13	13	NUM
cana-3303	231	3	]	]	SYM
cana-3303	231	4	gelfond	gelfond	NOUN
cana-3303	231	5	,	,	PUNCT
cana-3303	231	6	a.	a.	NOUN
cana-3303	231	7	o.	o.	PROPN
cana-3303	231	8	(	(	PUNCT
cana-3303	231	9	1967	1967	NUM
cana-3303	231	10	)	)	PUNCT
cana-3303	231	11	.	.	PUNCT
cana-3303	232	1	the	the	DET
cana-3303	232	2	calculus	calculus	NOUN
cana-3303	232	3	of	of	ADP
cana-3303	232	4	finite	finite	ADJ
cana-3303	232	5	differences	difference	NOUN
cana-3303	232	6	(	(	PUNCT
cana-3303	232	7	p.	p.	NOUN
cana-3303	232	8	376	376	NUM
cana-3303	232	9	)	)	PUNCT
cana-3303	232	10	.	.	PUNCT
cana-3303	233	1	moscow	moscow	PROPN
cana-3303	233	2	:	:	PUNCT
cana-3303	233	3	nauka	nauka	PROPN
cana-3303	233	4	.	.	PUNCT
cana-3303	234	1	[	[	X
cana-3303	234	2	14	14	NUM
cana-3303	234	3	]	]	PUNCT
cana-3303	234	4	grikomi	grikomi	NOUN
cana-3303	234	5	,	,	PUNCT
cana-3303	234	6	f.	f.	PROPN
cana-3303	234	7	(	(	PUNCT
cana-3303	234	8	1962	1962	NUM
cana-3303	234	9	)	)	PUNCT
cana-3303	234	10	.	.	PUNCT
cana-3303	235	1	differential	differential	ADJ
cana-3303	235	2	equations	equation	NOUN
cana-3303	235	3	il	il	PROPN
cana-3303	235	4	(	(	PUNCT
cana-3303	235	5	p.	p.	NOUN
cana-3303	235	6	352	352	NUM
cana-3303	235	7	)	)	PUNCT
cana-3303	235	8	.	.	PUNCT
cana-3303	236	1	moscow	moscow	PROPN
cana-3303	236	2	:	:	PUNCT
cana-3303	237	1	i.l	i.l	X
cana-3303	237	2	.	.	PUNCT
cana-3303	238	1	[	[	X
cana-3303	238	2	15	15	NUM
cana-3303	238	3	]	]	X
cana-3303	238	4	gurskiy	gurskiy	NOUN
cana-3303	238	5	,	,	PUNCT
cana-3303	238	6	a.	a.	PROPN
cana-3303	238	7	s.	s.	PROPN
cana-3303	238	8	,	,	PUNCT
cana-3303	238	9	&	&	CCONJ
cana-3303	238	10	sokolov	sokolov	PROPN
cana-3303	238	11	,	,	PUNCT
cana-3303	238	12	a.	a.	NOUN
cana-3303	238	13	p.	p.	NOUN
cana-3303	238	14	(	(	PUNCT
cana-3303	238	15	2010	2010	NUM
cana-3303	238	16	)	)	PUNCT
cana-3303	238	17	.	.	PUNCT
cana-3303	239	1	discrete	discrete	ADJ
cana-3303	239	2	additive	additive	NOUN
cana-3303	239	3	and	and	CCONJ
cana-3303	239	4	multiplicative	multiplicative	ADJ
cana-3303	239	5	derivative	derivative	ADJ
cana-3303	239	6	equations	equation	NOUN
cana-3303	239	7	:	:	PUNCT
cana-3303	239	8	theoretical	theoretical	ADJ
cana-3303	239	9	approaches	approach	NOUN
cana-3303	239	10	.	.	PUNCT
cana-3303	240	1	mathematical	mathematical	ADJ
cana-3303	240	2	methods	method	NOUN
cana-3303	240	3	in	in	ADP
cana-3303	240	4	the	the	DET
cana-3303	240	5	applied	apply	VERB
cana-3303	240	6	sciences	science	NOUN
cana-3303	240	7	,	,	PUNCT
cana-3303	240	8	33(12	33(12	NUM
cana-3303	240	9	)	)	PUNCT
cana-3303	240	10	,	,	PUNCT
cana-3303	240	11	1509	1509	NUM
cana-3303	240	12	-	-	SYM
cana-3303	240	13	1521	1521	NUM
cana-3303	240	14	.	.	PUNCT
cana-3303	241	1	[	[	X
cana-3303	241	2	16	16	NUM
cana-3303	241	3	]	]	X
cana-3303	241	4	hassani	hassani	PROPN
cana-3303	241	5	,	,	PUNCT
cana-3303	241	6	o.	o.	PROPN
cana-3303	241	7	h.	h.	PROPN
cana-3303	241	8	,	,	PUNCT
cana-3303	241	9	&	&	CCONJ
cana-3303	241	10	aliev	aliev	PROPN
cana-3303	241	11	,	,	PUNCT
cana-3303	241	12	n.	n.	NOUN
cana-3303	241	13	(	(	PUNCT
cana-3303	241	14	2008	2008	NUM
cana-3303	241	15	)	)	PUNCT
cana-3303	241	16	.	.	PUNCT
cana-3303	242	1	analytic	analytic	ADJ
cana-3303	242	2	approach	approach	NOUN
cana-3303	242	3	to	to	PART
cana-3303	242	4	solve	solve	VERB
cana-3303	242	5	specific	specific	ADJ
cana-3303	242	6	linear	linear	ADJ
cana-3303	242	7	and	and	CCONJ
cana-3303	242	8	nonlinear	nonlinear	ADJ
cana-3303	242	9	differential	differential	ADJ
cana-3303	242	10	equations	equation	NOUN
cana-3303	242	11	.	.	PUNCT
cana-3303	243	1	joint	joint	ADJ
cana-3303	243	2	mathematics	mathematics	PROPN
cana-3303	243	3	forum	forum	PROPN
cana-3303	243	4	journal	journal	PROPN
cana-3303	243	5	for	for	ADP
cana-3303	243	6	theory	theory	NOUN
cana-3303	243	7	and	and	CCONJ
cana-3303	243	8	applications	application	NOUN
cana-3303	243	9	,	,	PUNCT
cana-3303	243	10	3	3	NUM
cana-3303	243	11	,	,	PUNCT
cana-3303	243	12	1623–1631	1623–1631	NUM
cana-3303	243	13	.	.	PUNCT
cana-3303	244	1	[	[	X
cana-3303	244	2	17	17	NUM
cana-3303	244	3	]	]	X
cana-3303	244	4	mammadov	mammadov	PROPN
cana-3303	244	5	,	,	PUNCT
cana-3303	244	6	yu	yu	PROPN
cana-3303	244	7	.	.	PUNCT
cana-3303	244	8	a.	a.	PROPN
cana-3303	244	9	,	,	PUNCT
cana-3303	244	10	&	&	CCONJ
cana-3303	244	11	khankishiyev	khankishiyev	PROPN
cana-3303	244	12	,	,	PUNCT
cana-3303	244	13	z.	z.	PROPN
cana-3303	244	14	f.	f.	PROPN
cana-3303	244	15	(	(	PUNCT
cana-3303	244	16	2013	2013	NUM
cana-3303	244	17	)	)	PUNCT
cana-3303	244	18	.	.	PUNCT
cana-3303	245	1	differential	differential	ADJ
cana-3303	245	2	equations	equation	NOUN
cana-3303	245	3	(	(	PUNCT
cana-3303	245	4	p.	p.	NOUN
cana-3303	245	5	190	190	NUM
cana-3303	245	6	)	)	PUNCT
cana-3303	245	7	.	.	PUNCT
cana-3303	246	1	baku	baku	PROPN
cana-3303	246	2	.	.	PUNCT
cana-3303	247	1	[	[	X
cana-3303	247	2	18	18	NUM
cana-3303	247	3	]	]	X
cana-3303	247	4	mammadzade	mammadzade	NOUN
cana-3303	247	5	,	,	PUNCT
cana-3303	247	6	a.	a.	NOUN
cana-3303	247	7	(	(	PUNCT
cana-3303	247	8	2017	2017	NUM
cana-3303	247	9	)	)	PUNCT
cana-3303	247	10	.	.	PUNCT
cana-3303	248	1	properties	property	NOUN
cana-3303	248	2	of	of	ADP
cana-3303	248	3	discrete	discrete	ADJ
cana-3303	248	4	new	new	ADJ
cana-3303	248	5	derivative	derivative	NOUN
cana-3303	248	6	.	.	PUNCT
cana-3303	249	1	in	in	ADP
cana-3303	249	2	proceedings	proceeding	NOUN
cana-3303	249	3	of	of	ADP
cana-3303	249	4	the	the	DET
cana-3303	249	5	republican	republican	PROPN
cana-3303	249	6	scientific	scientific	ADJ
cana-3303	249	7	conference	conference	NOUN
cana-3303	249	8	of	of	ADP
cana-3303	249	9	young	young	ADJ
cana-3303	249	10	researchers	researcher	NOUN
cana-3303	249	11	on	on	ADP
cana-3303	249	12	“	"	PUNCT
cana-3303	249	13	modernizing	modernize	VERB
cana-3303	249	14	azerbaijan	azerbaijan	PROPN
cana-3303	249	15	:	:	PUNCT
cana-3303	249	16	the	the	DET
cana-3303	249	17	rise	rise	NOUN
cana-3303	249	18	of	of	ADP
cana-3303	249	19	the	the	DET
cana-3303	249	20	rise	rise	NOUN
cana-3303	249	21	”	"	PUNCT
cana-3303	249	22	(	(	PUNCT
cana-3303	249	23	p.	p.	NOUN
cana-3303	249	24	29	29	NUM
cana-3303	249	25	-	-	SYM
cana-3303	249	26	30	30	NUM
cana-3303	249	27	)	)	PUNCT
cana-3303	249	28	.	.	PUNCT
cana-3303	250	1	lankaran	lankaran	ADJ
cana-3303	250	2	.	.	PUNCT
cana-3303	251	1	[	[	X
cana-3303	251	2	19	19	NUM
cana-3303	251	3	]	]	X
cana-3303	251	4	smith	smith	PROPN
cana-3303	251	5	,	,	PUNCT
cana-3303	251	6	l.	l.	PROPN
cana-3303	251	7	m.	m.	PROPN
cana-3303	251	8	,	,	PUNCT
cana-3303	251	9	&	&	CCONJ
cana-3303	251	10	jones	jones	PROPN
cana-3303	251	11	,	,	PUNCT
cana-3303	251	12	r.	r.	PROPN
cana-3303	251	13	l.	l.	PROPN
cana-3303	251	14	(	(	PUNCT
cana-3303	251	15	2007	2007	NUM
cana-3303	251	16	)	)	PUNCT
cana-3303	251	17	.	.	PUNCT
cana-3303	252	1	boundary	boundary	ADJ
cana-3303	252	2	value	value	NOUN
cana-3303	252	3	problems	problem	NOUN
cana-3303	252	4	for	for	ADP
cana-3303	252	5	discrete	discrete	ADJ
cana-3303	252	6	derivatives	derivative	NOUN
cana-3303	252	7	and	and	CCONJ
cana-3303	252	8	integrals	integral	NOUN
cana-3303	252	9	.	.	PUNCT
cana-3303	253	1	siam	siam	PROPN
cana-3303	253	2	journal	journal	PROPN
cana-3303	253	3	on	on	ADP
cana-3303	253	4	mathematical	mathematical	ADJ
cana-3303	253	5	analysis	analysis	NOUN
cana-3303	253	6	,	,	PUNCT
cana-3303	253	7	35(4	35(4	NUM
cana-3303	253	8	)	)	PUNCT
cana-3303	253	9	,	,	PUNCT
cana-3303	253	10	781	781	NUM
cana-3303	253	11	-	-	SYM
cana-3303	253	12	795	795	NUM
cana-3303	253	13	.	.	PUNCT