id	sid	tid	token	lemma	pos
cana-3670	1	1	communications	communication	NOUN
cana-3670	1	2	on	on	ADP
cana-3670	1	3	applied	apply	VERB
cana-3670	1	4	nonlinear	nonlinear	ADJ
cana-3670	1	5	analysis	analysis	NOUN
cana-3670	1	6	issn	issn	NOUN
cana-3670	1	7	:	:	PUNCT
cana-3670	1	8	1074	1074	NUM
cana-3670	1	9	-	-	PUNCT
cana-3670	1	10	133x	133x	NUM
cana-3670	1	11	vol	vol	NOUN
cana-3670	1	12	32	32	NUM
cana-3670	1	13	no	no	NOUN
cana-3670	1	14	.	.	PUNCT
cana-3670	2	1	8s	8s	PROPN
cana-3670	2	2	(	(	PUNCT
cana-3670	2	3	2025	2025	NUM
cana-3670	2	4	)	)	PUNCT
cana-3670	2	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	2	6	271	271	NUM
cana-3670	2	7	a	a	DET
cana-3670	2	8	numerical	numerical	ADJ
cana-3670	2	9	approach	approach	NOUN
cana-3670	2	10	for	for	ADP
cana-3670	2	11	solving	solve	VERB
cana-3670	2	12	fuzzy	fuzzy	ADJ
cana-3670	2	13	differential	differential	ADJ
cana-3670	2	14	equation	equation	NOUN
cana-3670	2	15	using	use	VERB
cana-3670	2	16	enhanced	enhance	VERB
cana-3670	2	17	euler	euler	PROPN
cana-3670	2	18	’s	’s	PART
cana-3670	2	19	methods	method	NOUN
cana-3670	2	20	based	base	VERB
cana-3670	2	21	on	on	ADP
cana-3670	2	22	contra	contra	PROPN
cana-3670	2	23	harmonic	harmonic	PROPN
cana-3670	2	24	mean	mean	NOUN
cana-3670	2	25	and	and	CCONJ
cana-3670	2	26	centroidal	centroidal	PROPN
cana-3670	2	27	mean	mean	PROPN
cana-3670	2	28	a.	a.	NOUN
cana-3670	2	29	hari	hari	PROPN
cana-3670	2	30	ganesh1	ganesh1	PROPN
cana-3670	2	31	*	*	PROPN
cana-3670	2	32	,	,	PUNCT
cana-3670	2	33	r.	r.	PROPN
cana-3670	2	34	diwahar2	diwahar2	PROPN
cana-3670	2	35	1*assistant	1*assistant	NUM
cana-3670	2	36	professor	professor	NOUN
cana-3670	2	37	and	and	CCONJ
cana-3670	2	38	2research	2research	NUM
cana-3670	2	39	scholar	scholar	NOUN
cana-3670	2	40	(	(	PUNCT
cana-3670	2	41	full	full	ADJ
cana-3670	2	42	time	time	NOUN
cana-3670	2	43	)	)	PUNCT
cana-3670	2	44	,	,	PUNCT
cana-3670	2	45	pg	pg	PROPN
cana-3670	2	46	&	&	CCONJ
cana-3670	2	47	research	research	PROPN
cana-3670	2	48	department	department	PROPN
cana-3670	2	49	of	of	ADP
cana-3670	2	50	mathematic	mathematic	PROPN
cana-3670	2	51	,	,	PUNCT
cana-3670	2	52	poompuhar	poompuhar	PROPN
cana-3670	2	53	college	college	NOUN
cana-3670	2	54	(	(	PUNCT
cana-3670	2	55	autonomous	autonomous	ADJ
cana-3670	2	56	)	)	PUNCT
cana-3670	2	57	,	,	PUNCT
cana-3670	2	58	(	(	PUNCT
cana-3670	2	59	affiliated	affiliate	VERB
cana-3670	2	60	to	to	PART
cana-3670	2	61	bharathidasan	bharathidasan	VERB
cana-3670	2	62	university	university	NOUN
cana-3670	2	63	)	)	PUNCT
cana-3670	2	64	,	,	PUNCT
cana-3670	2	65	melaiyur	melaiyur	NOUN
cana-3670	2	66	–	–	PUNCT
cana-3670	2	67	609	609	NUM
cana-3670	2	68	107	107	NUM
cana-3670	2	69	,	,	PUNCT
cana-3670	2	70	mayiladuthurai	mayiladuthurai	X
cana-3670	2	71	(	(	PUNCT
cana-3670	2	72	dt	dt	PROPN
cana-3670	2	73	.	.	PUNCT
cana-3670	2	74	)	)	PUNCT
cana-3670	2	75	,	,	PUNCT
cana-3670	2	76	tamil	tamil	PROPN
cana-3670	2	77	nadu	nadu	PROPN
cana-3670	2	78	,	,	PUNCT
cana-3670	2	79	india	india	PROPN
cana-3670	2	80	.	.	PUNCT
cana-3670	3	1	article	article	PROPN
cana-3670	3	2	history	history	NOUN
cana-3670	3	3	:	:	PUNCT
cana-3670	3	4	received	receive	VERB
cana-3670	3	5	:	:	PUNCT
cana-3670	3	6	24	24	NUM
cana-3670	3	7	-	-	SYM
cana-3670	3	8	10	10	NUM
cana-3670	3	9	-	-	PUNCT
cana-3670	3	10	2024	2024	NUM
cana-3670	3	11	revised:25	revised:25	NOUN
cana-3670	3	12	-	-	SYM
cana-3670	3	13	11	11	NUM
cana-3670	3	14	-	-	PUNCT
cana-3670	3	15	2024	2024	NUM
cana-3670	3	16	accepted:27	accepted:27	PROPN
cana-3670	3	17	-	-	PUNCT
cana-3670	3	18	12	12	NUM
cana-3670	3	19	-	-	PUNCT
cana-3670	3	20	2024	2024	NUM
cana-3670	3	21	abstract	abstract	NOUN
cana-3670	3	22	:	:	PUNCT
cana-3670	3	23	objectives	objective	NOUN
cana-3670	3	24	:	:	PUNCT
cana-3670	3	25	fuzzy	fuzzy	ADJ
cana-3670	3	26	differential	differential	ADJ
cana-3670	3	27	equations	equation	NOUN
cana-3670	3	28	(	(	PUNCT
cana-3670	3	29	fdes	fde	NOUN
cana-3670	3	30	)	)	PUNCT
cana-3670	3	31	are	be	AUX
cana-3670	3	32	crucial	crucial	ADJ
cana-3670	3	33	for	for	ADP
cana-3670	3	34	modeling	model	VERB
cana-3670	3	35	dynamic	dynamic	ADJ
cana-3670	3	36	systems	system	NOUN
cana-3670	3	37	in	in	ADP
cana-3670	3	38	science	science	NOUN
cana-3670	3	39	,	,	PUNCT
cana-3670	3	40	economics	economic	NOUN
cana-3670	3	41	,	,	PUNCT
cana-3670	3	42	and	and	CCONJ
cana-3670	3	43	engineering	engineering	NOUN
cana-3670	3	44	due	due	ADP
cana-3670	3	45	to	to	ADP
cana-3670	3	46	their	their	PRON
cana-3670	3	47	unpredictable	unpredictable	ADJ
cana-3670	3	48	nature	nature	NOUN
cana-3670	3	49	and	and	CCONJ
cana-3670	3	50	the	the	DET
cana-3670	3	51	need	need	NOUN
cana-3670	3	52	for	for	ADP
cana-3670	3	53	numerical	numerical	ADJ
cana-3670	3	54	methods	method	NOUN
cana-3670	3	55	for	for	ADP
cana-3670	3	56	precise	precise	ADJ
cana-3670	3	57	solutions	solution	NOUN
cana-3670	3	58	.	.	PUNCT
cana-3670	4	1	the	the	DET
cana-3670	4	2	objective	objective	NOUN
cana-3670	4	3	of	of	ADP
cana-3670	4	4	this	this	DET
cana-3670	4	5	work	work	NOUN
cana-3670	4	6	is	be	AUX
cana-3670	4	7	to	to	PART
cana-3670	4	8	develop	develop	VERB
cana-3670	4	9	improved	improve	VERB
cana-3670	4	10	euler	euler	NOUN
cana-3670	4	11	's	's	PART
cana-3670	4	12	techniques	technique	NOUN
cana-3670	4	13	for	for	ADP
cana-3670	4	14	numerically	numerically	ADV
cana-3670	4	15	solving	solve	VERB
cana-3670	4	16	fuzzy	fuzzy	ADJ
cana-3670	4	17	differential	differential	ADJ
cana-3670	4	18	equations	equation	NOUN
cana-3670	4	19	(	(	PUNCT
cana-3670	4	20	fdes	fde	NOUN
cana-3670	4	21	)	)	PUNCT
cana-3670	4	22	in	in	ADP
cana-3670	4	23	order	order	NOUN
cana-3670	4	24	to	to	PART
cana-3670	4	25	produce	produce	VERB
cana-3670	4	26	approximate	approximate	ADJ
cana-3670	4	27	solutions	solution	NOUN
cana-3670	4	28	for	for	ADP
cana-3670	4	29	problems	problem	NOUN
cana-3670	4	30	that	that	PRON
cana-3670	4	31	are	be	AUX
cana-3670	4	32	too	too	ADV
cana-3670	4	33	complicated	complicated	ADJ
cana-3670	4	34	to	to	PART
cana-3670	4	35	be	be	AUX
cana-3670	4	36	stated	state	VERB
cana-3670	4	37	exactly	exactly	ADV
cana-3670	4	38	.	.	PUNCT
cana-3670	5	1	methods	method	NOUN
cana-3670	5	2	:	:	PUNCT
cana-3670	5	3	this	this	DET
cana-3670	5	4	study	study	NOUN
cana-3670	5	5	proposes	propose	VERB
cana-3670	5	6	improved	improved	ADJ
cana-3670	5	7	euler	euler	NOUN
cana-3670	5	8	's	's	PART
cana-3670	5	9	approaches	approach	NOUN
cana-3670	5	10	for	for	ADP
cana-3670	5	11	solving	solve	VERB
cana-3670	5	12	differential	differential	ADJ
cana-3670	5	13	equations	equation	NOUN
cana-3670	5	14	with	with	ADP
cana-3670	5	15	fuzzy	fuzzy	ADJ
cana-3670	5	16	initial	initial	ADJ
cana-3670	5	17	conditions	condition	NOUN
cana-3670	5	18	,	,	PUNCT
cana-3670	5	19	including	include	VERB
cana-3670	5	20	euler	euler	NOUN
cana-3670	5	21	's	's	PART
cana-3670	5	22	method	method	NOUN
cana-3670	5	23	,	,	PUNCT
cana-3670	5	24	modified	modify	VERB
cana-3670	5	25	euler	euler	NOUN
cana-3670	5	26	's	's	PART
cana-3670	5	27	method	method	NOUN
cana-3670	5	28	,	,	PUNCT
cana-3670	5	29	and	and	CCONJ
cana-3670	5	30	improved	improve	VERB
cana-3670	5	31	euler	euler	NOUN
cana-3670	5	32	's	's	PART
cana-3670	5	33	method	method	NOUN
cana-3670	5	34	employing	employ	VERB
cana-3670	5	35	contra	contra	PROPN
cana-3670	5	36	harmonic	harmonic	PROPN
cana-3670	5	37	mean	mean	NOUN
cana-3670	5	38	and	and	CCONJ
cana-3670	5	39	centroidal	centroidal	PROPN
cana-3670	5	40	mean	mean	NOUN
cana-3670	5	41	.	.	PUNCT
cana-3670	6	1	based	base	VERB
cana-3670	6	2	on	on	ADP
cana-3670	6	3	zadeh	zadeh	PROPN
cana-3670	6	4	's	's	PART
cana-3670	6	5	extension	extension	NOUN
cana-3670	6	6	concept	concept	NOUN
cana-3670	6	7	for	for	ADP
cana-3670	6	8	fuzzy	fuzzy	ADJ
cana-3670	6	9	sets	set	NOUN
cana-3670	6	10	,	,	PUNCT
cana-3670	6	11	we	we	PRON
cana-3670	6	12	extend	extend	VERB
cana-3670	6	13	euler	euler	NOUN
cana-3670	6	14	's	's	PART
cana-3670	6	15	classical	classical	ADJ
cana-3670	6	16	methods	method	NOUN
cana-3670	6	17	to	to	PART
cana-3670	6	18	address	address	VERB
cana-3670	6	19	this	this	DET
cana-3670	6	20	dependence	dependence	NOUN
cana-3670	6	21	problem	problem	NOUN
cana-3670	6	22	in	in	ADP
cana-3670	6	23	a	a	DET
cana-3670	6	24	fuzzy	fuzzy	ADJ
cana-3670	6	25	setting	setting	NOUN
cana-3670	6	26	.	.	PUNCT
cana-3670	7	1	findings	finding	NOUN
cana-3670	7	2	:	:	PUNCT
cana-3670	7	3	a	a	DET
cana-3670	7	4	numerical	numerical	ADJ
cana-3670	7	5	example	example	NOUN
cana-3670	7	6	that	that	PRON
cana-3670	7	7	contrasts	contrast	VERB
cana-3670	7	8	the	the	DET
cana-3670	7	9	suggested	suggest	VERB
cana-3670	7	10	approaches	approach	NOUN
cana-3670	7	11	with	with	ADP
cana-3670	7	12	the	the	DET
cana-3670	7	13	traditional	traditional	ADJ
cana-3670	7	14	euler	euler	NOUN
cana-3670	7	15	's	's	PART
cana-3670	7	16	method	method	NOUN
cana-3670	7	17	is	be	AUX
cana-3670	7	18	provided	provide	VERB
cana-3670	7	19	.	.	PUNCT
cana-3670	8	1	the	the	DET
cana-3670	8	2	efficiency	efficiency	NOUN
cana-3670	8	3	of	of	ADP
cana-3670	8	4	the	the	DET
cana-3670	8	5	suggested	suggest	VERB
cana-3670	8	6	methods	method	NOUN
cana-3670	8	7	is	be	AUX
cana-3670	8	8	shown	show	VERB
cana-3670	8	9	by	by	ADP
cana-3670	8	10	numerical	numerical	ADJ
cana-3670	8	11	solutions	solution	NOUN
cana-3670	8	12	and	and	CCONJ
cana-3670	8	13	their	their	PRON
cana-3670	8	14	geometrical	geometrical	ADJ
cana-3670	8	15	representations	representation	NOUN
cana-3670	8	16	.	.	PUNCT
cana-3670	9	1	furthermore	furthermore	ADV
cana-3670	9	2	,	,	PUNCT
cana-3670	9	3	the	the	DET
cana-3670	9	4	numerical	numerical	ADJ
cana-3670	9	5	example	example	NOUN
cana-3670	9	6	demonstrates	demonstrate	VERB
cana-3670	9	7	the	the	DET
cana-3670	9	8	acceptable	acceptable	ADJ
cana-3670	9	9	accuracy	accuracy	NOUN
cana-3670	9	10	of	of	ADP
cana-3670	9	11	the	the	DET
cana-3670	9	12	improved	improve	VERB
cana-3670	9	13	euler	euler	NOUN
cana-3670	9	14	's	's	PART
cana-3670	9	15	techniques	technique	NOUN
cana-3670	9	16	.	.	PUNCT
cana-3670	10	1	novelty	novelty	NOUN
cana-3670	10	2	:	:	PUNCT
cana-3670	10	3	in	in	ADP
cana-3670	10	4	this	this	DET
cana-3670	10	5	research	research	NOUN
cana-3670	10	6	proposal	proposal	NOUN
cana-3670	10	7	,	,	PUNCT
cana-3670	10	8	we	we	PRON
cana-3670	10	9	sought	seek	VERB
cana-3670	10	10	to	to	PART
cana-3670	10	11	solve	solve	VERB
cana-3670	10	12	first	first	ADJ
cana-3670	10	13	-	-	PUNCT
cana-3670	10	14	order	order	NOUN
cana-3670	10	15	differential	differential	ADJ
cana-3670	10	16	equations	equation	NOUN
cana-3670	10	17	utilizing	utilize	VERB
cana-3670	10	18	two	two	NUM
cana-3670	10	19	creative	creative	ADJ
cana-3670	10	20	approaches	approach	NOUN
cana-3670	10	21	:	:	PUNCT
cana-3670	10	22	the	the	DET
cana-3670	10	23	contra	contra	PROPN
cana-3670	10	24	harmonic	harmonic	PROPN
cana-3670	10	25	mean	mean	NOUN
cana-3670	10	26	and	and	CCONJ
cana-3670	10	27	the	the	DET
cana-3670	10	28	centroidal	centroidal	ADJ
cana-3670	10	29	mean	mean	NOUN
cana-3670	10	30	of	of	ADP
cana-3670	10	31	enhanced	enhance	VERB
cana-3670	10	32	euler	euler	NOUN
cana-3670	10	33	's	's	PART
cana-3670	10	34	approaches	approach	NOUN
cana-3670	10	35	for	for	ADP
cana-3670	10	36	fuzzy	fuzzy	ADJ
cana-3670	10	37	primary	primary	ADJ
cana-3670	10	38	value	value	NOUN
cana-3670	10	39	.	.	PUNCT
cana-3670	11	1	the	the	DET
cana-3670	11	2	suggested	suggest	VERB
cana-3670	11	3	methods	method	NOUN
cana-3670	11	4	yield	yield	VERB
cana-3670	11	5	great	great	ADJ
cana-3670	11	6	results	result	NOUN
cana-3670	11	7	and	and	CCONJ
cana-3670	11	8	are	be	AUX
cana-3670	11	9	quick	quick	ADJ
cana-3670	11	10	,	,	PUNCT
cana-3670	11	11	accurate	accurate	ADJ
cana-3670	11	12	,	,	PUNCT
cana-3670	11	13	and	and	CCONJ
cana-3670	11	14	easy	easy	ADJ
cana-3670	11	15	to	to	PART
cana-3670	11	16	use	use	VERB
cana-3670	11	17	.	.	PUNCT
cana-3670	12	1	keywords	keyword	NOUN
cana-3670	12	2	:	:	PUNCT
cana-3670	12	3	fuzzy	fuzzy	ADJ
cana-3670	12	4	initial	initial	ADJ
cana-3670	12	5	value	value	NOUN
cana-3670	12	6	problem	problem	NOUN
cana-3670	12	7	,	,	PUNCT
cana-3670	12	8	euler	euler	NOUN
cana-3670	12	9	’s	’s	PART
cana-3670	12	10	method	method	NOUN
cana-3670	12	11	,	,	PUNCT
cana-3670	12	12	modified	modified	PROPN
cana-3670	12	13	euler	euler	PROPN
cana-3670	12	14	’s	’s	PART
cana-3670	12	15	method	method	NOUN
cana-3670	12	16	,	,	PUNCT
cana-3670	12	17	improved	improve	VERB
cana-3670	12	18	euler	euler	NOUN
cana-3670	12	19	’s	’s	PART
cana-3670	12	20	method	method	NOUN
cana-3670	12	21	,	,	PUNCT
cana-3670	12	22	contra	contra	PROPN
cana-3670	12	23	harmonic	harmonic	PROPN
cana-3670	12	24	mean	mean	VERB
cana-3670	12	25	,	,	PUNCT
cana-3670	12	26	centroidal	centroidal	NOUN
cana-3670	12	27	mean	mean	NOUN
cana-3670	12	28	.	.	PUNCT
cana-3670	13	1	1	1	X
cana-3670	13	2	.	.	X
cana-3670	13	3	introduction	introduction	NOUN
cana-3670	13	4	:	:	PUNCT
cana-3670	13	5	since	since	SCONJ
cana-3670	13	6	most	most	ADJ
cana-3670	13	7	differential	differential	ADJ
cana-3670	13	8	equations	equation	NOUN
cana-3670	13	9	can	can	AUX
cana-3670	13	10	not	not	PART
cana-3670	13	11	be	be	AUX
cana-3670	13	12	solved	solve	VERB
cana-3670	13	13	directly	directly	ADV
cana-3670	13	14	and	and	CCONJ
cana-3670	13	15	must	must	AUX
cana-3670	13	16	instead	instead	ADV
cana-3670	13	17	be	be	AUX
cana-3670	13	18	approximated	approximate	VERB
cana-3670	13	19	by	by	ADP
cana-3670	13	20	approximation	approximation	NOUN
cana-3670	13	21	,	,	PUNCT
cana-3670	13	22	numerical	numerical	ADJ
cana-3670	13	23	methods	method	NOUN
cana-3670	13	24	are	be	AUX
cana-3670	13	25	crucial	crucial	ADJ
cana-3670	13	26	.	.	PUNCT
cana-3670	14	1	of	of	ADP
cana-3670	14	2	all	all	DET
cana-3670	14	3	the	the	DET
cana-3670	14	4	numerical	numerical	ADJ
cana-3670	14	5	techniques	technique	NOUN
cana-3670	14	6	,	,	PUNCT
cana-3670	14	7	euler	euler	PROPN
cana-3670	14	8	's	's	PART
cana-3670	14	9	method	method	NOUN
cana-3670	14	10	is	be	AUX
cana-3670	14	11	the	the	DET
cana-3670	14	12	most	most	ADV
cana-3670	14	13	straightforward	straightforward	ADJ
cana-3670	14	14	.	.	PUNCT
cana-3670	15	1	its	its	PRON
cana-3670	15	2	foundation	foundation	NOUN
cana-3670	15	3	is	be	AUX
cana-3670	15	4	using	use	VERB
cana-3670	15	5	a	a	DET
cana-3670	15	6	series	series	NOUN
cana-3670	15	7	of	of	ADP
cana-3670	15	8	successively	successively	ADV
cana-3670	15	9	calculated	calculate	VERB
cana-3670	15	10	tangent	tangent	ADJ
cana-3670	15	11	line	line	NOUN
cana-3670	15	12	approximations	approximation	NOUN
cana-3670	15	13	to	to	PART
cana-3670	15	14	approximate	approximate	VERB
cana-3670	15	15	the	the	DET
cana-3670	15	16	graph	graph	NOUN
cana-3670	15	17	of	of	ADP
cana-3670	15	18	a	a	DET
cana-3670	15	19	solution	solution	NOUN
cana-3670	15	20	,	,	PUNCT
cana-3670	15	21	y(x	y(x	PROPN
cana-3670	15	22	)	)	PUNCT
cana-3670	15	23	.	.	PUNCT
cana-3670	16	1	furthermore	furthermore	ADV
cana-3670	16	2	,	,	PUNCT
cana-3670	16	3	euler	euler	PROPN
cana-3670	16	4	's	's	PART
cana-3670	16	5	method	method	NOUN
cana-3670	16	6	is	be	AUX
cana-3670	16	7	the	the	DET
cana-3670	16	8	foundation	foundation	NOUN
cana-3670	16	9	for	for	ADP
cana-3670	16	10	the	the	DET
cana-3670	16	11	majority	majority	NOUN
cana-3670	16	12	of	of	ADP
cana-3670	16	13	other	other	ADJ
cana-3670	16	14	approaches	approach	NOUN
cana-3670	16	15	.	.	PUNCT
cana-3670	17	1	these	these	DET
cana-3670	17	2	days	day	NOUN
cana-3670	17	3	,	,	PUNCT
cana-3670	17	4	researchers	researcher	NOUN
cana-3670	17	5	are	be	AUX
cana-3670	17	6	interested	interested	ADJ
cana-3670	17	7	in	in	ADP
cana-3670	17	8	learning	learn	VERB
cana-3670	17	9	how	how	SCONJ
cana-3670	17	10	to	to	PART
cana-3670	17	11	use	use	VERB
cana-3670	17	12	the	the	DET
cana-3670	17	13	fuzzy	fuzzy	ADJ
cana-3670	17	14	version	version	NOUN
cana-3670	17	15	of	of	ADP
cana-3670	17	16	the	the	DET
cana-3670	17	17	euler	euler	NOUN
cana-3670	17	18	method	method	NOUN
cana-3670	17	19	to	to	PART
cana-3670	17	20	solve	solve	VERB
cana-3670	17	21	fuzzy	fuzzy	ADJ
cana-3670	17	22	differential	differential	ADJ
cana-3670	17	23	equations	equation	NOUN
cana-3670	18	1	[	[	X
cana-3670	18	2	1	1	NUM
cana-3670	18	3	,	,	PUNCT
cana-3670	18	4	2	2	NUM
cana-3670	18	5	]	]	PUNCT
cana-3670	18	6	.	.	PUNCT
cana-3670	19	1	a	a	DET
cana-3670	19	2	modified	modify	VERB
cana-3670	19	3	euler	euler	NOUN
cana-3670	19	4	technique	technique	NOUN
cana-3670	19	5	for	for	ADP
cana-3670	19	6	solving	solve	VERB
cana-3670	19	7	fuzzy	fuzzy	ADJ
cana-3670	19	8	differential	differential	ADJ
cana-3670	19	9	equations	equation	NOUN
cana-3670	19	10	was	be	AUX
cana-3670	19	11	described	describe	VERB
cana-3670	19	12	in	in	ADP
cana-3670	19	13	[	[	X
cana-3670	19	14	3	3	NUM
cana-3670	19	15	]	]	PUNCT
cana-3670	19	16	.	.	PUNCT
cana-3670	20	1	the	the	DET
cana-3670	20	2	graphical	graphical	ADJ
cana-3670	20	3	comparison	comparison	NOUN
cana-3670	20	4	of	of	ADP
cana-3670	20	5	the	the	DET
cana-3670	20	6	fourth	fourth	ADJ
cana-3670	20	7	order	order	NOUN
cana-3670	20	8	rk	rk	NOUN
cana-3670	20	9	technique	technique	NOUN
cana-3670	20	10	,	,	PUNCT
cana-3670	20	11	modified	modify	VERB
cana-3670	20	12	euler	euler	NOUN
cana-3670	20	13	's	's	PART
cana-3670	20	14	method	method	NOUN
cana-3670	20	15	and	and	CCONJ
cana-3670	20	16	euler	euler	NOUN
cana-3670	20	17	's	's	PART
cana-3670	20	18	method	method	NOUN
cana-3670	20	19	on	on	ADP
cana-3670	20	20	numerical	numerical	ADJ
cana-3670	20	21	solutions	solution	NOUN
cana-3670	20	22	of	of	ADP
cana-3670	20	23	the	the	DET
cana-3670	20	24	first	first	ADJ
cana-3670	20	25	order	order	NOUN
cana-3670	20	26	initial	initial	ADJ
cana-3670	20	27	communications	communication	NOUN
cana-3670	20	28	on	on	ADP
cana-3670	20	29	applied	apply	VERB
cana-3670	20	30	nonlinear	nonlinear	ADJ
cana-3670	20	31	analysis	analysis	NOUN
cana-3670	20	32	issn	issn	NOUN
cana-3670	20	33	:	:	PUNCT
cana-3670	20	34	1074	1074	NUM
cana-3670	20	35	-	-	PUNCT
cana-3670	20	36	133x	133x	NUM
cana-3670	20	37	vol	vol	NOUN
cana-3670	20	38	32	32	NUM
cana-3670	20	39	no	no	NOUN
cana-3670	20	40	.	.	PUNCT
cana-3670	21	1	8s	8s	PROPN
cana-3670	21	2	(	(	PUNCT
cana-3670	21	3	2025	2025	NUM
cana-3670	21	4	)	)	PUNCT
cana-3670	21	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	21	6	272	272	NUM
cana-3670	21	7	value	value	NOUN
cana-3670	21	8	problem	problem	NOUN
cana-3670	21	9	was	be	AUX
cana-3670	21	10	discussed	discuss	VERB
cana-3670	21	11	in	in	ADP
cana-3670	21	12	[	[	X
cana-3670	21	13	4	4	NUM
cana-3670	21	14	]	]	PUNCT
cana-3670	21	15	and	and	CCONJ
cana-3670	21	16	[	[	X
cana-3670	21	17	5	5	NUM
cana-3670	21	18	]	]	PUNCT
cana-3670	21	19	.	.	PUNCT
cana-3670	22	1	in	in	ADP
cana-3670	22	2	order	order	NOUN
cana-3670	22	3	to	to	PART
cana-3670	22	4	handle	handle	VERB
cana-3670	22	5	fuzzy	fuzzy	ADJ
cana-3670	22	6	initial	initial	ADJ
cana-3670	22	7	value	value	NOUN
cana-3670	22	8	problems	problem	NOUN
cana-3670	22	9	without	without	ADP
cana-3670	22	10	converting	convert	VERB
cana-3670	22	11	fuzzy	fuzzy	ADJ
cana-3670	22	12	differential	differential	ADJ
cana-3670	22	13	equations	equation	NOUN
cana-3670	22	14	into	into	ADP
cana-3670	22	15	a	a	DET
cana-3670	22	16	system	system	NOUN
cana-3670	22	17	of	of	ADP
cana-3670	22	18	crisp	crisp	ADJ
cana-3670	22	19	differential	differential	ADJ
cana-3670	22	20	equations	equation	NOUN
cana-3670	22	21	,	,	PUNCT
cana-3670	22	22	euler	euler	PROPN
cana-3670	22	23	's	's	PART
cana-3670	22	24	fuzzy	fuzzy	ADJ
cana-3670	22	25	form	form	NOUN
cana-3670	22	26	was	be	AUX
cana-3670	22	27	devised	devise	VERB
cana-3670	22	28	in	in	ADP
cana-3670	22	29	[	[	X
cana-3670	22	30	6	6	NUM
cana-3670	22	31	]	]	PUNCT
cana-3670	22	32	.	.	PUNCT
cana-3670	23	1	a	a	DET
cana-3670	23	2	numerical	numerical	ADJ
cana-3670	23	3	procedure	procedure	NOUN
cana-3670	23	4	based	base	VERB
cana-3670	23	5	on	on	ADP
cana-3670	23	6	a	a	DET
cana-3670	23	7	modified	modify	VERB
cana-3670	23	8	euler	euler	NOUN
cana-3670	23	9	's	's	PART
cana-3670	23	10	method	method	NOUN
cana-3670	23	11	was	be	AUX
cana-3670	23	12	used	use	VERB
cana-3670	23	13	to	to	PART
cana-3670	23	14	determine	determine	VERB
cana-3670	23	15	the	the	DET
cana-3670	23	16	numerical	numerical	ADJ
cana-3670	23	17	solution	solution	NOUN
cana-3670	23	18	of	of	ADP
cana-3670	23	19	a	a	DET
cana-3670	23	20	fuzzy	fuzzy	ADJ
cana-3670	23	21	ordinary	ordinary	ADJ
cana-3670	23	22	differential	differential	ADJ
cana-3670	23	23	equation	equation	NOUN
cana-3670	23	24	in	in	ADP
cana-3670	23	25	[	[	X
cana-3670	23	26	7	7	NUM
cana-3670	23	27	]	]	PUNCT
cana-3670	23	28	.	.	PUNCT
cana-3670	24	1	the	the	DET
cana-3670	24	2	codomain	codomain	NOUN
cana-3670	24	3	of	of	ADP
cana-3670	24	4	the	the	DET
cana-3670	24	5	membership	membership	NOUN
cana-3670	24	6	function	function	NOUN
cana-3670	24	7	of	of	ADP
cana-3670	24	8	the	the	DET
cana-3670	24	9	fuzzy	fuzzy	ADJ
cana-3670	24	10	topological	topological	ADJ
cana-3670	24	11	space	space	NOUN
cana-3670	24	12	was	be	AUX
cana-3670	24	13	extended	extend	VERB
cana-3670	24	14	from	from	ADP
cana-3670	24	15	[	[	X
cana-3670	24	16	0	0	NUM
cana-3670	24	17	,	,	PUNCT
cana-3670	24	18	1	1	NUM
cana-3670	24	19	]	]	PUNCT
cana-3670	24	20	to	to	ADP
cana-3670	24	21	the	the	DET
cana-3670	24	22	unit	unit	NOUN
cana-3670	24	23	disk	disk	NOUN
cana-3670	24	24	in	in	ADP
cana-3670	24	25	the	the	DET
cana-3670	24	26	complex	complex	ADJ
cana-3670	24	27	plane	plane	NOUN
cana-3670	24	28	in	in	ADP
cana-3670	24	29	order	order	NOUN
cana-3670	24	30	to	to	PART
cana-3670	24	31	achieve	achieve	VERB
cana-3670	24	32	the	the	DET
cana-3670	24	33	numerical	numerical	ADJ
cana-3670	24	34	solution	solution	NOUN
cana-3670	24	35	of	of	ADP
cana-3670	24	36	complex	complex	ADJ
cana-3670	24	37	fuzzy	fuzzy	ADJ
cana-3670	24	38	differential	differential	NOUN
cana-3670	24	39	equations	equation	NOUN
cana-3670	24	40	using	use	VERB
cana-3670	24	41	euler	euler	PROPN
cana-3670	24	42	and	and	CCONJ
cana-3670	24	43	taylor	taylor	PROPN
cana-3670	24	44	techniques	technique	NOUN
cana-3670	24	45	in	in	ADP
cana-3670	24	46	[	[	X
cana-3670	24	47	8	8	NUM
cana-3670	24	48	]	]	PUNCT
cana-3670	24	49	.	.	PUNCT
cana-3670	25	1	by	by	ADP
cana-3670	25	2	applying	apply	VERB
cana-3670	25	3	the	the	DET
cana-3670	25	4	harmonic	harmonic	ADJ
cana-3670	25	5	mean	mean	NOUN
cana-3670	25	6	and	and	CCONJ
cana-3670	25	7	cubic	cubic	ADJ
cana-3670	25	8	mean	mean	NOUN
cana-3670	25	9	of	of	ADP
cana-3670	25	10	euler	euler	PROPN
cana-3670	25	11	's	's	PART
cana-3670	25	12	modified	modify	VERB
cana-3670	25	13	technique	technique	NOUN
cana-3670	25	14	,	,	PUNCT
cana-3670	25	15	the	the	DET
cana-3670	25	16	first	first	ADJ
cana-3670	25	17	order	order	NOUN
cana-3670	25	18	differential	differential	ADJ
cana-3670	25	19	equations	equation	NOUN
cana-3670	25	20	were	be	AUX
cana-3670	25	21	solved	solve	VERB
cana-3670	25	22	in	in	ADP
cana-3670	25	23	[	[	X
cana-3670	25	24	9	9	NUM
cana-3670	25	25	]	]	PUNCT
cana-3670	25	26	.	.	PUNCT
cana-3670	26	1	a	a	DET
cana-3670	26	2	modified	modify	VERB
cana-3670	26	3	euler	euler	NOUN
cana-3670	26	4	method	method	NOUN
cana-3670	26	5	was	be	AUX
cana-3670	26	6	introduced	introduce	VERB
cana-3670	26	7	for	for	ADP
cana-3670	26	8	solving	solve	VERB
cana-3670	26	9	fuzzy	fuzzy	ADJ
cana-3670	26	10	differential	differential	ADJ
cana-3670	26	11	equations	equation	NOUN
cana-3670	26	12	under	under	ADP
cana-3670	26	13	generalized	generalize	VERB
cana-3670	26	14	differentiability	differentiability	NOUN
cana-3670	26	15	in	in	ADP
cana-3670	26	16	[	[	X
cana-3670	26	17	10	10	NUM
cana-3670	26	18	,	,	PUNCT
cana-3670	26	19	11	11	NUM
cana-3670	26	20	]	]	PUNCT
cana-3670	26	21	.	.	PUNCT
cana-3670	27	1	it	it	PRON
cana-3670	27	2	also	also	ADV
cana-3670	27	3	estimated	estimate	VERB
cana-3670	27	4	the	the	DET
cana-3670	27	5	differential	differential	ADJ
cana-3670	27	6	equations	equation	NOUN
cana-3670	27	7	by	by	ADP
cana-3670	27	8	using	use	VERB
cana-3670	27	9	two	two	NUM
cana-3670	27	10	stages	stage	NOUN
cana-3670	27	11	predictor	predictor	NOUN
cana-3670	27	12	–	–	PUNCT
cana-3670	27	13	corrector	corrector	NOUN
cana-3670	27	14	algorithm	algorithm	NOUN
cana-3670	27	15	with	with	ADP
cana-3670	27	16	local	local	ADJ
cana-3670	27	17	truncation	truncation	NOUN
cana-3670	27	18	error	error	NOUN
cana-3670	27	19	of	of	ADP
cana-3670	27	20	order	order	NOUN
cana-3670	27	21	two	two	NUM
cana-3670	27	22	.	.	PUNCT
cana-3670	28	1	two	two	NUM
cana-3670	28	2	standard	standard	ADJ
cana-3670	28	3	numerical	numerical	ADJ
cana-3670	28	4	methods	method	NOUN
cana-3670	28	5	such	such	ADJ
cana-3670	28	6	as	as	ADP
cana-3670	28	7	euler	euler	NOUN
cana-3670	28	8	and	and	CCONJ
cana-3670	28	9	runge	runge	NOUN
cana-3670	28	10	–	–	PUNCT
cana-3670	28	11	kutta	kutta	PROPN
cana-3670	28	12	fourth	fourth	ADJ
cana-3670	28	13	order	order	NOUN
cana-3670	28	14	methods	method	NOUN
cana-3670	28	15	were	be	AUX
cana-3670	28	16	presented	present	VERB
cana-3670	28	17	to	to	PART
cana-3670	28	18	solve	solve	VERB
cana-3670	28	19	second	second	ADJ
cana-3670	28	20	order	order	NOUN
cana-3670	28	21	initial	initial	ADJ
cana-3670	28	22	value	value	NOUN
cana-3670	28	23	problem	problem	NOUN
cana-3670	28	24	for	for	ADP
cana-3670	28	25	ordinary	ordinary	ADJ
cana-3670	28	26	differential	differential	ADJ
cana-3670	28	27	equation	equation	NOUN
cana-3670	28	28	in	in	ADP
cana-3670	28	29	[	[	X
cana-3670	28	30	12	12	NUM
cana-3670	28	31	,	,	PUNCT
cana-3670	28	32	13	13	NUM
cana-3670	28	33	]	]	PUNCT
cana-3670	28	34	.	.	PUNCT
cana-3670	29	1	in	in	ADP
cana-3670	29	2	order	order	NOUN
cana-3670	29	3	to	to	PART
cana-3670	29	4	improve	improve	VERB
cana-3670	29	5	the	the	DET
cana-3670	29	6	accuracy	accuracy	NOUN
cana-3670	29	7	of	of	ADP
cana-3670	29	8	numerical	numerical	ADJ
cana-3670	29	9	solution	solution	NOUN
cana-3670	29	10	of	of	ADP
cana-3670	29	11	fuzzy	fuzzy	ADJ
cana-3670	29	12	differential	differential	ADJ
cana-3670	29	13	equations	equation	NOUN
cana-3670	29	14	,	,	PUNCT
cana-3670	29	15	we	we	PRON
cana-3670	29	16	shall	shall	AUX
cana-3670	29	17	enhance	enhance	VERB
cana-3670	29	18	euler	euler	PROPN
cana-3670	29	19	's	's	PART
cana-3670	29	20	method	method	NOUN
cana-3670	29	21	,	,	PUNCT
cana-3670	29	22	modified	modify	VERB
cana-3670	29	23	euler	euler	NOUN
cana-3670	29	24	's	's	PART
cana-3670	29	25	method	method	NOUN
cana-3670	29	26	,	,	PUNCT
cana-3670	29	27	and	and	CCONJ
cana-3670	29	28	improved	improve	VERB
cana-3670	29	29	euler	euler	NOUN
cana-3670	29	30	's	's	PART
cana-3670	29	31	method	method	NOUN
cana-3670	29	32	based	base	VERB
cana-3670	29	33	on	on	ADP
cana-3670	29	34	contra	contra	PROPN
cana-3670	29	35	harmonic	harmonic	PROPN
cana-3670	29	36	mean	mean	NOUN
cana-3670	29	37	and	and	CCONJ
cana-3670	29	38	centroidal	centroidal	ADJ
cana-3670	29	39	mean	mean	VERB
cana-3670	29	40	in	in	ADP
cana-3670	29	41	this	this	DET
cana-3670	29	42	paper	paper	NOUN
cana-3670	29	43	.	.	PUNCT
cana-3670	30	1	the	the	DET
cana-3670	30	2	framework	framework	NOUN
cana-3670	30	3	of	of	ADP
cana-3670	30	4	this	this	DET
cana-3670	30	5	paper	paper	NOUN
cana-3670	30	6	is	be	AUX
cana-3670	30	7	as	as	SCONJ
cana-3670	30	8	follows	follow	VERB
cana-3670	30	9	:	:	PUNCT
cana-3670	30	10	we	we	PRON
cana-3670	30	11	review	review	VERB
cana-3670	30	12	some	some	DET
cana-3670	30	13	basic	basic	ADJ
cana-3670	30	14	ideas	idea	NOUN
cana-3670	30	15	about	about	ADP
cana-3670	30	16	fuzzy	fuzzy	ADJ
cana-3670	30	17	sets	set	NOUN
cana-3670	30	18	and	and	CCONJ
cana-3670	30	19	fuzzy	fuzzy	ADJ
cana-3670	30	20	integers	integer	NOUN
cana-3670	30	21	in	in	ADP
cana-3670	30	22	section	section	NOUN
cana-3670	30	23	2	2	NUM
cana-3670	30	24	.	.	PUNCT
cana-3670	31	1	in	in	ADP
cana-3670	31	2	section	section	NOUN
cana-3670	31	3	3	3	NUM
cana-3670	31	4	,	,	PUNCT
cana-3670	31	5	we	we	PRON
cana-3670	31	6	improve	improve	VERB
cana-3670	31	7	on	on	ADP
cana-3670	31	8	the	the	DET
cana-3670	31	9	conventional	conventional	ADJ
cana-3670	31	10	euler	euler	NOUN
cana-3670	31	11	's	's	PART
cana-3670	31	12	methods	method	NOUN
cana-3670	31	13	,	,	PUNCT
cana-3670	31	14	which	which	PRON
cana-3670	31	15	are	be	AUX
cana-3670	31	16	based	base	VERB
cana-3670	31	17	on	on	ADP
cana-3670	31	18	the	the	DET
cana-3670	31	19	centroidal	centroidal	ADJ
cana-3670	31	20	mean	mean	NOUN
cana-3670	31	21	and	and	CCONJ
cana-3670	31	22	contra	contra	PROPN
cana-3670	31	23	harmonic	harmonic	PROPN
cana-3670	31	24	mean	mean	VERB
cana-3670	31	25	,	,	PUNCT
cana-3670	31	26	to	to	PART
cana-3670	31	27	find	find	VERB
cana-3670	31	28	the	the	DET
cana-3670	31	29	numerical	numerical	ADJ
cana-3670	31	30	solutions	solution	NOUN
cana-3670	31	31	of	of	ADP
cana-3670	31	32	fuzzy	fuzzy	ADJ
cana-3670	31	33	differential	differential	ADJ
cana-3670	31	34	equations	equation	NOUN
cana-3670	31	35	with	with	ADP
cana-3670	31	36	lower	low	ADJ
cana-3670	31	37	error	error	NOUN
cana-3670	31	38	values	value	NOUN
cana-3670	31	39	.	.	PUNCT
cana-3670	32	1	section	section	NOUN
cana-3670	32	2	4	4	NUM
cana-3670	32	3	presents	present	VERB
cana-3670	32	4	the	the	DET
cana-3670	32	5	fuzzy	fuzzy	ADJ
cana-3670	32	6	initial	initial	ADJ
cana-3670	32	7	value	value	NOUN
cana-3670	32	8	problem	problem	NOUN
cana-3670	32	9	and	and	CCONJ
cana-3670	32	10	the	the	DET
cana-3670	32	11	fuzzy	fuzzy	ADJ
cana-3670	32	12	version	version	NOUN
cana-3670	32	13	of	of	ADP
cana-3670	32	14	the	the	DET
cana-3670	32	15	suggested	suggest	VERB
cana-3670	32	16	techniques	technique	NOUN
cana-3670	32	17	.	.	PUNCT
cana-3670	33	1	to	to	PART
cana-3670	33	2	demonstrate	demonstrate	VERB
cana-3670	33	3	the	the	DET
cana-3670	33	4	efficacy	efficacy	NOUN
cana-3670	33	5	of	of	ADP
cana-3670	33	6	our	our	PRON
cana-3670	33	7	approach	approach	NOUN
cana-3670	33	8	in	in	ADP
cana-3670	33	9	comparison	comparison	NOUN
cana-3670	33	10	to	to	ADP
cana-3670	33	11	the	the	DET
cana-3670	33	12	conventional	conventional	ADJ
cana-3670	33	13	euler	euler	PROPN
cana-3670	33	14	's	's	PART
cana-3670	33	15	method	method	NOUN
cana-3670	33	16	,	,	PUNCT
cana-3670	33	17	we	we	PRON
cana-3670	33	18	use	use	VERB
cana-3670	33	19	it	it	PRON
cana-3670	33	20	in	in	ADP
cana-3670	33	21	section	section	NOUN
cana-3670	33	22	5	5	NUM
cana-3670	33	23	to	to	PART
cana-3670	33	24	obtain	obtain	VERB
cana-3670	33	25	numerical	numerical	ADJ
cana-3670	33	26	solutions	solution	NOUN
cana-3670	33	27	for	for	ADP
cana-3670	33	28	the	the	DET
cana-3670	33	29	fuzzy	fuzzy	ADJ
cana-3670	33	30	initial	initial	ADJ
cana-3670	33	31	values	value	NOUN
cana-3670	33	32	problem	problem	NOUN
cana-3670	33	33	.	.	PUNCT
cana-3670	34	1	section	section	NOUN
cana-3670	34	2	6	6	NUM
cana-3670	34	3	offers	offer	VERB
cana-3670	34	4	a	a	DET
cana-3670	34	5	succinct	succinct	ADJ
cana-3670	34	6	conclusion	conclusion	NOUN
cana-3670	34	7	.	.	PUNCT
cana-3670	35	1	2	2	X
cana-3670	35	2	.	.	X
cana-3670	35	3	prologues	prologue	NOUN
cana-3670	35	4	:	:	PUNCT
cana-3670	35	5	some	some	DET
cana-3670	35	6	helpful	helpful	ADJ
cana-3670	35	7	definitions	definition	NOUN
cana-3670	35	8	of	of	ADP
cana-3670	35	9	fuzzy	fuzzy	ADJ
cana-3670	35	10	sets	set	NOUN
cana-3670	35	11	and	and	CCONJ
cana-3670	35	12	fuzzy	fuzzy	ADJ
cana-3670	35	13	numbers	number	NOUN
cana-3670	35	14	that	that	PRON
cana-3670	35	15	are	be	AUX
cana-3670	35	16	required	require	VERB
cana-3670	35	17	or	or	CCONJ
cana-3670	35	18	established	establish	VERB
cana-3670	35	19	in	in	ADP
cana-3670	35	20	this	this	DET
cana-3670	35	21	paper	paper	NOUN
cana-3670	35	22	will	will	AUX
cana-3670	35	23	be	be	AUX
cana-3670	35	24	introduced	introduce	VERB
cana-3670	35	25	in	in	ADP
cana-3670	35	26	this	this	DET
cana-3670	35	27	section	section	NOUN
cana-3670	35	28	.	.	PUNCT
cana-3670	36	1	definition	definition	NOUN
cana-3670	36	2	in	in	ADP
cana-3670	36	3	the	the	DET
cana-3670	36	4	universal	universal	ADJ
cana-3670	36	5	set	set	NOUN
cana-3670	36	6	x	x	NOUN
cana-3670	36	7	,	,	PUNCT
cana-3670	36	8	a	a	DET
cana-3670	36	9	fuzzy	fuzzy	ADJ
cana-3670	36	10	set	set	VERB
cana-3670	36	11	�	�	PROPN
cana-3670	36	12	̃	̃	PROPN
cana-3670	36	13	�	�	PROPN
cana-3670	36	14	is	be	AUX
cana-3670	36	15	defined	define	VERB
cana-3670	36	16	as	as	ADP
cana-3670	36	17	�	�	PROPN
cana-3670	36	18	̃	̃	PROPN
cana-3670	36	19	�	�	NOUN
cana-3670	36	20	=	=	SYM
cana-3670	36	21	{	{	PUNCT
cana-3670	36	22	(	(	PUNCT
cana-3670	36	23	𝒙	𝒙	PROPN
cana-3670	36	24	,	,	PUNCT
cana-3670	36	25	𝝁	𝝁	PROPN
cana-3670	36	26	�	�	PROPN
cana-3670	36	27	̃	̃	NOUN
cana-3670	36	28	�	�	PROPN
cana-3670	36	29	(𝒙));𝒙	(𝒙));𝒙	SYM
cana-3670	36	30	∈	∈	PROPN
cana-3670	36	31	𝑿	𝑿	PROPN
cana-3670	36	32	,	,	PUNCT
cana-3670	36	33	𝝁	𝝁	PROPN
cana-3670	36	34	�	�	PROPN
cana-3670	36	35	̃	̃	PROPN
cana-3670	36	36	�	�	NOUN
cana-3670	36	37	(𝒙	(𝒙	SYM
cana-3670	36	38	)	)	PUNCT
cana-3670	36	39	∈	∈	PROPN
cana-3670	36	40	(	(	PUNCT
cana-3670	36	41	𝟎	𝟎	PROPN
cana-3670	36	42	,	,	PUNCT
cana-3670	36	43	𝟏	𝟏	NUM
cana-3670	36	44	]	]	PUNCT
cana-3670	36	45	}	}	PUNCT
cana-3670	36	46	.	.	PUNCT
cana-3670	37	1	in	in	ADP
cana-3670	37	2	this	this	DET
cana-3670	37	3	case	case	NOUN
cana-3670	37	4	,	,	PUNCT
cana-3670	37	5	the	the	DET
cana-3670	37	6	membership	membership	NOUN
cana-3670	37	7	function	function	NOUN
cana-3670	37	8	's	's	PART
cana-3670	37	9	grade	grade	NOUN
cana-3670	37	10	is	be	AUX
cana-3670	37	11	𝝁	𝝁	PRON
cana-3670	37	12	�	�	PROPN
cana-3670	37	13	̃	̃	PROPN
cana-3670	37	14	�	�	NOUN
cana-3670	37	15	(𝒙	(𝒙	NUM
cana-3670	37	16	)	)	PUNCT
cana-3670	37	17	→	→	PUNCT
cana-3670	38	1	[	[	X
cana-3670	38	2	𝟎,𝟏	𝟎,𝟏	NOUN
cana-3670	38	3	]	]	PUNCT
cana-3670	38	4	,	,	PUNCT
cana-3670	38	5	and	and	CCONJ
cana-3670	38	6	the	the	DET
cana-3670	38	7	fuzzy	fuzzy	ADJ
cana-3670	38	8	set	set	VERB
cana-3670	38	9	�	�	PROPN
cana-3670	38	10	̃	̃	PROPN
cana-3670	38	11	�	�	PROPN
cana-3670	38	12	's	's	PART
cana-3670	38	13	grade	grade	NOUN
cana-3670	38	14	value	value	NOUN
cana-3670	38	15	of	of	ADP
cana-3670	38	16	𝒙	𝒙	PROPN
cana-3670	38	17	∈	∈	PROPN
cana-3670	38	18	𝑿	𝑿	PROPN
cana-3670	38	19	is	be	AUX
cana-3670	38	20	𝝁	𝝁	NUM
cana-3670	38	21	�	�	PROPN
cana-3670	38	22	̃	̃	PROPN
cana-3670	38	23	�	�	NOUN
cana-3670	38	24	(𝒙	(𝒙	NUM
cana-3670	38	25	)	)	PUNCT
cana-3670	38	26	.	.	PUNCT
cana-3670	39	1	definition	definition	NOUN
cana-3670	39	2	the	the	DET
cana-3670	39	3	crisp	crisp	ADJ
cana-3670	39	4	set𝐴𝒔	set𝐴𝒔	PROPN
cana-3670	39	5	that	that	PRON
cana-3670	39	6	includes	include	VERB
cana-3670	39	7	every	every	DET
cana-3670	39	8	element	element	NOUN
cana-3670	39	9	of	of	ADP
cana-3670	39	10	the	the	DET
cana-3670	39	11	universal	universal	ADJ
cana-3670	39	12	set	set	NOUN
cana-3670	39	13	x	x	PUNCT
cana-3670	39	14	whose	whose	DET
cana-3670	39	15	membership	membership	NOUN
cana-3670	39	16	grades	grade	NOUN
cana-3670	39	17	in	in	ADP
cana-3670	39	18	�	�	PROPN
cana-3670	39	19	̃	̃	PROPN
cana-3670	39	20	�	�	NOUN
cana-3670	39	21	are	be	AUX
cana-3670	39	22	higher	high	ADJ
cana-3670	39	23	than	than	ADP
cana-3670	39	24	or	or	CCONJ
cana-3670	39	25	equal	equal	ADJ
cana-3670	39	26	to	to	ADP
cana-3670	39	27	the	the	DET
cana-3670	39	28	designated	designate	VERB
cana-3670	39	29	value	value	NOUN
cana-3670	39	30	of	of	ADP
cana-3670	39	31	𝛼	𝛼	PROPN
cana-3670	39	32	is	be	AUX
cana-3670	39	33	the	the	DET
cana-3670	39	34	𝛼	𝛼	NOUN
cana-3670	39	35	-cut	-cut	NUM
cana-3670	39	36	of	of	ADP
cana-3670	39	37	a	a	DET
cana-3670	39	38	fuzzy	fuzzy	ADJ
cana-3670	39	39	set	set	VERB
cana-3670	39	40	�	�	PROPN
cana-3670	39	41	̃	̃	PROPN
cana-3670	39	42	�	�	PROPN
cana-3670	39	43	.	.	PUNCT
cana-3670	40	1	it	it	PRON
cana-3670	40	2	is	be	AUX
cana-3670	40	3	indicated	indicate	VERB
cana-3670	40	4	by	by	ADP
cana-3670	40	5	𝐴𝒔	𝐴𝒔	PROPN
cana-3670	40	6	=	=	PUNCT
cana-3670	40	7	{	{	PUNCT
cana-3670	40	8	𝒙	𝒙	PROPN
cana-3670	40	9	∈	∈	PROPN
cana-3670	40	10	𝑿	𝑿	NOUN
cana-3670	40	11	|	|	CCONJ
cana-3670	40	12	𝝁	𝝁	PROPN
cana-3670	40	13	�	�	PROPN
cana-3670	40	14	̃	̃	PROPN
cana-3670	40	15	�	�	NOUN
cana-3670	40	16	(𝒙	(𝒙	NUM
cana-3670	40	17	)	)	PUNCT
cana-3670	40	18	≥	≥	NOUN
cana-3670	40	19	𝒔	𝒔	NOUN
cana-3670	40	20	}	}	PUNCT
cana-3670	40	21	where	where	SCONJ
cana-3670	40	22	0	0	NUM
cana-3670	40	23	≤	≤	NUM
cana-3670	40	24	𝑠	𝑠	X
cana-3670	40	25	≤	≤	NUM
cana-3670	40	26	1	1	NUM
cana-3670	40	27	.	.	PUNCT
cana-3670	41	1	definition	definition	NOUN
cana-3670	41	2	the	the	DET
cana-3670	41	3	membership	membership	NOUN
cana-3670	41	4	function	function	NOUN
cana-3670	41	5	of	of	ADP
cana-3670	41	6	a	a	DET
cana-3670	41	7	fuzzy	fuzzy	ADJ
cana-3670	41	8	number	number	NOUN
cana-3670	41	9	�	�	PROPN
cana-3670	41	10	̃	̃	PROPN
cana-3670	41	11	�	�	PROPN
cana-3670	41	12	,	,	PUNCT
cana-3670	41	13	which	which	PRON
cana-3670	41	14	is	be	AUX
cana-3670	41	15	a	a	DET
cana-3670	41	16	subset	subset	NOUN
cana-3670	41	17	of	of	ADP
cana-3670	41	18	real	real	ADJ
cana-3670	41	19	line	line	NOUN
cana-3670	41	20	r	r	NOUN
cana-3670	41	21	,	,	PUNCT
cana-3670	41	22	has	have	VERB
cana-3670	41	23	the	the	DET
cana-3670	41	24	following	following	ADJ
cana-3670	41	25	characteristics	characteristic	NOUN
cana-3670	41	26	:	:	PUNCT
cana-3670	42	1	1	1	X
cana-3670	42	2	.	.	X
cana-3670	42	3	in	in	ADP
cana-3670	42	4	its	its	PRON
cana-3670	42	5	domain	domain	NOUN
cana-3670	42	6	,	,	PUNCT
cana-3670	42	7	𝝁	𝝁	PROPN
cana-3670	42	8	�	�	PROPN
cana-3670	42	9	̃	̃	NOUN
cana-3670	42	10	�	�	PROPN
cana-3670	42	11	(𝒙)is	(𝒙)is	NOUN
cana-3670	42	12	piecewise	piecewise	NOUN
cana-3670	42	13	continuous	continuous	ADJ
cana-3670	42	14	.	.	PUNCT
cana-3670	43	1	2	2	X
cana-3670	43	2	.	.	X
cana-3670	43	3	�	�	PROPN
cana-3670	43	4	̃	̃	PROPN
cana-3670	43	5	�	�	PROPN
cana-3670	43	6	is	be	AUX
cana-3670	43	7	normal	normal	ADJ
cana-3670	43	8	,	,	PUNCT
cana-3670	43	9	meaning	mean	VERB
cana-3670	43	10	that	that	SCONJ
cana-3670	43	11	a	a	DET
cana-3670	43	12	𝒙𝟎	𝒙𝟎	ADJ
cana-3670	43	13	∈	∈	NOUN
cana-3670	43	14	𝑿	𝑿	NOUN
cana-3670	43	15	such	such	ADJ
cana-3670	43	16	that	that	SCONJ
cana-3670	43	17	𝝁	𝝁	PROPN
cana-3670	43	18	�	�	PROPN
cana-3670	43	19	̃	̃	NOUN
cana-3670	43	20	�	�	PROPN
cana-3670	43	21	(𝒙𝟎	(𝒙𝟎	PUNCT
cana-3670	43	22	)	)	PUNCT
cana-3670	43	23	=	=	PUNCT
cana-3670	44	1	𝟏.	𝟏.	X
cana-3670	44	2	3	3	X
cana-3670	44	3	.	.	X
cana-3670	44	4	�	�	PROPN
cana-3670	44	5	̃	̃	PROPN
cana-3670	44	6	�	�	PROPN
cana-3670	44	7	is	be	AUX
cana-3670	44	8	convex	convex	ADJ
cana-3670	44	9	;	;	PUNCT
cana-3670	44	10	that	that	ADV
cana-3670	44	11	is	is	ADV
cana-3670	44	12	,	,	PUNCT
cana-3670	44	13	𝝁	𝝁	PROPN
cana-3670	44	14	�	�	PROPN
cana-3670	44	15	̃	̃	NOUN
cana-3670	44	16	�	�	PROPN
cana-3670	44	17	(𝝀	(𝝀	NUM
cana-3670	44	18	𝒙𝟏+	𝒙𝟏+	PROPN
cana-3670	44	19	(	(	PUNCT
cana-3670	44	20	𝟏	𝟏	NUM
cana-3670	44	21	−	−	PROPN
cana-3670	44	22	𝝀	𝝀	NOUN
cana-3670	44	23	)	)	PUNCT
cana-3670	44	24	𝒙𝟐	𝒙𝟐	NOUN
cana-3670	44	25	)	)	PUNCT
cana-3670	44	26	≥	≥	NOUN
cana-3670	44	27	𝒎𝒊𝒏(𝝁	𝒎𝒊𝒏(𝝁	PROPN
cana-3670	44	28	�	�	PROPN
cana-3670	44	29	̃	̃	PROPN
cana-3670	44	30	�	�	PROPN
cana-3670	44	31	(	(	PUNCT
cana-3670	44	32	𝒙𝟏),𝝁	𝒙𝟏),𝝁	X
cana-3670	44	33	�	�	PROPN
cana-3670	44	34	̃	̃	PROPN
cana-3670	44	35	�	�	PROPN
cana-3670	44	36	(	(	PUNCT
cana-3670	44	37	𝒙𝟐	𝒙𝟐	NOUN
cana-3670	44	38	)	)	PUNCT
cana-3670	44	39	)	)	PUNCT
cana-3670	44	40	and	and	CCONJ
cana-3670	44	41	∀	∀	NUM
cana-3670	44	42	𝒙𝟏	𝒙𝟏	NOUN
cana-3670	44	43	,	,	PUNCT
cana-3670	44	44	𝒙𝟐	𝒙𝟐	NOUN
cana-3670	44	45	∈	∈	NOUN
cana-3670	44	46	𝑿.	𝑿.	ADJ
cana-3670	44	47	definition	definition	NOUN
cana-3670	44	48	communications	communication	NOUN
cana-3670	44	49	on	on	ADP
cana-3670	44	50	applied	apply	VERB
cana-3670	44	51	nonlinear	nonlinear	ADJ
cana-3670	44	52	analysis	analysis	NOUN
cana-3670	44	53	issn	issn	NOUN
cana-3670	44	54	:	:	PUNCT
cana-3670	44	55	1074	1074	NUM
cana-3670	44	56	-	-	PUNCT
cana-3670	44	57	133x	133x	NUM
cana-3670	44	58	vol	vol	NOUN
cana-3670	44	59	32	32	NUM
cana-3670	44	60	no	no	NOUN
cana-3670	44	61	.	.	PUNCT
cana-3670	45	1	8s	8s	PROPN
cana-3670	45	2	(	(	PUNCT
cana-3670	45	3	2025	2025	NUM
cana-3670	45	4	)	)	PUNCT
cana-3670	45	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	45	6	273	273	NUM
cana-3670	45	7	the	the	DET
cana-3670	45	8	membership	membership	NOUN
cana-3670	45	9	function	function	NOUN
cana-3670	45	10	of	of	ADP
cana-3670	45	11	a	a	DET
cana-3670	45	12	triangular	triangular	NOUN
cana-3670	45	13	fuzzy	fuzzy	ADJ
cana-3670	45	14	number	number	NOUN
cana-3670	45	15	u~	u~	PUNCT
cana-3670	45	16	is	be	AUX
cana-3670	45	17	described	describe	VERB
cana-3670	45	18	as	as	SCONJ
cana-3670	45	19	follows	follow	VERB
cana-3670	45	20	.	.	PUNCT
cana-3670	46	1	it	it	PRON
cana-3670	46	2	may	may	AUX
cana-3670	46	3	be	be	AUX
cana-3670	46	4	expressed	express	VERB
cana-3670	46	5	by	by	ADP
cana-3670	46	6	a	a	DET
cana-3670	46	7	triplet	triplet	NOUN
cana-3670	46	8	(	(	PUNCT
cana-3670	46	9	𝜌1	𝜌1	NOUN
cana-3670	46	10	,	,	PUNCT
cana-3670	46	11	𝜌2	𝜌2	ADJ
cana-3670	46	12	,	,	PUNCT
cana-3670	46	13	𝜌3	𝜌3	PROPN
cana-3670	46	14	)	)	PUNCT
cana-3670	46	15	.	.	PUNCT
cana-3670	47	1	�	�	PROPN
cana-3670	47	2	̃	̃	PROPN
cana-3670	47	3	�	�	NOUN
cana-3670	47	4	(𝒙	(𝒙	NUM
cana-3670	47	5	)	)	PUNCT
cana-3670	48	1	=	=	SYM
cana-3670	48	2	{	{	PUNCT
cana-3670	48	3	𝟎	𝟎	PROPN
cana-3670	48	4	,	,	PUNCT
cana-3670	48	5	𝑖𝑓	𝑖𝑓	VERB
cana-3670	48	6	𝑥	𝑥	NOUN
cana-3670	48	7	<	<	X
cana-3670	48	8	𝜌1	𝜌1	NOUN
cana-3670	48	9	𝒙−	𝒙−	NOUN
cana-3670	48	10	𝜌1	𝜌1	PROPN
cana-3670	48	11	𝜌2−	𝜌2−	NOUN
cana-3670	48	12	𝜌1	𝜌1	NOUN
cana-3670	48	13	,	,	PUNCT
cana-3670	48	14	𝑖𝑓	𝑖𝑓	ADP
cana-3670	48	15	𝜌1	𝜌1	NOUN
cana-3670	48	16	≤	≤	NUM
cana-3670	48	17	𝑥	𝑥	DET
cana-3670	48	18	≤	≤	X
cana-3670	48	19	𝜌2	𝜌2	ADJ
cana-3670	48	20	𝜌3−	𝜌3−	NUM
cana-3670	48	21	𝑥	𝑥	PROPN
cana-3670	48	22	𝜌3−	𝜌3−	NUM
cana-3670	48	23	𝜌2	𝜌2	ADJ
cana-3670	48	24	,	,	PUNCT
cana-3670	48	25	𝑖𝑓	𝑖𝑓	ADP
cana-3670	48	26	𝜌2	𝜌2	ADJ
cana-3670	48	27	≤	≤	NUM
cana-3670	48	28	𝑥	𝑥	DET
cana-3670	48	29	≤	≤	NUM
cana-3670	48	30	𝜌3	𝜌3	PROPN
cana-3670	48	31	𝟎	𝟎	NUM
cana-3670	48	32	,	,	PUNCT
cana-3670	48	33	𝑖𝑓	𝑖𝑓	ADP
cana-3670	48	34	𝑥	𝑥	PRON
cana-3670	48	35	>	>	X
cana-3670	48	36	𝜌3	𝜌3	VERB
cana-3670	48	37	the	the	DET
cana-3670	48	38	𝑠	𝑠	PROPN
cana-3670	48	39	level	level	NOUN
cana-3670	48	40	of	of	ADP
cana-3670	48	41	the	the	DET
cana-3670	48	42	fuzzy	fuzzy	ADJ
cana-3670	48	43	number	number	NOUN
cana-3670	48	44	�	�	PROPN
cana-3670	48	45	̃	̃	PROPN
cana-3670	48	46	�	�	PROPN
cana-3670	48	47	is	be	AUX
cana-3670	48	48	�	�	PROPN
cana-3670	48	49	̃	̃	NOUN
cana-3670	48	50	�	�	NOUN
cana-3670	48	51	𝑠	𝑠	NOUN
cana-3670	48	52	=	=	PUNCT
cana-3670	49	1	[	[	X
cana-3670	49	2	𝜌1	𝜌1	NOUN
cana-3670	49	3	+	+	X
cana-3670	49	4	(	(	PUNCT
cana-3670	49	5	𝜌2−𝜌3)𝑠	𝜌2−𝜌3)𝑠	PROPN
cana-3670	49	6	,	,	PUNCT
cana-3670	49	7	𝜌3−	𝜌3−	NOUN
cana-3670	49	8	(	(	PUNCT
cana-3670	49	9	𝜌3−𝜌2)𝑠	𝜌3−𝜌2)𝑠	NOUN
cana-3670	49	10	]	]	PUNCT
cana-3670	49	11	for	for	ADP
cana-3670	49	12	any	any	DET
cana-3670	49	13	𝑠	𝑠	PROPN
cana-3670	49	14	∈	∈	PROPN
cana-3670	50	1	[	[	X
cana-3670	50	2	0,1	0,1	NUM
cana-3670	50	3	]	]	PUNCT
cana-3670	50	4	.	.	PUNCT
cana-3670	51	1	3	3	X
cana-3670	51	2	.	.	X
cana-3670	51	3	methodology	methodology	NOUN
cana-3670	51	4	this	this	DET
cana-3670	51	5	section	section	NOUN
cana-3670	51	6	will	will	AUX
cana-3670	51	7	cover	cover	VERB
cana-3670	51	8	the	the	DET
cana-3670	51	9	completion	completion	NOUN
cana-3670	51	10	of	of	ADP
cana-3670	51	11	the	the	DET
cana-3670	51	12	euler	euler	NOUN
cana-3670	51	13	,	,	PUNCT
cana-3670	51	14	modified	modify	VERB
cana-3670	51	15	euler	euler	NOUN
cana-3670	51	16	and	and	CCONJ
cana-3670	51	17	improved	improve	VERB
cana-3670	51	18	euler	euler	NOUN
cana-3670	51	19	approaches	approach	NOUN
cana-3670	51	20	based	base	VERB
cana-3670	51	21	on	on	ADP
cana-3670	51	22	contra	contra	PROPN
cana-3670	51	23	harmonic	harmonic	PROPN
cana-3670	51	24	mean	mean	NOUN
cana-3670	51	25	and	and	CCONJ
cana-3670	51	26	centroidal	centroidal	ADJ
cana-3670	51	27	mean	mean	VERB
cana-3670	51	28	for	for	ADP
cana-3670	51	29	solving	solve	VERB
cana-3670	51	30	the	the	DET
cana-3670	51	31	first	first	ADJ
cana-3670	51	32	order	order	NOUN
cana-3670	51	33	differential	differential	ADJ
cana-3670	51	34	equations	equation	NOUN
cana-3670	51	35	with	with	ADP
cana-3670	51	36	fuzzy	fuzzy	ADJ
cana-3670	51	37	initial	initial	ADJ
cana-3670	51	38	conditions	condition	NOUN
cana-3670	51	39	.	.	PUNCT
cana-3670	52	1	3.1	3.1	NUM
cana-3670	52	2	.	.	PUNCT
cana-3670	52	3	enhanced	enhance	VERB
cana-3670	52	4	euler	euler	PROPN
cana-3670	52	5	’s	’s	PART
cana-3670	52	6	methods	method	NOUN
cana-3670	52	7	based	base	VERB
cana-3670	52	8	on	on	ADP
cana-3670	52	9	contra	contra	PROPN
cana-3670	52	10	harmonic	harmonic	PROPN
cana-3670	52	11	mean	mean	NOUN
cana-3670	52	12	and	and	CCONJ
cana-3670	52	13	centroidal	centroidal	ADJ
cana-3670	52	14	mean	mean	NOUN
cana-3670	52	15	:	:	PUNCT
cana-3670	52	16	3.1.1	3.1.1	NUM
cana-3670	52	17	euler	euler	NOUN
cana-3670	52	18	’s	’s	PART
cana-3670	52	19	method	method	NOUN
cana-3670	52	20	based	base	VERB
cana-3670	52	21	on	on	ADP
cana-3670	52	22	contra	contra	PROPN
cana-3670	52	23	harmonic	harmonic	PROPN
cana-3670	52	24	mean	mean	NOUN
cana-3670	52	25	take	take	VERB
cana-3670	52	26	note	note	NOUN
cana-3670	52	27	of	of	ADP
cana-3670	52	28	a	a	DET
cana-3670	52	29	,	,	PUNCT
cana-3670	52	30	b	b	NOUN
cana-3670	52	31	,	,	PUNCT
cana-3670	52	32	and	and	CCONJ
cana-3670	52	33	c	c	NOUN
cana-3670	52	34	as	as	SCONJ
cana-3670	52	35	the	the	DET
cana-3670	52	36	contra	contra	PROPN
cana-3670	52	37	harmonic	harmonic	ADJ
cana-3670	52	38	sequence	sequence	NOUN
cana-3670	52	39	's	's	PART
cana-3670	52	40	components	component	NOUN
cana-3670	52	41	.	.	PUNCT
cana-3670	53	1	the	the	DET
cana-3670	53	2	formula	formula	NOUN
cana-3670	53	3	for	for	ADP
cana-3670	53	4	calculating	calculate	VERB
cana-3670	53	5	the	the	DET
cana-3670	53	6	contra	contra	PROPN
cana-3670	53	7	harmonic	harmonic	NOUN
cana-3670	53	8	mean	mean	VERB
cana-3670	53	9	is	be	AUX
cana-3670	53	10	𝑐	𝑐	NOUN
cana-3670	53	11	=	=	SYM
cana-3670	53	12	𝑎2+𝑏2	𝑎2+𝑏2	PROPN
cana-3670	53	13	𝑎+𝑏	𝑎+𝑏	NUM
cana-3670	53	14	.	.	PUNCT
cana-3670	54	1	next	next	ADV
cana-3670	54	2	,	,	PUNCT
cana-3670	54	3	the	the	DET
cana-3670	54	4	two	two	NUM
cana-3670	54	5	center	center	NOUN
cana-3670	54	6	points	point	NOUN
cana-3670	54	7	expected	expect	VERB
cana-3670	54	8	contra	contra	PROPN
cana-3670	54	9	harmonic	harmonic	PROPN
cana-3670	54	10	mean	mean	VERB
cana-3670	54	11	is	be	AUX
cana-3670	54	12	determined	determine	VERB
cana-3670	54	13	by	by	ADP
cana-3670	54	14	(	(	PUNCT
cana-3670	54	15	𝑡0	𝑡0	PROPN
cana-3670	54	16	2+𝑡1	2+𝑡1	NUM
cana-3670	54	17	2	2	NUM
cana-3670	54	18	𝑡0+𝑡1	𝑡0+𝑡1	PROPN
cana-3670	54	19	,	,	PUNCT
cana-3670	54	20	𝑢0	𝑢0	VERB
cana-3670	54	21	2+𝑢1	2+𝑢1	NUM
cana-3670	54	22	2	2	NUM
cana-3670	54	23	𝑢0+𝑢1	𝑢0+𝑢1	PROPN
cana-3670	54	24	)	)	PUNCT
cana-3670	54	25	.	.	PUNCT
cana-3670	55	1	the	the	DET
cana-3670	55	2	formula	formula	NOUN
cana-3670	55	3	may	may	AUX
cana-3670	55	4	be	be	AUX
cana-3670	55	5	written	write	VERB
cana-3670	55	6	as	as	ADP
cana-3670	55	7	(	(	PUNCT
cana-3670	55	8	𝑡0	𝑡0	PROPN
cana-3670	55	9	2+(𝑡0+ℎ	2+(𝑡0+ℎ	NUM
cana-3670	55	10	)	)	PUNCT
cana-3670	55	11	2	2	NUM
cana-3670	55	12	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	55	13	)	)	PUNCT
cana-3670	55	14	,	,	PUNCT
cana-3670	55	15	𝑢0	𝑢0	NOUN
cana-3670	55	16	2+(𝑢0+ℎ𝑓(𝑡0,𝑢0	2+(𝑢0+ℎ𝑓(𝑡0,𝑢0	NUM
cana-3670	55	17	)	)	PUNCT
cana-3670	55	18	)	)	PUNCT
cana-3670	55	19	2	2	NUM
cana-3670	55	20	𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	55	21	)	)	PUNCT
cana-3670	55	22	)	)	PUNCT
cana-3670	55	23	)	)	PUNCT
cana-3670	55	24	.	.	PUNCT
cana-3670	56	1	we	we	PRON
cana-3670	56	2	may	may	AUX
cana-3670	56	3	create	create	VERB
cana-3670	56	4	a	a	DET
cana-3670	56	5	new	new	ADJ
cana-3670	56	6	equation	equation	NOUN
cana-3670	56	7	denoted	denote	VERB
cana-3670	56	8	as	as	ADP
cana-3670	56	9	,	,	PUNCT
cana-3670	56	10	by	by	ADP
cana-3670	56	11	using	use	VERB
cana-3670	56	12	an	an	DET
cana-3670	56	13	equation	equation	NOUN
cana-3670	56	14	in	in	ADP
cana-3670	56	15	the	the	DET
cana-3670	56	16	vicinity	vicinity	NOUN
cana-3670	56	17	of	of	ADP
cana-3670	56	18	a	a	DET
cana-3670	56	19	point	point	NOUN
cana-3670	56	20	,	,	PUNCT
cana-3670	56	21	such	such	ADJ
cana-3670	56	22	as	as	ADP
cana-3670	56	23	𝑃(𝑡0	𝑃(𝑡0	PROPN
cana-3670	56	24	,	,	PUNCT
cana-3670	56	25	𝑢0	𝑢0	PROPN
cana-3670	56	26	)	)	PUNCT
cana-3670	56	27	,	,	PUNCT
cana-3670	56	28	and	and	CCONJ
cana-3670	56	29	the	the	DET
cana-3670	56	30	gradient	gradient	NOUN
cana-3670	56	31	up	up	ADP
cana-3670	56	32	the	the	DET
cana-3670	56	33	equation	equation	NOUN
cana-3670	56	34	.	.	PUNCT
cana-3670	57	1	𝑢1	𝑢1	PROPN
cana-3670	57	2	=	=	PROPN
cana-3670	57	3	𝑢0	𝑢0	PROPN
cana-3670	57	4	+	+	NUM
cana-3670	57	5	ℎ𝑓	ℎ𝑓	PROPN
cana-3670	57	6	(	(	PUNCT
cana-3670	57	7	𝑡0	𝑡0	PROPN
cana-3670	57	8	2+(𝑡0+ℎ	2+(𝑡0+ℎ	NUM
cana-3670	57	9	)	)	PUNCT
cana-3670	57	10	2	2	NUM
cana-3670	57	11	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	57	12	)	)	PUNCT
cana-3670	57	13	,	,	PUNCT
cana-3670	57	14	𝑢0	𝑢0	NOUN
cana-3670	57	15	2+(𝑢0+ℎ𝑓(𝑡0,𝑢0	2+(𝑢0+ℎ𝑓(𝑡0,𝑢0	NUM
cana-3670	57	16	)	)	PUNCT
cana-3670	57	17	)	)	PUNCT
cana-3670	57	18	2	2	NUM
cana-3670	57	19	𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	57	20	)	)	PUNCT
cana-3670	57	21	)	)	PUNCT
cana-3670	57	22	)	)	PUNCT
cana-3670	57	23	.	.	PUNCT
cana-3670	58	1	the	the	DET
cana-3670	58	2	euler	euler	NOUN
cana-3670	58	3	approach	approach	NOUN
cana-3670	58	4	will	will	AUX
cana-3670	58	5	be	be	AUX
cana-3670	58	6	more	more	ADV
cana-3670	58	7	stable	stable	ADJ
cana-3670	58	8	if	if	SCONJ
cana-3670	58	9	it	it	PRON
cana-3670	58	10	estimates	estimate	VERB
cana-3670	58	11	𝑢𝑛+1using	𝑢𝑛+1use	VERB
cana-3670	58	12	the	the	DET
cana-3670	58	13	modified	modified	ADJ
cana-3670	58	14	and	and	CCONJ
cana-3670	58	15	stable	stable	ADJ
cana-3670	58	16	slope	slope	NOUN
cana-3670	58	17	function	function	NOUN
cana-3670	58	18	at	at	ADP
cana-3670	58	19	the	the	DET
cana-3670	58	20	predicted	predict	VERB
cana-3670	58	21	center	center	NOUN
cana-3670	58	22	points	point	NOUN
cana-3670	58	23	of	of	ADP
cana-3670	58	24	(	(	PUNCT
cana-3670	58	25	𝑡0	𝑡0	PROPN
cana-3670	58	26	,	,	PUNCT
cana-3670	58	27	𝑢0	𝑢0	PROPN
cana-3670	58	28	)	)	PUNCT
cana-3670	58	29	and	and	CCONJ
cana-3670	58	30	(	(	PUNCT
cana-3670	58	31	𝑡1	𝑡1	NOUN
cana-3670	58	32	,	,	PUNCT
cana-3670	58	33	𝑢1	𝑢1	PROPN
cana-3670	58	34	)	)	PUNCT
cana-3670	58	35	.	.	PUNCT
cana-3670	59	1	the	the	DET
cana-3670	59	2	equation	equation	NOUN
cana-3670	59	3	stated	state	VERB
cana-3670	59	4	above	above	ADV
cana-3670	59	5	is	be	AUX
cana-3670	59	6	known	know	VERB
cana-3670	59	7	as	as	ADP
cana-3670	59	8	euler	euler	NOUN
cana-3670	59	9	's	's	PART
cana-3670	59	10	modified	modify	VERB
cana-3670	59	11	contra	contra	PROPN
cana-3670	59	12	harmonic	harmonic	PROPN
cana-3670	59	13	mean	mean	VERB
cana-3670	59	14	.	.	PUNCT
cana-3670	60	1	it	it	PRON
cana-3670	60	2	is	be	AUX
cana-3670	60	3	characterized	characterize	VERB
cana-3670	60	4	by	by	ADP
cana-3670	60	5	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	60	6	=	=	PUNCT
cana-3670	60	7	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	60	8	+	+	X
cana-3670	60	9	ℎ𝑓	ℎ𝑓	X
cana-3670	60	10	(	(	PUNCT
cana-3670	60	11	𝑡𝑛	𝑡𝑛	X
cana-3670	60	12	2	2	NUM
cana-3670	60	13	+	+	NOUN
cana-3670	60	14	(	(	PUNCT
cana-3670	60	15	𝑡𝑛+ℎ	𝑡𝑛+ℎ	NOUN
cana-3670	60	16	)	)	PUNCT
cana-3670	60	17	2	2	NUM
cana-3670	60	18	𝑡𝑛+(𝑡𝑛+ℎ	𝑡𝑛+(𝑡𝑛+ℎ	NOUN
cana-3670	60	19	)	)	PUNCT
cana-3670	60	20	,	,	PUNCT
cana-3670	60	21	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	60	22	2	2	NUM
cana-3670	60	23	+	+	NOUN
cana-3670	60	24	(	(	PUNCT
cana-3670	60	25	𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NUM
cana-3670	60	26	)	)	PUNCT
cana-3670	60	27	)	)	PUNCT
cana-3670	60	28	2	2	NUM
cana-3670	60	29	𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	60	30	)	)	PUNCT
cana-3670	60	31	)	)	PUNCT
cana-3670	60	32	)	)	PUNCT
cana-3670	60	33	.	.	PUNCT
cana-3670	61	1	3.1.2	3.1.2	NUM
cana-3670	61	2	modified	modify	VERB
cana-3670	61	3	euler	euler	PROPN
cana-3670	61	4	’s	’s	PART
cana-3670	61	5	method	method	NOUN
cana-3670	61	6	based	base	VERB
cana-3670	61	7	on	on	ADP
cana-3670	61	8	contra	contra	PROPN
cana-3670	61	9	harmonic	harmonic	PROPN
cana-3670	61	10	mean	mean	NOUN
cana-3670	61	11	take	take	VERB
cana-3670	61	12	note	note	NOUN
cana-3670	61	13	of	of	ADP
cana-3670	61	14	a	a	DET
cana-3670	61	15	,	,	PUNCT
cana-3670	61	16	b	b	NOUN
cana-3670	61	17	,	,	PUNCT
cana-3670	61	18	and	and	CCONJ
cana-3670	61	19	c	c	NOUN
cana-3670	61	20	as	as	SCONJ
cana-3670	61	21	the	the	DET
cana-3670	61	22	contra	contra	PROPN
cana-3670	61	23	harmonic	harmonic	ADJ
cana-3670	61	24	sequence	sequence	NOUN
cana-3670	61	25	's	's	PART
cana-3670	61	26	components	component	NOUN
cana-3670	61	27	.	.	PUNCT
cana-3670	62	1	the	the	DET
cana-3670	62	2	formula	formula	NOUN
cana-3670	62	3	for	for	ADP
cana-3670	62	4	calculating	calculate	VERB
cana-3670	62	5	the	the	DET
cana-3670	62	6	contra	contra	PROPN
cana-3670	62	7	harmonic	harmonic	NOUN
cana-3670	62	8	mean	mean	VERB
cana-3670	62	9	is	be	AUX
cana-3670	62	10	𝑐	𝑐	NOUN
cana-3670	62	11	=	=	SYM
cana-3670	62	12	𝑎2+𝑏2	𝑎2+𝑏2	PROPN
cana-3670	62	13	𝑎+𝑏	𝑎+𝑏	NUM
cana-3670	62	14	.	.	PUNCT
cana-3670	63	1	next	next	ADV
cana-3670	63	2	,	,	PUNCT
cana-3670	63	3	the	the	DET
cana-3670	63	4	two	two	NUM
cana-3670	63	5	center	center	NOUN
cana-3670	63	6	points	point	NOUN
cana-3670	63	7	expected	expect	VERB
cana-3670	63	8	contra	contra	PROPN
cana-3670	63	9	harmonic	harmonic	PROPN
cana-3670	63	10	mean	mean	VERB
cana-3670	63	11	is	be	AUX
cana-3670	63	12	determined	determine	VERB
cana-3670	63	13	by	by	ADP
cana-3670	63	14	(	(	PUNCT
cana-3670	63	15	𝑡0	𝑡0	PROPN
cana-3670	63	16	2+𝑡1	2+𝑡1	NUM
cana-3670	63	17	2	2	NUM
cana-3670	63	18	𝑡0+𝑡1	𝑡0+𝑡1	PROPN
cana-3670	63	19	,	,	PUNCT
cana-3670	63	20	𝑢0	𝑢0	VERB
cana-3670	63	21	2+𝑢1	2+𝑢1	NUM
cana-3670	63	22	2	2	NUM
cana-3670	63	23	𝑢0+𝑢1	𝑢0+𝑢1	PROPN
cana-3670	63	24	)	)	PUNCT
cana-3670	63	25	.	.	PUNCT
cana-3670	64	1	the	the	DET
cana-3670	64	2	formula	formula	NOUN
cana-3670	64	3	may	may	AUX
cana-3670	64	4	be	be	AUX
cana-3670	64	5	written	write	VERB
cana-3670	64	6	as	as	SCONJ
cana-3670	64	7	communications	communication	NOUN
cana-3670	64	8	on	on	ADP
cana-3670	64	9	applied	apply	VERB
cana-3670	64	10	nonlinear	nonlinear	ADJ
cana-3670	64	11	analysis	analysis	NOUN
cana-3670	64	12	issn	issn	NOUN
cana-3670	64	13	:	:	PUNCT
cana-3670	64	14	1074	1074	NUM
cana-3670	64	15	-	-	PUNCT
cana-3670	64	16	133x	133x	NUM
cana-3670	64	17	vol	vol	NOUN
cana-3670	64	18	32	32	NUM
cana-3670	64	19	no	no	NOUN
cana-3670	64	20	.	.	PUNCT
cana-3670	65	1	8s	8s	PROPN
cana-3670	65	2	(	(	PUNCT
cana-3670	65	3	2025	2025	NUM
cana-3670	65	4	)	)	PUNCT
cana-3670	65	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	65	6	274	274	NUM
cana-3670	65	7	(	(	PUNCT
cana-3670	65	8	𝑡0	𝑡0	NOUN
cana-3670	65	9	2+(𝑡0+ℎ	2+(𝑡0+ℎ	NUM
cana-3670	65	10	)	)	PUNCT
cana-3670	65	11	2	2	NUM
cana-3670	65	12	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	65	13	)	)	PUNCT
cana-3670	65	14	,	,	PUNCT
cana-3670	65	15	𝑢0	𝑢0	VERB
cana-3670	65	16	2+(𝑢0+ℎ𝑓(𝑡0	2+(𝑢0+ℎ𝑓(𝑡0	PROPN
cana-3670	65	17	+	+	NUM
cana-3670	65	18	ℎ	ℎ	PROPN
cana-3670	65	19	2	2	NUM
cana-3670	65	20	,	,	PUNCT
cana-3670	65	21	𝑢0	𝑢0	VERB
cana-3670	65	22	+	+	CCONJ
cana-3670	65	23	ℎ	ℎ	PROPN
cana-3670	65	24	2	2	NUM
cana-3670	65	25	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	65	26	)	)	PUNCT
cana-3670	65	27	)	)	PUNCT
cana-3670	65	28	)	)	PUNCT
cana-3670	65	29	2	2	NUM
cana-3670	65	30	𝑢0+(𝑢0+ℎ𝑓(𝑡0	𝑢0+(𝑢0+ℎ𝑓(𝑡0	NUM
cana-3670	65	31	+	+	NUM
cana-3670	65	32	ℎ	ℎ	SYM
cana-3670	65	33	2	2	NUM
cana-3670	65	34	,	,	PUNCT
cana-3670	65	35	𝑢0	𝑢0	VERB
cana-3670	65	36	+	+	CCONJ
cana-3670	65	37	ℎ	ℎ	PROPN
cana-3670	65	38	2	2	NUM
cana-3670	65	39	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADV
cana-3670	65	40	)	)	PUNCT
cana-3670	65	41	)	)	PUNCT
cana-3670	65	42	)	)	PUNCT
cana-3670	65	43	)	)	PUNCT
cana-3670	65	44	.	.	PUNCT
cana-3670	66	1	we	we	PRON
cana-3670	66	2	may	may	AUX
cana-3670	66	3	create	create	VERB
cana-3670	66	4	a	a	DET
cana-3670	66	5	new	new	ADJ
cana-3670	66	6	equation	equation	NOUN
cana-3670	66	7	denoted	denote	VERB
cana-3670	66	8	as	as	ADP
cana-3670	66	9	,	,	PUNCT
cana-3670	66	10	by	by	ADP
cana-3670	66	11	using	use	VERB
cana-3670	66	12	an	an	DET
cana-3670	66	13	equation	equation	NOUN
cana-3670	66	14	in	in	ADP
cana-3670	66	15	the	the	DET
cana-3670	66	16	vicinity	vicinity	NOUN
cana-3670	66	17	of	of	ADP
cana-3670	66	18	a	a	DET
cana-3670	66	19	point	point	NOUN
cana-3670	66	20	,	,	PUNCT
cana-3670	66	21	such	such	ADJ
cana-3670	66	22	as	as	ADP
cana-3670	66	23	𝑃(𝑡0	𝑃(𝑡0	PROPN
cana-3670	66	24	,	,	PUNCT
cana-3670	66	25	𝑢0	𝑢0	PROPN
cana-3670	66	26	)	)	PUNCT
cana-3670	66	27	,	,	PUNCT
cana-3670	66	28	and	and	CCONJ
cana-3670	66	29	the	the	DET
cana-3670	66	30	gradient	gradient	NOUN
cana-3670	66	31	up	up	ADP
cana-3670	66	32	the	the	DET
cana-3670	66	33	equation	equation	NOUN
cana-3670	66	34	.	.	PUNCT
cana-3670	67	1	𝑢1	𝑢1	PROPN
cana-3670	67	2	=	=	PROPN
cana-3670	67	3	𝑢0	𝑢0	PROPN
cana-3670	67	4	+	+	NUM
cana-3670	67	5	ℎ𝑓	ℎ𝑓	NOUN
cana-3670	67	6	(	(	PUNCT
cana-3670	67	7	𝑡0	𝑡0	NOUN
cana-3670	67	8	+	+	CCONJ
cana-3670	67	9	ℎ	ℎ	PROPN
cana-3670	67	10	2	2	NUM
cana-3670	67	11	,	,	PUNCT
cana-3670	67	12	𝑢0	𝑢0	VERB
cana-3670	67	13	+	+	CCONJ
cana-3670	67	14	ℎ	ℎ	PROPN
cana-3670	67	15	2	2	NUM
cana-3670	67	16	𝑓	𝑓	PROPN
cana-3670	67	17	(	(	PUNCT
cana-3670	67	18	𝑡0	𝑡0	PROPN
cana-3670	67	19	2+(𝑡0+ℎ	2+(𝑡0+ℎ	NUM
cana-3670	67	20	)	)	PUNCT
cana-3670	67	21	2	2	NUM
cana-3670	67	22	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	67	23	)	)	PUNCT
cana-3670	67	24	,	,	PUNCT
cana-3670	67	25	𝑢0	𝑢0	VERB
cana-3670	67	26	2+(𝑢0+ℎ𝑓(𝑡0	2+(𝑢0+ℎ𝑓(𝑡0	PROPN
cana-3670	67	27	+	+	NUM
cana-3670	67	28	ℎ	ℎ	PROPN
cana-3670	67	29	2	2	NUM
cana-3670	67	30	,	,	PUNCT
cana-3670	67	31	𝑢0	𝑢0	VERB
cana-3670	67	32	+	+	CCONJ
cana-3670	67	33	ℎ	ℎ	PROPN
cana-3670	67	34	2	2	NUM
cana-3670	67	35	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADV
cana-3670	67	36	)	)	PUNCT
cana-3670	67	37	)	)	PUNCT
cana-3670	67	38	)	)	PUNCT
cana-3670	68	1	2	2	NUM
cana-3670	68	2	𝑢0+(𝑢0+ℎ𝑓(𝑡0	𝑢0+(𝑢0+ℎ𝑓(𝑡0	NUM
cana-3670	68	3	+	+	NUM
cana-3670	68	4	ℎ	ℎ	SYM
cana-3670	68	5	2	2	NUM
cana-3670	68	6	,	,	PUNCT
cana-3670	68	7	𝑢0	𝑢0	VERB
cana-3670	68	8	+	+	CCONJ
cana-3670	68	9	ℎ	ℎ	PROPN
cana-3670	68	10	2	2	NUM
cana-3670	68	11	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	68	12	)	)	PUNCT
cana-3670	68	13	)	)	PUNCT
cana-3670	68	14	)	)	PUNCT
cana-3670	68	15	)	)	PUNCT
cana-3670	68	16	)	)	PUNCT
cana-3670	68	17	.	.	PUNCT
cana-3670	69	1	the	the	DET
cana-3670	69	2	euler	euler	NOUN
cana-3670	69	3	approach	approach	NOUN
cana-3670	69	4	will	will	AUX
cana-3670	69	5	be	be	AUX
cana-3670	69	6	more	more	ADV
cana-3670	69	7	stable	stable	ADJ
cana-3670	69	8	if	if	SCONJ
cana-3670	69	9	it	it	PRON
cana-3670	69	10	estimates	estimate	VERB
cana-3670	69	11	𝑢𝑛+1using	𝑢𝑛+1use	VERB
cana-3670	69	12	the	the	DET
cana-3670	69	13	modified	modified	ADJ
cana-3670	69	14	and	and	CCONJ
cana-3670	69	15	stable	stable	ADJ
cana-3670	69	16	slope	slope	NOUN
cana-3670	69	17	function	function	NOUN
cana-3670	69	18	at	at	ADP
cana-3670	69	19	the	the	DET
cana-3670	69	20	predicted	predict	VERB
cana-3670	69	21	center	center	NOUN
cana-3670	69	22	points	point	NOUN
cana-3670	69	23	of	of	ADP
cana-3670	69	24	(	(	PUNCT
cana-3670	69	25	𝑡0	𝑡0	PROPN
cana-3670	69	26	,	,	PUNCT
cana-3670	69	27	𝑢0	𝑢0	PROPN
cana-3670	69	28	)	)	PUNCT
cana-3670	69	29	and	and	CCONJ
cana-3670	69	30	(	(	PUNCT
cana-3670	69	31	𝑡1	𝑡1	NOUN
cana-3670	69	32	,	,	PUNCT
cana-3670	69	33	𝑢1	𝑢1	PROPN
cana-3670	69	34	)	)	PUNCT
cana-3670	69	35	.	.	PUNCT
cana-3670	70	1	the	the	DET
cana-3670	70	2	equation	equation	NOUN
cana-3670	70	3	stated	state	VERB
cana-3670	70	4	above	above	ADV
cana-3670	70	5	is	be	AUX
cana-3670	70	6	known	know	VERB
cana-3670	70	7	as	as	ADP
cana-3670	70	8	euler	euler	NOUN
cana-3670	70	9	's	's	PART
cana-3670	70	10	modified	modify	VERB
cana-3670	70	11	contra	contra	PROPN
cana-3670	70	12	harmonic	harmonic	PROPN
cana-3670	70	13	mean	mean	VERB
cana-3670	70	14	.	.	PUNCT
cana-3670	71	1	it	it	PRON
cana-3670	71	2	is	be	AUX
cana-3670	71	3	characterized	characterize	VERB
cana-3670	71	4	by	by	ADP
cana-3670	71	5	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	71	6	=	=	PUNCT
cana-3670	71	7	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	71	8	+	+	X
cana-3670	71	9	ℎ𝑓	ℎ𝑓	X
cana-3670	71	10	(	(	PUNCT
cana-3670	71	11	𝑡𝑛	𝑡𝑛	X
cana-3670	71	12	+	+	CCONJ
cana-3670	71	13	ℎ	ℎ	X
cana-3670	71	14	2	2	NUM
cana-3670	71	15	,	,	PUNCT
cana-3670	71	16	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	71	17	+	+	CCONJ
cana-3670	71	18	ℎ	ℎ	PROPN
cana-3670	71	19	2	2	NUM
cana-3670	71	20	𝑓	𝑓	PRON
cana-3670	71	21	(	(	PUNCT
cana-3670	71	22	𝑡𝑛	𝑡𝑛	X
cana-3670	71	23	2	2	NUM
cana-3670	71	24	+	+	NOUN
cana-3670	71	25	(	(	PUNCT
cana-3670	71	26	𝑡𝑛+ℎ	𝑡𝑛+ℎ	NOUN
cana-3670	71	27	)	)	PUNCT
cana-3670	71	28	2	2	NUM
cana-3670	71	29	𝑡𝑛+(𝑡𝑛+ℎ	𝑡𝑛+(𝑡𝑛+ℎ	NOUN
cana-3670	71	30	)	)	PUNCT
cana-3670	71	31	,	,	PUNCT
cana-3670	71	32	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	71	33	2	2	NUM
cana-3670	72	1	+	+	NOUN
cana-3670	72	2	(	(	PUNCT
cana-3670	72	3	𝑢𝑛+ℎ𝑓(𝑡𝑛+	𝑢𝑛+ℎ𝑓(𝑡𝑛+	ADP
cana-3670	72	4	ℎ	ℎ	PROPN
cana-3670	72	5	2	2	NUM
cana-3670	72	6	,	,	PUNCT
cana-3670	72	7	𝑢𝑛+	𝑢𝑛+	ADJ
cana-3670	72	8	ℎ	ℎ	PROPN
cana-3670	72	9	2	2	NUM
cana-3670	72	10	𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	72	11	)	)	PUNCT
cana-3670	72	12	)	)	PUNCT
cana-3670	72	13	)	)	PUNCT
cana-3670	73	1	2	2	NUM
cana-3670	73	2	𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛+	𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛+	PUNCT
cana-3670	73	3	ℎ	ℎ	X
cana-3670	73	4	2	2	NUM
cana-3670	73	5	,	,	PUNCT
cana-3670	73	6	𝑢𝑛+	𝑢𝑛+	ADJ
cana-3670	73	7	ℎ	ℎ	NOUN
cana-3670	73	8	2	2	NUM
cana-3670	73	9	𝑓(𝑡𝑛	𝑓(𝑡𝑛	NOUN
cana-3670	73	10	,	,	PUNCT
cana-3670	73	11	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	73	12	)	)	PUNCT
cana-3670	73	13	)	)	PUNCT
cana-3670	73	14	)	)	PUNCT
cana-3670	73	15	)	)	PUNCT
cana-3670	73	16	)	)	PUNCT
cana-3670	73	17	.	.	PUNCT
cana-3670	74	1	3.1.3	3.1.3	NUM
cana-3670	74	2	improved	improve	VERB
cana-3670	74	3	euler	euler	NOUN
cana-3670	74	4	’s	’s	PART
cana-3670	74	5	method	method	NOUN
cana-3670	74	6	based	base	VERB
cana-3670	74	7	on	on	ADP
cana-3670	74	8	contra	contra	PROPN
cana-3670	74	9	harmonic	harmonic	PROPN
cana-3670	74	10	mean	mean	NOUN
cana-3670	74	11	take	take	VERB
cana-3670	74	12	note	note	NOUN
cana-3670	74	13	of	of	ADP
cana-3670	74	14	a	a	DET
cana-3670	74	15	,	,	PUNCT
cana-3670	74	16	b	b	NOUN
cana-3670	74	17	,	,	PUNCT
cana-3670	74	18	and	and	CCONJ
cana-3670	74	19	c	c	NOUN
cana-3670	74	20	as	as	SCONJ
cana-3670	74	21	the	the	DET
cana-3670	74	22	contra	contra	PROPN
cana-3670	74	23	harmonic	harmonic	ADJ
cana-3670	74	24	sequence	sequence	NOUN
cana-3670	74	25	's	's	PART
cana-3670	74	26	components	component	NOUN
cana-3670	74	27	.	.	PUNCT
cana-3670	75	1	the	the	DET
cana-3670	75	2	formula	formula	NOUN
cana-3670	75	3	for	for	ADP
cana-3670	75	4	calculating	calculate	VERB
cana-3670	75	5	the	the	DET
cana-3670	75	6	contra	contra	PROPN
cana-3670	75	7	harmonic	harmonic	NOUN
cana-3670	75	8	mean	mean	VERB
cana-3670	75	9	is	be	AUX
cana-3670	75	10	𝑐	𝑐	NOUN
cana-3670	75	11	=	=	SYM
cana-3670	75	12	𝑎2+𝑏2	𝑎2+𝑏2	PROPN
cana-3670	75	13	𝑎+𝑏	𝑎+𝑏	NUM
cana-3670	75	14	.	.	PUNCT
cana-3670	76	1	next	next	ADV
cana-3670	76	2	,	,	PUNCT
cana-3670	76	3	the	the	DET
cana-3670	76	4	two	two	NUM
cana-3670	76	5	center	center	NOUN
cana-3670	76	6	points	point	NOUN
cana-3670	76	7	expected	expect	VERB
cana-3670	76	8	contra	contra	PROPN
cana-3670	76	9	harmonic	harmonic	PROPN
cana-3670	76	10	mean	mean	VERB
cana-3670	76	11	is	be	AUX
cana-3670	76	12	determined	determine	VERB
cana-3670	76	13	by	by	ADP
cana-3670	76	14	(	(	PUNCT
cana-3670	76	15	𝑡0	𝑡0	PROPN
cana-3670	76	16	2+𝑡1	2+𝑡1	NUM
cana-3670	76	17	2	2	NUM
cana-3670	76	18	𝑡0+𝑡1	𝑡0+𝑡1	PROPN
cana-3670	76	19	,	,	PUNCT
cana-3670	76	20	𝑢0	𝑢0	VERB
cana-3670	76	21	2+𝑢1	2+𝑢1	NUM
cana-3670	76	22	2	2	NUM
cana-3670	76	23	𝑢0+𝑢1	𝑢0+𝑢1	PROPN
cana-3670	76	24	)	)	PUNCT
cana-3670	76	25	.	.	PUNCT
cana-3670	77	1	the	the	DET
cana-3670	77	2	formula	formula	NOUN
cana-3670	77	3	may	may	AUX
cana-3670	77	4	be	be	AUX
cana-3670	77	5	written	write	VERB
cana-3670	77	6	as	as	ADP
cana-3670	77	7	(	(	PUNCT
cana-3670	77	8	𝑡0	𝑡0	PROPN
cana-3670	77	9	2+(𝑡0+ℎ	2+(𝑡0+ℎ	NUM
cana-3670	77	10	)	)	PUNCT
cana-3670	77	11	2	2	NUM
cana-3670	77	12	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	77	13	)	)	PUNCT
cana-3670	77	14	,	,	PUNCT
cana-3670	77	15	𝑢0	𝑢0	VERB
cana-3670	77	16	2+(𝑢0	2+(𝑢0	PROPN
cana-3670	77	17	+	+	CCONJ
cana-3670	77	18	ℎ	ℎ	SYM
cana-3670	77	19	2	2	NUM
cana-3670	77	20	(	(	PUNCT
cana-3670	77	21	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	PROPN
cana-3670	77	22	)	)	PUNCT
cana-3670	77	23	)	)	PUNCT
cana-3670	77	24	)	)	PUNCT
cana-3670	77	25	)	)	PUNCT
cana-3670	77	26	2	2	NUM
cana-3670	77	27	𝑢0+(𝑢0	𝑢0+(𝑢0	NOUN
cana-3670	77	28	+	+	SYM
cana-3670	77	29	ℎ	ℎ	SYM
cana-3670	77	30	2	2	NUM
cana-3670	77	31	(	(	PUNCT
cana-3670	77	32	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	77	33	)	)	PUNCT
cana-3670	77	34	+	+	ADJ
cana-3670	77	35	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	77	36	)	)	PUNCT
cana-3670	77	37	)	)	PUNCT
cana-3670	77	38	)	)	PUNCT
cana-3670	77	39	)	)	PUNCT
cana-3670	77	40	)	)	PUNCT
cana-3670	77	41	.	.	PUNCT
cana-3670	78	1	we	we	PRON
cana-3670	78	2	may	may	AUX
cana-3670	78	3	create	create	VERB
cana-3670	78	4	a	a	DET
cana-3670	78	5	new	new	ADJ
cana-3670	78	6	equation	equation	NOUN
cana-3670	78	7	denoted	denote	VERB
cana-3670	78	8	as	as	ADP
cana-3670	78	9	,	,	PUNCT
cana-3670	78	10	by	by	ADP
cana-3670	78	11	using	use	VERB
cana-3670	78	12	an	an	DET
cana-3670	78	13	equation	equation	NOUN
cana-3670	78	14	in	in	ADP
cana-3670	78	15	the	the	DET
cana-3670	78	16	vicinity	vicinity	NOUN
cana-3670	78	17	of	of	ADP
cana-3670	78	18	a	a	DET
cana-3670	78	19	point	point	NOUN
cana-3670	78	20	,	,	PUNCT
cana-3670	78	21	such	such	ADJ
cana-3670	78	22	as	as	ADP
cana-3670	78	23	𝑃(𝑡0	𝑃(𝑡0	PROPN
cana-3670	78	24	,	,	PUNCT
cana-3670	78	25	𝑢0	𝑢0	PROPN
cana-3670	78	26	)	)	PUNCT
cana-3670	78	27	,	,	PUNCT
cana-3670	78	28	and	and	CCONJ
cana-3670	78	29	the	the	DET
cana-3670	78	30	gradient	gradient	NOUN
cana-3670	78	31	up	up	ADP
cana-3670	78	32	the	the	DET
cana-3670	78	33	equation	equation	NOUN
cana-3670	78	34	.	.	PUNCT
cana-3670	79	1	𝑢1	𝑢1	PROPN
cana-3670	79	2	=	=	PROPN
cana-3670	79	3	𝑢0	𝑢0	PROPN
cana-3670	79	4	+	+	NUM
cana-3670	79	5	ℎ	ℎ	X
cana-3670	79	6	2	2	NUM
cana-3670	79	7	[	[	PUNCT
cana-3670	79	8	𝑓	𝑓	PROPN
cana-3670	79	9	(	(	PUNCT
cana-3670	79	10	𝑡0	𝑡0	PROPN
cana-3670	79	11	2+(𝑡0+ℎ	2+(𝑡0+ℎ	NUM
cana-3670	79	12	)	)	PUNCT
cana-3670	79	13	2	2	NUM
cana-3670	79	14	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	79	15	)	)	PUNCT
cana-3670	79	16	,	,	PUNCT
cana-3670	79	17	𝑢0	𝑢0	VERB
cana-3670	79	18	2+(𝑢0	2+(𝑢0	PROPN
cana-3670	79	19	+	+	CCONJ
cana-3670	79	20	ℎ	ℎ	SYM
cana-3670	79	21	2	2	NUM
cana-3670	79	22	(	(	PUNCT
cana-3670	79	23	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	PROPN
cana-3670	79	24	)	)	PUNCT
cana-3670	79	25	)	)	PUNCT
cana-3670	79	26	)	)	PUNCT
cana-3670	79	27	)	)	PUNCT
cana-3670	79	28	2	2	NUM
cana-3670	79	29	𝑢0+(𝑢0	𝑢0+(𝑢0	NOUN
cana-3670	79	30	+	+	SYM
cana-3670	79	31	ℎ	ℎ	SYM
cana-3670	79	32	2	2	NUM
cana-3670	79	33	(	(	PUNCT
cana-3670	79	34	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	79	35	)	)	PUNCT
cana-3670	79	36	+	+	ADJ
cana-3670	79	37	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	79	38	)	)	PUNCT
cana-3670	79	39	)	)	PUNCT
cana-3670	79	40	)	)	PUNCT
cana-3670	79	41	)	)	PUNCT
cana-3670	79	42	)	)	PUNCT
cana-3670	80	1	+	+	CCONJ
cana-3670	80	2	𝑓	𝑓	PRON
cana-3670	80	3	(	(	PUNCT
cana-3670	80	4	𝑡0	𝑡0	PROPN
cana-3670	80	5	+	+	CCONJ
cana-3670	80	6	ℎ	ℎ	PROPN
cana-3670	80	7	,	,	PUNCT
cana-3670	80	8	𝑢0	𝑢0	PROPN
cana-3670	80	9	+	+	CCONJ
cana-3670	80	10	ℎ	ℎ	PROPN
cana-3670	80	11	(	(	PUNCT
cana-3670	80	12	𝑡0	𝑡0	NOUN
cana-3670	80	13	2+(𝑡0+ℎ	2+(𝑡0+ℎ	NUM
cana-3670	80	14	)	)	PUNCT
cana-3670	80	15	2	2	NUM
cana-3670	80	16	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	80	17	)	)	PUNCT
cana-3670	80	18	,	,	PUNCT
cana-3670	80	19	𝑢0	𝑢0	VERB
cana-3670	80	20	2+(𝑢0	2+(𝑢0	PROPN
cana-3670	80	21	+	+	CCONJ
cana-3670	80	22	ℎ	ℎ	SYM
cana-3670	80	23	2	2	NUM
cana-3670	80	24	(	(	PUNCT
cana-3670	80	25	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	80	26	)	)	PUNCT
cana-3670	80	27	+	+	ADJ
cana-3670	80	28	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	80	29	)	)	PUNCT
cana-3670	80	30	)	)	PUNCT
cana-3670	80	31	)	)	PUNCT
cana-3670	80	32	)	)	PUNCT
cana-3670	80	33	2	2	NUM
cana-3670	80	34	𝑢0+(𝑢0	𝑢0+(𝑢0	NOUN
cana-3670	80	35	+	+	SYM
cana-3670	80	36	ℎ	ℎ	SYM
cana-3670	80	37	2	2	NUM
cana-3670	80	38	(	(	PUNCT
cana-3670	80	39	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	PROPN
cana-3670	80	40	)	)	PUNCT
cana-3670	80	41	)	)	PUNCT
cana-3670	80	42	)	)	PUNCT
cana-3670	80	43	)	)	PUNCT
cana-3670	80	44	)	)	PUNCT
cana-3670	80	45	)	)	PUNCT
cana-3670	80	46	]	]	PUNCT
cana-3670	80	47	.	.	PUNCT
cana-3670	81	1	communications	communication	NOUN
cana-3670	81	2	on	on	ADP
cana-3670	81	3	applied	apply	VERB
cana-3670	81	4	nonlinear	nonlinear	ADJ
cana-3670	81	5	analysis	analysis	NOUN
cana-3670	81	6	issn	issn	NOUN
cana-3670	81	7	:	:	PUNCT
cana-3670	81	8	1074	1074	NUM
cana-3670	81	9	-	-	PUNCT
cana-3670	81	10	133x	133x	NUM
cana-3670	81	11	vol	vol	NOUN
cana-3670	81	12	32	32	NUM
cana-3670	81	13	no	no	NOUN
cana-3670	81	14	.	.	PUNCT
cana-3670	82	1	8s	8s	PROPN
cana-3670	82	2	(	(	PUNCT
cana-3670	82	3	2025	2025	NUM
cana-3670	82	4	)	)	PUNCT
cana-3670	82	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	82	6	275	275	NUM
cana-3670	82	7	the	the	DET
cana-3670	82	8	euler	euler	NOUN
cana-3670	82	9	approach	approach	NOUN
cana-3670	82	10	will	will	AUX
cana-3670	82	11	be	be	AUX
cana-3670	82	12	more	more	ADV
cana-3670	82	13	stable	stable	ADJ
cana-3670	82	14	if	if	SCONJ
cana-3670	82	15	it	it	PRON
cana-3670	82	16	estimates	estimate	VERB
cana-3670	82	17	𝑢𝑛+1using	𝑢𝑛+1use	VERB
cana-3670	82	18	the	the	DET
cana-3670	82	19	modified	modified	ADJ
cana-3670	82	20	and	and	CCONJ
cana-3670	82	21	stable	stable	ADJ
cana-3670	82	22	slope	slope	NOUN
cana-3670	82	23	function	function	NOUN
cana-3670	82	24	at	at	ADP
cana-3670	82	25	the	the	DET
cana-3670	82	26	predicted	predict	VERB
cana-3670	82	27	center	center	NOUN
cana-3670	82	28	points	point	NOUN
cana-3670	82	29	of	of	ADP
cana-3670	82	30	(	(	PUNCT
cana-3670	82	31	𝑡0	𝑡0	PROPN
cana-3670	82	32	,	,	PUNCT
cana-3670	82	33	𝑢0	𝑢0	PROPN
cana-3670	82	34	)	)	PUNCT
cana-3670	82	35	and	and	CCONJ
cana-3670	82	36	(	(	PUNCT
cana-3670	82	37	𝑡1	𝑡1	NOUN
cana-3670	82	38	,	,	PUNCT
cana-3670	82	39	𝑢1	𝑢1	PROPN
cana-3670	82	40	)	)	PUNCT
cana-3670	82	41	.	.	PUNCT
cana-3670	83	1	the	the	DET
cana-3670	83	2	equation	equation	NOUN
cana-3670	83	3	stated	state	VERB
cana-3670	83	4	above	above	ADV
cana-3670	83	5	is	be	AUX
cana-3670	83	6	known	know	VERB
cana-3670	83	7	as	as	ADP
cana-3670	83	8	euler	euler	NOUN
cana-3670	83	9	's	's	PART
cana-3670	83	10	modified	modify	VERB
cana-3670	83	11	contra	contra	PROPN
cana-3670	83	12	harmonic	harmonic	PROPN
cana-3670	83	13	mean	mean	VERB
cana-3670	83	14	.	.	PUNCT
cana-3670	84	1	it	it	PRON
cana-3670	84	2	is	be	AUX
cana-3670	84	3	characterized	characterize	VERB
cana-3670	84	4	by	by	ADP
cana-3670	84	5	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	84	6	=	=	PUNCT
cana-3670	84	7	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	84	8	+	+	CCONJ
cana-3670	84	9	ℎ	ℎ	PROPN
cana-3670	84	10	2	2	NUM
cana-3670	84	11	[	[	PUNCT
cana-3670	84	12	𝑓	𝑓	X
cana-3670	84	13	(	(	PUNCT
cana-3670	84	14	𝑡𝑛	𝑡𝑛	ADJ
cana-3670	84	15	2	2	NUM
cana-3670	84	16	+	+	NOUN
cana-3670	84	17	(	(	PUNCT
cana-3670	84	18	𝑡𝑛+ℎ)2	𝑡𝑛+ℎ)2	PROPN
cana-3670	84	19	𝑡𝑛+(𝑡𝑛+ℎ	𝑡𝑛+(𝑡𝑛+ℎ	NOUN
cana-3670	84	20	)	)	PUNCT
cana-3670	84	21	,	,	PUNCT
cana-3670	84	22	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	84	23	2	2	NUM
cana-3670	84	24	+	+	NOUN
cana-3670	84	25	(	(	PUNCT
cana-3670	84	26	𝑢𝑛+	𝑢𝑛+	ADJ
cana-3670	84	27	ℎ	ℎ	X
cana-3670	84	28	2	2	NUM
cana-3670	84	29	(	(	PUNCT
cana-3670	84	30	𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	84	31	)	)	PUNCT
cana-3670	84	32	+	+	NOUN
cana-3670	84	33	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	84	34	)	)	PUNCT
cana-3670	84	35	)	)	PUNCT
cana-3670	84	36	)	)	PUNCT
cana-3670	84	37	)	)	PUNCT
cana-3670	84	38	2	2	NUM
cana-3670	84	39	𝑢𝑛+(𝑢𝑛+	𝑢𝑛+(𝑢𝑛+	NOUN
cana-3670	84	40	ℎ	ℎ	PROPN
cana-3670	84	41	2	2	NUM
cana-3670	84	42	(	(	PUNCT
cana-3670	84	43	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	84	44	)	)	PUNCT
cana-3670	84	45	)	)	PUNCT
cana-3670	84	46	)	)	PUNCT
cana-3670	84	47	)	)	PUNCT
cana-3670	84	48	)	)	PUNCT
cana-3670	85	1	+	+	CCONJ
cana-3670	85	2	𝑓	𝑓	X
cana-3670	85	3	(	(	PUNCT
cana-3670	85	4	𝑡𝑛	𝑡𝑛	X
cana-3670	85	5	+	+	NUM
cana-3670	85	6	ℎ	ℎ	PROPN
cana-3670	85	7	,	,	PUNCT
cana-3670	85	8	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	85	9	+	+	SYM
cana-3670	85	10	ℎ	ℎ	PROPN
cana-3670	85	11	(	(	PUNCT
cana-3670	85	12	𝑡𝑛	𝑡𝑛	NOUN
cana-3670	85	13	2	2	NUM
cana-3670	85	14	+	+	NOUN
cana-3670	85	15	(	(	PUNCT
cana-3670	85	16	𝑡𝑛+ℎ	𝑡𝑛+ℎ	NOUN
cana-3670	85	17	)	)	PUNCT
cana-3670	85	18	2	2	NUM
cana-3670	85	19	𝑡𝑛+(𝑡𝑛+ℎ	𝑡𝑛+(𝑡𝑛+ℎ	NOUN
cana-3670	85	20	)	)	PUNCT
cana-3670	85	21	,	,	PUNCT
cana-3670	85	22	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	85	23	2	2	NUM
cana-3670	85	24	+	+	NOUN
cana-3670	85	25	(	(	PUNCT
cana-3670	85	26	𝑢𝑛+	𝑢𝑛+	ADJ
cana-3670	85	27	ℎ	ℎ	X
cana-3670	85	28	2	2	NUM
cana-3670	85	29	(	(	PUNCT
cana-3670	85	30	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	85	31	)	)	PUNCT
cana-3670	85	32	)	)	PUNCT
cana-3670	85	33	)	)	PUNCT
cana-3670	85	34	)	)	PUNCT
cana-3670	85	35	2	2	NUM
cana-3670	85	36	𝑢𝑛+(𝑢𝑛+	𝑢𝑛+(𝑢𝑛+	NOUN
cana-3670	85	37	ℎ	ℎ	PROPN
cana-3670	85	38	2	2	NUM
cana-3670	85	39	(	(	PUNCT
cana-3670	85	40	𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	85	41	)	)	PUNCT
cana-3670	85	42	+	+	NOUN
cana-3670	85	43	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	85	44	)	)	PUNCT
cana-3670	85	45	)	)	PUNCT
cana-3670	85	46	)	)	PUNCT
cana-3670	85	47	)	)	PUNCT
cana-3670	85	48	)	)	PUNCT
cana-3670	85	49	)	)	PUNCT
cana-3670	85	50	]	]	PUNCT
cana-3670	85	51	.	.	PUNCT
cana-3670	86	1	3.1.4euler	3.1.4euler	NUM
cana-3670	86	2	’s	’s	PART
cana-3670	86	3	method	method	NOUN
cana-3670	86	4	based	base	VERB
cana-3670	86	5	on	on	ADP
cana-3670	86	6	centroidal	centroidal	ADJ
cana-3670	86	7	mean	mean	NOUN
cana-3670	86	8	assume	assume	VERB
cana-3670	86	9	that	that	SCONJ
cana-3670	86	10	the	the	DET
cana-3670	86	11	essential	essential	ADJ
cana-3670	86	12	components	component	NOUN
cana-3670	86	13	of	of	ADP
cana-3670	86	14	a	a	DET
cana-3670	86	15	centroidal	centroidal	ADJ
cana-3670	86	16	sequence	sequence	NOUN
cana-3670	86	17	are	be	AUX
cana-3670	86	18	a	a	DET
cana-3670	86	19	,	,	PUNCT
cana-3670	86	20	b	b	NOUN
cana-3670	86	21	,	,	PUNCT
cana-3670	86	22	and	and	CCONJ
cana-3670	86	23	c.	c.	NOUN
cana-3670	86	24	the	the	DET
cana-3670	86	25	formula	formula	NOUN
cana-3670	86	26	to	to	PART
cana-3670	86	27	compute	compute	VERB
cana-3670	86	28	the	the	DET
cana-3670	86	29	centroidal	centroidal	NOUN
cana-3670	86	30	mean	mean	NOUN
cana-3670	86	31	is	be	AUX
cana-3670	86	32	𝑐	𝑐	NOUN
cana-3670	86	33	=	=	SYM
cana-3670	86	34	2(𝑎2+𝑎𝑏+𝑏2	2(𝑎2+𝑎𝑏+𝑏2	NUM
cana-3670	86	35	)	)	PUNCT
cana-3670	86	36	3	3	NUM
cana-3670	86	37	(	(	PUNCT
cana-3670	86	38	𝑎+𝑏	𝑎+𝑏	NOUN
cana-3670	86	39	)	)	PUNCT
cana-3670	86	40	.	.	PUNCT
cana-3670	87	1	the	the	DET
cana-3670	87	2	expected	expect	VERB
cana-3670	87	3	centroidal	centroidal	ADJ
cana-3670	87	4	mean	mean	NOUN
cana-3670	87	5	of	of	ADP
cana-3670	87	6	the	the	DET
cana-3670	87	7	two	two	NUM
cana-3670	87	8	midpoints	midpoint	NOUN
cana-3670	87	9	is	be	AUX
cana-3670	87	10	therefore	therefore	ADV
cana-3670	87	11	given	give	VERB
cana-3670	87	12	as	as	ADP
cana-3670	87	13	(	(	PUNCT
cana-3670	87	14	2(𝑡0	2(𝑡0	NUM
cana-3670	87	15	2+𝑡0𝑡1+𝑡1	2+𝑡0𝑡1+𝑡1	NUM
cana-3670	87	16	2	2	X
cana-3670	87	17	)	)	PUNCT
cana-3670	87	18	3(𝑡0+𝑡1	3(𝑡0+𝑡1	NUM
cana-3670	87	19	)	)	PUNCT
cana-3670	87	20	,	,	PUNCT
cana-3670	87	21	2(𝑢0	2(𝑢0	NUM
cana-3670	88	1	2+𝑢0𝑢1+𝑢1	2+𝑢0𝑢1+𝑢1	NUM
cana-3670	88	2	2	2	NUM
cana-3670	88	3	)	)	PUNCT
cana-3670	88	4	3(𝑢0+𝑢1	3(𝑢0+𝑢1	NUM
cana-3670	88	5	)	)	PUNCT
cana-3670	88	6	)	)	PUNCT
cana-3670	88	7	.	.	PUNCT
cana-3670	89	1	the	the	DET
cana-3670	89	2	mathematical	mathematical	ADJ
cana-3670	89	3	equation	equation	NOUN
cana-3670	89	4	is	be	AUX
cana-3670	89	5	defined	define	VERB
cana-3670	89	6	as	as	ADP
cana-3670	89	7	(	(	PUNCT
cana-3670	89	8	2(𝑡0	2(𝑡0	NUM
cana-3670	89	9	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	NUM
cana-3670	89	10	)	)	PUNCT
cana-3670	89	11	2	2	NUM
cana-3670	89	12	)	)	PUNCT
cana-3670	89	13	3	3	NUM
cana-3670	89	14	(	(	PUNCT
cana-3670	89	15	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	89	16	)	)	PUNCT
cana-3670	89	17	)	)	PUNCT
cana-3670	89	18	,	,	PUNCT
cana-3670	89	19	2(𝑢0	2(𝑢0	NUM
cana-3670	89	20	2+𝑢0	2+𝑢0	NUM
cana-3670	89	21	(	(	PUNCT
cana-3670	89	22	𝑢0+ℎ𝑓(𝑡0,𝑢0))+(𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑢0+ℎ𝑓(𝑡0,𝑢0))+(𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	89	23	)	)	PUNCT
cana-3670	89	24	)	)	PUNCT
cana-3670	89	25	2	2	NUM
cana-3670	89	26	)	)	PUNCT
cana-3670	89	27	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	NUM
cana-3670	89	28	)	)	PUNCT
cana-3670	89	29	)	)	PUNCT
cana-3670	89	30	)	)	PUNCT
cana-3670	89	31	)	)	PUNCT
cana-3670	89	32	.	.	PUNCT
cana-3670	90	1	a	a	DET
cana-3670	90	2	new	new	ADJ
cana-3670	90	3	equation	equation	NOUN
cana-3670	90	4	can	can	AUX
cana-3670	90	5	be	be	AUX
cana-3670	90	6	created	create	VERB
cana-3670	90	7	and	and	CCONJ
cana-3670	90	8	provided	provide	VERB
cana-3670	90	9	as	as	SCONJ
cana-3670	90	10	follows	follow	VERB
cana-3670	90	11	if	if	SCONJ
cana-3670	90	12	an	an	DET
cana-3670	90	13	equation	equation	NOUN
cana-3670	90	14	goes	go	VERB
cana-3670	90	15	through	through	ADP
cana-3670	90	16	a	a	DET
cana-3670	90	17	point	point	NOUN
cana-3670	90	18	,	,	PUNCT
cana-3670	90	19	such	such	ADJ
cana-3670	90	20	as	as	ADP
cana-3670	90	21	𝑃(𝑡0	𝑃(𝑡0	PROPN
cana-3670	90	22	,	,	PUNCT
cana-3670	90	23	𝑢0	𝑢0	PROPN
cana-3670	90	24	)	)	PUNCT
cana-3670	90	25	in	in	ADP
cana-3670	90	26	the	the	DET
cana-3670	90	27	center	center	NOUN
cana-3670	90	28	of	of	ADP
cana-3670	90	29	the	the	DET
cana-3670	90	30	slope	slope	NOUN
cana-3670	90	31	throughout	throughout	ADP
cana-3670	90	32	the	the	DET
cana-3670	90	33	equation	equation	NOUN
cana-3670	90	34	.	.	PUNCT
cana-3670	91	1	𝑢1	𝑢1	PROPN
cana-3670	91	2	=	=	PROPN
cana-3670	91	3	𝑢0	𝑢0	PROPN
cana-3670	91	4	+	+	NUM
cana-3670	91	5	ℎ𝑓	ℎ𝑓	NOUN
cana-3670	91	6	(	(	PUNCT
cana-3670	91	7	2(𝑡0	2(𝑡0	NUM
cana-3670	91	8	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	NUM
cana-3670	91	9	)	)	PUNCT
cana-3670	91	10	2	2	NUM
cana-3670	91	11	)	)	PUNCT
cana-3670	91	12	3	3	NUM
cana-3670	91	13	(	(	PUNCT
cana-3670	91	14	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	91	15	)	)	PUNCT
cana-3670	91	16	)	)	PUNCT
cana-3670	91	17	,	,	PUNCT
cana-3670	91	18	2(𝑢0	2(𝑢0	NUM
cana-3670	91	19	2+𝑢0	2+𝑢0	NUM
cana-3670	91	20	(	(	PUNCT
cana-3670	91	21	𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑢0+ℎ𝑓(𝑡0,𝑢0	ADJ
cana-3670	91	22	)	)	PUNCT
cana-3670	91	23	)	)	PUNCT
cana-3670	92	1	+	+	ADJ
cana-3670	92	2	(	(	PUNCT
cana-3670	92	3	𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑢0+ℎ𝑓(𝑡0,𝑢0	ADJ
cana-3670	92	4	)	)	PUNCT
cana-3670	92	5	)	)	PUNCT
cana-3670	92	6	2	2	NUM
cana-3670	92	7	)	)	PUNCT
cana-3670	92	8	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0,𝑢0	NUM
cana-3670	92	9	)	)	PUNCT
cana-3670	92	10	)	)	PUNCT
cana-3670	92	11	)	)	PUNCT
cana-3670	92	12	)	)	PUNCT
cana-3670	92	13	.	.	PUNCT
cana-3670	93	1	the	the	DET
cana-3670	93	2	results	result	NOUN
cana-3670	93	3	of	of	ADP
cana-3670	93	4	the	the	DET
cana-3670	93	5	modified	modify	VERB
cana-3670	93	6	euler	euler	NOUN
cana-3670	93	7	's	's	PART
cana-3670	93	8	approach	approach	NOUN
cana-3670	93	9	are	be	AUX
cana-3670	93	10	now	now	ADV
cana-3670	93	11	certain	certain	ADJ
cana-3670	93	12	and	and	CCONJ
cana-3670	93	13	accurate	accurate	ADJ
cana-3670	93	14	.	.	PUNCT
cana-3670	94	1	the	the	DET
cana-3670	94	2	centroidal	centroidal	ADJ
cana-3670	94	3	mean	mean	NOUN
cana-3670	94	4	of	of	ADP
cana-3670	94	5	the	the	DET
cana-3670	94	6	modified	modify	VERB
cana-3670	94	7	euler	euler	NOUN
cana-3670	94	8	's	's	PART
cana-3670	94	9	method	method	NOUN
cana-3670	94	10	is	be	AUX
cana-3670	94	11	the	the	DET
cana-3670	94	12	name	name	NOUN
cana-3670	94	13	given	give	VERB
cana-3670	94	14	to	to	ADP
cana-3670	94	15	the	the	DET
cana-3670	94	16	preceding	precede	VERB
cana-3670	94	17	equation	equation	NOUN
cana-3670	94	18	.	.	PUNCT
cana-3670	95	1	it	it	PRON
cana-3670	95	2	is	be	AUX
cana-3670	95	3	possible	possible	ADJ
cana-3670	95	4	to	to	PART
cana-3670	95	5	write	write	VERB
cana-3670	95	6	it	it	PRON
cana-3670	95	7	as	as	ADP
cana-3670	95	8	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	95	9	=	=	PUNCT
cana-3670	95	10	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	95	11	+	+	X
cana-3670	95	12	ℎ𝑓	ℎ𝑓	NOUN
cana-3670	95	13	(	(	PUNCT
cana-3670	95	14	2(𝑡𝑛	2(𝑡𝑛	NUM
cana-3670	95	15	2	2	NUM
cana-3670	95	16	+	+	NOUN
cana-3670	95	17	𝑡𝑛(𝑡𝑛+ℎ)+(𝑡𝑛+ℎ)2	𝑡𝑛(𝑡𝑛+ℎ)+(𝑡𝑛+ℎ)2	NOUN
cana-3670	95	18	)	)	PUNCT
cana-3670	95	19	3(𝑡𝑛+(𝑡𝑛+ℎ	3(𝑡𝑛+(𝑡𝑛+ℎ	NUM
cana-3670	95	20	)	)	PUNCT
cana-3670	95	21	)	)	PUNCT
cana-3670	95	22	,	,	PUNCT
cana-3670	95	23	2(𝑢𝑛	2(𝑢𝑛	NUM
cana-3670	95	24	2	2	NUM
cana-3670	95	25	+	+	NOUN
cana-3670	95	26	𝑢𝑛(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛))+(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑢𝑛(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛))+(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	X
cana-3670	95	27	)	)	PUNCT
cana-3670	95	28	)	)	PUNCT
cana-3670	95	29	2	2	NUM
cana-3670	95	30	)	)	PUNCT
cana-3670	95	31	3(𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	3(𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NUM
cana-3670	95	32	)	)	PUNCT
cana-3670	95	33	)	)	PUNCT
cana-3670	95	34	)	)	PUNCT
cana-3670	95	35	)	)	PUNCT
cana-3670	95	36	.	.	PUNCT
cana-3670	96	1	3.1.5	3.1.5	NUM
cana-3670	96	2	modified	modify	VERB
cana-3670	96	3	euler	euler	PROPN
cana-3670	96	4	’s	’s	PART
cana-3670	96	5	method	method	NOUN
cana-3670	96	6	based	base	VERB
cana-3670	96	7	on	on	ADP
cana-3670	96	8	centroidal	centroidal	ADJ
cana-3670	96	9	mean	mean	NOUN
cana-3670	96	10	assume	assume	VERB
cana-3670	96	11	that	that	SCONJ
cana-3670	96	12	the	the	DET
cana-3670	96	13	essential	essential	ADJ
cana-3670	96	14	components	component	NOUN
cana-3670	96	15	of	of	ADP
cana-3670	96	16	a	a	DET
cana-3670	96	17	centroidal	centroidal	ADJ
cana-3670	96	18	sequence	sequence	NOUN
cana-3670	96	19	are	be	AUX
cana-3670	96	20	a	a	DET
cana-3670	96	21	,	,	PUNCT
cana-3670	96	22	b	b	NOUN
cana-3670	96	23	,	,	PUNCT
cana-3670	96	24	and	and	CCONJ
cana-3670	96	25	c.	c.	NOUN
cana-3670	96	26	the	the	DET
cana-3670	96	27	formula	formula	NOUN
cana-3670	96	28	to	to	PART
cana-3670	96	29	compute	compute	VERB
cana-3670	96	30	the	the	DET
cana-3670	96	31	centroidal	centroidal	NOUN
cana-3670	96	32	mean	mean	NOUN
cana-3670	96	33	is	be	AUX
cana-3670	96	34	𝑐	𝑐	NOUN
cana-3670	96	35	=	=	SYM
cana-3670	96	36	2(𝑎2+𝑎𝑏+𝑏2	2(𝑎2+𝑎𝑏+𝑏2	NUM
cana-3670	96	37	)	)	PUNCT
cana-3670	96	38	3	3	NUM
cana-3670	96	39	(	(	PUNCT
cana-3670	96	40	𝑎+𝑏	𝑎+𝑏	NOUN
cana-3670	96	41	)	)	PUNCT
cana-3670	96	42	.	.	PUNCT
cana-3670	97	1	the	the	DET
cana-3670	97	2	expected	expect	VERB
cana-3670	97	3	centroidal	centroidal	ADJ
cana-3670	97	4	mean	mean	NOUN
cana-3670	97	5	of	of	ADP
cana-3670	97	6	the	the	DET
cana-3670	97	7	two	two	NUM
cana-3670	97	8	midpoints	midpoint	NOUN
cana-3670	97	9	is	be	AUX
cana-3670	97	10	therefore	therefore	ADV
cana-3670	97	11	given	give	VERB
cana-3670	97	12	as	as	ADP
cana-3670	97	13	(	(	PUNCT
cana-3670	97	14	2(𝑡0	2(𝑡0	NUM
cana-3670	97	15	2+𝑡0𝑡1+𝑡1	2+𝑡0𝑡1+𝑡1	NUM
cana-3670	97	16	2	2	X
cana-3670	97	17	)	)	PUNCT
cana-3670	97	18	3(𝑡0+𝑡1	3(𝑡0+𝑡1	NUM
cana-3670	97	19	)	)	PUNCT
cana-3670	97	20	,	,	PUNCT
cana-3670	97	21	2(𝑢0	2(𝑢0	NUM
cana-3670	98	1	2+𝑢0𝑢1+𝑢1	2+𝑢0𝑢1+𝑢1	NUM
cana-3670	98	2	2	2	NUM
cana-3670	98	3	)	)	PUNCT
cana-3670	98	4	3(𝑢0+𝑢1	3(𝑢0+𝑢1	NUM
cana-3670	98	5	)	)	PUNCT
cana-3670	98	6	)	)	PUNCT
cana-3670	98	7	.	.	PUNCT
cana-3670	99	1	communications	communication	NOUN
cana-3670	99	2	on	on	ADP
cana-3670	99	3	applied	apply	VERB
cana-3670	99	4	nonlinear	nonlinear	ADJ
cana-3670	99	5	analysis	analysis	NOUN
cana-3670	99	6	issn	issn	NOUN
cana-3670	99	7	:	:	PUNCT
cana-3670	99	8	1074	1074	NUM
cana-3670	99	9	-	-	PUNCT
cana-3670	99	10	133x	133x	NUM
cana-3670	99	11	vol	vol	NOUN
cana-3670	99	12	32	32	NUM
cana-3670	99	13	no	no	NOUN
cana-3670	99	14	.	.	PUNCT
cana-3670	100	1	8s	8s	PROPN
cana-3670	100	2	(	(	PUNCT
cana-3670	100	3	2025	2025	NUM
cana-3670	100	4	)	)	PUNCT
cana-3670	100	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	100	6	276	276	NUM
cana-3670	100	7	the	the	DET
cana-3670	100	8	mathematical	mathematical	ADJ
cana-3670	100	9	equation	equation	NOUN
cana-3670	100	10	is	be	AUX
cana-3670	100	11	defined	define	VERB
cana-3670	100	12	as	as	ADP
cana-3670	100	13	(	(	PUNCT
cana-3670	100	14	2(𝑡0	2(𝑡0	NUM
cana-3670	100	15	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	NUM
cana-3670	100	16	)	)	PUNCT
cana-3670	100	17	2	2	NUM
cana-3670	100	18	)	)	PUNCT
cana-3670	100	19	3(𝑡0+(𝑡0+ℎ	3(𝑡0+(𝑡0+ℎ	NUM
cana-3670	100	20	)	)	PUNCT
cana-3670	100	21	)	)	PUNCT
cana-3670	100	22	,	,	PUNCT
cana-3670	100	23	2(𝑢0	2(𝑢0	NUM
cana-3670	100	24	2+𝑢0(𝑢0+ℎ𝑓(𝑡0	2+𝑢0(𝑢0+ℎ𝑓(𝑡0	NUM
cana-3670	100	25	+	+	CCONJ
cana-3670	100	26	ℎ	ℎ	PROPN
cana-3670	100	27	2	2	NUM
cana-3670	100	28	,	,	PUNCT
cana-3670	100	29	𝑢0	𝑢0	VERB
cana-3670	100	30	+	+	CCONJ
cana-3670	100	31	ℎ	ℎ	PROPN
cana-3670	100	32	2	2	NUM
cana-3670	100	33	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADV
cana-3670	100	34	)	)	PUNCT
cana-3670	100	35	)	)	PUNCT
cana-3670	100	36	)	)	PUNCT
cana-3670	101	1	+	+	ADJ
cana-3670	101	2	(	(	PUNCT
cana-3670	101	3	𝑢0+ℎ𝑓(𝑡0	𝑢0+ℎ𝑓(𝑡0	ADJ
cana-3670	101	4	+	+	NOUN
cana-3670	101	5	ℎ	ℎ	SYM
cana-3670	101	6	2	2	NUM
cana-3670	101	7	,	,	PUNCT
cana-3670	101	8	𝑢0	𝑢0	VERB
cana-3670	101	9	+	+	CCONJ
cana-3670	101	10	ℎ	ℎ	PROPN
cana-3670	101	11	2	2	NUM
cana-3670	101	12	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	101	13	)	)	PUNCT
cana-3670	101	14	)	)	PUNCT
cana-3670	101	15	)	)	PUNCT
cana-3670	102	1	2	2	X
cana-3670	102	2	)	)	PUNCT
cana-3670	103	1	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0	NUM
cana-3670	103	2	+	+	NUM
cana-3670	103	3	ℎ	ℎ	SYM
cana-3670	103	4	2	2	NUM
cana-3670	103	5	,	,	PUNCT
cana-3670	103	6	𝑢0	𝑢0	VERB
cana-3670	103	7	+	+	CCONJ
cana-3670	103	8	ℎ	ℎ	PROPN
cana-3670	103	9	2	2	NUM
cana-3670	103	10	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	103	11	)	)	PUNCT
cana-3670	103	12	)	)	PUNCT
cana-3670	103	13	)	)	PUNCT
cana-3670	103	14	)	)	PUNCT
cana-3670	103	15	)	)	PUNCT
cana-3670	103	16	.	.	PUNCT
cana-3670	104	1	a	a	DET
cana-3670	104	2	new	new	ADJ
cana-3670	104	3	equation	equation	NOUN
cana-3670	104	4	can	can	AUX
cana-3670	104	5	be	be	AUX
cana-3670	104	6	created	create	VERB
cana-3670	104	7	and	and	CCONJ
cana-3670	104	8	provided	provide	VERB
cana-3670	104	9	as	as	SCONJ
cana-3670	104	10	follows	follow	VERB
cana-3670	104	11	if	if	SCONJ
cana-3670	104	12	an	an	DET
cana-3670	104	13	equation	equation	NOUN
cana-3670	104	14	goes	go	VERB
cana-3670	104	15	through	through	ADP
cana-3670	104	16	a	a	DET
cana-3670	104	17	point	point	NOUN
cana-3670	104	18	,	,	PUNCT
cana-3670	104	19	such	such	ADJ
cana-3670	104	20	as	as	ADP
cana-3670	104	21	𝑃(𝑡0	𝑃(𝑡0	PROPN
cana-3670	104	22	,	,	PUNCT
cana-3670	104	23	𝑢0	𝑢0	PROPN
cana-3670	104	24	)	)	PUNCT
cana-3670	104	25	in	in	ADP
cana-3670	104	26	the	the	DET
cana-3670	104	27	center	center	NOUN
cana-3670	104	28	of	of	ADP
cana-3670	104	29	the	the	DET
cana-3670	104	30	slope	slope	NOUN
cana-3670	104	31	throughout	throughout	ADP
cana-3670	104	32	the	the	DET
cana-3670	104	33	equation	equation	NOUN
cana-3670	104	34	.	.	PUNCT
cana-3670	105	1	𝑢1	𝑢1	PROPN
cana-3670	105	2	=	=	PROPN
cana-3670	105	3	𝑢0	𝑢0	PROPN
cana-3670	105	4	+	+	NUM
cana-3670	105	5	ℎ𝑓	ℎ𝑓	NOUN
cana-3670	105	6	(	(	PUNCT
cana-3670	105	7	𝑡0	𝑡0	NOUN
cana-3670	105	8	+	+	CCONJ
cana-3670	105	9	ℎ	ℎ	PROPN
cana-3670	105	10	2	2	NUM
cana-3670	105	11	,	,	PUNCT
cana-3670	105	12	𝑢0	𝑢0	VERB
cana-3670	105	13	+	+	CCONJ
cana-3670	105	14	ℎ	ℎ	PROPN
cana-3670	105	15	2	2	NUM
cana-3670	105	16	𝑓	𝑓	PROPN
cana-3670	105	17	(	(	PUNCT
cana-3670	105	18	2(𝑡0	2(𝑡0	NUM
cana-3670	105	19	2+𝑡0	2+𝑡0	NUM
cana-3670	105	20	(	(	PUNCT
cana-3670	105	21	𝑡0+ℎ)+(𝑡0+ℎ	𝑡0+ℎ)+(𝑡0+ℎ	NOUN
cana-3670	105	22	)	)	PUNCT
cana-3670	105	23	2	2	NUM
cana-3670	105	24	)	)	PUNCT
cana-3670	105	25	3(𝑡0+(𝑡0+ℎ	3(𝑡0+(𝑡0+ℎ	NUM
cana-3670	105	26	)	)	PUNCT
cana-3670	105	27	)	)	PUNCT
cana-3670	105	28	,	,	PUNCT
cana-3670	105	29	2(𝑢0	2(𝑢0	NUM
cana-3670	105	30	2+𝑢0(𝑢0+ℎ𝑓(𝑡0	2+𝑢0(𝑢0+ℎ𝑓(𝑡0	NUM
cana-3670	105	31	+	+	CCONJ
cana-3670	105	32	ℎ	ℎ	PROPN
cana-3670	105	33	2	2	NUM
cana-3670	105	34	,	,	PUNCT
cana-3670	105	35	𝑢0	𝑢0	VERB
cana-3670	105	36	+	+	CCONJ
cana-3670	105	37	ℎ	ℎ	PROPN
cana-3670	105	38	2	2	NUM
cana-3670	105	39	𝑓(𝑡0,𝑢0)))+(𝑢0+ℎ𝑓(𝑡0	𝑓(𝑡0,𝑢0)))+(𝑢0+ℎ𝑓(𝑡0	NOUN
cana-3670	105	40	+	+	NUM
cana-3670	105	41	ℎ	ℎ	SYM
cana-3670	105	42	2	2	NUM
cana-3670	105	43	,	,	PUNCT
cana-3670	105	44	𝑢0	𝑢0	VERB
cana-3670	105	45	+	+	CCONJ
cana-3670	105	46	ℎ	ℎ	PROPN
cana-3670	105	47	2	2	NUM
cana-3670	105	48	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	105	49	)	)	PUNCT
cana-3670	105	50	)	)	PUNCT
cana-3670	105	51	)	)	PUNCT
cana-3670	106	1	2	2	X
cana-3670	106	2	)	)	PUNCT
cana-3670	107	1	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0	3(𝑢0+(𝑢0+ℎ𝑓(𝑡0	NUM
cana-3670	107	2	+	+	NUM
cana-3670	107	3	ℎ	ℎ	SYM
cana-3670	107	4	2	2	NUM
cana-3670	107	5	,	,	PUNCT
cana-3670	107	6	𝑢0	𝑢0	VERB
cana-3670	107	7	+	+	CCONJ
cana-3670	107	8	ℎ	ℎ	PROPN
cana-3670	107	9	2	2	NUM
cana-3670	107	10	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADV
cana-3670	107	11	)	)	PUNCT
cana-3670	107	12	)	)	PUNCT
cana-3670	107	13	)	)	PUNCT
cana-3670	107	14	)	)	PUNCT
cana-3670	107	15	)	)	PUNCT
cana-3670	107	16	)	)	PUNCT
cana-3670	107	17	.	.	PUNCT
cana-3670	108	1	the	the	DET
cana-3670	108	2	results	result	NOUN
cana-3670	108	3	of	of	ADP
cana-3670	108	4	the	the	DET
cana-3670	108	5	modified	modify	VERB
cana-3670	108	6	euler	euler	NOUN
cana-3670	108	7	's	's	PART
cana-3670	108	8	approach	approach	NOUN
cana-3670	108	9	are	be	AUX
cana-3670	108	10	now	now	ADV
cana-3670	108	11	certain	certain	ADJ
cana-3670	108	12	and	and	CCONJ
cana-3670	108	13	accurate	accurate	ADJ
cana-3670	108	14	.	.	PUNCT
cana-3670	109	1	the	the	DET
cana-3670	109	2	centroidal	centroidal	ADJ
cana-3670	109	3	mean	mean	NOUN
cana-3670	109	4	of	of	ADP
cana-3670	109	5	the	the	DET
cana-3670	109	6	modified	modify	VERB
cana-3670	109	7	euler	euler	NOUN
cana-3670	109	8	's	's	PART
cana-3670	109	9	method	method	NOUN
cana-3670	109	10	is	be	AUX
cana-3670	109	11	the	the	DET
cana-3670	109	12	name	name	NOUN
cana-3670	109	13	given	give	VERB
cana-3670	109	14	to	to	ADP
cana-3670	109	15	the	the	DET
cana-3670	109	16	preceding	precede	VERB
cana-3670	109	17	equation	equation	NOUN
cana-3670	109	18	.	.	PUNCT
cana-3670	110	1	it	it	PRON
cana-3670	110	2	is	be	AUX
cana-3670	110	3	possible	possible	ADJ
cana-3670	110	4	to	to	PART
cana-3670	110	5	write	write	VERB
cana-3670	110	6	it	it	PRON
cana-3670	110	7	as	as	ADP
cana-3670	110	8	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	110	9	=	=	PUNCT
cana-3670	110	10	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	110	11	+	+	X
cana-3670	110	12	ℎ𝑓	ℎ𝑓	X
cana-3670	110	13	(	(	PUNCT
cana-3670	110	14	𝑡𝑛	𝑡𝑛	X
cana-3670	110	15	+	+	CCONJ
cana-3670	110	16	ℎ	ℎ	X
cana-3670	110	17	2	2	NUM
cana-3670	110	18	,	,	PUNCT
cana-3670	110	19	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	110	20	+	+	CCONJ
cana-3670	110	21	ℎ	ℎ	PROPN
cana-3670	110	22	2	2	NUM
cana-3670	110	23	𝑓	𝑓	PROPN
cana-3670	110	24	(	(	PUNCT
cana-3670	110	25	2(𝑡𝑛	2(𝑡𝑛	NUM
cana-3670	110	26	2	2	NUM
cana-3670	110	27	+	+	NUM
cana-3670	110	28	𝑡𝑛	𝑡𝑛	NOUN
cana-3670	110	29	(	(	PUNCT
cana-3670	110	30	𝑡𝑛+ℎ)+(𝑡𝑛+ℎ	𝑡𝑛+ℎ)+(𝑡𝑛+ℎ	NOUN
cana-3670	110	31	)	)	PUNCT
cana-3670	110	32	2	2	NUM
cana-3670	110	33	)	)	PUNCT
cana-3670	110	34	3(𝑡𝑛+(𝑡𝑛+ℎ	3(𝑡𝑛+(𝑡𝑛+ℎ	NUM
cana-3670	110	35	)	)	PUNCT
cana-3670	110	36	)	)	PUNCT
cana-3670	110	37	,	,	PUNCT
cana-3670	110	38	2(𝑢𝑛	2(𝑢𝑛	NUM
cana-3670	110	39	2	2	NUM
cana-3670	110	40	+	+	NOUN
cana-3670	110	41	𝑢𝑛(𝑢𝑛+ℎ𝑓(𝑡𝑛+	𝑢𝑛(𝑢𝑛+ℎ𝑓(𝑡𝑛+	NUM
cana-3670	110	42	ℎ	ℎ	ADP
cana-3670	110	43	2	2	NUM
cana-3670	110	44	,	,	PUNCT
cana-3670	110	45	𝑢𝑛+	𝑢𝑛+	ADJ
cana-3670	110	46	ℎ	ℎ	NOUN
cana-3670	110	47	2	2	NUM
cana-3670	110	48	𝑓(𝑡𝑛,𝑢𝑛)))+(𝑢𝑛+ℎ𝑓(𝑡𝑛+	𝑓(𝑡𝑛,𝑢𝑛)))+(𝑢𝑛+ℎ𝑓(𝑡𝑛+	NOUN
cana-3670	110	49	ℎ	ℎ	SYM
cana-3670	110	50	2	2	NUM
cana-3670	110	51	,	,	PUNCT
cana-3670	110	52	𝑢𝑛+	𝑢𝑛+	ADJ
cana-3670	110	53	ℎ	ℎ	NOUN
cana-3670	110	54	2	2	NUM
cana-3670	110	55	𝑓(𝑡𝑛	𝑓(𝑡𝑛	NOUN
cana-3670	110	56	,	,	PUNCT
cana-3670	110	57	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	110	58	)	)	PUNCT
cana-3670	110	59	)	)	PUNCT
cana-3670	110	60	)	)	PUNCT
cana-3670	110	61	2	2	X
cana-3670	110	62	)	)	PUNCT
cana-3670	110	63	3(𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛+	3(𝑢𝑛+(𝑢𝑛+ℎ𝑓(𝑡𝑛+	NOUN
cana-3670	110	64	ℎ	ℎ	ADP
cana-3670	110	65	2	2	NUM
cana-3670	110	66	,	,	PUNCT
cana-3670	110	67	𝑢𝑛+	𝑢𝑛+	ADJ
cana-3670	110	68	ℎ	ℎ	PROPN
cana-3670	110	69	2	2	NUM
cana-3670	110	70	𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	110	71	)	)	PUNCT
cana-3670	110	72	)	)	PUNCT
cana-3670	110	73	)	)	PUNCT
cana-3670	110	74	)	)	PUNCT
cana-3670	110	75	)	)	PUNCT
cana-3670	110	76	)	)	PUNCT
cana-3670	110	77	.	.	PUNCT
cana-3670	111	1	3.1.6	3.1.6	NUM
cana-3670	111	2	improved	improve	VERB
cana-3670	111	3	euler	euler	NOUN
cana-3670	111	4	’s	’s	PART
cana-3670	111	5	method	method	NOUN
cana-3670	111	6	based	base	VERB
cana-3670	111	7	on	on	ADP
cana-3670	111	8	centroidal	centroidal	ADJ
cana-3670	111	9	mean	mean	NOUN
cana-3670	111	10	assume	assume	VERB
cana-3670	111	11	that	that	SCONJ
cana-3670	111	12	the	the	DET
cana-3670	111	13	essential	essential	ADJ
cana-3670	111	14	components	component	NOUN
cana-3670	111	15	of	of	ADP
cana-3670	111	16	a	a	DET
cana-3670	111	17	centroidal	centroidal	ADJ
cana-3670	111	18	sequence	sequence	NOUN
cana-3670	111	19	are	be	AUX
cana-3670	111	20	a	a	DET
cana-3670	111	21	,	,	PUNCT
cana-3670	111	22	b	b	NOUN
cana-3670	111	23	,	,	PUNCT
cana-3670	111	24	and	and	CCONJ
cana-3670	111	25	c.	c.	NOUN
cana-3670	111	26	the	the	DET
cana-3670	111	27	formula	formula	NOUN
cana-3670	111	28	to	to	PART
cana-3670	111	29	compute	compute	VERB
cana-3670	111	30	the	the	DET
cana-3670	111	31	centroidal	centroidal	NOUN
cana-3670	111	32	mean	mean	NOUN
cana-3670	111	33	is	be	AUX
cana-3670	111	34	𝑐	𝑐	NOUN
cana-3670	111	35	=	=	SYM
cana-3670	111	36	2(𝑎2+𝑎𝑏+𝑏2	2(𝑎2+𝑎𝑏+𝑏2	NUM
cana-3670	111	37	)	)	PUNCT
cana-3670	111	38	3	3	NUM
cana-3670	111	39	(	(	PUNCT
cana-3670	111	40	𝑎+𝑏	𝑎+𝑏	NOUN
cana-3670	111	41	)	)	PUNCT
cana-3670	111	42	.	.	PUNCT
cana-3670	112	1	the	the	DET
cana-3670	112	2	expected	expect	VERB
cana-3670	112	3	centroidal	centroidal	ADJ
cana-3670	112	4	mean	mean	NOUN
cana-3670	112	5	of	of	ADP
cana-3670	112	6	the	the	DET
cana-3670	112	7	two	two	NUM
cana-3670	112	8	midpoints	midpoint	NOUN
cana-3670	112	9	is	be	AUX
cana-3670	112	10	therefore	therefore	ADV
cana-3670	112	11	given	give	VERB
cana-3670	112	12	as	as	ADP
cana-3670	112	13	(	(	PUNCT
cana-3670	112	14	2(𝑡0	2(𝑡0	NUM
cana-3670	112	15	2+𝑡0𝑡1+𝑡1	2+𝑡0𝑡1+𝑡1	NUM
cana-3670	112	16	2	2	X
cana-3670	112	17	)	)	PUNCT
cana-3670	112	18	3(𝑡0+𝑡1	3(𝑡0+𝑡1	NUM
cana-3670	112	19	)	)	PUNCT
cana-3670	112	20	,	,	PUNCT
cana-3670	112	21	2(𝑢0	2(𝑢0	NUM
cana-3670	113	1	2+𝑢0𝑢1+𝑢1	2+𝑢0𝑢1+𝑢1	NUM
cana-3670	113	2	2	2	NUM
cana-3670	113	3	)	)	PUNCT
cana-3670	113	4	3(𝑢0+𝑢1	3(𝑢0+𝑢1	NUM
cana-3670	113	5	)	)	PUNCT
cana-3670	113	6	)	)	PUNCT
cana-3670	113	7	.	.	PUNCT
cana-3670	114	1	the	the	DET
cana-3670	114	2	mathematical	mathematical	ADJ
cana-3670	114	3	equation	equation	NOUN
cana-3670	114	4	is	be	AUX
cana-3670	114	5	defined	define	VERB
cana-3670	114	6	as	as	ADP
cana-3670	114	7	(	(	PUNCT
cana-3670	114	8	2(𝑡0	2(𝑡0	NUM
cana-3670	114	9	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	NUM
cana-3670	114	10	)	)	PUNCT
cana-3670	114	11	2	2	NUM
cana-3670	114	12	)	)	PUNCT
cana-3670	114	13	3	3	NUM
cana-3670	114	14	(	(	PUNCT
cana-3670	114	15	𝑡0+(𝑡0+ℎ	𝑡0+(𝑡0+ℎ	PROPN
cana-3670	114	16	)	)	PUNCT
cana-3670	114	17	)	)	PUNCT
cana-3670	114	18	,	,	PUNCT
cana-3670	114	19	2(𝑢0	2(𝑢0	NUM
cana-3670	114	20	2+𝑢0(𝑢0	2+𝑢0(𝑢0	NUM
cana-3670	114	21	+	+	CCONJ
cana-3670	114	22	ℎ	ℎ	SYM
cana-3670	114	23	2	2	NUM
cana-3670	114	24	(	(	PUNCT
cana-3670	114	25	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	PROPN
cana-3670	114	26	)	)	PUNCT
cana-3670	114	27	)	)	PUNCT
cana-3670	114	28	)	)	PUNCT
cana-3670	114	29	)	)	PUNCT
cana-3670	115	1	+	+	PROPN
cana-3670	115	2	(	(	PUNCT
cana-3670	115	3	𝑢0	𝑢0	PROPN
cana-3670	115	4	+	+	CCONJ
cana-3670	115	5	ℎ	ℎ	PROPN
cana-3670	115	6	2	2	NUM
cana-3670	115	7	(	(	PUNCT
cana-3670	115	8	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	PROPN
cana-3670	115	9	)	)	PUNCT
cana-3670	115	10	)	)	PUNCT
cana-3670	115	11	)	)	PUNCT
cana-3670	115	12	)	)	PUNCT
cana-3670	115	13	2	2	X
cana-3670	115	14	)	)	PUNCT
cana-3670	115	15	3(𝑢0+(𝑢0	3(𝑢0+(𝑢0	NOUN
cana-3670	115	16	+	+	SYM
cana-3670	115	17	ℎ	ℎ	SYM
cana-3670	115	18	2	2	NUM
cana-3670	115	19	(	(	PUNCT
cana-3670	115	20	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	115	21	)	)	PUNCT
cana-3670	115	22	+	+	ADJ
cana-3670	115	23	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	115	24	)	)	PUNCT
cana-3670	115	25	)	)	PUNCT
cana-3670	115	26	)	)	PUNCT
cana-3670	115	27	)	)	PUNCT
cana-3670	115	28	)	)	PUNCT
cana-3670	115	29	)	)	PUNCT
cana-3670	115	30	.	.	PUNCT
cana-3670	116	1	a	a	DET
cana-3670	116	2	new	new	ADJ
cana-3670	116	3	equation	equation	NOUN
cana-3670	116	4	can	can	AUX
cana-3670	116	5	be	be	AUX
cana-3670	116	6	created	create	VERB
cana-3670	116	7	and	and	CCONJ
cana-3670	116	8	provided	provide	VERB
cana-3670	116	9	as	as	SCONJ
cana-3670	116	10	follows	follow	VERB
cana-3670	116	11	if	if	SCONJ
cana-3670	116	12	an	an	DET
cana-3670	116	13	equation	equation	NOUN
cana-3670	116	14	goes	go	VERB
cana-3670	116	15	through	through	ADP
cana-3670	116	16	a	a	DET
cana-3670	116	17	point	point	NOUN
cana-3670	116	18	,	,	PUNCT
cana-3670	116	19	such	such	ADJ
cana-3670	116	20	as	as	ADP
cana-3670	116	21	𝑃(𝑡0	𝑃(𝑡0	PROPN
cana-3670	116	22	,	,	PUNCT
cana-3670	116	23	𝑢0	𝑢0	PROPN
cana-3670	116	24	)	)	PUNCT
cana-3670	116	25	in	in	ADP
cana-3670	116	26	the	the	DET
cana-3670	116	27	center	center	NOUN
cana-3670	116	28	of	of	ADP
cana-3670	116	29	the	the	DET
cana-3670	116	30	slope	slope	NOUN
cana-3670	116	31	throughout	throughout	ADP
cana-3670	116	32	the	the	DET
cana-3670	116	33	equation	equation	NOUN
cana-3670	116	34	.	.	PUNCT
cana-3670	117	1	communications	communication	NOUN
cana-3670	117	2	on	on	ADP
cana-3670	117	3	applied	apply	VERB
cana-3670	117	4	nonlinear	nonlinear	ADJ
cana-3670	117	5	analysis	analysis	NOUN
cana-3670	117	6	issn	issn	NOUN
cana-3670	117	7	:	:	PUNCT
cana-3670	117	8	1074	1074	NUM
cana-3670	117	9	-	-	PUNCT
cana-3670	117	10	133x	133x	NUM
cana-3670	117	11	vol	vol	NOUN
cana-3670	117	12	32	32	NUM
cana-3670	117	13	no	no	NOUN
cana-3670	117	14	.	.	PUNCT
cana-3670	118	1	8s	8s	PROPN
cana-3670	118	2	(	(	PUNCT
cana-3670	118	3	2025	2025	NUM
cana-3670	118	4	)	)	PUNCT
cana-3670	118	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	118	6	277	277	NUM
cana-3670	118	7	𝑢1	𝑢1	NOUN
cana-3670	118	8	=	=	PROPN
cana-3670	118	9	𝑢0	𝑢0	PROPN
cana-3670	118	10	+	+	NUM
cana-3670	118	11	ℎ	ℎ	X
cana-3670	118	12	2	2	NUM
cana-3670	118	13	[	[	PUNCT
cana-3670	118	14	𝑓	𝑓	X
cana-3670	118	15	(	(	PUNCT
cana-3670	118	16	2(𝑡0	2(𝑡0	NUM
cana-3670	118	17	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	NUM
cana-3670	118	18	)	)	PUNCT
cana-3670	118	19	2	2	NUM
cana-3670	118	20	)	)	PUNCT
cana-3670	118	21	3(𝑡0+(𝑡0+ℎ	3(𝑡0+(𝑡0+ℎ	NUM
cana-3670	118	22	)	)	PUNCT
cana-3670	118	23	)	)	PUNCT
cana-3670	118	24	,	,	PUNCT
cana-3670	118	25	2(𝑢0	2(𝑢0	NUM
cana-3670	118	26	2+𝑢0(𝑢0	2+𝑢0(𝑢0	NUM
cana-3670	118	27	+	+	CCONJ
cana-3670	118	28	ℎ	ℎ	SYM
cana-3670	118	29	2	2	NUM
cana-3670	118	30	(	(	PUNCT
cana-3670	118	31	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADV
cana-3670	118	32	)	)	PUNCT
cana-3670	119	1	+	+	NOUN
cana-3670	119	2	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0))))+(𝑢0	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0))))+(𝑢0	PROPN
cana-3670	119	3	+	+	SYM
cana-3670	119	4	ℎ	ℎ	X
cana-3670	119	5	2	2	NUM
cana-3670	119	6	(	(	PUNCT
cana-3670	119	7	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	119	8	)	)	PUNCT
cana-3670	119	9	+	+	ADJ
cana-3670	119	10	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	119	11	)	)	PUNCT
cana-3670	119	12	)	)	PUNCT
cana-3670	119	13	)	)	PUNCT
cana-3670	119	14	)	)	PUNCT
cana-3670	120	1	2	2	X
cana-3670	120	2	)	)	PUNCT
cana-3670	120	3	3(𝑢0+(𝑢0	3(𝑢0+(𝑢0	NOUN
cana-3670	120	4	+	+	SYM
cana-3670	120	5	ℎ	ℎ	SYM
cana-3670	120	6	2	2	NUM
cana-3670	120	7	(	(	PUNCT
cana-3670	120	8	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	PROPN
cana-3670	120	9	)	)	PUNCT
cana-3670	120	10	)	)	PUNCT
cana-3670	120	11	)	)	PUNCT
cana-3670	120	12	)	)	PUNCT
cana-3670	120	13	)	)	PUNCT
cana-3670	120	14	)	)	PUNCT
cana-3670	121	1	+	+	CCONJ
cana-3670	121	2	𝑓	𝑓	PRON
cana-3670	121	3	(	(	PUNCT
cana-3670	121	4	𝑡0	𝑡0	PROPN
cana-3670	121	5	+	+	CCONJ
cana-3670	121	6	ℎ	ℎ	PROPN
cana-3670	121	7	,	,	PUNCT
cana-3670	121	8	𝑢0	𝑢0	VERB
cana-3670	121	9	+	+	ADJ
cana-3670	121	10	ℎ	ℎ	NOUN
cana-3670	121	11	(	(	PUNCT
cana-3670	121	12	2(𝑡0	2(𝑡0	NUM
cana-3670	121	13	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	2+𝑡0(𝑡0+ℎ)+(𝑡0+ℎ	NUM
cana-3670	121	14	)	)	PUNCT
cana-3670	121	15	2	2	NUM
cana-3670	121	16	)	)	PUNCT
cana-3670	121	17	3(𝑡0+(𝑡0+ℎ	3(𝑡0+(𝑡0+ℎ	NUM
cana-3670	121	18	)	)	PUNCT
cana-3670	121	19	)	)	PUNCT
cana-3670	121	20	,	,	PUNCT
cana-3670	121	21	2(𝑢0	2(𝑢0	NUM
cana-3670	121	22	2+𝑢0(𝑢0	2+𝑢0(𝑢0	NUM
cana-3670	121	23	+	+	CCONJ
cana-3670	121	24	ℎ	ℎ	SYM
cana-3670	121	25	2	2	NUM
cana-3670	121	26	(	(	PUNCT
cana-3670	121	27	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0))))+(𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0))))+(𝑢0	X
cana-3670	121	28	+	+	NOUN
cana-3670	121	29	ℎ	ℎ	SYM
cana-3670	121	30	2	2	NUM
cana-3670	121	31	(	(	PUNCT
cana-3670	121	32	𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0	ADJ
cana-3670	121	33	)	)	PUNCT
cana-3670	121	34	+	+	ADJ
cana-3670	121	35	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	NOUN
cana-3670	121	36	)	)	PUNCT
cana-3670	121	37	)	)	PUNCT
cana-3670	121	38	)	)	PUNCT
cana-3670	121	39	)	)	PUNCT
cana-3670	121	40	2	2	X
cana-3670	121	41	)	)	PUNCT
cana-3670	121	42	3(𝑢0+(𝑢0	3(𝑢0+(𝑢0	NOUN
cana-3670	122	1	+	+	SYM
cana-3670	122	2	ℎ	ℎ	SYM
cana-3670	122	3	2	2	NUM
cana-3670	122	4	(	(	PUNCT
cana-3670	122	5	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	𝑓(𝑡0,𝑢0)+𝑓(𝑡0+ℎ,𝑢0+ℎ𝑓(𝑡0,𝑢0	PROPN
cana-3670	122	6	)	)	PUNCT
cana-3670	122	7	)	)	PUNCT
cana-3670	122	8	)	)	PUNCT
cana-3670	122	9	)	)	PUNCT
cana-3670	122	10	)	)	PUNCT
cana-3670	122	11	)	)	PUNCT
cana-3670	122	12	)	)	PUNCT
cana-3670	122	13	]	]	PUNCT
cana-3670	122	14	.	.	PUNCT
cana-3670	123	1	the	the	DET
cana-3670	123	2	results	result	NOUN
cana-3670	123	3	of	of	ADP
cana-3670	123	4	the	the	DET
cana-3670	123	5	modified	modify	VERB
cana-3670	123	6	euler	euler	NOUN
cana-3670	123	7	's	's	PART
cana-3670	123	8	approach	approach	NOUN
cana-3670	123	9	are	be	AUX
cana-3670	123	10	now	now	ADV
cana-3670	123	11	certain	certain	ADJ
cana-3670	123	12	and	and	CCONJ
cana-3670	123	13	accurate	accurate	ADJ
cana-3670	123	14	.	.	PUNCT
cana-3670	124	1	the	the	DET
cana-3670	124	2	centroidal	centroidal	ADJ
cana-3670	124	3	mean	mean	NOUN
cana-3670	124	4	of	of	ADP
cana-3670	124	5	the	the	DET
cana-3670	124	6	modified	modify	VERB
cana-3670	124	7	euler	euler	NOUN
cana-3670	124	8	's	's	PART
cana-3670	124	9	method	method	NOUN
cana-3670	124	10	is	be	AUX
cana-3670	124	11	the	the	DET
cana-3670	124	12	name	name	NOUN
cana-3670	124	13	given	give	VERB
cana-3670	124	14	to	to	ADP
cana-3670	124	15	the	the	DET
cana-3670	124	16	preceding	precede	VERB
cana-3670	124	17	equation	equation	NOUN
cana-3670	124	18	.	.	PUNCT
cana-3670	125	1	it	it	PRON
cana-3670	125	2	is	be	AUX
cana-3670	125	3	possible	possible	ADJ
cana-3670	125	4	to	to	PART
cana-3670	125	5	write	write	VERB
cana-3670	125	6	it	it	PRON
cana-3670	125	7	as	as	ADP
cana-3670	125	8	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	125	9	=	=	PUNCT
cana-3670	125	10	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	125	11	+	+	CCONJ
cana-3670	125	12	ℎ	ℎ	PROPN
cana-3670	125	13	2	2	NUM
cana-3670	125	14	[	[	PUNCT
cana-3670	125	15	𝑓	𝑓	PROPN
cana-3670	125	16	(	(	PUNCT
cana-3670	125	17	2(𝑡𝑛	2(𝑡𝑛	NUM
cana-3670	125	18	2+𝑡𝑛(𝑡𝑛+ℎ)+(𝑡𝑛+ℎ	2+𝑡𝑛(𝑡𝑛+ℎ)+(𝑡𝑛+ℎ	NUM
cana-3670	125	19	)	)	PUNCT
cana-3670	125	20	2	2	NUM
cana-3670	125	21	)	)	PUNCT
cana-3670	125	22	3(𝑡𝑛+(𝑡𝑛+ℎ	3(𝑡𝑛+(𝑡𝑛+ℎ	NUM
cana-3670	125	23	)	)	PUNCT
cana-3670	125	24	)	)	PUNCT
cana-3670	125	25	,	,	PUNCT
cana-3670	125	26	2(𝑢𝑛	2(𝑢𝑛	NUM
cana-3670	125	27	2	2	NUM
cana-3670	126	1	+	+	NOUN
cana-3670	126	2	𝑢𝑛(𝑢𝑛+	𝑢𝑛(𝑢𝑛+	PROPN
cana-3670	126	3	ℎ	ℎ	ADP
cana-3670	126	4	2	2	NUM
cana-3670	126	5	(	(	PUNCT
cana-3670	126	6	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛))))+(𝑢𝑛+	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛))))+(𝑢𝑛+	PROPN
cana-3670	126	7	ℎ	ℎ	NOUN
cana-3670	126	8	2	2	NUM
cana-3670	126	9	(	(	PUNCT
cana-3670	126	10	𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	126	11	)	)	PUNCT
cana-3670	126	12	+	+	NOUN
cana-3670	126	13	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	126	14	)	)	PUNCT
cana-3670	126	15	)	)	PUNCT
cana-3670	126	16	)	)	PUNCT
cana-3670	126	17	)	)	PUNCT
cana-3670	127	1	2	2	X
cana-3670	127	2	)	)	PUNCT
cana-3670	127	3	3(𝑢𝑛+(𝑢𝑛+	3(𝑢𝑛+(𝑢𝑛+	NUM
cana-3670	127	4	ℎ	ℎ	NOUN
cana-3670	127	5	2	2	NUM
cana-3670	127	6	(	(	PUNCT
cana-3670	127	7	𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	127	8	)	)	PUNCT
cana-3670	127	9	+	+	NOUN
cana-3670	127	10	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	127	11	)	)	PUNCT
cana-3670	127	12	)	)	PUNCT
cana-3670	127	13	)	)	PUNCT
cana-3670	127	14	)	)	PUNCT
cana-3670	127	15	)	)	PUNCT
cana-3670	127	16	)	)	PUNCT
cana-3670	128	1	+	+	CCONJ
cana-3670	128	2	𝑓	𝑓	X
cana-3670	128	3	(	(	PUNCT
cana-3670	128	4	𝑡𝑛	𝑡𝑛	X
cana-3670	128	5	+	+	NUM
cana-3670	128	6	ℎ	ℎ	PROPN
cana-3670	128	7	,	,	PUNCT
cana-3670	128	8	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	128	9	+	+	X
cana-3670	128	10	ℎ	ℎ	PROPN
cana-3670	128	11	(	(	PUNCT
cana-3670	128	12	2(𝑡𝑛	2(𝑡𝑛	NUM
cana-3670	128	13	2	2	NUM
cana-3670	128	14	+	+	NOUN
cana-3670	128	15	𝑡𝑛(𝑡𝑛+ℎ)+(𝑡𝑛+ℎ	𝑡𝑛(𝑡𝑛+ℎ)+(𝑡𝑛+ℎ	NOUN
cana-3670	128	16	)	)	PUNCT
cana-3670	128	17	2	2	NUM
cana-3670	128	18	)	)	PUNCT
cana-3670	128	19	3(𝑡𝑛+(𝑡𝑛+ℎ	3(𝑡𝑛+(𝑡𝑛+ℎ	NUM
cana-3670	128	20	)	)	PUNCT
cana-3670	128	21	)	)	PUNCT
cana-3670	128	22	,	,	PUNCT
cana-3670	128	23	2(𝑢𝑛	2(𝑢𝑛	NUM
cana-3670	128	24	2	2	NUM
cana-3670	128	25	+	+	NOUN
cana-3670	128	26	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	128	27	(	(	PUNCT
cana-3670	128	28	𝑢𝑛+	𝑢𝑛+	NOUN
cana-3670	128	29	ℎ	ℎ	X
cana-3670	128	30	2	2	NUM
cana-3670	128	31	(	(	PUNCT
cana-3670	128	32	𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	128	33	)	)	PUNCT
cana-3670	129	1	+	+	NOUN
cana-3670	129	2	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛))))+(𝑢𝑛+	𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛))))+(𝑢𝑛+	NOUN
cana-3670	129	3	ℎ	ℎ	X
cana-3670	129	4	2	2	NUM
cana-3670	129	5	(	(	PUNCT
cana-3670	129	6	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑓(𝑡𝑛,𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	NOUN
cana-3670	129	7	)	)	PUNCT
cana-3670	129	8	)	)	PUNCT
cana-3670	129	9	)	)	PUNCT
cana-3670	129	10	)	)	PUNCT
cana-3670	129	11	2	2	X
cana-3670	129	12	)	)	PUNCT
cana-3670	129	13	3(𝑢𝑛+(𝑢𝑛+	3(𝑢𝑛+(𝑢𝑛+	NUM
cana-3670	129	14	ℎ	ℎ	NOUN
cana-3670	129	15	2	2	NUM
cana-3670	129	16	(	(	PUNCT
cana-3670	129	17	𝑓(𝑡𝑛	𝑓(𝑡𝑛	ADJ
cana-3670	129	18	,	,	PUNCT
cana-3670	129	19	𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	𝑢𝑛)+𝑓(𝑡𝑛+ℎ,𝑢𝑛+ℎ𝑓(𝑡𝑛,𝑢𝑛	PROPN
cana-3670	129	20	)	)	PUNCT
cana-3670	129	21	)	)	PUNCT
cana-3670	129	22	)	)	PUNCT
cana-3670	129	23	)	)	PUNCT
cana-3670	129	24	)	)	PUNCT
cana-3670	129	25	)	)	PUNCT
cana-3670	129	26	)	)	PUNCT
cana-3670	129	27	]	]	PUNCT
cana-3670	129	28	.	.	PUNCT
cana-3670	130	1	3.2	3.2	NUM
cana-3670	130	2	.	.	PUNCT
cana-3670	130	3	fuzzy	fuzzy	ADJ
cana-3670	130	4	initial	initial	ADJ
cana-3670	130	5	value	value	NOUN
cana-3670	130	6	problem	problem	NOUN
cana-3670	130	7	3.2.1	3.2.1	NUM
cana-3670	130	8	.	.	PUNCT
cana-3670	130	9	technique	technique	NOUN
cana-3670	130	10	for	for	ADP
cana-3670	130	11	precisely	precisely	ADV
cana-3670	130	12	solving	solve	VERB
cana-3670	130	13	fivp	fivp	NOUN
cana-3670	130	14	look	look	VERB
cana-3670	130	15	over	over	ADP
cana-3670	130	16	the	the	DET
cana-3670	130	17	fivp	fivp	NOUN
cana-3670	130	18	𝑢′(𝑡	𝑢′(𝑡	PROPN
cana-3670	130	19	)	)	PUNCT
cana-3670	130	20	=	=	PRON
cana-3670	130	21	{	{	PUNCT
cana-3670	130	22	𝑓(𝑡	𝑓(𝑡	PROPN
cana-3670	130	23	,	,	PUNCT
cana-3670	130	24	𝑢	𝑢	NOUN
cana-3670	130	25	)	)	PUNCT
cana-3670	130	26	𝑢(𝑡0	𝑢(𝑡0	PROPN
cana-3670	130	27	)	)	PUNCT
cana-3670	131	1	=	=	NOUN
cana-3670	131	2	(	(	PUNCT
cana-3670	131	3	𝑢0,𝑢0	𝑢0,𝑢0	PROPN
cana-3670	131	4	𝑚	𝑚	PROPN
cana-3670	131	5	,	,	PUNCT
cana-3670	131	6	𝑢0	𝑢0	PROPN
cana-3670	131	7	)	)	PUNCT
cana-3670	131	8	in	in	ADP
cana-3670	131	9	this	this	DET
cana-3670	131	10	case	case	NOUN
cana-3670	131	11	,	,	PUNCT
cana-3670	131	12	a	a	DET
cana-3670	131	13	fuzzy	fuzzy	ADJ
cana-3670	131	14	initial	initial	ADJ
cana-3670	131	15	condition	condition	NOUN
cana-3670	131	16	is	be	AUX
cana-3670	131	17	expressed	express	VERB
cana-3670	131	18	in	in	ADP
cana-3670	131	19	terms	term	NOUN
cana-3670	131	20	of	of	ADP
cana-3670	131	21	fuzzy	fuzzy	ADJ
cana-3670	131	22	triangles	triangle	NOUN
cana-3670	131	23	.	.	PUNCT
cana-3670	132	1	the	the	DET
cana-3670	132	2	fivp	fivp	NOUN
cana-3670	132	3	’s	’s	PART
cana-3670	132	4	solution	solution	NOUN
cana-3670	132	5	is	be	AUX
cana-3670	132	6	found	find	VERB
cana-3670	132	7	by	by	ADP
cana-3670	132	8	using	use	VERB
cana-3670	132	9	fuzzy	fuzzy	ADJ
cana-3670	132	10	initial	initial	ADJ
cana-3670	132	11	conditions	condition	NOUN
cana-3670	132	12	.	.	PUNCT
cana-3670	133	1	3.2.2	3.2.2	NUM
cana-3670	133	2	.	.	PUNCT
cana-3670	133	3	proposed	propose	VERB
cana-3670	133	4	methods	method	NOUN
cana-3670	133	5	to	to	PART
cana-3670	133	6	fivp	fivp	VERB
cana-3670	133	7	here	here	ADV
cana-3670	133	8	,	,	PUNCT
cana-3670	133	9	we	we	PRON
cana-3670	133	10	investigate	investigate	VERB
cana-3670	133	11	the	the	DET
cana-3670	133	12	fuzzy	fuzzy	ADJ
cana-3670	133	13	initial	initial	ADJ
cana-3670	133	14	value	value	NOUN
cana-3670	133	15	problem	problem	NOUN
cana-3670	133	16	as	as	ADP
cana-3670	133	17	𝑢′(𝑡	𝑢′(𝑡	NOUN
cana-3670	133	18	)	)	PUNCT
cana-3670	133	19	=	=	NOUN
cana-3670	133	20	{	{	PUNCT
cana-3670	133	21	𝑓(𝑡	𝑓(𝑡	PROPN
cana-3670	133	22	,	,	PUNCT
cana-3670	133	23	𝑢	𝑢	NOUN
cana-3670	133	24	)	)	PUNCT
cana-3670	133	25	𝑢(𝑡0	𝑢(𝑡0	PROPN
cana-3670	133	26	)	)	PUNCT
cana-3670	133	27	=	=	NOUN
cana-3670	133	28	(	(	PUNCT
cana-3670	133	29	𝑢0,𝑢0	𝑢0,𝑢0	PROPN
cana-3670	133	30	𝑚	𝑚	PROPN
cana-3670	133	31	,	,	PUNCT
cana-3670	133	32	𝑢0	𝑢0	PROPN
cana-3670	133	33	)	)	PUNCT
cana-3670	133	34	communications	communication	NOUN
cana-3670	133	35	on	on	ADP
cana-3670	133	36	applied	apply	VERB
cana-3670	133	37	nonlinear	nonlinear	ADJ
cana-3670	133	38	analysis	analysis	NOUN
cana-3670	133	39	issn	issn	NOUN
cana-3670	133	40	:	:	PUNCT
cana-3670	133	41	1074	1074	NUM
cana-3670	133	42	-	-	PUNCT
cana-3670	133	43	133x	133x	NUM
cana-3670	133	44	vol	vol	NOUN
cana-3670	133	45	32	32	NUM
cana-3670	133	46	no	no	NOUN
cana-3670	133	47	.	.	PUNCT
cana-3670	134	1	8s	8s	PROPN
cana-3670	134	2	(	(	PUNCT
cana-3670	134	3	2025	2025	NUM
cana-3670	134	4	)	)	PUNCT
cana-3670	134	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	134	6	278	278	NUM
cana-3670	134	7	using	use	VERB
cana-3670	134	8	an	an	DET
cana-3670	134	9	s	s	NOUN
cana-3670	134	10	-	-	PUNCT
cana-3670	134	11	cut	cut	VERB
cana-3670	134	12	method	method	NOUN
cana-3670	134	13	,	,	PUNCT
cana-3670	134	14	the	the	DET
cana-3670	134	15	triangular	triangular	NOUN
cana-3670	134	16	fuzzy	fuzzy	ADJ
cana-3670	134	17	initial	initial	ADJ
cana-3670	134	18	condition	condition	NOUN
cana-3670	134	19	may	may	AUX
cana-3670	134	20	be	be	AUX
cana-3670	134	21	represented	represent	VERB
cana-3670	134	22	as	as	ADP
cana-3670	134	23	[	[	X
cana-3670	134	24	(	(	PUNCT
cana-3670	134	25	𝑢𝑚	𝑢𝑚	NOUN
cana-3670	134	26	−𝑢0)𝑠+	−𝑢0)𝑠+	ADP
cana-3670	134	27	𝑢0,𝑢0	𝑢0,𝑢0	PROPN
cana-3670	134	28	+	+	CCONJ
cana-3670	134	29	(	(	PUNCT
cana-3670	134	30	𝑢	𝑢	X
cana-3670	134	31	𝑚	𝑚	ADP
cana-3670	134	32	−	−	PROPN
cana-3670	134	33	𝑢0)𝑠],0	𝑢0)𝑠],0	SYM
cana-3670	134	34	≤	≤	NUM
cana-3670	134	35	𝑠	𝑠	PRON
cana-3670	134	36	≤	≤	NUM
cana-3670	134	37	1	1	NUM
cana-3670	134	38	.	.	PUNCT
cana-3670	135	1	the	the	DET
cana-3670	135	2	fivp	fivp	NOUN
cana-3670	135	3	has	have	AUX
cana-3670	135	4	been	be	AUX
cana-3670	135	5	intended	intend	VERB
cana-3670	135	6	to	to	PART
cana-3670	135	7	be	be	AUX
cana-3670	135	8	solved	solve	VERB
cana-3670	135	9	using	use	VERB
cana-3670	135	10	the	the	DET
cana-3670	135	11	enhanced	enhance	VERB
cana-3670	135	12	euler	euler	NOUN
cana-3670	135	13	's	's	PART
cana-3670	135	14	approaches	approach	NOUN
cana-3670	135	15	.	.	PUNCT
cana-3670	136	1	now	now	ADV
cana-3670	136	2	,	,	PUNCT
cana-3670	136	3	the	the	DET
cana-3670	136	4	enhanced	enhance	VERB
cana-3670	136	5	euler	euler	NOUN
cana-3670	136	6	's	's	PART
cana-3670	136	7	approaches	approach	NOUN
cana-3670	136	8	are	be	AUX
cana-3670	136	9	used	use	VERB
cana-3670	136	10	to	to	PART
cana-3670	136	11	assess	assess	VERB
cana-3670	136	12	all	all	DET
cana-3670	136	13	new	new	ADJ
cana-3670	136	14	upper	upper	ADJ
cana-3670	136	15	and	and	CCONJ
cana-3670	136	16	lower	lower	ADV
cana-3670	136	17	bound	bind	VERB
cana-3670	136	18	feasible	feasible	ADJ
cana-3670	136	19	permutations	permutation	NOUN
cana-3670	136	20	.	.	PUNCT
cana-3670	137	1	the	the	DET
cana-3670	137	2	results	result	NOUN
cana-3670	137	3	of	of	ADP
cana-3670	137	4	calculating	calculate	VERB
cana-3670	137	5	the	the	DET
cana-3670	137	6	grid	grid	NOUN
cana-3670	137	7	points	point	NOUN
cana-3670	137	8	as	as	SCONJ
cana-3670	137	9	t	t	PROPN
cana-3670	137	10	are	be	AUX
cana-3670	137	11	displayed	display	VERB
cana-3670	137	12	below	below	ADV
cana-3670	137	13	.	.	PUNCT
cana-3670	138	1	𝑢	𝑢	PROPN
cana-3670	138	2	𝑛+1	𝑛+1	PROPN
cana-3670	138	3	(	(	PUNCT
cana-3670	138	4	1	1	NUM
cana-3670	138	5	)	)	PUNCT
cana-3670	138	6	(	(	PUNCT
cana-3670	138	7	𝑡𝑛+1	𝑡𝑛+1	X
cana-3670	138	8	:	:	SYM
cana-3670	138	9	𝑠	𝑠	X
cana-3670	138	10	)	)	PUNCT
cana-3670	138	11	=	=	SYM
cana-3670	138	12	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	138	13	:	:	PUNCT
cana-3670	138	14	𝑠	𝑠	NUM
cana-3670	138	15	)	)	PUNCT
cana-3670	138	16	+	+	CCONJ
cana-3670	138	17	𝐹[𝑢(𝑡𝑛	𝐹[𝑢(𝑡𝑛	NUM
cana-3670	138	18	:	:	PUNCT
cana-3670	138	19	𝑠	𝑠	X
cana-3670	138	20	)	)	PUNCT
cana-3670	138	21	]	]	PUNCT
cana-3670	138	22	,	,	PUNCT
cana-3670	138	23	𝑢𝑛+1	𝑢𝑛+1	NUM
cana-3670	138	24	(	(	PUNCT
cana-3670	138	25	1	1	NUM
cana-3670	138	26	)	)	PUNCT
cana-3670	138	27	(	(	PUNCT
cana-3670	138	28	𝑡𝑛+1	𝑡𝑛+1	X
cana-3670	138	29	:	:	SYM
cana-3670	138	30	𝑠	𝑠	X
cana-3670	138	31	)	)	PUNCT
cana-3670	138	32	=	=	SYM
cana-3670	138	33	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	138	34	:	:	PUNCT
cana-3670	138	35	𝑠	𝑠	NUM
cana-3670	138	36	)	)	PUNCT
cana-3670	138	37	+	+	CCONJ
cana-3670	138	38	𝐺[𝑢(𝑡𝑛	𝐺[𝑢(𝑡𝑛	ADJ
cana-3670	138	39	:	:	PUNCT
cana-3670	138	40	𝑠	𝑠	X
cana-3670	138	41	)	)	PUNCT
cana-3670	138	42	]	]	PUNCT
cana-3670	138	43	,	,	PUNCT
cana-3670	138	44	𝑢	𝑢	PROPN
cana-3670	138	45	𝑛+1	𝑛+1	PROPN
cana-3670	138	46	(	(	PUNCT
cana-3670	138	47	2	2	NUM
cana-3670	138	48	)	)	PUNCT
cana-3670	138	49	(	(	PUNCT
cana-3670	138	50	𝑡𝑛+1	𝑡𝑛+1	X
cana-3670	138	51	:	:	SYM
cana-3670	138	52	𝑠	𝑠	X
cana-3670	138	53	)	)	PUNCT
cana-3670	138	54	=	=	SYM
cana-3670	138	55	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	138	56	:	:	PUNCT
cana-3670	138	57	𝑠	𝑠	NUM
cana-3670	138	58	)	)	PUNCT
cana-3670	138	59	+	+	CCONJ
cana-3670	138	60	𝐺[𝑢(𝑡𝑛	𝐺[𝑢(𝑡𝑛	ADJ
cana-3670	138	61	:	:	PUNCT
cana-3670	138	62	𝑠	𝑠	X
cana-3670	138	63	)	)	PUNCT
cana-3670	138	64	]	]	PUNCT
cana-3670	138	65	,	,	PUNCT
cana-3670	138	66	𝑢𝑛+1	𝑢𝑛+1	NUM
cana-3670	138	67	(	(	PUNCT
cana-3670	138	68	2	2	NUM
cana-3670	138	69	)	)	PUNCT
cana-3670	138	70	(	(	PUNCT
cana-3670	138	71	𝑡𝑛+1	𝑡𝑛+1	X
cana-3670	138	72	:	:	SYM
cana-3670	138	73	𝑠	𝑠	X
cana-3670	138	74	)	)	PUNCT
cana-3670	138	75	=	=	SYM
cana-3670	138	76	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	138	77	:	:	PUNCT
cana-3670	138	78	𝑠	𝑠	NUM
cana-3670	138	79	)	)	PUNCT
cana-3670	138	80	+	+	CCONJ
cana-3670	138	81	𝐹[𝑢(𝑡𝑛	𝐹[𝑢(𝑡𝑛	NUM
cana-3670	138	82	:	:	PUNCT
cana-3670	138	83	𝑠	𝑠	X
cana-3670	138	84	)	)	PUNCT
cana-3670	138	85	]	]	PUNCT
cana-3670	138	86	,	,	PUNCT
cana-3670	138	87	then	then	ADV
cana-3670	138	88	,	,	PUNCT
cana-3670	138	89	we	we	PRON
cana-3670	138	90	take	take	VERB
cana-3670	138	91	the	the	DET
cana-3670	138	92	values	value	NOUN
cana-3670	138	93	from	from	ADP
cana-3670	138	94	lowest	low	ADJ
cana-3670	138	95	to	to	ADP
cana-3670	138	96	maximum	maximum	ADJ
cana-3670	138	97	,	,	PUNCT
cana-3670	138	98	which	which	PRON
cana-3670	138	99	are	be	AUX
cana-3670	138	100	meant	mean	VERB
cana-3670	138	101	for	for	ADP
cana-3670	138	102	the	the	DET
cana-3670	138	103	inferior	inferior	ADJ
cana-3670	138	104	and	and	CCONJ
cana-3670	138	105	superior	superior	ADJ
cana-3670	138	106	values	value	NOUN
cana-3670	138	107	of	of	ADP
cana-3670	138	108	the	the	DET
cana-3670	138	109	variable	variable	ADJ
cana-3670	138	110	u	u	NOUN
cana-3670	138	111	in	in	ADP
cana-3670	138	112	order	order	NOUN
cana-3670	138	113	for	for	SCONJ
cana-3670	138	114	it	it	PRON
cana-3670	138	115	to	to	PART
cana-3670	138	116	yield	yield	VERB
cana-3670	138	117	more	more	ADV
cana-3670	138	118	precise	precise	ADJ
cana-3670	138	119	and	and	CCONJ
cana-3670	138	120	superior	superior	ADJ
cana-3670	138	121	answers	answer	NOUN
cana-3670	138	122	.	.	PUNCT
cana-3670	139	1	𝑢𝑛+1	𝑢𝑛+1	NUM
cana-3670	139	2	=	=	PUNCT
cana-3670	139	3	𝑚𝑖𝑛{𝑢(𝑡𝑛	𝑚𝑖𝑛{𝑢(𝑡𝑛	ADJ
cana-3670	139	4	:	:	PUNCT
cana-3670	139	5	𝑠	𝑠	X
cana-3670	139	6	)	)	PUNCT
cana-3670	139	7	+	+	CCONJ
cana-3670	139	8	𝐹[𝑢(𝑡𝑛	𝐹[𝑢(𝑡𝑛	ADV
cana-3670	139	9	:	:	PUNCT
cana-3670	139	10	𝑠)],𝑢(𝑡𝑛	𝑠)],𝑢(𝑡𝑛	ADJ
cana-3670	139	11	:	:	PUNCT
cana-3670	139	12	𝑠	𝑠	NUM
cana-3670	139	13	)	)	PUNCT
cana-3670	139	14	+	+	CCONJ
cana-3670	139	15	𝐺[𝑢(𝑡𝑛	𝐺[𝑢(𝑡𝑛	ADJ
cana-3670	139	16	:	:	PUNCT
cana-3670	139	17	𝑠	𝑠	X
cana-3670	139	18	)	)	PUNCT
cana-3670	139	19	]	]	PUNCT
cana-3670	139	20	}	}	PUNCT
cana-3670	139	21	,	,	PUNCT
cana-3670	139	22	𝑢𝑛+1	𝑢𝑛+1	X
cana-3670	139	23	=	=	SYM
cana-3670	139	24	𝑚𝑎𝑥{𝑢(𝑡𝑛	𝑚𝑎𝑥{𝑢(𝑡𝑛	ADJ
cana-3670	139	25	:	:	PUNCT
cana-3670	139	26	𝑠	𝑠	X
cana-3670	139	27	)	)	PUNCT
cana-3670	140	1	+	+	CCONJ
cana-3670	140	2	𝐺[𝑢(𝑡𝑛	𝐺[𝑢(𝑡𝑛	ADJ
cana-3670	140	3	:	:	PUNCT
cana-3670	140	4	𝑠)],𝑢(𝑡𝑛	𝑠)],𝑢(𝑡𝑛	ADJ
cana-3670	140	5	:	:	PUNCT
cana-3670	140	6	𝑠	𝑠	NUM
cana-3670	140	7	)	)	PUNCT
cana-3670	141	1	+	+	CCONJ
cana-3670	141	2	𝐹[𝑢(𝑡𝑛	𝐹[𝑢(𝑡𝑛	NUM
cana-3670	141	3	:	:	PUNCT
cana-3670	141	4	𝑠	𝑠	X
cana-3670	141	5	)	)	PUNCT
cana-3670	141	6	]	]	PUNCT
cana-3670	141	7	}	}	PUNCT
cana-3670	141	8	.	.	PUNCT
cana-3670	142	1	3.2.3	3.2.3	X
cana-3670	142	2	.	.	PUNCT
cana-3670	142	3	enhanced	enhance	VERB
cana-3670	142	4	euler	euler	PROPN
cana-3670	142	5	’s	’s	PART
cana-3670	142	6	methods	method	NOUN
cana-3670	142	7	based	base	VERB
cana-3670	142	8	on	on	ADP
cana-3670	142	9	contra	contra	PROPN
cana-3670	142	10	harmonic	harmonic	PROPN
cana-3670	142	11	mean	mean	NOUN
cana-3670	142	12	and	and	CCONJ
cana-3670	142	13	centroidal	centroidal	PROPN
cana-3670	142	14	mean	mean	VERB
cana-3670	142	15	to	to	ADP
cana-3670	142	16	fivp	fivp	PROPN
cana-3670	142	17	3.2.3.1	3.2.3.1	PROPN
cana-3670	142	18	.	.	PROPN
cana-3670	142	19	euler	euler	PROPN
cana-3670	142	20	’s	’s	PART
cana-3670	142	21	method	method	NOUN
cana-3670	142	22	based	base	VERB
cana-3670	142	23	on	on	ADP
cana-3670	142	24	contra	contra	PROPN
cana-3670	142	25	harmonic	harmonic	PROPN
cana-3670	142	26	mean	mean	VERB
cana-3670	142	27	to	to	PART
cana-3670	142	28	fivp	fivp	VERB
cana-3670	142	29	the	the	DET
cana-3670	142	30	modified	modify	VERB
cana-3670	142	31	euler	euler	PROPN
cana-3670	142	32	's	's	PART
cana-3670	142	33	method	method	NOUN
cana-3670	142	34	's	's	PART
cana-3670	142	35	expected	expect	VERB
cana-3670	142	36	contra	contra	PROPN
cana-3670	142	37	harmonic	harmonic	PROPN
cana-3670	142	38	mean	mean	NOUN
cana-3670	142	39	is	be	AUX
cana-3670	142	40	precisely	precisely	ADV
cana-3670	142	41	defined	define	VERB
cana-3670	142	42	by	by	ADP
cana-3670	142	43	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	142	44	=	=	PUNCT
cana-3670	142	45	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	142	46	:	:	PUNCT
cana-3670	142	47	𝑠	𝑠	NUM
cana-3670	142	48	)	)	PUNCT
cana-3670	143	1	+	+	CCONJ
cana-3670	143	2	𝐹(𝑢(𝑡𝑛	𝐹(𝑢(𝑡𝑛	NUM
cana-3670	143	3	:	:	PUNCT
cana-3670	143	4	𝑠	𝑠	NUM
cana-3670	143	5	)	)	PUNCT
cana-3670	143	6	)	)	PUNCT
cana-3670	143	7	,	,	PUNCT
cana-3670	143	8	𝑢𝑛+1	𝑢𝑛+1	X
cana-3670	143	9	=	=	SYM
cana-3670	143	10	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	143	11	:	:	PUNCT
cana-3670	143	12	𝑠	𝑠	X
cana-3670	143	13	)	)	PUNCT
cana-3670	143	14	+	+	CCONJ
cana-3670	143	15	𝐺(𝑢(𝑡𝑛	𝐺(𝑢(𝑡𝑛	X
cana-3670	143	16	:	:	PUNCT
cana-3670	143	17	𝑠	𝑠	X
cana-3670	143	18	)	)	PUNCT
cana-3670	143	19	)	)	PUNCT
cana-3670	143	20	.	.	PUNCT
cana-3670	144	1	where	where	SCONJ
cana-3670	144	2	𝐹	𝐹	PROPN
cana-3670	144	3	=	=	SYM
cana-3670	144	4	ℎ𝑓	ℎ𝑓	X
cana-3670	144	5	(	(	PUNCT
cana-3670	144	6	(	(	PUNCT
cana-3670	144	7	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	144	8	)	)	PUNCT
cana-3670	144	9	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	144	10	)	)	PUNCT
cana-3670	144	11	2	2	NUM
cana-3670	144	12	(	(	PUNCT
cana-3670	144	13	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	144	14	)	)	PUNCT
cana-3670	144	15	,	,	PUNCT
cana-3670	144	16	(	(	PUNCT
cana-3670	144	17	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	144	18	)	)	PUNCT
cana-3670	144	19	2	2	NUM
cana-3670	145	1	+	+	NOUN
cana-3670	145	2	(	(	PUNCT
cana-3670	145	3	(	(	PUNCT
cana-3670	145	4	𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	145	5	)	)	PUNCT
cana-3670	145	6	)	)	PUNCT
cana-3670	145	7	)	)	PUNCT
cana-3670	145	8	2	2	NUM
cana-3670	145	9	(	(	PUNCT
cana-3670	145	10	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	145	11	)	)	PUNCT
cana-3670	145	12	)	)	PUNCT
cana-3670	145	13	)	)	PUNCT
cana-3670	145	14	)	)	PUNCT
cana-3670	145	15	,	,	PUNCT
cana-3670	145	16	𝐺	𝐺	PROPN
cana-3670	145	17	=	=	SYM
cana-3670	145	18	ℎ𝑓	ℎ𝑓	X
cana-3670	145	19	(	(	PUNCT
cana-3670	145	20	(	(	PUNCT
cana-3670	145	21	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	145	22	)	)	PUNCT
cana-3670	145	23	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	145	24	)	)	PUNCT
cana-3670	145	25	2	2	NUM
cana-3670	145	26	(	(	PUNCT
cana-3670	145	27	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	145	28	)	)	PUNCT
cana-3670	145	29	,	,	PUNCT
cana-3670	145	30	(	(	PUNCT
cana-3670	145	31	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	145	32	)	)	PUNCT
cana-3670	145	33	2+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	2+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	NUM
cana-3670	145	34	)	)	PUNCT
cana-3670	145	35	)	)	PUNCT
cana-3670	145	36	)	)	PUNCT
cana-3670	145	37	2	2	NUM
cana-3670	145	38	(	(	PUNCT
cana-3670	145	39	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	145	40	)	)	PUNCT
cana-3670	145	41	)	)	PUNCT
cana-3670	145	42	)	)	PUNCT
cana-3670	145	43	)	)	PUNCT
cana-3670	145	44	.	.	PUNCT
cana-3670	146	1	3.2.3.2	3.2.3.2	NUM
cana-3670	146	2	.	.	PUNCT
cana-3670	146	3	modified	modified	PROPN
cana-3670	146	4	euler	euler	PROPN
cana-3670	146	5	’s	’s	PART
cana-3670	146	6	method	method	NOUN
cana-3670	146	7	based	base	VERB
cana-3670	146	8	on	on	ADP
cana-3670	146	9	contra	contra	PROPN
cana-3670	146	10	harmonic	harmonic	PROPN
cana-3670	146	11	mean	mean	VERB
cana-3670	146	12	to	to	PART
cana-3670	146	13	fivp	fivp	VERB
cana-3670	146	14	the	the	DET
cana-3670	146	15	modified	modify	VERB
cana-3670	146	16	euler	euler	PROPN
cana-3670	146	17	's	's	PART
cana-3670	146	18	method	method	NOUN
cana-3670	146	19	's	's	PART
cana-3670	146	20	expected	expect	VERB
cana-3670	146	21	contra	contra	PROPN
cana-3670	146	22	harmonic	harmonic	PROPN
cana-3670	146	23	mean	mean	NOUN
cana-3670	146	24	is	be	AUX
cana-3670	146	25	precisely	precisely	ADV
cana-3670	146	26	defined	define	VERB
cana-3670	146	27	by	by	ADP
cana-3670	146	28	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	146	29	=	=	PUNCT
cana-3670	146	30	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	146	31	:	:	PUNCT
cana-3670	146	32	𝑠	𝑠	NUM
cana-3670	146	33	)	)	PUNCT
cana-3670	147	1	+	+	CCONJ
cana-3670	147	2	𝐹(𝑢(𝑡𝑛	𝐹(𝑢(𝑡𝑛	NUM
cana-3670	147	3	:	:	PUNCT
cana-3670	147	4	𝑠	𝑠	NUM
cana-3670	147	5	)	)	PUNCT
cana-3670	147	6	)	)	PUNCT
cana-3670	147	7	,	,	PUNCT
cana-3670	147	8	𝑢𝑛+1	𝑢𝑛+1	X
cana-3670	147	9	=	=	SYM
cana-3670	147	10	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	147	11	:	:	PUNCT
cana-3670	147	12	𝑠	𝑠	X
cana-3670	147	13	)	)	PUNCT
cana-3670	148	1	+	+	CCONJ
cana-3670	148	2	𝐺(𝑢(𝑡𝑛	𝐺(𝑢(𝑡𝑛	X
cana-3670	148	3	:	:	PUNCT
cana-3670	148	4	𝑠	𝑠	X
cana-3670	148	5	)	)	PUNCT
cana-3670	148	6	)	)	PUNCT
cana-3670	148	7	.	.	PUNCT
cana-3670	149	1	where	where	SCONJ
cana-3670	149	2	𝐹	𝐹	PROPN
cana-3670	149	3	=	=	SYM
cana-3670	149	4	ℎ𝑓	ℎ𝑓	X
cana-3670	149	5	(	(	PUNCT
cana-3670	149	6	(	(	PUNCT
cana-3670	149	7	𝑡𝑛	𝑡𝑛	NUM
cana-3670	149	8	:	:	SYM
cana-3670	149	9	𝑠	𝑠	NUM
cana-3670	149	10	)	)	PUNCT
cana-3670	149	11	+	+	CCONJ
cana-3670	149	12	ℎ	ℎ	X
cana-3670	149	13	2	2	NUM
cana-3670	149	14	,	,	PUNCT
cana-3670	149	15	(	(	PUNCT
cana-3670	149	16	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	149	17	:	:	SYM
cana-3670	149	18	𝑠	𝑠	NUM
cana-3670	149	19	)	)	PUNCT
cana-3670	149	20	+	+	CCONJ
cana-3670	149	21	ℎ	ℎ	X
cana-3670	149	22	2	2	NUM
cana-3670	149	23	𝑓	𝑓	PRON
cana-3670	149	24	(	(	PUNCT
cana-3670	149	25	(	(	PUNCT
cana-3670	149	26	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	149	27	)	)	PUNCT
cana-3670	149	28	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	149	29	)	)	PUNCT
cana-3670	149	30	2	2	NUM
cana-3670	149	31	(	(	PUNCT
cana-3670	149	32	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	149	33	)	)	PUNCT
cana-3670	149	34	,	,	PUNCT
cana-3670	149	35	(	(	PUNCT
cana-3670	149	36	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	149	37	)	)	PUNCT
cana-3670	149	38	2	2	NUM
cana-3670	150	1	+	+	NOUN
cana-3670	151	1	(	(	PUNCT
cana-3670	152	1	(	(	PUNCT
cana-3670	152	2	𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	PROPN
cana-3670	152	3	ℎ	ℎ	PROPN
cana-3670	152	4	2	2	NUM
cana-3670	152	5	,	,	PUNCT
cana-3670	152	6	(	(	PUNCT
cana-3670	152	7	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	PROPN
cana-3670	152	8	ℎ	ℎ	X
cana-3670	152	9	2	2	NUM
cana-3670	152	10	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	152	11	)	)	PUNCT
cana-3670	152	12	)	)	PUNCT
cana-3670	152	13	)	)	PUNCT
cana-3670	152	14	)	)	PUNCT
cana-3670	152	15	2	2	NUM
cana-3670	152	16	(	(	PUNCT
cana-3670	152	17	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	ADJ
cana-3670	152	18	ℎ	ℎ	X
cana-3670	152	19	2	2	NUM
cana-3670	152	20	,	,	PUNCT
cana-3670	152	21	(	(	PUNCT
cana-3670	152	22	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	PROPN
cana-3670	152	23	ℎ	ℎ	X
cana-3670	152	24	2	2	NUM
cana-3670	152	25	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	152	26	)	)	PUNCT
cana-3670	152	27	)	)	PUNCT
cana-3670	152	28	)	)	PUNCT
cana-3670	152	29	)	)	PUNCT
cana-3670	152	30	)	)	PUNCT
cana-3670	152	31	)	)	PUNCT
cana-3670	152	32	,	,	PUNCT
cana-3670	152	33	communications	communication	NOUN
cana-3670	152	34	on	on	ADP
cana-3670	152	35	applied	apply	VERB
cana-3670	152	36	nonlinear	nonlinear	ADJ
cana-3670	152	37	analysis	analysis	NOUN
cana-3670	152	38	issn	issn	NOUN
cana-3670	152	39	:	:	PUNCT
cana-3670	152	40	1074	1074	NUM
cana-3670	152	41	-	-	PUNCT
cana-3670	152	42	133x	133x	NUM
cana-3670	152	43	vol	vol	NOUN
cana-3670	152	44	32	32	NUM
cana-3670	152	45	no	no	NOUN
cana-3670	152	46	.	.	PUNCT
cana-3670	153	1	8s	8s	PROPN
cana-3670	153	2	(	(	PUNCT
cana-3670	153	3	2025	2025	NUM
cana-3670	153	4	)	)	PUNCT
cana-3670	153	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	153	6	279	279	NUM
cana-3670	153	7	𝐺	𝐺	NOUN
cana-3670	153	8	=	=	SYM
cana-3670	153	9	ℎ𝑓	ℎ𝑓	X
cana-3670	153	10	(	(	PUNCT
cana-3670	153	11	(	(	PUNCT
cana-3670	153	12	𝑡𝑛	𝑡𝑛	NUM
cana-3670	153	13	:	:	SYM
cana-3670	153	14	𝑠	𝑠	NUM
cana-3670	153	15	)	)	PUNCT
cana-3670	154	1	+	+	CCONJ
cana-3670	154	2	ℎ	ℎ	X
cana-3670	154	3	2	2	NUM
cana-3670	154	4	,	,	PUNCT
cana-3670	154	5	(	(	PUNCT
cana-3670	154	6	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	154	7	:	:	SYM
cana-3670	154	8	𝑠	𝑠	NUM
cana-3670	154	9	)	)	PUNCT
cana-3670	155	1	+	+	CCONJ
cana-3670	155	2	ℎ	ℎ	X
cana-3670	155	3	2	2	NUM
cana-3670	155	4	𝑓	𝑓	PRON
cana-3670	155	5	(	(	PUNCT
cana-3670	155	6	(	(	PUNCT
cana-3670	155	7	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	155	8	)	)	PUNCT
cana-3670	155	9	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	155	10	)	)	PUNCT
cana-3670	155	11	2	2	NUM
cana-3670	155	12	(	(	PUNCT
cana-3670	155	13	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	155	14	)	)	PUNCT
cana-3670	155	15	,	,	PUNCT
cana-3670	155	16	(	(	PUNCT
cana-3670	155	17	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	155	18	)	)	PUNCT
cana-3670	155	19	2+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	2+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	NUM
cana-3670	155	20	ℎ	ℎ	ADP
cana-3670	155	21	2	2	NUM
cana-3670	155	22	,	,	PUNCT
cana-3670	155	23	(	(	PUNCT
cana-3670	155	24	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	PROPN
cana-3670	155	25	ℎ	ℎ	X
cana-3670	155	26	2	2	NUM
cana-3670	155	27	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	155	28	)	)	PUNCT
cana-3670	155	29	)	)	PUNCT
cana-3670	155	30	)	)	PUNCT
cana-3670	155	31	)	)	PUNCT
cana-3670	155	32	2	2	NUM
cana-3670	155	33	(	(	PUNCT
cana-3670	155	34	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	ADJ
cana-3670	155	35	ℎ	ℎ	X
cana-3670	155	36	2	2	NUM
cana-3670	155	37	,	,	PUNCT
cana-3670	155	38	(	(	PUNCT
cana-3670	155	39	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	PROPN
cana-3670	155	40	ℎ	ℎ	X
cana-3670	155	41	2	2	NUM
cana-3670	155	42	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	155	43	)	)	PUNCT
cana-3670	155	44	)	)	PUNCT
cana-3670	155	45	)	)	PUNCT
cana-3670	155	46	)	)	PUNCT
cana-3670	155	47	)	)	PUNCT
cana-3670	155	48	)	)	PUNCT
cana-3670	155	49	.	.	PUNCT
cana-3670	156	1	3.2.3.3	3.2.3.3	X
cana-3670	156	2	.	.	NUM
cana-3670	156	3	improved	improve	VERB
cana-3670	156	4	euler	euler	PROPN
cana-3670	156	5	’s	’s	PART
cana-3670	156	6	method	method	NOUN
cana-3670	156	7	based	base	VERB
cana-3670	156	8	on	on	ADP
cana-3670	156	9	contra	contra	PROPN
cana-3670	156	10	harmonic	harmonic	PROPN
cana-3670	156	11	mean	mean	VERB
cana-3670	156	12	to	to	PART
cana-3670	156	13	fivp	fivp	VERB
cana-3670	156	14	the	the	DET
cana-3670	156	15	improved	improve	VERB
cana-3670	156	16	euler	euler	NOUN
cana-3670	156	17	's	's	PART
cana-3670	156	18	method	method	NOUN
cana-3670	156	19	's	's	PART
cana-3670	156	20	expected	expect	VERB
cana-3670	156	21	contra	contra	PROPN
cana-3670	156	22	harmonic	harmonic	PROPN
cana-3670	156	23	mean	mean	NOUN
cana-3670	156	24	is	be	AUX
cana-3670	156	25	precisely	precisely	ADV
cana-3670	156	26	defined	define	VERB
cana-3670	156	27	by	by	ADP
cana-3670	156	28	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	156	29	=	=	PUNCT
cana-3670	156	30	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	156	31	:	:	PUNCT
cana-3670	156	32	𝑠	𝑠	NUM
cana-3670	156	33	)	)	PUNCT
cana-3670	157	1	+	+	CCONJ
cana-3670	157	2	𝐹(𝑢(𝑡𝑛	𝐹(𝑢(𝑡𝑛	NUM
cana-3670	157	3	:	:	PUNCT
cana-3670	157	4	𝑠	𝑠	NUM
cana-3670	157	5	)	)	PUNCT
cana-3670	157	6	)	)	PUNCT
cana-3670	157	7	,	,	PUNCT
cana-3670	157	8	𝑢𝑛+1	𝑢𝑛+1	X
cana-3670	157	9	=	=	SYM
cana-3670	157	10	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	157	11	:	:	PUNCT
cana-3670	157	12	𝑠	𝑠	X
cana-3670	157	13	)	)	PUNCT
cana-3670	158	1	+	+	CCONJ
cana-3670	158	2	𝐺(𝑢(𝑡𝑛	𝐺(𝑢(𝑡𝑛	X
cana-3670	158	3	:	:	PUNCT
cana-3670	158	4	𝑠	𝑠	X
cana-3670	158	5	)	)	PUNCT
cana-3670	158	6	)	)	PUNCT
cana-3670	158	7	.	.	PUNCT
cana-3670	159	1	where	where	SCONJ
cana-3670	159	2	𝐹	𝐹	PROPN
cana-3670	159	3	=	=	SYM
cana-3670	159	4	ℎ	ℎ	X
cana-3670	159	5	2	2	NUM
cana-3670	159	6	[	[	PUNCT
cana-3670	159	7	𝑓	𝑓	X
cana-3670	159	8	(	(	PUNCT
cana-3670	159	9	(	(	PUNCT
cana-3670	159	10	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	159	11	)	)	PUNCT
cana-3670	159	12	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	159	13	)	)	PUNCT
cana-3670	159	14	2	2	NUM
cana-3670	159	15	(	(	PUNCT
cana-3670	159	16	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	159	17	)	)	PUNCT
cana-3670	159	18	,	,	PUNCT
cana-3670	159	19	(	(	PUNCT
cana-3670	159	20	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	159	21	)	)	PUNCT
cana-3670	159	22	2	2	NUM
cana-3670	159	23	+	+	NOUN
cana-3670	159	24	(	(	PUNCT
cana-3670	159	25	(	(	PUNCT
cana-3670	159	26	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	X
cana-3670	159	27	ℎ	ℎ	X
cana-3670	159	28	2	2	NUM
cana-3670	159	29	(	(	PUNCT
cana-3670	159	30	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	159	31	)	)	PUNCT
cana-3670	159	32	,	,	PUNCT
cana-3670	159	33	(	(	PUNCT
cana-3670	159	34	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	PROPN
cana-3670	159	35	)	)	PUNCT
cana-3670	159	36	)	)	PUNCT
cana-3670	159	37	)	)	PUNCT
cana-3670	159	38	)	)	PUNCT
cana-3670	159	39	)	)	PUNCT
cana-3670	159	40	2	2	NUM
cana-3670	159	41	(	(	PUNCT
cana-3670	159	42	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	ADJ
cana-3670	159	43	ℎ	ℎ	X
cana-3670	159	44	2	2	NUM
cana-3670	159	45	(	(	PUNCT
cana-3670	159	46	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	159	47	)	)	PUNCT
cana-3670	159	48	,	,	PUNCT
cana-3670	159	49	(	(	PUNCT
cana-3670	159	50	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	PROPN
cana-3670	159	51	)	)	PUNCT
cana-3670	159	52	)	)	PUNCT
cana-3670	159	53	)	)	PUNCT
cana-3670	159	54	)	)	PUNCT
cana-3670	159	55	)	)	PUNCT
cana-3670	159	56	)	)	PUNCT
cana-3670	160	1	+	+	ADP
cana-3670	160	2	𝑓	𝑓	DET
cana-3670	160	3	(	(	PUNCT
cana-3670	160	4	(	(	PUNCT
cana-3670	160	5	𝑡𝑛	𝑡𝑛	NUM
cana-3670	160	6	:	:	SYM
cana-3670	160	7	𝑠	𝑠	NUM
cana-3670	160	8	)	)	PUNCT
cana-3670	160	9	+	+	NUM
cana-3670	160	10	ℎ	ℎ	NOUN
cana-3670	160	11	,	,	PUNCT
cana-3670	160	12	(	(	PUNCT
cana-3670	160	13	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	160	14	:	:	SYM
cana-3670	160	15	𝑠	𝑠	NUM
cana-3670	160	16	)	)	PUNCT
cana-3670	160	17	+	+	CCONJ
cana-3670	160	18	ℎ	ℎ	X
cana-3670	160	19	(	(	PUNCT
cana-3670	160	20	(	(	PUNCT
cana-3670	160	21	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	160	22	)	)	PUNCT
cana-3670	160	23	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	160	24	)	)	PUNCT
cana-3670	160	25	2	2	NUM
cana-3670	160	26	(	(	PUNCT
cana-3670	160	27	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	160	28	)	)	PUNCT
cana-3670	160	29	,	,	PUNCT
cana-3670	160	30	(	(	PUNCT
cana-3670	160	31	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	160	32	)	)	PUNCT
cana-3670	160	33	2	2	NUM
cana-3670	160	34	+	+	NOUN
cana-3670	160	35	(	(	PUNCT
cana-3670	160	36	(	(	PUNCT
cana-3670	160	37	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	X
cana-3670	160	38	ℎ	ℎ	X
cana-3670	160	39	2	2	NUM
cana-3670	160	40	(	(	PUNCT
cana-3670	160	41	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	160	42	)	)	PUNCT
cana-3670	160	43	,	,	PUNCT
cana-3670	160	44	(	(	PUNCT
cana-3670	160	45	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	PROPN
cana-3670	160	46	)	)	PUNCT
cana-3670	160	47	)	)	PUNCT
cana-3670	160	48	)	)	PUNCT
cana-3670	160	49	)	)	PUNCT
cana-3670	160	50	)	)	PUNCT
cana-3670	160	51	2	2	NUM
cana-3670	160	52	(	(	PUNCT
cana-3670	160	53	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	ADJ
cana-3670	160	54	ℎ	ℎ	X
cana-3670	160	55	2	2	NUM
cana-3670	160	56	(	(	PUNCT
cana-3670	160	57	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	160	58	)	)	PUNCT
cana-3670	160	59	,	,	PUNCT
cana-3670	160	60	(	(	PUNCT
cana-3670	160	61	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	PROPN
cana-3670	160	62	)	)	PUNCT
cana-3670	160	63	)	)	PUNCT
cana-3670	160	64	)	)	PUNCT
cana-3670	160	65	)	)	PUNCT
cana-3670	160	66	)	)	PUNCT
cana-3670	160	67	)	)	PUNCT
cana-3670	160	68	)	)	PUNCT
cana-3670	160	69	]	]	PUNCT
cana-3670	160	70	,	,	PUNCT
cana-3670	160	71	𝐺	𝐺	NOUN
cana-3670	160	72	=	=	PUNCT
cana-3670	160	73	ℎ	ℎ	X
cana-3670	160	74	2	2	NUM
cana-3670	160	75	[	[	PUNCT
cana-3670	160	76	𝑓	𝑓	X
cana-3670	160	77	(	(	PUNCT
cana-3670	160	78	(	(	PUNCT
cana-3670	160	79	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	160	80	)	)	PUNCT
cana-3670	160	81	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	160	82	)	)	PUNCT
cana-3670	160	83	2	2	NUM
cana-3670	160	84	(	(	PUNCT
cana-3670	160	85	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	160	86	)	)	PUNCT
cana-3670	160	87	,	,	PUNCT
cana-3670	160	88	(	(	PUNCT
cana-3670	160	89	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	160	90	)	)	PUNCT
cana-3670	160	91	2+((𝑢𝑛:𝑠)+	2+((𝑢𝑛:𝑠)+	NUM
cana-3670	160	92	ℎ	ℎ	SYM
cana-3670	160	93	2	2	NUM
cana-3670	160	94	(	(	PUNCT
cana-3670	160	95	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	160	96	)	)	PUNCT
cana-3670	160	97	)	)	PUNCT
cana-3670	160	98	)	)	PUNCT
cana-3670	160	99	)	)	PUNCT
cana-3670	160	100	)	)	PUNCT
cana-3670	160	101	2	2	NUM
cana-3670	160	102	(	(	PUNCT
cana-3670	160	103	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	ADJ
cana-3670	160	104	ℎ	ℎ	X
cana-3670	160	105	2	2	NUM
cana-3670	160	106	(	(	PUNCT
cana-3670	160	107	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	160	108	)	)	PUNCT
cana-3670	160	109	)	)	PUNCT
cana-3670	160	110	)	)	PUNCT
cana-3670	160	111	)	)	PUNCT
cana-3670	160	112	)	)	PUNCT
cana-3670	160	113	)	)	PUNCT
cana-3670	161	1	+	+	ADP
cana-3670	161	2	𝑓	𝑓	DET
cana-3670	161	3	(	(	PUNCT
cana-3670	161	4	(	(	PUNCT
cana-3670	161	5	𝑡𝑛	𝑡𝑛	NUM
cana-3670	161	6	:	:	SYM
cana-3670	161	7	𝑠	𝑠	NUM
cana-3670	161	8	)	)	PUNCT
cana-3670	161	9	+	+	NUM
cana-3670	161	10	ℎ	ℎ	NOUN
cana-3670	161	11	,	,	PUNCT
cana-3670	161	12	(	(	PUNCT
cana-3670	161	13	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	161	14	:	:	SYM
cana-3670	161	15	𝑠	𝑠	NUM
cana-3670	161	16	)	)	PUNCT
cana-3670	162	1	+	+	CCONJ
cana-3670	162	2	ℎ	ℎ	X
cana-3670	162	3	(	(	PUNCT
cana-3670	162	4	(	(	PUNCT
cana-3670	162	5	𝑡𝑛:𝑠	𝑡𝑛:𝑠	NOUN
cana-3670	162	6	)	)	PUNCT
cana-3670	162	7	2+((𝑡𝑛:𝑠)+ℎ	2+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	162	8	)	)	PUNCT
cana-3670	162	9	2	2	NUM
cana-3670	162	10	(	(	PUNCT
cana-3670	162	11	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	PROPN
cana-3670	162	12	)	)	PUNCT
cana-3670	162	13	,	,	PUNCT
cana-3670	162	14	(	(	PUNCT
cana-3670	162	15	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	162	16	)	)	PUNCT
cana-3670	162	17	2+((𝑢𝑛:𝑠)+	2+((𝑢𝑛:𝑠)+	NUM
cana-3670	162	18	ℎ	ℎ	SYM
cana-3670	162	19	2	2	NUM
cana-3670	162	20	(	(	PUNCT
cana-3670	162	21	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	162	22	)	)	PUNCT
cana-3670	162	23	,	,	PUNCT
cana-3670	162	24	(	(	PUNCT
cana-3670	162	25	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	PROPN
cana-3670	162	26	)	)	PUNCT
cana-3670	162	27	,	,	PUNCT
cana-3670	162	28	(	(	PUNCT
cana-3670	162	29	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	162	30	)	)	PUNCT
cana-3670	162	31	)	)	PUNCT
cana-3670	162	32	)	)	PUNCT
cana-3670	162	33	)	)	PUNCT
cana-3670	162	34	)	)	PUNCT
cana-3670	162	35	2	2	NUM
cana-3670	162	36	(	(	PUNCT
cana-3670	162	37	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	ADJ
cana-3670	162	38	ℎ	ℎ	X
cana-3670	162	39	2	2	NUM
cana-3670	162	40	(	(	PUNCT
cana-3670	162	41	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	PROPN
cana-3670	162	42	)	)	PUNCT
cana-3670	162	43	,	,	PUNCT
cana-3670	162	44	(	(	PUNCT
cana-3670	162	45	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	162	46	)	)	PUNCT
cana-3670	162	47	)	)	PUNCT
cana-3670	162	48	)	)	PUNCT
cana-3670	162	49	)	)	PUNCT
cana-3670	162	50	)	)	PUNCT
cana-3670	162	51	)	)	PUNCT
cana-3670	162	52	)	)	PUNCT
cana-3670	162	53	]	]	PUNCT
cana-3670	162	54	.	.	PUNCT
cana-3670	163	1	3.2.3.4	3.2.3.4	X
cana-3670	163	2	.	.	X
cana-3670	163	3	euler	euler	PROPN
cana-3670	163	4	’s	’s	PART
cana-3670	163	5	method	method	NOUN
cana-3670	163	6	based	base	VERB
cana-3670	163	7	on	on	ADP
cana-3670	163	8	centroidal	centroidal	PROPN
cana-3670	163	9	mean	mean	VERB
cana-3670	163	10	the	the	DET
cana-3670	163	11	modified	modify	VERB
cana-3670	163	12	euler	euler	NOUN
cana-3670	163	13	's	's	PART
cana-3670	163	14	method	method	NOUN
cana-3670	163	15	's	's	PART
cana-3670	163	16	expected	expect	VERB
cana-3670	163	17	centroidal	centroidal	NOUN
cana-3670	163	18	mean	mean	NOUN
cana-3670	163	19	is	be	AUX
cana-3670	163	20	precisely	precisely	ADV
cana-3670	163	21	defined	define	VERB
cana-3670	163	22	by	by	ADP
cana-3670	163	23	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	163	24	=	=	PUNCT
cana-3670	163	25	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	163	26	:	:	PUNCT
cana-3670	163	27	𝑠	𝑠	NUM
cana-3670	163	28	)	)	PUNCT
cana-3670	163	29	+	+	CCONJ
cana-3670	163	30	𝐹(𝑢(𝑡𝑛	𝐹(𝑢(𝑡𝑛	NUM
cana-3670	163	31	:	:	PUNCT
cana-3670	163	32	𝑠	𝑠	NUM
cana-3670	163	33	)	)	PUNCT
cana-3670	163	34	)	)	PUNCT
cana-3670	163	35	,	,	PUNCT
cana-3670	163	36	𝑢𝑛+1	𝑢𝑛+1	X
cana-3670	163	37	=	=	SYM
cana-3670	163	38	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	163	39	:	:	PUNCT
cana-3670	163	40	𝑠	𝑠	X
cana-3670	163	41	)	)	PUNCT
cana-3670	163	42	+	+	CCONJ
cana-3670	163	43	𝐺(𝑢(𝑡𝑛	𝐺(𝑢(𝑡𝑛	X
cana-3670	163	44	:	:	PUNCT
cana-3670	163	45	𝑠	𝑠	NUM
cana-3670	163	46	)	)	PUNCT
cana-3670	163	47	)	)	PUNCT
cana-3670	163	48	.	.	PUNCT
cana-3670	164	1	communications	communication	NOUN
cana-3670	164	2	on	on	ADP
cana-3670	164	3	applied	apply	VERB
cana-3670	164	4	nonlinear	nonlinear	ADJ
cana-3670	164	5	analysis	analysis	NOUN
cana-3670	164	6	issn	issn	NOUN
cana-3670	164	7	:	:	PUNCT
cana-3670	164	8	1074	1074	NUM
cana-3670	164	9	-	-	PUNCT
cana-3670	164	10	133x	133x	NUM
cana-3670	164	11	vol	vol	NOUN
cana-3670	164	12	32	32	NUM
cana-3670	164	13	no	no	NOUN
cana-3670	164	14	.	.	PUNCT
cana-3670	165	1	8s	8s	PROPN
cana-3670	165	2	(	(	PUNCT
cana-3670	165	3	2025	2025	NUM
cana-3670	165	4	)	)	PUNCT
cana-3670	165	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	165	6	280	280	NUM
cana-3670	165	7	where	where	SCONJ
cana-3670	165	8	𝐹	𝐹	PROPN
cana-3670	165	9	=	=	SYM
cana-3670	165	10	ℎ𝑓	ℎ𝑓	X
cana-3670	165	11	(	(	PUNCT
cana-3670	165	12	2((𝑡𝑛:𝑠	2((𝑡𝑛:𝑠	NUM
cana-3670	165	13	)	)	PUNCT
cana-3670	165	14	2+(𝑡𝑛:𝑠	2+(𝑡𝑛:𝑠	NUM
cana-3670	165	15	)	)	PUNCT
cana-3670	165	16	(	(	PUNCT
cana-3670	165	17	(	(	PUNCT
cana-3670	165	18	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	X
cana-3670	165	19	)	)	PUNCT
cana-3670	165	20	2	2	NUM
cana-3670	165	21	)	)	PUNCT
cana-3670	165	22	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	165	23	)	)	PUNCT
cana-3670	165	24	)	)	PUNCT
cana-3670	165	25	,	,	PUNCT
cana-3670	165	26	2((𝑢𝑛:𝑠	2((𝑢𝑛:𝑠	NUM
cana-3670	165	27	)	)	PUNCT
cana-3670	165	28	2	2	NUM
cana-3670	166	1	+	+	NOUN
cana-3670	166	2	(	(	PUNCT
cana-3670	166	3	𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠)))+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠)))+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	166	4	)	)	PUNCT
cana-3670	166	5	,	,	PUNCT
cana-3670	166	6	(	(	PUNCT
cana-3670	166	7	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	166	8	)	)	PUNCT
cana-3670	166	9	)	)	PUNCT
cana-3670	166	10	)	)	PUNCT
cana-3670	166	11	2	2	X
cana-3670	166	12	)	)	PUNCT
cana-3670	166	13	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	NUM
cana-3670	166	14	)	)	PUNCT
cana-3670	166	15	,	,	PUNCT
cana-3670	166	16	(	(	PUNCT
cana-3670	166	17	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	166	18	)	)	PUNCT
cana-3670	166	19	)	)	PUNCT
cana-3670	166	20	)	)	PUNCT
cana-3670	166	21	)	)	PUNCT
cana-3670	166	22	)	)	PUNCT
cana-3670	167	1	,	,	PUNCT
cana-3670	167	2	𝐺	𝐺	PROPN
cana-3670	167	3	=	=	SYM
cana-3670	167	4	ℎ𝑓	ℎ𝑓	PROPN
cana-3670	167	5	(	(	PUNCT
cana-3670	167	6	2((𝑡𝑛:𝑠	2((𝑡𝑛:𝑠	NUM
cana-3670	167	7	)	)	PUNCT
cana-3670	167	8	2+(𝑡𝑛:𝑠	2+(𝑡𝑛:𝑠	NUM
cana-3670	167	9	)	)	PUNCT
cana-3670	167	10	(	(	PUNCT
cana-3670	167	11	(	(	PUNCT
cana-3670	167	12	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	X
cana-3670	167	13	)	)	PUNCT
cana-3670	167	14	2	2	NUM
cana-3670	167	15	)	)	PUNCT
cana-3670	167	16	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	167	17	)	)	PUNCT
cana-3670	167	18	)	)	PUNCT
cana-3670	167	19	,	,	PUNCT
cana-3670	167	20	2((𝑢𝑛:𝑠	2((𝑢𝑛:𝑠	NUM
cana-3670	167	21	)	)	PUNCT
cana-3670	167	22	2+(𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	2+(𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	NUM
cana-3670	167	23	)	)	PUNCT
cana-3670	167	24	,	,	PUNCT
cana-3670	167	25	(	(	PUNCT
cana-3670	167	26	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	167	27	)	)	PUNCT
cana-3670	167	28	)	)	PUNCT
cana-3670	167	29	)	)	PUNCT
cana-3670	168	1	+	+	X
cana-3670	168	2	(	(	PUNCT
cana-3670	168	3	(	(	PUNCT
cana-3670	168	4	𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	NOUN
cana-3670	168	5	)	)	PUNCT
cana-3670	168	6	,	,	PUNCT
cana-3670	168	7	(	(	PUNCT
cana-3670	168	8	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	168	9	)	)	PUNCT
cana-3670	168	10	)	)	PUNCT
cana-3670	168	11	)	)	PUNCT
cana-3670	168	12	2	2	X
cana-3670	168	13	)	)	PUNCT
cana-3670	168	14	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠	NUM
cana-3670	168	15	)	)	PUNCT
cana-3670	168	16	,	,	PUNCT
cana-3670	168	17	(	(	PUNCT
cana-3670	168	18	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	168	19	)	)	PUNCT
cana-3670	168	20	)	)	PUNCT
cana-3670	168	21	)	)	PUNCT
cana-3670	168	22	)	)	PUNCT
cana-3670	168	23	)	)	PUNCT
cana-3670	168	24	.	.	PUNCT
cana-3670	169	1	3.2.3.5	3.2.3.5	NUM
cana-3670	169	2	.	.	PUNCT
cana-3670	169	3	modified	modified	PROPN
cana-3670	169	4	euler	euler	PROPN
cana-3670	169	5	’s	’s	PART
cana-3670	169	6	method	method	NOUN
cana-3670	169	7	based	base	VERB
cana-3670	169	8	on	on	ADP
cana-3670	169	9	centroidal	centroidal	PROPN
cana-3670	169	10	mean	mean	VERB
cana-3670	169	11	the	the	DET
cana-3670	169	12	modified	modify	VERB
cana-3670	169	13	euler	euler	NOUN
cana-3670	169	14	's	's	PART
cana-3670	169	15	method	method	NOUN
cana-3670	169	16	's	's	PART
cana-3670	169	17	expected	expect	VERB
cana-3670	169	18	centroidal	centroidal	NOUN
cana-3670	169	19	mean	mean	NOUN
cana-3670	169	20	is	be	AUX
cana-3670	169	21	precisely	precisely	ADV
cana-3670	169	22	defined	define	VERB
cana-3670	169	23	by	by	ADP
cana-3670	169	24	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	169	25	=	=	PUNCT
cana-3670	169	26	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	169	27	:	:	PUNCT
cana-3670	169	28	𝑠	𝑠	NUM
cana-3670	169	29	)	)	PUNCT
cana-3670	169	30	+	+	CCONJ
cana-3670	169	31	𝐹(𝑢(𝑡𝑛	𝐹(𝑢(𝑡𝑛	NUM
cana-3670	169	32	:	:	PUNCT
cana-3670	169	33	𝑠	𝑠	NUM
cana-3670	169	34	)	)	PUNCT
cana-3670	169	35	)	)	PUNCT
cana-3670	169	36	,	,	PUNCT
cana-3670	169	37	𝑢𝑛+1	𝑢𝑛+1	X
cana-3670	169	38	=	=	SYM
cana-3670	169	39	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	169	40	:	:	PUNCT
cana-3670	169	41	𝑠	𝑠	X
cana-3670	169	42	)	)	PUNCT
cana-3670	169	43	+	+	CCONJ
cana-3670	169	44	𝐺(𝑢(𝑡𝑛	𝐺(𝑢(𝑡𝑛	X
cana-3670	169	45	:	:	PUNCT
cana-3670	169	46	𝑠	𝑠	X
cana-3670	169	47	)	)	PUNCT
cana-3670	169	48	)	)	PUNCT
cana-3670	169	49	.	.	PUNCT
cana-3670	170	1	where	where	SCONJ
cana-3670	170	2	𝐹	𝐹	PROPN
cana-3670	170	3	=	=	SYM
cana-3670	170	4	ℎ𝑓	ℎ𝑓	X
cana-3670	170	5	(	(	PUNCT
cana-3670	170	6	(	(	PUNCT
cana-3670	170	7	𝑡𝑛	𝑡𝑛	X
cana-3670	170	8	:	:	PUNCT
cana-3670	170	9	𝑠	𝑠	X
cana-3670	170	10	)	)	PUNCT
cana-3670	170	11	+	+	CCONJ
cana-3670	170	12	ℎ	ℎ	X
cana-3670	170	13	2	2	NUM
cana-3670	170	14	,	,	PUNCT
cana-3670	170	15	(	(	PUNCT
cana-3670	170	16	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	170	17	:	:	PUNCT
cana-3670	170	18	𝑠	𝑠	X
cana-3670	170	19	)	)	PUNCT
cana-3670	170	20	+	+	CCONJ
cana-3670	170	21	ℎ	ℎ	X
cana-3670	170	22	2	2	NUM
cana-3670	170	23	𝑓	𝑓	PRON
cana-3670	170	24	(	(	PUNCT
cana-3670	170	25	2((𝑡𝑛:𝑠	2((𝑡𝑛:𝑠	NUM
cana-3670	170	26	)	)	PUNCT
cana-3670	170	27	2+(𝑡𝑛:𝑠	2+(𝑡𝑛:𝑠	NUM
cana-3670	170	28	)	)	PUNCT
cana-3670	170	29	(	(	PUNCT
cana-3670	170	30	(	(	PUNCT
cana-3670	170	31	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	X
cana-3670	170	32	)	)	PUNCT
cana-3670	170	33	2	2	NUM
cana-3670	170	34	)	)	PUNCT
cana-3670	170	35	3	3	NUM
cana-3670	170	36	(	(	PUNCT
cana-3670	170	37	(	(	PUNCT
cana-3670	170	38	𝑡𝑛	𝑡𝑛	X
cana-3670	170	39	:	:	PUNCT
cana-3670	170	40	𝑠)+((𝑡𝑛:𝑠)+ℎ	𝑠)+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	170	41	)	)	PUNCT
cana-3670	170	42	)	)	PUNCT
cana-3670	170	43	,	,	PUNCT
cana-3670	170	44	2((𝑢𝑛:𝑠	2((𝑢𝑛:𝑠	NUM
cana-3670	170	45	)	)	PUNCT
cana-3670	170	46	2	2	NUM
cana-3670	171	1	+	+	NOUN
cana-3670	171	2	(	(	PUNCT
cana-3670	171	3	𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	ADJ
cana-3670	171	4	ℎ	ℎ	PROPN
cana-3670	171	5	2	2	NUM
cana-3670	171	6	,	,	PUNCT
cana-3670	171	7	(	(	PUNCT
cana-3670	171	8	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	PROPN
cana-3670	171	9	ℎ	ℎ	PROPN
cana-3670	171	10	2	2	NUM
cana-3670	171	11	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	171	12	)	)	PUNCT
cana-3670	171	13	,	,	PUNCT
cana-3670	171	14	(	(	PUNCT
cana-3670	171	15	𝑢𝑛:𝑠))))+((𝑢𝑛:𝑠)+ℎ𝑓((𝑢𝑛:𝑠)+	𝑢𝑛:𝑠))))+((𝑢𝑛:𝑠)+ℎ𝑓((𝑢𝑛:𝑠)+	X
cana-3670	171	16	ℎ	ℎ	X
cana-3670	171	17	2	2	NUM
cana-3670	171	18	,	,	PUNCT
cana-3670	171	19	(	(	PUNCT
cana-3670	171	20	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	PROPN
cana-3670	171	21	ℎ	ℎ	PROPN
cana-3670	171	22	2	2	NUM
cana-3670	171	23	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	171	24	)	)	PUNCT
cana-3670	171	25	,	,	PUNCT
cana-3670	171	26	(	(	PUNCT
cana-3670	171	27	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	171	28	)	)	PUNCT
cana-3670	171	29	)	)	PUNCT
cana-3670	171	30	)	)	PUNCT
cana-3670	171	31	)	)	PUNCT
cana-3670	171	32	2	2	X
cana-3670	171	33	)	)	PUNCT
cana-3670	171	34	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠)+	NUM
cana-3670	171	35	ℎ	ℎ	SYM
cana-3670	171	36	2	2	NUM
cana-3670	171	37	,	,	PUNCT
cana-3670	171	38	(	(	PUNCT
cana-3670	171	39	𝑢𝑛:𝑠)+	𝑢𝑛:𝑠)+	PROPN
cana-3670	171	40	ℎ	ℎ	PROPN
cana-3670	171	41	2	2	NUM
cana-3670	171	42	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	171	43	)	)	PUNCT
cana-3670	171	44	,	,	PUNCT
cana-3670	171	45	(	(	PUNCT
cana-3670	171	46	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	171	47	)	)	PUNCT
cana-3670	171	48	)	)	PUNCT
cana-3670	171	49	)	)	PUNCT
cana-3670	171	50	)	)	PUNCT
cana-3670	171	51	)	)	PUNCT
cana-3670	171	52	)	)	PUNCT
cana-3670	171	53	)	)	PUNCT
cana-3670	171	54	,	,	PUNCT
cana-3670	171	55	𝐺	𝐺	PROPN
cana-3670	171	56	=	=	SYM
cana-3670	171	57	ℎ𝑓	ℎ𝑓	X
cana-3670	171	58	(	(	PUNCT
cana-3670	171	59	(	(	PUNCT
cana-3670	171	60	𝑡𝑛	𝑡𝑛	X
cana-3670	171	61	:	:	PUNCT
cana-3670	171	62	𝑠	𝑠	X
cana-3670	171	63	)	)	PUNCT
cana-3670	171	64	+	+	CCONJ
cana-3670	171	65	ℎ	ℎ	X
cana-3670	171	66	2	2	NUM
cana-3670	171	67	,	,	PUNCT
cana-3670	171	68	(	(	PUNCT
cana-3670	171	69	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	171	70	:	:	PUNCT
cana-3670	171	71	𝑠	𝑠	X
cana-3670	171	72	)	)	PUNCT
cana-3670	171	73	+	+	CCONJ
cana-3670	171	74	ℎ	ℎ	X
cana-3670	171	75	2	2	NUM
cana-3670	171	76	𝑓	𝑓	PRON
cana-3670	171	77	(	(	PUNCT
cana-3670	171	78	2((𝑡𝑛:𝑠	2((𝑡𝑛:𝑠	NUM
cana-3670	171	79	)	)	PUNCT
cana-3670	171	80	2+(𝑡𝑛:𝑠	2+(𝑡𝑛:𝑠	NUM
cana-3670	171	81	)	)	PUNCT
cana-3670	171	82	(	(	PUNCT
cana-3670	171	83	(	(	PUNCT
cana-3670	171	84	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	X
cana-3670	171	85	)	)	PUNCT
cana-3670	171	86	2	2	NUM
cana-3670	171	87	)	)	PUNCT
cana-3670	171	88	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	171	89	)	)	PUNCT
cana-3670	171	90	)	)	PUNCT
cana-3670	171	91	,	,	PUNCT
cana-3670	171	92	2((𝑢𝑛:𝑠	2((𝑢𝑛:𝑠	NUM
cana-3670	171	93	)	)	PUNCT
cana-3670	171	94	2+(𝑢𝑛:𝑠	2+(𝑢𝑛:𝑠	NUM
cana-3670	171	95	)	)	PUNCT
cana-3670	171	96	(	(	PUNCT
cana-3670	171	97	(	(	PUNCT
cana-3670	171	98	𝑢𝑛:𝑠	𝑢𝑛:𝑠	NOUN
cana-3670	171	99	)	)	PUNCT
cana-3670	171	100	+	+	NOUN
cana-3670	171	101	ℎ𝑓((𝑡𝑛:𝑠)+	ℎ𝑓((𝑡𝑛:𝑠)+	NOUN
cana-3670	171	102	ℎ	ℎ	NOUN
cana-3670	171	103	2	2	NUM
cana-3670	171	104	,	,	PUNCT
cana-3670	171	105	(	(	PUNCT
cana-3670	171	106	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	171	107	)	)	PUNCT
cana-3670	171	108	+	+	NUM
cana-3670	171	109	ℎ	ℎ	SYM
cana-3670	171	110	2	2	NUM
cana-3670	171	111	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	171	112	)	)	PUNCT
cana-3670	171	113	,	,	PUNCT
cana-3670	171	114	(	(	PUNCT
cana-3670	171	115	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	171	116	)	)	PUNCT
cana-3670	171	117	)	)	PUNCT
cana-3670	171	118	)	)	PUNCT
cana-3670	171	119	)	)	PUNCT
cana-3670	172	1	+	+	X
cana-3670	172	2	(	(	PUNCT
cana-3670	172	3	(	(	PUNCT
cana-3670	172	4	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	172	5	)	)	PUNCT
cana-3670	172	6	+	+	NOUN
cana-3670	172	7	ℎ𝑓((𝑡𝑛:𝑠)+	ℎ𝑓((𝑡𝑛:𝑠)+	NOUN
cana-3670	172	8	ℎ	ℎ	NOUN
cana-3670	172	9	2	2	NUM
cana-3670	172	10	,	,	PUNCT
cana-3670	172	11	(	(	PUNCT
cana-3670	172	12	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	172	13	)	)	PUNCT
cana-3670	172	14	+	+	NUM
cana-3670	172	15	ℎ	ℎ	SYM
cana-3670	172	16	2	2	NUM
cana-3670	172	17	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	172	18	)	)	PUNCT
cana-3670	172	19	,	,	PUNCT
cana-3670	172	20	(	(	PUNCT
cana-3670	172	21	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	172	22	)	)	PUNCT
cana-3670	172	23	)	)	PUNCT
cana-3670	172	24	)	)	PUNCT
cana-3670	172	25	)	)	PUNCT
cana-3670	172	26	2	2	NUM
cana-3670	172	27	)	)	PUNCT
cana-3670	172	28	3((𝑢𝑛:𝑠	3((𝑢𝑛:𝑠	NUM
cana-3670	172	29	)	)	PUNCT
cana-3670	173	1	+	+	ADJ
cana-3670	173	2	(	(	PUNCT
cana-3670	173	3	(	(	PUNCT
cana-3670	173	4	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	173	5	)	)	PUNCT
cana-3670	173	6	+	+	NOUN
cana-3670	173	7	ℎ𝑓((𝑡𝑛:𝑠)+	ℎ𝑓((𝑡𝑛:𝑠)+	NOUN
cana-3670	173	8	ℎ	ℎ	NOUN
cana-3670	173	9	2	2	NUM
cana-3670	173	10	,	,	PUNCT
cana-3670	173	11	(	(	PUNCT
cana-3670	173	12	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	173	13	)	)	PUNCT
cana-3670	173	14	+	+	NUM
cana-3670	173	15	ℎ	ℎ	SYM
cana-3670	173	16	2	2	NUM
cana-3670	173	17	𝑓((𝑡𝑛:𝑠	𝑓((𝑡𝑛:𝑠	ADJ
cana-3670	173	18	)	)	PUNCT
cana-3670	173	19	,	,	PUNCT
cana-3670	173	20	(	(	PUNCT
cana-3670	173	21	𝑢𝑛:𝑠	𝑢𝑛:𝑠	PROPN
cana-3670	173	22	)	)	PUNCT
cana-3670	173	23	)	)	PUNCT
cana-3670	173	24	)	)	PUNCT
cana-3670	173	25	)	)	PUNCT
cana-3670	173	26	)	)	PUNCT
cana-3670	173	27	)	)	PUNCT
cana-3670	173	28	)	)	PUNCT
cana-3670	173	29	.	.	PUNCT
cana-3670	174	1	3.2.3.6	3.2.3.6	X
cana-3670	174	2	.	.	PUNCT
cana-3670	174	3	improved	improved	PROPN
cana-3670	174	4	euler	euler	PROPN
cana-3670	174	5	’s	’s	PART
cana-3670	174	6	method	method	NOUN
cana-3670	174	7	based	base	VERB
cana-3670	174	8	on	on	ADP
cana-3670	174	9	centroidal	centroidal	PROPN
cana-3670	174	10	mean	mean	VERB
cana-3670	174	11	the	the	DET
cana-3670	174	12	improved	improve	VERB
cana-3670	174	13	euler	euler	NOUN
cana-3670	174	14	's	's	PART
cana-3670	174	15	method	method	NOUN
cana-3670	174	16	's	's	PART
cana-3670	174	17	expected	expect	VERB
cana-3670	174	18	centroidal	centroidal	NOUN
cana-3670	174	19	mean	mean	NOUN
cana-3670	174	20	is	be	AUX
cana-3670	174	21	precisely	precisely	ADV
cana-3670	174	22	defined	define	VERB
cana-3670	174	23	by	by	ADP
cana-3670	174	24	𝑢𝑛+1	𝑢𝑛+1	PROPN
cana-3670	174	25	=	=	PUNCT
cana-3670	174	26	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	174	27	:	:	PUNCT
cana-3670	174	28	𝑠	𝑠	NUM
cana-3670	174	29	)	)	PUNCT
cana-3670	174	30	+	+	CCONJ
cana-3670	174	31	𝐹(𝑢(𝑡𝑛	𝐹(𝑢(𝑡𝑛	NUM
cana-3670	174	32	:	:	PUNCT
cana-3670	174	33	𝑠	𝑠	NUM
cana-3670	174	34	)	)	PUNCT
cana-3670	174	35	)	)	PUNCT
cana-3670	174	36	,	,	PUNCT
cana-3670	174	37	𝑢𝑛+1	𝑢𝑛+1	X
cana-3670	174	38	=	=	SYM
cana-3670	174	39	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	174	40	:	:	PUNCT
cana-3670	174	41	𝑠	𝑠	X
cana-3670	174	42	)	)	PUNCT
cana-3670	174	43	+	+	CCONJ
cana-3670	174	44	𝐺(𝑢(𝑡𝑛	𝐺(𝑢(𝑡𝑛	X
cana-3670	174	45	:	:	PUNCT
cana-3670	174	46	𝑠	𝑠	X
cana-3670	174	47	)	)	PUNCT
cana-3670	174	48	)	)	PUNCT
cana-3670	174	49	.	.	PUNCT
cana-3670	175	1	where	where	SCONJ
cana-3670	175	2	communications	communication	NOUN
cana-3670	175	3	on	on	ADP
cana-3670	175	4	applied	apply	VERB
cana-3670	175	5	nonlinear	nonlinear	ADJ
cana-3670	175	6	analysis	analysis	NOUN
cana-3670	175	7	issn	issn	NOUN
cana-3670	175	8	:	:	PUNCT
cana-3670	175	9	1074	1074	NUM
cana-3670	175	10	-	-	PUNCT
cana-3670	175	11	133x	133x	NUM
cana-3670	175	12	vol	vol	NOUN
cana-3670	175	13	32	32	NUM
cana-3670	175	14	no	no	NOUN
cana-3670	175	15	.	.	PUNCT
cana-3670	176	1	8s	8s	PROPN
cana-3670	176	2	(	(	PUNCT
cana-3670	176	3	2025	2025	NUM
cana-3670	176	4	)	)	PUNCT
cana-3670	176	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	176	6	281	281	NUM
cana-3670	176	7	f	f	NOUN
cana-3670	176	8	=	=	NOUN
cana-3670	176	9	h	h	NOUN
cana-3670	176	10	2	2	NUM
cana-3670	176	11	[	[	PUNCT
cana-3670	176	12	f	f	X
cana-3670	176	13	(	(	PUNCT
cana-3670	176	14	2((tn	2((tn	NUM
cana-3670	176	15	:	:	PUNCT
cana-3670	176	16	s	s	X
cana-3670	176	17	)	)	PUNCT
cana-3670	176	18	2+(tn	2+(tn	NOUN
cana-3670	176	19	:	:	PUNCT
cana-3670	176	20	s)((tn	s)((tn	NOUN
cana-3670	176	21	:	:	PUNCT
cana-3670	176	22	s)+h)+((tn	s)+h)+((tn	ADJ
cana-3670	176	23	:	:	PUNCT
cana-3670	176	24	s)+h	s)+h	NOUN
cana-3670	176	25	)	)	PUNCT
cana-3670	176	26	2	2	NUM
cana-3670	176	27	)	)	PUNCT
cana-3670	176	28	3((tn	3((tn	NUM
cana-3670	176	29	:	:	PUNCT
cana-3670	176	30	s)+((tn	s)+((tn	PROPN
cana-3670	176	31	:	:	PUNCT
cana-3670	176	32	s)+h	s)+h	NUM
cana-3670	176	33	)	)	PUNCT
cana-3670	176	34	)	)	PUNCT
cana-3670	176	35	,	,	PUNCT
cana-3670	176	36	2((un	2((un	NUM
cana-3670	176	37	:	:	PUNCT
cana-3670	176	38	s	s	X
cana-3670	176	39	)	)	PUNCT
cana-3670	176	40	2	2	NUM
cana-3670	177	1	+	+	NOUN
cana-3670	177	2	(	(	PUNCT
cana-3670	177	3	un	un	PROPN
cana-3670	177	4	:	:	PUNCT
cana-3670	177	5	s)((un	s)((un	NOUN
cana-3670	177	6	:	:	PUNCT
cana-3670	177	7	s)+	s)+	ADJ
cana-3670	177	8	h	h	NOUN
cana-3670	177	9	2	2	NUM
cana-3670	177	10	(	(	PUNCT
cana-3670	177	11	f((tn	f((tn	NOUN
cana-3670	177	12	:	:	PUNCT
cana-3670	177	13	s),(un	s),(un	PROPN
cana-3670	177	14	:	:	PUNCT
cana-3670	177	15	s))+f((tn	s))+f((tn	PROPN
cana-3670	177	16	:	:	PUNCT
cana-3670	177	17	s)+h,(un	s)+h,(un	NOUN
cana-3670	177	18	:	:	PUNCT
cana-3670	177	19	s)+hf((tn	s)+hf((tn	ADJ
cana-3670	177	20	:	:	PUNCT
cana-3670	177	21	s),(un	s),(un	PROPN
cana-3670	177	22	:	:	PUNCT
cana-3670	177	23	s)))))+((un	s)))))+((un	NUM
cana-3670	177	24	:	:	PUNCT
cana-3670	177	25	s)+	s)+	ADJ
cana-3670	177	26	h	h	NOUN
cana-3670	177	27	2	2	NUM
cana-3670	177	28	(	(	PUNCT
cana-3670	177	29	f((tn	f((tn	NOUN
cana-3670	177	30	:	:	PUNCT
cana-3670	177	31	s),(un	s),(un	PROPN
cana-3670	177	32	:	:	PUNCT
cana-3670	177	33	s))+f((tn	s))+f((tn	PROPN
cana-3670	177	34	:	:	PUNCT
cana-3670	177	35	s)+h,(un	s)+h,(un	NOUN
cana-3670	177	36	:	:	PUNCT
cana-3670	177	37	s)+hf((tn	s)+hf((tn	ADJ
cana-3670	177	38	:	:	PUNCT
cana-3670	177	39	s),(un	s),(un	PROPN
cana-3670	177	40	:	:	PUNCT
cana-3670	177	41	s	s	X
cana-3670	177	42	)	)	PUNCT
cana-3670	177	43	)	)	PUNCT
cana-3670	177	44	)	)	PUNCT
cana-3670	177	45	)	)	PUNCT
cana-3670	177	46	)	)	PUNCT
cana-3670	177	47	2	2	X
cana-3670	177	48	)	)	PUNCT
cana-3670	177	49	3((un	3((un	NUM
cana-3670	177	50	:	:	PUNCT
cana-3670	177	51	s)+((un	s)+((un	NUM
cana-3670	177	52	:	:	PUNCT
cana-3670	177	53	s)+	s)+	ADJ
cana-3670	177	54	h	h	NOUN
cana-3670	177	55	2	2	NUM
cana-3670	177	56	(	(	PUNCT
cana-3670	177	57	f((tn	f((tn	NOUN
cana-3670	177	58	:	:	PUNCT
cana-3670	177	59	s),(un	s),(un	PROPN
cana-3670	177	60	:	:	PUNCT
cana-3670	177	61	s))+f((tn	s))+f((tn	PROPN
cana-3670	177	62	:	:	PUNCT
cana-3670	177	63	s)+h,(un	s)+h,(un	NOUN
cana-3670	177	64	:	:	PUNCT
cana-3670	177	65	s)+hf((tn	s)+hf((tn	ADJ
cana-3670	177	66	:	:	PUNCT
cana-3670	177	67	s),(un	s),(un	PROPN
cana-3670	177	68	:	:	PUNCT
cana-3670	177	69	s	s	X
cana-3670	177	70	)	)	PUNCT
cana-3670	177	71	)	)	PUNCT
cana-3670	177	72	)	)	PUNCT
cana-3670	177	73	)	)	PUNCT
cana-3670	177	74	)	)	PUNCT
cana-3670	177	75	)	)	PUNCT
cana-3670	177	76	)	)	PUNCT
cana-3670	178	1	+	+	CCONJ
cana-3670	178	2	f	f	X
cana-3670	178	3	(	(	PUNCT
cana-3670	178	4	(	(	PUNCT
cana-3670	178	5	tn	tn	NOUN
cana-3670	178	6	:	:	PUNCT
cana-3670	178	7	s	s	X
cana-3670	178	8	)	)	PUNCT
cana-3670	178	9	+	+	CCONJ
cana-3670	178	10	h	h	NOUN
cana-3670	178	11	,	,	PUNCT
cana-3670	178	12	(	(	PUNCT
cana-3670	178	13	un	un	PROPN
cana-3670	178	14	:	:	SYM
cana-3670	178	15	s	s	X
cana-3670	178	16	)	)	PUNCT
cana-3670	179	1	+	+	NUM
cana-3670	179	2	h	h	NOUN
cana-3670	179	3	(	(	PUNCT
cana-3670	179	4	2((tn	2((tn	NUM
cana-3670	179	5	:	:	PUNCT
cana-3670	179	6	s	s	X
cana-3670	179	7	)	)	PUNCT
cana-3670	179	8	2+(tn	2+(tn	NOUN
cana-3670	179	9	:	:	PUNCT
cana-3670	179	10	s)((tn	s)((tn	NOUN
cana-3670	179	11	:	:	PUNCT
cana-3670	179	12	s)+h)+((tn	s)+h)+((tn	ADJ
cana-3670	179	13	:	:	PUNCT
cana-3670	179	14	s)+h	s)+h	NOUN
cana-3670	179	15	)	)	PUNCT
cana-3670	179	16	2	2	NUM
cana-3670	179	17	)	)	PUNCT
cana-3670	179	18	3((tn	3((tn	NUM
cana-3670	179	19	:	:	PUNCT
cana-3670	179	20	s)+((tn	s)+((tn	PROPN
cana-3670	179	21	:	:	PUNCT
cana-3670	179	22	s)+h	s)+h	NUM
cana-3670	179	23	)	)	PUNCT
cana-3670	179	24	)	)	PUNCT
cana-3670	179	25	,	,	PUNCT
cana-3670	179	26	2((un	2((un	NUM
cana-3670	179	27	:	:	PUNCT
cana-3670	179	28	s	s	X
cana-3670	179	29	)	)	PUNCT
cana-3670	179	30	2	2	NUM
cana-3670	180	1	+	+	NOUN
cana-3670	180	2	(	(	PUNCT
cana-3670	180	3	un	un	PROPN
cana-3670	180	4	:	:	PUNCT
cana-3670	180	5	s)((un	s)((un	NOUN
cana-3670	180	6	:	:	PUNCT
cana-3670	180	7	s)+	s)+	ADJ
cana-3670	180	8	h	h	NOUN
cana-3670	180	9	2	2	NUM
cana-3670	180	10	(	(	PUNCT
cana-3670	180	11	f((tn	f((tn	NOUN
cana-3670	180	12	:	:	PUNCT
cana-3670	180	13	s),(un	s),(un	PROPN
cana-3670	180	14	:	:	PUNCT
cana-3670	180	15	s))+f((tn	s))+f((tn	PROPN
cana-3670	180	16	:	:	PUNCT
cana-3670	180	17	s)+h,(un	s)+h,(un	NOUN
cana-3670	180	18	:	:	PUNCT
cana-3670	180	19	s)+hf((tn	s)+hf((tn	ADJ
cana-3670	180	20	:	:	PUNCT
cana-3670	180	21	s),(un	s),(un	PROPN
cana-3670	180	22	:	:	PUNCT
cana-3670	180	23	s)))))+((un	s)))))+((un	NUM
cana-3670	180	24	:	:	PUNCT
cana-3670	180	25	s)+	s)+	ADJ
cana-3670	180	26	h	h	NOUN
cana-3670	180	27	2	2	NUM
cana-3670	180	28	(	(	PUNCT
cana-3670	180	29	f((tn	f((tn	NOUN
cana-3670	180	30	:	:	PUNCT
cana-3670	180	31	s),(un	s),(un	PROPN
cana-3670	180	32	:	:	PUNCT
cana-3670	180	33	s))+f((tn	s))+f((tn	PROPN
cana-3670	180	34	:	:	PUNCT
cana-3670	180	35	s)+h,(un	s)+h,(un	NOUN
cana-3670	180	36	:	:	PUNCT
cana-3670	180	37	s)+hf((tn	s)+hf((tn	ADJ
cana-3670	180	38	:	:	PUNCT
cana-3670	180	39	s),(un	s),(un	PROPN
cana-3670	180	40	:	:	PUNCT
cana-3670	180	41	s	s	X
cana-3670	180	42	)	)	PUNCT
cana-3670	180	43	)	)	PUNCT
cana-3670	180	44	)	)	PUNCT
cana-3670	180	45	)	)	PUNCT
cana-3670	180	46	)	)	PUNCT
cana-3670	180	47	2	2	X
cana-3670	180	48	)	)	PUNCT
cana-3670	180	49	3((un	3((un	NUM
cana-3670	180	50	:	:	PUNCT
cana-3670	180	51	s)+((un	s)+((un	NUM
cana-3670	180	52	:	:	PUNCT
cana-3670	180	53	s)+	s)+	ADJ
cana-3670	180	54	h	h	NOUN
cana-3670	180	55	2	2	NUM
cana-3670	180	56	(	(	PUNCT
cana-3670	180	57	f((tn	f((tn	NOUN
cana-3670	180	58	:	:	PUNCT
cana-3670	180	59	s),(un	s),(un	PROPN
cana-3670	180	60	:	:	PUNCT
cana-3670	180	61	s))+f((tn	s))+f((tn	PROPN
cana-3670	180	62	:	:	PUNCT
cana-3670	180	63	s)+h,(un	s)+h,(un	NOUN
cana-3670	180	64	:	:	PUNCT
cana-3670	180	65	s)+hf((tn	s)+hf((tn	ADJ
cana-3670	180	66	:	:	PUNCT
cana-3670	180	67	s),(un	s),(un	PROPN
cana-3670	180	68	:	:	PUNCT
cana-3670	180	69	s	s	X
cana-3670	180	70	)	)	PUNCT
cana-3670	180	71	)	)	PUNCT
cana-3670	180	72	)	)	PUNCT
cana-3670	180	73	)	)	PUNCT
cana-3670	180	74	)	)	PUNCT
cana-3670	180	75	)	)	PUNCT
cana-3670	180	76	)	)	PUNCT
cana-3670	180	77	)	)	PUNCT
cana-3670	180	78	]	]	PUNCT
cana-3670	180	79	.	.	PUNCT
cana-3670	181	1	𝐺	𝐺	NOUN
cana-3670	182	1	=	=	PUNCT
cana-3670	182	2	ℎ	ℎ	X
cana-3670	182	3	2	2	NUM
cana-3670	182	4	[	[	PUNCT
cana-3670	182	5	𝑓	𝑓	X
cana-3670	182	6	(	(	PUNCT
cana-3670	182	7	2((𝑡𝑛:𝑠	2((𝑡𝑛:𝑠	NUM
cana-3670	182	8	)	)	PUNCT
cana-3670	182	9	2+(𝑡𝑛:𝑠)((𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	2+(𝑡𝑛:𝑠)((𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	NOUN
cana-3670	182	10	)	)	PUNCT
cana-3670	182	11	2	2	NUM
cana-3670	182	12	)	)	PUNCT
cana-3670	182	13	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	182	14	)	)	PUNCT
cana-3670	182	15	)	)	PUNCT
cana-3670	182	16	,	,	PUNCT
cana-3670	182	17	2((𝑢𝑛:𝑠	2((𝑢𝑛:𝑠	NUM
cana-3670	182	18	)	)	PUNCT
cana-3670	182	19	2+(𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+	2+(𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+	NUM
cana-3670	182	20	ℎ	ℎ	SYM
cana-3670	182	21	2	2	NUM
cana-3670	182	22	(	(	PUNCT
cana-3670	182	23	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠)))))+((𝑢𝑛:𝑠)+	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠)))))+((𝑢𝑛:𝑠)+	ADJ
cana-3670	182	24	ℎ	ℎ	X
cana-3670	182	25	2	2	NUM
cana-3670	182	26	(	(	PUNCT
cana-3670	182	27	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	182	28	)	)	PUNCT
cana-3670	182	29	)	)	PUNCT
cana-3670	182	30	)	)	PUNCT
cana-3670	182	31	)	)	PUNCT
cana-3670	182	32	)	)	PUNCT
cana-3670	182	33	2	2	X
cana-3670	182	34	)	)	PUNCT
cana-3670	182	35	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	NUM
cana-3670	182	36	ℎ	ℎ	ADP
cana-3670	182	37	2	2	NUM
cana-3670	182	38	(	(	PUNCT
cana-3670	182	39	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	182	40	)	)	PUNCT
cana-3670	182	41	)	)	PUNCT
cana-3670	182	42	)	)	PUNCT
cana-3670	182	43	)	)	PUNCT
cana-3670	182	44	)	)	PUNCT
cana-3670	182	45	)	)	PUNCT
cana-3670	182	46	)	)	PUNCT
cana-3670	183	1	+	+	CCONJ
cana-3670	183	2	𝑓	𝑓	X
cana-3670	183	3	(	(	PUNCT
cana-3670	183	4	(	(	PUNCT
cana-3670	183	5	𝑡𝑛	𝑡𝑛	NUM
cana-3670	183	6	:	:	PUNCT
cana-3670	183	7	𝑠)+	𝑠)+	PROPN
cana-3670	183	8	ℎ	ℎ	PROPN
cana-3670	183	9	,	,	PUNCT
cana-3670	183	10	(	(	PUNCT
cana-3670	183	11	𝑢𝑛	𝑢𝑛	NOUN
cana-3670	183	12	:	:	SYM
cana-3670	183	13	𝑠	𝑠	NUM
cana-3670	183	14	)	)	PUNCT
cana-3670	183	15	+	+	CCONJ
cana-3670	183	16	ℎ	ℎ	PROPN
cana-3670	183	17	(	(	PUNCT
cana-3670	183	18	2((𝑡𝑛:𝑠	2((𝑡𝑛:𝑠	NUM
cana-3670	183	19	)	)	PUNCT
cana-3670	183	20	2+(𝑡𝑛:𝑠)((𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	2+(𝑡𝑛:𝑠)((𝑡𝑛:𝑠)+ℎ)+((𝑡𝑛:𝑠)+ℎ	NOUN
cana-3670	183	21	)	)	PUNCT
cana-3670	183	22	2	2	NUM
cana-3670	183	23	)	)	PUNCT
cana-3670	183	24	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	3((𝑡𝑛:𝑠)+((𝑡𝑛:𝑠)+ℎ	NUM
cana-3670	183	25	)	)	PUNCT
cana-3670	183	26	)	)	PUNCT
cana-3670	183	27	,	,	PUNCT
cana-3670	183	28	2((𝑢𝑛:𝑠	2((𝑢𝑛:𝑠	NUM
cana-3670	183	29	)	)	PUNCT
cana-3670	183	30	2+(𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+	2+(𝑢𝑛:𝑠)((𝑢𝑛:𝑠)+	NUM
cana-3670	183	31	ℎ	ℎ	SYM
cana-3670	183	32	2	2	NUM
cana-3670	183	33	(	(	PUNCT
cana-3670	183	34	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠)))))+((𝑢𝑛:𝑠)+	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠)))))+((𝑢𝑛:𝑠)+	ADJ
cana-3670	183	35	ℎ	ℎ	X
cana-3670	183	36	2	2	NUM
cana-3670	183	37	(	(	PUNCT
cana-3670	183	38	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	183	39	)	)	PUNCT
cana-3670	183	40	)	)	PUNCT
cana-3670	183	41	)	)	PUNCT
cana-3670	183	42	)	)	PUNCT
cana-3670	183	43	)	)	PUNCT
cana-3670	183	44	2	2	X
cana-3670	183	45	)	)	PUNCT
cana-3670	183	46	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	3((𝑢𝑛:𝑠)+((𝑢𝑛:𝑠)+	NUM
cana-3670	183	47	ℎ	ℎ	ADP
cana-3670	183	48	2	2	NUM
cana-3670	183	49	(	(	PUNCT
cana-3670	183	50	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠))+𝑓((𝑡𝑛:𝑠)+ℎ,(𝑢𝑛:𝑠)+ℎ𝑓((𝑡𝑛:𝑠),(𝑢𝑛:𝑠	ADJ
cana-3670	183	51	)	)	PUNCT
cana-3670	183	52	)	)	PUNCT
cana-3670	183	53	)	)	PUNCT
cana-3670	183	54	)	)	PUNCT
cana-3670	183	55	)	)	PUNCT
cana-3670	183	56	)	)	PUNCT
cana-3670	183	57	)	)	PUNCT
cana-3670	183	58	)	)	PUNCT
cana-3670	183	59	]	]	PUNCT
cana-3670	183	60	.	.	PUNCT
cana-3670	184	1	4	4	X
cana-3670	184	2	.	.	NUM
cana-3670	184	3	findings	finding	NOUN
cana-3670	184	4	and	and	CCONJ
cana-3670	184	5	discussions	discussion	NOUN
cana-3670	184	6	with	with	ADP
cana-3670	184	7	numerical	numerical	ADJ
cana-3670	184	8	illustrations	illustration	NOUN
cana-3670	184	9	think	think	VERB
cana-3670	184	10	about	about	ADP
cana-3670	184	11	the	the	DET
cana-3670	184	12	differential	differential	ADJ
cana-3670	184	13	equation	equation	NOUN
cana-3670	184	14	whose	whose	DET
cana-3670	184	15	fuzzy	fuzzy	ADJ
cana-3670	184	16	initial	initial	ADJ
cana-3670	184	17	value	value	NOUN
cana-3670	184	18	problem	problem	NOUN
cana-3670	184	19	is	be	AUX
cana-3670	184	20	there	there	ADV
cana-3670	184	21	.	.	PUNCT
cana-3670	185	1	𝑢′(𝑡	𝑢′(𝑡	X
cana-3670	185	2	)	)	PUNCT
cana-3670	185	3	=	=	SYM
cana-3670	185	4	−𝑡3𝑢	−𝑡3𝑢	PROPN
cana-3670	185	5	,	,	PUNCT
cana-3670	185	6	𝑢(0	𝑢(0	PROPN
cana-3670	185	7	)	)	PUNCT
cana-3670	185	8	=	=	PUNCT
cana-3670	185	9	(	(	PUNCT
cana-3670	185	10	0.75	0.75	NUM
cana-3670	185	11	+	+	NUM
cana-3670	185	12	0.25𝑠,1.125−	0.25𝑠,1.125−	NOUN
cana-3670	185	13	0.125𝑠	0.125𝑠	NOUN
cana-3670	185	14	)	)	PUNCT
cana-3670	185	15	𝑊ℎ𝑒𝑟𝑒	𝑊ℎ𝑒𝑟𝑒	PROPN
cana-3670	185	16	0	0	NUM
cana-3670	185	17	≤	≤	NUM
cana-3670	186	1	𝑠	𝑠	X
cana-3670	186	2	≤	≤	NUM
cana-3670	186	3	1	1	NUM
cana-3670	186	4	.	.	PUNCT
cana-3670	187	1	solution	solution	NOUN
cana-3670	187	2	:	:	PUNCT
cana-3670	187	3	the	the	DET
cana-3670	187	4	exact	exact	ADJ
cana-3670	187	5	solution	solution	NOUN
cana-3670	187	6	is	be	AUX
cana-3670	187	7	provided	provide	VERB
cana-3670	187	8	by	by	ADP
cana-3670	187	9	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	187	10	:	:	PUNCT
cana-3670	187	11	𝑠	𝑠	NUM
cana-3670	187	12	)	)	PUNCT
cana-3670	187	13	=	=	SYM
cana-3670	187	14	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	187	15	:	:	PUNCT
cana-3670	187	16	𝑠)𝑒	𝑠)𝑒	NOUN
cana-3670	187	17	−	−	PROPN
cana-3670	187	18	𝑡4	𝑡4	PROPN
cana-3670	187	19	4	4	NUM
cana-3670	187	20	and	and	CCONJ
cana-3670	187	21	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	187	22	:	:	PUNCT
cana-3670	187	23	𝑠	𝑠	NUM
cana-3670	187	24	)	)	PUNCT
cana-3670	187	25	=	=	SYM
cana-3670	187	26	𝑢(𝑡𝑛	𝑢(𝑡𝑛	NUM
cana-3670	187	27	:	:	PUNCT
cana-3670	187	28	𝑠)𝑒	𝑠)𝑒	NOUN
cana-3670	187	29	−	−	PROPN
cana-3670	187	30	𝑡4	𝑡4	PROPN
cana-3670	187	31	4	4	NUM
cana-3670	187	32	,	,	PUNCT
cana-3670	187	33	after	after	ADP
cana-3670	187	34	that	that	PRON
cana-3670	187	35	,	,	PUNCT
cana-3670	187	36	the	the	DET
cana-3670	187	37	solutions	solution	NOUN
cana-3670	187	38	at	at	ADP
cana-3670	187	39	1	1	NUM
cana-3670	187	40	=	=	NOUN
cana-3670	187	41	t	t	NOUN
cana-3670	187	42	,	,	PUNCT
cana-3670	187	43	𝑢(1	𝑢(1	NOUN
cana-3670	187	44	:	:	PUNCT
cana-3670	187	45	𝑠	𝑠	X
cana-3670	187	46	)	)	PUNCT
cana-3670	187	47	=	=	PUNCT
cana-3670	188	1	[	[	X
cana-3670	188	2	(	(	PUNCT
cana-3670	188	3	0.75	0.75	NUM
cana-3670	188	4	+	+	NUM
cana-3670	188	5	0.25𝑠)𝑒−0.25	0.25𝑠)𝑒−0.25	NOUN
cana-3670	188	6	,	,	PUNCT
cana-3670	188	7	(	(	PUNCT
cana-3670	188	8	1.125−	1.125−	NUM
cana-3670	188	9	0.125𝑠)𝑒−0.25],0	0.125𝑠)𝑒−0.25],0	NUM
cana-3670	188	10	≤	≤	NUM
cana-3670	188	11	𝑠	𝑠	X
cana-3670	188	12	≤	≤	NUM
cana-3670	188	13	1	1	NUM
cana-3670	188	14	.	.	PUNCT
cana-3670	189	1	(	(	PUNCT
cana-3670	189	2	a	a	X
cana-3670	189	3	)	)	PUNCT
cana-3670	189	4	enhanced	enhance	VERB
cana-3670	189	5	euler	euler	PROPN
cana-3670	189	6	’s	’s	PART
cana-3670	189	7	methods	method	NOUN
cana-3670	189	8	based	base	VERB
cana-3670	189	9	on	on	ADP
cana-3670	189	10	contra	contra	PROPN
cana-3670	189	11	harmonic	harmonic	PROPN
cana-3670	189	12	mean	mean	NOUN
cana-3670	189	13	and	and	CCONJ
cana-3670	189	14	centroidal	centroidal	ADJ
cana-3670	189	15	mean	mean	NOUN
cana-3670	189	16	:	:	PUNCT
cana-3670	189	17	euler	euler	PROPN
cana-3670	189	18	's	's	PART
cana-3670	189	19	approach	approach	NOUN
cana-3670	189	20	provides	provide	VERB
cana-3670	189	21	a	a	DET
cana-3670	189	22	precise	precise	ADJ
cana-3670	189	23	and	and	CCONJ
cana-3670	189	24	fairly	fairly	ADV
cana-3670	189	25	accurate	accurate	ADJ
cana-3670	189	26	result	result	NOUN
cana-3670	189	27	.	.	PUNCT
cana-3670	190	1	below	below	ADV
cana-3670	190	2	is	be	AUX
cana-3670	190	3	the	the	DET
cana-3670	190	4	contra	contra	PROPN
cana-3670	190	5	harmonic	harmonic	PROPN
cana-3670	190	6	mean	mean	NOUN
cana-3670	190	7	and	and	CCONJ
cana-3670	190	8	centroidal	centroidal	ADJ
cana-3670	190	9	mean	mean	NOUN
cana-3670	190	10	of	of	ADP
cana-3670	190	11	the	the	DET
cana-3670	190	12	euler	euler	NOUN
cana-3670	190	13	’s	’s	PART
cana-3670	190	14	method	method	NOUN
cana-3670	190	15	,	,	PUNCT
cana-3670	190	16	modified	modified	PROPN
cana-3670	190	17	euler	euler	PROPN
cana-3670	190	18	’s	’s	PART
cana-3670	190	19	method	method	NOUN
cana-3670	190	20	and	and	CCONJ
cana-3670	190	21	improved	improve	VERB
cana-3670	190	22	euler	euler	NOUN
cana-3670	190	23	’s	’s	PART
cana-3670	190	24	method	method	NOUN
cana-3670	190	25	using	use	VERB
cana-3670	190	26	ℎ	ℎ	X
cana-3670	190	27	=	=	SYM
cana-3670	190	28	0.1	0.1	NUM
cana-3670	190	29	:	:	PUNCT
cana-3670	190	30	(	(	PUNCT
cana-3670	190	31	a1	a1	PROPN
cana-3670	190	32	)	)	PUNCT
cana-3670	190	33	euler	euler	NOUN
cana-3670	190	34	’s	’s	PART
cana-3670	190	35	method	method	NOUN
cana-3670	190	36	based	base	VERB
cana-3670	190	37	on	on	ADP
cana-3670	190	38	contra	contra	PROPN
cana-3670	190	39	harmonic	harmonic	PROPN
cana-3670	190	40	mean	mean	NOUN
cana-3670	190	41	and	and	CCONJ
cana-3670	190	42	centroidal	centroidal	ADJ
cana-3670	190	43	mean	mean	NOUN
cana-3670	190	44	:	:	PUNCT
cana-3670	190	45	communications	communication	NOUN
cana-3670	190	46	on	on	ADP
cana-3670	190	47	applied	apply	VERB
cana-3670	190	48	nonlinear	nonlinear	ADJ
cana-3670	190	49	analysis	analysis	NOUN
cana-3670	190	50	issn	issn	NOUN
cana-3670	190	51	:	:	PUNCT
cana-3670	190	52	1074	1074	NUM
cana-3670	190	53	-	-	PUNCT
cana-3670	190	54	133x	133x	NUM
cana-3670	190	55	vol	vol	NOUN
cana-3670	190	56	32	32	NUM
cana-3670	190	57	no	no	NOUN
cana-3670	190	58	.	.	PUNCT
cana-3670	191	1	8s	8s	PROPN
cana-3670	191	2	(	(	PUNCT
cana-3670	191	3	2025	2025	NUM
cana-3670	191	4	)	)	PUNCT
cana-3670	191	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	191	6	282	282	NUM
cana-3670	191	7	the	the	DET
cana-3670	191	8	approximate	approximate	ADJ
cana-3670	191	9	solutions	solution	NOUN
cana-3670	191	10	obtained	obtain	VERB
cana-3670	191	11	by	by	ADP
cana-3670	191	12	contra	contra	PROPN
cana-3670	191	13	harmonic	harmonic	PROPN
cana-3670	191	14	mean	mean	NOUN
cana-3670	191	15	and	and	CCONJ
cana-3670	191	16	centroidal	centroidal	ADJ
cana-3670	191	17	mean	mean	NOUN
cana-3670	191	18	of	of	ADP
cana-3670	191	19	euler	euler	PROPN
cana-3670	191	20	’s	’s	PART
cana-3670	191	21	method	method	NOUN
cana-3670	191	22	for	for	ADP
cana-3670	191	23	different	different	ADJ
cana-3670	191	24	values	value	NOUN
cana-3670	191	25	of	of	ADP
cana-3670	191	26	𝑠	𝑠	PROPN
cana-3670	191	27	∈	∈	PROPN
cana-3670	192	1	[	[	X
cana-3670	192	2	0,1	0,1	NUM
cana-3670	192	3	]	]	PUNCT
cana-3670	192	4	when	when	SCONJ
cana-3670	192	5	ℎ	ℎ	X
cana-3670	192	6	=	=	SYM
cana-3670	192	7	0.1	0.1	NUM
cana-3670	192	8	are	be	AUX
cana-3670	192	9	given	give	VERB
cana-3670	192	10	below	below	ADP
cana-3670	192	11	:	:	PUNCT
cana-3670	192	12	table	table	NOUN
cana-3670	192	13	1	1	NUM
cana-3670	192	14	:	:	PUNCT
cana-3670	192	15	numerical	numerical	ADJ
cana-3670	192	16	solutions	solution	NOUN
cana-3670	192	17	for	for	ADP
cana-3670	192	18	euler	euler	PROPN
cana-3670	192	19	’s	’s	PART
cana-3670	192	20	method	method	NOUN
cana-3670	192	21	using	use	VERB
cana-3670	192	22	contra	contra	PROPN
cana-3670	192	23	harmonic	harmonic	PROPN
cana-3670	192	24	mean	mean	NOUN
cana-3670	192	25	and	and	CCONJ
cana-3670	192	26	centroidal	centroidal	PROPN
cana-3670	192	27	mean	mean	VERB
cana-3670	192	28	with	with	ADP
cana-3670	192	29	𝒉	𝒉	NOUN
cana-3670	192	30	=	=	PUNCT
cana-3670	192	31	0.1	0.1	NUM
cana-3670	192	32	s	s	NOUN
cana-3670	192	33	t	t	NOUN
cana-3670	192	34	exact	exact	ADJ
cana-3670	192	35	solutions	solution	NOUN
cana-3670	192	36	euler	euler	PROPN
cana-3670	192	37	’s	’s	PART
cana-3670	192	38	method	method	PROPN
cana-3670	192	39	euler	euler	PROPN
cana-3670	192	40	’s	’s	PART
cana-3670	192	41	method	method	NOUN
cana-3670	192	42	using	use	VERB
cana-3670	192	43	contra	contra	PROPN
cana-3670	192	44	harmonic	harmonic	PROPN
cana-3670	192	45	mean	mean	PROPN
cana-3670	192	46	euler	euler	PROPN
cana-3670	192	47	’s	’s	PART
cana-3670	192	48	method	method	NOUN
cana-3670	192	49	using	use	VERB
cana-3670	192	50	centroidal	centroidal	ADJ
cana-3670	192	51	mean	mean	NOUN
cana-3670	192	52	low	low	ADJ
cana-3670	193	1	er	er	INTJ
cana-3670	194	1	uppe	uppe	PROPN
cana-3670	194	2	r	r	PROPN
cana-3670	194	3	lowe	lowe	PROPN
cana-3670	194	4	r	r	NOUN
cana-3670	194	5	uppe	uppe	PROPN
cana-3670	194	6	r	r	NOUN
cana-3670	194	7	lower	low	ADJ
cana-3670	194	8	upper	upper	ADJ
cana-3670	194	9	lower	lower	ADV
cana-3670	194	10	upper	upper	ADJ
cana-3670	194	11	𝟎	𝟎	NUM
cana-3670	194	12	𝟏	𝟏	NUM
cana-3670	194	13	0.584101	0.584101	NUM
cana-3670	194	14	0.876151	0.876151	NUM
cana-3670	194	15	0.548463	0.548463	NUM
cana-3670	194	16	0.822694	0.822694	NUM
cana-3670	194	17	0.581800	0.581800	NUM
cana-3670	194	18	0.872700	0.872700	NUM
cana-3670	194	19	0.583415	0.583415	NUM
cana-3670	194	20	0.875123	0.875123	NUM
cana-3670	194	21	𝟎.	𝟎.	PUNCT
cana-3670	194	22	𝟏	𝟏	NUM
cana-3670	194	23	𝟏	𝟏	NUM
cana-3670	194	24	0.603571	0.603571	NUM
cana-3670	195	1	0.866416	0.866416	NUM
cana-3670	195	2	0.566745	0.566745	NUM
cana-3670	195	3	0.813553	0.813553	NUM
cana-3670	195	4	0.601193	0.601193	NUM
cana-3670	195	5	0.863003	0.863003	NUM
cana-3670	195	6	0.602862	0.602862	NUM
cana-3670	195	7	0.865399	0.865399	NUM
cana-3670	195	8	𝟎.	𝟎.	PUNCT
cana-3670	195	9	𝟐	𝟐	NUM
cana-3670	195	10	𝟏	𝟏	NUM
cana-3670	195	11	0.623041	0.623041	NUM
cana-3670	195	12	0.856681	0.856681	NUM
cana-3670	195	13	0.585027	0.585027	NUM
cana-3670	195	14	0.804412	0.804412	NUM
cana-3670	195	15	0.620587	0.620587	NUM
cana-3670	195	16	0.853307	0.853307	NUM
cana-3670	195	17	0.622310	0.622310	NUM
cana-3670	195	18	0.855676	0.855676	NUM
cana-3670	195	19	𝟎.	𝟎.	PUNCT
cana-3670	195	20	𝟑	𝟑	NUM
cana-3670	195	21	𝟏	𝟏	NUM
cana-3670	195	22	0.642511	0.642511	NUM
cana-3670	195	23	0.846946	0.846946	NUM
cana-3670	195	24	0.603309	0.603309	NUM
cana-3670	195	25	0.795271	0.795271	NUM
cana-3670	195	26	0.639980	0.639980	NUM
cana-3670	195	27	0.843610	0.843610	NUM
cana-3670	195	28	0.641757	0.641757	NUM
cana-3670	195	29	0.845952	0.845952	NUM
cana-3670	195	30	𝟎.	𝟎.	SYM
cana-3670	195	31	𝟒	𝟒	NUM
cana-3670	195	32	𝟏	𝟏	NUM
cana-3670	195	33	0.661981	0.661981	NUM
cana-3670	195	34	0.837211	0.837211	NUM
cana-3670	195	35	0.621591	0.621591	NUM
cana-3670	195	36	0.786130	0.786130	NUM
cana-3670	195	37	0.659373	0.659373	NUM
cana-3670	195	38	0.833913	0.833913	NUM
cana-3670	195	39	0.661204	0.661204	NUM
cana-3670	195	40	0.836229	0.836229	NUM
cana-3670	195	41	𝟎.	𝟎.	X
cana-3670	195	42	𝟓	𝟓	NUM
cana-3670	195	43	𝟏	𝟏	NUM
cana-3670	195	44	0.681451	0.681451	NUM
cana-3670	195	45	0.827476	0.827476	NUM
cana-3670	195	46	0.639873	0.639873	NUM
cana-3670	195	47	0.776989	0.776989	NUM
cana-3670	195	48	0.678767	0.678767	NUM
cana-3670	195	49	0.824217	0.824217	NUM
cana-3670	195	50	0.680651	0.680651	NUM
cana-3670	195	51	0.826505	0.826505	NUM
cana-3670	195	52	𝟎.	𝟎.	PUNCT
cana-3670	195	53	𝟔	𝟔	NUM
cana-3670	195	54	𝟏	𝟏	NUM
cana-3670	195	55	0.700921	0.700921	NUM
cana-3670	195	56	0.817741	0.817741	NUM
cana-3670	195	57	0.658155	0.658155	NUM
cana-3670	195	58	0.767848	0.767848	NUM
cana-3670	195	59	0.698160	0.698160	NUM
cana-3670	195	60	0.814520	0.814520	NUM
cana-3670	195	61	0.700098	0.700098	NUM
cana-3670	195	62	0.816781	0.816781	NUM
cana-3670	195	63	𝟎.	𝟎.	PUNCT
cana-3670	195	64	𝟕	𝟕	NUM
cana-3670	195	65	𝟏	𝟏	NUM
cana-3670	195	66	0.720391	0.720391	NUM
cana-3670	195	67	0.808006	0.808006	NUM
cana-3670	195	68	0.676437	0.676437	NUM
cana-3670	195	69	0.758707	0.758707	NUM
cana-3670	195	70	0.717553	0.717553	NUM
cana-3670	195	71	0.804823	0.804823	NUM
cana-3670	195	72	0.719546	0.719546	NUM
cana-3670	195	73	0.807058	0.807058	NUM
cana-3670	195	74	𝟎.	𝟎.	X
cana-3670	195	75	𝟖	𝟖	NUM
cana-3670	195	76	𝟏	𝟏	NUM
cana-3670	195	77	0.739861	0.739861	NUM
cana-3670	195	78	0.798271	0.798271	NUM
cana-3670	195	79	0.694720	0.694720	NUM
cana-3670	195	80	0.749566	0.749566	NUM
cana-3670	195	81	0.736947	0.736947	NUM
cana-3670	195	82	0.795127	0.795127	NUM
cana-3670	195	83	0.738993	0.738993	NUM
cana-3670	195	84	0.797334	0.797334	NUM
cana-3670	195	85	𝟎.	𝟎.	PUNCT
cana-3670	195	86	𝟗	𝟗	NUM
cana-3670	195	87	𝟏	𝟏	NUM
cana-3670	195	88	0.759331	0.759331	NUM
cana-3670	195	89	0.788536	0.788536	NUM
cana-3670	195	90	0.713002	0.713002	NUM
cana-3670	195	91	0.740425	0.740425	NUM
cana-3670	195	92	0.756340	0.756340	NUM
cana-3670	195	93	0.785430	0.785430	NUM
cana-3670	195	94	0.758440	0.758440	NUM
cana-3670	195	95	0.787611	0.787611	NUM
cana-3670	195	96	𝟏	𝟏	NUM
cana-3670	195	97	𝟏	𝟏	NUM
cana-3670	195	98	0.778801	0.778801	NUM
cana-3670	195	99	0.778801	0.778801	NUM
cana-3670	195	100	0.731284	0.731284	NUM
cana-3670	195	101	0.731284	0.731284	NUM
cana-3670	195	102	0.775733	0.775733	NUM
cana-3670	195	103	0.775733	0.775733	NUM
cana-3670	195	104	0.777887	0.777887	NUM
cana-3670	195	105	0.777887	0.777887	NUM
cana-3670	195	106	(	(	PUNCT
cana-3670	195	107	a2	a2	PROPN
cana-3670	195	108	)	)	PUNCT
cana-3670	195	109	modified	modify	VERB
cana-3670	195	110	euler	euler	NOUN
cana-3670	195	111	methodbased	methodbase	VERB
cana-3670	195	112	on	on	ADP
cana-3670	195	113	contra	contra	PROPN
cana-3670	195	114	harmonic	harmonic	PROPN
cana-3670	195	115	mean	mean	NOUN
cana-3670	195	116	and	and	CCONJ
cana-3670	195	117	centroidal	centroidal	ADJ
cana-3670	195	118	mean	mean	NOUN
cana-3670	195	119	:	:	PUNCT
cana-3670	195	120	the	the	DET
cana-3670	195	121	approximate	approximate	ADJ
cana-3670	195	122	solutions	solution	NOUN
cana-3670	195	123	obtained	obtain	VERB
cana-3670	195	124	by	by	ADP
cana-3670	195	125	contra	contra	PROPN
cana-3670	195	126	harmonic	harmonic	PROPN
cana-3670	195	127	mean	mean	NOUN
cana-3670	195	128	and	and	CCONJ
cana-3670	195	129	centroidal	centroidal	ADJ
cana-3670	195	130	mean	mean	NOUN
cana-3670	195	131	of	of	ADP
cana-3670	195	132	modified	modified	PROPN
cana-3670	195	133	euler	euler	PROPN
cana-3670	195	134	’s	’s	PART
cana-3670	195	135	method	method	NOUN
cana-3670	195	136	for	for	ADP
cana-3670	195	137	different	different	ADJ
cana-3670	195	138	values	value	NOUN
cana-3670	195	139	of	of	ADP
cana-3670	195	140	𝑠	𝑠	PROPN
cana-3670	195	141	∈	∈	PROPN
cana-3670	196	1	[	[	X
cana-3670	196	2	0,1	0,1	NUM
cana-3670	196	3	]	]	PUNCT
cana-3670	196	4	when	when	SCONJ
cana-3670	196	5	ℎ	ℎ	X
cana-3670	196	6	=	=	SYM
cana-3670	196	7	0.1	0.1	NUM
cana-3670	196	8	are	be	AUX
cana-3670	196	9	given	give	VERB
cana-3670	196	10	below	below	ADP
cana-3670	196	11	:	:	PUNCT
cana-3670	196	12	table2	table2	PROPN
cana-3670	196	13	:	:	PUNCT
cana-3670	196	14	numerical	numerical	ADJ
cana-3670	196	15	solutions	solution	NOUN
cana-3670	196	16	for	for	ADP
cana-3670	196	17	modified	modified	PROPN
cana-3670	196	18	euler	euler	PROPN
cana-3670	196	19	’s	’s	PART
cana-3670	196	20	method	method	NOUN
cana-3670	196	21	using	use	VERB
cana-3670	196	22	contra	contra	PROPN
cana-3670	196	23	harmonic	harmonic	PROPN
cana-3670	196	24	mean	mean	NOUN
cana-3670	196	25	and	and	CCONJ
cana-3670	196	26	centroidal	centroidal	PROPN
cana-3670	196	27	mean	mean	VERB
cana-3670	196	28	with	with	ADP
cana-3670	196	29	𝒉	𝒉	NOUN
cana-3670	196	30	=	=	SYM
cana-3670	196	31	𝟎.𝟏	𝟎.𝟏	NUM
cana-3670	196	32	s	s	NOUN
cana-3670	196	33	t	t	NOUN
cana-3670	196	34	exact	exact	ADJ
cana-3670	196	35	solutions	solution	NOUN
cana-3670	196	36	euler	euler	PROPN
cana-3670	196	37	’s	’s	PART
cana-3670	196	38	method	method	PROPN
cana-3670	196	39	modified	modified	PROPN
cana-3670	196	40	euler	euler	PROPN
cana-3670	196	41	’s	’s	PART
cana-3670	196	42	method	method	NOUN
cana-3670	196	43	using	use	VERB
cana-3670	196	44	contra	contra	PROPN
cana-3670	196	45	harmonic	harmonic	PROPN
cana-3670	196	46	mean	mean	PROPN
cana-3670	196	47	modified	modified	PROPN
cana-3670	196	48	euler	euler	PROPN
cana-3670	196	49	’s	’s	PART
cana-3670	196	50	method	method	NOUN
cana-3670	196	51	using	use	VERB
cana-3670	196	52	centroidal	centroidal	ADJ
cana-3670	196	53	mean	mean	NOUN
cana-3670	196	54	low	low	ADJ
cana-3670	197	1	er	er	INTJ
cana-3670	198	1	uppe	uppe	PROPN
cana-3670	198	2	r	r	PROPN
cana-3670	198	3	lowe	lowe	PROPN
cana-3670	198	4	r	r	NOUN
cana-3670	198	5	uppe	uppe	PROPN
cana-3670	198	6	r	r	NOUN
cana-3670	198	7	lower	low	ADJ
cana-3670	198	8	upper	upper	ADJ
cana-3670	198	9	lower	lower	ADV
cana-3670	198	10	upper	upper	ADJ
cana-3670	198	11	𝟎	𝟎	NUM
cana-3670	198	12	𝟏	𝟏	NUM
cana-3670	198	13	0.584101	0.584101	NUM
cana-3670	198	14	0.876151	0.876151	NUM
cana-3670	198	15	0.548463	0.548463	NUM
cana-3670	198	16	0.822694	0.822694	NUM
cana-3670	198	17	0.584834	0.584834	NUM
cana-3670	198	18	0.877251	0.877251	NUM
cana-3670	198	19	0.584812	0.584812	NUM
cana-3670	198	20	0.877217	0.877217	NUM
cana-3670	198	21	𝟎.	𝟎.	PUNCT
cana-3670	198	22	𝟏	𝟏	NUM
cana-3670	198	23	𝟏	𝟏	NUM
cana-3670	198	24	0.603571	0.603571	NUM
cana-3670	199	1	0.866416	0.866416	NUM
cana-3670	199	2	0.566745	0.566745	NUM
cana-3670	199	3	0.813553	0.813553	NUM
cana-3670	199	4	0.604329	0.604329	NUM
cana-3670	199	5	0.867504	0.867504	NUM
cana-3670	199	6	0.604305	0.604305	NUM
cana-3670	199	7	0.867471	0.867471	NUM
cana-3670	199	8	𝟎.	𝟎.	PUNCT
cana-3670	199	9	𝟐	𝟐	NUM
cana-3670	199	10	𝟏	𝟏	NUM
cana-3670	199	11	0.623041	0.623041	NUM
cana-3670	199	12	0.856681	0.856681	NUM
cana-3670	199	13	0.585027	0.585027	NUM
cana-3670	199	14	0.804412	0.804412	NUM
cana-3670	199	15	0.623823	0.623823	NUM
cana-3670	199	16	0.857757	0.857757	NUM
cana-3670	199	17	0.623799	0.623799	NUM
cana-3670	199	18	0.857724	0.857724	NUM
cana-3670	199	19	𝟎.	𝟎.	PUNCT
cana-3670	199	20	𝟑	𝟑	NUM
cana-3670	199	21	𝟏	𝟏	NUM
cana-3670	199	22	0.642511	0.642511	NUM
cana-3670	199	23	0.846946	0.846946	NUM
cana-3670	199	24	0.603309	0.603309	NUM
cana-3670	199	25	0.795271	0.795271	NUM
cana-3670	200	1	0.643318	0.643318	NUM
cana-3670	200	2	0.848010	0.848010	NUM
cana-3670	200	3	0.643293	0.643293	NUM
cana-3670	200	4	0.847977	0.847977	NUM
cana-3670	200	5	𝟎.	𝟎.	SYM
cana-3670	200	6	𝟒	𝟒	NUM
cana-3670	200	7	𝟏	𝟏	NUM
cana-3670	200	8	0.661981	0.661981	NUM
cana-3670	200	9	0.837211	0.837211	NUM
cana-3670	200	10	0.621591	0.621591	NUM
cana-3670	200	11	0.786130	0.786130	NUM
cana-3670	200	12	0.662812	0.662812	NUM
cana-3670	200	13	0.838262	0.838262	NUM
cana-3670	200	14	0.662787	0.662787	NUM
cana-3670	200	15	0.838230	0.838230	NUM
cana-3670	200	16	𝟎.	𝟎.	PUNCT
cana-3670	200	17	𝟓	𝟓	NUM
cana-3670	200	18	𝟏	𝟏	NUM
cana-3670	200	19	0.681451	0.681451	NUM
cana-3670	200	20	0.827476	0.827476	NUM
cana-3670	200	21	0.639873	0.639873	NUM
cana-3670	200	22	0.776989	0.776989	NUM
cana-3670	200	23	0.682307	0.682307	NUM
cana-3670	200	24	0.828515	0.828515	NUM
cana-3670	200	25	0.682280	0.682280	NUM
cana-3670	200	26	0.828483	0.828483	NUM
cana-3670	200	27	𝟎.	𝟎.	PUNCT
cana-3670	200	28	𝟔	𝟔	NUM
cana-3670	200	29	𝟏	𝟏	NUM
cana-3670	200	30	0.700921	0.700921	NUM
cana-3670	200	31	0.817741	0.817741	NUM
cana-3670	200	32	0.658155	0.658155	NUM
cana-3670	200	33	0.767848	0.767848	NUM
cana-3670	200	34	0.701801	0.701801	NUM
cana-3670	200	35	0.818768	0.818768	NUM
cana-3670	200	36	0.701774	0.701774	NUM
cana-3670	200	37	0.818736	0.818736	NUM
cana-3670	200	38	𝟎.	𝟎.	NOUN
cana-3670	200	39	𝟕	𝟕	NUM
cana-3670	200	40	𝟏	𝟏	NUM
cana-3670	200	41	0.720391	0.720391	NUM
cana-3670	200	42	0.808006	0.808006	NUM
cana-3670	200	43	0.676437	0.676437	NUM
cana-3670	200	44	0.758707	0.758707	NUM
cana-3670	200	45	0.721295	0.721295	NUM
cana-3670	200	46	0.809021	0.809021	NUM
cana-3670	200	47	0.721268	0.721268	NUM
cana-3670	200	48	0.808989	0.808989	NUM
cana-3670	200	49	communications	communication	NOUN
cana-3670	200	50	on	on	ADP
cana-3670	200	51	applied	apply	VERB
cana-3670	200	52	nonlinear	nonlinear	ADJ
cana-3670	200	53	analysis	analysis	NOUN
cana-3670	200	54	issn	issn	NOUN
cana-3670	200	55	:	:	PUNCT
cana-3670	200	56	1074	1074	NUM
cana-3670	200	57	-	-	PUNCT
cana-3670	200	58	133x	133x	NUM
cana-3670	200	59	vol	vol	NOUN
cana-3670	200	60	32	32	NUM
cana-3670	200	61	no	no	NOUN
cana-3670	200	62	.	.	PUNCT
cana-3670	201	1	8s	8s	PROPN
cana-3670	201	2	(	(	PUNCT
cana-3670	201	3	2025	2025	NUM
cana-3670	201	4	)	)	PUNCT
cana-3670	201	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	201	6	283	283	NUM
cana-3670	201	7	𝟎.	𝟎.	X
cana-3670	201	8	𝟖	𝟖	NUM
cana-3670	201	9	𝟏	𝟏	NUM
cana-3670	201	10	0.739861	0.739861	NUM
cana-3670	201	11	0.798271	0.798271	NUM
cana-3670	201	12	0.694720	0.694720	NUM
cana-3670	202	1	0.749566	0.749566	NUM
cana-3670	202	2	0.740790	0.740790	NUM
cana-3670	202	3	0.799273	0.799273	NUM
cana-3670	202	4	0.740761	0.740761	NUM
cana-3670	202	5	0.799243	0.799243	NUM
cana-3670	202	6	𝟎.	𝟎.	PUNCT
cana-3670	202	7	𝟗	𝟗	NUM
cana-3670	202	8	𝟏	𝟏	NUM
cana-3670	202	9	0.759331	0.759331	NUM
cana-3670	202	10	0.788536	0.788536	NUM
cana-3670	202	11	0.713002	0.713002	NUM
cana-3670	202	12	0.740425	0.740425	NUM
cana-3670	202	13	0.760284	0.760284	NUM
cana-3670	202	14	0.789526	0.789526	NUM
cana-3670	202	15	0.760255	0.760255	NUM
cana-3670	202	16	0.789496	0.789496	NUM
cana-3670	202	17	𝟏	𝟏	NUM
cana-3670	202	18	𝟏	𝟏	NUM
cana-3670	202	19	0.778801	0.778801	NUM
cana-3670	202	20	0.778801	0.778801	NUM
cana-3670	202	21	0.731284	0.731284	NUM
cana-3670	202	22	0.731284	0.731284	NUM
cana-3670	202	23	0.779779	0.779779	NUM
cana-3670	203	1	0.779779	0.779779	NUM
cana-3670	204	1	0.779749	0.779749	NUM
cana-3670	204	2	0.779749	0.779749	NUM
cana-3670	204	3	(	(	PUNCT
cana-3670	204	4	a3	a3	NOUN
cana-3670	204	5	)	)	PUNCT
cana-3670	204	6	improved	improve	VERB
cana-3670	204	7	euler	euler	NOUN
cana-3670	204	8	method	method	NOUN
cana-3670	204	9	based	base	VERB
cana-3670	204	10	on	on	ADP
cana-3670	204	11	contra	contra	PROPN
cana-3670	204	12	harmonic	harmonic	PROPN
cana-3670	204	13	mean	mean	NOUN
cana-3670	204	14	and	and	CCONJ
cana-3670	204	15	centroidal	centroidal	ADJ
cana-3670	204	16	mean	mean	NOUN
cana-3670	204	17	:	:	PUNCT
cana-3670	205	1	the	the	DET
cana-3670	205	2	approximate	approximate	ADJ
cana-3670	205	3	solutions	solution	NOUN
cana-3670	205	4	obtained	obtain	VERB
cana-3670	205	5	by	by	ADP
cana-3670	205	6	contra	contra	PROPN
cana-3670	205	7	harmonic	harmonic	PROPN
cana-3670	205	8	mean	mean	NOUN
cana-3670	205	9	and	and	CCONJ
cana-3670	205	10	centroidal	centroidal	ADJ
cana-3670	205	11	mean	mean	NOUN
cana-3670	205	12	of	of	ADP
cana-3670	205	13	improved	improve	VERB
cana-3670	205	14	euler	euler	NOUN
cana-3670	205	15	’s	’s	PART
cana-3670	205	16	method	method	NOUN
cana-3670	205	17	for	for	ADP
cana-3670	205	18	different	different	ADJ
cana-3670	205	19	values	value	NOUN
cana-3670	205	20	of	of	ADP
cana-3670	205	21	𝑠	𝑠	PROPN
cana-3670	205	22	∈	∈	PROPN
cana-3670	206	1	[	[	X
cana-3670	206	2	0,1	0,1	NUM
cana-3670	206	3	]	]	PUNCT
cana-3670	206	4	when	when	SCONJ
cana-3670	206	5	ℎ	ℎ	X
cana-3670	206	6	=	=	SYM
cana-3670	206	7	0.1	0.1	NUM
cana-3670	206	8	are	be	AUX
cana-3670	206	9	given	give	VERB
cana-3670	206	10	below	below	ADV
cana-3670	206	11	:	:	PUNCT
cana-3670	206	12	table3	table3	PROPN
cana-3670	206	13	:	:	PUNCT
cana-3670	206	14	numerical	numerical	ADJ
cana-3670	206	15	solutions	solution	NOUN
cana-3670	206	16	for	for	ADP
cana-3670	206	17	improved	improve	VERB
cana-3670	206	18	euler	euler	NOUN
cana-3670	206	19	’s	’s	PART
cana-3670	206	20	method	method	NOUN
cana-3670	206	21	using	use	VERB
cana-3670	206	22	contra	contra	PROPN
cana-3670	206	23	harmonic	harmonic	PROPN
cana-3670	206	24	mean	mean	NOUN
cana-3670	206	25	and	and	CCONJ
cana-3670	206	26	centroidal	centroidal	PROPN
cana-3670	206	27	mean	mean	VERB
cana-3670	206	28	with	with	ADP
cana-3670	206	29	𝒉	𝒉	NOUN
cana-3670	206	30	=	=	SYM
cana-3670	206	31	𝟎.𝟏	𝟎.𝟏	NUM
cana-3670	206	32	s	s	NOUN
cana-3670	206	33	t	t	NOUN
cana-3670	206	34	exact	exact	ADJ
cana-3670	206	35	solutions	solution	NOUN
cana-3670	206	36	euler	euler	PROPN
cana-3670	206	37	’s	’s	PART
cana-3670	206	38	method	method	NOUN
cana-3670	206	39	improved	improve	VERB
cana-3670	206	40	euler	euler	NOUN
cana-3670	206	41	’s	’s	PART
cana-3670	206	42	method	method	NOUN
cana-3670	206	43	using	use	VERB
cana-3670	206	44	contra	contra	PROPN
cana-3670	206	45	harmonic	harmonic	PROPN
cana-3670	206	46	mean	mean	VERB
cana-3670	206	47	improved	improve	VERB
cana-3670	206	48	euler	euler	PROPN
cana-3670	206	49	’s	’s	PART
cana-3670	206	50	method	method	NOUN
cana-3670	206	51	using	use	VERB
cana-3670	207	1	centroidal	centroidal	NOUN
cana-3670	207	2	mean	mean	NOUN
cana-3670	207	3	lo	lo	NOUN
cana-3670	208	1	we	we	PRON
cana-3670	208	2	r	r	VERB
cana-3670	208	3	upp	upp	INTJ
cana-3670	208	4	er	er	INTJ
cana-3670	208	5	low	low	ADJ
cana-3670	208	6	er	er	INTJ
cana-3670	208	7	upp	upp	INTJ
cana-3670	208	8	er	er	INTJ
cana-3670	208	9	lower	low	ADJ
cana-3670	208	10	upper	upper	ADJ
cana-3670	208	11	lower	low	ADJ
cana-3670	208	12	upper	upper	ADJ
cana-3670	208	13	𝟎	𝟎	NUM
cana-3670	208	14	𝟏	𝟏	NUM
cana-3670	208	15	0.584101	0.584101	NUM
cana-3670	208	16	0.876151	0.876151	NUM
cana-3670	208	17	0.548463	0.548463	NUM
cana-3670	208	18	0.822694	0.822694	NUM
cana-3670	208	19	0.570348	0.570348	NUM
cana-3670	208	20	0.855521	0.855521	NUM
cana-3670	208	21	0.571118	0.571118	NUM
cana-3670	208	22	0.856677	0.856677	NUM
cana-3670	208	23	𝟎.	𝟎.	PUNCT
cana-3670	208	24	𝟏	𝟏	NUM
cana-3670	208	25	𝟏	𝟏	NUM
cana-3670	208	26	0.603571	0.603571	NUM
cana-3670	209	1	0.866416	0.866416	NUM
cana-3670	209	2	0.566745	0.566745	NUM
cana-3670	209	3	0.813553	0.813553	NUM
cana-3670	209	4	0.589359	0.589359	NUM
cana-3670	209	5	0.846016	0.846016	NUM
cana-3670	209	6	0.590156	0.590156	NUM
cana-3670	209	7	0.847159	0.847159	NUM
cana-3670	209	8	𝟎.	𝟎.	PUNCT
cana-3670	209	9	𝟐	𝟐	NUM
cana-3670	209	10	𝟏	𝟏	NUM
cana-3670	209	11	0.623041	0.623041	NUM
cana-3670	209	12	0.856681	0.856681	NUM
cana-3670	209	13	0.585027	0.585027	NUM
cana-3670	210	1	0.804412	0.804412	NUM
cana-3670	210	2	0.608371	0.608371	NUM
cana-3670	210	3	0.836510	0.836510	NUM
cana-3670	210	4	0.609193	0.609193	NUM
cana-3670	210	5	0.837640	0.837640	NUM
cana-3670	210	6	𝟎.	𝟎.	SYM
cana-3670	210	7	𝟑	𝟑	NUM
cana-3670	210	8	𝟏	𝟏	NUM
cana-3670	210	9	0.642511	0.642511	NUM
cana-3670	210	10	0.846946	0.846946	NUM
cana-3670	210	11	0.603309	0.603309	NUM
cana-3670	210	12	0.795271	0.795271	NUM
cana-3670	210	13	0.627382	0.627382	NUM
cana-3670	210	14	0.827004	0.827004	NUM
cana-3670	210	15	0.628230	0.628230	NUM
cana-3670	210	16	0.828122	0.828122	NUM
cana-3670	210	17	𝟎.	𝟎.	SYM
cana-3670	210	18	𝟒	𝟒	NUM
cana-3670	210	19	𝟏	𝟏	NUM
cana-3670	210	20	0.661981	0.661981	NUM
cana-3670	210	21	0.837211	0.837211	NUM
cana-3670	210	22	0.621591	0.621591	NUM
cana-3670	210	23	0.786130	0.786130	NUM
cana-3670	210	24	0.646394	0.646394	NUM
cana-3670	210	25	0.817498	0.817498	NUM
cana-3670	210	26	0.647267	0.647267	NUM
cana-3670	210	27	0.818603	0.818603	NUM
cana-3670	210	28	𝟎.	𝟎.	PUNCT
cana-3670	210	29	𝟓	𝟓	NUM
cana-3670	210	30	𝟏	𝟏	NUM
cana-3670	210	31	0.681451	0.681451	NUM
cana-3670	210	32	0.827476	0.827476	NUM
cana-3670	210	33	0.639873	0.639873	NUM
cana-3670	210	34	0.776989	0.776989	NUM
cana-3670	210	35	0.665406	0.665406	NUM
cana-3670	210	36	0.807992	0.807992	NUM
cana-3670	210	37	0.666305	0.666305	NUM
cana-3670	210	38	0.809084	0.809084	NUM
cana-3670	210	39	𝟎.	𝟎.	PUNCT
cana-3670	210	40	𝟔	𝟔	NUM
cana-3670	210	41	𝟏	𝟏	NUM
cana-3670	210	42	0.700921	0.700921	NUM
cana-3670	210	43	0.817741	0.817741	NUM
cana-3670	210	44	0.658155	0.658155	NUM
cana-3670	210	45	0.767848	0.767848	NUM
cana-3670	210	46	0.684417	0.684417	NUM
cana-3670	210	47	0.798487	0.798487	NUM
cana-3670	210	48	0.685342	0.685342	NUM
cana-3670	210	49	0.799566	0.799566	NUM
cana-3670	210	50	𝟎.	𝟎.	PUNCT
cana-3670	210	51	𝟕	𝟕	NUM
cana-3670	210	52	𝟏	𝟏	NUM
cana-3670	210	53	0.720391	0.720391	NUM
cana-3670	210	54	0.808006	0.808006	NUM
cana-3670	210	55	0.676437	0.676437	NUM
cana-3670	210	56	0.758707	0.758707	NUM
cana-3670	210	57	0.703429	0.703429	NUM
cana-3670	211	1	0.788981	0.788981	NUM
cana-3670	211	2	0.704379	0.704379	NUM
cana-3670	211	3	0.790047	0.790047	NUM
cana-3670	211	4	𝟎.	𝟎.	PUNCT
cana-3670	211	5	𝟖	𝟖	NUM
cana-3670	211	6	𝟏	𝟏	NUM
cana-3670	211	7	0.739861	0.739861	NUM
cana-3670	211	8	0.798271	0.798271	NUM
cana-3670	211	9	0.694720	0.694720	NUM
cana-3670	211	10	0.749566	0.749566	NUM
cana-3670	211	11	0.722440	0.722440	NUM
cana-3670	211	12	0.779475	0.779475	NUM
cana-3670	211	13	0.723417	0.723417	NUM
cana-3670	211	14	0.780528	0.780528	NUM
cana-3670	211	15	𝟎.	𝟎.	PUNCT
cana-3670	211	16	𝟗	𝟗	NUM
cana-3670	211	17	𝟏	𝟏	NUM
cana-3670	211	18	0.759331	0.759331	NUM
cana-3670	211	19	0.788536	0.788536	NUM
cana-3670	211	20	0.713002	0.713002	NUM
cana-3670	211	21	0.740425	0.740425	NUM
cana-3670	211	22	0.741452	0.741452	NUM
cana-3670	211	23	0.769969	0.769969	NUM
cana-3670	211	24	0.742454	0.742454	NUM
cana-3670	211	25	0.771010	0.771010	NUM
cana-3670	211	26	𝟏	𝟏	NUM
cana-3670	211	27	𝟏	𝟏	NUM
cana-3670	211	28	0.778801	0.778801	NUM
cana-3670	211	29	0.778801	0.778801	NUM
cana-3670	211	30	0.731284	0.731284	NUM
cana-3670	211	31	0.731284	0.731284	NUM
cana-3670	211	32	0.760463	0.760463	NUM
cana-3670	211	33	0.760463	0.760463	NUM
cana-3670	211	34	0.761491	0.761491	NUM
cana-3670	211	35	0.761491	0.761491	NUM
cana-3670	211	36	(	(	PUNCT
cana-3670	211	37	b	b	NOUN
cana-3670	211	38	)	)	PUNCT
cana-3670	211	39	graphical	graphical	ADJ
cana-3670	211	40	representation	representation	NOUN
cana-3670	211	41	of	of	ADP
cana-3670	211	42	error	error	NOUN
cana-3670	211	43	analysis	analysis	NOUN
cana-3670	211	44	for	for	ADP
cana-3670	211	45	enhanced	enhance	VERB
cana-3670	211	46	euler	euler	PROPN
cana-3670	211	47	’s	’s	PART
cana-3670	211	48	methods	method	NOUN
cana-3670	211	49	based	base	VERB
cana-3670	211	50	on	on	ADP
cana-3670	211	51	contra	contra	PROPN
cana-3670	211	52	harmonic	harmonic	PROPN
cana-3670	211	53	mean	mean	NOUN
cana-3670	211	54	and	and	CCONJ
cana-3670	211	55	centroidal	centroidal	PROPN
cana-3670	211	56	mean	mean	VERB
cana-3670	211	57	with	with	ADP
cana-3670	211	58	ℎ	ℎ	PROPN
cana-3670	211	59	=	=	SYM
cana-3670	211	60	0.1	0.1	NUM
cana-3670	211	61	&	&	CCONJ
cana-3670	211	62	ℎ	ℎ	PROPN
cana-3670	211	63	=	=	NOUN
cana-3670	211	64	0.001	0.001	NUM
cana-3670	211	65	:	:	PUNCT
cana-3670	211	66	the	the	DET
cana-3670	211	67	graphical	graphical	ADJ
cana-3670	211	68	solutions	solution	NOUN
cana-3670	211	69	obtained	obtain	VERB
cana-3670	211	70	by	by	ADP
cana-3670	211	71	contra	contra	PROPN
cana-3670	211	72	harmonic	harmonic	PROPN
cana-3670	211	73	mean	mean	NOUN
cana-3670	211	74	and	and	CCONJ
cana-3670	211	75	centroidal	centroidal	ADJ
cana-3670	211	76	mean	mean	NOUN
cana-3670	211	77	of	of	ADP
cana-3670	211	78	enhanced	enhance	VERB
cana-3670	211	79	euler	euler	PROPN
cana-3670	211	80	’s	’s	PART
cana-3670	211	81	method	method	NOUN
cana-3670	211	82	for	for	ADP
cana-3670	211	83	different	different	ADJ
cana-3670	211	84	values	value	NOUN
cana-3670	211	85	of	of	ADP
cana-3670	211	86	𝑠	𝑠	PROPN
cana-3670	211	87	∈	∈	PROPN
cana-3670	211	88	[	[	X
cana-3670	211	89	0,1	0,1	NUM
cana-3670	211	90	]	]	X
cana-3670	211	91	withℎ	withℎ	NOUN
cana-3670	211	92	=	=	SYM
cana-3670	211	93	0.1	0.1	NUM
cana-3670	211	94	&	&	CCONJ
cana-3670	211	95	ℎ	ℎ	PROPN
cana-3670	211	96	=	=	SYM
cana-3670	211	97	0.001	0.001	NUM
cana-3670	211	98	are	be	AUX
cana-3670	211	99	given	give	VERB
cana-3670	211	100	below	below	ADV
cana-3670	211	101	:	:	PUNCT
cana-3670	211	102	figure	figure	VERB
cana-3670	211	103	4	4	NUM
cana-3670	211	104	:	:	PUNCT
cana-3670	211	105	graphical	graphical	ADJ
cana-3670	211	106	representation	representation	NOUN
cana-3670	211	107	of	of	ADP
cana-3670	211	108	error	error	NOUN
cana-3670	211	109	analysis	analysis	NOUN
cana-3670	211	110	for	for	ADP
cana-3670	211	111	enhanced	enhance	VERB
cana-3670	211	112	euler	euler	PROPN
cana-3670	211	113	’s	’s	PART
cana-3670	211	114	method	method	NOUN
cana-3670	211	115	using	use	VERB
cana-3670	211	116	contra	contra	PROPN
cana-3670	211	117	harmonic	harmonic	PROPN
cana-3670	211	118	mean	mean	NOUN
cana-3670	211	119	and	and	CCONJ
cana-3670	211	120	centroidal	centroidal	PROPN
cana-3670	211	121	mean	mean	VERB
cana-3670	211	122	with	with	ADP
cana-3670	211	123	h	h	NOUN
cana-3670	211	124	=	=	SYM
cana-3670	211	125	0.1	0.1	NUM
cana-3670	211	126	figure	figure	NOUN
cana-3670	211	127	5	5	NUM
cana-3670	211	128	:	:	PUNCT
cana-3670	211	129	graphical	graphical	ADJ
cana-3670	211	130	representation	representation	NOUN
cana-3670	211	131	of	of	ADP
cana-3670	211	132	error	error	NOUN
cana-3670	211	133	analysis	analysis	NOUN
cana-3670	211	134	for	for	ADP
cana-3670	211	135	enhanced	enhance	VERB
cana-3670	211	136	euler	euler	PROPN
cana-3670	211	137	’s	’s	PART
cana-3670	211	138	method	method	NOUN
cana-3670	211	139	using	use	VERB
cana-3670	211	140	contra	contra	PROPN
cana-3670	211	141	harmonic	harmonic	PROPN
cana-3670	211	142	mean	mean	NOUN
cana-3670	211	143	and	and	CCONJ
cana-3670	211	144	centroidal	centroidal	PROPN
cana-3670	211	145	mean	mean	VERB
cana-3670	211	146	with	with	ADP
cana-3670	211	147	𝒉	𝒉	NOUN
cana-3670	211	148	=	=	PUNCT
cana-3670	211	149	𝟎.𝟎𝟎𝟏	𝟎.𝟎𝟎𝟏	NUM
cana-3670	211	150	communications	communication	NOUN
cana-3670	211	151	on	on	ADP
cana-3670	211	152	applied	apply	VERB
cana-3670	211	153	nonlinear	nonlinear	ADJ
cana-3670	211	154	analysis	analysis	NOUN
cana-3670	211	155	issn	issn	NOUN
cana-3670	211	156	:	:	PUNCT
cana-3670	211	157	1074	1074	NUM
cana-3670	211	158	-	-	PUNCT
cana-3670	211	159	133x	133x	NUM
cana-3670	211	160	vol	vol	NOUN
cana-3670	211	161	32	32	NUM
cana-3670	211	162	no	no	NOUN
cana-3670	211	163	.	.	PUNCT
cana-3670	212	1	8s	8s	PROPN
cana-3670	212	2	(	(	PUNCT
cana-3670	212	3	2025	2025	NUM
cana-3670	212	4	)	)	PUNCT
cana-3670	212	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-3670	212	6	284	284	NUM
cana-3670	212	7	the	the	DET
cana-3670	212	8	fuzzy	fuzzy	ADJ
cana-3670	212	9	initial	initial	ADJ
cana-3670	212	10	value	value	NOUN
cana-3670	212	11	problem	problem	NOUN
cana-3670	212	12	has	have	AUX
cana-3670	212	13	been	be	AUX
cana-3670	212	14	solved	solve	VERB
cana-3670	212	15	by	by	ADP
cana-3670	212	16	proposed	propose	VERB
cana-3670	212	17	enhanced	enhanced	PROPN
cana-3670	212	18	euler	euler	PROPN
cana-3670	212	19	’s	’s	PART
cana-3670	212	20	methods	method	NOUN
cana-3670	212	21	such	such	ADJ
cana-3670	212	22	as	as	ADP
cana-3670	212	23	euler	euler	PROPN
cana-3670	212	24	’s	’s	PART
cana-3670	212	25	method	method	NOUN
cana-3670	212	26	,	,	PUNCT
cana-3670	212	27	modified	modified	PROPN
cana-3670	212	28	euler	euler	PROPN
cana-3670	212	29	’s	’s	PART
cana-3670	212	30	method	method	NOUN
cana-3670	212	31	and	and	CCONJ
cana-3670	212	32	improved	improve	VERB
cana-3670	212	33	euler	euler	NOUN
cana-3670	212	34	’s	’s	PART
cana-3670	212	35	method	method	NOUN
cana-3670	212	36	using	use	VERB
cana-3670	212	37	contra	contra	PROPN
cana-3670	212	38	harmonic	harmonic	PROPN
cana-3670	212	39	mean	mean	NOUN
cana-3670	212	40	and	and	CCONJ
cana-3670	212	41	centroidal	centroidal	PROPN
cana-3670	212	42	mean	mean	NOUN
cana-3670	212	43	.	.	PUNCT
cana-3670	213	1	the	the	DET
cana-3670	213	2	above	above	ADJ
cana-3670	213	3	tables	table	NOUN
cana-3670	213	4	and	and	CCONJ
cana-3670	213	5	figures	figure	NOUN
cana-3670	213	6	show	show	VERB
cana-3670	213	7	the	the	DET
cana-3670	213	8	numerical	numerical	ADJ
cana-3670	213	9	solutions	solution	NOUN
cana-3670	213	10	obtained	obtain	VERB
cana-3670	213	11	by	by	ADP
cana-3670	213	12	the	the	DET
cana-3670	213	13	proposed	propose	VERB
cana-3670	213	14	methods	method	NOUN
cana-3670	213	15	for	for	ADP
cana-3670	213	16	different	different	ADJ
cana-3670	213	17	values	value	NOUN
cana-3670	213	18	of	of	ADP
cana-3670	213	19	𝑠	𝑠	PROPN
cana-3670	213	20	∈	∈	PROPN
cana-3670	214	1	[	[	X
cana-3670	214	2	0,1	0,1	NUM
cana-3670	214	3	]	]	PUNCT
cana-3670	214	4	when	when	SCONJ
cana-3670	214	5	ℎ	ℎ	X
cana-3670	214	6	=	=	SYM
cana-3670	214	7	0.1	0.1	NUM
cana-3670	214	8	and	and	CCONJ
cana-3670	214	9	ℎ	ℎ	X
cana-3670	214	10	=	=	NOUN
cana-3670	214	11	0.001	0.001	NUM
cana-3670	214	12	.	.	PUNCT
cana-3670	215	1	it	it	PRON
cana-3670	215	2	also	also	ADV
cana-3670	215	3	shows	show	VERB
cana-3670	215	4	the	the	DET
cana-3670	215	5	solutions	solution	NOUN
cana-3670	215	6	obtained	obtain	VERB
cana-3670	215	7	by	by	ADP
cana-3670	215	8	euler	euler	PROPN
cana-3670	215	9	’s	’s	PART
cana-3670	215	10	method	method	NOUN
cana-3670	215	11	in	in	ADP
cana-3670	215	12	order	order	NOUN
cana-3670	215	13	to	to	PART
cana-3670	215	14	analyse	analyse	VERB
cana-3670	215	15	the	the	DET
cana-3670	215	16	effectiveness	effectiveness	NOUN
cana-3670	215	17	of	of	ADP
cana-3670	215	18	proposed	propose	VERB
cana-3670	215	19	methods	method	NOUN
cana-3670	215	20	.	.	PUNCT
cana-3670	216	1	the	the	DET
cana-3670	216	2	analysis	analysis	NOUN
cana-3670	216	3	concluded	conclude	VERB
cana-3670	216	4	that	that	SCONJ
cana-3670	216	5	the	the	DET
cana-3670	216	6	proposed	propose	VERB
cana-3670	216	7	methods	method	NOUN
cana-3670	216	8	give	give	VERB
cana-3670	216	9	the	the	DET
cana-3670	216	10	very	very	ADV
cana-3670	216	11	close	close	ADJ
cana-3670	216	12	solutions	solution	NOUN
cana-3670	216	13	to	to	ADP
cana-3670	216	14	the	the	DET
cana-3670	216	15	exact	exact	ADJ
cana-3670	216	16	values	value	NOUN
cana-3670	216	17	other	other	ADJ
cana-3670	216	18	than	than	ADP
cana-3670	216	19	euler	euler	PROPN
cana-3670	216	20	’s	’s	PART
cana-3670	216	21	method	method	NOUN
cana-3670	216	22	.	.	PUNCT
cana-3670	217	1	5	5	X
cana-3670	217	2	.	.	X
cana-3670	217	3	conclusion	conclusion	NOUN
cana-3670	217	4	:	:	PUNCT
cana-3670	217	5	in	in	ADP
cana-3670	217	6	order	order	NOUN
cana-3670	217	7	to	to	PART
cana-3670	217	8	solve	solve	VERB
cana-3670	217	9	fuzzy	fuzzy	ADJ
cana-3670	217	10	initial	initial	ADJ
cana-3670	217	11	value	value	NOUN
cana-3670	217	12	problems	problem	NOUN
cana-3670	217	13	based	base	VERB
cana-3670	217	14	on	on	ADP
cana-3670	217	15	contra	contra	PROPN
cana-3670	217	16	harmonic	harmonic	PROPN
cana-3670	217	17	mean	mean	NOUN
cana-3670	217	18	and	and	CCONJ
cana-3670	217	19	centroidal	centroidal	PROPN
cana-3670	217	20	mean	mean	VERB
cana-3670	217	21	,	,	PUNCT
cana-3670	217	22	this	this	DET
cana-3670	217	23	research	research	NOUN
cana-3670	217	24	proposed	propose	VERB
cana-3670	217	25	three	three	NUM
cana-3670	217	26	enhanced	enhanced	ADJ
cana-3670	217	27	numerical	numerical	ADJ
cana-3670	217	28	techniques	technique	NOUN
cana-3670	217	29	of	of	ADP
cana-3670	217	30	euler	euler	PROPN
cana-3670	217	31	,	,	PUNCT
cana-3670	217	32	modified	modify	VERB
cana-3670	217	33	euler	euler	NOUN
cana-3670	217	34	,	,	PUNCT
cana-3670	217	35	and	and	CCONJ
cana-3670	217	36	improved	improved	ADJ
cana-3670	217	37	euler	euler	NOUN
cana-3670	217	38	.	.	PUNCT
cana-3670	218	1	by	by	ADP
cana-3670	218	2	comparing	compare	VERB
cana-3670	218	3	the	the	DET
cana-3670	218	4	example	example	NOUN
cana-3670	218	5	's	's	PART
cana-3670	218	6	answers	answer	NOUN
cana-3670	218	7	,	,	PUNCT
cana-3670	218	8	it	it	PRON
cana-3670	218	9	was	be	AUX
cana-3670	218	10	shown	show	VERB
cana-3670	218	11	that	that	SCONJ
cana-3670	218	12	the	the	DET
cana-3670	218	13	three	three	NUM
cana-3670	218	14	enhanced	enhanced	ADJ
cana-3670	218	15	methods	method	NOUN
cana-3670	218	16	outperform	outperform	VERB
cana-3670	218	17	the	the	DET
cana-3670	218	18	conventional	conventional	ADJ
cana-3670	218	19	euler	euler	NOUN
cana-3670	218	20	's	's	PART
cana-3670	218	21	method	method	NOUN
cana-3670	218	22	.	.	PUNCT
cana-3670	219	1	references	reference	NOUN
cana-3670	219	2	:	:	PUNCT
cana-3670	220	1	[	[	X
cana-3670	220	2	1	1	NUM
cana-3670	220	3	]	]	PUNCT
cana-3670	220	4	allahviranloo	allahviranloo	NOUN
cana-3670	220	5	,	,	PUNCT
cana-3670	220	6	t.	t.	NOUN
cana-3670	220	7	,	,	PUNCT
cana-3670	220	8	noeiaghdam	noeiaghdam	PROPN
cana-3670	220	9	,	,	PUNCT
cana-3670	220	10	z.	z.	PROPN
cana-3670	220	11	,	,	PUNCT
cana-3670	220	12	noeiaghdam	noeiaghdam	PROPN
cana-3670	220	13	,	,	PUNCT
cana-3670	220	14	s.	s.	PROPN
cana-3670	220	15	&	&	CCONJ
cana-3670	220	16	juan	juan	PROPN
cana-3670	220	17	,	,	PUNCT
cana-3670	220	18	j.n	j.n	PROPN
cana-3670	220	19	.	.	PROPN
cana-3670	221	1	a	a	DET
cana-3670	221	2	fuzy	fuzy	ADJ
cana-3670	221	3	method	method	NOUN
cana-3670	221	4	for	for	ADP
cana-3670	221	5	solving	solve	VERB
cana-3670	221	6	fuzzy	fuzzy	ADJ
cana-3670	221	7	fractional	fractional	ADJ
cana-3670	221	8	differential	differential	NOUN
cana-3670	221	9	equation	equation	NOUN
cana-3670	221	10	based	base	VERB
cana-3670	221	11	on	on	ADP
cana-3670	221	12	the	the	DET
cana-3670	221	13	generalized	generalize	VERB
cana-3670	221	14	fuzzy	fuzzy	ADJ
cana-3670	221	15	taylor	taylor	PROPN
cana-3670	221	16	expansion	expansion	NOUN
cana-3670	221	17	.	.	PUNCT
cana-3670	222	1	mathematics	mathematic	NOUN
cana-3670	222	2	.	.	PUNCT
cana-3670	223	1	2020	2020	NUM
cana-3670	223	2	;	;	PUNCT
cana-3670	223	3	8(12):1	8(12):1	NUM
cana-3670	223	4	-	-	SYM
cana-3670	223	5	24	24	NUM
cana-3670	223	6	.	.	PUNCT
cana-3670	223	7	available	available	ADJ
cana-3670	223	8	from	from	ADP
cana-3670	223	9	:	:	PUNCT
cana-3670	223	10	https://doi.org/10.3390/math8122166	https://doi.org/10.3390/math8122166	PROPN
cana-3670	223	11	.	.	PUNCT
cana-3670	224	1	[	[	X
cana-3670	224	2	2	2	X
cana-3670	224	3	]	]	PUNCT
cana-3670	224	4	you	you	PRON
cana-3670	224	5	.	.	PUNCT
cana-3670	224	6	,	,	PUNCT
cana-3670	224	7	c.	c.	PROPN
cana-3670	224	8	,	,	PUNCT
cana-3670	224	9	cheng	cheng	PROPN
cana-3670	224	10	.	.	PROPN
cana-3670	224	11	,	,	PUNCT
cana-3670	224	12	y.	y.	PROPN
cana-3670	224	13	&	&	CCONJ
cana-3670	224	14	ma	ma	PROPN
cana-3670	224	15	,	,	PUNCT
cana-3670	224	16	h.	h.	PROPN
cana-3670	224	17	stability	stability	PROPN
cana-3670	224	18	of	of	ADP
cana-3670	224	19	euler	euler	PROPN
cana-3670	224	20	method	method	NOUN
cana-3670	224	21	for	for	ADP
cana-3670	224	22	fuzzy	fuzzy	ADJ
cana-3670	224	23	differential	differential	ADJ
cana-3670	224	24	equation	equation	NOUN
cana-3670	224	25	.	.	PUNCT
cana-3670	225	1	symmetry	symmetry	PROPN
cana-3670	225	2	.	.	PUNCT
cana-3670	226	1	2022	2022	NUM
cana-3670	226	2	;	;	PUNCT
cana-3670	226	3	14(16	14(16	NUM
cana-3670	226	4	)	)	PUNCT
cana-3670	226	5	,	,	PUNCT
cana-3670	226	6	1	1	NUM
cana-3670	226	7	-	-	SYM
cana-3670	226	8	13	13	NUM
cana-3670	226	9	.	.	PUNCT
cana-3670	227	1	available	available	ADJ
cana-3670	227	2	from	from	ADP
cana-3670	227	3	:	:	PUNCT
cana-3670	227	4	https://doi.org/10.3390/sym14061279	https://doi.org/10.3390/sym14061279	NOUN
cana-3670	227	5	.	.	PUNCT
cana-3670	228	1	[	[	X
cana-3670	228	2	3	3	X
cana-3670	228	3	]	]	PUNCT
cana-3670	228	4	ahmady	ahmady	NOUN
cana-3670	228	5	,	,	PUNCT
cana-3670	228	6	n.	n.	NOUN
cana-3670	228	7	,	,	PUNCT
cana-3670	228	8	allahviranloo	allahviranloo	NOUN
cana-3670	228	9	,	,	PUNCT
cana-3670	228	10	t.	t.	PROPN
cana-3670	228	11	&	&	CCONJ
cana-3670	228	12	ahmady	ahmady	PROPN
cana-3670	228	13	,	,	PUNCT
cana-3670	228	14	e.	e.	PROPN
cana-3670	228	15	a	a	DET
cana-3670	228	16	modified	modify	VERB
cana-3670	228	17	euler	euler	NOUN
cana-3670	228	18	method	method	NOUN
cana-3670	228	19	for	for	ADP
cana-3670	228	20	solving	solve	VERB
cana-3670	228	21	fuzzy	fuzzy	ADJ
cana-3670	228	22	differential	differential	ADJ
cana-3670	228	23	equations	equation	NOUN
cana-3670	228	24	under	under	ADP
cana-3670	228	25	generalized	generalized	ADJ
cana-3670	228	26	differentiability	differentiability	NOUN
cana-3670	228	27	.	.	PUNCT
cana-3670	229	1	computational	computational	ADJ
cana-3670	229	2	and	and	CCONJ
cana-3670	229	3	applied	applied	ADJ
cana-3670	229	4	mathematics	mathematic	NOUN
cana-3670	229	5	.	.	PUNCT
cana-3670	230	1	2020	2020	NUM
cana-3670	230	2	;	;	PUNCT
cana-3670	230	3	39(2	39(2	NUM
cana-3670	230	4	):	):	PUNCT
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cana-3670	232	42	.	.	PUNCT
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cana-3670	234	8	.	.	PUNCT
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cana-3670	235	3	:	:	PUNCT
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cana-3670	237	9	)	)	PUNCT
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cana-3670	244	4	):	):	PUNCT
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cana-3670	244	8	.	.	PUNCT
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