id	sid	tid	token	lemma	pos
cana-941	1	1	communications	communication	NOUN
cana-941	1	2	on	on	ADP
cana-941	1	3	applied	apply	VERB
cana-941	1	4	nonlinear	nonlinear	ADJ
cana-941	1	5	analysis	analysis	NOUN
cana-941	1	6	issn	issn	NOUN
cana-941	1	7	:	:	PUNCT
cana-941	1	8	1074	1074	NUM
cana-941	1	9	-	-	PUNCT
cana-941	1	10	133x	133x	NUM
cana-941	1	11	vol	vol	NOUN
cana-941	1	12	31	31	NUM
cana-941	1	13	no	no	NOUN
cana-941	1	14	.	.	PUNCT
cana-941	2	1	4s	4s	NUM
cana-941	2	2	(	(	PUNCT
cana-941	2	3	2024	2024	NUM
cana-941	2	4	)	)	PUNCT
cana-941	2	5	440	440	NUM
cana-941	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	3	2	analytical	analytical	ADJ
cana-941	3	3	study	study	NOUN
cana-941	3	4	of	of	ADP
cana-941	3	5	coupled	couple	VERB
cana-941	3	6	schrödinger	schrödinger	ADJ
cana-941	3	7	equations	equation	NOUN
cana-941	3	8	with	with	ADP
cana-941	3	9	fractional	fractional	ADJ
cana-941	3	10	damping	damping	NOUN
cana-941	3	11	:	:	PUNCT
cana-941	3	12	decay	decay	VERB
cana-941	3	13	solutions	solution	NOUN
cana-941	3	14	and	and	CCONJ
cana-941	3	15	stability	stability	NOUN
cana-941	3	16	analysis	analysis	NOUN
cana-941	3	17	dr	dr	PROPN
cana-941	3	18	.	.	PROPN
cana-941	3	19	eric	eric	PROPN
cana-941	3	20	howard1	howard1	PROPN
cana-941	3	21	,	,	PUNCT
cana-941	3	22	vikas	vikas	PROPN
cana-941	3	23	kumar2	kumar2	PROPN
cana-941	3	24	,	,	PUNCT
cana-941	3	25	dr	dr	PROPN
cana-941	3	26	nand	nand	PROPN
cana-941	3	27	kumar3	kumar3	PROPN
cana-941	3	28	1	1	NUM
cana-941	3	29	department	department	NOUN
cana-941	3	30	of	of	ADP
cana-941	3	31	physics	physics	PROPN
cana-941	3	32	and	and	CCONJ
cana-941	3	33	astronomy	astronomy	NOUN
cana-941	3	34	,	,	PUNCT
cana-941	3	35	macquarie	macquarie	PROPN
cana-941	3	36	university	university	PROPN
cana-941	3	37	,	,	PUNCT
cana-941	3	38	australia.email	australia.email	PROPN
cana-941	4	1	i	i	PROPN
cana-941	4	2	d	d	PROPN
cana-941	4	3	eric.howard@mq.edu.au	eric.howard@mq.edu.au	PROPN
cana-941	4	4	.	.	PUNCT
cana-941	5	1	orcid	orcid	NOUN
cana-941	5	2	:	:	PUNCT
cana-941	5	3	0000	0000	NUM
cana-941	5	4	-	-	PUNCT
cana-941	5	5	0002	0002	NUM
cana-941	5	6	-	-	PUNCT
cana-941	5	7	8133	8133	NUM
cana-941	5	8	-	-	PUNCT
cana-941	5	9	8323	8323	NUM
cana-941	5	10	.	.	PUNCT
cana-941	6	1	2	2	NUM
cana-941	6	2	assistant	assistant	NOUN
cana-941	6	3	professor	professor	NOUN
cana-941	6	4	,	,	PUNCT
cana-941	6	5	school	school	NOUN
cana-941	6	6	of	of	ADP
cana-941	6	7	applied	applied	ADJ
cana-941	6	8	and	and	CCONJ
cana-941	6	9	life	life	NOUN
cana-941	6	10	sciences	science	NOUN
cana-941	6	11	,	,	PUNCT
cana-941	6	12	uttaranchal	uttaranchal	PROPN
cana-941	6	13	university	university	NOUN
cana-941	6	14	,	,	PUNCT
cana-941	6	15	dehradun	dehradun	PROPN
cana-941	6	16	,	,	PUNCT
cana-941	6	17	248007	248007	NUM
cana-941	6	18	,	,	PUNCT
cana-941	6	19	india	india	PROPN
cana-941	6	20	,	,	PUNCT
cana-941	6	21	orcid	orcid	PROPN
cana-941	6	22	:	:	PUNCT
cana-941	6	23	0000	0000	NUM
cana-941	6	24	-	-	PUNCT
cana-941	6	25	0003	0003	NUM
cana-941	6	26	-	-	PUNCT
cana-941	6	27	3170	3170	NUM
cana-941	6	28	-	-	PUNCT
cana-941	6	29	7933	7933	NUM
cana-941	6	30	.	.	PUNCT
cana-941	7	1	3	3	NUM
cana-941	7	2	assistant	assistant	NOUN
cana-941	7	3	professor	professor	NOUN
cana-941	7	4	,	,	PUNCT
cana-941	7	5	department	department	NOUN
cana-941	7	6	of	of	ADP
cana-941	7	7	physics	physics	PROPN
cana-941	7	8	,	,	PUNCT
cana-941	7	9	jawaharlal	jawaharlal	PROPN
cana-941	7	10	nehru	nehru	PROPN
cana-941	7	11	memorial	memorial	PROPN
cana-941	7	12	pg	pg	PROPN
cana-941	7	13	college	college	PROPN
cana-941	7	14	,	,	PUNCT
cana-941	7	15	barabanki	barabanki	PROPN
cana-941	7	16	,	,	PUNCT
cana-941	7	17	up	up	ADP
cana-941	7	18	,	,	PUNCT
cana-941	7	19	india.email	india.email	PROPN
cana-941	7	20	:	:	PUNCT
cana-941	7	21	nandkumar350@gmail.com	nandkumar350@gmail.com	X
cana-941	7	22	orcid	orcid	NOUN
cana-941	7	23	:	:	PUNCT
cana-941	7	24	0009	0009	NUM
cana-941	7	25	-	-	SYM
cana-941	7	26	0004	0004	NUM
cana-941	7	27	-	-	PUNCT
cana-941	7	28	3139	3139	NUM
cana-941	7	29	-	-	PUNCT
cana-941	7	30	3385	3385	NUM
cana-941	7	31	.	.	PUNCT
cana-941	8	1	article	article	NOUN
cana-941	8	2	history	history	NOUN
cana-941	8	3	:	:	PUNCT
cana-941	8	4	received	receive	VERB
cana-941	8	5	:	:	PUNCT
cana-941	8	6	18	18	NUM
cana-941	8	7	-	-	PUNCT
cana-941	8	8	04	04	NUM
cana-941	8	9	-	-	PUNCT
cana-941	8	10	2024	2024	NUM
cana-941	8	11	revised	revise	VERB
cana-941	8	12	:	:	PUNCT
cana-941	8	13	10	10	NUM
cana-941	8	14	-	-	SYM
cana-941	8	15	06	06	NUM
cana-941	8	16	-	-	PUNCT
cana-941	8	17	2024	2024	NUM
cana-941	8	18	accepted	accept	VERB
cana-941	8	19	:	:	PUNCT
cana-941	8	20	23	23	NUM
cana-941	8	21	-	-	SYM
cana-941	8	22	06	06	NUM
cana-941	8	23	-	-	PUNCT
cana-941	8	24	2024	2024	NUM
cana-941	8	25	abstract	abstract	NOUN
cana-941	8	26	:	:	PUNCT
cana-941	8	27	coupled	couple	VERB
cana-941	8	28	schrödinger	schrödinger	ADJ
cana-941	8	29	equations	equation	NOUN
cana-941	8	30	with	with	ADP
cana-941	8	31	fractional	fractional	ADJ
cana-941	8	32	damping	damping	NOUN
cana-941	8	33	represent	represent	VERB
cana-941	8	34	a	a	DET
cana-941	8	35	complex	complex	ADJ
cana-941	8	36	yet	yet	CCONJ
cana-941	8	37	fascinating	fascinating	ADJ
cana-941	8	38	area	area	NOUN
cana-941	8	39	of	of	ADP
cana-941	8	40	study	study	NOUN
cana-941	8	41	in	in	ADP
cana-941	8	42	quantum	quantum	ADJ
cana-941	8	43	mechanics	mechanic	NOUN
cana-941	8	44	.	.	PUNCT
cana-941	9	1	this	this	DET
cana-941	9	2	research	research	NOUN
cana-941	9	3	paper	paper	NOUN
cana-941	9	4	delves	delf	NOUN
cana-941	9	5	into	into	ADP
cana-941	9	6	the	the	DET
cana-941	9	7	analytical	analytical	ADJ
cana-941	9	8	investigation	investigation	NOUN
cana-941	9	9	of	of	ADP
cana-941	9	10	such	such	ADJ
cana-941	9	11	systems	system	NOUN
cana-941	9	12	,	,	PUNCT
cana-941	9	13	focusing	focus	VERB
cana-941	9	14	on	on	ADP
cana-941	9	15	deriving	derive	VERB
cana-941	9	16	decay	decay	NOUN
cana-941	9	17	solutions	solution	NOUN
cana-941	9	18	and	and	CCONJ
cana-941	9	19	conducting	conduct	VERB
cana-941	9	20	stability	stability	NOUN
cana-941	9	21	analysis	analysis	NOUN
cana-941	9	22	.	.	PUNCT
cana-941	10	1	by	by	ADP
cana-941	10	2	combining	combine	VERB
cana-941	10	3	theoretical	theoretical	ADJ
cana-941	10	4	frameworks	framework	NOUN
cana-941	10	5	with	with	ADP
cana-941	10	6	numerical	numerical	ADJ
cana-941	10	7	methods	method	NOUN
cana-941	10	8	,	,	PUNCT
cana-941	10	9	we	we	PRON
cana-941	10	10	explore	explore	VERB
cana-941	10	11	the	the	DET
cana-941	10	12	behavior	behavior	NOUN
cana-941	10	13	of	of	ADP
cana-941	10	14	these	these	DET
cana-941	10	15	coupled	couple	VERB
cana-941	10	16	equations	equation	NOUN
cana-941	10	17	under	under	ADP
cana-941	10	18	varying	vary	VERB
cana-941	10	19	parameters	parameter	NOUN
cana-941	10	20	and	and	CCONJ
cana-941	10	21	conditions	condition	NOUN
cana-941	10	22	.	.	PUNCT
cana-941	11	1	through	through	ADP
cana-941	11	2	a	a	DET
cana-941	11	3	thorough	thorough	ADJ
cana-941	11	4	analysis	analysis	NOUN
cana-941	11	5	,	,	PUNCT
cana-941	11	6	we	we	PRON
cana-941	11	7	aim	aim	VERB
cana-941	11	8	to	to	PART
cana-941	11	9	deepen	deepen	VERB
cana-941	11	10	our	our	PRON
cana-941	11	11	understanding	understanding	NOUN
cana-941	11	12	of	of	ADP
cana-941	11	13	the	the	DET
cana-941	11	14	dynamics	dynamic	NOUN
cana-941	11	15	and	and	CCONJ
cana-941	11	16	stability	stability	NOUN
cana-941	11	17	properties	property	NOUN
cana-941	11	18	of	of	ADP
cana-941	11	19	quantum	quantum	NOUN
cana-941	11	20	systems	system	NOUN
cana-941	11	21	subject	subject	ADJ
cana-941	11	22	to	to	ADP
cana-941	11	23	fractional	fractional	ADJ
cana-941	11	24	damping	damping	NOUN
cana-941	11	25	,	,	PUNCT
cana-941	11	26	with	with	ADP
cana-941	11	27	potential	potential	ADJ
cana-941	11	28	implications	implication	NOUN
cana-941	11	29	for	for	ADP
cana-941	11	30	diverse	diverse	ADJ
cana-941	11	31	fields	field	NOUN
cana-941	11	32	ranging	range	VERB
cana-941	11	33	from	from	ADP
cana-941	11	34	quantum	quantum	ADJ
cana-941	11	35	mechanics	mechanic	NOUN
cana-941	11	36	to	to	ADP
cana-941	11	37	condensed	condense	VERB
cana-941	11	38	matter	matter	NOUN
cana-941	11	39	physics	physics	NOUN
cana-941	11	40	.	.	PUNCT
cana-941	12	1	keywords	keyword	NOUN
cana-941	12	2	:	:	PUNCT
cana-941	12	3	coupled	couple	VERB
cana-941	12	4	schrödinger	schrödinger	ADJ
cana-941	12	5	equations	equation	NOUN
cana-941	12	6	,	,	PUNCT
cana-941	12	7	equations	equation	NOUN
cana-941	12	8	with	with	ADP
cana-941	12	9	fractional	fractional	ADJ
cana-941	12	10	damping	damping	NOUN
cana-941	12	11	,	,	PUNCT
cana-941	12	12	quantum	quantum	NOUN
cana-941	12	13	mechanics	mechanic	NOUN
cana-941	12	14	.	.	PUNCT
cana-941	13	1	1	1	X
cana-941	13	2	.	.	X
cana-941	13	3	introduction	introduction	NOUN
cana-941	13	4	coupled	couple	VERB
cana-941	13	5	schrödinger	schrödinger	ADJ
cana-941	13	6	equations	equation	NOUN
cana-941	13	7	arise	arise	VERB
cana-941	13	8	in	in	ADP
cana-941	13	9	many	many	ADJ
cana-941	13	10	areas	area	NOUN
cana-941	13	11	of	of	ADP
cana-941	13	12	physics	physics	NOUN
cana-941	13	13	,	,	PUNCT
cana-941	13	14	describing	describe	VERB
cana-941	13	15	the	the	DET
cana-941	13	16	evolution	evolution	NOUN
cana-941	13	17	of	of	ADP
cana-941	13	18	quantum	quantum	NOUN
cana-941	13	19	systems	system	NOUN
cana-941	13	20	composed	compose	VERB
cana-941	13	21	of	of	ADP
cana-941	13	22	multiple	multiple	ADJ
cana-941	13	23	interacting	interact	VERB
cana-941	13	24	particles	particle	NOUN
cana-941	13	25	.	.	PUNCT
cana-941	14	1	when	when	SCONJ
cana-941	14	2	coupled	couple	VERB
cana-941	14	3	with	with	ADP
cana-941	14	4	fractional	fractional	ADJ
cana-941	14	5	damping	damping	NOUN
cana-941	14	6	,	,	PUNCT
cana-941	14	7	these	these	DET
cana-941	14	8	equations	equation	NOUN
cana-941	14	9	present	present	VERB
cana-941	14	10	intriguing	intriguing	ADJ
cana-941	14	11	challenges	challenge	NOUN
cana-941	14	12	and	and	CCONJ
cana-941	14	13	opportunities	opportunity	NOUN
cana-941	14	14	for	for	ADP
cana-941	14	15	study	study	NOUN
cana-941	14	16	.	.	PUNCT
cana-941	15	1	in	in	ADP
cana-941	15	2	this	this	DET
cana-941	15	3	paper	paper	NOUN
cana-941	15	4	,	,	PUNCT
cana-941	15	5	we	we	PRON
cana-941	15	6	embark	embark	VERB
cana-941	15	7	on	on	ADP
cana-941	15	8	an	an	DET
cana-941	15	9	analytical	analytical	ADJ
cana-941	15	10	journey	journey	NOUN
cana-941	15	11	to	to	PART
cana-941	15	12	investigate	investigate	VERB
cana-941	15	13	the	the	DET
cana-941	15	14	decay	decay	NOUN
cana-941	15	15	solutions	solution	NOUN
cana-941	15	16	and	and	CCONJ
cana-941	15	17	stability	stability	NOUN
cana-941	15	18	properties	property	NOUN
cana-941	15	19	of	of	ADP
cana-941	15	20	such	such	ADJ
cana-941	15	21	systems	system	NOUN
cana-941	15	22	.	.	PUNCT
cana-941	16	1	by	by	ADP
cana-941	16	2	elucidating	elucidate	VERB
cana-941	16	3	the	the	DET
cana-941	16	4	dynamics	dynamic	NOUN
cana-941	16	5	of	of	ADP
cana-941	16	6	coupled	couple	VERB
cana-941	16	7	quantum	quantum	NOUN
cana-941	16	8	systems	system	NOUN
cana-941	16	9	under	under	ADP
cana-941	16	10	fractional	fractional	ADJ
cana-941	16	11	damping	damping	NOUN
cana-941	16	12	,	,	PUNCT
cana-941	16	13	we	we	PRON
cana-941	16	14	aim	aim	VERB
cana-941	16	15	to	to	PART
cana-941	16	16	contribute	contribute	VERB
cana-941	16	17	to	to	ADP
cana-941	16	18	the	the	DET
cana-941	16	19	broader	broad	ADJ
cana-941	16	20	understanding	understanding	NOUN
cana-941	16	21	of	of	ADP
cana-941	16	22	quantum	quantum	ADJ
cana-941	16	23	phenomena	phenomenon	NOUN
cana-941	16	24	and	and	CCONJ
cana-941	16	25	their	their	PRON
cana-941	16	26	practical	practical	ADJ
cana-941	16	27	implications	implication	NOUN
cana-941	16	28	.	.	PUNCT
cana-941	17	1	2	2	X
cana-941	17	2	.	.	X
cana-941	17	3	literature	literature	NOUN
cana-941	17	4	review	review	VERB
cana-941	17	5	the	the	DET
cana-941	17	6	study	study	NOUN
cana-941	17	7	of	of	ADP
cana-941	17	8	coupled	couple	VERB
cana-941	17	9	schrödinger	schrödinger	ADJ
cana-941	17	10	equations	equation	NOUN
cana-941	17	11	with	with	ADP
cana-941	17	12	fractional	fractional	ADJ
cana-941	17	13	damping	damping	NOUN
cana-941	17	14	has	have	AUX
cana-941	17	15	garnered	garner	VERB
cana-941	17	16	significant	significant	ADJ
cana-941	17	17	attention	attention	NOUN
cana-941	17	18	in	in	ADP
cana-941	17	19	recent	recent	ADJ
cana-941	17	20	years	year	NOUN
cana-941	17	21	,	,	PUNCT
cana-941	17	22	drawing	draw	VERB
cana-941	17	23	upon	upon	SCONJ
cana-941	17	24	a	a	DET
cana-941	17	25	rich	rich	ADJ
cana-941	17	26	body	body	NOUN
cana-941	17	27	of	of	ADP
cana-941	17	28	literature	literature	NOUN
cana-941	17	29	spanning	span	VERB
cana-941	17	30	various	various	ADJ
cana-941	17	31	disciplines	discipline	NOUN
cana-941	17	32	within	within	ADP
cana-941	17	33	physics	physics	NOUN
cana-941	17	34	and	and	CCONJ
cana-941	17	35	applied	apply	VERB
cana-941	17	36	mathematics	mathematic	NOUN
cana-941	17	37	.	.	PUNCT
cana-941	18	1	this	this	DET
cana-941	18	2	section	section	NOUN
cana-941	18	3	provides	provide	VERB
cana-941	18	4	a	a	DET
cana-941	18	5	synthesis	synthesis	NOUN
cana-941	18	6	of	of	ADP
cana-941	18	7	key	key	ADJ
cana-941	18	8	findings	finding	NOUN
cana-941	18	9	from	from	ADP
cana-941	18	10	previous	previous	ADJ
cana-941	18	11	research	research	NOUN
cana-941	18	12	,	,	PUNCT
cana-941	18	13	highlighting	highlight	VERB
cana-941	18	14	methodologies	methodology	NOUN
cana-941	18	15	,	,	PUNCT
cana-941	18	16	theoretical	theoretical	ADJ
cana-941	18	17	frameworks	framework	NOUN
cana-941	18	18	,	,	PUNCT
cana-941	18	19	and	and	CCONJ
cana-941	18	20	experimental	experimental	ADJ
cana-941	18	21	observations	observation	NOUN
cana-941	18	22	,	,	PUNCT
cana-941	18	23	while	while	SCONJ
cana-941	18	24	also	also	ADV
cana-941	18	25	identifying	identify	VERB
cana-941	18	26	gaps	gap	NOUN
cana-941	18	27	and	and	CCONJ
cana-941	18	28	opportunities	opportunity	NOUN
cana-941	18	29	for	for	ADP
cana-941	18	30	further	further	ADJ
cana-941	18	31	investigation	investigation	NOUN
cana-941	18	32	.	.	PUNCT
cana-941	19	1	2.1	2.1	NUM
cana-941	19	2	.	.	PUNCT
cana-941	19	3	fractional	fractional	ADJ
cana-941	19	4	calculus	calculus	NOUN
cana-941	19	5	in	in	ADP
cana-941	19	6	quantum	quantum	ADJ
cana-941	19	7	mechanics	mechanic	NOUN
cana-941	19	8	:	:	PUNCT
cana-941	19	9	fractional	fractional	ADJ
cana-941	19	10	calculus	calculus	NOUN
cana-941	19	11	has	have	AUX
cana-941	19	12	emerged	emerge	VERB
cana-941	19	13	as	as	ADP
cana-941	19	14	a	a	DET
cana-941	19	15	powerful	powerful	ADJ
cana-941	19	16	tool	tool	NOUN
cana-941	19	17	for	for	ADP
cana-941	19	18	modeling	model	VERB
cana-941	19	19	complex	complex	ADJ
cana-941	19	20	systems	system	NOUN
cana-941	19	21	exhibiting	exhibit	VERB
cana-941	19	22	non	non	ADJ
cana-941	19	23	-	-	ADJ
cana-941	19	24	local	local	ADJ
cana-941	19	25	or	or	CCONJ
cana-941	19	26	memory	memory	NOUN
cana-941	19	27	-	-	PUNCT
cana-941	19	28	dependent	dependent	ADJ
cana-941	19	29	behavior	behavior	NOUN
cana-941	19	30	.	.	PUNCT
cana-941	20	1	in	in	ADP
cana-941	20	2	the	the	DET
cana-941	20	3	context	context	NOUN
cana-941	20	4	of	of	ADP
cana-941	20	5	quantum	quantum	ADJ
cana-941	20	6	mechanics	mechanic	NOUN
cana-941	20	7	,	,	PUNCT
cana-941	20	8	fractional	fractional	ADJ
cana-941	20	9	derivatives	derivative	NOUN
cana-941	20	10	offer	offer	VERB
cana-941	20	11	a	a	DET
cana-941	20	12	means	mean	NOUN
cana-941	20	13	to	to	PART
cana-941	20	14	incorporate	incorporate	VERB
cana-941	20	15	non	non	ADJ
cana-941	20	16	-	-	ADJ
cana-941	20	17	local	local	ADJ
cana-941	20	18	effects	effect	NOUN
cana-941	20	19	into	into	ADP
cana-941	20	20	the	the	DET
cana-941	20	21	mailto:eric.howard@mq.edu.au	mailto:eric.howard@mq.edu.au	PROPN
cana-941	20	22	communications	communication	NOUN
cana-941	20	23	on	on	ADP
cana-941	20	24	applied	apply	VERB
cana-941	20	25	nonlinear	nonlinear	ADJ
cana-941	20	26	analysis	analysis	NOUN
cana-941	20	27	issn	issn	NOUN
cana-941	20	28	:	:	PUNCT
cana-941	20	29	1074	1074	NUM
cana-941	20	30	-	-	PUNCT
cana-941	20	31	133x	133x	NUM
cana-941	20	32	vol	vol	NOUN
cana-941	20	33	31	31	NUM
cana-941	20	34	no	no	NOUN
cana-941	20	35	.	.	PUNCT
cana-941	21	1	4s	4s	NUM
cana-941	21	2	(	(	PUNCT
cana-941	21	3	2024	2024	NUM
cana-941	21	4	)	)	PUNCT
cana-941	21	5	441	441	NUM
cana-941	21	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	21	7	schrödinger	schrödinger	ADJ
cana-941	21	8	equation	equation	NOUN
cana-941	21	9	,	,	PUNCT
cana-941	21	10	allowing	allow	VERB
cana-941	21	11	for	for	ADP
cana-941	21	12	a	a	DET
cana-941	21	13	more	more	ADV
cana-941	21	14	comprehensive	comprehensive	ADJ
cana-941	21	15	description	description	NOUN
cana-941	21	16	of	of	ADP
cana-941	21	17	physical	physical	ADJ
cana-941	21	18	phenomena	phenomenon	NOUN
cana-941	21	19	.	.	PUNCT
cana-941	22	1	several	several	ADJ
cana-941	22	2	studies	study	NOUN
cana-941	22	3	have	have	AUX
cana-941	22	4	explored	explore	VERB
cana-941	22	5	the	the	DET
cana-941	22	6	application	application	NOUN
cana-941	22	7	of	of	ADP
cana-941	22	8	fractional	fractional	ADJ
cana-941	22	9	calculus	calculus	NOUN
cana-941	22	10	in	in	ADP
cana-941	22	11	quantum	quantum	ADJ
cana-941	22	12	mechanics	mechanic	NOUN
cana-941	22	13	,	,	PUNCT
cana-941	22	14	elucidating	elucidate	VERB
cana-941	22	15	its	its	PRON
cana-941	22	16	utility	utility	NOUN
cana-941	22	17	in	in	ADP
cana-941	22	18	addressing	address	VERB
cana-941	22	19	phenomena	phenomenon	NOUN
cana-941	22	20	such	such	ADJ
cana-941	22	21	as	as	ADP
cana-941	22	22	anomalous	anomalous	ADJ
cana-941	22	23	diffusion	diffusion	NOUN
cana-941	22	24	and	and	CCONJ
cana-941	22	25	fractional	fractional	ADJ
cana-941	22	26	quantum	quantum	ADJ
cana-941	22	27	mechanics	mechanic	NOUN
cana-941	22	28	.	.	PUNCT
cana-941	23	1	equation	equation	NOUN
cana-941	23	2	1	1	NUM
cana-941	23	3	:	:	PUNCT
cana-941	23	4	𝑖ℏ	𝑖ℏ	X
cana-941	23	5	𝜕𝜓	𝜕𝜓	NOUN
cana-941	23	6	𝜕𝑡	𝜕𝑡	NOUN
cana-941	23	7	=	=	SYM
cana-941	24	1	𝐻𝜓	𝐻𝜓	PROPN
cana-941	24	2	−	−	PROPN
cana-941	24	3	𝛾(−𝛥)𝛼𝜓	𝛾(−𝛥)𝛼𝜓	NOUN
cana-941	24	4	equation	equation	NOUN
cana-941	24	5	2	2	NUM
cana-941	24	6	:	:	PUNCT
cana-941	24	7	(	(	PUNCT
cana-941	24	8	−𝛥)𝛼𝜓	−𝛥)𝛼𝜓	PROPN
cana-941	24	9	=	=	SYM
cana-941	24	10	∫	∫	PROPN
cana-941	24	11	₀ˣ	₀ˣ	NOUN
cana-941	24	12	(	(	PUNCT
cana-941	24	13	𝑥	𝑥	NOUN
cana-941	24	14	−	−	PROPN
cana-941	24	15	𝑡)𝛼	𝑡)𝛼	NOUN
cana-941	24	16	−	−	PROPN
cana-941	24	17	1	1	NUM
cana-941	24	18	𝜕²𝜓	𝜕²𝜓	PROPN
cana-941	24	19	𝜕𝑡²	𝜕𝑡²	PROPN
cana-941	24	20	(	(	PUNCT
cana-941	24	21	𝑡	𝑡	PROPN
cana-941	24	22	)	)	PUNCT
cana-941	24	23	𝑑𝑡	𝑑𝑡	ADP
cana-941	24	24	2.2	2.2	NUM
cana-941	24	25	.	.	PUNCT
cana-941	24	26	coupled	couple	VERB
cana-941	24	27	schrödinger	schrödinger	ADJ
cana-941	24	28	equations	equation	NOUN
cana-941	24	29	and	and	CCONJ
cana-941	24	30	quantum	quantum	NOUN
cana-941	24	31	systems	system	NOUN
cana-941	24	32	:	:	PUNCT
cana-941	24	33	the	the	DET
cana-941	24	34	study	study	NOUN
cana-941	24	35	of	of	ADP
cana-941	24	36	coupled	couple	VERB
cana-941	24	37	schrödinger	schrödinger	ADJ
cana-941	24	38	equations	equation	NOUN
cana-941	24	39	is	be	AUX
cana-941	24	40	central	central	ADJ
cana-941	24	41	to	to	ADP
cana-941	24	42	understanding	understand	VERB
cana-941	24	43	the	the	DET
cana-941	24	44	behavior	behavior	NOUN
cana-941	24	45	of	of	ADP
cana-941	24	46	quantum	quantum	NOUN
cana-941	24	47	systems	system	NOUN
cana-941	24	48	composed	compose	VERB
cana-941	24	49	of	of	ADP
cana-941	24	50	multiple	multiple	ADJ
cana-941	24	51	interacting	interact	VERB
cana-941	24	52	particles	particle	NOUN
cana-941	24	53	.	.	PUNCT
cana-941	25	1	such	such	ADJ
cana-941	25	2	systems	system	NOUN
cana-941	25	3	arise	arise	VERB
cana-941	25	4	in	in	ADP
cana-941	25	5	various	various	ADJ
cana-941	25	6	contexts	contexts	NOUN
cana-941	25	7	,	,	PUNCT
cana-941	25	8	including	include	VERB
cana-941	25	9	atomic	atomic	ADJ
cana-941	25	10	and	and	CCONJ
cana-941	25	11	molecular	molecular	ADJ
cana-941	25	12	physics	physics	NOUN
cana-941	25	13	,	,	PUNCT
cana-941	25	14	quantum	quantum	NOUN
cana-941	25	15	optics	optic	NOUN
cana-941	25	16	,	,	PUNCT
cana-941	25	17	and	and	CCONJ
cana-941	25	18	condensed	condense	VERB
cana-941	25	19	matter	matter	NOUN
cana-941	25	20	physics	physics	NOUN
cana-941	25	21	.	.	PUNCT
cana-941	26	1	previous	previous	ADJ
cana-941	26	2	research	research	NOUN
cana-941	26	3	has	have	AUX
cana-941	26	4	investigated	investigate	VERB
cana-941	26	5	diverse	diverse	ADJ
cana-941	26	6	aspects	aspect	NOUN
cana-941	26	7	of	of	ADP
cana-941	26	8	coupled	couple	VERB
cana-941	26	9	quantum	quantum	NOUN
cana-941	26	10	systems	system	NOUN
cana-941	26	11	,	,	PUNCT
cana-941	26	12	ranging	range	VERB
cana-941	26	13	from	from	ADP
cana-941	26	14	the	the	DET
cana-941	26	15	formation	formation	NOUN
cana-941	26	16	of	of	ADP
cana-941	26	17	bound	bind	VERB
cana-941	26	18	states	state	NOUN
cana-941	26	19	to	to	ADP
cana-941	26	20	the	the	DET
cana-941	26	21	emergence	emergence	NOUN
cana-941	26	22	of	of	ADP
cana-941	26	23	quantum	quantum	ADJ
cana-941	26	24	entanglement	entanglement	NOUN
cana-941	26	25	and	and	CCONJ
cana-941	26	26	collective	collective	ADJ
cana-941	26	27	phenomena	phenomenon	NOUN
cana-941	26	28	.	.	PUNCT
cana-941	27	1	equation	equation	NOUN
cana-941	27	2	3	3	NUM
cana-941	27	3	:	:	SYM
cana-941	27	4	𝑖ℏ	𝑖ℏ	ADV
cana-941	27	5	𝜕𝜓₁/𝜕𝑡	𝜕𝜓₁/𝜕𝑡	NOUN
cana-941	27	6	=	=	PUNCT
cana-941	27	7	𝐻₁𝜓₁	𝐻₁𝜓₁	ADJ
cana-941	27	8	−	−	PROPN
cana-941	27	9	𝛾₁(−𝛥)^𝛼₁𝜓₁	𝛾₁(−𝛥)^𝛼₁𝜓₁	NOUN
cana-941	27	10	+	+	CCONJ
cana-941	27	11	𝑔₁₂𝜓₂	𝑔₁₂𝜓₂	NOUN
cana-941	27	12	(	(	PUNCT
cana-941	27	13	−𝛥)^𝛼₂𝜓₂	−𝛥)^𝛼₂𝜓₂	NOUN
cana-941	27	14	=	=	SYM
cana-941	27	15	∫	∫	PROPN
cana-941	27	16	₀ˣ	₀ˣ	NOUN
cana-941	27	17	(	(	PUNCT
cana-941	27	18	𝑥²	𝑥²	VERB
cana-941	27	19	−	−	PROPN
cana-941	27	20	𝑡²)^𝛼₂	𝑡²)^𝛼₂	PROPN
cana-941	27	21	−	−	PROPN
cana-941	27	22	1	1	NUM
cana-941	27	23	𝜕²𝜓₂/𝜕𝑡²(𝑡	𝜕²𝜓₂/𝜕𝑡²(𝑡	NOUN
cana-941	27	24	)	)	PUNCT
cana-941	27	25	𝑑𝑡	𝑑𝑡	ADP
cana-941	27	26	𝑖ℏ	𝑖ℏ	X
cana-941	27	27	𝜕𝜓₂/𝜕𝑡	𝜕𝜓₂/𝜕𝑡	PROPN
cana-941	27	28	=	=	PUNCT
cana-941	27	29	𝐻₂𝜓₂	𝐻₂𝜓₂	ADJ
cana-941	27	30	−	−	PROPN
cana-941	27	31	𝛾₂(−𝛥)^𝛼₂𝜓₂	𝛾₂(−𝛥)^𝛼₂𝜓₂	X
cana-941	28	1	+	+	NUM
cana-941	28	2	𝑔₂₁𝜓₁	𝑔₂₁𝜓₁	PROPN
cana-941	28	3	2.3	2.3	NUM
cana-941	28	4	.	.	PUNCT
cana-941	29	1	fractional	fractional	ADJ
cana-941	29	2	damping	damp	VERB
cana-941	29	3	and	and	CCONJ
cana-941	29	4	dissipative	dissipative	ADJ
cana-941	29	5	dynamics	dynamic	NOUN
cana-941	29	6	:	:	PUNCT
cana-941	29	7	the	the	DET
cana-941	29	8	introduction	introduction	NOUN
cana-941	29	9	of	of	ADP
cana-941	29	10	fractional	fractional	ADJ
cana-941	29	11	damping	damp	VERB
cana-941	29	12	into	into	ADP
cana-941	29	13	quantum	quantum	ADJ
cana-941	29	14	mechanical	mechanical	ADJ
cana-941	29	15	systems	system	NOUN
cana-941	29	16	has	have	AUX
cana-941	29	17	provided	provide	VERB
cana-941	29	18	insights	insight	NOUN
cana-941	29	19	into	into	ADP
cana-941	29	20	dissipative	dissipative	ADJ
cana-941	29	21	dynamics	dynamic	NOUN
cana-941	29	22	and	and	CCONJ
cana-941	29	23	decoherence	decoherence	NOUN
cana-941	29	24	phenomena	phenomenon	NOUN
cana-941	29	25	.	.	PUNCT
cana-941	30	1	fractional	fractional	ADJ
cana-941	30	2	damping	damp	VERB
cana-941	30	3	models	model	NOUN
cana-941	30	4	,	,	PUNCT
cana-941	30	5	which	which	PRON
cana-941	30	6	generalize	generalize	VERB
cana-941	30	7	traditional	traditional	ADJ
cana-941	30	8	damping	damp	VERB
cana-941	30	9	terms	term	NOUN
cana-941	30	10	,	,	PUNCT
cana-941	30	11	offer	offer	VERB
cana-941	30	12	a	a	DET
cana-941	30	13	framework	framework	NOUN
cana-941	30	14	for	for	ADP
cana-941	30	15	studying	study	VERB
cana-941	30	16	the	the	DET
cana-941	30	17	influence	influence	NOUN
cana-941	30	18	of	of	ADP
cana-941	30	19	environmental	environmental	ADJ
cana-941	30	20	interactions	interaction	NOUN
cana-941	30	21	on	on	ADP
cana-941	30	22	quantum	quantum	ADJ
cana-941	30	23	coherence	coherence	NOUN
cana-941	30	24	and	and	CCONJ
cana-941	30	25	the	the	DET
cana-941	30	26	emergence	emergence	NOUN
cana-941	30	27	of	of	ADP
cana-941	30	28	classical	classical	ADJ
cana-941	30	29	behavior	behavior	NOUN
cana-941	30	30	from	from	ADP
cana-941	30	31	quantum	quantum	ADJ
cana-941	30	32	systems	system	NOUN
cana-941	30	33	.	.	PUNCT
cana-941	31	1	previous	previous	ADJ
cana-941	31	2	studies	study	NOUN
cana-941	31	3	have	have	AUX
cana-941	31	4	investigated	investigate	VERB
cana-941	31	5	the	the	DET
cana-941	31	6	effects	effect	NOUN
cana-941	31	7	of	of	ADP
cana-941	31	8	fractional	fractional	ADJ
cana-941	31	9	damping	damp	VERB
cana-941	31	10	on	on	ADP
cana-941	31	11	various	various	ADJ
cana-941	31	12	quantum	quantum	ADJ
cana-941	31	13	processes	process	NOUN
cana-941	31	14	,	,	PUNCT
cana-941	31	15	including	include	VERB
cana-941	31	16	relaxation	relaxation	NOUN
cana-941	31	17	dynamics	dynamic	NOUN
cana-941	31	18	,	,	PUNCT
cana-941	31	19	energy	energy	NOUN
cana-941	31	20	transfer	transfer	NOUN
cana-941	31	21	,	,	PUNCT
cana-941	31	22	and	and	CCONJ
cana-941	31	23	decoherence	decoherence	NOUN
cana-941	31	24	.	.	PUNCT
cana-941	32	1	equation	equation	NOUN
cana-941	32	2	4	4	NUM
cana-941	32	3	:	:	PUNCT
cana-941	32	4	𝛾𝑒𝑓𝑓	𝛾𝑒𝑓𝑓	ADJ
cana-941	32	5	=	=	SYM
cana-941	32	6	𝛾(−𝛥)𝑎	𝛾(−𝛥)𝑎	PROPN
cana-941	32	7	2.4	2.4	NUM
cana-941	32	8	.	.	PUNCT
cana-941	33	1	numerical	numerical	ADJ
cana-941	33	2	and	and	CCONJ
cana-941	33	3	analytical	analytical	ADJ
cana-941	33	4	techniques	technique	NOUN
cana-941	33	5	for	for	ADP
cana-941	33	6	fractional	fractional	ADJ
cana-941	33	7	equations	equation	NOUN
cana-941	33	8	:	:	PUNCT
cana-941	33	9	analyzing	analyzing	NOUN
cana-941	33	10	coupled	couple	VERB
cana-941	33	11	schrödinger	schrödinger	ADJ
cana-941	33	12	equations	equation	NOUN
cana-941	33	13	with	with	ADP
cana-941	33	14	fractional	fractional	ADJ
cana-941	33	15	damping	damping	NOUN
cana-941	33	16	often	often	ADV
cana-941	33	17	necessitates	necessitate	VERB
cana-941	33	18	a	a	DET
cana-941	33	19	combination	combination	NOUN
cana-941	33	20	of	of	ADP
cana-941	33	21	numerical	numerical	ADJ
cana-941	33	22	and	and	CCONJ
cana-941	33	23	analytical	analytical	ADJ
cana-941	33	24	techniques	technique	NOUN
cana-941	33	25	.	.	PUNCT
cana-941	34	1	numerical	numerical	ADJ
cana-941	34	2	methods	method	NOUN
cana-941	34	3	such	such	ADJ
cana-941	34	4	as	as	ADP
cana-941	34	5	finite	finite	ADJ
cana-941	34	6	difference	difference	NOUN
cana-941	34	7	,	,	PUNCT
cana-941	34	8	spectral	spectral	ADJ
cana-941	34	9	,	,	PUNCT
cana-941	34	10	and	and	CCONJ
cana-941	34	11	monte	monte	PROPN
cana-941	34	12	carlo	carlo	PROPN
cana-941	34	13	simulations	simulation	NOUN
cana-941	34	14	are	be	AUX
cana-941	34	15	commonly	commonly	ADV
cana-941	34	16	employed	employ	VERB
cana-941	34	17	to	to	PART
cana-941	34	18	solve	solve	VERB
cana-941	34	19	fractional	fractional	ADJ
cana-941	34	20	differential	differential	ADJ
cana-941	34	21	equations	equation	NOUN
cana-941	34	22	numerically	numerically	ADV
cana-941	34	23	,	,	PUNCT
cana-941	34	24	allowing	allow	VERB
cana-941	34	25	for	for	ADP
cana-941	34	26	the	the	DET
cana-941	34	27	investigation	investigation	NOUN
cana-941	34	28	of	of	ADP
cana-941	34	29	complex	complex	ADJ
cana-941	34	30	quantum	quantum	ADJ
cana-941	34	31	systems	system	NOUN
cana-941	34	32	beyond	beyond	ADP
cana-941	34	33	analytical	analytical	ADJ
cana-941	34	34	solutions	solution	NOUN
cana-941	34	35	.	.	PUNCT
cana-941	35	1	furthermore	furthermore	ADV
cana-941	35	2	,	,	PUNCT
cana-941	35	3	analytical	analytical	ADJ
cana-941	35	4	approaches	approach	NOUN
cana-941	35	5	such	such	ADJ
cana-941	35	6	as	as	ADP
cana-941	35	7	perturbation	perturbation	NOUN
cana-941	35	8	theory	theory	NOUN
cana-941	35	9	,	,	PUNCT
cana-941	35	10	asymptotic	asymptotic	ADJ
cana-941	35	11	analysis	analysis	NOUN
cana-941	35	12	,	,	PUNCT
cana-941	35	13	and	and	CCONJ
cana-941	35	14	integral	integral	ADJ
cana-941	35	15	transforms	transform	NOUN
cana-941	35	16	provide	provide	VERB
cana-941	35	17	valuable	valuable	ADJ
cana-941	35	18	insights	insight	NOUN
cana-941	35	19	into	into	ADP
cana-941	35	20	the	the	DET
cana-941	35	21	qualitative	qualitative	ADJ
cana-941	35	22	behavior	behavior	NOUN
cana-941	35	23	and	and	CCONJ
cana-941	35	24	stability	stability	NOUN
cana-941	35	25	properties	property	NOUN
cana-941	35	26	of	of	ADP
cana-941	35	27	fractional	fractional	ADJ
cana-941	35	28	systems	system	NOUN
cana-941	35	29	.	.	PUNCT
cana-941	36	1	numerical	numerical	ADJ
cana-941	36	2	methods	method	NOUN
cana-941	36	3	:	:	PUNCT
cana-941	36	4	finite	finite	ADJ
cana-941	36	5	difference	difference	NOUN
cana-941	36	6	method	method	NOUN
cana-941	36	7	:	:	PUNCT
cana-941	36	8	the	the	DET
cana-941	36	9	finite	finite	ADJ
cana-941	36	10	difference	difference	NOUN
cana-941	36	11	method	method	NOUN
cana-941	36	12	discretizes	discretize	VERB
cana-941	36	13	fractional	fractional	ADJ
cana-941	36	14	derivatives	derivative	NOUN
cana-941	36	15	,	,	PUNCT
cana-941	36	16	providing	provide	VERB
cana-941	36	17	an	an	DET
cana-941	36	18	approximation	approximation	NOUN
cana-941	36	19	that	that	PRON
cana-941	36	20	enables	enable	VERB
cana-941	36	21	numerical	numerical	ADJ
cana-941	36	22	solutions	solution	NOUN
cana-941	36	23	.	.	PUNCT
cana-941	37	1	through	through	ADP
cana-941	37	2	the	the	DET
cana-941	37	3	use	use	NOUN
cana-941	37	4	of	of	ADP
cana-941	37	5	weighted	weight	VERB
cana-941	37	6	sums	sum	NOUN
cana-941	37	7	,	,	PUNCT
cana-941	37	8	the	the	DET
cana-941	37	9	method	method	NOUN
cana-941	37	10	effectively	effectively	ADV
cana-941	37	11	captures	capture	VERB
cana-941	37	12	the	the	DET
cana-941	37	13	fractional	fractional	ADJ
cana-941	37	14	behavior	behavior	NOUN
cana-941	37	15	:	:	PUNCT
cana-941	37	16	communications	communication	NOUN
cana-941	37	17	on	on	ADP
cana-941	37	18	applied	apply	VERB
cana-941	37	19	nonlinear	nonlinear	ADJ
cana-941	37	20	analysis	analysis	NOUN
cana-941	37	21	issn	issn	NOUN
cana-941	37	22	:	:	PUNCT
cana-941	37	23	1074	1074	NUM
cana-941	37	24	-	-	PUNCT
cana-941	37	25	133x	133x	NUM
cana-941	37	26	vol	vol	NOUN
cana-941	37	27	31	31	NUM
cana-941	37	28	no	no	NOUN
cana-941	37	29	.	.	PUNCT
cana-941	38	1	4s	4s	NUM
cana-941	38	2	(	(	PUNCT
cana-941	38	3	2024	2024	NUM
cana-941	38	4	)	)	PUNCT
cana-941	38	5	442	442	NUM
cana-941	38	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	39	1	∂𝛼𝜓	∂𝛼𝜓	PROPN
cana-941	39	2	∂𝑡𝛼	∂𝑡𝛼	PROPN
cana-941	39	3	≈	≈	PROPN
cana-941	39	4	1	1	NUM
cana-941	39	5	δ𝑡𝛼	δ𝑡𝛼	ADV
cana-941	39	6	∑	∑	ADP
cana-941	39	7	𝑤𝑘𝜓(𝑡𝑘	𝑤𝑘𝜓(𝑡𝑘	NOUN
cana-941	39	8	)	)	PUNCT
cana-941	39	9	n	n	NOUN
cana-941	39	10	𝑘=0	𝑘=0	ADP
cana-941	39	11	spectral	spectral	ADJ
cana-941	39	12	methods	method	NOUN
cana-941	39	13	:	:	PUNCT
cana-941	39	14	spectral	spectral	ADJ
cana-941	39	15	methods	method	NOUN
cana-941	39	16	exploit	exploit	VERB
cana-941	39	17	fourier	fourier	NOUN
cana-941	39	18	transforms	transform	VERB
cana-941	39	19	to	to	PART
cana-941	39	20	handle	handle	VERB
cana-941	39	21	fractional	fractional	ADJ
cana-941	39	22	differential	differential	ADJ
cana-941	39	23	equations	equation	NOUN
cana-941	39	24	.	.	PUNCT
cana-941	40	1	by	by	ADP
cana-941	40	2	transforming	transform	VERB
cana-941	40	3	the	the	DET
cana-941	40	4	equations	equation	NOUN
cana-941	40	5	into	into	ADP
cana-941	40	6	the	the	DET
cana-941	40	7	frequency	frequency	NOUN
cana-941	40	8	domain	domain	NOUN
cana-941	40	9	,	,	PUNCT
cana-941	40	10	spectral	spectral	ADJ
cana-941	40	11	methods	method	NOUN
cana-941	40	12	render	render	VERB
cana-941	40	13	the	the	DET
cana-941	40	14	problem	problem	NOUN
cana-941	40	15	tractable	tractable	ADJ
cana-941	40	16	in	in	ADP
cana-941	40	17	algebraic	algebraic	ADJ
cana-941	40	18	terms	term	NOUN
cana-941	40	19	:	:	PUNCT
cana-941	40	20	𝐹	𝐹	PROPN
cana-941	40	21	−	−	PROPN
cana-941	40	22	1[(𝜔𝑖)𝛼𝐹[𝜓(𝑡	1[(𝜔𝑖)𝛼𝐹[𝜓(𝑡	NUM
cana-941	40	23	)	)	PUNCT
cana-941	40	24	]	]	PUNCT
cana-941	40	25	]	]	X
cana-941	40	26	=	=	SYM
cana-941	40	27	𝜓(𝑡	𝜓(𝑡	PROPN
cana-941	40	28	)	)	PUNCT
cana-941	40	29	monte	monte	PROPN
cana-941	40	30	carlo	carlo	PROPN
cana-941	40	31	simulations	simulation	NOUN
cana-941	40	32	:	:	PUNCT
cana-941	40	33	monte	monte	PROPN
cana-941	40	34	carlo	carlo	PROPN
cana-941	40	35	simulations	simulation	NOUN
cana-941	40	36	introduce	introduce	VERB
cana-941	40	37	randomness	randomness	NOUN
cana-941	40	38	into	into	ADP
cana-941	40	39	the	the	DET
cana-941	40	40	computational	computational	ADJ
cana-941	40	41	process	process	NOUN
cana-941	40	42	.	.	PUNCT
cana-941	41	1	these	these	DET
cana-941	41	2	simulations	simulation	NOUN
cana-941	41	3	,	,	PUNCT
cana-941	41	4	while	while	SCONJ
cana-941	41	5	computationally	computationally	ADV
cana-941	41	6	intensive	intensive	ADJ
cana-941	41	7	,	,	PUNCT
cana-941	41	8	provide	provide	VERB
cana-941	41	9	a	a	DET
cana-941	41	10	stochastic	stochastic	ADJ
cana-941	41	11	approach	approach	NOUN
cana-941	41	12	for	for	ADP
cana-941	41	13	tackling	tackle	VERB
cana-941	41	14	fractional	fractional	ADJ
cana-941	41	15	systems	system	NOUN
cana-941	41	16	:	:	PUNCT
cana-941	41	17	𝜓(𝑡𝑛	𝜓(𝑡𝑛	SYM
cana-941	41	18	+	+	SYM
cana-941	41	19	1	1	X
cana-941	41	20	)	)	PUNCT
cana-941	41	21	=	=	SYM
cana-941	41	22	𝜓(𝑡𝑛	𝜓(𝑡𝑛	X
cana-941	41	23	)	)	PUNCT
cana-941	41	24	+	+	SYM
cana-941	41	25	𝛥𝑡𝐻(𝑡𝑛)𝜓(𝑡𝑛	𝛥𝑡𝐻(𝑡𝑛)𝜓(𝑡𝑛	NUM
cana-941	41	26	)	)	PUNCT
cana-941	42	1	+	+	CCONJ
cana-941	42	2	𝛥𝑊	𝛥𝑊	PROPN
cana-941	42	3	analytical	analytical	ADJ
cana-941	42	4	approaches	approach	NOUN
cana-941	42	5	:	:	PUNCT
cana-941	42	6	perturbation	perturbation	NOUN
cana-941	42	7	theory	theory	NOUN
cana-941	42	8	:	:	PUNCT
cana-941	42	9	perturbation	perturbation	NOUN
cana-941	42	10	theory	theory	NOUN
cana-941	42	11	offers	offer	VERB
cana-941	42	12	a	a	DET
cana-941	42	13	systematic	systematic	ADJ
cana-941	42	14	framework	framework	NOUN
cana-941	42	15	for	for	ADP
cana-941	42	16	analyzing	analyze	VERB
cana-941	42	17	coupled	couple	VERB
cana-941	42	18	systems	system	NOUN
cana-941	42	19	by	by	ADP
cana-941	42	20	treating	treat	VERB
cana-941	42	21	the	the	DET
cana-941	42	22	coupling	coupling	NOUN
cana-941	42	23	terms	term	NOUN
cana-941	42	24	as	as	ADP
cana-941	42	25	perturbations	perturbation	NOUN
cana-941	42	26	.	.	PUNCT
cana-941	43	1	it	it	PRON
cana-941	43	2	provides	provide	VERB
cana-941	43	3	an	an	DET
cana-941	43	4	approximate	approximate	ADJ
cana-941	43	5	solution	solution	NOUN
cana-941	43	6	that	that	PRON
cana-941	43	7	elucidates	elucidate	VERB
cana-941	43	8	the	the	DET
cana-941	43	9	impact	impact	NOUN
cana-941	43	10	of	of	ADP
cana-941	43	11	perturbations	perturbation	NOUN
cana-941	43	12	:	:	PUNCT
cana-941	43	13	𝜓(𝑡	𝜓(𝑡	PROPN
cana-941	43	14	)	)	PUNCT
cana-941	43	15	=	=	PUNCT
cana-941	43	16	𝜓0(𝑡	𝜓0(𝑡	PROPN
cana-941	43	17	)	)	PUNCT
cana-941	43	18	+	+	NUM
cana-941	43	19	𝜖𝜓1(𝑡	𝜖𝜓1(𝑡	SYM
cana-941	43	20	)	)	PUNCT
cana-941	43	21	+	+	CCONJ
cana-941	43	22	𝑂(𝜖2	𝑂(𝜖2	ADJ
cana-941	43	23	)	)	PUNCT
cana-941	43	24	asymptotic	asymptotic	ADJ
cana-941	43	25	analysis	analysis	NOUN
cana-941	43	26	:	:	PUNCT
cana-941	43	27	asymptotic	asymptotic	ADJ
cana-941	43	28	analysis	analysis	NOUN
cana-941	43	29	explores	explore	VERB
cana-941	43	30	the	the	DET
cana-941	43	31	long	long	ADJ
cana-941	43	32	-	-	PUNCT
cana-941	43	33	term	term	NOUN
cana-941	43	34	behavior	behavior	NOUN
cana-941	43	35	of	of	ADP
cana-941	43	36	systems	system	NOUN
cana-941	43	37	by	by	ADP
cana-941	43	38	deriving	derive	VERB
cana-941	43	39	asymptotic	asymptotic	ADJ
cana-941	43	40	expansions	expansion	NOUN
cana-941	43	41	.	.	PUNCT
cana-941	44	1	these	these	DET
cana-941	44	2	expansions	expansion	NOUN
cana-941	44	3	unveil	unveil	VERB
cana-941	44	4	dominant	dominant	ADJ
cana-941	44	5	trends	trend	NOUN
cana-941	44	6	and	and	CCONJ
cana-941	44	7	stability	stability	NOUN
cana-941	44	8	properties	property	NOUN
cana-941	44	9	:	:	PUNCT
cana-941	44	10	𝜓(𝑡	𝜓(𝑡	ADJ
cana-941	44	11	)	)	PUNCT
cana-941	44	12	∼	∼	NOUN
cana-941	44	13	𝑒𝜆𝑡	𝑒𝜆𝑡	NOUN
cana-941	44	14	integral	integral	ADJ
cana-941	44	15	transforms	transform	NOUN
cana-941	44	16	:	:	PUNCT
cana-941	44	17	integral	integral	ADJ
cana-941	44	18	transforms	transform	NOUN
cana-941	44	19	,	,	PUNCT
cana-941	44	20	such	such	ADJ
cana-941	44	21	as	as	ADP
cana-941	44	22	laplace	laplace	NOUN
cana-941	44	23	transforms	transform	VERB
cana-941	44	24	,	,	PUNCT
cana-941	44	25	facilitate	facilitate	VERB
cana-941	44	26	the	the	DET
cana-941	44	27	conversion	conversion	NOUN
cana-941	44	28	of	of	ADP
cana-941	44	29	fractional	fractional	ADJ
cana-941	44	30	differential	differential	ADJ
cana-941	44	31	equations	equation	NOUN
cana-941	44	32	into	into	ADP
cana-941	44	33	algebraic	algebraic	ADJ
cana-941	44	34	forms	form	NOUN
cana-941	44	35	.	.	PUNCT
cana-941	45	1	this	this	DET
cana-941	45	2	transformation	transformation	NOUN
cana-941	45	3	simplifies	simplify	VERB
cana-941	45	4	the	the	DET
cana-941	45	5	solution	solution	NOUN
cana-941	45	6	process	process	NOUN
cana-941	45	7	and	and	CCONJ
cana-941	45	8	enables	enable	VERB
cana-941	45	9	analytical	analytical	ADJ
cana-941	45	10	insights	insight	NOUN
cana-941	45	11	:	:	PUNCT
cana-941	45	12	𝐿[𝜓(𝑡	𝐿[𝜓(𝑡	PROPN
cana-941	45	13	)	)	PUNCT
cana-941	45	14	]	]	PUNCT
cana-941	46	1	=	=	SYM
cana-941	46	2	∫	∫	PROPN
cana-941	46	3	𝑒−𝑠𝑡𝜓(𝑡	𝑒−𝑠𝑡𝜓(𝑡	PROPN
cana-941	46	4	)	)	PUNCT
cana-941	46	5	∞	∞	PROPN
cana-941	46	6	0	0	NUM
cana-941	46	7	𝑑𝑡	𝑑𝑡	ADP
cana-941	46	8	these	these	DET
cana-941	46	9	methodologies	methodology	NOUN
cana-941	46	10	provide	provide	VERB
cana-941	46	11	a	a	DET
cana-941	46	12	comprehensive	comprehensive	ADJ
cana-941	46	13	toolkit	toolkit	NOUN
cana-941	46	14	for	for	ADP
cana-941	46	15	investigating	investigate	VERB
cana-941	46	16	the	the	DET
cana-941	46	17	behavior	behavior	NOUN
cana-941	46	18	and	and	CCONJ
cana-941	46	19	stability	stability	NOUN
cana-941	46	20	properties	property	NOUN
cana-941	46	21	of	of	ADP
cana-941	46	22	fractional	fractional	ADJ
cana-941	46	23	quantum	quantum	NOUN
cana-941	46	24	systems	system	NOUN
cana-941	46	25	.	.	PUNCT
cana-941	47	1	their	their	PRON
cana-941	47	2	combined	combine	VERB
cana-941	47	3	application	application	NOUN
cana-941	47	4	ensures	ensure	VERB
cana-941	47	5	a	a	DET
cana-941	47	6	thorough	thorough	ADJ
cana-941	47	7	exploration	exploration	NOUN
cana-941	47	8	of	of	ADP
cana-941	47	9	the	the	DET
cana-941	47	10	complex	complex	ADJ
cana-941	47	11	dynamics	dynamic	NOUN
cana-941	47	12	inherent	inherent	ADJ
cana-941	47	13	in	in	ADP
cana-941	47	14	coupled	couple	VERB
cana-941	47	15	schrödinger	schrödinger	ADJ
cana-941	47	16	equations	equation	NOUN
cana-941	47	17	with	with	ADP
cana-941	47	18	fractional	fractional	ADJ
cana-941	47	19	damping	damping	NOUN
cana-941	47	20	.	.	PUNCT
cana-941	48	1	2.5	2.5	NUM
cana-941	48	2	.	.	PUNCT
cana-941	49	1	experimental	experimental	ADJ
cana-941	49	2	observations	observation	NOUN
cana-941	49	3	and	and	CCONJ
cana-941	49	4	quantum	quantum	NOUN
cana-941	49	5	technologies	technology	NOUN
cana-941	49	6	:	:	PUNCT
cana-941	49	7	experimental	experimental	ADJ
cana-941	49	8	studies	study	NOUN
cana-941	49	9	play	play	VERB
cana-941	49	10	a	a	DET
cana-941	49	11	crucial	crucial	ADJ
cana-941	49	12	role	role	NOUN
cana-941	49	13	in	in	ADP
cana-941	49	14	validating	validate	VERB
cana-941	49	15	theoretical	theoretical	ADJ
cana-941	49	16	models	model	NOUN
cana-941	49	17	and	and	CCONJ
cana-941	49	18	exploring	explore	VERB
cana-941	49	19	the	the	DET
cana-941	49	20	implications	implication	NOUN
cana-941	49	21	of	of	ADP
cana-941	49	22	fractional	fractional	ADJ
cana-941	49	23	damping	damping	NOUN
cana-941	49	24	in	in	ADP
cana-941	49	25	real	real	ADJ
cana-941	49	26	-	-	PUNCT
cana-941	49	27	world	world	NOUN
cana-941	49	28	quantum	quantum	NOUN
cana-941	49	29	systems	system	NOUN
cana-941	49	30	.	.	PUNCT
cana-941	50	1	advances	advance	NOUN
cana-941	50	2	in	in	ADP
cana-941	50	3	experimental	experimental	ADJ
cana-941	50	4	techniques	technique	NOUN
cana-941	50	5	,	,	PUNCT
cana-941	50	6	such	such	ADJ
cana-941	50	7	as	as	ADP
cana-941	50	8	cold	cold	ADJ
cana-941	50	9	atom	atom	NOUN
cana-941	50	10	systems	system	NOUN
cana-941	50	11	,	,	PUNCT
cana-941	50	12	trapped	trap	VERB
cana-941	50	13	ions	ion	NOUN
cana-941	50	14	,	,	PUNCT
cana-941	50	15	and	and	CCONJ
cana-941	50	16	superconducting	superconducting	ADJ
cana-941	50	17	circuits	circuit	NOUN
cana-941	50	18	,	,	PUNCT
cana-941	50	19	have	have	AUX
cana-941	50	20	enabled	enable	VERB
cana-941	50	21	researchers	researcher	NOUN
cana-941	50	22	to	to	PART
cana-941	50	23	probe	probe	VERB
cana-941	50	24	the	the	DET
cana-941	50	25	effects	effect	NOUN
cana-941	50	26	of	of	ADP
cana-941	50	27	fractional	fractional	ADJ
cana-941	50	28	damping	damp	VERB
cana-941	50	29	on	on	ADP
cana-941	50	30	quantum	quantum	ADJ
cana-941	50	31	coherence	coherence	NOUN
cana-941	50	32	,	,	PUNCT
cana-941	50	33	quantum	quantum	NOUN
cana-941	50	34	transport	transport	NOUN
cana-941	50	35	,	,	PUNCT
cana-941	50	36	and	and	CCONJ
cana-941	50	37	quantum	quantum	NOUN
cana-941	50	38	information	information	NOUN
cana-941	50	39	processing	processing	NOUN
cana-941	50	40	.	.	PUNCT
cana-941	51	1	these	these	DET
cana-941	51	2	experimental	experimental	ADJ
cana-941	51	3	observations	observation	NOUN
cana-941	51	4	provide	provide	VERB
cana-941	51	5	valuable	valuable	ADJ
cana-941	51	6	feedback	feedback	NOUN
cana-941	51	7	for	for	ADP
cana-941	51	8	refining	refine	VERB
cana-941	51	9	theoretical	theoretical	ADJ
cana-941	51	10	models	model	NOUN
cana-941	51	11	and	and	CCONJ
cana-941	51	12	advancing	advance	VERB
cana-941	51	13	our	our	PRON
cana-941	51	14	understanding	understanding	NOUN
cana-941	51	15	of	of	ADP
cana-941	51	16	coupled	couple	VERB
cana-941	51	17	quantum	quantum	NOUN
cana-941	51	18	systems	system	NOUN
cana-941	51	19	under	under	ADP
cana-941	51	20	fractional	fractional	ADJ
cana-941	51	21	damping	damping	NOUN
cana-941	51	22	.	.	PUNCT
cana-941	52	1	communications	communication	NOUN
cana-941	52	2	on	on	ADP
cana-941	52	3	applied	apply	VERB
cana-941	52	4	nonlinear	nonlinear	ADJ
cana-941	52	5	analysis	analysis	NOUN
cana-941	52	6	issn	issn	NOUN
cana-941	52	7	:	:	PUNCT
cana-941	52	8	1074	1074	NUM
cana-941	52	9	-	-	PUNCT
cana-941	52	10	133x	133x	NUM
cana-941	52	11	vol	vol	NOUN
cana-941	52	12	31	31	NUM
cana-941	52	13	no	no	NOUN
cana-941	52	14	.	.	PUNCT
cana-941	53	1	4s	4s	NUM
cana-941	53	2	(	(	PUNCT
cana-941	53	3	2024	2024	NUM
cana-941	53	4	)	)	PUNCT
cana-941	53	5	443	443	NUM
cana-941	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	53	7	this	this	DET
cana-941	53	8	literature	literature	NOUN
cana-941	53	9	review	review	NOUN
cana-941	53	10	underscores	underscore	VERB
cana-941	53	11	the	the	DET
cana-941	53	12	multifaceted	multifaceted	ADJ
cana-941	53	13	nature	nature	NOUN
cana-941	53	14	of	of	ADP
cana-941	53	15	research	research	NOUN
cana-941	53	16	on	on	ADP
cana-941	53	17	coupled	couple	VERB
cana-941	53	18	schrödinger	schrödinger	ADJ
cana-941	53	19	equations	equation	NOUN
cana-941	53	20	with	with	ADP
cana-941	53	21	fractional	fractional	ADJ
cana-941	53	22	damping	damping	NOUN
cana-941	53	23	,	,	PUNCT
cana-941	53	24	encompassing	encompass	VERB
cana-941	53	25	theoretical	theoretical	ADJ
cana-941	53	26	developments	development	NOUN
cana-941	53	27	,	,	PUNCT
cana-941	53	28	numerical	numerical	ADJ
cana-941	53	29	simulations	simulation	NOUN
cana-941	53	30	,	,	PUNCT
cana-941	53	31	experimental	experimental	ADJ
cana-941	53	32	observations	observation	NOUN
cana-941	53	33	,	,	PUNCT
cana-941	53	34	and	and	CCONJ
cana-941	53	35	their	their	PRON
cana-941	53	36	interplay	interplay	NOUN
cana-941	53	37	in	in	ADP
cana-941	53	38	advancing	advance	VERB
cana-941	53	39	our	our	PRON
cana-941	53	40	understanding	understanding	NOUN
cana-941	53	41	of	of	ADP
cana-941	53	42	quantum	quantum	ADJ
cana-941	53	43	dynamics	dynamic	NOUN
cana-941	53	44	and	and	CCONJ
cana-941	53	45	its	its	PRON
cana-941	53	46	applications	application	NOUN
cana-941	53	47	.	.	PUNCT
cana-941	54	1	while	while	SCONJ
cana-941	54	2	significant	significant	ADJ
cana-941	54	3	progress	progress	NOUN
cana-941	54	4	has	have	AUX
cana-941	54	5	been	be	AUX
cana-941	54	6	made	make	VERB
cana-941	54	7	,	,	PUNCT
cana-941	54	8	there	there	PRON
cana-941	54	9	remain	remain	VERB
cana-941	54	10	ample	ample	ADJ
cana-941	54	11	opportunities	opportunity	NOUN
cana-941	54	12	for	for	ADP
cana-941	54	13	further	further	ADJ
cana-941	54	14	investigation	investigation	NOUN
cana-941	54	15	,	,	PUNCT
cana-941	54	16	particularly	particularly	ADV
cana-941	54	17	in	in	ADP
cana-941	54	18	elucidating	elucidate	VERB
cana-941	54	19	the	the	DET
cana-941	54	20	intricate	intricate	ADJ
cana-941	54	21	interplay	interplay	NOUN
cana-941	54	22	between	between	ADP
cana-941	54	23	fractional	fractional	ADJ
cana-941	54	24	damping	damping	NOUN
cana-941	54	25	,	,	PUNCT
cana-941	54	26	quantum	quantum	ADJ
cana-941	54	27	coherence	coherence	NOUN
cana-941	54	28	,	,	PUNCT
cana-941	54	29	and	and	CCONJ
cana-941	54	30	complex	complex	ADJ
cana-941	54	31	quantum	quantum	ADJ
cana-941	54	32	phenomena	phenomenon	NOUN
cana-941	54	33	.	.	PUNCT
cana-941	55	1	3	3	X
cana-941	55	2	.	.	X
cana-941	55	3	theoretical	theoretical	ADJ
cana-941	55	4	background	background	NOUN
cana-941	55	5	:	:	PUNCT
cana-941	55	6	the	the	DET
cana-941	55	7	theoretical	theoretical	ADJ
cana-941	55	8	foundation	foundation	NOUN
cana-941	55	9	of	of	ADP
cana-941	55	10	the	the	DET
cana-941	55	11	research	research	NOUN
cana-941	55	12	revolves	revolve	VERB
cana-941	55	13	around	around	ADV
cana-941	55	14	coupled	couple	VERB
cana-941	55	15	schrödinger	schrödinger	ADJ
cana-941	55	16	equations	equation	NOUN
cana-941	55	17	with	with	ADP
cana-941	55	18	fractional	fractional	ADJ
cana-941	55	19	damping	damping	NOUN
cana-941	55	20	,	,	PUNCT
cana-941	55	21	combining	combine	VERB
cana-941	55	22	principles	principle	NOUN
cana-941	55	23	from	from	ADP
cana-941	55	24	quantum	quantum	ADJ
cana-941	55	25	mechanics	mechanic	NOUN
cana-941	55	26	and	and	CCONJ
cana-941	55	27	fractional	fractional	ADJ
cana-941	55	28	calculus	calculus	NOUN
cana-941	55	29	.	.	PUNCT
cana-941	56	1	this	this	DET
cana-941	56	2	section	section	NOUN
cana-941	56	3	provides	provide	VERB
cana-941	56	4	an	an	DET
cana-941	56	5	overview	overview	NOUN
cana-941	56	6	of	of	ADP
cana-941	56	7	the	the	DET
cana-941	56	8	key	key	ADJ
cana-941	56	9	theoretical	theoretical	ADJ
cana-941	56	10	concepts	concept	NOUN
cana-941	56	11	necessary	necessary	ADJ
cana-941	56	12	for	for	ADP
cana-941	56	13	understanding	understand	VERB
cana-941	56	14	the	the	DET
cana-941	56	15	dynamics	dynamic	NOUN
cana-941	56	16	of	of	ADP
cana-941	56	17	such	such	ADJ
cana-941	56	18	systems	system	NOUN
cana-941	56	19	.	.	PUNCT
cana-941	57	1	3.1	3.1	NUM
cana-941	57	2	.	.	PUNCT
cana-941	58	1	schrödinger	schrödinger	ADJ
cana-941	58	2	equation	equation	NOUN
cana-941	58	3	:	:	PUNCT
cana-941	58	4	central	central	ADJ
cana-941	58	5	to	to	ADP
cana-941	58	6	quantum	quantum	ADJ
cana-941	58	7	mechanics	mechanic	NOUN
cana-941	58	8	is	be	AUX
cana-941	58	9	the	the	DET
cana-941	58	10	schrödinger	schrödinger	ADJ
cana-941	58	11	equation	equation	NOUN
cana-941	58	12	,	,	PUNCT
cana-941	58	13	which	which	PRON
cana-941	58	14	describes	describe	VERB
cana-941	58	15	the	the	DET
cana-941	58	16	time	time	NOUN
cana-941	58	17	evolution	evolution	NOUN
cana-941	58	18	of	of	ADP
cana-941	58	19	quantum	quantum	ADJ
cana-941	58	20	states	state	NOUN
cana-941	58	21	.	.	PUNCT
cana-941	59	1	for	for	ADP
cana-941	59	2	a	a	DET
cana-941	59	3	single	single	ADJ
cana-941	59	4	particle	particle	NOUN
cana-941	59	5	,	,	PUNCT
cana-941	59	6	the	the	DET
cana-941	59	7	time	time	NOUN
cana-941	59	8	-	-	PUNCT
cana-941	59	9	dependent	dependent	ADJ
cana-941	59	10	schrödinger	schrödinger	ADJ
cana-941	59	11	equation	equation	NOUN
cana-941	59	12	is	be	AUX
cana-941	59	13	given	give	VERB
cana-941	59	14	by	by	ADP
cana-941	59	15	:	:	PUNCT
cana-941	59	16	𝑖ℏ	𝑖ℏ	PROPN
cana-941	59	17	𝜕𝑡	𝜕𝑡	PROPN
cana-941	59	18	𝜕𝜓	𝜕𝜓	NOUN
cana-941	59	19	=	=	PUNCT
cana-941	60	1	𝐻𝜓	𝐻𝜓	PROPN
cana-941	60	2	where	where	SCONJ
cana-941	60	3	ℏ	ℏ	PROPN
cana-941	60	4	is	be	AUX
cana-941	60	5	the	the	DET
cana-941	60	6	reduced	reduce	VERB
cana-941	60	7	planck	planck	NOUN
cana-941	60	8	constant	constant	ADJ
cana-941	60	9	,	,	PUNCT
cana-941	60	10	ψ	ψ	X
cana-941	60	11	is	be	AUX
cana-941	60	12	the	the	DET
cana-941	60	13	wave	wave	NOUN
cana-941	60	14	function	function	NOUN
cana-941	60	15	of	of	ADP
cana-941	60	16	the	the	DET
cana-941	60	17	particle	particle	NOUN
cana-941	60	18	,	,	PUNCT
cana-941	60	19	t	t	PROPN
cana-941	60	20	is	be	AUX
cana-941	60	21	time	time	NOUN
cana-941	60	22	,	,	PUNCT
cana-941	60	23	and	and	CCONJ
cana-941	60	24	h	h	NOUN
cana-941	60	25	is	be	AUX
cana-941	60	26	the	the	DET
cana-941	60	27	hamiltonian	hamiltonian	ADJ
cana-941	60	28	operator	operator	NOUN
cana-941	60	29	representing	represent	VERB
cana-941	60	30	the	the	DET
cana-941	60	31	total	total	ADJ
cana-941	60	32	energy	energy	NOUN
cana-941	60	33	of	of	ADP
cana-941	60	34	the	the	DET
cana-941	60	35	system	system	NOUN
cana-941	60	36	.	.	PUNCT
cana-941	61	1	3.2	3.2	NUM
cana-941	61	2	.	.	PUNCT
cana-941	61	3	fractional	fractional	ADJ
cana-941	61	4	calculus	calculus	NOUN
cana-941	61	5	:	:	PUNCT
cana-941	61	6	fractional	fractional	ADJ
cana-941	61	7	calculus	calculus	NOUN
cana-941	61	8	extends	extend	VERB
cana-941	61	9	the	the	DET
cana-941	61	10	concepts	concept	NOUN
cana-941	61	11	of	of	ADP
cana-941	61	12	differentiation	differentiation	NOUN
cana-941	61	13	and	and	CCONJ
cana-941	61	14	integration	integration	NOUN
cana-941	61	15	to	to	ADP
cana-941	61	16	non	non	ADJ
cana-941	61	17	-	-	ADJ
cana-941	61	18	integer	integer	ADJ
cana-941	61	19	orders	order	NOUN
cana-941	61	20	,	,	PUNCT
cana-941	61	21	allowing	allow	VERB
cana-941	61	22	for	for	ADP
cana-941	61	23	the	the	DET
cana-941	61	24	description	description	NOUN
cana-941	61	25	of	of	ADP
cana-941	61	26	systems	system	NOUN
cana-941	61	27	with	with	ADP
cana-941	61	28	memory	memory	NOUN
cana-941	61	29	and	and	CCONJ
cana-941	61	30	non	non	ADJ
cana-941	61	31	-	-	ADJ
cana-941	61	32	local	local	ADJ
cana-941	61	33	effects	effect	NOUN
cana-941	61	34	.	.	PUNCT
cana-941	62	1	the	the	DET
cana-941	62	2	fractional	fractional	ADJ
cana-941	62	3	derivative	derivative	NOUN
cana-941	62	4	of	of	ADP
cana-941	62	5	order	order	NOUN
cana-941	62	6	α	α	NOUN
cana-941	62	7	of	of	ADP
cana-941	62	8	a	a	DET
cana-941	62	9	function	function	NOUN
cana-941	62	10	f(t	f(t	NOUN
cana-941	62	11	)	)	PUNCT
cana-941	62	12	is	be	AUX
cana-941	62	13	defined	define	VERB
cana-941	62	14	as	as	ADP
cana-941	62	15	:	:	PUNCT
cana-941	62	16	𝑑𝛼	𝑑𝛼	NOUN
cana-941	62	17	𝑑𝑡𝛼	𝑑𝑡𝛼	VERB
cana-941	62	18	𝑓(𝑡	𝑓(𝑡	PROPN
cana-941	62	19	)	)	PUNCT
cana-941	62	20	=	=	SYM
cana-941	62	21	𝛤(𝑛	𝛤(𝑛	PROPN
cana-941	62	22	−	−	NOUN
cana-941	62	23	𝛼)1∫	𝛼)1∫	ADJ
cana-941	62	24	0𝑡(𝑡	0𝑡(𝑡	NOUN
cana-941	63	1	−	−	NOUN
cana-941	64	1	𝑠)𝛼−1	𝑠)𝛼−1	NOUN
cana-941	64	2	𝑑𝑛	𝑑𝑛	NOUN
cana-941	64	3	𝑑𝑠𝑛	𝑑𝑠𝑛	NOUN
cana-941	64	4	𝑓(𝑠)𝑑𝑠	𝑓(𝑠)𝑑𝑠	PROPN
cana-941	64	5	3.3	3.3	NUM
cana-941	64	6	.	.	PUNCT
cana-941	65	1	coupled	couple	VERB
cana-941	65	2	schrödinger	schrödinger	ADJ
cana-941	65	3	equations	equation	NOUN
cana-941	65	4	:	:	PUNCT
cana-941	65	5	when	when	SCONJ
cana-941	65	6	considering	consider	VERB
cana-941	65	7	systems	system	NOUN
cana-941	65	8	composed	compose	VERB
cana-941	65	9	of	of	ADP
cana-941	65	10	multiple	multiple	ADJ
cana-941	65	11	interacting	interacting	ADJ
cana-941	65	12	particles	particle	NOUN
cana-941	65	13	,	,	PUNCT
cana-941	65	14	coupled	couple	VERB
cana-941	65	15	schrödinger	schrödinger	ADJ
cana-941	65	16	equations	equation	NOUN
cana-941	65	17	are	be	AUX
cana-941	65	18	employed	employ	VERB
cana-941	65	19	.	.	PUNCT
cana-941	66	1	for	for	ADP
cana-941	66	2	two	two	NUM
cana-941	66	3	particles	particle	NOUN
cana-941	66	4	,	,	PUNCT
cana-941	66	5	the	the	DET
cana-941	66	6	coupled	couple	VERB
cana-941	66	7	system	system	NOUN
cana-941	66	8	can	can	AUX
cana-941	66	9	be	be	AUX
cana-941	66	10	described	describe	VERB
cana-941	66	11	by	by	ADP
cana-941	66	12	the	the	DET
cana-941	66	13	following	follow	VERB
cana-941	66	14	set	set	NOUN
cana-941	66	15	of	of	ADP
cana-941	66	16	equations	equation	NOUN
cana-941	66	17	:	:	PUNCT
cana-941	66	18	𝑖ℏ	𝑖ℏ	PUNCT
cana-941	67	1	𝜕𝜓₁	𝜕𝜓₁	PROPN
cana-941	67	2	𝜕𝑡	𝜕𝑡	PROPN
cana-941	67	3	=	=	PUNCT
cana-941	67	4	𝐻₁𝜓₁	𝐻₁𝜓₁	PROPN
cana-941	67	5	−	−	PROPN
cana-941	67	6	𝛾₁(−𝛥)^𝛼₁𝜓₁	𝛾₁(−𝛥)^𝛼₁𝜓₁	NOUN
cana-941	67	7	+	+	CCONJ
cana-941	67	8	𝑔₁₂𝜓₂	𝑔₁₂𝜓₂	NOUN
cana-941	67	9	𝑖ℏ	𝑖ℏ	NOUN
cana-941	67	10	𝜕𝜓₂	𝜕𝜓₂	VERB
cana-941	68	1	𝜕𝑡	𝜕𝑡	NOUN
cana-941	68	2	=	=	SYM
cana-941	68	3	𝐻₂𝜓₂	𝐻₂𝜓₂	ADJ
cana-941	68	4	−	−	PROPN
cana-941	68	5	𝛾₂(−𝛥)^𝛼₂𝜓₂	𝛾₂(−𝛥)^𝛼₂𝜓₂	X
cana-941	68	6	+	+	NUM
cana-941	68	7	𝑔₂₁𝜓₁	𝑔₂₁𝜓₁	PROPN
cana-941	68	8	3.4	3.4	NUM
cana-941	68	9	.	.	PUNCT
cana-941	69	1	physical	physical	ADJ
cana-941	69	2	interpretation	interpretation	NOUN
cana-941	69	3	:	:	PUNCT
cana-941	69	4	each	each	DET
cana-941	69	5	term	term	NOUN
cana-941	69	6	in	in	ADP
cana-941	69	7	the	the	DET
cana-941	69	8	coupled	couple	VERB
cana-941	69	9	schrödinger	schrödinger	ADJ
cana-941	69	10	equations	equation	NOUN
cana-941	69	11	has	have	VERB
cana-941	69	12	a	a	DET
cana-941	69	13	physical	physical	ADJ
cana-941	69	14	interpretation	interpretation	NOUN
cana-941	69	15	.	.	PUNCT
cana-941	70	1	the	the	DET
cana-941	70	2	first	first	ADJ
cana-941	70	3	term	term	NOUN
cana-941	70	4	represents	represent	VERB
cana-941	70	5	the	the	DET
cana-941	70	6	evolution	evolution	NOUN
cana-941	70	7	of	of	ADP
cana-941	70	8	the	the	DET
cana-941	70	9	particle	particle	NOUN
cana-941	70	10	's	's	PART
cana-941	70	11	wave	wave	NOUN
cana-941	70	12	function	function	NOUN
cana-941	70	13	under	under	ADP
cana-941	70	14	its	its	PRON
cana-941	70	15	own	own	ADJ
cana-941	70	16	hamiltonian	hamiltonian	NOUN
cana-941	70	17	,	,	PUNCT
cana-941	70	18	while	while	SCONJ
cana-941	70	19	the	the	DET
cana-941	70	20	second	second	ADJ
cana-941	70	21	term	term	NOUN
cana-941	70	22	accounts	account	VERB
cana-941	70	23	for	for	ADP
cana-941	70	24	fractional	fractional	ADJ
cana-941	70	25	damping	damp	VERB
cana-941	70	26	effects	effect	NOUN
cana-941	70	27	.	.	PUNCT
cana-941	71	1	the	the	DET
cana-941	71	2	last	last	ADJ
cana-941	71	3	term	term	NOUN
cana-941	71	4	captures	capture	VERB
cana-941	71	5	the	the	DET
cana-941	71	6	interaction	interaction	NOUN
cana-941	71	7	between	between	ADP
cana-941	71	8	the	the	DET
cana-941	71	9	particles	particle	NOUN
cana-941	71	10	,	,	PUNCT
cana-941	71	11	mediated	mediate	VERB
cana-941	71	12	by	by	ADP
cana-941	71	13	the	the	DET
cana-941	71	14	coupling	coupling	NOUN
cana-941	71	15	coefficients	coefficient	NOUN
cana-941	71	16	.	.	PUNCT
cana-941	72	1	understanding	understand	VERB
cana-941	72	2	these	these	DET
cana-941	72	3	theoretical	theoretical	ADJ
cana-941	72	4	concepts	concept	NOUN
cana-941	72	5	is	be	AUX
cana-941	72	6	essential	essential	ADJ
cana-941	72	7	for	for	ADP
cana-941	72	8	formulating	formulate	VERB
cana-941	72	9	and	and	CCONJ
cana-941	72	10	analyzing	analyze	VERB
cana-941	72	11	the	the	DET
cana-941	72	12	dynamics	dynamic	NOUN
cana-941	72	13	of	of	ADP
cana-941	72	14	coupled	couple	VERB
cana-941	72	15	quantum	quantum	NOUN
cana-941	72	16	systems	system	NOUN
cana-941	72	17	with	with	ADP
cana-941	72	18	fractional	fractional	ADJ
cana-941	72	19	damping	damping	NOUN
cana-941	72	20	.	.	PUNCT
cana-941	73	1	in	in	ADP
cana-941	73	2	the	the	DET
cana-941	73	3	subsequent	subsequent	ADJ
cana-941	73	4	sections	section	NOUN
cana-941	73	5	,	,	PUNCT
cana-941	73	6	we	we	PRON
cana-941	73	7	will	will	AUX
cana-941	73	8	delve	delve	VERB
cana-941	73	9	into	into	ADP
cana-941	73	10	the	the	DET
cana-941	73	11	formulation	formulation	NOUN
cana-941	73	12	of	of	ADP
cana-941	73	13	the	the	DET
cana-941	73	14	coupled	couple	VERB
cana-941	73	15	equations	equation	NOUN
cana-941	73	16	and	and	CCONJ
cana-941	73	17	explore	explore	VERB
cana-941	73	18	analytical	analytical	ADJ
cana-941	73	19	and	and	CCONJ
cana-941	73	20	numerical	numerical	ADJ
cana-941	73	21	techniques	technique	NOUN
cana-941	73	22	for	for	ADP
cana-941	73	23	investigating	investigate	VERB
cana-941	73	24	their	their	PRON
cana-941	73	25	behavior	behavior	NOUN
cana-941	73	26	.	.	PUNCT
cana-941	74	1	communications	communication	NOUN
cana-941	74	2	on	on	ADP
cana-941	74	3	applied	apply	VERB
cana-941	74	4	nonlinear	nonlinear	ADJ
cana-941	74	5	analysis	analysis	NOUN
cana-941	74	6	issn	issn	NOUN
cana-941	74	7	:	:	PUNCT
cana-941	74	8	1074	1074	NUM
cana-941	74	9	-	-	PUNCT
cana-941	74	10	133x	133x	NUM
cana-941	74	11	vol	vol	NOUN
cana-941	74	12	31	31	NUM
cana-941	74	13	no	no	NOUN
cana-941	74	14	.	.	PUNCT
cana-941	75	1	4s	4s	NUM
cana-941	75	2	(	(	PUNCT
cana-941	75	3	2024	2024	NUM
cana-941	75	4	)	)	PUNCT
cana-941	75	5	444	444	NUM
cana-941	75	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	75	7	4	4	NUM
cana-941	75	8	.	.	PUNCT
cana-941	75	9	formulation	formulation	NOUN
cana-941	75	10	of	of	ADP
cana-941	75	11	coupled	couple	VERB
cana-941	75	12	schrödinger	schrödinger	ADJ
cana-941	75	13	equations	equation	NOUN
cana-941	75	14	with	with	ADP
cana-941	75	15	fractional	fractional	ADJ
cana-941	75	16	damping	damping	NOUN
cana-941	75	17	:	:	PUNCT
cana-941	75	18	in	in	ADP
cana-941	75	19	this	this	DET
cana-941	75	20	section	section	NOUN
cana-941	75	21	,	,	PUNCT
cana-941	75	22	we	we	PRON
cana-941	75	23	outline	outline	VERB
cana-941	75	24	the	the	DET
cana-941	75	25	derivation	derivation	NOUN
cana-941	75	26	and	and	CCONJ
cana-941	75	27	formulation	formulation	NOUN
cana-941	75	28	of	of	ADP
cana-941	75	29	the	the	DET
cana-941	75	30	coupled	couple	VERB
cana-941	75	31	schrödinger	schrödinger	ADJ
cana-941	75	32	equations	equation	NOUN
cana-941	75	33	with	with	ADP
cana-941	75	34	fractional	fractional	ADJ
cana-941	75	35	damping	damping	NOUN
cana-941	75	36	.	.	PUNCT
cana-941	76	1	we	we	PRON
cana-941	76	2	detail	detail	VERB
cana-941	76	3	the	the	DET
cana-941	76	4	physical	physical	ADJ
cana-941	76	5	interpretation	interpretation	NOUN
cana-941	76	6	of	of	ADP
cana-941	76	7	each	each	DET
cana-941	76	8	term	term	NOUN
cana-941	76	9	in	in	ADP
cana-941	76	10	the	the	DET
cana-941	76	11	equations	equation	NOUN
cana-941	76	12	and	and	CCONJ
cana-941	76	13	discuss	discuss	VERB
cana-941	76	14	any	any	DET
cana-941	76	15	assumptions	assumption	NOUN
cana-941	76	16	made	make	VERB
cana-941	76	17	during	during	ADP
cana-941	76	18	the	the	DET
cana-941	76	19	formulation	formulation	NOUN
cana-941	76	20	process	process	NOUN
cana-941	76	21	.	.	PUNCT
cana-941	77	1	through	through	ADP
cana-941	77	2	a	a	DET
cana-941	77	3	systematic	systematic	ADJ
cana-941	77	4	approach	approach	NOUN
cana-941	77	5	,	,	PUNCT
cana-941	77	6	we	we	PRON
cana-941	77	7	present	present	VERB
cana-941	77	8	a	a	DET
cana-941	77	9	rigorous	rigorous	ADJ
cana-941	77	10	mathematical	mathematical	ADJ
cana-941	77	11	description	description	NOUN
cana-941	77	12	of	of	ADP
cana-941	77	13	the	the	DET
cana-941	77	14	coupled	couple	VERB
cana-941	77	15	system	system	NOUN
cana-941	77	16	under	under	ADP
cana-941	77	17	investigation	investigation	NOUN
cana-941	77	18	.	.	PUNCT
cana-941	78	1	term	term	NOUN
cana-941	78	2	physical	physical	ADJ
cana-941	78	3	interpretation	interpretation	NOUN
cana-941	78	4	𝑖ℏ	𝑖ℏ	ADP
cana-941	78	5	𝜕𝜓1	𝜕𝜓1	ADJ
cana-941	78	6	𝜕𝑡	𝜕𝑡	ADJ
cana-941	78	7	evolution	evolution	NOUN
cana-941	78	8	of	of	ADP
cana-941	78	9	the	the	DET
cana-941	78	10	wave	wave	NOUN
cana-941	78	11	function	function	NOUN
cana-941	78	12	ψ1	ψ1	NOUN
cana-941	78	13	under	under	ADP
cana-941	78	14	the	the	DET
cana-941	78	15	hamiltonian	hamiltonian	ADJ
cana-941	78	16	h1	h1	PROPN
cana-941	78	17	−𝛾1(−𝛥)𝛼1	−𝛾1(−𝛥)𝛼1	ADJ
cana-941	78	18	𝜓1	𝜓1	NOUN
cana-941	78	19	fractional	fractional	ADJ
cana-941	78	20	damping	damp	VERB
cana-941	78	21	effect	effect	NOUN
cana-941	78	22	on	on	ADP
cana-941	78	23	ψ1	ψ1	NOUN
cana-941	78	24	due	due	ADJ
cana-941	78	25	to	to	ADP
cana-941	78	26	external	external	ADJ
cana-941	78	27	influences	influence	NOUN
cana-941	78	28	𝑔12𝜓2	𝑔12𝜓2	NOUN
cana-941	78	29	interaction	interaction	NOUN
cana-941	78	30	term	term	NOUN
cana-941	78	31	representing	represent	VERB
cana-941	78	32	the	the	DET
cana-941	78	33	influence	influence	NOUN
cana-941	78	34	of	of	ADP
cana-941	78	35	particle	particle	NOUN
cana-941	78	36	ψ2	ψ2	NOUN
cana-941	78	37	on	on	ADP
cana-941	78	38	ψ1	ψ1	ADJ
cana-941	78	39	𝑖ℏ	𝑖ℏ	PUNCT
cana-941	78	40	∂𝜓2	∂𝜓2	PROPN
cana-941	78	41	𝜕𝑡	𝜕𝑡	ADJ
cana-941	78	42	evolution	evolution	NOUN
cana-941	78	43	of	of	ADP
cana-941	78	44	the	the	DET
cana-941	78	45	wave	wave	NOUN
cana-941	78	46	function	function	NOUN
cana-941	78	47	ψ2	ψ2	NOUN
cana-941	78	48	under	under	ADP
cana-941	78	49	the	the	DET
cana-941	78	50	hamiltonian	hamiltonian	ADJ
cana-941	78	51	h2	h2	PROPN
cana-941	78	52	−𝛾2(−𝛥)𝛼2𝜓2	−𝛾2(−𝛥)𝛼2𝜓2	PROPN
cana-941	78	53	fractional	fractional	ADJ
cana-941	78	54	damping	damp	VERB
cana-941	78	55	effect	effect	NOUN
cana-941	78	56	on	on	ADP
cana-941	78	57	ψ2	ψ2	NOUN
cana-941	78	58	due	due	ADJ
cana-941	78	59	to	to	ADP
cana-941	78	60	external	external	ADJ
cana-941	78	61	influences	influence	NOUN
cana-941	78	62	𝑔21𝜓1	𝑔21𝜓1	NOUN
cana-941	78	63	interaction	interaction	NOUN
cana-941	78	64	term	term	NOUN
cana-941	78	65	representing	represent	VERB
cana-941	78	66	the	the	DET
cana-941	78	67	influence	influence	NOUN
cana-941	78	68	of	of	ADP
cana-941	78	69	particle	particle	NOUN
cana-941	78	70	ψ1	ψ1	NOUN
cana-941	78	71	on	on	ADP
cana-941	78	72	ψ2	ψ2	NOUN
cana-941	78	73	assumptions	assumption	NOUN
cana-941	78	74	:	:	PUNCT
cana-941	78	75	1	1	X
cana-941	78	76	.	.	X
cana-941	78	77	linear	linear	ADJ
cana-941	78	78	coupling	coupling	NOUN
cana-941	78	79	between	between	ADP
cana-941	78	80	particles	particle	NOUN
cana-941	78	81	,	,	PUNCT
cana-941	78	82	neglecting	neglect	VERB
cana-941	78	83	higher	high	ADJ
cana-941	78	84	-	-	PUNCT
cana-941	78	85	order	order	NOUN
cana-941	78	86	interaction	interaction	NOUN
cana-941	78	87	terms	term	NOUN
cana-941	78	88	.	.	PUNCT
cana-941	79	1	2	2	X
cana-941	79	2	.	.	NOUN
cana-941	79	3	isotropic	isotropic	NOUN
cana-941	79	4	and	and	CCONJ
cana-941	79	5	homogeneous	homogeneous	ADJ
cana-941	79	6	fractional	fractional	ADJ
cana-941	79	7	damping	damping	NOUN
cana-941	79	8	affecting	affect	VERB
cana-941	79	9	both	both	DET
cana-941	79	10	particles	particle	NOUN
cana-941	79	11	equally	equally	ADV
cana-941	79	12	in	in	ADP
cana-941	79	13	all	all	DET
cana-941	79	14	directions	direction	NOUN
cana-941	79	15	.	.	PUNCT
cana-941	80	1	mathematical	mathematical	ADJ
cana-941	80	2	description	description	NOUN
cana-941	80	3	:	:	PUNCT
cana-941	80	4	the	the	DET
cana-941	80	5	coupled	couple	VERB
cana-941	80	6	schrödinger	schrödinger	ADJ
cana-941	80	7	equations	equation	NOUN
cana-941	80	8	with	with	ADP
cana-941	80	9	fractional	fractional	ADJ
cana-941	80	10	damping	damping	NOUN
cana-941	80	11	constitute	constitute	VERB
cana-941	80	12	a	a	DET
cana-941	80	13	set	set	NOUN
cana-941	80	14	of	of	ADP
cana-941	80	15	partial	partial	ADJ
cana-941	80	16	differential	differential	ADJ
cana-941	80	17	equations	equation	NOUN
cana-941	80	18	(	(	PUNCT
cana-941	80	19	pdes	pde	NOUN
cana-941	80	20	)	)	PUNCT
cana-941	80	21	governing	govern	VERB
cana-941	80	22	the	the	DET
cana-941	80	23	system	system	NOUN
cana-941	80	24	's	's	PART
cana-941	80	25	dynamics	dynamic	NOUN
cana-941	80	26	.	.	PUNCT
cana-941	81	1	these	these	DET
cana-941	81	2	equations	equation	NOUN
cana-941	81	3	can	can	AUX
cana-941	81	4	be	be	AUX
cana-941	81	5	solved	solve	VERB
cana-941	81	6	numerically	numerically	ADV
cana-941	81	7	or	or	CCONJ
cana-941	81	8	analytically	analytically	ADV
cana-941	81	9	to	to	PART
cana-941	81	10	explore	explore	VERB
cana-941	81	11	the	the	DET
cana-941	81	12	behavior	behavior	NOUN
cana-941	81	13	of	of	ADP
cana-941	81	14	the	the	DET
cana-941	81	15	coupled	couple	VERB
cana-941	81	16	quantum	quantum	NOUN
cana-941	81	17	system	system	NOUN
cana-941	81	18	under	under	ADP
cana-941	81	19	various	various	ADJ
cana-941	81	20	conditions	condition	NOUN
cana-941	81	21	.	.	PUNCT
cana-941	82	1	this	this	DET
cana-941	82	2	tabular	tabular	PROPN
cana-941	82	3	format	format	NOUN
cana-941	82	4	provides	provide	VERB
cana-941	82	5	a	a	DET
cana-941	82	6	clear	clear	ADJ
cana-941	82	7	overview	overview	NOUN
cana-941	82	8	of	of	ADP
cana-941	82	9	the	the	DET
cana-941	82	10	physical	physical	ADJ
cana-941	82	11	interpretation	interpretation	NOUN
cana-941	82	12	of	of	ADP
cana-941	82	13	each	each	DET
cana-941	82	14	term	term	NOUN
cana-941	82	15	in	in	ADP
cana-941	82	16	the	the	DET
cana-941	82	17	coupled	couple	VERB
cana-941	82	18	schrödinger	schrödinger	ADJ
cana-941	82	19	equations	equation	NOUN
cana-941	82	20	with	with	ADP
cana-941	82	21	fractional	fractional	ADJ
cana-941	82	22	damping	damping	NOUN
cana-941	82	23	,	,	PUNCT
cana-941	82	24	as	as	ADV
cana-941	82	25	well	well	ADV
cana-941	82	26	as	as	ADP
cana-941	82	27	the	the	DET
cana-941	82	28	underlying	underlying	ADJ
cana-941	82	29	assumptions	assumption	NOUN
cana-941	82	30	made	make	VERB
cana-941	82	31	during	during	ADP
cana-941	82	32	the	the	DET
cana-941	82	33	formulation	formulation	NOUN
cana-941	82	34	process	process	NOUN
cana-941	82	35	.	.	PUNCT
cana-941	83	1	5	5	X
cana-941	83	2	.	.	X
cana-941	83	3	analytical	analytical	ADJ
cana-941	83	4	solutions	solution	NOUN
cana-941	83	5	for	for	ADP
cana-941	83	6	decay	decay	NOUN
cana-941	83	7	:	:	PUNCT
cana-941	83	8	in	in	ADP
cana-941	83	9	this	this	DET
cana-941	83	10	section	section	NOUN
cana-941	83	11	,	,	PUNCT
cana-941	83	12	we	we	PRON
cana-941	83	13	explore	explore	VERB
cana-941	83	14	analytical	analytical	ADJ
cana-941	83	15	techniques	technique	NOUN
cana-941	83	16	for	for	ADP
cana-941	83	17	deriving	derive	VERB
cana-941	83	18	decay	decay	NOUN
cana-941	83	19	solutions	solution	NOUN
cana-941	83	20	of	of	ADP
cana-941	83	21	the	the	DET
cana-941	83	22	coupled	couple	VERB
cana-941	83	23	schrödinger	schrödinger	ADJ
cana-941	83	24	equations	equation	NOUN
cana-941	83	25	with	with	ADP
cana-941	83	26	fractional	fractional	ADJ
cana-941	83	27	damping	damping	NOUN
cana-941	83	28	.	.	PUNCT
cana-941	84	1	we	we	PRON
cana-941	84	2	analyze	analyze	VERB
cana-941	84	3	the	the	DET
cana-941	84	4	behavior	behavior	NOUN
cana-941	84	5	of	of	ADP
cana-941	84	6	solutions	solution	NOUN
cana-941	84	7	over	over	ADP
cana-941	84	8	time	time	NOUN
cana-941	84	9	,	,	PUNCT
cana-941	84	10	considering	consider	VERB
cana-941	84	11	various	various	ADJ
cana-941	84	12	parameters	parameter	NOUN
cana-941	84	13	and	and	CCONJ
cana-941	84	14	initial	initial	ADJ
cana-941	84	15	conditions	condition	NOUN
cana-941	84	16	.	.	PUNCT
cana-941	85	1	through	through	ADP
cana-941	85	2	mathematical	mathematical	ADJ
cana-941	85	3	rigor	rigor	NOUN
cana-941	85	4	and	and	CCONJ
cana-941	85	5	analytical	analytical	ADJ
cana-941	85	6	insights	insight	NOUN
cana-941	85	7	,	,	PUNCT
cana-941	85	8	we	we	PRON
cana-941	85	9	uncover	uncover	VERB
cana-941	85	10	the	the	DET
cana-941	85	11	underlying	underlie	VERB
cana-941	85	12	decay	decay	NOUN
cana-941	85	13	mechanisms	mechanism	NOUN
cana-941	85	14	governing	govern	VERB
cana-941	85	15	the	the	DET
cana-941	85	16	dynamics	dynamic	NOUN
cana-941	85	17	of	of	ADP
cana-941	85	18	the	the	DET
cana-941	85	19	coupled	couple	VERB
cana-941	85	20	quantum	quantum	NOUN
cana-941	85	21	system	system	NOUN
cana-941	85	22	.	.	PUNCT
cana-941	86	1	example	example	NOUN
cana-941	86	2	1	1	NUM
cana-941	86	3	:	:	PUNCT
cana-941	86	4	consider	consider	VERB
cana-941	86	5	a	a	DET
cana-941	86	6	system	system	NOUN
cana-941	86	7	of	of	ADP
cana-941	86	8	coupled	couple	VERB
cana-941	86	9	schrödinger	schrödinger	ADJ
cana-941	86	10	equations	equation	NOUN
cana-941	86	11	with	with	ADP
cana-941	86	12	fractional	fractional	ADJ
cana-941	86	13	damping	damping	NOUN
cana-941	86	14	:	:	PUNCT
cana-941	86	15	𝑖ℏ	𝑖ℏ	PUNCT
cana-941	87	1	𝜕𝜓₁	𝜕𝜓₁	PROPN
cana-941	87	2	𝜕𝑡	𝜕𝑡	PROPN
cana-941	88	1	=	=	X
cana-941	88	2	(	(	PUNCT
cana-941	88	3	𝐻₁	𝐻₁	PROPN
cana-941	88	4	−	−	PROPN
cana-941	88	5	𝒊ℏγ1	𝒊ℏγ1	PROPN
cana-941	88	6	𝑫𝛼1	𝑫𝛼1	NOUN
cana-941	88	7	)	)	PUNCT
cana-941	88	8	𝜓₁	𝜓₁	NOUN
cana-941	88	9	+	+	CCONJ
cana-941	88	10	𝑔₁₂𝜓₂	𝑔₁₂𝜓₂	NOUN
cana-941	88	11	𝑖ℏ	𝑖ℏ	ADP
cana-941	88	12	𝜕𝜓₁	𝜕𝜓₁	PROPN
cana-941	88	13	𝜕𝑡	𝜕𝑡	NOUN
cana-941	89	1	=	=	X
cana-941	89	2	(	(	PUNCT
cana-941	89	3	𝐻₂	𝐻₂	PROPN
cana-941	89	4	−	−	PROPN
cana-941	89	5	𝒊ℏγ2	𝒊ℏγ2	PROPN
cana-941	89	6	𝑫𝛼2	𝑫𝛼2	NOUN
cana-941	89	7	)	)	PUNCT
cana-941	89	8	𝜓₂	𝜓₂	PROPN
cana-941	89	9	+	+	CCONJ
cana-941	89	10	𝑔₂₁𝜓₁	𝑔₂₁𝜓₁	NUM
cana-941	90	1	where	where	SCONJ
cana-941	90	2	:	:	PUNCT
cana-941	90	3	•	•	NOUN
cana-941	90	4	1ψ1	1ψ1	NUM
cana-941	90	5	and	and	CCONJ
cana-941	90	6	2ψ2	2ψ2	NUM
cana-941	90	7	are	be	AUX
cana-941	90	8	the	the	DET
cana-941	90	9	wave	wave	NOUN
cana-941	90	10	functions	function	NOUN
cana-941	90	11	of	of	ADP
cana-941	90	12	two	two	NUM
cana-941	90	13	coupled	couple	VERB
cana-941	90	14	particles	particle	NOUN
cana-941	90	15	,	,	PUNCT
cana-941	90	16	•	•	ADP
cana-941	90	17	h1	h1	NOUN
cana-941	90	18	and	and	CCONJ
cana-941	90	19	h2	h2	NOUN
cana-941	90	20	are	be	AUX
cana-941	90	21	the	the	DET
cana-941	90	22	corresponding	corresponding	ADJ
cana-941	90	23	hamiltonians	hamiltonian	NOUN
cana-941	90	24	,	,	PUNCT
cana-941	90	25	•	•	NUM
cana-941	90	26	γ1	γ1	NOUN
cana-941	90	27	and	and	CCONJ
cana-941	90	28	γ2	γ2	PROPN
cana-941	90	29	are	be	AUX
cana-941	90	30	fractional	fractional	ADJ
cana-941	90	31	damping	damp	VERB
cana-941	90	32	coefficients	coefficient	NOUN
cana-941	90	33	,	,	PUNCT
cana-941	90	34	•	•	NUM
cana-941	90	35	α1	α1	PROPN
cana-941	90	36	and	and	CCONJ
cana-941	90	37	α2	α2	NOUN
cana-941	90	38	are	be	AUX
cana-941	90	39	fractional	fractional	ADJ
cana-941	90	40	orders	order	NOUN
cana-941	90	41	of	of	ADP
cana-941	90	42	the	the	DET
cana-941	90	43	damping	damping	NOUN
cana-941	90	44	,	,	PUNCT
cana-941	90	45	•	•	ADP
cana-941	90	46	g	g	PROPN
cana-941	90	47	12	12	NUM
cana-941	90	48	and	and	CCONJ
cana-941	90	49	g21	g21	PROPN
cana-941	90	50	are	be	AUX
cana-941	90	51	coupling	couple	VERB
cana-941	90	52	coefficients	coefficient	NOUN
cana-941	90	53	,	,	PUNCT
cana-941	90	54	communications	communication	NOUN
cana-941	90	55	on	on	ADP
cana-941	90	56	applied	apply	VERB
cana-941	90	57	nonlinear	nonlinear	ADJ
cana-941	90	58	analysis	analysis	NOUN
cana-941	90	59	issn	issn	NOUN
cana-941	90	60	:	:	PUNCT
cana-941	90	61	1074	1074	NUM
cana-941	90	62	-	-	PUNCT
cana-941	90	63	133x	133x	NUM
cana-941	90	64	vol	vol	NOUN
cana-941	90	65	31	31	NUM
cana-941	90	66	no	no	NOUN
cana-941	90	67	.	.	PUNCT
cana-941	91	1	4s	4s	NUM
cana-941	91	2	(	(	PUNCT
cana-941	91	3	2024	2024	NUM
cana-941	91	4	)	)	PUNCT
cana-941	91	5	445	445	NUM
cana-941	91	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	91	7	•	•	NOUN
cana-941	91	8	d	d	NOUN
cana-941	91	9	α	α	PROPN
cana-941	91	10	denotes	denote	VERB
cana-941	91	11	the	the	DET
cana-941	91	12	fractional	fractional	ADJ
cana-941	91	13	derivative	derivative	ADJ
cana-941	91	14	operator	operator	NOUN
cana-941	91	15	.	.	PUNCT
cana-941	92	1	term	term	NOUN
cana-941	92	2	physical	physical	ADJ
cana-941	92	3	interpretation	interpretation	NOUN
cana-941	92	4	ψ₁	ψ₁	NOUN
cana-941	92	5	,	,	PUNCT
cana-941	92	6	ψ₂	ψ₂	NOUN
cana-941	92	7	wave	wave	NOUN
cana-941	92	8	functions	function	NOUN
cana-941	92	9	of	of	ADP
cana-941	92	10	coupled	couple	VERB
cana-941	92	11	particles	particle	NOUN
cana-941	92	12	h₁	h₁	NOUN
cana-941	92	13	,	,	PUNCT
cana-941	92	14	h₂	h₂	PROPN
cana-941	92	15	hamiltonians	hamiltonian	NOUN
cana-941	92	16	of	of	ADP
cana-941	92	17	the	the	DET
cana-941	92	18	particles	particle	NOUN
cana-941	92	19	γ₁	γ₁	NOUN
cana-941	92	20	,	,	PUNCT
cana-941	92	21	γ₂	γ₂	ADJ
cana-941	92	22	fractional	fractional	ADJ
cana-941	92	23	damping	damp	VERB
cana-941	92	24	coefficients	coefficient	NOUN
cana-941	92	25	α₁	α₁	NOUN
cana-941	92	26	,	,	PUNCT
cana-941	92	27	α₂	α₂	NOUN
cana-941	92	28	fractional	fractional	ADJ
cana-941	92	29	orders	order	NOUN
cana-941	92	30	of	of	ADP
cana-941	92	31	the	the	DET
cana-941	92	32	damping	damp	VERB
cana-941	92	33	g₁₂	g₁₂	NOUN
cana-941	92	34	,	,	PUNCT
cana-941	92	35	g₂₁	g₂₁	NOUN
cana-941	92	36	coupling	coupling	NOUN
cana-941	92	37	coefficients	coefficient	VERB
cana-941	92	38	𝒟ᵅ	𝒟ᵅ	PROPN
cana-941	92	39	fractional	fractional	ADJ
cana-941	92	40	derivative	derivative	ADJ
cana-941	92	41	operator	operator	NOUN
cana-941	92	42	6	6	NUM
cana-941	92	43	.	.	PUNCT
cana-941	93	1	numerical	numerical	ADJ
cana-941	93	2	methods	method	NOUN
cana-941	93	3	:	:	PUNCT
cana-941	93	4	in	in	ADP
cana-941	93	5	this	this	DET
cana-941	93	6	section	section	NOUN
cana-941	93	7	,	,	PUNCT
cana-941	93	8	we	we	PRON
cana-941	93	9	describe	describe	VERB
cana-941	93	10	the	the	DET
cana-941	93	11	numerical	numerical	ADJ
cana-941	93	12	methods	method	NOUN
cana-941	93	13	employed	employ	VERB
cana-941	93	14	to	to	PART
cana-941	93	15	solve	solve	VERB
cana-941	93	16	the	the	DET
cana-941	93	17	coupled	couple	VERB
cana-941	93	18	equations	equation	NOUN
cana-941	93	19	numerically	numerically	ADV
cana-941	93	20	.	.	PUNCT
cana-941	94	1	we	we	PRON
cana-941	94	2	discuss	discuss	VERB
cana-941	94	3	implementation	implementation	NOUN
cana-941	94	4	details	detail	NOUN
cana-941	94	5	,	,	PUNCT
cana-941	94	6	computational	computational	ADJ
cana-941	94	7	considerations	consideration	NOUN
cana-941	94	8	,	,	PUNCT
cana-941	94	9	and	and	CCONJ
cana-941	94	10	validation	validation	NOUN
cana-941	94	11	strategies	strategy	NOUN
cana-941	94	12	.	.	PUNCT
cana-941	95	1	by	by	ADP
cana-941	95	2	leveraging	leverage	VERB
cana-941	95	3	numerical	numerical	ADJ
cana-941	95	4	simulations	simulation	NOUN
cana-941	95	5	,	,	PUNCT
cana-941	95	6	we	we	PRON
cana-941	95	7	complement	complement	VERB
cana-941	95	8	our	our	PRON
cana-941	95	9	analytical	analytical	ADJ
cana-941	95	10	findings	finding	NOUN
cana-941	95	11	and	and	CCONJ
cana-941	95	12	provide	provide	VERB
cana-941	95	13	insights	insight	NOUN
cana-941	95	14	into	into	ADP
cana-941	95	15	the	the	DET
cana-941	95	16	behavior	behavior	NOUN
cana-941	95	17	of	of	ADP
cana-941	95	18	the	the	DET
cana-941	95	19	system	system	NOUN
cana-941	95	20	beyond	beyond	ADP
cana-941	95	21	analytical	analytical	ADJ
cana-941	95	22	solutions	solution	NOUN
cana-941	95	23	.	.	PUNCT
cana-941	96	1	7	7	X
cana-941	96	2	.	.	X
cana-941	96	3	stability	stability	NOUN
cana-941	96	4	analysis	analysis	NOUN
cana-941	96	5	:	:	PUNCT
cana-941	96	6	this	this	DET
cana-941	96	7	section	section	NOUN
cana-941	96	8	focuses	focus	VERB
cana-941	96	9	on	on	ADP
cana-941	96	10	the	the	DET
cana-941	96	11	stability	stability	NOUN
cana-941	96	12	analysis	analysis	NOUN
cana-941	96	13	of	of	ADP
cana-941	96	14	solutions	solution	NOUN
cana-941	96	15	obtained	obtain	VERB
cana-941	96	16	from	from	ADP
cana-941	96	17	both	both	CCONJ
cana-941	96	18	analytical	analytical	ADJ
cana-941	96	19	and	and	CCONJ
cana-941	96	20	numerical	numerical	ADJ
cana-941	96	21	approaches	approach	NOUN
cana-941	96	22	.	.	PUNCT
cana-941	97	1	we	we	PRON
cana-941	97	2	examine	examine	VERB
cana-941	97	3	stability	stability	NOUN
cana-941	97	4	criteria	criterion	NOUN
cana-941	97	5	,	,	PUNCT
cana-941	97	6	stability	stability	NOUN
cana-941	97	7	regions	region	NOUN
cana-941	97	8	in	in	ADP
cana-941	97	9	parameter	parameter	NOUN
cana-941	97	10	space	space	NOUN
cana-941	97	11	,	,	PUNCT
cana-941	97	12	and	and	CCONJ
cana-941	97	13	the	the	DET
cana-941	97	14	influence	influence	NOUN
cana-941	97	15	of	of	ADP
cana-941	97	16	system	system	NOUN
cana-941	97	17	parameters	parameter	NOUN
cana-941	97	18	on	on	ADP
cana-941	97	19	stability	stability	NOUN
cana-941	97	20	properties	property	NOUN
cana-941	97	21	.	.	PUNCT
cana-941	98	1	through	through	ADP
cana-941	98	2	rigorous	rigorous	ADJ
cana-941	98	3	analysis	analysis	NOUN
cana-941	98	4	,	,	PUNCT
cana-941	98	5	we	we	PRON
cana-941	98	6	elucidate	elucidate	VERB
cana-941	98	7	the	the	DET
cana-941	98	8	stability	stability	NOUN
cana-941	98	9	characteristics	characteristic	NOUN
cana-941	98	10	of	of	ADP
cana-941	98	11	the	the	DET
cana-941	98	12	coupled	couple	VERB
cana-941	98	13	quantum	quantum	NOUN
cana-941	98	14	system	system	NOUN
cana-941	98	15	under	under	ADP
cana-941	98	16	fractional	fractional	ADJ
cana-941	98	17	damping	damping	NOUN
cana-941	98	18	.	.	PUNCT
cana-941	99	1	example	example	NOUN
cana-941	99	2	2	2	NUM
cana-941	99	3	:	:	PUNCT
cana-941	99	4	consider	consider	VERB
cana-941	99	5	the	the	DET
cana-941	99	6	stability	stability	NOUN
cana-941	99	7	analysis	analysis	NOUN
cana-941	99	8	of	of	ADP
cana-941	99	9	the	the	DET
cana-941	99	10	coupled	couple	VERB
cana-941	99	11	schrödinger	schrödinger	ADJ
cana-941	99	12	equations	equation	NOUN
cana-941	99	13	presented	present	VERB
cana-941	99	14	earlier	early	ADV
cana-941	99	15	.	.	PUNCT
cana-941	100	1	we	we	PRON
cana-941	100	2	analyze	analyze	VERB
cana-941	100	3	the	the	DET
cana-941	100	4	eigenvalues	eigenvalue	NOUN
cana-941	100	5	of	of	ADP
cana-941	100	6	the	the	DET
cana-941	100	7	linearized	linearize	VERB
cana-941	100	8	system	system	NOUN
cana-941	100	9	to	to	PART
cana-941	100	10	determine	determine	VERB
cana-941	100	11	the	the	DET
cana-941	100	12	stability	stability	NOUN
cana-941	100	13	of	of	ADP
cana-941	100	14	the	the	DET
cana-941	100	15	equilibrium	equilibrium	NOUN
cana-941	100	16	points	point	NOUN
cana-941	100	17	.	.	PUNCT
cana-941	101	1	the	the	DET
cana-941	101	2	stability	stability	NOUN
cana-941	101	3	criteria	criterion	NOUN
cana-941	101	4	can	can	AUX
cana-941	101	5	be	be	AUX
cana-941	101	6	expressed	express	VERB
cana-941	101	7	as	as	ADP
cana-941	101	8	:	:	PUNCT
cana-941	101	9	𝑅𝑒(𝜆𝑖	𝑅𝑒(𝜆𝑖	X
cana-941	101	10	)	)	PUNCT
cana-941	101	11	<	<	X
cana-941	101	12	0	0	NUM
cana-941	101	13	,	,	PUNCT
cana-941	101	14	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-941	101	15	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-941	101	16	𝑖	𝑖	X
cana-941	101	17	by	by	ADP
cana-941	101	18	computing	compute	VERB
cana-941	101	19	the	the	DET
cana-941	101	20	eigenvalues	eigenvalue	NOUN
cana-941	101	21	for	for	ADP
cana-941	101	22	various	various	ADJ
cana-941	101	23	parameter	parameter	NOUN
cana-941	101	24	values	value	NOUN
cana-941	101	25	,	,	PUNCT
cana-941	101	26	we	we	PRON
cana-941	101	27	determine	determine	VERB
cana-941	101	28	the	the	DET
cana-941	101	29	stability	stability	NOUN
cana-941	101	30	regions	region	NOUN
cana-941	101	31	in	in	ADP
cana-941	101	32	the	the	DET
cana-941	101	33	parameter	parameter	NOUN
cana-941	101	34	space	space	NOUN
cana-941	101	35	,	,	PUNCT
cana-941	101	36	providing	provide	VERB
cana-941	101	37	valuable	valuable	ADJ
cana-941	101	38	insights	insight	NOUN
cana-941	101	39	into	into	ADP
cana-941	101	40	the	the	DET
cana-941	101	41	system	system	NOUN
cana-941	101	42	's	's	PART
cana-941	101	43	behavior	behavior	NOUN
cana-941	101	44	under	under	ADP
cana-941	101	45	different	different	ADJ
cana-941	101	46	conditions	condition	NOUN
cana-941	101	47	.	.	PUNCT
cana-941	102	1	example	example	NOUN
cana-941	102	2	3	3	NUM
cana-941	102	3	:	:	PUNCT
cana-941	103	1	dissipation	dissipation	NOUN
cana-941	103	2	effects	effect	NOUN
cana-941	103	3	on	on	ADP
cana-941	103	4	quantum	quantum	NOUN
cana-941	103	5	tunneling	tunneling	NOUN
cana-941	103	6	in	in	ADP
cana-941	103	7	this	this	DET
cana-941	103	8	example	example	NOUN
cana-941	103	9	,	,	PUNCT
cana-941	103	10	we	we	PRON
cana-941	103	11	examine	examine	VERB
cana-941	103	12	the	the	DET
cana-941	103	13	influence	influence	NOUN
cana-941	103	14	of	of	ADP
cana-941	103	15	fractional	fractional	ADJ
cana-941	103	16	damping	damp	VERB
cana-941	103	17	on	on	ADP
cana-941	103	18	quantum	quantum	ADJ
cana-941	103	19	tunneling	tunneling	NOUN
cana-941	103	20	phenomena	phenomenon	NOUN
cana-941	103	21	in	in	ADP
cana-941	103	22	coupled	couple	VERB
cana-941	103	23	schrödinger	schrödinger	ADJ
cana-941	103	24	equations	equation	NOUN
cana-941	103	25	.	.	PUNCT
cana-941	104	1	quantum	quantum	PROPN
cana-941	104	2	tunneling	tunneling	NOUN
cana-941	104	3	is	be	AUX
cana-941	104	4	a	a	DET
cana-941	104	5	fundamental	fundamental	ADJ
cana-941	104	6	quantum	quantum	ADJ
cana-941	104	7	mechanical	mechanical	ADJ
cana-941	104	8	process	process	NOUN
cana-941	104	9	where	where	SCONJ
cana-941	104	10	a	a	DET
cana-941	104	11	particle	particle	NOUN
cana-941	104	12	penetrates	penetrate	VERB
cana-941	104	13	through	through	ADP
cana-941	104	14	a	a	DET
cana-941	104	15	classically	classically	ADV
cana-941	104	16	forbidden	forbid	VERB
cana-941	104	17	energy	energy	NOUN
cana-941	104	18	barrier	barrier	NOUN
cana-941	104	19	.	.	PUNCT
cana-941	105	1	fractional	fractional	ADJ
cana-941	105	2	damping	damp	VERB
cana-941	105	3	can	can	AUX
cana-941	105	4	significantly	significantly	ADV
cana-941	105	5	affect	affect	VERB
cana-941	105	6	tunneling	tunneling	NOUN
cana-941	105	7	probabilities	probability	NOUN
cana-941	105	8	and	and	CCONJ
cana-941	105	9	dynamics	dynamic	NOUN
cana-941	105	10	.	.	PUNCT
cana-941	106	1	consider	consider	VERB
cana-941	106	2	a	a	DET
cana-941	106	3	coupled	couple	VERB
cana-941	106	4	system	system	NOUN
cana-941	106	5	with	with	ADP
cana-941	106	6	two	two	NUM
cana-941	106	7	particles	particle	NOUN
cana-941	106	8	described	describe	VERB
cana-941	106	9	by	by	ADP
cana-941	106	10	the	the	DET
cana-941	106	11	schrödinger	schrödinger	ADJ
cana-941	106	12	equations	equation	NOUN
cana-941	106	13	:	:	PUNCT
cana-941	106	14	𝑖ℏ	𝑖ℏ	PUNCT
cana-941	107	1	𝜕𝜓₁	𝜕𝜓₁	PROPN
cana-941	107	2	𝜕𝑡	𝜕𝑡	PROPN
cana-941	107	3	=	=	PUNCT
cana-941	107	4	𝐻₁𝜓₁	𝐻₁𝜓₁	PROPN
cana-941	107	5	−	−	PROPN
cana-941	107	6	𝛾₁(−𝛥)^𝛼₁𝜓₁	𝛾₁(−𝛥)^𝛼₁𝜓₁	NOUN
cana-941	107	7	+	+	CCONJ
cana-941	107	8	𝑔₁₂𝜓₂	𝑔₁₂𝜓₂	NOUN
cana-941	107	9	𝑖ℏ	𝑖ℏ	NOUN
cana-941	107	10	𝜕𝜓₂	𝜕𝜓₂	VERB
cana-941	108	1	𝜕𝑡	𝜕𝑡	NOUN
cana-941	108	2	=	=	SYM
cana-941	108	3	𝐻₂𝜓₂	𝐻₂𝜓₂	ADJ
cana-941	108	4	−	−	PROPN
cana-941	108	5	𝛾₂(−𝛥)^𝛼₂𝜓₂	𝛾₂(−𝛥)^𝛼₂𝜓₂	X
cana-941	108	6	+	+	NUM
cana-941	108	7	𝑔₂₁𝜓₁	𝑔₂₁𝜓₁	NOUN
cana-941	108	8	where	where	SCONJ
cana-941	108	9	ψ₁h₁	ψ₁h₁	X
cana-941	108	10	and	and	CCONJ
cana-941	108	11	ψ₂h₂	ψ₂h₂	X
cana-941	108	12	are	be	AUX
cana-941	108	13	the	the	DET
cana-941	108	14	hamiltonians	hamiltonian	NOUN
cana-941	108	15	,	,	PUNCT
cana-941	108	16	γ₁	γ₁	NOUN
cana-941	108	17	and	and	CCONJ
cana-941	108	18	γ₂	γ₂	NOUN
cana-941	108	19	are	be	AUX
cana-941	108	20	the	the	DET
cana-941	108	21	fractional	fractional	ADJ
cana-941	108	22	damping	damp	VERB
cana-941	108	23	coefficients	coefficient	NOUN
cana-941	108	24	,	,	PUNCT
cana-941	108	25	α₁	α₁	NOUN
cana-941	108	26	and	and	CCONJ
cana-941	108	27	α₂	α₂	NOUN
cana-941	108	28	are	be	AUX
cana-941	108	29	the	the	DET
cana-941	108	30	fractional	fractional	ADJ
cana-941	108	31	orders	order	NOUN
cana-941	108	32	,	,	PUNCT
cana-941	108	33	and	and	CCONJ
cana-941	108	34	g₁₂	g₁₂	NOUN
cana-941	108	35	and	and	CCONJ
cana-941	108	36	g₂₁	g₂₁	NOUN
cana-941	108	37	are	be	AUX
cana-941	108	38	the	the	DET
cana-941	108	39	coupling	coupling	NOUN
cana-941	108	40	coefficients	coefficient	NOUN
cana-941	108	41	.	.	PUNCT
cana-941	109	1	we	we	PRON
cana-941	109	2	analyze	analyze	VERB
cana-941	109	3	how	how	SCONJ
cana-941	109	4	fractional	fractional	ADJ
cana-941	109	5	damping	damping	NOUN
cana-941	109	6	affects	affect	VERB
cana-941	109	7	the	the	DET
cana-941	109	8	tunneling	tunneling	NOUN
cana-941	109	9	behavior	behavior	NOUN
cana-941	109	10	of	of	ADP
cana-941	109	11	the	the	DET
cana-941	109	12	particles	particle	NOUN
cana-941	109	13	through	through	ADP
cana-941	109	14	a	a	DET
cana-941	109	15	potential	potential	ADJ
cana-941	109	16	barrier	barrier	NOUN
cana-941	109	17	.	.	PUNCT
cana-941	110	1	by	by	ADP
cana-941	110	2	varying	vary	VERB
cana-941	110	3	the	the	DET
cana-941	110	4	damping	damp	VERB
cana-941	110	5	coefficients	coefficient	NOUN
cana-941	110	6	and	and	CCONJ
cana-941	110	7	fractional	fractional	ADJ
cana-941	110	8	orders	order	NOUN
cana-941	110	9	,	,	PUNCT
cana-941	110	10	we	we	PRON
cana-941	110	11	observe	observe	VERB
cana-941	110	12	changes	change	NOUN
cana-941	110	13	in	in	ADP
cana-941	110	14	tunneling	tunnel	VERB
cana-941	110	15	communications	communication	NOUN
cana-941	110	16	on	on	ADP
cana-941	110	17	applied	apply	VERB
cana-941	110	18	nonlinear	nonlinear	ADJ
cana-941	110	19	analysis	analysis	NOUN
cana-941	110	20	issn	issn	NOUN
cana-941	110	21	:	:	PUNCT
cana-941	110	22	1074	1074	NUM
cana-941	110	23	-	-	PUNCT
cana-941	110	24	133x	133x	NUM
cana-941	110	25	vol	vol	NOUN
cana-941	110	26	31	31	NUM
cana-941	110	27	no	no	NOUN
cana-941	110	28	.	.	PUNCT
cana-941	111	1	4s	4s	NUM
cana-941	111	2	(	(	PUNCT
cana-941	111	3	2024	2024	NUM
cana-941	111	4	)	)	PUNCT
cana-941	111	5	446	446	NUM
cana-941	111	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	111	7	probabilities	probability	NOUN
cana-941	111	8	and	and	CCONJ
cana-941	111	9	tunneling	tunneling	NOUN
cana-941	111	10	rates	rate	NOUN
cana-941	111	11	.	.	PUNCT
cana-941	112	1	this	this	DET
cana-941	112	2	analysis	analysis	NOUN
cana-941	112	3	provides	provide	VERB
cana-941	112	4	insights	insight	NOUN
cana-941	112	5	into	into	ADP
cana-941	112	6	the	the	DET
cana-941	112	7	interplay	interplay	NOUN
cana-941	112	8	between	between	ADP
cana-941	112	9	dissipation	dissipation	NOUN
cana-941	112	10	and	and	CCONJ
cana-941	112	11	quantum	quantum	NOUN
cana-941	112	12	tunneling	tunneling	NOUN
cana-941	112	13	in	in	ADP
cana-941	112	14	coupled	couple	VERB
cana-941	112	15	quantum	quantum	NOUN
cana-941	112	16	systems	system	NOUN
cana-941	112	17	.	.	PUNCT
cana-941	113	1	example	example	NOUN
cana-941	113	2	4	4	NUM
cana-941	113	3	:	:	PUNCT
cana-941	113	4	nonlinear	nonlinear	ADJ
cana-941	113	5	coupling	coupling	NOUN
cana-941	113	6	effects	effect	NOUN
cana-941	113	7	in	in	ADP
cana-941	113	8	this	this	DET
cana-941	113	9	example	example	NOUN
cana-941	113	10	,	,	PUNCT
cana-941	113	11	we	we	PRON
cana-941	113	12	explore	explore	VERB
cana-941	113	13	the	the	DET
cana-941	113	14	impact	impact	NOUN
cana-941	113	15	of	of	ADP
cana-941	113	16	nonlinear	nonlinear	ADJ
cana-941	113	17	coupling	coupling	NOUN
cana-941	113	18	terms	term	NOUN
cana-941	113	19	on	on	ADP
cana-941	113	20	the	the	DET
cana-941	113	21	dynamics	dynamic	NOUN
cana-941	113	22	of	of	ADP
cana-941	113	23	coupled	couple	VERB
cana-941	113	24	schrödinger	schrödinger	ADJ
cana-941	113	25	equations	equation	NOUN
cana-941	113	26	with	with	ADP
cana-941	113	27	fractional	fractional	ADJ
cana-941	113	28	damping	damping	NOUN
cana-941	113	29	.	.	PUNCT
cana-941	114	1	nonlinear	nonlinear	ADJ
cana-941	114	2	couplings	coupling	NOUN
cana-941	114	3	arise	arise	VERB
cana-941	114	4	in	in	ADP
cana-941	114	5	many	many	ADJ
cana-941	114	6	physical	physical	ADJ
cana-941	114	7	systems	system	NOUN
cana-941	114	8	and	and	CCONJ
cana-941	114	9	can	can	AUX
cana-941	114	10	lead	lead	VERB
cana-941	114	11	to	to	ADP
cana-941	114	12	rich	rich	ADJ
cana-941	114	13	and	and	CCONJ
cana-941	114	14	complex	complex	ADJ
cana-941	114	15	behavior	behavior	NOUN
cana-941	114	16	,	,	PUNCT
cana-941	114	17	including	include	VERB
cana-941	114	18	chaos	chaos	NOUN
cana-941	114	19	and	and	CCONJ
cana-941	114	20	pattern	pattern	NOUN
cana-941	114	21	formation	formation	NOUN
cana-941	114	22	.	.	PUNCT
cana-941	115	1	consider	consider	VERB
cana-941	115	2	the	the	DET
cana-941	115	3	coupled	couple	VERB
cana-941	115	4	schrödinger	schrödinger	ADJ
cana-941	115	5	equations	equation	NOUN
cana-941	115	6	with	with	ADP
cana-941	115	7	nonlinear	nonlinear	ADJ
cana-941	115	8	coupling	coupling	NOUN
cana-941	115	9	terms	term	NOUN
cana-941	115	10	:	:	PUNCT
cana-941	115	11	𝑖ℏ	𝑖ℏ	PUNCT
cana-941	116	1	𝜕𝜓₁	𝜕𝜓₁	PROPN
cana-941	116	2	𝜕𝑡	𝜕𝑡	PROPN
cana-941	117	1	=	=	PUNCT
cana-941	117	2	𝐻₁𝜓₁	𝐻₁𝜓₁	PROPN
cana-941	117	3	−	−	ADP
cana-941	117	4	𝛾₁(−𝛥)𝛼₁𝜓₁	𝛾₁(−𝛥)𝛼₁𝜓₁	NOUN
cana-941	117	5	+	+	CCONJ
cana-941	117	6	𝑔₁₂𝜓₂	𝑔₁₂𝜓₂	NOUN
cana-941	118	1	𝑖ℏ	𝑖ℏ	NOUN
cana-941	118	2	𝜕𝜓₂	𝜕𝜓₂	VERB
cana-941	118	3	𝜕𝑡	𝜕𝑡	NOUN
cana-941	118	4	=	=	SYM
cana-941	118	5	𝐻₂𝜓₂	𝐻₂𝜓₂	PROPN
cana-941	118	6	−	−	PROPN
cana-941	118	7	𝛾₂(−𝛥)𝛼₂𝜓₂	𝛾₂(−𝛥)𝛼₂𝜓₂	AUX
cana-941	118	8	+	+	CCONJ
cana-941	118	9	𝑔₂₁𝜓₁	𝑔₂₁𝜓₁	ADJ
cana-941	118	10	table	table	NOUN
cana-941	118	11	1	1	NUM
cana-941	118	12	:	:	PUNCT
cana-941	118	13	summary	summary	NOUN
cana-941	118	14	of	of	ADP
cana-941	118	15	parameters	parameter	NOUN
cana-941	118	16	parameter	parameter	NOUN
cana-941	118	17	description	description	NOUN
cana-941	118	18	ψ₁	ψ₁	NOUN
cana-941	118	19	,	,	PUNCT
cana-941	118	20	ψ₂	ψ₂	NOUN
cana-941	118	21	wave	wave	NOUN
cana-941	118	22	functions	function	NOUN
cana-941	118	23	of	of	ADP
cana-941	118	24	the	the	DET
cana-941	118	25	coupled	couple	VERB
cana-941	118	26	particles	particle	NOUN
cana-941	118	27	h₁	h₁	NOUN
cana-941	118	28	,	,	PUNCT
cana-941	118	29	h₂	h₂	PROPN
cana-941	118	30	hamiltonians	hamiltonian	NOUN
cana-941	118	31	of	of	ADP
cana-941	118	32	the	the	DET
cana-941	118	33	particles	particle	NOUN
cana-941	118	34	γ₁	γ₁	NOUN
cana-941	118	35	,	,	PUNCT
cana-941	118	36	γ₂	γ₂	ADJ
cana-941	118	37	fractional	fractional	ADJ
cana-941	118	38	damping	damp	VERB
cana-941	118	39	coefficients	coefficient	NOUN
cana-941	118	40	a₁	a₁	NOUN
cana-941	118	41	,	,	PUNCT
cana-941	118	42	a₂	a₂	NOUN
cana-941	118	43	fractional	fractional	ADJ
cana-941	118	44	orders	order	NOUN
cana-941	118	45	g₁₂	g₁₂	NOUN
cana-941	118	46	,	,	PUNCT
cana-941	118	47	g₂₁	g₂₁	ADJ
cana-941	118	48	coupling	coupling	NOUN
cana-941	118	49	coefficients	coefficient	NOUN
cana-941	118	50	we	we	PRON
cana-941	118	51	investigate	investigate	VERB
cana-941	118	52	how	how	SCONJ
cana-941	118	53	nonlinear	nonlinear	ADJ
cana-941	118	54	coupling	coupling	NOUN
cana-941	118	55	terms	term	NOUN
cana-941	118	56	affect	affect	VERB
cana-941	118	57	the	the	DET
cana-941	118	58	stability	stability	NOUN
cana-941	118	59	and	and	CCONJ
cana-941	118	60	dynamics	dynamic	NOUN
cana-941	118	61	of	of	ADP
cana-941	118	62	the	the	DET
cana-941	118	63	coupled	couple	VERB
cana-941	118	64	system	system	NOUN
cana-941	118	65	.	.	PUNCT
cana-941	119	1	through	through	ADP
cana-941	119	2	numerical	numerical	ADJ
cana-941	119	3	simulations	simulation	NOUN
cana-941	119	4	and	and	CCONJ
cana-941	119	5	stability	stability	NOUN
cana-941	119	6	analysis	analysis	NOUN
cana-941	119	7	,	,	PUNCT
cana-941	119	8	we	we	PRON
cana-941	119	9	analyze	analyze	VERB
cana-941	119	10	the	the	DET
cana-941	119	11	emergence	emergence	NOUN
cana-941	119	12	of	of	ADP
cana-941	119	13	nonlinear	nonlinear	ADJ
cana-941	119	14	phenomena	phenomenon	NOUN
cana-941	119	15	such	such	ADJ
cana-941	119	16	as	as	ADP
cana-941	119	17	self	self	NOUN
cana-941	119	18	-	-	PUNCT
cana-941	119	19	trapping	trap	VERB
cana-941	119	20	,	,	PUNCT
cana-941	119	21	soliton	soliton	NOUN
cana-941	119	22	formation	formation	NOUN
cana-941	119	23	,	,	PUNCT
cana-941	119	24	and	and	CCONJ
cana-941	119	25	modulational	modulational	ADJ
cana-941	119	26	instability	instability	NOUN
cana-941	119	27	.	.	PUNCT
cana-941	120	1	this	this	DET
cana-941	120	2	example	example	NOUN
cana-941	120	3	highlights	highlight	VERB
cana-941	120	4	the	the	DET
cana-941	120	5	importance	importance	NOUN
cana-941	120	6	of	of	ADP
cana-941	120	7	considering	consider	VERB
cana-941	120	8	nonlinear	nonlinear	ADJ
cana-941	120	9	effects	effect	NOUN
cana-941	120	10	in	in	ADP
cana-941	120	11	the	the	DET
cana-941	120	12	study	study	NOUN
cana-941	120	13	of	of	ADP
cana-941	120	14	coupled	couple	VERB
cana-941	120	15	quantum	quantum	NOUN
cana-941	120	16	systems	system	NOUN
cana-941	120	17	with	with	ADP
cana-941	120	18	fractional	fractional	ADJ
cana-941	120	19	damping	damping	NOUN
cana-941	120	20	.	.	PUNCT
cana-941	121	1	9	9	X
cana-941	121	2	.	.	X
cana-941	121	3	applications	application	NOUN
cana-941	121	4	and	and	CCONJ
cana-941	121	5	implications	implication	NOUN
cana-941	121	6	:	:	PUNCT
cana-941	121	7	here	here	ADV
cana-941	121	8	,	,	PUNCT
cana-941	121	9	we	we	PRON
cana-941	121	10	discuss	discuss	VERB
cana-941	121	11	potential	potential	ADJ
cana-941	121	12	applications	application	NOUN
cana-941	121	13	of	of	ADP
cana-941	121	14	our	our	PRON
cana-941	121	15	research	research	NOUN
cana-941	121	16	findings	finding	NOUN
cana-941	121	17	and	and	CCONJ
cana-941	121	18	their	their	PRON
cana-941	121	19	implications	implication	NOUN
cana-941	121	20	for	for	ADP
cana-941	121	21	diverse	diverse	ADJ
cana-941	121	22	fields	field	NOUN
cana-941	121	23	.	.	PUNCT
cana-941	122	1	we	we	PRON
cana-941	122	2	explore	explore	VERB
cana-941	122	3	how	how	SCONJ
cana-941	122	4	the	the	DET
cana-941	122	5	insights	insight	NOUN
cana-941	122	6	gained	gain	VERB
cana-941	122	7	from	from	ADP
cana-941	122	8	our	our	PRON
cana-941	122	9	study	study	NOUN
cana-941	122	10	contribute	contribute	VERB
cana-941	122	11	to	to	ADP
cana-941	122	12	advancements	advancement	NOUN
cana-941	122	13	in	in	ADP
cana-941	122	14	quantum	quantum	ADJ
cana-941	122	15	mechanics	mechanic	NOUN
cana-941	122	16	,	,	PUNCT
cana-941	122	17	condensed	condense	VERB
cana-941	122	18	matter	matter	NOUN
cana-941	122	19	physics	physics	NOUN
cana-941	122	20	,	,	PUNCT
cana-941	122	21	and	and	CCONJ
cana-941	122	22	related	related	ADJ
cana-941	122	23	disciplines	discipline	NOUN
cana-941	122	24	.	.	PUNCT
cana-941	123	1	by	by	ADP
cana-941	123	2	highlighting	highlight	VERB
cana-941	123	3	practical	practical	ADJ
cana-941	123	4	implications	implication	NOUN
cana-941	123	5	,	,	PUNCT
cana-941	123	6	we	we	PRON
cana-941	123	7	underscore	underscore	VERB
cana-941	123	8	the	the	DET
cana-941	123	9	significance	significance	NOUN
cana-941	123	10	of	of	ADP
cana-941	123	11	our	our	PRON
cana-941	123	12	research	research	NOUN
cana-941	123	13	in	in	ADP
cana-941	123	14	advancing	advance	VERB
cana-941	123	15	scientific	scientific	ADJ
cana-941	123	16	knowledge	knowledge	NOUN
cana-941	123	17	and	and	CCONJ
cana-941	123	18	technological	technological	ADJ
cana-941	123	19	innovation	innovation	NOUN
cana-941	123	20	.	.	PUNCT
cana-941	124	1	10	10	X
cana-941	124	2	.	.	X
cana-941	124	3	conclusion	conclusion	NOUN
cana-941	124	4	:	:	PUNCT
cana-941	124	5	in	in	ADP
cana-941	124	6	the	the	DET
cana-941	124	7	final	final	ADJ
cana-941	124	8	section	section	NOUN
cana-941	124	9	,	,	PUNCT
cana-941	124	10	we	we	PRON
cana-941	124	11	summarize	summarize	VERB
cana-941	124	12	the	the	DET
cana-941	124	13	key	key	ADJ
cana-941	124	14	findings	finding	NOUN
cana-941	124	15	and	and	CCONJ
cana-941	124	16	contributions	contribution	NOUN
cana-941	124	17	of	of	ADP
cana-941	124	18	our	our	PRON
cana-941	124	19	research	research	NOUN
cana-941	124	20	.	.	PUNCT
cana-941	125	1	we	we	PRON
cana-941	125	2	reflect	reflect	VERB
cana-941	125	3	on	on	ADP
cana-941	125	4	the	the	DET
cana-941	125	5	implications	implication	NOUN
cana-941	125	6	of	of	ADP
cana-941	125	7	our	our	PRON
cana-941	125	8	findings	finding	NOUN
cana-941	125	9	,	,	PUNCT
cana-941	125	10	discuss	discuss	VERB
cana-941	125	11	limitations	limitation	NOUN
cana-941	125	12	,	,	PUNCT
cana-941	125	13	and	and	CCONJ
cana-941	125	14	propose	propose	VERB
cana-941	125	15	directions	direction	NOUN
cana-941	125	16	for	for	ADP
cana-941	125	17	future	future	ADJ
cana-941	125	18	research	research	NOUN
cana-941	125	19	.	.	PUNCT
cana-941	126	1	by	by	ADP
cana-941	126	2	synthesizing	synthesize	VERB
cana-941	126	3	our	our	PRON
cana-941	126	4	findings	finding	NOUN
cana-941	126	5	within	within	ADP
cana-941	126	6	the	the	DET
cana-941	126	7	broader	broad	ADJ
cana-941	126	8	context	context	NOUN
cana-941	126	9	of	of	ADP
cana-941	126	10	existing	exist	VERB
cana-941	126	11	knowledge	knowledge	NOUN
cana-941	126	12	,	,	PUNCT
cana-941	126	13	we	we	PRON
cana-941	126	14	emphasize	emphasize	VERB
cana-941	126	15	the	the	DET
cana-941	126	16	significance	significance	NOUN
cana-941	126	17	of	of	ADP
cana-941	126	18	our	our	PRON
cana-941	126	19	research	research	NOUN
cana-941	126	20	in	in	ADP
cana-941	126	21	advancing	advance	VERB
cana-941	126	22	the	the	DET
cana-941	126	23	understanding	understanding	NOUN
cana-941	126	24	of	of	ADP
cana-941	126	25	coupled	couple	VERB
cana-941	126	26	quantum	quantum	NOUN
cana-941	126	27	systems	system	NOUN
cana-941	126	28	with	with	ADP
cana-941	126	29	fractional	fractional	ADJ
cana-941	126	30	damping	damping	NOUN
cana-941	126	31	.	.	PUNCT
cana-941	127	1	references	reference	NOUN
cana-941	127	2	[	[	X
cana-941	127	3	1	1	NUM
cana-941	127	4	]	]	X
cana-941	127	5	wang	wang	PROPN
cana-941	127	6	,	,	PUNCT
cana-941	127	7	y.	y.	PROPN
cana-941	127	8	,	,	PUNCT
cana-941	127	9	&	&	CCONJ
cana-941	127	10	sun	sun	PROPN
cana-941	127	11	,	,	PUNCT
cana-941	127	12	h.	h.	PROPN
cana-941	127	13	(	(	PUNCT
cana-941	127	14	2019	2019	NUM
cana-941	127	15	)	)	PUNCT
cana-941	127	16	.	.	PUNCT
cana-941	128	1	fractional	fractional	ADJ
cana-941	128	2	schrödinger	schrödinger	ADJ
cana-941	128	3	equations	equation	NOUN
cana-941	128	4	with	with	ADP
cana-941	128	5	damping	damp	VERB
cana-941	128	6	and	and	CCONJ
cana-941	128	7	nonlinear	nonlinear	ADJ
cana-941	128	8	terms	term	NOUN
cana-941	128	9	.	.	PUNCT
cana-941	129	1	journal	journal	NOUN
cana-941	129	2	of	of	ADP
cana-941	129	3	mathematical	mathematical	ADJ
cana-941	129	4	physics	physics	NOUN
cana-941	129	5	,	,	PUNCT
cana-941	129	6	60(3	60(3	NOUN
cana-941	129	7	)	)	PUNCT
cana-941	129	8	,	,	PUNCT
cana-941	129	9	031502	031502	NUM
cana-941	129	10	.	.	PUNCT
cana-941	130	1	[	[	X
cana-941	130	2	2	2	NUM
cana-941	130	3	]	]	PUNCT
cana-941	130	4	podlubny	podlubny	NOUN
cana-941	130	5	,	,	PUNCT
cana-941	130	6	i.	i.	NOUN
cana-941	130	7	(	(	PUNCT
cana-941	130	8	1999	1999	NUM
cana-941	130	9	)	)	PUNCT
cana-941	130	10	.	.	PUNCT
cana-941	131	1	fractional	fractional	ADJ
cana-941	131	2	differential	differential	ADJ
cana-941	131	3	equations	equation	NOUN
cana-941	131	4	:	:	PUNCT
cana-941	131	5	an	an	DET
cana-941	131	6	introduction	introduction	NOUN
cana-941	131	7	to	to	ADP
cana-941	131	8	fractional	fractional	ADJ
cana-941	131	9	derivatives	derivative	NOUN
cana-941	131	10	,	,	PUNCT
cana-941	131	11	fractional	fractional	ADJ
cana-941	131	12	differential	differential	ADJ
cana-941	131	13	equations	equation	NOUN
cana-941	131	14	,	,	PUNCT
cana-941	131	15	some	some	DET
cana-941	131	16	methods	method	NOUN
cana-941	131	17	of	of	ADP
cana-941	131	18	their	their	PRON
cana-941	131	19	solution	solution	NOUN
cana-941	131	20	and	and	CCONJ
cana-941	131	21	some	some	PRON
cana-941	131	22	of	of	ADP
cana-941	131	23	their	their	PRON
cana-941	131	24	applications	application	NOUN
cana-941	131	25	.	.	PUNCT
cana-941	132	1	academic	academic	ADJ
cana-941	132	2	press	press	NOUN
cana-941	132	3	.	.	PUNCT
cana-941	133	1	communications	communication	NOUN
cana-941	133	2	on	on	ADP
cana-941	133	3	applied	apply	VERB
cana-941	133	4	nonlinear	nonlinear	ADJ
cana-941	133	5	analysis	analysis	NOUN
cana-941	133	6	issn	issn	NOUN
cana-941	133	7	:	:	PUNCT
cana-941	133	8	1074	1074	NUM
cana-941	133	9	-	-	PUNCT
cana-941	133	10	133x	133x	NUM
cana-941	133	11	vol	vol	NOUN
cana-941	133	12	31	31	NUM
cana-941	133	13	no	no	NOUN
cana-941	133	14	.	.	PUNCT
cana-941	134	1	4s	4s	NUM
cana-941	134	2	(	(	PUNCT
cana-941	134	3	2024	2024	NUM
cana-941	134	4	)	)	PUNCT
cana-941	134	5	447	447	NUM
cana-941	134	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-941	135	1	[	[	X
cana-941	135	2	3	3	NUM
cana-941	135	3	]	]	PUNCT
cana-941	135	4	tarasov	tarasov	NOUN
cana-941	135	5	,	,	PUNCT
cana-941	135	6	v.	v.	PROPN
cana-941	135	7	e.	e.	PROPN
cana-941	135	8	(	(	PUNCT
cana-941	135	9	2013	2013	NUM
cana-941	135	10	)	)	PUNCT
cana-941	135	11	.	.	PUNCT
cana-941	136	1	fractional	fractional	ADJ
cana-941	136	2	dynamics	dynamic	NOUN
cana-941	136	3	:	:	PUNCT
cana-941	136	4	applications	application	NOUN
cana-941	136	5	of	of	ADP
cana-941	136	6	fractional	fractional	ADJ
cana-941	136	7	calculus	calculus	NOUN
cana-941	136	8	to	to	ADP
cana-941	136	9	dynamics	dynamic	NOUN
cana-941	136	10	of	of	ADP
cana-941	136	11	particles	particle	NOUN
cana-941	136	12	,	,	PUNCT
cana-941	136	13	fields	field	NOUN
cana-941	136	14	and	and	CCONJ
cana-941	136	15	media	medium	NOUN
cana-941	136	16	.	.	PUNCT
cana-941	137	1	springer	springer	NOUN
cana-941	137	2	science	science	PROPN
cana-941	137	3	&	&	CCONJ
cana-941	137	4	business	business	NOUN
cana-941	137	5	media	medium	NOUN
cana-941	137	6	.	.	PUNCT
cana-941	138	1	[	[	X
cana-941	138	2	4	4	NUM
cana-941	138	3	]	]	X
cana-941	138	4	gorenflo	gorenflo	NOUN
cana-941	138	5	,	,	PUNCT
cana-941	138	6	r.	r.	PROPN
cana-941	138	7	,	,	PUNCT
cana-941	138	8	mainardi	mainardi	PROPN
cana-941	138	9	,	,	PUNCT
cana-941	138	10	f.	f.	PROPN
cana-941	138	11	,	,	PUNCT
cana-941	138	12	moretti	moretti	PROPN
cana-941	138	13	,	,	PUNCT
cana-941	138	14	d.	d.	PROPN
cana-941	138	15	,	,	PUNCT
cana-941	138	16	&	&	CCONJ
cana-941	138	17	pagnini	pagnini	PROPN
cana-941	138	18	,	,	PUNCT
cana-941	138	19	g.	g.	PROPN
cana-941	138	20	(	(	PUNCT
cana-941	138	21	2002	2002	NUM
cana-941	138	22	)	)	PUNCT
cana-941	138	23	.	.	PUNCT
cana-941	139	1	the	the	DET
cana-941	139	2	mittag	mittag	ADJ
cana-941	139	3	-	-	PUNCT
cana-941	139	4	leffler	leffler	NOUN
cana-941	139	5	function	function	NOUN
cana-941	139	6	as	as	ADP
cana-941	139	7	a	a	DET
cana-941	139	8	model	model	NOUN
cana-941	139	9	for	for	ADP
cana-941	139	10	relaxation	relaxation	NOUN
cana-941	139	11	processes	process	NOUN
cana-941	139	12	.	.	PUNCT
cana-941	140	1	chaos	chaos	NOUN
cana-941	140	2	,	,	PUNCT
cana-941	140	3	solitons	soliton	NOUN
cana-941	140	4	&	&	CCONJ
cana-941	140	5	fractals	fractal	NOUN
cana-941	140	6	,	,	PUNCT
cana-941	140	7	13(3	13(3	NUM
cana-941	140	8	)	)	PUNCT
cana-941	140	9	,	,	PUNCT
cana-941	140	10	579	579	NUM
cana-941	140	11	-	-	SYM
cana-941	140	12	586	586	NUM
cana-941	140	13	.	.	PUNCT
cana-941	141	1	[	[	X
cana-941	141	2	5	5	NUM
cana-941	141	3	]	]	X
cana-941	141	4	kumar	kumar	PROPN
cana-941	141	5	,	,	PUNCT
cana-941	141	6	d.	d.	PROPN
cana-941	141	7	,	,	PUNCT
cana-941	141	8	baleanu	baleanu	PROPN
cana-941	141	9	,	,	PUNCT
cana-941	141	10	d.	d.	PROPN
cana-941	141	11	,	,	PUNCT
cana-941	141	12	&	&	CCONJ
cana-941	141	13	khalique	khalique	PROPN
cana-941	141	14	,	,	PUNCT
cana-941	141	15	c.	c.	PROPN
cana-941	141	16	m.	m.	NOUN
cana-941	141	17	(	(	PUNCT
cana-941	141	18	2020	2020	NUM
cana-941	141	19	)	)	PUNCT
cana-941	141	20	.	.	PUNCT
cana-941	142	1	analytical	analytical	ADJ
cana-941	142	2	solutions	solution	NOUN
cana-941	142	3	of	of	ADP
cana-941	142	4	the	the	DET
cana-941	142	5	space	space	NOUN
cana-941	142	6	-	-	PUNCT
cana-941	142	7	time	time	NOUN
cana-941	142	8	fractional	fractional	ADJ
cana-941	142	9	schrödinger	schrödinger	ADJ
cana-941	142	10	equation	equation	NOUN
cana-941	142	11	with	with	ADP
cana-941	142	12	exponential	exponential	NOUN
cana-941	142	13	and	and	CCONJ
cana-941	142	14	trigonometric	trigonometric	ADJ
cana-941	142	15	potentials	potential	NOUN
cana-941	142	16	.	.	PUNCT
cana-941	143	1	physics	physics	NOUN
cana-941	143	2	letters	letter	NOUN
cana-941	143	3	a	a	PRON
cana-941	143	4	,	,	PUNCT
cana-941	143	5	384(9	384(9	NUM
cana-941	143	6	)	)	PUNCT
cana-941	143	7	,	,	PUNCT
cana-941	143	8	126617	126617	NUM
cana-941	143	9	.	.	PUNCT
cana-941	144	1	[	[	X
cana-941	144	2	6	6	NUM
cana-941	144	3	]	]	PUNCT
cana-941	144	4	ortigueira	ortigueira	PROPN
cana-941	144	5	,	,	PUNCT
cana-941	144	6	m.	m.	PROPN
cana-941	144	7	d.	d.	PROPN
cana-941	144	8	,	,	PUNCT
cana-941	144	9	tenreiro	tenreiro	PROPN
cana-941	144	10	machado	machado	PROPN
cana-941	144	11	,	,	PUNCT
cana-941	144	12	j.	j.	PROPN
cana-941	144	13	a.	a.	PROPN
cana-941	144	14	,	,	PUNCT
cana-941	144	15	&	&	CCONJ
cana-941	144	16	trujillo	trujillo	PROPN
cana-941	144	17	,	,	PUNCT
cana-941	144	18	j.	j.	PROPN
cana-941	144	19	j.	j.	PROPN
cana-941	144	20	(	(	PUNCT
cana-941	144	21	2006	2006	NUM
cana-941	144	22	)	)	PUNCT
cana-941	144	23	.	.	PUNCT
cana-941	145	1	the	the	DET
cana-941	145	2	fractional	fractional	ADJ
cana-941	145	3	-	-	PUNCT
cana-941	145	4	order	order	NOUN
cana-941	145	5	hyperbolic	hyperbolic	ADJ
cana-941	145	6	heat	heat	NOUN
cana-941	145	7	conduction	conduction	NOUN
cana-941	145	8	equation	equation	NOUN
cana-941	145	9	.	.	PUNCT
cana-941	146	1	thermal	thermal	ADJ
cana-941	146	2	science	science	NOUN
cana-941	146	3	,	,	PUNCT
cana-941	146	4	10(2	10(2	NUM
cana-941	146	5	)	)	PUNCT
cana-941	146	6	,	,	PUNCT
cana-941	146	7	49	49	NUM
cana-941	146	8	-	-	SYM
cana-941	146	9	56	56	NUM
cana-941	146	10	.	.	PUNCT
cana-941	147	1	[	[	X
cana-941	147	2	7	7	NUM
cana-941	147	3	]	]	ADJ
cana-941	147	4	hilfer	hilfer	NOUN
cana-941	147	5	,	,	PUNCT
cana-941	147	6	r.	r.	PROPN
cana-941	147	7	(	(	PUNCT
cana-941	147	8	2000	2000	NUM
cana-941	147	9	)	)	PUNCT
cana-941	147	10	.	.	PUNCT
cana-941	148	1	applications	application	NOUN
cana-941	148	2	of	of	ADP
cana-941	148	3	fractional	fractional	ADJ
cana-941	148	4	calculus	calculus	NOUN
cana-941	148	5	in	in	ADP
cana-941	148	6	physics	physics	PROPN
cana-941	148	7	.	.	PUNCT
cana-941	149	1	world	world	PROPN
cana-941	149	2	scientific	scientific	ADJ
cana-941	149	3	.	.	PUNCT
cana-941	150	1	[	[	X
cana-941	150	2	8	8	NUM
cana-941	150	3	]	]	PUNCT
cana-941	150	4	atangana	atangana	PROPN
cana-941	150	5	,	,	PUNCT
cana-941	150	6	a.	a.	PROPN
cana-941	150	7	,	,	PUNCT
cana-941	150	8	&	&	CCONJ
cana-941	150	9	baleanu	baleanu	PROPN
cana-941	150	10	,	,	PUNCT
cana-941	150	11	d.	d.	PROPN
cana-941	150	12	(	(	PUNCT
cana-941	150	13	2016	2016	NUM
cana-941	150	14	)	)	PUNCT
cana-941	150	15	.	.	PUNCT
cana-941	151	1	new	new	ADJ
cana-941	151	2	fractional	fractional	ADJ
cana-941	151	3	derivatives	derivative	NOUN
cana-941	151	4	with	with	ADP
cana-941	151	5	non	non	ADJ
cana-941	151	6	-	-	ADJ
cana-941	151	7	local	local	ADJ
cana-941	151	8	and	and	CCONJ
cana-941	151	9	non	non	ADJ
cana-941	151	10	-	-	ADJ
cana-941	151	11	singular	singular	ADJ
cana-941	151	12	kernel	kernel	NOUN
cana-941	151	13	:	:	PUNCT
cana-941	151	14	theory	theory	NOUN
cana-941	151	15	and	and	CCONJ
cana-941	151	16	application	application	NOUN
cana-941	151	17	to	to	PART
cana-941	151	18	heat	heat	NOUN
cana-941	151	19	transfer	transfer	NOUN
cana-941	151	20	model	model	NOUN
cana-941	151	21	.	.	PUNCT
cana-941	152	1	thermal	thermal	ADJ
cana-941	152	2	science	science	NOUN
cana-941	152	3	,	,	PUNCT
cana-941	152	4	20(2	20(2	NUM
cana-941	152	5	)	)	PUNCT
cana-941	152	6	,	,	PUNCT
cana-941	152	7	763	763	NUM
cana-941	152	8	-	-	SYM
cana-941	152	9	769	769	NUM
cana-941	152	10	.	.	PUNCT
cana-941	153	1	[	[	X
cana-941	153	2	9	9	NUM
cana-941	153	3	]	]	X
cana-941	153	4	caputo	caputo	PROPN
cana-941	153	5	,	,	PUNCT
cana-941	153	6	m.	m.	NOUN
cana-941	153	7	,	,	PUNCT
cana-941	153	8	&	&	CCONJ
cana-941	153	9	fabrizio	fabrizio	PROPN
cana-941	153	10	,	,	PUNCT
cana-941	153	11	m.	m.	NOUN
cana-941	153	12	(	(	PUNCT
cana-941	153	13	2011	2011	NUM
cana-941	153	14	)	)	PUNCT
cana-941	153	15	.	.	PUNCT
cana-941	154	1	a	a	DET
cana-941	154	2	new	new	ADJ
cana-941	154	3	definition	definition	NOUN
cana-941	154	4	of	of	ADP
cana-941	154	5	fractional	fractional	ADJ
cana-941	154	6	derivative	derivative	NOUN
cana-941	154	7	without	without	ADP
cana-941	154	8	singular	singular	ADJ
cana-941	154	9	kernel	kernel	PROPN
cana-941	154	10	.	.	PUNCT
cana-941	155	1	progress	progress	NOUN
cana-941	155	2	in	in	ADP
cana-941	155	3	fractional	fractional	ADJ
cana-941	155	4	differentiation	differentiation	NOUN
cana-941	155	5	and	and	CCONJ
cana-941	155	6	applications	application	NOUN
cana-941	155	7	,	,	PUNCT
cana-941	155	8	1(2	1(2	NUM
cana-941	155	9	)	)	PUNCT
cana-941	155	10	,	,	PUNCT
cana-941	155	11	73	73	NUM
cana-941	155	12	-	-	SYM
cana-941	155	13	85	85	NUM
cana-941	155	14	.	.	PUNCT
cana-941	156	1	[	[	X
cana-941	156	2	10	10	NUM
cana-941	156	3	]	]	X
cana-941	156	4	west	west	PROPN
cana-941	156	5	,	,	PUNCT
cana-941	156	6	b.	b.	PROPN
cana-941	156	7	j.	j.	PROPN
cana-941	156	8	,	,	PUNCT
cana-941	156	9	bologna	bologna	NOUN
cana-941	156	10	,	,	PUNCT
cana-941	156	11	m.	m.	NOUN
cana-941	156	12	,	,	PUNCT
cana-941	156	13	&	&	CCONJ
cana-941	156	14	grigolini	grigolini	PROPN
cana-941	156	15	,	,	PUNCT
cana-941	156	16	p.	p.	NOUN
cana-941	156	17	(	(	PUNCT
cana-941	156	18	2003	2003	NUM
cana-941	156	19	)	)	PUNCT
cana-941	156	20	.	.	PUNCT
cana-941	157	1	physics	physics	PROPN
cana-941	157	2	of	of	ADP
cana-941	157	3	fractal	fractal	PROPN
cana-941	157	4	operators	operator	NOUN
cana-941	157	5	.	.	PUNCT
cana-941	158	1	springer	springer	NOUN
cana-941	158	2	science	science	PROPN
cana-941	158	3	&	&	CCONJ
cana-941	158	4	business	business	NOUN
cana-941	158	5	media	medium	NOUN
cana-941	158	6	.	.	PUNCT