id	sid	tid	token	lemma	pos
ejde-627	1	1	electronic	electronic	ADJ
ejde-627	1	2	journal	journal	NOUN
ejde-627	1	3	of	of	ADP
ejde-627	1	4	differential	differential	ADJ
ejde-627	1	5	equations	equation	NOUN
ejde-627	1	6	,	,	PUNCT
ejde-627	1	7	vol	vol	NOUN
ejde-627	1	8	.	.	NOUN
ejde-627	1	9	2024	2024	NUM
ejde-627	1	10	(	(	PUNCT
ejde-627	1	11	2024	2024	NUM
ejde-627	1	12	)	)	PUNCT
ejde-627	1	13	,	,	PUNCT
ejde-627	1	14	no	no	INTJ
ejde-627	1	15	.	.	NOUN
ejde-627	1	16	28	28	NUM
ejde-627	1	17	,	,	PUNCT
ejde-627	1	18	pp	pp	ADJ
ejde-627	1	19	.	.	PUNCT
ejde-627	2	1	1–8	1–8	X
ejde-627	2	2	.	.	PUNCT
ejde-627	2	3	issn	issn	PROPN
ejde-627	2	4	:	:	PUNCT
ejde-627	2	5	1072	1072	NUM
ejde-627	2	6	-	-	SYM
ejde-627	2	7	6691	6691	NUM
ejde-627	2	8	.	.	PUNCT
ejde-627	3	1	url	url	PROPN
ejde-627	3	2	:	:	PUNCT
ejde-627	3	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-627	3	4	,	,	PUNCT
ejde-627	3	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-627	3	6	doi	doi	PROPN
ejde-627	3	7	:	:	PUNCT
ejde-627	3	8	10.58997	10.58997	NUM
ejde-627	3	9	/	/	SYM
ejde-627	3	10	ejde.2024.28	ejde.2024.28	NOUN
ejde-627	3	11	solutions	solution	NOUN
ejde-627	3	12	of	of	ADP
ejde-627	3	13	linear	linear	PROPN
ejde-627	3	14	and	and	CCONJ
ejde-627	3	15	non	non	ADJ
ejde-627	3	16	-	-	ADJ
ejde-627	3	17	linear	linear	ADJ
ejde-627	3	18	partial	partial	ADJ
ejde-627	3	19	differential	differential	NOUN
ejde-627	3	20	equations	equation	NOUN
ejde-627	3	21	by	by	ADP
ejde-627	3	22	means	mean	NOUN
ejde-627	3	23	of	of	ADP
ejde-627	3	24	tensor	tensor	NOUN
ejde-627	3	25	product	product	NOUN
ejde-627	3	26	theory	theory	NOUN
ejde-627	3	27	of	of	ADP
ejde-627	3	28	banach	banach	NOUN
ejde-627	3	29	spaces	space	NOUN
ejde-627	3	30	waseem	waseem	PROPN
ejde-627	3	31	ghazi	ghazi	PROPN
ejde-627	3	32	alshanti	alshanti	PROPN
ejde-627	3	33	communicated	communicate	VERB
ejde-627	3	34	by	by	ADP
ejde-627	3	35	jerome	jerome	PROPN
ejde-627	3	36	a.	a.	PROPN
ejde-627	3	37	goldstein	goldstein	PROPN
ejde-627	3	38	abstract	abstract	NOUN
ejde-627	3	39	.	.	PUNCT
ejde-627	4	1	in	in	ADP
ejde-627	4	2	this	this	DET
ejde-627	4	3	article	article	NOUN
ejde-627	4	4	,	,	PUNCT
ejde-627	4	5	we	we	PRON
ejde-627	4	6	introduce	introduce	VERB
ejde-627	4	7	an	an	DET
ejde-627	4	8	analytical	analytical	ADJ
ejde-627	4	9	method	method	NOUN
ejde-627	4	10	for	for	ADP
ejde-627	4	11	solving	solve	VERB
ejde-627	4	12	both	both	CCONJ
ejde-627	4	13	non	non	ADJ
ejde-627	4	14	-	-	ADJ
ejde-627	4	15	separable	separable	ADJ
ejde-627	4	16	linear	linear	ADJ
ejde-627	4	17	and	and	CCONJ
ejde-627	4	18	non	non	ADJ
ejde-627	4	19	-	-	ADJ
ejde-627	4	20	linear	linear	ADJ
ejde-627	4	21	partial	partial	ADJ
ejde-627	4	22	differential	differential	NOUN
ejde-627	4	23	equations	equation	NOUN
ejde-627	4	24	,	,	PUNCT
ejde-627	4	25	for	for	ADP
ejde-627	4	26	which	which	DET
ejde-627	4	27	separation	separation	NOUN
ejde-627	4	28	of	of	ADP
ejde-627	4	29	variables	variable	NOUN
ejde-627	4	30	method	method	NOUN
ejde-627	4	31	does	do	AUX
ejde-627	4	32	not	not	PART
ejde-627	4	33	work	work	VERB
ejde-627	4	34	.	.	PUNCT
ejde-627	5	1	this	this	DET
ejde-627	5	2	method	method	NOUN
ejde-627	5	3	is	be	AUX
ejde-627	5	4	based	base	VERB
ejde-627	5	5	on	on	ADP
ejde-627	5	6	the	the	DET
ejde-627	5	7	theory	theory	NOUN
ejde-627	5	8	of	of	ADP
ejde-627	5	9	tensor	tensor	NOUN
ejde-627	5	10	product	product	NOUN
ejde-627	5	11	in	in	ADP
ejde-627	5	12	banach	banach	NOUN
ejde-627	5	13	spaces	space	NOUN
ejde-627	5	14	coupled	couple	VERB
ejde-627	5	15	with	with	ADP
ejde-627	5	16	some	some	DET
ejde-627	5	17	properties	property	NOUN
ejde-627	5	18	of	of	ADP
ejde-627	5	19	atoms	atom	NOUN
ejde-627	5	20	operators	operator	NOUN
ejde-627	5	21	.	.	PUNCT
ejde-627	6	1	we	we	PRON
ejde-627	6	2	provide	provide	VERB
ejde-627	6	3	some	some	DET
ejde-627	6	4	illustrative	illustrative	ADJ
ejde-627	6	5	examples	example	NOUN
ejde-627	6	6	.	.	PUNCT
ejde-627	7	1	1	1	X
ejde-627	7	2	.	.	X
ejde-627	7	3	introduction	introduction	NOUN
ejde-627	7	4	converting	convert	VERB
ejde-627	7	5	a	a	DET
ejde-627	7	6	real	real	ADJ
ejde-627	7	7	-	-	PUNCT
ejde-627	7	8	world	world	NOUN
ejde-627	7	9	problem	problem	NOUN
ejde-627	7	10	into	into	ADP
ejde-627	7	11	ordinary	ordinary	ADJ
ejde-627	7	12	or	or	CCONJ
ejde-627	7	13	partial	partial	ADJ
ejde-627	7	14	differential	differential	NOUN
ejde-627	7	15	equations	equation	NOUN
ejde-627	7	16	is	be	AUX
ejde-627	7	17	the	the	DET
ejde-627	7	18	essential	essential	ADJ
ejde-627	7	19	task	task	NOUN
ejde-627	7	20	for	for	ADP
ejde-627	7	21	mathematical	mathematical	ADJ
ejde-627	7	22	modeling	modeling	NOUN
ejde-627	7	23	[	[	X
ejde-627	7	24	4	4	NUM
ejde-627	7	25	,	,	PUNCT
ejde-627	7	26	5	5	NUM
ejde-627	7	27	,	,	PUNCT
ejde-627	7	28	7	7	NUM
ejde-627	7	29	]	]	PUNCT
ejde-627	7	30	.	.	PUNCT
ejde-627	8	1	differential	differential	ADJ
ejde-627	8	2	equations	equation	NOUN
ejde-627	8	3	,	,	PUNCT
ejde-627	8	4	ordinary	ordinary	ADJ
ejde-627	8	5	or	or	CCONJ
ejde-627	8	6	partial	partial	ADJ
ejde-627	8	7	,	,	PUNCT
ejde-627	8	8	are	be	AUX
ejde-627	8	9	equations	equation	NOUN
ejde-627	8	10	relating	relate	VERB
ejde-627	8	11	unknown	unknown	ADJ
ejde-627	8	12	functions	function	NOUN
ejde-627	8	13	and	and	CCONJ
ejde-627	8	14	some	some	PRON
ejde-627	8	15	of	of	ADP
ejde-627	8	16	their	their	PRON
ejde-627	8	17	derivatives	derivative	NOUN
ejde-627	8	18	.	.	PUNCT
ejde-627	9	1	there	there	PRON
ejde-627	9	2	are	be	VERB
ejde-627	9	3	plenty	plenty	NOUN
ejde-627	9	4	of	of	ADP
ejde-627	9	5	examples	example	NOUN
ejde-627	9	6	of	of	ADP
ejde-627	9	7	their	their	PRON
ejde-627	9	8	use	use	NOUN
ejde-627	9	9	in	in	ADP
ejde-627	9	10	economics	economic	NOUN
ejde-627	9	11	,	,	PUNCT
ejde-627	9	12	physics	physics	NOUN
ejde-627	9	13	,	,	PUNCT
ejde-627	9	14	chemistry	chemistry	NOUN
ejde-627	9	15	,	,	PUNCT
ejde-627	9	16	and	and	CCONJ
ejde-627	9	17	biology	biology	NOUN
ejde-627	9	18	.	.	PUNCT
ejde-627	10	1	in	in	ADP
ejde-627	10	2	general	general	ADJ
ejde-627	10	3	,	,	PUNCT
ejde-627	10	4	to	to	PART
ejde-627	10	5	find	find	VERB
ejde-627	10	6	analytical	analytical	ADJ
ejde-627	10	7	or	or	CCONJ
ejde-627	10	8	numerical	numerical	ADJ
ejde-627	10	9	solutions	solution	NOUN
ejde-627	10	10	of	of	ADP
ejde-627	10	11	partial	partial	ADJ
ejde-627	10	12	differential	differential	NOUN
ejde-627	10	13	equations	equation	NOUN
ejde-627	10	14	is	be	AUX
ejde-627	10	15	not	not	PART
ejde-627	10	16	straightforward	straightforward	ADJ
ejde-627	10	17	.	.	PUNCT
ejde-627	11	1	there	there	PRON
ejde-627	11	2	are	be	VERB
ejde-627	11	3	no	no	DET
ejde-627	11	4	generally	generally	ADV
ejde-627	11	5	applicable	applicable	ADJ
ejde-627	11	6	methods	method	NOUN
ejde-627	11	7	or	or	CCONJ
ejde-627	11	8	approaches	approach	NOUN
ejde-627	11	9	to	to	PART
ejde-627	11	10	solve	solve	VERB
ejde-627	11	11	all	all	DET
ejde-627	11	12	partial	partial	ADJ
ejde-627	11	13	differential	differential	ADJ
ejde-627	11	14	equations	equation	NOUN
ejde-627	11	15	of	of	ADP
ejde-627	11	16	a	a	DET
ejde-627	11	17	given	give	VERB
ejde-627	11	18	order	order	NOUN
ejde-627	11	19	,	,	PUNCT
ejde-627	11	20	even	even	ADV
ejde-627	11	21	the	the	DET
ejde-627	11	22	classes	class	NOUN
ejde-627	11	23	for	for	ADP
ejde-627	11	24	which	which	PRON
ejde-627	11	25	we	we	PRON
ejde-627	11	26	have	have	VERB
ejde-627	11	27	general	general	ADJ
ejde-627	11	28	analytic	analytic	ADJ
ejde-627	11	29	methods	method	NOUN
ejde-627	11	30	of	of	ADP
ejde-627	11	31	solution	solution	NOUN
ejde-627	11	32	are	be	AUX
ejde-627	11	33	quite	quite	ADV
ejde-627	11	34	limited	limited	ADJ
ejde-627	11	35	[	[	X
ejde-627	11	36	8	8	NUM
ejde-627	11	37	]	]	PUNCT
ejde-627	11	38	.	.	PUNCT
ejde-627	12	1	thus	thus	ADV
ejde-627	12	2	,	,	PUNCT
ejde-627	12	3	we	we	PRON
ejde-627	12	4	have	have	VERB
ejde-627	12	5	to	to	PART
ejde-627	12	6	study	study	VERB
ejde-627	12	7	fairly	fairly	ADV
ejde-627	12	8	small	small	ADJ
ejde-627	12	9	classes	class	NOUN
ejde-627	12	10	of	of	ADP
ejde-627	12	11	partial	partial	ADJ
ejde-627	12	12	differential	differential	ADJ
ejde-627	12	13	equations	equation	NOUN
ejde-627	12	14	individually	individually	ADV
ejde-627	12	15	.	.	PUNCT
ejde-627	13	1	one	one	NUM
ejde-627	13	2	of	of	ADP
ejde-627	13	3	the	the	DET
ejde-627	13	4	most	most	ADV
ejde-627	13	5	popular	popular	ADJ
ejde-627	13	6	methods	method	NOUN
ejde-627	13	7	for	for	ADP
ejde-627	13	8	solving	solve	VERB
ejde-627	13	9	specific	specific	ADJ
ejde-627	13	10	types	type	NOUN
ejde-627	13	11	of	of	ADP
ejde-627	13	12	partial	partial	ADJ
ejde-627	13	13	differential	differential	ADJ
ejde-627	13	14	equations	equation	NOUN
ejde-627	13	15	is	be	AUX
ejde-627	13	16	the	the	DET
ejde-627	13	17	method	method	NOUN
ejde-627	13	18	of	of	ADP
ejde-627	13	19	separation	separation	NOUN
ejde-627	13	20	of	of	ADP
ejde-627	13	21	variables	variable	NOUN
ejde-627	13	22	.	.	PUNCT
ejde-627	14	1	this	this	DET
ejde-627	14	2	method	method	NOUN
ejde-627	14	3	is	be	AUX
ejde-627	14	4	based	base	VERB
ejde-627	14	5	on	on	ADP
ejde-627	14	6	the	the	DET
ejde-627	14	7	idea	idea	NOUN
ejde-627	14	8	that	that	SCONJ
ejde-627	14	9	the	the	DET
ejde-627	14	10	solution	solution	NOUN
ejde-627	14	11	of	of	ADP
ejde-627	14	12	the	the	DET
ejde-627	14	13	equation	equation	NOUN
ejde-627	14	14	is	be	AUX
ejde-627	14	15	separable	separable	ADJ
ejde-627	14	16	,	,	PUNCT
ejde-627	14	17	that	that	ADV
ejde-627	14	18	is	is	ADV
ejde-627	14	19	,	,	PUNCT
ejde-627	14	20	the	the	DET
ejde-627	14	21	final	final	ADJ
ejde-627	14	22	solution	solution	NOUN
ejde-627	14	23	can	can	AUX
ejde-627	14	24	be	be	AUX
ejde-627	14	25	represented	represent	VERB
ejde-627	14	26	as	as	ADP
ejde-627	14	27	the	the	DET
ejde-627	14	28	product	product	NOUN
ejde-627	14	29	of	of	ADP
ejde-627	14	30	several	several	ADJ
ejde-627	14	31	functions	function	NOUN
ejde-627	14	32	,	,	PUNCT
ejde-627	14	33	each	each	PRON
ejde-627	14	34	of	of	ADP
ejde-627	14	35	which	which	PRON
ejde-627	14	36	only	only	ADV
ejde-627	14	37	depends	depend	VERB
ejde-627	14	38	on	on	ADP
ejde-627	14	39	one	one	NUM
ejde-627	14	40	independent	independent	ADJ
ejde-627	14	41	variable	variable	NOUN
ejde-627	14	42	.	.	PUNCT
ejde-627	15	1	however	however	ADV
ejde-627	15	2	,	,	PUNCT
ejde-627	15	3	the	the	DET
ejde-627	15	4	procedure	procedure	NOUN
ejde-627	15	5	of	of	ADP
ejde-627	15	6	separation	separation	NOUN
ejde-627	15	7	of	of	ADP
ejde-627	15	8	variables	variable	NOUN
ejde-627	15	9	does	do	AUX
ejde-627	15	10	have	have	VERB
ejde-627	15	11	some	some	DET
ejde-627	15	12	limitations	limitation	NOUN
ejde-627	15	13	,	,	PUNCT
ejde-627	15	14	it	it	PRON
ejde-627	15	15	works	work	VERB
ejde-627	15	16	in	in	ADP
ejde-627	15	17	very	very	ADV
ejde-627	15	18	special	special	ADJ
ejde-627	15	19	cases	case	NOUN
ejde-627	15	20	involving	involve	VERB
ejde-627	15	21	equations	equation	NOUN
ejde-627	15	22	with	with	ADP
ejde-627	15	23	high	high	ADJ
ejde-627	15	24	degree	degree	NOUN
ejde-627	15	25	of	of	ADP
ejde-627	15	26	symmetry	symmetry	NOUN
ejde-627	15	27	.	.	PUNCT
ejde-627	16	1	also	also	ADV
ejde-627	16	2	,	,	PUNCT
ejde-627	16	3	to	to	PART
ejde-627	16	4	be	be	AUX
ejde-627	16	5	able	able	ADJ
ejde-627	16	6	to	to	PART
ejde-627	16	7	apply	apply	VERB
ejde-627	16	8	this	this	DET
ejde-627	16	9	method	method	NOUN
ejde-627	16	10	,	,	PUNCT
ejde-627	16	11	many	many	ADJ
ejde-627	16	12	constraints	constraint	NOUN
ejde-627	16	13	need	need	VERB
ejde-627	16	14	to	to	PART
ejde-627	16	15	be	be	AUX
ejde-627	16	16	imposed	impose	VERB
ejde-627	16	17	on	on	ADP
ejde-627	16	18	the	the	DET
ejde-627	16	19	coefficients	coefficient	NOUN
ejde-627	16	20	expressions	expression	NOUN
ejde-627	16	21	as	as	ADV
ejde-627	16	22	well	well	ADV
ejde-627	16	23	as	as	ADP
ejde-627	16	24	the	the	DET
ejde-627	16	25	companion	companion	NOUN
ejde-627	16	26	initial	initial	ADJ
ejde-627	16	27	and	and	CCONJ
ejde-627	16	28	boundary	boundary	ADJ
ejde-627	16	29	conditions	condition	NOUN
ejde-627	16	30	.	.	PUNCT
ejde-627	17	1	some	some	DET
ejde-627	17	2	other	other	ADJ
ejde-627	17	3	analytical	analytical	ADJ
ejde-627	17	4	and	and	CCONJ
ejde-627	17	5	numerical	numerical	ADJ
ejde-627	17	6	techniques	technique	NOUN
ejde-627	17	7	of	of	ADP
ejde-627	17	8	solving	solve	VERB
ejde-627	17	9	partial	partial	ADJ
ejde-627	17	10	differential	differential	ADJ
ejde-627	17	11	equations	equation	NOUN
ejde-627	17	12	are	be	AUX
ejde-627	17	13	fourier	fourier	ADJ
ejde-627	17	14	transform	transform	NOUN
ejde-627	17	15	,	,	PUNCT
ejde-627	17	16	laplace	laplace	NOUN
ejde-627	17	17	transform	transform	NOUN
ejde-627	17	18	,	,	PUNCT
ejde-627	17	19	green	green	PROPN
ejde-627	17	20	’s	’s	PART
ejde-627	17	21	functions	function	NOUN
ejde-627	17	22	,	,	PUNCT
ejde-627	17	23	2020	2020	NUM
ejde-627	17	24	mathematics	mathematic	NOUN
ejde-627	17	25	subject	subject	ADJ
ejde-627	17	26	classification	classification	NOUN
ejde-627	17	27	.	.	PUNCT
ejde-627	18	1	35e99	35e99	NUM
ejde-627	18	2	,	,	PUNCT
ejde-627	18	3	47b01	47b01	NUM
ejde-627	18	4	.	.	PUNCT
ejde-627	19	1	key	key	ADJ
ejde-627	19	2	words	word	NOUN
ejde-627	19	3	and	and	CCONJ
ejde-627	19	4	phrases	phrase	NOUN
ejde-627	19	5	.	.	PUNCT
ejde-627	20	1	partial	partial	ADJ
ejde-627	20	2	differential	differential	NOUN
ejde-627	20	3	equations	equation	NOUN
ejde-627	20	4	;	;	PUNCT
ejde-627	20	5	tensor	tensor	NOUN
ejde-627	20	6	product	product	NOUN
ejde-627	20	7	of	of	ADP
ejde-627	20	8	banach	banach	NOUN
ejde-627	20	9	spaces	space	NOUN
ejde-627	20	10	;	;	PUNCT
ejde-627	20	11	atomic	atomic	ADJ
ejde-627	20	12	solution	solution	NOUN
ejde-627	20	13	.	.	PUNCT
ejde-627	21	1	©	©	PROPN
ejde-627	21	2	2024	2024	NUM
ejde-627	21	3	.	.	PUNCT
ejde-627	22	1	this	this	DET
ejde-627	22	2	work	work	NOUN
ejde-627	22	3	is	be	AUX
ejde-627	22	4	licensed	license	VERB
ejde-627	22	5	under	under	ADP
ejde-627	22	6	a	a	DET
ejde-627	22	7	cc	cc	NOUN
ejde-627	22	8	by	by	ADP
ejde-627	22	9	4.0	4.0	NUM
ejde-627	22	10	license	license	NOUN
ejde-627	22	11	.	.	PUNCT
ejde-627	23	1	submitted	submit	VERB
ejde-627	23	2	august	august	PROPN
ejde-627	23	3	21	21	NUM
ejde-627	23	4	,	,	PUNCT
ejde-627	23	5	2023	2023	NUM
ejde-627	23	6	.	.	PUNCT
ejde-627	24	1	published	publish	VERB
ejde-627	24	2	march	march	PROPN
ejde-627	24	3	29	29	NUM
ejde-627	24	4	,	,	PUNCT
ejde-627	24	5	2024	2024	NUM
ejde-627	24	6	.	.	PUNCT
ejde-627	24	7	1	1	NUM
ejde-627	24	8	2	2	NUM
ejde-627	24	9	w.	w.	NOUN
ejde-627	24	10	g.	g.	PROPN
ejde-627	24	11	alshanti	alshanti	PROPN
ejde-627	25	1	ejde-2024/28	ejde-2024/28	PROPN
ejde-627	25	2	finite	finite	VERB
ejde-627	25	3	element	element	NOUN
ejde-627	25	4	method	method	NOUN
ejde-627	25	5	(	(	PUNCT
ejde-627	25	6	fem	fem	PROPN
ejde-627	25	7	)	)	PUNCT
ejde-627	25	8	,	,	PUNCT
ejde-627	25	9	finite	finite	ADJ
ejde-627	25	10	volume	volume	NOUN
ejde-627	25	11	methods	method	NOUN
ejde-627	25	12	(	(	PUNCT
ejde-627	25	13	fvm	fvm	NOUN
ejde-627	25	14	)	)	PUNCT
ejde-627	25	15	,	,	PUNCT
ejde-627	25	16	and	and	CCONJ
ejde-627	25	17	finite	finite	ADJ
ejde-627	25	18	difference	difference	NOUN
ejde-627	25	19	methods	method	NOUN
ejde-627	25	20	(	(	PUNCT
ejde-627	25	21	fdm)[12	fdm)[12	NOUN
ejde-627	25	22	]	]	X
ejde-627	25	23	.	.	PUNCT
ejde-627	26	1	in	in	ADP
ejde-627	26	2	2010	2010	NUM
ejde-627	26	3	,	,	PUNCT
ejde-627	26	4	khalil	khalil	PROPN
ejde-627	27	1	[	[	X
ejde-627	27	2	10	10	NUM
ejde-627	27	3	]	]	PUNCT
ejde-627	27	4	introduced	introduce	VERB
ejde-627	27	5	a	a	DET
ejde-627	27	6	novel	novel	ADJ
ejde-627	27	7	approach	approach	NOUN
ejde-627	27	8	to	to	PART
ejde-627	27	9	handle	handle	VERB
ejde-627	27	10	differential	differential	ADJ
ejde-627	27	11	equations	equation	NOUN
ejde-627	27	12	for	for	ADP
ejde-627	27	13	both	both	CCONJ
ejde-627	27	14	ordinary	ordinary	ADJ
ejde-627	27	15	and	and	CCONJ
ejde-627	27	16	fractional	fractional	ADJ
ejde-627	27	17	orders	order	NOUN
ejde-627	27	18	.	.	PUNCT
ejde-627	28	1	the	the	DET
ejde-627	28	2	new	new	ADJ
ejde-627	28	3	technique	technique	NOUN
ejde-627	28	4	is	be	AUX
ejde-627	28	5	based	base	VERB
ejde-627	28	6	on	on	ADP
ejde-627	28	7	the	the	DET
ejde-627	28	8	theory	theory	NOUN
ejde-627	28	9	of	of	ADP
ejde-627	28	10	tensor	tensor	NOUN
ejde-627	28	11	product	product	NOUN
ejde-627	28	12	of	of	ADP
ejde-627	28	13	banach	banach	NOUN
ejde-627	28	14	spaces	space	NOUN
ejde-627	28	15	,	,	PUNCT
ejde-627	28	16	and	and	CCONJ
ejde-627	28	17	it	it	PRON
ejde-627	28	18	can	can	AUX
ejde-627	28	19	be	be	AUX
ejde-627	28	20	utilized	utilize	VERB
ejde-627	28	21	for	for	ADP
ejde-627	28	22	obtaining	obtain	VERB
ejde-627	28	23	the	the	DET
ejde-627	28	24	so	so	ADV
ejde-627	28	25	-	-	PUNCT
ejde-627	28	26	called	call	VERB
ejde-627	28	27	atomic	atomic	ADJ
ejde-627	28	28	solutions	solution	NOUN
ejde-627	28	29	of	of	ADP
ejde-627	28	30	the	the	DET
ejde-627	28	31	differential	differential	ADJ
ejde-627	28	32	equation	equation	NOUN
ejde-627	29	1	[	[	X
ejde-627	29	2	1	1	NUM
ejde-627	29	3	,	,	PUNCT
ejde-627	29	4	2	2	NUM
ejde-627	29	5	,	,	PUNCT
ejde-627	29	6	3	3	NUM
ejde-627	29	7	]	]	PUNCT
ejde-627	29	8	.	.	PUNCT
ejde-627	30	1	before	before	SCONJ
ejde-627	30	2	we	we	PRON
ejde-627	30	3	introduce	introduce	VERB
ejde-627	30	4	our	our	PRON
ejde-627	30	5	main	main	ADJ
ejde-627	30	6	result	result	NOUN
ejde-627	30	7	for	for	ADP
ejde-627	30	8	atomic	atomic	ADJ
ejde-627	30	9	solutions	solution	NOUN
ejde-627	30	10	of	of	ADP
ejde-627	30	11	partial	partial	ADJ
ejde-627	30	12	differential	differential	ADJ
ejde-627	30	13	equations	equation	NOUN
ejde-627	30	14	with	with	ADP
ejde-627	30	15	ordinary	ordinary	ADJ
ejde-627	30	16	orders	order	NOUN
ejde-627	30	17	,	,	PUNCT
ejde-627	30	18	we	we	PRON
ejde-627	30	19	commence	commence	VERB
ejde-627	30	20	with	with	ADP
ejde-627	30	21	some	some	DET
ejde-627	30	22	required	require	VERB
ejde-627	30	23	materials	material	NOUN
ejde-627	30	24	that	that	PRON
ejde-627	30	25	are	be	AUX
ejde-627	30	26	reported	report	VERB
ejde-627	30	27	in	in	ADP
ejde-627	30	28	[	[	X
ejde-627	30	29	6	6	NUM
ejde-627	30	30	,	,	PUNCT
ejde-627	30	31	11	11	NUM
ejde-627	30	32	,	,	PUNCT
ejde-627	30	33	13	13	NUM
ejde-627	30	34	,	,	PUNCT
ejde-627	30	35	14	14	NUM
ejde-627	30	36	,	,	PUNCT
ejde-627	30	37	15	15	NUM
ejde-627	30	38	]	]	PUNCT
ejde-627	30	39	.	.	PUNCT
ejde-627	31	1	2	2	X
ejde-627	31	2	.	.	X
ejde-627	31	3	atoms	atom	NOUN
ejde-627	31	4	operators	operator	NOUN
ejde-627	31	5	in	in	ADP
ejde-627	31	6	this	this	DET
ejde-627	31	7	section	section	NOUN
ejde-627	31	8	,	,	PUNCT
ejde-627	31	9	we	we	PRON
ejde-627	31	10	introduce	introduce	VERB
ejde-627	31	11	some	some	DET
ejde-627	31	12	preliminaries	preliminary	NOUN
ejde-627	31	13	related	relate	VERB
ejde-627	31	14	to	to	ADP
ejde-627	31	15	the	the	DET
ejde-627	31	16	main	main	ADJ
ejde-627	31	17	result	result	NOUN
ejde-627	31	18	.	.	PUNCT
ejde-627	32	1	definition	definition	NOUN
ejde-627	32	2	2.1	2.1	NUM
ejde-627	32	3	.	.	PUNCT
ejde-627	33	1	let	let	VERB
ejde-627	33	2	x	x	PRON
ejde-627	33	3	and	and	CCONJ
ejde-627	33	4	y	y	PROPN
ejde-627	33	5	be	be	AUX
ejde-627	33	6	two	two	NUM
ejde-627	33	7	banach	banach	NOUN
ejde-627	33	8	spaces	space	NOUN
ejde-627	33	9	and	and	CCONJ
ejde-627	33	10	x∗	x∗	PROPN
ejde-627	33	11	is	be	AUX
ejde-627	33	12	the	the	DET
ejde-627	33	13	dual	dual	ADJ
ejde-627	33	14	space	space	NOUN
ejde-627	33	15	of	of	ADP
ejde-627	33	16	x.	x.	NOUN
ejde-627	33	17	for	for	ADP
ejde-627	33	18	x	x	SYM
ejde-627	33	19	∈	∈	PROPN
ejde-627	33	20	x	x	X
ejde-627	33	21	and	and	CCONJ
ejde-627	33	22	y	y	PROPN
ejde-627	33	23	∈	∈	PROPN
ejde-627	33	24	y	y	PROPN
ejde-627	33	25	,	,	PUNCT
ejde-627	33	26	the	the	DET
ejde-627	33	27	operator	operator	NOUN
ejde-627	33	28	t	t	NOUN
ejde-627	33	29	:	:	PUNCT
ejde-627	33	30	x∗	x∗	PROPN
ejde-627	33	31	→	→	SYM
ejde-627	33	32	y	y	PROPN
ejde-627	33	33	,	,	PUNCT
ejde-627	33	34	defined	define	VERB
ejde-627	33	35	by	by	ADP
ejde-627	33	36	t	t	PROPN
ejde-627	33	37	(	(	PUNCT
ejde-627	33	38	x∗	x∗	PROPN
ejde-627	33	39	)	)	PUNCT
ejde-627	34	1	=	=	SYM
ejde-627	34	2	x∗(x)y	x∗(x)y	PROPN
ejde-627	34	3	=	=	PUNCT
ejde-627	34	4	⟨x	⟨x	VERB
ejde-627	34	5	,	,	PUNCT
ejde-627	34	6	x∗⟩y	x∗⟩y	PROPN
ejde-627	34	7	,	,	PUNCT
ejde-627	34	8	is	be	AUX
ejde-627	34	9	bounded	bound	VERB
ejde-627	34	10	one	one	NUM
ejde-627	34	11	rank	rank	NOUN
ejde-627	34	12	linear	linear	NOUN
ejde-627	34	13	operator	operator	NOUN
ejde-627	34	14	since	since	SCONJ
ejde-627	34	15	the	the	DET
ejde-627	34	16	range	range	NOUN
ejde-627	34	17	of	of	ADP
ejde-627	34	18	t	t	PROPN
ejde-627	34	19	is	be	AUX
ejde-627	34	20	the	the	DET
ejde-627	34	21	span	span	NOUN
ejde-627	34	22	of	of	ADP
ejde-627	34	23	y.	y.	NOUN
ejde-627	34	24	we	we	PRON
ejde-627	34	25	write	write	VERB
ejde-627	34	26	x⊗	x⊗	PROPN
ejde-627	34	27	y	y	PROPN
ejde-627	34	28	for	for	ADP
ejde-627	34	29	t	t	PROPN
ejde-627	34	30	and	and	CCONJ
ejde-627	34	31	such	such	ADJ
ejde-627	34	32	operators	operator	NOUN
ejde-627	34	33	are	be	AUX
ejde-627	34	34	called	call	VERB
ejde-627	34	35	atoms	atom	NOUN
ejde-627	34	36	.	.	PUNCT
ejde-627	35	1	for	for	ADP
ejde-627	35	2	example	example	NOUN
ejde-627	35	3	,	,	PUNCT
ejde-627	35	4	let	let	VERB
ejde-627	35	5	x	x	PRON
ejde-627	35	6	and	and	CCONJ
ejde-627	35	7	y	y	PROPN
ejde-627	35	8	be	be	AUX
ejde-627	35	9	two	two	NUM
ejde-627	35	10	banach	banach	NOUN
ejde-627	35	11	spaces	space	NOUN
ejde-627	35	12	such	such	ADJ
ejde-627	35	13	that	that	SCONJ
ejde-627	35	14	x	x	X
ejde-627	35	15	=	=	PUNCT
ejde-627	35	16	y	y	PROPN
ejde-627	35	17	=	=	SYM
ejde-627	35	18	c[0	c[0	PROPN
ejde-627	35	19	,	,	PUNCT
ejde-627	35	20	1	1	NUM
ejde-627	35	21	]	]	PUNCT
ejde-627	35	22	,	,	PUNCT
ejde-627	35	23	with	with	ADP
ejde-627	35	24	the	the	DET
ejde-627	35	25	norm	norm	NOUN
ejde-627	35	26	∥f∥	∥f∥	PROPN
ejde-627	35	27	=	=	SYM
ejde-627	35	28	sup	sup	NOUN
ejde-627	35	29	t∈[0,1	t∈[0,1	NOUN
ejde-627	35	30	]	]	PUNCT
ejde-627	36	1	|f(t)|	|f(t)|	ADJ
ejde-627	36	2	for	for	ADP
ejde-627	36	3	any	any	DET
ejde-627	36	4	f	f	PROPN
ejde-627	36	5	∈	∈	PROPN
ejde-627	36	6	c[0	c[0	PROPN
ejde-627	36	7	,	,	PUNCT
ejde-627	36	8	1	1	NUM
ejde-627	36	9	]	]	PUNCT
ejde-627	36	10	.	.	PUNCT
ejde-627	37	1	define	define	VERB
ejde-627	37	2	x∗	x∗	PROPN
ejde-627	37	3	to	to	PART
ejde-627	37	4	be	be	AUX
ejde-627	37	5	the	the	DET
ejde-627	37	6	space	space	NOUN
ejde-627	37	7	of	of	ADP
ejde-627	37	8	all	all	DET
ejde-627	37	9	regular	regular	ADJ
ejde-627	37	10	borel	borel	NOUN
ejde-627	37	11	measures	measure	NOUN
ejde-627	37	12	on	on	ADP
ejde-627	37	13	[	[	X
ejde-627	37	14	0	0	NUM
ejde-627	37	15	,	,	PUNCT
ejde-627	37	16	1	1	NUM
ejde-627	37	17	]	]	PUNCT
ejde-627	37	18	.	.	PUNCT
ejde-627	38	1	so	so	ADV
ejde-627	38	2	,	,	PUNCT
ejde-627	38	3	if	if	SCONJ
ejde-627	38	4	µ	µ	PRON
ejde-627	38	5	∈	∈	NOUN
ejde-627	38	6	x∗	x∗	NOUN
ejde-627	38	7	,	,	PUNCT
ejde-627	38	8	then	then	ADV
ejde-627	38	9	µ(f	µ(f	PROPN
ejde-627	38	10	)	)	PUNCT
ejde-627	39	1	=	=	PUNCT
ejde-627	39	2	∫	∫	PROPN
ejde-627	39	3	1	1	NUM
ejde-627	39	4	0	0	NUM
ejde-627	39	5	f	f	PROPN
ejde-627	39	6	dµ.	dµ.	PROPN
ejde-627	39	7	now	now	ADV
ejde-627	39	8	,	,	PUNCT
ejde-627	39	9	let	let	VERB
ejde-627	39	10	f	f	PROPN
ejde-627	39	11	∈	∈	PROPN
ejde-627	39	12	x	x	X
ejde-627	39	13	,	,	PUNCT
ejde-627	39	14	g	g	PROPN
ejde-627	39	15	∈	∈	PROPN
ejde-627	39	16	y	y	PROPN
ejde-627	39	17	,	,	PUNCT
ejde-627	39	18	and	and	CCONJ
ejde-627	39	19	µ	µ	PRON
ejde-627	39	20	∈	∈	NOUN
ejde-627	39	21	x∗.	x∗.	PUNCT
ejde-627	40	1	then	then	ADV
ejde-627	40	2	(	(	PUNCT
ejde-627	40	3	f	f	PROPN
ejde-627	40	4	⊗	⊗	PROPN
ejde-627	40	5	g)µ	g)µ	VERB
ejde-627	40	6	=	=	PUNCT
ejde-627	40	7	µ(f)g	µ(f)g	X
ejde-627	40	8	=	=	PRON
ejde-627	40	9	g	g	PROPN
ejde-627	40	10	∫	∫	PROPN
ejde-627	40	11	1	1	NUM
ejde-627	40	12	0	0	NUM
ejde-627	40	13	f(t	f(t	NOUN
ejde-627	40	14	)	)	PUNCT
ejde-627	40	15	dµ(t	dµ(t	NOUN
ejde-627	40	16	)	)	PUNCT
ejde-627	40	17	.	.	PUNCT
ejde-627	41	1	further	far	ADV
ejde-627	41	2	,	,	PUNCT
ejde-627	41	3	f	f	PROPN
ejde-627	41	4	⊗	⊗	PROPN
ejde-627	41	5	g	g	PROPN
ejde-627	41	6	is	be	AUX
ejde-627	41	7	a	a	DET
ejde-627	41	8	bounded	bounded	ADJ
ejde-627	41	9	linear	linear	ADJ
ejde-627	41	10	operator	operator	NOUN
ejde-627	41	11	.	.	PUNCT
ejde-627	42	1	also	also	ADV
ejde-627	42	2	∥(f	∥(f	VERB
ejde-627	42	3	⊗	⊗	PROPN
ejde-627	42	4	g)(µ)∥	g)(µ)∥	PROPN
ejde-627	42	5	=	=	SYM
ejde-627	42	6	|µ(f)|∥g∥	|µ(f)|∥g∥	PROPN
ejde-627	42	7	≤	≤	PROPN
ejde-627	42	8	∥µ∥∥f∥∥g∥	∥µ∥∥f∥∥g∥	PROPN
ejde-627	42	9	for	for	SCONJ
ejde-627	42	10	all	all	DET
ejde-627	42	11	µ	µ	PRON
ejde-627	42	12	∈	∈	PROPN
ejde-627	42	13	x∗.	x∗.	NOUN
ejde-627	42	14	atoms	atom	NOUN
ejde-627	42	15	are	be	AUX
ejde-627	42	16	used	use	VERB
ejde-627	42	17	in	in	ADP
ejde-627	42	18	theory	theory	NOUN
ejde-627	42	19	of	of	ADP
ejde-627	42	20	best	good	ADJ
ejde-627	42	21	approximation	approximation	NOUN
ejde-627	42	22	in	in	ADP
ejde-627	42	23	banach	banach	NOUN
ejde-627	42	24	spaces	space	NOUN
ejde-627	42	25	and	and	CCONJ
ejde-627	42	26	they	they	PRON
ejde-627	42	27	are	be	AUX
ejde-627	42	28	considered	consider	VERB
ejde-627	42	29	among	among	ADP
ejde-627	42	30	the	the	DET
ejde-627	42	31	fundamental	fundamental	ADJ
ejde-627	42	32	ingredients	ingredient	NOUN
ejde-627	42	33	in	in	ADP
ejde-627	42	34	the	the	DET
ejde-627	42	35	theory	theory	NOUN
ejde-627	42	36	of	of	ADP
ejde-627	42	37	tensor	tensor	NOUN
ejde-627	42	38	product	product	NOUN
ejde-627	42	39	.	.	PUNCT
ejde-627	43	1	one	one	NUM
ejde-627	43	2	of	of	ADP
ejde-627	43	3	the	the	DET
ejde-627	43	4	known	know	VERB
ejde-627	43	5	results	result	NOUN
ejde-627	43	6	that	that	PRON
ejde-627	43	7	we	we	PRON
ejde-627	43	8	need	need	VERB
ejde-627	43	9	in	in	ADP
ejde-627	43	10	our	our	PRON
ejde-627	43	11	paper	paper	NOUN
ejde-627	43	12	can	can	AUX
ejde-627	43	13	be	be	AUX
ejde-627	43	14	presented	present	VERB
ejde-627	43	15	in	in	ADP
ejde-627	43	16	theorem	theorem	NOUN
ejde-627	43	17	[	[	X
ejde-627	43	18	9	9	NUM
ejde-627	43	19	]	]	PUNCT
ejde-627	43	20	below	below	ADV
ejde-627	43	21	,	,	PUNCT
ejde-627	43	22	which	which	PRON
ejde-627	43	23	guarantees	guarantee	VERB
ejde-627	43	24	that	that	SCONJ
ejde-627	43	25	if	if	SCONJ
ejde-627	43	26	the	the	DET
ejde-627	43	27	sum	sum	NOUN
ejde-627	43	28	of	of	ADP
ejde-627	43	29	two	two	NUM
ejde-627	43	30	atoms	atom	NOUN
ejde-627	43	31	is	be	AUX
ejde-627	43	32	an	an	DET
ejde-627	43	33	atom	atom	NOUN
ejde-627	43	34	,	,	PUNCT
ejde-627	43	35	then	then	ADV
ejde-627	43	36	either	either	CCONJ
ejde-627	43	37	the	the	DET
ejde-627	43	38	first	first	ADJ
ejde-627	43	39	components	component	NOUN
ejde-627	43	40	are	be	AUX
ejde-627	43	41	dependent	dependent	ADJ
ejde-627	43	42	or	or	CCONJ
ejde-627	43	43	the	the	DET
ejde-627	43	44	second	second	ADJ
ejde-627	43	45	ones	one	NOUN
ejde-627	43	46	are	be	AUX
ejde-627	43	47	dependent	dependent	ADJ
ejde-627	43	48	.	.	PUNCT
ejde-627	44	1	theorem	theorem	VERB
ejde-627	44	2	2.2	2.2	NUM
ejde-627	44	3	(	(	PUNCT
ejde-627	44	4	[	[	X
ejde-627	44	5	9	9	NUM
ejde-627	44	6	]	]	PUNCT
ejde-627	44	7	)	)	PUNCT
ejde-627	44	8	.	.	PUNCT
ejde-627	45	1	let	let	VERB
ejde-627	45	2	x1	x1	PROPN
ejde-627	45	3	⊗	⊗	PROPN
ejde-627	45	4	y1	y1	PROPN
ejde-627	46	1	and	and	CCONJ
ejde-627	47	1	x2	x2	PROPN
ejde-627	47	2	⊗	⊗	PROPN
ejde-627	47	3	y2	y2	INTJ
ejde-627	47	4	be	be	VERB
ejde-627	47	5	two	two	NUM
ejde-627	47	6	nonzero	nonzero	NOUN
ejde-627	47	7	atoms	atom	NOUN
ejde-627	47	8	in	in	ADP
ejde-627	47	9	x	x	PROPN
ejde-627	47	10	⊗	⊗	PROPN
ejde-627	47	11	y	y	PROPN
ejde-627	47	12	such	such	ADJ
ejde-627	47	13	that	that	SCONJ
ejde-627	47	14	x1	x1	PROPN
ejde-627	47	15	⊗	⊗	PROPN
ejde-627	47	16	y1	y1	PROPN
ejde-627	48	1	+	+	CCONJ
ejde-627	48	2	x2	x2	PROPN
ejde-627	49	1	⊗	⊗	ADJ
ejde-627	49	2	y2	y2	PROPN
ejde-627	49	3	=	=	SYM
ejde-627	50	1	x3	x3	PROPN
ejde-627	50	2	⊗	⊗	PROPN
ejde-627	50	3	y3	y3	PROPN
ejde-627	50	4	.	.	PUNCT
ejde-627	51	1	then	then	ADV
ejde-627	51	2	either	either	CCONJ
ejde-627	51	3	x1	x1	PROPN
ejde-627	51	4	=	=	PUNCT
ejde-627	52	1	x2	x2	PROPN
ejde-627	52	2	=	=	PUNCT
ejde-627	52	3	x3	x3	ADJ
ejde-627	52	4	or	or	CCONJ
ejde-627	52	5	y1	y1	NOUN
ejde-627	52	6	=	=	PUNCT
ejde-627	52	7	y2	y2	NOUN
ejde-627	52	8	=	=	SYM
ejde-627	52	9	y3	y3	PROPN
ejde-627	52	10	.	.	PUNCT
ejde-627	53	1	this	this	PRON
ejde-627	53	2	leads	lead	VERB
ejde-627	53	3	us	we	PRON
ejde-627	53	4	to	to	ADP
ejde-627	53	5	the	the	DET
ejde-627	53	6	following	follow	VERB
ejde-627	53	7	interesting	interesting	ADJ
ejde-627	53	8	theorem	theorem	NOUN
ejde-627	53	9	that	that	PRON
ejde-627	53	10	lies	lie	VERB
ejde-627	53	11	at	at	ADP
ejde-627	53	12	the	the	DET
ejde-627	53	13	heart	heart	NOUN
ejde-627	53	14	of	of	ADP
ejde-627	53	15	functional	functional	ADJ
ejde-627	53	16	analysis	analysis	NOUN
ejde-627	53	17	as	as	ADV
ejde-627	53	18	well	well	ADV
ejde-627	53	19	as	as	ADP
ejde-627	53	20	approximation	approximation	NOUN
ejde-627	53	21	theory	theory	NOUN
ejde-627	53	22	and	and	CCONJ
ejde-627	53	23	guarantee	guarantee	VERB
ejde-627	53	24	that	that	SCONJ
ejde-627	53	25	any	any	DET
ejde-627	53	26	continuous	continuous	ADJ
ejde-627	53	27	function	function	NOUN
ejde-627	53	28	of	of	ADP
ejde-627	53	29	several	several	ADJ
ejde-627	53	30	variables	variable	NOUN
ejde-627	53	31	can	can	AUX
ejde-627	53	32	be	be	AUX
ejde-627	53	33	written	write	VERB
ejde-627	53	34	as	as	ADP
ejde-627	53	35	a	a	DET
ejde-627	53	36	sum	sum	NOUN
ejde-627	53	37	of	of	ADP
ejde-627	53	38	products	product	NOUN
ejde-627	53	39	of	of	ADP
ejde-627	53	40	continuous	continuous	ADJ
ejde-627	53	41	separated	separate	VERB
ejde-627	53	42	functions	function	NOUN
ejde-627	53	43	[	[	X
ejde-627	53	44	6	6	NUM
ejde-627	53	45	]	]	PUNCT
ejde-627	53	46	.	.	PUNCT
ejde-627	54	1	theorem	theorem	VERB
ejde-627	54	2	2.3	2.3	NUM
ejde-627	54	3	(	(	PUNCT
ejde-627	54	4	[	[	X
ejde-627	54	5	6	6	NUM
ejde-627	54	6	]	]	PUNCT
ejde-627	54	7	)	)	PUNCT
ejde-627	54	8	.	.	PUNCT
ejde-627	55	1	let	let	VERB
ejde-627	55	2	i	i	PRON
ejde-627	55	3	,	,	PUNCT
ejde-627	55	4	j	j	PROPN
ejde-627	55	5	be	be	VERB
ejde-627	55	6	two	two	NUM
ejde-627	55	7	compact	compact	ADJ
ejde-627	55	8	intervals	interval	NOUN
ejde-627	55	9	,	,	PUNCT
ejde-627	55	10	and	and	CCONJ
ejde-627	55	11	c(i	c(i	NOUN
ejde-627	55	12	)	)	PUNCT
ejde-627	55	13	,	,	PUNCT
ejde-627	55	14	c(j	c(j	PROPN
ejde-627	55	15	)	)	PUNCT
ejde-627	55	16	,	,	PUNCT
ejde-627	55	17	and	and	CCONJ
ejde-627	55	18	c(i×	c(i×	PROPN
ejde-627	55	19	j	j	PROPN
ejde-627	55	20	)	)	PUNCT
ejde-627	55	21	be	be	VERB
ejde-627	55	22	the	the	DET
ejde-627	55	23	spaces	space	NOUN
ejde-627	55	24	of	of	ADP
ejde-627	55	25	continuous	continuous	ADJ
ejde-627	55	26	functions	function	NOUN
ejde-627	55	27	on	on	ADP
ejde-627	55	28	i	i	PROPN
ejde-627	55	29	,	,	PUNCT
ejde-627	55	30	j	j	PROPN
ejde-627	55	31	,	,	PUNCT
ejde-627	55	32	and	and	CCONJ
ejde-627	56	1	i	i	PRON
ejde-627	56	2	×	×	VERB
ejde-627	56	3	j	j	PROPN
ejde-627	56	4	,	,	PUNCT
ejde-627	56	5	respectively	respectively	ADV
ejde-627	56	6	.	.	PUNCT
ejde-627	57	1	then	then	ADV
ejde-627	57	2	every	every	DET
ejde-627	57	3	f	f	PROPN
ejde-627	57	4	∈	∈	PROPN
ejde-627	57	5	c(i	c(i	VERB
ejde-627	57	6	×	×	PROPN
ejde-627	57	7	j	j	NOUN
ejde-627	57	8	)	)	PUNCT
ejde-627	57	9	can	can	AUX
ejde-627	57	10	be	be	AUX
ejde-627	57	11	written	write	VERB
ejde-627	57	12	in	in	ADP
ejde-627	57	13	the	the	DET
ejde-627	57	14	form	form	NOUN
ejde-627	57	15	f(x	f(x	PROPN
ejde-627	57	16	,	,	PUNCT
ejde-627	57	17	y	y	NOUN
ejde-627	57	18	)	)	PUNCT
ejde-627	57	19	=	=	NOUN
ejde-627	58	1	∑∞	∑∞	NOUN
ejde-627	58	2	i=1	i=1	X
ejde-627	58	3	ui(x)vi(y	ui(x)vi(y	PROPN
ejde-627	58	4	)	)	PUNCT
ejde-627	58	5	,	,	PUNCT
ejde-627	58	6	where	where	SCONJ
ejde-627	58	7	ui(x	ui(x	X
ejde-627	58	8	)	)	PUNCT
ejde-627	58	9	∈	∈	PROPN
ejde-627	58	10	c(i	c(i	PROPN
ejde-627	58	11	)	)	PUNCT
ejde-627	58	12	and	and	CCONJ
ejde-627	58	13	vi(y	vi(y	NOUN
ejde-627	58	14	)	)	PUNCT
ejde-627	58	15	∈	∈	PROPN
ejde-627	58	16	c(j	c(j	PROPN
ejde-627	58	17	)	)	PUNCT
ejde-627	58	18	.	.	PUNCT
ejde-627	59	1	3	3	X
ejde-627	59	2	.	.	X
ejde-627	59	3	general	general	ADJ
ejde-627	59	4	scheme	scheme	NOUN
ejde-627	59	5	for	for	ADP
ejde-627	59	6	the	the	DET
ejde-627	59	7	atomic	atomic	ADJ
ejde-627	59	8	solution	solution	NOUN
ejde-627	59	9	method	method	NOUN
ejde-627	59	10	consider	consider	VERB
ejde-627	59	11	the	the	DET
ejde-627	59	12	second	second	ADJ
ejde-627	59	13	order	order	NOUN
ejde-627	59	14	non	non	ADJ
ejde-627	59	15	-	-	ADJ
ejde-627	59	16	linear	linear	ADJ
ejde-627	59	17	partial	partial	ADJ
ejde-627	59	18	differential	differential	NOUN
ejde-627	59	19	equation	equation	NOUN
ejde-627	59	20	uxy(x	uxy(x	NOUN
ejde-627	59	21	,	,	PUNCT
ejde-627	59	22	y	y	NOUN
ejde-627	59	23	)	)	PUNCT
ejde-627	60	1	+	+	CCONJ
ejde-627	60	2	u(x	u(x	NOUN
ejde-627	60	3	,	,	PUNCT
ejde-627	60	4	y)uy(x	y)uy(x	NUM
ejde-627	60	5	,	,	PUNCT
ejde-627	60	6	y	y	NOUN
ejde-627	60	7	)	)	PUNCT
ejde-627	60	8	=	=	SYM
ejde-627	60	9	ux(x	ux(x	PROPN
ejde-627	60	10	,	,	PUNCT
ejde-627	60	11	y	y	PROPN
ejde-627	60	12	)	)	PUNCT
ejde-627	60	13	+	+	CCONJ
ejde-627	60	14	u2(x	u2(x	PROPN
ejde-627	60	15	,	,	PUNCT
ejde-627	60	16	y	y	PROPN
ejde-627	60	17	)	)	PUNCT
ejde-627	60	18	,	,	PUNCT
ejde-627	60	19	(	(	PUNCT
ejde-627	60	20	3.1	3.1	NUM
ejde-627	60	21	)	)	PUNCT
ejde-627	60	22	where	where	SCONJ
ejde-627	60	23	u(x	u(x	NOUN
ejde-627	60	24	,	,	PUNCT
ejde-627	60	25	y	y	NOUN
ejde-627	60	26	)	)	PUNCT
ejde-627	60	27	is	be	AUX
ejde-627	60	28	an	an	DET
ejde-627	60	29	unknown	unknown	ADJ
ejde-627	60	30	function	function	NOUN
ejde-627	60	31	and	and	CCONJ
ejde-627	60	32	subjected	subject	VERB
ejde-627	60	33	to	to	ADP
ejde-627	60	34	the	the	DET
ejde-627	60	35	following	following	ADJ
ejde-627	60	36	conditions	condition	NOUN
ejde-627	60	37	:	:	PUNCT
ejde-627	60	38	u(0	u(0	NOUN
ejde-627	60	39	,	,	PUNCT
ejde-627	60	40	0	0	NUM
ejde-627	60	41	)	)	PUNCT
ejde-627	61	1	=	=	SYM
ejde-627	61	2	1	1	NUM
ejde-627	61	3	,	,	PUNCT
ejde-627	61	4	ux(0	ux(0	PROPN
ejde-627	61	5	,	,	PUNCT
ejde-627	61	6	0	0	NUM
ejde-627	61	7	)	)	PUNCT
ejde-627	61	8	=	=	SYM
ejde-627	61	9	1	1	NUM
ejde-627	61	10	,	,	PUNCT
ejde-627	61	11	uy(0	uy(0	PROPN
ejde-627	61	12	,	,	PUNCT
ejde-627	61	13	0	0	NUM
ejde-627	61	14	)	)	PUNCT
ejde-627	61	15	=	=	SYM
ejde-627	62	1	1	1	X
ejde-627	62	2	.	.	PUNCT
ejde-627	62	3	(	(	PUNCT
ejde-627	62	4	3.2	3.2	NUM
ejde-627	62	5	)	)	PUNCT
ejde-627	62	6	ejde-2024/28	ejde-2024/28	PROPN
ejde-627	62	7	solutions	solution	NOUN
ejde-627	62	8	of	of	ADP
ejde-627	62	9	linear	linear	PROPN
ejde-627	62	10	and	and	CCONJ
ejde-627	62	11	non	non	ADJ
ejde-627	62	12	-	-	ADJ
ejde-627	62	13	linear	linear	ADJ
ejde-627	62	14	pdes	pde	NOUN
ejde-627	62	15	3	3	NUM
ejde-627	62	16	clearly	clearly	ADV
ejde-627	62	17	,	,	PUNCT
ejde-627	62	18	equation	equation	NOUN
ejde-627	62	19	(	(	PUNCT
ejde-627	62	20	3.1	3.1	NUM
ejde-627	62	21	)	)	PUNCT
ejde-627	62	22	is	be	AUX
ejde-627	62	23	non	non	ADJ
ejde-627	62	24	-	-	ADJ
ejde-627	62	25	separable	separable	ADJ
ejde-627	62	26	as	as	SCONJ
ejde-627	62	27	we	we	PRON
ejde-627	62	28	can	can	AUX
ejde-627	62	29	not	not	PART
ejde-627	62	30	move	move	VERB
ejde-627	62	31	the	the	DET
ejde-627	62	32	x	x	NOUN
ejde-627	62	33	-	-	NOUN
ejde-627	62	34	terms	term	NOUN
ejde-627	62	35	to	to	ADP
ejde-627	62	36	one	one	NUM
ejde-627	62	37	side	side	NOUN
ejde-627	62	38	and	and	CCONJ
ejde-627	62	39	the	the	DET
ejde-627	62	40	y	y	NOUN
ejde-627	62	41	-	-	PUNCT
ejde-627	62	42	terms	term	NOUN
ejde-627	62	43	to	to	ADP
ejde-627	62	44	the	the	DET
ejde-627	62	45	other	other	ADJ
ejde-627	62	46	.	.	PUNCT
ejde-627	63	1	hence	hence	ADV
ejde-627	63	2	,	,	PUNCT
ejde-627	63	3	the	the	DET
ejde-627	63	4	method	method	NOUN
ejde-627	63	5	of	of	ADP
ejde-627	63	6	separation	separation	NOUN
ejde-627	63	7	of	of	ADP
ejde-627	63	8	variables	variable	NOUN
ejde-627	63	9	does	do	AUX
ejde-627	63	10	not	not	PART
ejde-627	63	11	work	work	VERB
ejde-627	63	12	.	.	PUNCT
ejde-627	64	1	from	from	ADP
ejde-627	64	2	(	(	PUNCT
ejde-627	64	3	3.2	3.2	NUM
ejde-627	64	4	)	)	PUNCT
ejde-627	64	5	,	,	PUNCT
ejde-627	64	6	we	we	PRON
ejde-627	64	7	can	can	AUX
ejde-627	64	8	assume	assume	VERB
ejde-627	64	9	without	without	ADP
ejde-627	64	10	loss	loss	NOUN
ejde-627	64	11	of	of	ADP
ejde-627	64	12	generality	generality	NOUN
ejde-627	64	13	that	that	PRON
ejde-627	64	14	p	p	X
ejde-627	64	15	(	(	PUNCT
ejde-627	64	16	0	0	NUM
ejde-627	64	17	)	)	PUNCT
ejde-627	64	18	=	=	SYM
ejde-627	64	19	q(0	q(0	NOUN
ejde-627	64	20	)	)	PUNCT
ejde-627	64	21	=	=	PUNCT
ejde-627	65	1	p	p	X
ejde-627	65	2	′(0	′(0	PROPN
ejde-627	65	3	)	)	PUNCT
ejde-627	66	1	=	=	SYM
ejde-627	66	2	q′(0	q′(0	PROPN
ejde-627	66	3	)	)	PUNCT
ejde-627	66	4	=	=	SYM
ejde-627	66	5	1	1	X
ejde-627	66	6	.	.	PUNCT
ejde-627	66	7	(	(	PUNCT
ejde-627	66	8	3.3	3.3	NUM
ejde-627	66	9	)	)	PUNCT
ejde-627	66	10	according	accord	VERB
ejde-627	66	11	to	to	ADP
ejde-627	66	12	theorem	theorem	ADJ
ejde-627	66	13	2.3	2.3	NUM
ejde-627	66	14	,	,	PUNCT
ejde-627	66	15	we	we	PRON
ejde-627	66	16	start	start	VERB
ejde-627	66	17	our	our	PRON
ejde-627	66	18	approach	approach	NOUN
ejde-627	66	19	by	by	ADP
ejde-627	66	20	assuming	assume	VERB
ejde-627	66	21	that	that	SCONJ
ejde-627	66	22	u(x	u(x	NOUN
ejde-627	66	23	,	,	PUNCT
ejde-627	66	24	y	y	NOUN
ejde-627	66	25	)	)	PUNCT
ejde-627	66	26	=	=	SYM
ejde-627	67	1	p	p	X
ejde-627	67	2	(	(	PUNCT
ejde-627	67	3	x)q(y	x)q(y	PROPN
ejde-627	67	4	)	)	PUNCT
ejde-627	67	5	.	.	PUNCT
ejde-627	68	1	(	(	PUNCT
ejde-627	68	2	3.4	3.4	NUM
ejde-627	68	3	)	)	PUNCT
ejde-627	68	4	now	now	ADV
ejde-627	68	5	,	,	PUNCT
ejde-627	68	6	we	we	PRON
ejde-627	68	7	substitute	substitute	VERB
ejde-627	68	8	(	(	PUNCT
ejde-627	68	9	3.4	3.4	NUM
ejde-627	68	10	)	)	PUNCT
ejde-627	68	11	into	into	ADP
ejde-627	68	12	the	the	DET
ejde-627	68	13	main	main	ADJ
ejde-627	68	14	partial	partial	ADJ
ejde-627	68	15	differential	differential	NOUN
ejde-627	68	16	equation	equation	NOUN
ejde-627	68	17	(	(	PUNCT
ejde-627	68	18	3.1	3.1	NUM
ejde-627	68	19	)	)	PUNCT
ejde-627	68	20	.	.	PUNCT
ejde-627	69	1	hence	hence	ADV
ejde-627	69	2	,	,	PUNCT
ejde-627	69	3	p	p	PROPN
ejde-627	69	4	′(x)q′(y	′(x)q′(y	PROPN
ejde-627	69	5	)	)	PUNCT
ejde-627	70	1	+	+	CCONJ
ejde-627	71	1	p	p	X
ejde-627	71	2	(	(	PUNCT
ejde-627	71	3	x)q(y)p	x)q(y)p	PROPN
ejde-627	71	4	(	(	PUNCT
ejde-627	71	5	x)q′(y	x)q′(y	PROPN
ejde-627	71	6	)	)	PUNCT
ejde-627	71	7	=	=	SYM
ejde-627	71	8	p	p	NOUN
ejde-627	71	9	′(x)q(y	′(x)q(y	NOUN
ejde-627	71	10	)	)	PUNCT
ejde-627	72	1	+	+	CCONJ
ejde-627	72	2	p	p	NOUN
ejde-627	72	3	2(x)q2(y	2(x)q2(y	NUM
ejde-627	72	4	)	)	PUNCT
ejde-627	72	5	.	.	PUNCT
ejde-627	73	1	(	(	PUNCT
ejde-627	73	2	3.5	3.5	NUM
ejde-627	73	3	)	)	PUNCT
ejde-627	73	4	equivalently	equivalently	ADV
ejde-627	73	5	,	,	PUNCT
ejde-627	73	6	(	(	PUNCT
ejde-627	73	7	3.5	3.5	NUM
ejde-627	73	8	)	)	PUNCT
ejde-627	73	9	can	can	AUX
ejde-627	73	10	be	be	AUX
ejde-627	73	11	written	write	VERB
ejde-627	73	12	as	as	ADP
ejde-627	73	13	p	p	NOUN
ejde-627	73	14	′(x)q′(y	′(x)q′(y	PROPN
ejde-627	73	15	)	)	PUNCT
ejde-627	74	1	+	+	CCONJ
ejde-627	75	1	p	p	NOUN
ejde-627	75	2	2(x)q(y)[q′(y)−q(y	2(x)q(y)[q′(y)−q(y	NUM
ejde-627	75	3	)	)	PUNCT
ejde-627	75	4	]	]	PUNCT
ejde-627	76	1	=	=	PUNCT
ejde-627	76	2	p	p	NOUN
ejde-627	76	3	′(x)q(y	′(x)q(y	NOUN
ejde-627	76	4	)	)	PUNCT
ejde-627	76	5	(	(	PUNCT
ejde-627	76	6	3.6	3.6	NUM
ejde-627	76	7	)	)	PUNCT
ejde-627	76	8	clearly	clearly	ADV
ejde-627	76	9	,	,	PUNCT
ejde-627	76	10	each	each	DET
ejde-627	76	11	term	term	NOUN
ejde-627	76	12	of	of	ADP
ejde-627	76	13	(	(	PUNCT
ejde-627	76	14	3.6	3.6	NUM
ejde-627	76	15	)	)	PUNCT
ejde-627	76	16	is	be	AUX
ejde-627	76	17	just	just	ADV
ejde-627	76	18	a	a	DET
ejde-627	76	19	product	product	NOUN
ejde-627	76	20	of	of	ADP
ejde-627	76	21	two	two	NUM
ejde-627	76	22	functions	function	NOUN
ejde-627	76	23	one	one	NUM
ejde-627	76	24	of	of	ADP
ejde-627	76	25	them	they	PRON
ejde-627	76	26	only	only	ADV
ejde-627	76	27	in	in	ADP
ejde-627	76	28	x	x	X
ejde-627	76	29	and	and	CCONJ
ejde-627	76	30	the	the	DET
ejde-627	76	31	other	other	ADJ
ejde-627	76	32	one	one	NOUN
ejde-627	76	33	only	only	ADV
ejde-627	76	34	in	in	ADP
ejde-627	76	35	y.	y.	PROPN
ejde-627	76	36	therefore	therefore	ADV
ejde-627	76	37	,	,	PUNCT
ejde-627	76	38	in	in	ADP
ejde-627	76	39	tensor	tensor	NOUN
ejde-627	76	40	product	product	NOUN
ejde-627	76	41	form	form	NOUN
ejde-627	76	42	,	,	PUNCT
ejde-627	76	43	(	(	PUNCT
ejde-627	76	44	3.6	3.6	NUM
ejde-627	76	45	)	)	PUNCT
ejde-627	76	46	can	can	AUX
ejde-627	76	47	be	be	AUX
ejde-627	76	48	written	write	VERB
ejde-627	76	49	as	as	ADP
ejde-627	76	50	p	p	PROPN
ejde-627	76	51	′(x)⊗q′(y	′(x)⊗q′(y	PROPN
ejde-627	76	52	)	)	PUNCT
ejde-627	77	1	+	+	CCONJ
ejde-627	78	1	p	p	X
ejde-627	78	2	2(x)⊗	2(x)⊗	NUM
ejde-627	79	1	[	[	X
ejde-627	79	2	q(y)q′(y)−q2(y	q(y)q′(y)−q2(y	X
ejde-627	79	3	)	)	PUNCT
ejde-627	79	4	]	]	PUNCT
ejde-627	80	1	=	=	PUNCT
ejde-627	80	2	p	p	NOUN
ejde-627	80	3	′(x)⊗q(y	′(x)⊗q(y	NOUN
ejde-627	80	4	)	)	PUNCT
ejde-627	80	5	.	.	PUNCT
ejde-627	81	1	(	(	PUNCT
ejde-627	81	2	3.7	3.7	NUM
ejde-627	81	3	)	)	PUNCT
ejde-627	81	4	this	this	PRON
ejde-627	81	5	implies	imply	VERB
ejde-627	81	6	that	that	SCONJ
ejde-627	81	7	the	the	DET
ejde-627	81	8	sum	sum	NOUN
ejde-627	81	9	of	of	ADP
ejde-627	81	10	two	two	NUM
ejde-627	81	11	atoms	atom	NOUN
ejde-627	81	12	is	be	AUX
ejde-627	81	13	an	an	DET
ejde-627	81	14	atom	atom	NOUN
ejde-627	81	15	.	.	PUNCT
ejde-627	82	1	by	by	ADP
ejde-627	82	2	theorem	theorem	NOUN
ejde-627	82	3	2.2	2.2	NUM
ejde-627	82	4	,	,	PUNCT
ejde-627	82	5	we	we	PRON
ejde-627	82	6	have	have	VERB
ejde-627	82	7	the	the	DET
ejde-627	82	8	following	follow	VERB
ejde-627	82	9	two	two	NUM
ejde-627	82	10	cases	case	NOUN
ejde-627	82	11	:	:	PUNCT
ejde-627	82	12	(	(	PUNCT
ejde-627	82	13	i	i	NOUN
ejde-627	82	14	)	)	PUNCT
ejde-627	82	15	p	p	NOUN
ejde-627	82	16	′(x	′(x	NOUN
ejde-627	82	17	)	)	PUNCT
ejde-627	82	18	=	=	PUNCT
ejde-627	83	1	p	p	ADJ
ejde-627	83	2	2(x	2(x	NUM
ejde-627	83	3	)	)	PUNCT
ejde-627	84	1	=	=	NOUN
ejde-627	84	2	p	p	NOUN
ejde-627	84	3	′(x	′(x	NOUN
ejde-627	84	4	)	)	PUNCT
ejde-627	84	5	,	,	PUNCT
ejde-627	84	6	(	(	PUNCT
ejde-627	84	7	ii	ii	NOUN
ejde-627	84	8	)	)	PUNCT
ejde-627	84	9	q′(y	q′(y	PROPN
ejde-627	84	10	)	)	PUNCT
ejde-627	85	1	=	=	PUNCT
ejde-627	86	1	[	[	X
ejde-627	86	2	q(y)q′(y)−q2(y	q(y)q′(y)−q2(y	NOUN
ejde-627	86	3	)	)	PUNCT
ejde-627	86	4	]	]	PUNCT
ejde-627	86	5	=	=	PUNCT
ejde-627	86	6	q(y	q(y	X
ejde-627	86	7	)	)	PUNCT
ejde-627	86	8	.	.	PUNCT
ejde-627	87	1	case	case	NOUN
ejde-627	87	2	(	(	PUNCT
ejde-627	87	3	i	i	NOUN
ejde-627	87	4	):	):	PUNCT
ejde-627	87	5	this	this	DET
ejde-627	87	6	case	case	NOUN
ejde-627	87	7	has	have	VERB
ejde-627	87	8	only	only	ADV
ejde-627	87	9	the	the	DET
ejde-627	87	10	situation	situation	NOUN
ejde-627	87	11	p	p	NOUN
ejde-627	87	12	′(x	′(x	NOUN
ejde-627	87	13	)	)	PUNCT
ejde-627	87	14	=	=	PUNCT
ejde-627	88	1	p	p	PRON
ejde-627	88	2	2(x	2(x	NUM
ejde-627	88	3	)	)	PUNCT
ejde-627	88	4	which	which	PRON
ejde-627	88	5	can	can	AUX
ejde-627	88	6	be	be	AUX
ejde-627	88	7	solved	solve	VERB
ejde-627	88	8	for	for	ADP
ejde-627	88	9	p	p	PROPN
ejde-627	88	10	(	(	PUNCT
ejde-627	88	11	x	x	NOUN
ejde-627	88	12	)	)	PUNCT
ejde-627	88	13	,	,	PUNCT
ejde-627	88	14	by	by	ADP
ejde-627	88	15	taking	take	VERB
ejde-627	88	16	into	into	ADP
ejde-627	88	17	account	account	NOUN
ejde-627	88	18	(	(	PUNCT
ejde-627	88	19	3.3	3.3	NUM
ejde-627	88	20	)	)	PUNCT
ejde-627	88	21	,	,	PUNCT
ejde-627	88	22	as	as	SCONJ
ejde-627	88	23	p	p	X
ejde-627	88	24	(	(	PUNCT
ejde-627	88	25	x	x	NOUN
ejde-627	88	26	)	)	PUNCT
ejde-627	88	27	=	=	SYM
ejde-627	88	28	1	1	NUM
ejde-627	88	29	1−	1−	NUM
ejde-627	88	30	x	x	X
ejde-627	88	31	.	.	PUNCT
ejde-627	89	1	(	(	PUNCT
ejde-627	89	2	3.8	3.8	NUM
ejde-627	89	3	)	)	PUNCT
ejde-627	89	4	our	our	PRON
ejde-627	89	5	next	next	ADJ
ejde-627	89	6	step	step	NOUN
ejde-627	89	7	is	be	AUX
ejde-627	89	8	to	to	PART
ejde-627	89	9	substitute	substitute	VERB
ejde-627	89	10	(	(	PUNCT
ejde-627	89	11	3.8	3.8	NUM
ejde-627	89	12	)	)	PUNCT
ejde-627	89	13	into	into	ADP
ejde-627	89	14	(	(	PUNCT
ejde-627	89	15	3.5	3.5	NUM
ejde-627	89	16	)	)	PUNCT
ejde-627	89	17	taking	take	VERB
ejde-627	89	18	into	into	ADP
ejde-627	89	19	account	account	NOUN
ejde-627	89	20	that	that	SCONJ
ejde-627	89	21	p	p	NOUN
ejde-627	89	22	′(x	′(x	NOUN
ejde-627	89	23	)	)	PUNCT
ejde-627	89	24	=	=	PUNCT
ejde-627	89	25	p	p	PRON
ejde-627	89	26	2(x	2(x	NUM
ejde-627	89	27	)	)	PUNCT
ejde-627	89	28	.	.	PUNCT
ejde-627	90	1	this	this	DET
ejde-627	90	2	yields	yield	NOUN
ejde-627	90	3	q′(y	q′(y	X
ejde-627	90	4	)	)	PUNCT
ejde-627	91	1	+	+	CCONJ
ejde-627	92	1	[	[	X
ejde-627	92	2	q(y)q′(y)−q2(y	q(y)q′(y)−q2(y	NOUN
ejde-627	92	3	)	)	PUNCT
ejde-627	92	4	]	]	PUNCT
ejde-627	92	5	=	=	PUNCT
ejde-627	92	6	q(y	q(y	X
ejde-627	92	7	)	)	PUNCT
ejde-627	92	8	,	,	PUNCT
ejde-627	92	9	(	(	PUNCT
ejde-627	92	10	3.9	3.9	NUM
ejde-627	92	11	)	)	PUNCT
ejde-627	92	12	or	or	CCONJ
ejde-627	92	13	equivalently	equivalently	ADV
ejde-627	92	14	,	,	PUNCT
ejde-627	92	15	(	(	PUNCT
ejde-627	92	16	q′(y)−q(y))(1	q′(y)−q(y))(1	NOUN
ejde-627	92	17	+	+	ADJ
ejde-627	92	18	q(y	q(y	NOUN
ejde-627	92	19	)	)	PUNCT
ejde-627	92	20	)	)	PUNCT
ejde-627	93	1	=	=	SYM
ejde-627	93	2	0	0	X
ejde-627	93	3	.	.	PUNCT
ejde-627	93	4	(	(	PUNCT
ejde-627	93	5	3.10	3.10	NUM
ejde-627	93	6	)	)	PUNCT
ejde-627	93	7	so	so	ADV
ejde-627	93	8	,	,	PUNCT
ejde-627	93	9	by	by	ADP
ejde-627	93	10	referring	refer	VERB
ejde-627	93	11	to	to	ADP
ejde-627	93	12	(	(	PUNCT
ejde-627	93	13	3.3	3.3	NUM
ejde-627	93	14	)	)	PUNCT
ejde-627	93	15	,	,	PUNCT
ejde-627	93	16	case	case	NOUN
ejde-627	93	17	(	(	PUNCT
ejde-627	93	18	i	i	NOUN
ejde-627	93	19	)	)	PUNCT
ejde-627	93	20	gives	give	VERB
ejde-627	93	21	two	two	NUM
ejde-627	93	22	values	value	NOUN
ejde-627	93	23	for	for	ADP
ejde-627	93	24	q(y	q(y	NOUN
ejde-627	93	25	)	)	PUNCT
ejde-627	93	26	,	,	PUNCT
ejde-627	93	27	namely	namely	ADV
ejde-627	93	28	,	,	PUNCT
ejde-627	93	29	q(y	q(y	PROPN
ejde-627	93	30	)	)	PUNCT
ejde-627	93	31	=	=	SYM
ejde-627	93	32	−1	−1	NOUN
ejde-627	93	33	and	and	CCONJ
ejde-627	93	34	q(y	q(y	PROPN
ejde-627	93	35	)	)	PUNCT
ejde-627	93	36	=	=	SYM
ejde-627	94	1	ey	ey	NOUN
ejde-627	94	2	.	.	PROPN
ejde-627	94	3	hence	hence	ADV
ejde-627	94	4	,	,	PUNCT
ejde-627	94	5	by	by	ADP
ejde-627	94	6	taking	take	VERB
ejde-627	94	7	into	into	ADP
ejde-627	94	8	account	account	NOUN
ejde-627	94	9	these	these	DET
ejde-627	94	10	two	two	NUM
ejde-627	94	11	values	value	NOUN
ejde-627	94	12	of	of	ADP
ejde-627	94	13	q(y	q(y	NOUN
ejde-627	94	14	)	)	PUNCT
ejde-627	94	15	together	together	ADV
ejde-627	94	16	with	with	ADP
ejde-627	94	17	both	both	DET
ejde-627	94	18	(	(	PUNCT
ejde-627	94	19	3.4	3.4	NUM
ejde-627	94	20	)	)	PUNCT
ejde-627	94	21	and	and	CCONJ
ejde-627	94	22	(	(	PUNCT
ejde-627	94	23	3.8	3.8	NUM
ejde-627	94	24	)	)	PUNCT
ejde-627	94	25	,	,	PUNCT
ejde-627	94	26	case	case	NOUN
ejde-627	94	27	(	(	PUNCT
ejde-627	94	28	i	i	NOUN
ejde-627	94	29	)	)	PUNCT
ejde-627	94	30	generates	generate	VERB
ejde-627	94	31	two	two	NUM
ejde-627	94	32	atomic	atomic	ADJ
ejde-627	94	33	solutions	solution	NOUN
ejde-627	94	34	:	:	PUNCT
ejde-627	94	35	u11(x	u11(x	NOUN
ejde-627	94	36	,	,	PUNCT
ejde-627	94	37	y	y	NOUN
ejde-627	94	38	)	)	PUNCT
ejde-627	94	39	=	=	SYM
ejde-627	94	40	1	1	NUM
ejde-627	94	41	x−	x−	PROPN
ejde-627	94	42	1	1	NUM
ejde-627	94	43	,	,	PUNCT
ejde-627	94	44	u12(x	u12(x	PROPN
ejde-627	94	45	,	,	PUNCT
ejde-627	94	46	y	y	PROPN
ejde-627	94	47	)	)	PUNCT
ejde-627	94	48	=	=	PUNCT
ejde-627	95	1	ey	ey	PRON
ejde-627	95	2	1−	1−	NUM
ejde-627	95	3	x	x	NOUN
ejde-627	95	4	.	.	PUNCT
ejde-627	96	1	(	(	PUNCT
ejde-627	96	2	3.11	3.11	NUM
ejde-627	96	3	)	)	PUNCT
ejde-627	96	4	case	case	NOUN
ejde-627	96	5	(	(	PUNCT
ejde-627	96	6	ii	ii	NUM
ejde-627	96	7	):	):	PUNCT
ejde-627	96	8	this	this	DET
ejde-627	96	9	case	case	NOUN
ejde-627	96	10	has	have	VERB
ejde-627	96	11	the	the	DET
ejde-627	96	12	following	follow	VERB
ejde-627	96	13	three	three	NUM
ejde-627	96	14	situations	situation	NOUN
ejde-627	96	15	:	:	PUNCT
ejde-627	96	16	(	(	PUNCT
ejde-627	96	17	a	a	X
ejde-627	96	18	)	)	PUNCT
ejde-627	96	19	q′(y	q′(y	PROPN
ejde-627	96	20	)	)	PUNCT
ejde-627	96	21	=	=	SYM
ejde-627	96	22	q(y)q′(y)−q2(y	q(y)q′(y)−q2(y	NOUN
ejde-627	96	23	)	)	PUNCT
ejde-627	96	24	,	,	PUNCT
ejde-627	96	25	(	(	PUNCT
ejde-627	96	26	b	b	X
ejde-627	96	27	)	)	PUNCT
ejde-627	96	28	q′(y	q′(y	PROPN
ejde-627	96	29	)	)	PUNCT
ejde-627	96	30	=	=	SYM
ejde-627	96	31	q(y	q(y	PROPN
ejde-627	96	32	)	)	PUNCT
ejde-627	96	33	,	,	PUNCT
ejde-627	96	34	(	(	PUNCT
ejde-627	96	35	c	c	NOUN
ejde-627	96	36	)	)	PUNCT
ejde-627	96	37	q(y	q(y	NOUN
ejde-627	96	38	)	)	PUNCT
ejde-627	96	39	=	=	SYM
ejde-627	96	40	q(y)q′(y)−q2(y	q(y)q′(y)−q2(y	NOUN
ejde-627	96	41	)	)	PUNCT
ejde-627	96	42	.	.	PUNCT
ejde-627	97	1	again	again	ADV
ejde-627	97	2	by	by	ADP
ejde-627	97	3	(	(	PUNCT
ejde-627	97	4	3.3	3.3	NUM
ejde-627	97	5	)	)	PUNCT
ejde-627	97	6	the	the	DET
ejde-627	97	7	last	last	ADJ
ejde-627	97	8	two	two	NUM
ejde-627	97	9	situations	situation	NOUN
ejde-627	97	10	(	(	PUNCT
ejde-627	97	11	b	b	NOUN
ejde-627	97	12	)	)	PUNCT
ejde-627	97	13	and	and	CCONJ
ejde-627	97	14	(	(	PUNCT
ejde-627	97	15	c	c	NOUN
ejde-627	97	16	)	)	PUNCT
ejde-627	97	17	yield	yield	NOUN
ejde-627	97	18	,	,	PUNCT
ejde-627	97	19	respectively	respectively	ADV
ejde-627	97	20	,	,	PUNCT
ejde-627	97	21	q(y	q(y	PROPN
ejde-627	97	22	)	)	PUNCT
ejde-627	97	23	=	=	SYM
ejde-627	97	24	ey	ey	NOUN
ejde-627	97	25	and	and	CCONJ
ejde-627	97	26	q(y	q(y	PROPN
ejde-627	97	27	)	)	PUNCT
ejde-627	97	28	=	=	PUNCT
ejde-627	97	29	2ey	2ey	NOUN
ejde-627	97	30	−	−	NOUN
ejde-627	97	31	1	1	X
ejde-627	97	32	.	.	PUNCT
ejde-627	98	1	therefore	therefore	ADV
ejde-627	98	2	,	,	PUNCT
ejde-627	98	3	according	accord	VERB
ejde-627	98	4	to	to	ADP
ejde-627	98	5	theorem	theorem	VERB
ejde-627	98	6	2.2	2.2	NUM
ejde-627	98	7	there	there	PRON
ejde-627	98	8	is	be	VERB
ejde-627	98	9	no	no	DET
ejde-627	98	10	atomic	atomic	ADJ
ejde-627	98	11	solution	solution	NOUN
ejde-627	98	12	for	for	ADP
ejde-627	98	13	case	case	NOUN
ejde-627	98	14	(	(	PUNCT
ejde-627	98	15	ii	ii	NOUN
ejde-627	98	16	)	)	PUNCT
ejde-627	98	17	and	and	CCONJ
ejde-627	98	18	hence	hence	ADV
ejde-627	98	19	,	,	PUNCT
ejde-627	98	20	this	this	DET
ejde-627	98	21	case	case	NOUN
ejde-627	98	22	does	do	AUX
ejde-627	98	23	not	not	PART
ejde-627	98	24	admit	admit	VERB
ejde-627	98	25	an	an	DET
ejde-627	98	26	atomic	atomic	ADJ
ejde-627	98	27	solution	solution	NOUN
ejde-627	98	28	.	.	PUNCT
ejde-627	99	1	the	the	DET
ejde-627	99	2	two	two	NUM
ejde-627	99	3	atomic	atomic	ADJ
ejde-627	99	4	solutions	solution	NOUN
ejde-627	99	5	u11(x	u11(x	NOUN
ejde-627	99	6	,	,	PUNCT
ejde-627	99	7	y	y	PROPN
ejde-627	99	8	)	)	PUNCT
ejde-627	99	9	and	and	CCONJ
ejde-627	99	10	u12(x	u12(x	PROPN
ejde-627	99	11	,	,	PUNCT
ejde-627	99	12	y	y	NOUN
ejde-627	99	13	)	)	PUNCT
ejde-627	99	14	in	in	ADP
ejde-627	99	15	(	(	PUNCT
ejde-627	99	16	3.11	3.11	NUM
ejde-627	99	17	)	)	PUNCT
ejde-627	99	18	of	of	ADP
ejde-627	99	19	problem	problem	NOUN
ejde-627	99	20	(	(	PUNCT
ejde-627	99	21	3.1	3.1	NUM
ejde-627	99	22	)	)	PUNCT
ejde-627	99	23	are	be	AUX
ejde-627	99	24	displayed	display	VERB
ejde-627	99	25	,	,	PUNCT
ejde-627	99	26	in	in	ADP
ejde-627	99	27	figures	figure	NOUN
ejde-627	99	28	1	1	NUM
ejde-627	99	29	and	and	CCONJ
ejde-627	100	1	2	2	NUM
ejde-627	100	2	.	.	NOUN
ejde-627	100	3	4	4	NUM
ejde-627	100	4	w.	w.	PROPN
ejde-627	100	5	g.	g.	PROPN
ejde-627	100	6	alshanti	alshanti	PROPN
ejde-627	100	7	ejde-2024/28	ejde-2024/28	PROPN
ejde-627	100	8	figure	figure	VERB
ejde-627	100	9	1	1	NUM
ejde-627	100	10	.	.	PUNCT
ejde-627	101	1	first	first	ADJ
ejde-627	101	2	atomic	atomic	ADJ
ejde-627	101	3	solution	solution	NOUN
ejde-627	101	4	u11(x	u11(x	NOUN
ejde-627	101	5	,	,	PUNCT
ejde-627	101	6	y	y	PROPN
ejde-627	101	7	)	)	PUNCT
ejde-627	101	8	of	of	ADP
ejde-627	101	9	problem	problem	NOUN
ejde-627	101	10	(	(	PUNCT
ejde-627	101	11	3.1	3.1	NUM
ejde-627	101	12	)	)	PUNCT
ejde-627	101	13	.	.	PUNCT
ejde-627	102	1	4	4	X
ejde-627	102	2	.	.	X
ejde-627	102	3	applications	application	NOUN
ejde-627	102	4	in	in	ADP
ejde-627	102	5	this	this	DET
ejde-627	102	6	section	section	NOUN
ejde-627	102	7	,	,	PUNCT
ejde-627	102	8	we	we	PRON
ejde-627	102	9	provide	provide	VERB
ejde-627	102	10	two	two	NUM
ejde-627	102	11	examples	example	NOUN
ejde-627	102	12	for	for	ADP
ejde-627	102	13	solving	solve	VERB
ejde-627	102	14	non	non	ADJ
ejde-627	102	15	-	-	ADJ
ejde-627	102	16	separable	separable	ADJ
ejde-627	102	17	linear	linear	ADJ
ejde-627	102	18	partial	partial	ADJ
ejde-627	102	19	differential	differential	NOUN
ejde-627	102	20	equations	equation	NOUN
ejde-627	102	21	utilizing	utilize	VERB
ejde-627	102	22	the	the	DET
ejde-627	102	23	atomic	atomic	ADJ
ejde-627	102	24	method	method	NOUN
ejde-627	102	25	.	.	PUNCT
ejde-627	103	1	example	example	NOUN
ejde-627	103	2	4.1	4.1	NUM
ejde-627	103	3	.	.	PUNCT
ejde-627	104	1	consider	consider	VERB
ejde-627	104	2	the	the	DET
ejde-627	104	3	initial	initial	ADJ
ejde-627	104	4	value	value	NOUN
ejde-627	104	5	problem	problem	NOUN
ejde-627	104	6	f(x)ux	f(x)ux	PROPN
ejde-627	105	1	+	+	CCONJ
ejde-627	105	2	uy	uy	NOUN
ejde-627	105	3	=	=	PUNCT
ejde-627	105	4	g(y)u	g(y)u	PROPN
ejde-627	105	5	,	,	PUNCT
ejde-627	105	6	(	(	PUNCT
ejde-627	105	7	4.1	4.1	NUM
ejde-627	105	8	)	)	PUNCT
ejde-627	105	9	where	where	SCONJ
ejde-627	105	10	u(x	u(x	NOUN
ejde-627	105	11	,	,	PUNCT
ejde-627	105	12	y	y	NOUN
ejde-627	105	13	)	)	PUNCT
ejde-627	105	14	is	be	AUX
ejde-627	105	15	the	the	DET
ejde-627	105	16	unknown	unknown	ADJ
ejde-627	105	17	function	function	NOUN
ejde-627	105	18	,	,	PUNCT
ejde-627	105	19	and	and	CCONJ
ejde-627	105	20	f(x	f(x	PROPN
ejde-627	105	21	)	)	PUNCT
ejde-627	105	22	and	and	CCONJ
ejde-627	105	23	g(y	g(y	NOUN
ejde-627	105	24	)	)	PUNCT
ejde-627	105	25	are	be	AUX
ejde-627	105	26	given	give	VERB
ejde-627	105	27	functions	function	NOUN
ejde-627	105	28	.	.	PUNCT
ejde-627	106	1	by	by	ADP
ejde-627	106	2	substituting	substitute	VERB
ejde-627	106	3	u(x	u(x	NOUN
ejde-627	106	4	,	,	PUNCT
ejde-627	106	5	y	y	NOUN
ejde-627	106	6	)	)	PUNCT
ejde-627	106	7	=	=	SYM
ejde-627	106	8	p	p	X
ejde-627	106	9	(	(	PUNCT
ejde-627	106	10	x)q(y	x)q(y	PROPN
ejde-627	106	11	)	)	PUNCT
ejde-627	106	12	into	into	ADP
ejde-627	106	13	(	(	PUNCT
ejde-627	106	14	4.1	4.1	NUM
ejde-627	106	15	)	)	PUNCT
ejde-627	106	16	we	we	PRON
ejde-627	106	17	obtain	obtain	VERB
ejde-627	106	18	f(x)p	f(x)p	NOUN
ejde-627	106	19	′(x)q(y	′(x)q(y	ADJ
ejde-627	106	20	)	)	PUNCT
ejde-627	107	1	+	+	CCONJ
ejde-627	107	2	p	p	X
ejde-627	107	3	(	(	PUNCT
ejde-627	107	4	x)q′(y	x)q′(y	PROPN
ejde-627	107	5	)	)	PUNCT
ejde-627	107	6	=	=	SYM
ejde-627	107	7	g(y)p	g(y)p	PROPN
ejde-627	107	8	(	(	PUNCT
ejde-627	107	9	x)q(y	x)q(y	PROPN
ejde-627	107	10	)	)	PUNCT
ejde-627	107	11	.	.	PUNCT
ejde-627	108	1	(	(	PUNCT
ejde-627	108	2	4.2	4.2	NUM
ejde-627	108	3	)	)	PUNCT
ejde-627	108	4	therefore	therefore	ADV
ejde-627	108	5	,	,	PUNCT
ejde-627	108	6	in	in	ADP
ejde-627	108	7	tensor	tensor	NOUN
ejde-627	108	8	product	product	NOUN
ejde-627	108	9	form	form	NOUN
ejde-627	108	10	,	,	PUNCT
ejde-627	108	11	(	(	PUNCT
ejde-627	108	12	4.2	4.2	NUM
ejde-627	108	13	)	)	PUNCT
ejde-627	108	14	becomes	become	VERB
ejde-627	108	15	f(x)p	f(x)p	PROPN
ejde-627	108	16	′(x)⊗q(y	′(x)⊗q(y	ADJ
ejde-627	108	17	)	)	PUNCT
ejde-627	109	1	+	+	CCONJ
ejde-627	110	1	p	p	X
ejde-627	110	2	(	(	PUNCT
ejde-627	110	3	x)⊗q′(y	x)⊗q′(y	NOUN
ejde-627	110	4	)	)	PUNCT
ejde-627	110	5	=	=	SYM
ejde-627	111	1	p	p	X
ejde-627	111	2	(	(	PUNCT
ejde-627	111	3	x)⊗	x)⊗	PROPN
ejde-627	111	4	g(y)q(y	g(y)q(y	NOUN
ejde-627	111	5	)	)	PUNCT
ejde-627	111	6	,	,	PUNCT
ejde-627	111	7	(	(	PUNCT
ejde-627	111	8	4.3	4.3	NUM
ejde-627	111	9	)	)	PUNCT
ejde-627	111	10	by	by	ADP
ejde-627	111	11	theorem	theorem	NOUN
ejde-627	111	12	2.2	2.2	NUM
ejde-627	111	13	,	,	PUNCT
ejde-627	111	14	we	we	PRON
ejde-627	111	15	have	have	VERB
ejde-627	111	16	the	the	DET
ejde-627	111	17	following	follow	VERB
ejde-627	111	18	two	two	NUM
ejde-627	111	19	cases	case	NOUN
ejde-627	111	20	:	:	PUNCT
ejde-627	111	21	(	(	PUNCT
ejde-627	111	22	i	i	NOUN
ejde-627	111	23	)	)	PUNCT
ejde-627	111	24	f(x)p	f(x)p	NOUN
ejde-627	111	25	′(x	′(x	NOUN
ejde-627	111	26	)	)	PUNCT
ejde-627	111	27	=	=	SYM
ejde-627	112	1	p	p	X
ejde-627	112	2	(	(	PUNCT
ejde-627	112	3	x	x	NOUN
ejde-627	112	4	)	)	PUNCT
ejde-627	112	5	=	=	SYM
ejde-627	112	6	p	p	X
ejde-627	112	7	(	(	PUNCT
ejde-627	112	8	x	x	NOUN
ejde-627	112	9	)	)	PUNCT
ejde-627	112	10	,	,	PUNCT
ejde-627	112	11	(	(	PUNCT
ejde-627	112	12	ii	ii	NOUN
ejde-627	112	13	)	)	PUNCT
ejde-627	112	14	q(y	q(y	PROPN
ejde-627	112	15	)	)	PUNCT
ejde-627	112	16	=	=	SYM
ejde-627	112	17	q′(y	q′(y	X
ejde-627	112	18	)	)	PUNCT
ejde-627	112	19	=	=	NOUN
ejde-627	112	20	g(y)q(y	g(y)q(y	NOUN
ejde-627	112	21	)	)	PUNCT
ejde-627	112	22	.	.	PUNCT
ejde-627	113	1	for	for	ADP
ejde-627	113	2	case	case	NOUN
ejde-627	113	3	(	(	PUNCT
ejde-627	113	4	i	i	NOUN
ejde-627	113	5	)	)	PUNCT
ejde-627	113	6	we	we	PRON
ejde-627	113	7	have	have	VERB
ejde-627	113	8	only	only	ADV
ejde-627	113	9	one	one	NUM
ejde-627	113	10	situation	situation	NOUN
ejde-627	113	11	,	,	PUNCT
ejde-627	113	12	f(x)p	f(x)p	NOUN
ejde-627	113	13	′(x	′(x	NOUN
ejde-627	113	14	)	)	PUNCT
ejde-627	114	1	=	=	SYM
ejde-627	114	2	p	p	X
ejde-627	114	3	(	(	PUNCT
ejde-627	114	4	x	x	X
ejde-627	114	5	)	)	PUNCT
ejde-627	114	6	which	which	PRON
ejde-627	114	7	yields	yield	VERB
ejde-627	114	8	p	p	X
ejde-627	114	9	(	(	PUNCT
ejde-627	114	10	x	x	NOUN
ejde-627	114	11	)	)	PUNCT
ejde-627	114	12	=	=	SYM
ejde-627	114	13	c1e	c1e	NOUN
ejde-627	114	14	∫	∫	PROPN
ejde-627	114	15	1	1	NUM
ejde-627	114	16	/	/	SYM
ejde-627	114	17	f(x	f(x	PROPN
ejde-627	114	18	)	)	PUNCT
ejde-627	114	19	dx	dx	PROPN
ejde-627	114	20	(	(	PUNCT
ejde-627	114	21	4.4	4.4	NUM
ejde-627	114	22	)	)	PUNCT
ejde-627	114	23	ejde-2024/28	ejde-2024/28	PROPN
ejde-627	114	24	solutions	solution	NOUN
ejde-627	114	25	of	of	ADP
ejde-627	114	26	linear	linear	PROPN
ejde-627	114	27	and	and	CCONJ
ejde-627	114	28	non	non	ADJ
ejde-627	114	29	-	-	ADJ
ejde-627	114	30	linear	linear	ADJ
ejde-627	114	31	pdes	pde	NOUN
ejde-627	114	32	5	5	NUM
ejde-627	114	33	figure	figure	NOUN
ejde-627	114	34	2	2	NUM
ejde-627	114	35	.	.	NOUN
ejde-627	114	36	second	second	ADJ
ejde-627	114	37	atomic	atomic	ADJ
ejde-627	114	38	solution	solution	NOUN
ejde-627	114	39	u12(x	u12(x	PROPN
ejde-627	114	40	,	,	PUNCT
ejde-627	114	41	y	y	NOUN
ejde-627	114	42	)	)	PUNCT
ejde-627	114	43	of	of	ADP
ejde-627	114	44	problem	problem	NOUN
ejde-627	114	45	(	(	PUNCT
ejde-627	114	46	3.1	3.1	NUM
ejde-627	114	47	)	)	PUNCT
ejde-627	114	48	.	.	PUNCT
ejde-627	115	1	where	where	SCONJ
ejde-627	115	2	c1	c1	PROPN
ejde-627	115	3	is	be	AUX
ejde-627	115	4	a	a	DET
ejde-627	115	5	constant	constant	ADJ
ejde-627	115	6	.	.	PUNCT
ejde-627	116	1	therefore	therefore	ADV
ejde-627	116	2	,	,	PUNCT
ejde-627	116	3	an	an	DET
ejde-627	116	4	atomic	atomic	ADJ
ejde-627	116	5	solution	solution	NOUN
ejde-627	116	6	exists	exist	VERB
ejde-627	116	7	for	for	ADP
ejde-627	116	8	case	case	NOUN
ejde-627	116	9	(	(	PUNCT
ejde-627	116	10	i	i	NOUN
ejde-627	116	11	)	)	PUNCT
ejde-627	116	12	,	,	PUNCT
ejde-627	116	13	and	and	CCONJ
ejde-627	116	14	it	it	PRON
ejde-627	116	15	can	can	AUX
ejde-627	116	16	be	be	AUX
ejde-627	116	17	obtained	obtain	VERB
ejde-627	116	18	by	by	ADP
ejde-627	116	19	substituting	substitute	VERB
ejde-627	116	20	(	(	PUNCT
ejde-627	116	21	4.4	4.4	NUM
ejde-627	116	22	)	)	PUNCT
ejde-627	116	23	into	into	ADP
ejde-627	116	24	(	(	PUNCT
ejde-627	116	25	4.2	4.2	NUM
ejde-627	116	26	)	)	PUNCT
ejde-627	116	27	,	,	PUNCT
ejde-627	116	28	q′(y	q′(y	X
ejde-627	116	29	)	)	PUNCT
ejde-627	117	1	+	+	CCONJ
ejde-627	118	1	[	[	X
ejde-627	118	2	1−	1−	NUM
ejde-627	118	3	g(y)]q(y	g(y)]q(y	PROPN
ejde-627	118	4	)	)	PUNCT
ejde-627	118	5	=	=	SYM
ejde-627	119	1	0	0	X
ejde-627	119	2	.	.	PUNCT
ejde-627	120	1	(	(	PUNCT
ejde-627	120	2	4.5	4.5	NUM
ejde-627	120	3	)	)	PUNCT
ejde-627	120	4	this	this	PRON
ejde-627	120	5	is	be	AUX
ejde-627	120	6	a	a	DET
ejde-627	120	7	linear	linear	ADJ
ejde-627	120	8	ordinary	ordinary	ADJ
ejde-627	120	9	differential	differential	ADJ
ejde-627	120	10	equation	equation	NOUN
ejde-627	120	11	which	which	PRON
ejde-627	120	12	has	have	VERB
ejde-627	120	13	solution	solution	NOUN
ejde-627	120	14	q(y	q(y	NOUN
ejde-627	120	15	)	)	PUNCT
ejde-627	121	1	=	=	SYM
ejde-627	121	2	c2e	c2e	NOUN
ejde-627	121	3	∫	∫	PROPN
ejde-627	121	4	(	(	PUNCT
ejde-627	121	5	g(y)−1)dy	g(y)−1)dy	PROPN
ejde-627	121	6	.	.	PUNCT
ejde-627	122	1	(	(	PUNCT
ejde-627	122	2	4.6	4.6	NUM
ejde-627	122	3	)	)	PUNCT
ejde-627	122	4	hence	hence	ADV
ejde-627	122	5	,	,	PUNCT
ejde-627	122	6	referring	refer	VERB
ejde-627	122	7	to	to	ADP
ejde-627	122	8	(	(	PUNCT
ejde-627	122	9	4.4	4.4	NUM
ejde-627	122	10	)	)	PUNCT
ejde-627	122	11	and	and	CCONJ
ejde-627	122	12	(	(	PUNCT
ejde-627	122	13	4.6	4.6	NUM
ejde-627	122	14	)	)	PUNCT
ejde-627	122	15	,	,	PUNCT
ejde-627	122	16	the	the	DET
ejde-627	122	17	first	first	ADJ
ejde-627	122	18	atomic	atomic	ADJ
ejde-627	122	19	solution	solution	NOUN
ejde-627	122	20	for	for	ADP
ejde-627	122	21	case	case	NOUN
ejde-627	122	22	(	(	PUNCT
ejde-627	122	23	i	i	NOUN
ejde-627	122	24	)	)	PUNCT
ejde-627	122	25	is	be	AUX
ejde-627	122	26	u1(x	u1(x	PROPN
ejde-627	122	27	,	,	PUNCT
ejde-627	122	28	y	y	NOUN
ejde-627	122	29	)	)	PUNCT
ejde-627	122	30	=	=	SYM
ejde-627	123	1	c∗e	c∗e	NOUN
ejde-627	123	2	∫	∫	NOUN
ejde-627	123	3	1	1	NUM
ejde-627	123	4	/	/	SYM
ejde-627	123	5	f(x	f(x	PROPN
ejde-627	123	6	)	)	PUNCT
ejde-627	123	7	dx+	dx+	NOUN
ejde-627	123	8	∫	∫	PROPN
ejde-627	123	9	(	(	PUNCT
ejde-627	123	10	g(y)−1	g(y)−1	NOUN
ejde-627	123	11	)	)	PUNCT
ejde-627	123	12	dy	dy	NOUN
ejde-627	123	13	,	,	PUNCT
ejde-627	123	14	(	(	PUNCT
ejde-627	123	15	4.7	4.7	NUM
ejde-627	123	16	)	)	PUNCT
ejde-627	123	17	where	where	SCONJ
ejde-627	123	18	c∗	c∗	NOUN
ejde-627	123	19	=	=	SYM
ejde-627	123	20	c1c2	c1c2	NOUN
ejde-627	123	21	.	.	NOUN
ejde-627	124	1	for	for	ADP
ejde-627	124	2	case	case	NOUN
ejde-627	124	3	(	(	PUNCT
ejde-627	124	4	ii	ii	NOUN
ejde-627	124	5	)	)	PUNCT
ejde-627	124	6	,	,	PUNCT
ejde-627	124	7	we	we	PRON
ejde-627	124	8	have	have	VERB
ejde-627	124	9	three	three	NUM
ejde-627	124	10	situations	situation	NOUN
ejde-627	124	11	:	:	PUNCT
ejde-627	124	12	(	(	PUNCT
ejde-627	124	13	a	a	X
ejde-627	124	14	)	)	PUNCT
ejde-627	124	15	q(y	q(y	PROPN
ejde-627	124	16	)	)	PUNCT
ejde-627	124	17	=	=	SYM
ejde-627	124	18	q′(y	q′(y	X
ejde-627	124	19	)	)	PUNCT
ejde-627	124	20	,	,	PUNCT
ejde-627	124	21	(	(	PUNCT
ejde-627	124	22	b	b	NOUN
ejde-627	124	23	)	)	PUNCT
ejde-627	124	24	q(y	q(y	NOUN
ejde-627	124	25	)	)	PUNCT
ejde-627	124	26	=	=	NOUN
ejde-627	124	27	g(y)q(y	g(y)q(y	NOUN
ejde-627	124	28	)	)	PUNCT
ejde-627	124	29	,	,	PUNCT
ejde-627	124	30	(	(	PUNCT
ejde-627	124	31	c	c	X
ejde-627	124	32	)	)	PUNCT
ejde-627	124	33	q′(y	q′(y	X
ejde-627	124	34	)	)	PUNCT
ejde-627	124	35	=	=	NOUN
ejde-627	124	36	g(y)q(y	g(y)q(y	NOUN
ejde-627	124	37	)	)	PUNCT
ejde-627	124	38	.	.	PUNCT
ejde-627	125	1	according	accord	VERB
ejde-627	125	2	to	to	ADP
ejde-627	125	3	theorem	theorem	ADJ
ejde-627	125	4	2.2	2.2	NUM
ejde-627	125	5	,	,	PUNCT
ejde-627	125	6	to	to	PART
ejde-627	125	7	obtain	obtain	VERB
ejde-627	125	8	an	an	DET
ejde-627	125	9	atomic	atomic	ADJ
ejde-627	125	10	solution	solution	NOUN
ejde-627	125	11	for	for	ADP
ejde-627	125	12	case	case	NOUN
ejde-627	125	13	(	(	PUNCT
ejde-627	125	14	ii	ii	NOUN
ejde-627	125	15	)	)	PUNCT
ejde-627	125	16	,	,	PUNCT
ejde-627	125	17	we	we	PRON
ejde-627	125	18	require	require	VERB
ejde-627	125	19	q(y	q(y	NOUN
ejde-627	125	20	)	)	PUNCT
ejde-627	125	21	=	=	SYM
ejde-627	125	22	q′(y	q′(y	X
ejde-627	125	23	)	)	PUNCT
ejde-627	125	24	=	=	SYM
ejde-627	125	25	c3e	c3e	PUNCT
ejde-627	125	26	y	y	PROPN
ejde-627	125	27	and	and	CCONJ
ejde-627	125	28	g(y	g(y	PROPN
ejde-627	125	29	)	)	PUNCT
ejde-627	125	30	=	=	SYM
ejde-627	125	31	1	1	NUM
ejde-627	125	32	,	,	PUNCT
ejde-627	125	33	(	(	PUNCT
ejde-627	125	34	4.8	4.8	NUM
ejde-627	125	35	)	)	PUNCT
ejde-627	125	36	6	6	NUM
ejde-627	125	37	w.	w.	NOUN
ejde-627	125	38	g.	g.	PROPN
ejde-627	125	39	alshanti	alshanti	VERB
ejde-627	125	40	ejde-2024/28	ejde-2024/28	PROPN
ejde-627	125	41	where	where	SCONJ
ejde-627	125	42	c3	c3	PROPN
ejde-627	125	43	is	be	AUX
ejde-627	125	44	a	a	DET
ejde-627	125	45	constant	constant	ADJ
ejde-627	125	46	.	.	PUNCT
ejde-627	126	1	now	now	ADV
ejde-627	126	2	,	,	PUNCT
ejde-627	126	3	on	on	ADP
ejde-627	126	4	substituting	substitute	VERB
ejde-627	126	5	(	(	PUNCT
ejde-627	126	6	4.8	4.8	NUM
ejde-627	126	7	)	)	PUNCT
ejde-627	126	8	into	into	ADP
ejde-627	126	9	(	(	PUNCT
ejde-627	126	10	4.2	4.2	NUM
ejde-627	126	11	)	)	PUNCT
ejde-627	126	12	,	,	PUNCT
ejde-627	126	13	we	we	PRON
ejde-627	126	14	have	have	VERB
ejde-627	126	15	f(x)p	f(x)p	NOUN
ejde-627	126	16	′(x	′(x	NOUN
ejde-627	126	17	)	)	PUNCT
ejde-627	126	18	=	=	SYM
ejde-627	127	1	0	0	X
ejde-627	127	2	.	.	PUNCT
ejde-627	128	1	(	(	PUNCT
ejde-627	128	2	4.9	4.9	NUM
ejde-627	128	3	)	)	PUNCT
ejde-627	128	4	therefore	therefore	ADV
ejde-627	128	5	,	,	PUNCT
ejde-627	128	6	p	p	X
ejde-627	128	7	(	(	PUNCT
ejde-627	128	8	x	x	NOUN
ejde-627	128	9	)	)	PUNCT
ejde-627	128	10	=	=	SYM
ejde-627	128	11	c4	c4	NOUN
ejde-627	128	12	(	(	PUNCT
ejde-627	128	13	a	a	DET
ejde-627	128	14	constant	constant	ADJ
ejde-627	128	15	)	)	PUNCT
ejde-627	128	16	and	and	CCONJ
ejde-627	128	17	hence	hence	ADV
ejde-627	128	18	,	,	PUNCT
ejde-627	128	19	the	the	DET
ejde-627	128	20	second	second	ADJ
ejde-627	128	21	atomic	atomic	ADJ
ejde-627	128	22	solution	solution	NOUN
ejde-627	128	23	associated	associate	VERB
ejde-627	128	24	with	with	ADP
ejde-627	128	25	case	case	NOUN
ejde-627	128	26	(	(	PUNCT
ejde-627	128	27	ii	ii	NOUN
ejde-627	128	28	)	)	PUNCT
ejde-627	128	29	,	,	PUNCT
ejde-627	128	30	can	can	AUX
ejde-627	128	31	be	be	AUX
ejde-627	128	32	achieved	achieve	VERB
ejde-627	128	33	by	by	ADP
ejde-627	128	34	compiling	compile	VERB
ejde-627	128	35	both	both	CCONJ
ejde-627	128	36	the	the	DET
ejde-627	128	37	value	value	NOUN
ejde-627	128	38	of	of	ADP
ejde-627	128	39	p	p	NOUN
ejde-627	128	40	(	(	PUNCT
ejde-627	128	41	x	x	X
ejde-627	128	42	)	)	PUNCT
ejde-627	128	43	from	from	ADP
ejde-627	128	44	(	(	PUNCT
ejde-627	128	45	4.9	4.9	NUM
ejde-627	128	46	)	)	PUNCT
ejde-627	128	47	and	and	CCONJ
ejde-627	128	48	the	the	DET
ejde-627	128	49	obtained	obtain	VERB
ejde-627	128	50	value	value	NOUN
ejde-627	128	51	of	of	ADP
ejde-627	128	52	q(x	q(x	NOUN
ejde-627	128	53	)	)	PUNCT
ejde-627	128	54	from	from	ADP
ejde-627	128	55	(	(	PUNCT
ejde-627	128	56	4.8	4.8	NUM
ejde-627	128	57	)	)	PUNCT
ejde-627	128	58	as	as	SCONJ
ejde-627	128	59	follows	follow	VERB
ejde-627	128	60	u2(x	u2(x	PRON
ejde-627	128	61	,	,	PUNCT
ejde-627	128	62	y	y	NOUN
ejde-627	128	63	)	)	PUNCT
ejde-627	129	1	=	=	VERB
ejde-627	129	2	c∗∗e	c∗∗e	NOUN
ejde-627	129	3	y	y	PROPN
ejde-627	129	4	,	,	PUNCT
ejde-627	129	5	(	(	PUNCT
ejde-627	129	6	4.10	4.10	NUM
ejde-627	129	7	)	)	PUNCT
ejde-627	129	8	where	where	SCONJ
ejde-627	129	9	c∗∗	c∗∗	PRON
ejde-627	129	10	=	=	PUNCT
ejde-627	129	11	c3c4	c3c4	PROPN
ejde-627	129	12	.	.	PROPN
ejde-627	129	13	example	example	NOUN
ejde-627	129	14	4.2	4.2	NUM
ejde-627	129	15	.	.	PUNCT
ejde-627	129	16	consider	consider	VERB
ejde-627	129	17	the	the	DET
ejde-627	129	18	partial	partial	ADJ
ejde-627	129	19	differential	differential	NOUN
ejde-627	129	20	equation	equation	NOUN
ejde-627	129	21	uxxy	uxxy	NOUN
ejde-627	129	22	+	+	CCONJ
ejde-627	129	23	uxyy	uxyy	PROPN
ejde-627	129	24	=	=	SYM
ejde-627	129	25	yu	yu	PROPN
ejde-627	129	26	,	,	PUNCT
ejde-627	129	27	(	(	PUNCT
ejde-627	129	28	4.11	4.11	NUM
ejde-627	129	29	)	)	PUNCT
ejde-627	129	30	where	where	SCONJ
ejde-627	129	31	u(x	u(x	NOUN
ejde-627	129	32	,	,	PUNCT
ejde-627	129	33	y	y	NOUN
ejde-627	129	34	)	)	PUNCT
ejde-627	129	35	is	be	AUX
ejde-627	129	36	the	the	DET
ejde-627	129	37	unknown	unknown	ADJ
ejde-627	129	38	function	function	NOUN
ejde-627	129	39	.	.	PUNCT
ejde-627	130	1	the	the	DET
ejde-627	130	2	following	follow	VERB
ejde-627	130	3	conditions	condition	NOUN
ejde-627	130	4	are	be	AUX
ejde-627	130	5	imposed	impose	VERB
ejde-627	130	6	on	on	ADP
ejde-627	130	7	u	u	NOUN
ejde-627	130	8	:	:	PUNCT
ejde-627	130	9	u(0	u(0	PROPN
ejde-627	130	10	,	,	PUNCT
ejde-627	130	11	0	0	NUM
ejde-627	130	12	)	)	PUNCT
ejde-627	130	13	=	=	SYM
ejde-627	130	14	1	1	NUM
ejde-627	130	15	,	,	PUNCT
ejde-627	130	16	ux(0	ux(0	PROPN
ejde-627	130	17	,	,	PUNCT
ejde-627	130	18	0	0	NUM
ejde-627	130	19	)	)	PUNCT
ejde-627	130	20	=	=	SYM
ejde-627	130	21	1	1	NUM
ejde-627	130	22	,	,	PUNCT
ejde-627	130	23	uy(0	uy(0	PROPN
ejde-627	130	24	,	,	PUNCT
ejde-627	130	25	0	0	NUM
ejde-627	130	26	)	)	PUNCT
ejde-627	130	27	=	=	SYM
ejde-627	131	1	1	1	X
ejde-627	131	2	.	.	PUNCT
ejde-627	131	3	(	(	PUNCT
ejde-627	131	4	4.12	4.12	NUM
ejde-627	131	5	)	)	PUNCT
ejde-627	131	6	by	by	ADP
ejde-627	131	7	substituting	substitute	VERB
ejde-627	131	8	u(x	u(x	NOUN
ejde-627	131	9	,	,	PUNCT
ejde-627	131	10	y	y	NOUN
ejde-627	131	11	)	)	PUNCT
ejde-627	131	12	=	=	SYM
ejde-627	132	1	p	p	X
ejde-627	132	2	(	(	PUNCT
ejde-627	132	3	x)q(y	x)q(y	PROPN
ejde-627	132	4	)	)	PUNCT
ejde-627	132	5	into	into	ADP
ejde-627	132	6	(	(	PUNCT
ejde-627	132	7	4.11	4.11	NUM
ejde-627	132	8	)	)	PUNCT
ejde-627	132	9	we	we	PRON
ejde-627	132	10	obtain	obtain	VERB
ejde-627	132	11	p	p	PROPN
ejde-627	132	12	′′(x)q′(y	′′(x)q′(y	NOUN
ejde-627	132	13	)	)	PUNCT
ejde-627	133	1	+	+	CCONJ
ejde-627	133	2	p	p	NOUN
ejde-627	133	3	′(x)q′′(y	′(x)q′′(y	PROPN
ejde-627	133	4	)	)	PUNCT
ejde-627	134	1	=	=	SYM
ejde-627	134	2	yp	yp	PROPN
ejde-627	134	3	(	(	PUNCT
ejde-627	134	4	x)q(y	x)q(y	PROPN
ejde-627	134	5	)	)	PUNCT
ejde-627	134	6	.	.	PUNCT
ejde-627	135	1	(	(	PUNCT
ejde-627	135	2	4.13	4.13	NUM
ejde-627	135	3	)	)	PUNCT
ejde-627	135	4	therefore	therefore	ADV
ejde-627	135	5	,	,	PUNCT
ejde-627	135	6	in	in	ADP
ejde-627	135	7	tensor	tensor	NOUN
ejde-627	135	8	product	product	NOUN
ejde-627	135	9	form	form	NOUN
ejde-627	135	10	,	,	PUNCT
ejde-627	135	11	(	(	PUNCT
ejde-627	135	12	4.13	4.13	NUM
ejde-627	135	13	)	)	PUNCT
ejde-627	135	14	becomes	become	VERB
ejde-627	135	15	p	p	PROPN
ejde-627	135	16	′′(x)⊗q′(y	′′(x)⊗q′(y	PROPN
ejde-627	135	17	)	)	PUNCT
ejde-627	136	1	+	+	CCONJ
ejde-627	136	2	p	p	PROPN
ejde-627	136	3	′(x)⊗q′′(y	′(x)⊗q′′(y	NOUN
ejde-627	136	4	)	)	PUNCT
ejde-627	136	5	=	=	SYM
ejde-627	137	1	p	p	X
ejde-627	137	2	(	(	PUNCT
ejde-627	137	3	x)⊗	x)⊗	PROPN
ejde-627	137	4	yq(y	yq(y	PROPN
ejde-627	137	5	)	)	PUNCT
ejde-627	137	6	.	.	PUNCT
ejde-627	138	1	(	(	PUNCT
ejde-627	138	2	4.14	4.14	NUM
ejde-627	138	3	)	)	PUNCT
ejde-627	138	4	using	use	VERB
ejde-627	138	5	theorem	theorem	NOUN
ejde-627	138	6	2.2	2.2	NUM
ejde-627	138	7	,	,	PUNCT
ejde-627	138	8	we	we	PRON
ejde-627	138	9	have	have	VERB
ejde-627	138	10	one	one	NUM
ejde-627	138	11	of	of	ADP
ejde-627	138	12	the	the	DET
ejde-627	138	13	following	follow	VERB
ejde-627	138	14	two	two	NUM
ejde-627	138	15	cases	case	NOUN
ejde-627	138	16	:	:	PUNCT
ejde-627	138	17	(	(	PUNCT
ejde-627	138	18	i	i	NOUN
ejde-627	138	19	)	)	PUNCT
ejde-627	138	20	p	p	NOUN
ejde-627	138	21	′′(x	′′(x	NOUN
ejde-627	138	22	)	)	PUNCT
ejde-627	138	23	=	=	PUNCT
ejde-627	138	24	p	p	NOUN
ejde-627	138	25	′(x	′(x	NOUN
ejde-627	138	26	)	)	PUNCT
ejde-627	139	1	=	=	SYM
ejde-627	139	2	p	p	X
ejde-627	139	3	(	(	PUNCT
ejde-627	139	4	x	x	NOUN
ejde-627	139	5	)	)	PUNCT
ejde-627	139	6	,	,	PUNCT
ejde-627	139	7	(	(	PUNCT
ejde-627	139	8	ii	ii	NOUN
ejde-627	139	9	)	)	PUNCT
ejde-627	139	10	q′′(y	q′′(y	NOUN
ejde-627	139	11	)	)	PUNCT
ejde-627	139	12	=	=	SYM
ejde-627	139	13	q′(y	q′(y	X
ejde-627	139	14	)	)	PUNCT
ejde-627	139	15	=	=	SYM
ejde-627	139	16	yq(y	yq(y	NOUN
ejde-627	139	17	)	)	PUNCT
ejde-627	139	18	.	.	PUNCT
ejde-627	140	1	for	for	ADP
ejde-627	140	2	case	case	NOUN
ejde-627	140	3	(	(	PUNCT
ejde-627	140	4	i	i	NOUN
ejde-627	140	5	)	)	PUNCT
ejde-627	140	6	,	,	PUNCT
ejde-627	140	7	we	we	PRON
ejde-627	140	8	have	have	VERB
ejde-627	140	9	the	the	DET
ejde-627	140	10	following	follow	VERB
ejde-627	140	11	three	three	NUM
ejde-627	140	12	situations	situation	NOUN
ejde-627	140	13	:	:	PUNCT
ejde-627	140	14	(	(	PUNCT
ejde-627	140	15	a	a	X
ejde-627	140	16	)	)	PUNCT
ejde-627	140	17	p	p	NOUN
ejde-627	140	18	′′(x	′′(x	NOUN
ejde-627	140	19	)	)	PUNCT
ejde-627	140	20	=	=	PUNCT
ejde-627	141	1	p	p	NOUN
ejde-627	141	2	′(x	′(x	NOUN
ejde-627	141	3	)	)	PUNCT
ejde-627	141	4	,	,	PUNCT
ejde-627	141	5	(	(	PUNCT
ejde-627	141	6	b	b	X
ejde-627	141	7	)	)	PUNCT
ejde-627	141	8	p	p	NOUN
ejde-627	141	9	′′(x	′′(x	NOUN
ejde-627	141	10	)	)	PUNCT
ejde-627	141	11	=	=	SYM
ejde-627	142	1	p	p	X
ejde-627	142	2	(	(	PUNCT
ejde-627	142	3	x	x	NOUN
ejde-627	142	4	)	)	PUNCT
ejde-627	142	5	,	,	PUNCT
ejde-627	142	6	(	(	PUNCT
ejde-627	142	7	c	c	X
ejde-627	142	8	)	)	PUNCT
ejde-627	142	9	p	p	NOUN
ejde-627	142	10	′(x	′(x	NOUN
ejde-627	142	11	)	)	PUNCT
ejde-627	142	12	=	=	SYM
ejde-627	143	1	p	p	X
ejde-627	143	2	(	(	PUNCT
ejde-627	143	3	x	x	NOUN
ejde-627	143	4	)	)	PUNCT
ejde-627	143	5	.	.	PUNCT
ejde-627	144	1	from	from	ADP
ejde-627	144	2	(	(	PUNCT
ejde-627	144	3	4.12	4.12	NUM
ejde-627	144	4	)	)	PUNCT
ejde-627	144	5	,	,	PUNCT
ejde-627	144	6	we	we	PRON
ejde-627	144	7	can	can	AUX
ejde-627	144	8	assume	assume	VERB
ejde-627	144	9	,	,	PUNCT
ejde-627	144	10	without	without	ADP
ejde-627	144	11	loss	loss	NOUN
ejde-627	144	12	of	of	ADP
ejde-627	144	13	generality	generality	NOUN
ejde-627	144	14	,	,	PUNCT
ejde-627	144	15	that	that	SCONJ
ejde-627	144	16	p	p	X
ejde-627	144	17	(	(	PUNCT
ejde-627	144	18	0	0	NUM
ejde-627	144	19	)	)	PUNCT
ejde-627	144	20	=	=	SYM
ejde-627	144	21	q(0	q(0	NOUN
ejde-627	144	22	)	)	PUNCT
ejde-627	145	1	=	=	PUNCT
ejde-627	146	1	p	p	NOUN
ejde-627	146	2	′(x	′(x	NOUN
ejde-627	146	3	)	)	PUNCT
ejde-627	146	4	=	=	SYM
ejde-627	146	5	q′(y	q′(y	X
ejde-627	146	6	)	)	PUNCT
ejde-627	146	7	=	=	SYM
ejde-627	147	1	1	1	X
ejde-627	147	2	.	.	PUNCT
ejde-627	147	3	(	(	PUNCT
ejde-627	147	4	4.15	4.15	NUM
ejde-627	147	5	)	)	PUNCT
ejde-627	147	6	thus	thus	ADV
ejde-627	147	7	,	,	PUNCT
ejde-627	147	8	all	all	DET
ejde-627	147	9	three	three	NUM
ejde-627	147	10	situations	situation	NOUN
ejde-627	147	11	(	(	PUNCT
ejde-627	147	12	a	a	X
ejde-627	147	13	)	)	PUNCT
ejde-627	147	14	,	,	PUNCT
ejde-627	147	15	(	(	PUNCT
ejde-627	147	16	b	b	NOUN
ejde-627	147	17	)	)	PUNCT
ejde-627	147	18	,	,	PUNCT
ejde-627	147	19	and	and	CCONJ
ejde-627	147	20	(	(	PUNCT
ejde-627	147	21	c	c	NOUN
ejde-627	147	22	)	)	PUNCT
ejde-627	147	23	yield	yield	VERB
ejde-627	147	24	the	the	DET
ejde-627	147	25	same	same	ADJ
ejde-627	147	26	result	result	NOUN
ejde-627	147	27	,	,	PUNCT
ejde-627	147	28	that	that	ADV
ejde-627	147	29	is	is	ADV
ejde-627	147	30	p	p	X
ejde-627	147	31	(	(	PUNCT
ejde-627	147	32	x	x	NOUN
ejde-627	147	33	)	)	PUNCT
ejde-627	147	34	=	=	SYM
ejde-627	148	1	ex	ex	X
ejde-627	148	2	.	.	PUNCT
ejde-627	148	3	(	(	PUNCT
ejde-627	148	4	4.16	4.16	NUM
ejde-627	148	5	)	)	PUNCT
ejde-627	148	6	by	by	ADP
ejde-627	148	7	substituting	substitute	VERB
ejde-627	148	8	(	(	PUNCT
ejde-627	148	9	4.16	4.16	NUM
ejde-627	148	10	)	)	PUNCT
ejde-627	148	11	into	into	ADP
ejde-627	148	12	(	(	PUNCT
ejde-627	148	13	4.13	4.13	NUM
ejde-627	148	14	)	)	PUNCT
ejde-627	148	15	we	we	PRON
ejde-627	148	16	obtain	obtain	VERB
ejde-627	148	17	q′′(y	q′′(y	NOUN
ejde-627	148	18	)	)	PUNCT
ejde-627	149	1	+	+	ADP
ejde-627	149	2	q′(y)−	q′(y)−	NOUN
ejde-627	149	3	yq(y	yq(y	NOUN
ejde-627	149	4	)	)	PUNCT
ejde-627	149	5	=	=	SYM
ejde-627	149	6	0	0	NUM
ejde-627	149	7	which	which	PRON
ejde-627	149	8	is	be	AUX
ejde-627	149	9	a	a	DET
ejde-627	149	10	linear	linear	ADJ
ejde-627	149	11	ordinary	ordinary	ADJ
ejde-627	149	12	differential	differential	ADJ
ejde-627	149	13	equation	equation	NOUN
ejde-627	149	14	with	with	ADP
ejde-627	149	15	variable	variable	ADJ
ejde-627	149	16	coefficients	coefficient	NOUN
ejde-627	149	17	and	and	CCONJ
ejde-627	149	18	can	can	AUX
ejde-627	149	19	be	be	AUX
ejde-627	149	20	solved	solve	VERB
ejde-627	149	21	by	by	ADP
ejde-627	149	22	implementing	implement	VERB
ejde-627	149	23	the	the	DET
ejde-627	149	24	power	power	NOUN
ejde-627	149	25	series	series	NOUN
ejde-627	149	26	method	method	VERB
ejde-627	149	27	such	such	ADJ
ejde-627	149	28	that	that	SCONJ
ejde-627	149	29	q(y	q(y	NOUN
ejde-627	149	30	)	)	PUNCT
ejde-627	149	31	=	=	PUNCT
ejde-627	149	32	∑∞	∑∞	NOUN
ejde-627	149	33	n=0	n=0	NUM
ejde-627	149	34	kny	kny	NOUN
ejde-627	149	35	n	n	CCONJ
ejde-627	149	36	as	as	SCONJ
ejde-627	149	37	follows	follow	VERB
ejde-627	149	38	:	:	PUNCT
ejde-627	149	39	q(y	q(y	X
ejde-627	149	40	)	)	PUNCT
ejde-627	149	41	=	=	SYM
ejde-627	149	42	k0	k0	PROPN
ejde-627	149	43	[	[	PUNCT
ejde-627	149	44	1	1	NUM
ejde-627	149	45	+	+	NUM
ejde-627	149	46	y3	y3	NOUN
ejde-627	149	47	2	2	NUM
ejde-627	149	48	·	·	SYM
ejde-627	149	49	3	3	NUM
ejde-627	149	50	−	−	NOUN
ejde-627	149	51	y4	y4	ADJ
ejde-627	149	52	2	2	NUM
ejde-627	149	53	·	·	SYM
ejde-627	149	54	3	3	NUM
ejde-627	149	55	·	·	SYM
ejde-627	149	56	4	4	NUM
ejde-627	149	57	+	+	NUM
ejde-627	149	58	y5	y5	NOUN
ejde-627	149	59	2	2	NUM
ejde-627	149	60	·	·	SYM
ejde-627	149	61	3	3	NUM
ejde-627	149	62	·	·	SYM
ejde-627	149	63	4	4	NUM
ejde-627	149	64	·	·	SYM
ejde-627	149	65	5	5	NUM
ejde-627	149	66	+	+	CCONJ
ejde-627	149	67	·	·	PUNCT
ejde-627	149	68	·	·	PUNCT
ejde-627	149	69	·	·	PUNCT
ejde-627	149	70	]	]	PUNCT
ejde-627	150	1	+	+	CCONJ
ejde-627	150	2	k1	k1	X
ejde-627	150	3	[	[	PUNCT
ejde-627	150	4	y	y	NOUN
ejde-627	150	5	−	−	PROPN
ejde-627	151	1	y2	y2	NOUN
ejde-627	151	2	2	2	NUM
ejde-627	152	1	+	+	NOUN
ejde-627	152	2	y3	y3	NOUN
ejde-627	152	3	2	2	NUM
ejde-627	152	4	·	·	SYM
ejde-627	152	5	3	3	NUM
ejde-627	152	6	+	+	CCONJ
ejde-627	152	7	y4	y4	ADJ
ejde-627	152	8	2	2	NUM
ejde-627	152	9	·	·	SYM
ejde-627	152	10	3	3	NUM
ejde-627	152	11	·	·	SYM
ejde-627	152	12	4	4	NUM
ejde-627	152	13	−	−	NOUN
ejde-627	152	14	·	·	PUNCT
ejde-627	152	15	·	·	PUNCT
ejde-627	152	16	·	·	PUNCT
ejde-627	152	17	]	]	PUNCT
ejde-627	153	1	=	=	X
ejde-627	153	2	:	:	PUNCT
ejde-627	153	3	k0q1	k0q1	PROPN
ejde-627	153	4	+	+	CCONJ
ejde-627	153	5	k1q2	k1q2	X
ejde-627	153	6	.	.	PUNCT
ejde-627	153	7	(	(	PUNCT
ejde-627	153	8	4.17	4.17	NUM
ejde-627	153	9	)	)	PUNCT
ejde-627	153	10	hence	hence	ADV
ejde-627	153	11	,	,	PUNCT
ejde-627	153	12	from	from	ADP
ejde-627	153	13	(	(	PUNCT
ejde-627	153	14	4.16	4.16	NUM
ejde-627	153	15	)	)	PUNCT
ejde-627	153	16	and	and	CCONJ
ejde-627	153	17	(	(	PUNCT
ejde-627	153	18	4.17	4.17	NUM
ejde-627	153	19	)	)	PUNCT
ejde-627	153	20	,	,	PUNCT
ejde-627	153	21	the	the	DET
ejde-627	153	22	first	first	ADJ
ejde-627	153	23	atomic	atomic	ADJ
ejde-627	153	24	solution	solution	NOUN
ejde-627	153	25	with	with	ADP
ejde-627	153	26	respect	respect	NOUN
ejde-627	153	27	to	to	ADP
ejde-627	153	28	case	case	NOUN
ejde-627	153	29	(	(	PUNCT
ejde-627	153	30	i	i	NOUN
ejde-627	153	31	)	)	PUNCT
ejde-627	153	32	is	be	AUX
ejde-627	153	33	u1(x	u1(x	PROPN
ejde-627	153	34	,	,	PUNCT
ejde-627	153	35	y	y	NOUN
ejde-627	153	36	)	)	PUNCT
ejde-627	153	37	=	=	PUNCT
ejde-627	154	1	ex(k0q1	ex(k0q1	PROPN
ejde-627	154	2	+	+	CCONJ
ejde-627	154	3	k1q2	k1q2	NOUN
ejde-627	154	4	)	)	PUNCT
ejde-627	154	5	,	,	PUNCT
ejde-627	154	6	(	(	PUNCT
ejde-627	154	7	4.18	4.18	NUM
ejde-627	154	8	)	)	PUNCT
ejde-627	154	9	where	where	SCONJ
ejde-627	154	10	q1	q1	PROPN
ejde-627	154	11	and	and	CCONJ
ejde-627	154	12	q2	q2	NOUN
ejde-627	154	13	are	be	AUX
ejde-627	154	14	given	give	VERB
ejde-627	154	15	in	in	ADP
ejde-627	154	16	(	(	PUNCT
ejde-627	154	17	4.17	4.17	NUM
ejde-627	154	18	)	)	PUNCT
ejde-627	154	19	.	.	PUNCT
ejde-627	155	1	for	for	ADP
ejde-627	155	2	case	case	NOUN
ejde-627	155	3	(	(	PUNCT
ejde-627	155	4	ii	ii	NOUN
ejde-627	155	5	)	)	PUNCT
ejde-627	155	6	,	,	PUNCT
ejde-627	155	7	we	we	PRON
ejde-627	155	8	have	have	VERB
ejde-627	155	9	three	three	NUM
ejde-627	155	10	situations	situation	NOUN
ejde-627	155	11	,	,	PUNCT
ejde-627	155	12	(	(	PUNCT
ejde-627	155	13	a	a	X
ejde-627	155	14	)	)	PUNCT
ejde-627	155	15	q′′(y	q′′(y	NOUN
ejde-627	155	16	)	)	PUNCT
ejde-627	155	17	=	=	SYM
ejde-627	155	18	q′(y	q′(y	X
ejde-627	155	19	)	)	PUNCT
ejde-627	155	20	,	,	PUNCT
ejde-627	155	21	(	(	PUNCT
ejde-627	155	22	b	b	X
ejde-627	155	23	)	)	PUNCT
ejde-627	155	24	q′′(y	q′′(y	NOUN
ejde-627	155	25	)	)	PUNCT
ejde-627	155	26	=	=	SYM
ejde-627	155	27	yq(y	yq(y	NOUN
ejde-627	155	28	)	)	PUNCT
ejde-627	155	29	,	,	PUNCT
ejde-627	155	30	ejde-2024/28	ejde-2024/28	DET
ejde-627	155	31	solutions	solution	NOUN
ejde-627	155	32	of	of	ADP
ejde-627	155	33	linear	linear	PROPN
ejde-627	155	34	and	and	CCONJ
ejde-627	155	35	non	non	ADJ
ejde-627	155	36	-	-	ADJ
ejde-627	155	37	linear	linear	ADJ
ejde-627	155	38	pdes	pde	NOUN
ejde-627	155	39	7	7	NUM
ejde-627	155	40	(	(	PUNCT
ejde-627	155	41	c	c	NOUN
ejde-627	155	42	)	)	PUNCT
ejde-627	155	43	q′(y	q′(y	X
ejde-627	155	44	)	)	PUNCT
ejde-627	155	45	=	=	SYM
ejde-627	155	46	yq(y	yq(y	NOUN
ejde-627	155	47	)	)	PUNCT
ejde-627	155	48	.	.	PUNCT
ejde-627	156	1	by	by	ADP
ejde-627	156	2	assuming	assume	VERB
ejde-627	156	3	both	both	DET
ejde-627	156	4	situations	situation	NOUN
ejde-627	156	5	(	(	PUNCT
ejde-627	156	6	a	a	X
ejde-627	156	7	)	)	PUNCT
ejde-627	156	8	and	and	CCONJ
ejde-627	156	9	(	(	PUNCT
ejde-627	156	10	b	b	NOUN
ejde-627	156	11	)	)	PUNCT
ejde-627	156	12	,	,	PUNCT
ejde-627	156	13	we	we	PRON
ejde-627	156	14	have	have	VERB
ejde-627	156	15	q′(y	q′(y	PRON
ejde-627	156	16	)	)	PUNCT
ejde-627	156	17	=	=	SYM
ejde-627	156	18	yq(y	yq(y	NOUN
ejde-627	156	19	)	)	PUNCT
ejde-627	156	20	which	which	PRON
ejde-627	156	21	together	together	ADV
ejde-627	156	22	with	with	ADP
ejde-627	156	23	conditions	condition	NOUN
ejde-627	156	24	(	(	PUNCT
ejde-627	156	25	4.15	4.15	NUM
ejde-627	156	26	)	)	PUNCT
ejde-627	156	27	yields	yield	NOUN
ejde-627	156	28	,	,	PUNCT
ejde-627	156	29	q(y	q(y	PROPN
ejde-627	156	30	)	)	PUNCT
ejde-627	156	31	=	=	PUNCT
ejde-627	156	32	ey	ey	X
ejde-627	156	33	2/2	2/2	NUM
ejde-627	156	34	.	.	PUNCT
ejde-627	157	1	but	but	CCONJ
ejde-627	157	2	situation	situation	NOUN
ejde-627	157	3	(	(	PUNCT
ejde-627	157	4	a	a	X
ejde-627	157	5	)	)	PUNCT
ejde-627	157	6	gives	give	VERB
ejde-627	157	7	q(y	q(y	NOUN
ejde-627	157	8	)	)	PUNCT
ejde-627	157	9	=	=	SYM
ejde-627	158	1	ey	ey	PROPN
ejde-627	158	2	.	.	PROPN
ejde-627	158	3	therefore	therefore	ADV
ejde-627	158	4	,	,	PUNCT
ejde-627	158	5	according	accord	VERB
ejde-627	158	6	to	to	ADP
ejde-627	158	7	theorem	theorem	VERB
ejde-627	158	8	2.2	2.2	NUM
ejde-627	158	9	there	there	PRON
ejde-627	158	10	is	be	VERB
ejde-627	158	11	no	no	DET
ejde-627	158	12	atomic	atomic	ADJ
ejde-627	158	13	solution	solution	NOUN
ejde-627	158	14	for	for	ADP
ejde-627	158	15	case	case	NOUN
ejde-627	158	16	(	(	PUNCT
ejde-627	158	17	ii	ii	NOUN
ejde-627	158	18	)	)	PUNCT
ejde-627	158	19	and	and	CCONJ
ejde-627	158	20	hence	hence	ADV
ejde-627	158	21	,	,	PUNCT
ejde-627	158	22	this	this	DET
ejde-627	158	23	case	case	NOUN
ejde-627	158	24	does	do	AUX
ejde-627	158	25	not	not	PART
ejde-627	158	26	admit	admit	VERB
ejde-627	158	27	an	an	DET
ejde-627	158	28	atomic	atomic	ADJ
ejde-627	158	29	solution	solution	NOUN
ejde-627	158	30	.	.	PUNCT
ejde-627	159	1	therefore	therefore	ADV
ejde-627	159	2	,	,	PUNCT
ejde-627	159	3	(	(	PUNCT
ejde-627	159	4	4.11	4.11	NUM
ejde-627	159	5	)	)	PUNCT
ejde-627	159	6	has	have	VERB
ejde-627	159	7	only	only	ADV
ejde-627	159	8	one	one	NUM
ejde-627	159	9	atomic	atomic	ADJ
ejde-627	159	10	solution	solution	NOUN
ejde-627	159	11	,	,	PUNCT
ejde-627	159	12	that	that	ADV
ejde-627	159	13	is	is	ADV
ejde-627	159	14	(	(	PUNCT
ejde-627	159	15	4.18	4.18	NUM
ejde-627	159	16	)	)	PUNCT
ejde-627	159	17	.	.	PUNCT
ejde-627	160	1	5	5	X
ejde-627	160	2	.	.	X
ejde-627	160	3	conclusions	conclusion	NOUN
ejde-627	160	4	this	this	DET
ejde-627	160	5	article	article	NOUN
ejde-627	160	6	has	have	AUX
ejde-627	160	7	introduced	introduce	VERB
ejde-627	160	8	a	a	DET
ejde-627	160	9	new	new	ADJ
ejde-627	160	10	analytical	analytical	ADJ
ejde-627	160	11	method	method	NOUN
ejde-627	160	12	for	for	ADP
ejde-627	160	13	handling	handle	VERB
ejde-627	160	14	non	non	ADJ
ejde-627	160	15	-	-	ADJ
ejde-627	160	16	separable	separable	ADJ
ejde-627	160	17	,	,	PUNCT
ejde-627	160	18	linear	linear	ADJ
ejde-627	160	19	and	and	CCONJ
ejde-627	160	20	non	non	ADJ
ejde-627	160	21	-	-	ADJ
ejde-627	160	22	linear	linear	ADJ
ejde-627	160	23	partial	partial	ADJ
ejde-627	160	24	differential	differential	NOUN
ejde-627	160	25	equations	equation	NOUN
ejde-627	160	26	via	via	ADP
ejde-627	160	27	atomic	atomic	ADJ
ejde-627	160	28	solutions	solution	NOUN
ejde-627	160	29	method	method	NOUN
ejde-627	160	30	.	.	PUNCT
ejde-627	161	1	the	the	DET
ejde-627	161	2	theory	theory	NOUN
ejde-627	161	3	of	of	ADP
ejde-627	161	4	tensor	tensor	NOUN
ejde-627	161	5	product	product	NOUN
ejde-627	161	6	of	of	ADP
ejde-627	161	7	banach	banach	NOUN
ejde-627	161	8	spaces	space	NOUN
ejde-627	161	9	coupled	couple	VERB
ejde-627	161	10	with	with	ADP
ejde-627	161	11	some	some	DET
ejde-627	161	12	properties	property	NOUN
ejde-627	161	13	of	of	ADP
ejde-627	161	14	atoms	atom	NOUN
ejde-627	161	15	operators	operator	NOUN
ejde-627	161	16	have	have	AUX
ejde-627	161	17	been	be	AUX
ejde-627	161	18	utilized	utilize	VERB
ejde-627	161	19	for	for	ADP
ejde-627	161	20	achieving	achieve	VERB
ejde-627	161	21	such	such	DET
ejde-627	161	22	a	a	DET
ejde-627	161	23	notion	notion	NOUN
ejde-627	161	24	.	.	PUNCT
ejde-627	162	1	for	for	ADP
ejde-627	162	2	atomic	atomic	ADJ
ejde-627	162	3	solution	solution	NOUN
ejde-627	162	4	method	method	NOUN
ejde-627	162	5	,	,	PUNCT
ejde-627	162	6	we	we	PRON
ejde-627	162	7	emphasize	emphasize	VERB
ejde-627	162	8	the	the	DET
ejde-627	162	9	following	following	ADJ
ejde-627	162	10	points	point	NOUN
ejde-627	162	11	:	:	PUNCT
ejde-627	163	1	1	1	X
ejde-627	163	2	.	.	X
ejde-627	163	3	in	in	ADP
ejde-627	163	4	most	most	ADJ
ejde-627	163	5	cases	case	NOUN
ejde-627	163	6	,	,	PUNCT
ejde-627	163	7	the	the	DET
ejde-627	163	8	atomic	atomic	ADJ
ejde-627	163	9	solution	solution	NOUN
ejde-627	163	10	approach	approach	NOUN
ejde-627	163	11	can	can	AUX
ejde-627	163	12	provide	provide	VERB
ejde-627	163	13	an	an	DET
ejde-627	163	14	exact	exact	ADJ
ejde-627	163	15	solutions	solution	NOUN
ejde-627	163	16	to	to	ADP
ejde-627	163	17	non	non	ADJ
ejde-627	163	18	-	-	ADJ
ejde-627	163	19	separable	separable	ADJ
ejde-627	163	20	,	,	PUNCT
ejde-627	163	21	non	non	ADJ
ejde-627	163	22	-	-	ADJ
ejde-627	163	23	homogeneous	homogeneous	ADJ
ejde-627	163	24	,	,	PUNCT
ejde-627	163	25	and	and	CCONJ
ejde-627	163	26	non	non	ADJ
ejde-627	163	27	-	-	ADJ
ejde-627	163	28	linear	linear	ADJ
ejde-627	163	29	partial	partial	ADJ
ejde-627	163	30	differential	differential	NOUN
ejde-627	163	31	equations	equation	NOUN
ejde-627	163	32	when	when	SCONJ
ejde-627	163	33	the	the	DET
ejde-627	163	34	method	method	NOUN
ejde-627	163	35	of	of	ADP
ejde-627	163	36	separation	separation	NOUN
ejde-627	163	37	of	of	ADP
ejde-627	163	38	variables	variable	NOUN
ejde-627	163	39	does	do	AUX
ejde-627	163	40	not	not	PART
ejde-627	163	41	work	work	VERB
ejde-627	163	42	.	.	PUNCT
ejde-627	164	1	2	2	X
ejde-627	164	2	.	.	X
ejde-627	164	3	it	it	PRON
ejde-627	164	4	is	be	AUX
ejde-627	164	5	not	not	PART
ejde-627	164	6	necessary	necessary	ADJ
ejde-627	164	7	that	that	SCONJ
ejde-627	164	8	each	each	DET
ejde-627	164	9	case	case	NOUN
ejde-627	164	10	reported	report	VERB
ejde-627	164	11	in	in	ADP
ejde-627	164	12	theorem	theorem	ADJ
ejde-627	164	13	2.3	2.3	NUM
ejde-627	164	14	admits	admit	VERB
ejde-627	164	15	an	an	DET
ejde-627	164	16	atomic	atomic	ADJ
ejde-627	164	17	solution	solution	NOUN
ejde-627	164	18	.	.	PUNCT
ejde-627	165	1	this	this	PRON
ejde-627	165	2	means	mean	VERB
ejde-627	165	3	that	that	SCONJ
ejde-627	165	4	the	the	DET
ejde-627	165	5	three	three	NUM
ejde-627	165	6	situations	situation	NOUN
ejde-627	165	7	of	of	ADP
ejde-627	165	8	each	each	DET
ejde-627	165	9	case	case	NOUN
ejde-627	165	10	have	have	VERB
ejde-627	165	11	to	to	PART
ejde-627	165	12	provide	provide	VERB
ejde-627	165	13	the	the	DET
ejde-627	165	14	same	same	ADJ
ejde-627	165	15	result	result	NOUN
ejde-627	165	16	;	;	PUNCT
ejde-627	165	17	otherwise	otherwise	ADV
ejde-627	165	18	,	,	PUNCT
ejde-627	165	19	there	there	PRON
ejde-627	165	20	is	be	VERB
ejde-627	165	21	no	no	DET
ejde-627	165	22	atomic	atomic	ADJ
ejde-627	165	23	solution	solution	NOUN
ejde-627	165	24	.	.	PUNCT
ejde-627	166	1	acknowledgments	acknowledgment	NOUN
ejde-627	166	2	.	.	PUNCT
ejde-627	167	1	the	the	DET
ejde-627	167	2	author	author	NOUN
ejde-627	167	3	would	would	AUX
ejde-627	167	4	like	like	VERB
ejde-627	167	5	to	to	PART
ejde-627	167	6	thank	thank	VERB
ejde-627	167	7	professor	professor	PROPN
ejde-627	167	8	r.	r.	PROPN
ejde-627	167	9	khalil	khalil	PROPN
ejde-627	167	10	for	for	ADP
ejde-627	167	11	introducing	introduce	VERB
ejde-627	167	12	him	he	PRON
ejde-627	167	13	to	to	ADP
ejde-627	167	14	the	the	DET
ejde-627	167	15	subject	subject	NOUN
ejde-627	167	16	of	of	ADP
ejde-627	167	17	atomic	atomic	ADJ
ejde-627	167	18	solution	solution	NOUN
ejde-627	167	19	.	.	PUNCT
ejde-627	168	1	the	the	DET
ejde-627	168	2	author	author	NOUN
ejde-627	168	3	is	be	AUX
ejde-627	168	4	thankful	thankful	ADJ
ejde-627	168	5	to	to	ADP
ejde-627	168	6	the	the	DET
ejde-627	168	7	editor	editor	NOUN
ejde-627	168	8	professor	professor	PROPN
ejde-627	168	9	j.	j.	PROPN
ejde-627	168	10	goldstein	goldstein	PROPN
ejde-627	168	11	for	for	ADP
ejde-627	168	12	his	his	PRON
ejde-627	168	13	support	support	NOUN
ejde-627	168	14	and	and	CCONJ
ejde-627	168	15	kindness	kindness	NOUN
ejde-627	168	16	.	.	PUNCT
ejde-627	169	1	also	also	ADV
ejde-627	169	2	,	,	PUNCT
ejde-627	169	3	the	the	DET
ejde-627	169	4	author	author	NOUN
ejde-627	169	5	would	would	AUX
ejde-627	169	6	like	like	VERB
ejde-627	169	7	to	to	PART
ejde-627	169	8	extend	extend	VERB
ejde-627	169	9	his	his	PRON
ejde-627	169	10	sincere	sincere	ADJ
ejde-627	169	11	gratitude	gratitude	NOUN
ejde-627	169	12	to	to	ADP
ejde-627	169	13	the	the	DET
ejde-627	169	14	anonymous	anonymous	ADJ
ejde-627	169	15	reviewers	reviewer	NOUN
ejde-627	169	16	for	for	ADP
ejde-627	169	17	their	their	PRON
ejde-627	169	18	valuable	valuable	ADJ
ejde-627	169	19	comments	comment	NOUN
ejde-627	169	20	and	and	CCONJ
ejde-627	169	21	suggestions	suggestion	NOUN
ejde-627	169	22	.	.	PUNCT
ejde-627	170	1	references	reference	NOUN
ejde-627	170	2	[	[	X
ejde-627	170	3	1	1	NUM
ejde-627	170	4	]	]	PUNCT
ejde-627	170	5	w.	w.	PROPN
ejde-627	170	6	g.	g.	PROPN
ejde-627	170	7	alshanti	alshanti	PROPN
ejde-627	170	8	,	,	PUNCT
ejde-627	170	9	i.	i.	PROPN
ejde-627	170	10	m.	m.	PROPN
ejde-627	170	11	batiha	batiha	PROPN
ejde-627	170	12	,	,	PUNCT
ejde-627	170	13	a.	a.	NOUN
ejde-627	170	14	alshanty	alshanty	PROPN
ejde-627	170	15	;	;	PUNCT
ejde-627	170	16	atomic	atomic	ADJ
ejde-627	170	17	solutions	solution	NOUN
ejde-627	170	18	of	of	ADP
ejde-627	170	19	partial	partial	ADJ
ejde-627	170	20	differential	differential	ADJ
ejde-627	170	21	equations	equation	NOUN
ejde-627	170	22	via	via	ADP
ejde-627	170	23	tensor	tensor	NOUN
ejde-627	170	24	product	product	NOUN
ejde-627	170	25	theory	theory	NOUN
ejde-627	170	26	of	of	ADP
ejde-627	170	27	banach	banach	NOUN
ejde-627	170	28	spaces	space	NOUN
ejde-627	170	29	.	.	PUNCT
ejde-627	171	1	contemp	contemp	NOUN
ejde-627	171	2	.	.	PUNCT
ejde-627	172	1	math	math	NOUN
ejde-627	172	2	.	.	PUNCT
ejde-627	173	1	,	,	PUNCT
ejde-627	173	2	4	4	NUM
ejde-627	173	3	2	2	NUM
ejde-627	173	4	(	(	PUNCT
ejde-627	173	5	2023	2023	NUM
ejde-627	173	6	)	)	PUNCT
ejde-627	173	7	,	,	PUNCT
ejde-627	173	8	286–295	286–295	NUM
ejde-627	173	9	.	.	PUNCT
ejde-627	174	1	[	[	X
ejde-627	174	2	2	2	NUM
ejde-627	174	3	]	]	PUNCT
ejde-627	174	4	w.	w.	PROPN
ejde-627	174	5	g.	g.	PROPN
ejde-627	174	6	alshanti	alshanti	PROPN
ejde-627	174	7	,	,	PUNCT
ejde-627	174	8	i.	i.	PROPN
ejde-627	174	9	m.	m.	PROPN
ejde-627	174	10	batiha	batiha	PROPN
ejde-627	174	11	,	,	PUNCT
ejde-627	174	12	m.	m.	NOUN
ejde-627	174	13	abu	abu	PROPN
ejde-627	174	14	hammad	hammad	PROPN
ejde-627	174	15	,	,	PUNCT
ejde-627	174	16	r.	r.	PROPN
ejde-627	174	17	khalil	khalil	PROPN
ejde-627	174	18	;	;	PUNCT
ejde-627	174	19	a	a	DET
ejde-627	174	20	novel	novel	ADJ
ejde-627	174	21	analytical	analytical	ADJ
ejde-627	174	22	approach	approach	NOUN
ejde-627	174	23	for	for	ADP
ejde-627	174	24	solving	solve	VERB
ejde-627	174	25	partial	partial	ADJ
ejde-627	174	26	differential	differential	ADJ
ejde-627	174	27	equations	equation	NOUN
ejde-627	174	28	via	via	ADP
ejde-627	174	29	a	a	DET
ejde-627	174	30	tensor	tensor	NOUN
ejde-627	174	31	product	product	NOUN
ejde-627	174	32	theory	theory	NOUN
ejde-627	174	33	of	of	ADP
ejde-627	174	34	banach	banach	NOUN
ejde-627	174	35	spaces	space	NOUN
ejde-627	174	36	.	.	PUNCT
ejde-627	175	1	partial	partial	ADJ
ejde-627	175	2	differ	differ	VERB
ejde-627	175	3	.	.	PUNCT
ejde-627	176	1	equ	equ	PROPN
ejde-627	176	2	.	.	PUNCT
ejde-627	176	3	appl	appl	PROPN
ejde-627	176	4	.	.	PROPN
ejde-627	176	5	math	math	PROPN
ejde-627	176	6	.	.	PUNCT
ejde-627	177	1	,	,	PUNCT
ejde-627	177	2	8	8	NUM
ejde-627	177	3	(	(	PUNCT
ejde-627	177	4	2023	2023	NUM
ejde-627	177	5	)	)	PUNCT
ejde-627	177	6	,	,	PUNCT
ejde-627	177	7	100531	100531	NUM
ejde-627	177	8	.	.	PUNCT
ejde-627	178	1	[	[	X
ejde-627	178	2	3	3	X
ejde-627	178	3	]	]	X
ejde-627	178	4	w.	w.	PROPN
ejde-627	178	5	g.	g.	PROPN
ejde-627	178	6	alshanti	alshanti	PROPN
ejde-627	178	7	,	,	PUNCT
ejde-627	178	8	i.	i.	PROPN
ejde-627	178	9	m.	m.	PROPN
ejde-627	178	10	batiha	batiha	PROPN
ejde-627	178	11	,	,	PUNCT
ejde-627	178	12	a.	a.	PROPN
ejde-627	178	13	alshanty	alshanty	PROPN
ejde-627	178	14	,	,	PUNCT
ejde-627	178	15	r.	r.	PROPN
ejde-627	178	16	khalil	khalil	PROPN
ejde-627	178	17	;	;	PUNCT
ejde-627	178	18	tensor	tensor	NOUN
ejde-627	178	19	product	product	NOUN
ejde-627	178	20	of	of	ADP
ejde-627	178	21	banach	banach	NOUN
ejde-627	178	22	spaces	space	NOUN
ejde-627	178	23	and	and	CCONJ
ejde-627	178	24	atomic	atomic	ADJ
ejde-627	178	25	solutions	solution	NOUN
ejde-627	178	26	of	of	ADP
ejde-627	178	27	partial	partial	ADJ
ejde-627	178	28	differential	differential	NOUN
ejde-627	178	29	equations	equation	NOUN
ejde-627	178	30	.	.	PUNCT
ejde-627	179	1	int	int	NOUN
ejde-627	179	2	.	.	PUNCT
ejde-627	180	1	j.	j.	PROPN
ejde-627	180	2	math	math	PROPN
ejde-627	180	3	.	.	PUNCT
ejde-627	181	1	comput	comput	NOUN
ejde-627	181	2	.	.	PUNCT
ejde-627	182	1	sci	sci	PROPN
ejde-627	182	2	.	.	PROPN
ejde-627	182	3	,	,	PUNCT
ejde-627	182	4	19(3	19(3	NUM
ejde-627	182	5	)	)	PUNCT
ejde-627	182	6	(	(	PUNCT
ejde-627	182	7	2024	2024	NUM
ejde-627	182	8	)	)	PUNCT
ejde-627	182	9	,	,	PUNCT
ejde-627	182	10	903	903	NUM
ejde-627	182	11	-	-	SYM
ejde-627	182	12	16	16	NUM
ejde-627	182	13	.	.	PUNCT
ejde-627	183	1	[	[	X
ejde-627	183	2	4	4	NUM
ejde-627	183	3	]	]	X
ejde-627	183	4	i.	i.	NOUN
ejde-627	183	5	batiha	batiha	PROPN
ejde-627	183	6	,	,	PUNCT
ejde-627	183	7	s.	s.	PROPN
ejde-627	183	8	njadat	njadat	PROPN
ejde-627	183	9	,	,	PUNCT
ejde-627	183	10	r.	r.	PROPN
ejde-627	183	11	batyha	batyha	PROPN
ejde-627	183	12	,	,	PUNCT
ejde-627	183	13	a.	a.	NOUN
ejde-627	183	14	zraiqat	zraiqat	PROPN
ejde-627	183	15	,	,	PUNCT
ejde-627	183	16	a.	a.	NOUN
ejde-627	183	17	dababneh	dababneh	PROPN
ejde-627	183	18	,	,	PUNCT
ejde-627	183	19	s.	s.	PROPN
ejde-627	183	20	momani	momani	PROPN
ejde-627	183	21	;	;	PUNCT
ejde-627	183	22	design	design	NOUN
ejde-627	183	23	fractionalorder	fractionalorder	NOUN
ejde-627	183	24	pid	pid	NOUN
ejde-627	183	25	controllers	controller	NOUN
ejde-627	183	26	for	for	ADP
ejde-627	183	27	single	single	ADJ
ejde-627	183	28	-	-	PUNCT
ejde-627	183	29	joint	joint	ADJ
ejde-627	183	30	robot	robot	NOUN
ejde-627	183	31	arm	arm	NOUN
ejde-627	183	32	model	model	PROPN
ejde-627	183	33	.	.	PUNCT
ejde-627	184	1	int	int	NOUN
ejde-627	184	2	.	.	PUNCT
ejde-627	185	1	j.	j.	PROPN
ejde-627	185	2	advance	advance	VERB
ejde-627	185	3	soft	soft	ADJ
ejde-627	185	4	compu	compu	PROPN
ejde-627	185	5	.	.	PUNCT
ejde-627	186	1	appl	appl	PROPN
ejde-627	186	2	.	.	PROPN
ejde-627	186	3	,	,	PUNCT
ejde-627	186	4	14(2	14(2	NUM
ejde-627	186	5	)	)	PUNCT
ejde-627	186	6	(	(	PUNCT
ejde-627	186	7	2022	2022	NUM
ejde-627	186	8	)	)	PUNCT
ejde-627	186	9	,	,	PUNCT
ejde-627	186	10	97–114	97–114	NUM
ejde-627	186	11	.	.	PUNCT
ejde-627	187	1	[	[	X
ejde-627	187	2	5	5	NUM
ejde-627	187	3	]	]	PUNCT
ejde-627	187	4	i.	i.	NOUN
ejde-627	187	5	batiha	batiha	PROPN
ejde-627	187	6	,	,	PUNCT
ejde-627	187	7	j.	j.	PROPN
ejde-627	187	8	oudetallah	oudetallah	PROPN
ejde-627	187	9	,	,	PUNCT
ejde-627	187	10	a.	a.	NOUN
ejde-627	187	11	ouannas	ouannas	PROPN
ejde-627	187	12	,	,	PUNCT
ejde-627	187	13	a.	a.	PROPN
ejde-627	187	14	al	al	PROPN
ejde-627	187	15	-	-	PUNCT
ejde-627	187	16	nana	nana	PROPN
ejde-627	187	17	,	,	PUNCT
ejde-627	187	18	i.	i.	PROPN
ejde-627	187	19	jebril	jebril	NOUN
ejde-627	187	20	;	;	PUNCT
ejde-627	187	21	tuning	tune	VERB
ejde-627	187	22	the	the	DET
ejde-627	187	23	fractional	fractional	ADJ
ejde-627	187	24	-	-	PUNCT
ejde-627	187	25	order	order	NOUN
ejde-627	187	26	pidcontroller	pidcontroller	NOUN
ejde-627	187	27	for	for	ADP
ejde-627	187	28	blood	blood	NOUN
ejde-627	187	29	glucose	glucose	NOUN
ejde-627	187	30	level	level	NOUN
ejde-627	187	31	of	of	ADP
ejde-627	187	32	diabetic	diabetic	ADJ
ejde-627	187	33	patients	patient	NOUN
ejde-627	187	34	.	.	PUNCT
ejde-627	188	1	int	int	NOUN
ejde-627	188	2	.	.	PUNCT
ejde-627	189	1	j.	j.	PROPN
ejde-627	189	2	advance	advance	VERB
ejde-627	189	3	soft	soft	ADJ
ejde-627	189	4	compu	compu	PROPN
ejde-627	189	5	.	.	PUNCT
ejde-627	190	1	appl	appl	PROPN
ejde-627	190	2	.	.	PROPN
ejde-627	190	3	,	,	PUNCT
ejde-627	190	4	13(2	13(2	PROPN
ejde-627	190	5	)	)	PUNCT
ejde-627	190	6	(	(	PUNCT
ejde-627	190	7	2021	2021	NUM
ejde-627	190	8	)	)	PUNCT
ejde-627	190	9	,	,	PUNCT
ejde-627	190	10	1–10	1–10	NOUN
ejde-627	190	11	.	.	PUNCT
ejde-627	191	1	[	[	X
ejde-627	191	2	6	6	NUM
ejde-627	191	3	]	]	PUNCT
ejde-627	191	4	j.	j.	PROPN
ejde-627	191	5	diestel	diestel	PROPN
ejde-627	191	6	,	,	PUNCT
ejde-627	191	7	j.j	j.j	PROPN
ejde-627	191	8	.	.	PROPN
ejde-627	191	9	uhl	uhl	PROPN
ejde-627	191	10	,	,	PUNCT
ejde-627	191	11	jr	jr	PROPN
ejde-627	191	12	;	;	PUNCT
ejde-627	191	13	vector	vector	NOUN
ejde-627	191	14	measures	measure	NOUN
ejde-627	191	15	,	,	PUNCT
ejde-627	191	16	american	american	PROPN
ejde-627	191	17	mathematical	mathematical	ADJ
ejde-627	191	18	society	society	NOUN
ejde-627	191	19	,	,	PUNCT
ejde-627	191	20	providence	providence	NOUN
ejde-627	191	21	,	,	PUNCT
ejde-627	191	22	rhode	rhode	NOUN
ejde-627	191	23	island	island	NOUN
ejde-627	191	24	,	,	PUNCT
ejde-627	191	25	1977	1977	NUM
ejde-627	191	26	.	.	PUNCT
ejde-627	192	1	[	[	X
ejde-627	192	2	7	7	X
ejde-627	192	3	]	]	X
ejde-627	192	4	l.	l.	PROPN
ejde-627	192	5	evans	evans	PROPN
ejde-627	192	6	;	;	PUNCT
ejde-627	192	7	partial	partial	ADJ
ejde-627	192	8	differential	differential	NOUN
ejde-627	192	9	equations	equation	NOUN
ejde-627	192	10	,	,	PUNCT
ejde-627	192	11	american	american	PROPN
ejde-627	192	12	mathematical	mathematical	ADJ
ejde-627	192	13	society	society	NOUN
ejde-627	192	14	,	,	PUNCT
ejde-627	192	15	providence	providence	NOUN
ejde-627	192	16	,	,	PUNCT
ejde-627	192	17	rhode	rhode	NOUN
ejde-627	192	18	island	island	NOUN
ejde-627	192	19	,	,	PUNCT
ejde-627	192	20	2010	2010	NUM
ejde-627	192	21	.	.	PUNCT
ejde-627	193	1	[	[	X
ejde-627	193	2	8	8	NUM
ejde-627	193	3	]	]	X
ejde-627	193	4	g.	g.	PROPN
ejde-627	193	5	evans	evans	PROPN
ejde-627	193	6	,	,	PUNCT
ejde-627	193	7	j.	j.	PROPN
ejde-627	193	8	blackledge	blackledge	PROPN
ejde-627	193	9	,	,	PUNCT
ejde-627	193	10	p.	p.	PROPN
ejde-627	193	11	yardley	yardley	PROPN
ejde-627	193	12	;	;	PUNCT
ejde-627	193	13	analytic	analytic	ADJ
ejde-627	193	14	methods	method	NOUN
ejde-627	193	15	for	for	ADP
ejde-627	193	16	partial	partial	ADJ
ejde-627	193	17	differential	differential	NOUN
ejde-627	193	18	equations	equation	NOUN
ejde-627	193	19	,	,	PUNCT
ejde-627	193	20	springer	springer	NOUN
ejde-627	193	21	science	science	NOUN
ejde-627	193	22	and	and	CCONJ
ejde-627	193	23	business	business	NOUN
ejde-627	193	24	media	medium	NOUN
ejde-627	193	25	,	,	PUNCT
ejde-627	193	26	berlin	berlin	PROPN
ejde-627	193	27	,	,	PUNCT
ejde-627	193	28	heidelberg	heidelberg	PROPN
ejde-627	193	29	,	,	PUNCT
ejde-627	193	30	2012	2012	NUM
ejde-627	193	31	.	.	PUNCT
ejde-627	194	1	[	[	X
ejde-627	194	2	9	9	NUM
ejde-627	194	3	]	]	PUNCT
ejde-627	194	4	r.	r.	PROPN
ejde-627	194	5	khalil	khalil	PROPN
ejde-627	194	6	;	;	PUNCT
ejde-627	194	7	isometries	isometry	NOUN
ejde-627	194	8	of	of	ADP
ejde-627	194	9	lp∗⊗lp	lp∗⊗lp	PROPN
ejde-627	194	10	.	.	PUNCT
ejde-627	195	1	tam	tam	PROPN
ejde-627	195	2	.	.	PUNCT
ejde-627	196	1	j.	j.	PROPN
ejde-627	196	2	math	math	PROPN
ejde-627	196	3	..	..	PROPN
ejde-627	196	4	16	16	NUM
ejde-627	196	5	(	(	PUNCT
ejde-627	196	6	1985	1985	NUM
ejde-627	196	7	)	)	PUNCT
ejde-627	196	8	,	,	PUNCT
ejde-627	196	9	77–85	77–85	NUM
ejde-627	196	10	.	.	PUNCT
ejde-627	197	1	[	[	X
ejde-627	197	2	10	10	NUM
ejde-627	197	3	]	]	X
ejde-627	197	4	r.	r.	PROPN
ejde-627	197	5	khalil	khalil	PROPN
ejde-627	197	6	,	,	PUNCT
ejde-627	197	7	l.	l.	PROPN
ejde-627	197	8	abdullah	abdullah	PROPN
ejde-627	197	9	;	;	PUNCT
ejde-627	197	10	atomic	atomic	ADJ
ejde-627	197	11	solution	solution	NOUN
ejde-627	197	12	of	of	ADP
ejde-627	197	13	certain	certain	ADJ
ejde-627	197	14	inverse	inverse	NOUN
ejde-627	197	15	problems	problem	NOUN
ejde-627	197	16	.	.	PUNCT
ejde-627	198	1	eur	eur	PROPN
ejde-627	198	2	.	.	PUNCT
ejde-627	199	1	j.	j.	PROPN
ejde-627	199	2	pure	pure	PROPN
ejde-627	199	3	appl	appl	PROPN
ejde-627	199	4	.	.	PUNCT
ejde-627	200	1	math	math	PROPN
ejde-627	200	2	..	..	PUNCT
ejde-627	200	3	34	34	NUM
ejde-627	200	4	(	(	PUNCT
ejde-627	200	5	2010	2010	NUM
ejde-627	200	6	)	)	PUNCT
ejde-627	200	7	,	,	PUNCT
ejde-627	200	8	725–729	725–729	NUM
ejde-627	200	9	.	.	PUNCT
ejde-627	201	1	[	[	X
ejde-627	201	2	11	11	NUM
ejde-627	201	3	]	]	X
ejde-627	201	4	w.	w.	PROPN
ejde-627	201	5	light	light	PROPN
ejde-627	201	6	,	,	PUNCT
ejde-627	201	7	e.	e.	PROPN
ejde-627	201	8	w.	w.	PROPN
ejde-627	201	9	cheney	cheney	PROPN
ejde-627	201	10	;	;	PUNCT
ejde-627	201	11	approximation	approximation	NOUN
ejde-627	201	12	theory	theory	NOUN
ejde-627	201	13	in	in	ADP
ejde-627	201	14	tensor	tensor	NOUN
ejde-627	201	15	product	product	NOUN
ejde-627	201	16	spaces	space	NOUN
ejde-627	201	17	,	,	PUNCT
ejde-627	201	18	springer	springer	NOUN
ejde-627	201	19	science	science	NOUN
ejde-627	201	20	and	and	CCONJ
ejde-627	201	21	business	business	NOUN
ejde-627	201	22	media	medium	NOUN
ejde-627	201	23	,	,	PUNCT
ejde-627	201	24	berlin	berlin	PROPN
ejde-627	201	25	,	,	PUNCT
ejde-627	201	26	heidelberg	heidelberg	PROPN
ejde-627	201	27	,	,	PUNCT
ejde-627	201	28	1985	1985	NUM
ejde-627	201	29	.	.	PUNCT
ejde-627	201	30	8	8	NUM
ejde-627	201	31	w.	w.	PROPN
ejde-627	201	32	g.	g.	PROPN
ejde-627	201	33	alshanti	alshanti	VERB
ejde-627	201	34	ejde-2024/28	ejde-2024/28	X
ejde-627	202	1	[	[	X
ejde-627	202	2	12	12	NUM
ejde-627	202	3	]	]	PUNCT
ejde-627	202	4	j.	j.	PROPN
ejde-627	202	5	peiró	peiró	PROPN
ejde-627	202	6	,	,	PUNCT
ejde-627	202	7	s.	s.	PROPN
ejde-627	202	8	sherwin	sherwin	PROPN
ejde-627	202	9	;	;	PUNCT
ejde-627	202	10	finite	finite	ADJ
ejde-627	202	11	difference	difference	NOUN
ejde-627	202	12	,	,	PUNCT
ejde-627	202	13	finite	finite	PROPN
ejde-627	202	14	element	element	NOUN
ejde-627	202	15	and	and	CCONJ
ejde-627	202	16	finite	finite	ADJ
ejde-627	202	17	volume	volume	NOUN
ejde-627	202	18	methods	method	NOUN
ejde-627	202	19	for	for	ADP
ejde-627	202	20	partial	partial	ADJ
ejde-627	202	21	differential	differential	ADJ
ejde-627	202	22	equations	equation	NOUN
ejde-627	202	23	.	.	PUNCT
ejde-627	203	1	in	in	ADP
ejde-627	203	2	handbook	handbook	NOUN
ejde-627	203	3	of	of	ADP
ejde-627	203	4	materials	material	NOUN
ejde-627	203	5	modeling	modeling	NOUN
ejde-627	203	6	:	:	PUNCT
ejde-627	203	7	methods	method	NOUN
ejde-627	203	8	,	,	PUNCT
ejde-627	203	9	dordrecht	dordrecht	PROPN
ejde-627	203	10	:	:	PUNCT
ejde-627	203	11	springer	springer	PROPN
ejde-627	203	12	netherlands	netherlands	PROPN
ejde-627	203	13	,	,	PUNCT
ejde-627	203	14	2005	2005	NUM
ejde-627	203	15	.	.	PUNCT
ejde-627	204	1	2415–2446	2415–2446	NUM
ejde-627	204	2	[	[	X
ejde-627	204	3	13	13	NUM
ejde-627	204	4	]	]	X
ejde-627	204	5	r.	r.	PROPN
ejde-627	204	6	ryan	ryan	PROPN
ejde-627	204	7	;	;	PUNCT
ejde-627	204	8	introduction	introduction	NOUN
ejde-627	204	9	to	to	ADP
ejde-627	204	10	tensor	tensor	NOUN
ejde-627	204	11	products	product	NOUN
ejde-627	204	12	of	of	ADP
ejde-627	204	13	banach	banach	NOUN
ejde-627	204	14	spaces	space	NOUN
ejde-627	204	15	,	,	PUNCT
ejde-627	204	16	springer	springer	NOUN
ejde-627	204	17	,	,	PUNCT
ejde-627	204	18	london	london	PROPN
ejde-627	204	19	,	,	PUNCT
ejde-627	204	20	2002	2002	NUM
ejde-627	204	21	.	.	PUNCT
ejde-627	205	1	[	[	X
ejde-627	205	2	14	14	NUM
ejde-627	205	3	]	]	X
ejde-627	205	4	f.	f.	PROPN
ejde-627	205	5	c.	c.	PROPN
ejde-627	205	6	sánchez	sánchez	PROPN
ejde-627	205	7	,	,	PUNCT
ejde-627	205	8	r.	r.	PROPN
ejde-627	205	9	garćıa	garćıa	PROPN
ejde-627	205	10	;	;	PUNCT
ejde-627	205	11	the	the	DET
ejde-627	205	12	bidual	bidual	NOUN
ejde-627	205	13	of	of	ADP
ejde-627	205	14	a	a	DET
ejde-627	205	15	tensor	tensor	NOUN
ejde-627	205	16	product	product	NOUN
ejde-627	205	17	of	of	ADP
ejde-627	205	18	banach	banach	NOUN
ejde-627	205	19	spaces	space	NOUN
ejde-627	205	20	.	.	PUNCT
ejde-627	206	1	rev	rev	PROPN
ejde-627	206	2	.	.	PROPN
ejde-627	206	3	mat	mat	PROPN
ejde-627	206	4	.	.	PUNCT
ejde-627	206	5	iberoam	iberoam	PROPN
ejde-627	206	6	.	.	PUNCT
ejde-627	207	1	213	213	NUM
ejde-627	207	2	(	(	PUNCT
ejde-627	207	3	2005	2005	NUM
ejde-627	207	4	)	)	PUNCT
ejde-627	207	5	,	,	PUNCT
ejde-627	207	6	843–861	843–861	NUM
ejde-627	207	7	.	.	PUNCT
ejde-627	208	1	[	[	X
ejde-627	208	2	15	15	NUM
ejde-627	208	3	]	]	X
ejde-627	208	4	r.	r.	PROPN
ejde-627	208	5	schatten	schatten	PROPN
ejde-627	208	6	;	;	PUNCT
ejde-627	208	7	a	a	DET
ejde-627	208	8	theory	theory	NOUN
ejde-627	208	9	of	of	ADP
ejde-627	208	10	cross	cross	NOUN
ejde-627	208	11	-	-	NOUN
ejde-627	208	12	spaces	space	NOUN
ejde-627	208	13	,	,	PUNCT
ejde-627	208	14	princeton	princeton	PROPN
ejde-627	208	15	university	university	PROPN
ejde-627	208	16	press	press	NOUN
ejde-627	208	17	,	,	PUNCT
ejde-627	208	18	new	new	PROPN
ejde-627	208	19	jersey	jersey	PROPN
ejde-627	208	20	,	,	PUNCT
ejde-627	208	21	united	united	PROPN
ejde-627	208	22	states	states	PROPN
ejde-627	208	23	,	,	PUNCT
ejde-627	208	24	1950	1950	NUM
ejde-627	208	25	.	.	PUNCT
ejde-627	209	1	waseem	waseem	PROPN
ejde-627	209	2	ghazi	ghazi	PROPN
ejde-627	209	3	alshanti	alshanti	PROPN
ejde-627	209	4	department	department	PROPN
ejde-627	209	5	of	of	ADP
ejde-627	209	6	mathematics	mathematics	PROPN
ejde-627	209	7	,	,	PUNCT
ejde-627	209	8	al	al	PROPN
ejde-627	209	9	zaytoonah	zaytoonah	PROPN
ejde-627	209	10	university	university	PROPN
ejde-627	209	11	of	of	ADP
ejde-627	209	12	jordan	jordan	PROPN
ejde-627	209	13	,	,	PUNCT
ejde-627	209	14	queen	queen	PROPN
ejde-627	209	15	alia	alia	PROPN
ejde-627	209	16	airport	airport	PROPN
ejde-627	209	17	st	st	PROPN
ejde-627	210	1	594	594	NUM
ejde-627	210	2	,	,	PUNCT
ejde-627	210	3	amman	amman	PROPN
ejde-627	210	4	11733	11733	NUM
ejde-627	210	5	,	,	PUNCT
ejde-627	210	6	jordan	jordan	PROPN
ejde-627	210	7	email	email	PROPN
ejde-627	210	8	address	address	PROPN
ejde-627	210	9	:	:	PUNCT
ejde-627	210	10	w.alshanti@zuj.edu.jo	w.alshanti@zuj.edu.jo	PROPN
ejde-627	210	11	1	1	X
ejde-627	210	12	.	.	PUNCT
ejde-627	210	13	introduction	introduction	NOUN
ejde-627	210	14	2	2	NUM
ejde-627	210	15	.	.	PUNCT
ejde-627	210	16	atoms	atom	NOUN
ejde-627	210	17	operators	operator	NOUN
ejde-627	210	18	3	3	X
ejde-627	210	19	.	.	PUNCT
ejde-627	210	20	general	general	ADJ
ejde-627	210	21	scheme	scheme	NOUN
ejde-627	210	22	for	for	ADP
ejde-627	210	23	the	the	DET
ejde-627	210	24	atomic	atomic	ADJ
ejde-627	210	25	solution	solution	NOUN
ejde-627	210	26	method	method	NOUN
ejde-627	210	27	4	4	NUM
ejde-627	210	28	.	.	PUNCT
ejde-627	210	29	applications	application	NOUN
ejde-627	210	30	5	5	NUM
ejde-627	210	31	.	.	PUNCT
ejde-627	211	1	conclusions	conclusion	NOUN
ejde-627	211	2	acknowledgments	acknowledgment	NOUN
ejde-627	211	3	references	reference	NOUN