id sid tid token lemma pos cana-4512 1 1 communications communication NOUN cana-4512 1 2 on on ADP cana-4512 1 3 applied apply VERB cana-4512 1 4 nonlinear nonlinear ADJ cana-4512 1 5 analysis analysis NOUN cana-4512 1 6 issn issn NOUN cana-4512 1 7 : : PUNCT cana-4512 1 8 1074 1074 NUM cana-4512 1 9 - - PUNCT cana-4512 1 10 133x 133x NUM cana-4512 1 11 vol vol NOUN cana-4512 1 12 32 32 NUM cana-4512 1 13 no no NOUN cana-4512 1 14 . . PUNCT cana-4512 2 1 9s 9s NUM cana-4512 2 2 ( ( PUNCT cana-4512 2 3 2025 2025 NUM cana-4512 2 4 ) ) PUNCT cana-4512 2 5 2271 2271 NUM cana-4512 2 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 2 7 fredholm fredholm NOUN cana-4512 2 8 integral integral ADJ cana-4512 2 9 equation equation NOUN cana-4512 2 10 solving solve VERB cana-4512 2 11 by by ADP cana-4512 2 12 different different ADJ cana-4512 2 13 methods method NOUN cana-4512 2 14 poonamjagtap1 poonamjagtap1 NOUN cana-4512 2 15 , , PUNCT cana-4512 2 16 avinash avinash PROPN cana-4512 2 17 khambayat2 khambayat2 PROPN cana-4512 3 1 1research 1research NUM cana-4512 3 2 scholars scholar NOUN cana-4512 3 3 , , PUNCT cana-4512 3 4 sos sos PROPN cana-4512 3 5 , , PUNCT cana-4512 3 6 sandip sandip PROPN cana-4512 3 7 university university NOUN cana-4512 3 8 nashik nashik PROPN cana-4512 3 9 , , PUNCT cana-4512 3 10 maharashtra maharashtra PROPN cana-4512 3 11 , , PUNCT cana-4512 3 12 india india PROPN cana-4512 3 13 poonamjagtap2008@gmail.com poonamjagtap2008@gmail.com X cana-4512 4 1 2sos 2sos NUM cana-4512 4 2 , , PUNCT cana-4512 4 3 sandip sandip PROPN cana-4512 4 4 university university NOUN cana-4512 4 5 , , PUNCT cana-4512 4 6 nashik nashik PROPN cana-4512 4 7 , , PUNCT cana-4512 4 8 maharashtra maharashtra PROPN cana-4512 4 9 , , PUNCT cana-4512 4 10 india india PROPN cana-4512 4 11 avinash.khambayat@sandipuniversity.edu.in avinash.khambayat@sandipuniversity.edu.in PROPN cana-4512 4 12 article article PROPN cana-4512 4 13 history history NOUN cana-4512 4 14 : : PUNCT cana-4512 4 15 received receive VERB cana-4512 4 16 : : PUNCT cana-4512 4 17 12 12 NUM cana-4512 4 18 - - SYM cana-4512 4 19 01 01 NUM cana-4512 4 20 - - PUNCT cana-4512 4 21 2025 2025 NUM cana-4512 4 22 revised revise VERB cana-4512 4 23 : : PUNCT cana-4512 4 24 15 15 NUM cana-4512 4 25 - - NUM cana-4512 4 26 02 02 NUM cana-4512 4 27 - - PUNCT cana-4512 4 28 2025 2025 NUM cana-4512 4 29 accepted accept VERB cana-4512 4 30 : : PUNCT cana-4512 4 31 01 01 NUM cana-4512 4 32 - - SYM cana-4512 4 33 03 03 NUM cana-4512 4 34 - - PUNCT cana-4512 4 35 2025 2025 NUM cana-4512 4 36 abstract abstract NOUN cana-4512 4 37 : : PUNCT cana-4512 4 38 for for ADP cana-4512 4 39 many many ADJ cana-4512 4 40 branches branch NOUN cana-4512 4 41 of of ADP cana-4512 4 42 applied apply VERB cana-4512 4 43 mathematics mathematic NOUN cana-4512 4 44 , , PUNCT cana-4512 4 45 integral integral ADJ cana-4512 4 46 equations equation NOUN cana-4512 4 47 have have AUX cana-4512 4 48 proven prove VERB cana-4512 4 49 essential essential ADJ cana-4512 4 50 tools tool NOUN cana-4512 4 51 . . PUNCT cana-4512 5 1 this this DET cana-4512 5 2 article article NOUN cana-4512 5 3 examines examine VERB cana-4512 5 4 various various ADJ cana-4512 5 5 numerical numerical ADJ cana-4512 5 6 techniques technique NOUN cana-4512 5 7 for for ADP cana-4512 5 8 resolving resolve VERB cana-4512 5 9 fredholm fredholm ADJ cana-4512 5 10 integral integral ADJ cana-4512 5 11 problems problem NOUN cana-4512 5 12 . . PUNCT cana-4512 6 1 the the DET cana-4512 6 2 purpose purpose NOUN cana-4512 6 3 is be AUX cana-4512 6 4 to to PART cana-4512 6 5 classify classify VERB cana-4512 6 6 chosen choose VERB cana-4512 6 7 techniques technique NOUN cana-4512 6 8 and and CCONJ cana-4512 6 9 evaluate evaluate VERB cana-4512 6 10 their their PRON cana-4512 6 11 efficacy efficacy NOUN cana-4512 6 12 . . PUNCT cana-4512 7 1 integral integral ADJ cana-4512 7 2 equations equation NOUN cana-4512 7 3 are be AUX cana-4512 7 4 one one NUM cana-4512 7 5 of of ADP cana-4512 7 6 the the DET cana-4512 7 7 most most ADV cana-4512 7 8 significant significant ADJ cana-4512 7 9 subfields subfield NOUN cana-4512 7 10 of of ADP cana-4512 7 11 mathematical mathematical ADJ cana-4512 7 12 analysis analysis NOUN cana-4512 7 13 in in ADP cana-4512 7 14 a a DET cana-4512 7 15 number number NOUN cana-4512 7 16 of of ADP cana-4512 7 17 branches branch NOUN cana-4512 7 18 of of ADP cana-4512 7 19 mathematical mathematical ADJ cana-4512 7 20 physics physics NOUN cana-4512 7 21 and and CCONJ cana-4512 7 22 mechanics mechanic NOUN cana-4512 7 23 . . PUNCT cana-4512 8 1 this this DET cana-4512 8 2 research research NOUN cana-4512 8 3 will will AUX cana-4512 8 4 discuss discuss VERB cana-4512 8 5 the the DET cana-4512 8 6 fredholm fredholm ADJ cana-4512 8 7 integral integral ADJ cana-4512 8 8 equation equation NOUN cana-4512 8 9 , , PUNCT cana-4512 8 10 its its PRON cana-4512 8 11 solutions solution NOUN cana-4512 8 12 , , PUNCT cana-4512 8 13 and and CCONJ cana-4512 8 14 its its PRON cana-4512 8 15 uses use NOUN cana-4512 8 16 . . PUNCT cana-4512 9 1 keywords keyword NOUN cana-4512 9 2 : : PUNCT cana-4512 9 3 fredholm fredholm VERB cana-4512 9 4 integral integral ADJ cana-4512 9 5 equations equation NOUN cana-4512 9 6 ( ( PUNCT cana-4512 9 7 fie fie NOUN cana-4512 9 8 ) ) PUNCT cana-4512 9 9 , , PUNCT cana-4512 9 10 first first ADJ cana-4512 9 11 kind kind NOUN cana-4512 9 12 , , PUNCT cana-4512 9 13 second second ADJ cana-4512 9 14 kind.direct kind.direct ADJ cana-4512 9 15 computation computation NOUN cana-4512 9 16 method method NOUN cana-4512 9 17 ( ( PUNCT cana-4512 9 18 dcm dcm PROPN cana-4512 9 19 ) ) PUNCT cana-4512 9 20 . . PUNCT cana-4512 10 1 introduction introduction NOUN cana-4512 10 2 : : PUNCT cana-4512 10 3 numerous numerous ADJ cana-4512 10 4 scientific scientific ADJ cana-4512 10 5 and and CCONJ cana-4512 10 6 engineering engineering NOUN cana-4512 10 7 fields field NOUN cana-4512 10 8 naturally naturally ADV cana-4512 10 9 involve involve VERB cana-4512 10 10 integral integral ADJ cana-4512 10 11 equations equation NOUN cana-4512 10 12 [ [ X cana-4512 10 13 1–5 1–5 X cana-4512 10 14 ] ] X cana-4512 10 15 . . PUNCT cana-4512 11 1 a a DET cana-4512 11 2 computer computer NOUN cana-4512 11 3 approach approach NOUN cana-4512 11 4 to to ADP cana-4512 11 5 solving solve VERB cana-4512 11 6 integral integral ADJ cana-4512 11 7 equations equation NOUN cana-4512 11 8 is be AUX cana-4512 11 9 essential essential ADJ cana-4512 11 10 in in ADP cana-4512 11 11 scientific scientific ADJ cana-4512 11 12 research research NOUN cana-4512 11 13 . . PUNCT cana-4512 12 1 integral integral ADJ cana-4512 12 2 equations equation NOUN cana-4512 12 3 are be AUX cana-4512 12 4 widely widely ADV cana-4512 12 5 used use VERB cana-4512 12 6 in in ADP cana-4512 12 7 many many ADJ cana-4512 12 8 different different ADJ cana-4512 12 9 fields field NOUN cana-4512 12 10 , , PUNCT cana-4512 12 11 including include VERB cana-4512 12 12 : : PUNCT cana-4512 12 13 , , PUNCT cana-4512 12 14 electricity electricity NOUN cana-4512 12 15 and and CCONJ cana-4512 12 16 magnetism magnetism NOUN cana-4512 12 17 , , PUNCT cana-4512 12 18 kinetic kinetic ADJ cana-4512 12 19 theory theory NOUN cana-4512 12 20 of of ADP cana-4512 12 21 gases gas NOUN cana-4512 12 22 , , PUNCT cana-4512 12 23 geophysics geophysic NOUN cana-4512 12 24 and and CCONJ cana-4512 12 25 biology biology NOUN cana-4512 12 26 , , PUNCT cana-4512 12 27 radiation radiation NOUN cana-4512 12 28 , , PUNCT cana-4512 12 29 optimization optimization NOUN cana-4512 12 30 , , PUNCT cana-4512 12 31 mathematical mathematical ADJ cana-4512 12 32 economics economic NOUN cana-4512 12 33 , , PUNCT cana-4512 12 34 population population NOUN cana-4512 12 35 genetics genetic NOUN cana-4512 12 36 , , PUNCT cana-4512 12 37 queuing queue VERB cana-4512 12 38 theory theory NOUN cana-4512 12 39 , , PUNCT cana-4512 12 40 medicine medicine NOUN cana-4512 12 41 , , PUNCT cana-4512 12 42 optimal optimal ADJ cana-4512 12 43 control control NOUN cana-4512 12 44 systems system NOUN cana-4512 12 45 , , PUNCT cana-4512 12 46 acoustics acoustic NOUN cana-4512 12 47 , , PUNCT cana-4512 12 48 fluid fluid ADJ cana-4512 12 49 mechanics mechanic NOUN cana-4512 12 50 , , PUNCT cana-4512 12 51 steady steady ADJ cana-4512 12 52 state state NOUN cana-4512 12 53 heat heat NOUN cana-4512 12 54 conduction conduction NOUN cana-4512 12 55 , , PUNCT cana-4512 12 56 fracture fracture NOUN cana-4512 12 57 mechanics mechanic NOUN cana-4512 12 58 , , PUNCT cana-4512 12 59 radiative radiative ADJ cana-4512 12 60 heat heat NOUN cana-4512 12 61 transfer transfer NOUN cana-4512 12 62 problems problem NOUN cana-4512 12 63 , , PUNCT cana-4512 12 64 quantum quantum NOUN cana-4512 12 65 mechanics mechanic NOUN cana-4512 12 66 , , PUNCT cana-4512 12 67 communication communication NOUN cana-4512 12 68 theory theory NOUN cana-4512 12 69 and and CCONJ cana-4512 12 70 many many ADJ cana-4512 12 71 more more ADJ cana-4512 12 72 . . PUNCT cana-4512 13 1 an an DET cana-4512 13 2 integral integral ADJ cana-4512 13 3 equation equation NOUN cana-4512 13 4 of of ADP cana-4512 13 5 extreme extreme ADJ cana-4512 13 6 significance significance NOUN cana-4512 13 7 is be AUX cana-4512 13 8 the the DET cana-4512 13 9 fredholm fredholm ADJ cana-4512 13 10 integral integral ADJ cana-4512 13 11 equation equation NOUN cana-4512 13 12 . . PUNCT cana-4512 14 1 equations equation NOUN cana-4512 14 2 that that PRON cana-4512 14 3 arise arise VERB cana-4512 14 4 from from ADP cana-4512 14 5 the the DET cana-4512 14 6 transformation transformation NOUN cana-4512 14 7 of of ADP cana-4512 14 8 points point NOUN cana-4512 14 9 in in ADP cana-4512 14 10 a a DET cana-4512 14 11 certain certain ADJ cana-4512 14 12 vector vector NOUN cana-4512 14 13 space space NOUN cana-4512 14 14 of of ADP cana-4512 14 15 integrable integrable ADJ cana-4512 14 16 functions function NOUN cana-4512 14 17 to to PART cana-4512 14 18 points point NOUN cana-4512 14 19 in in ADP cana-4512 14 20 the the DET cana-4512 14 21 same same ADJ cana-4512 14 22 space space NOUN cana-4512 14 23 using use VERB cana-4512 14 24 particular particular ADJ cana-4512 14 25 integral integral ADJ cana-4512 14 26 operators operator NOUN cana-4512 14 27 are be AUX cana-4512 14 28 known know VERB cana-4512 14 29 as as ADP cana-4512 14 30 integral integral ADJ cana-4512 14 31 equations equation NOUN cana-4512 14 32 . . PUNCT cana-4512 15 1 there there PRON cana-4512 15 2 are be VERB cana-4512 15 3 a a DET cana-4512 15 4 number number NOUN cana-4512 15 5 of of ADP cana-4512 15 6 approximation approximation NOUN cana-4512 15 7 techniques technique NOUN cana-4512 15 8 for for ADP cana-4512 15 9 solving solve VERB cana-4512 15 10 integral integral ADJ cana-4512 15 11 equations equation NOUN cana-4512 15 12 .. .. PUNCT cana-4512 15 13 solving solve VERB cana-4512 15 14 integral integral ADJ cana-4512 15 15 equations equation NOUN cana-4512 15 16 computationally computationally ADV cana-4512 15 17 is be AUX cana-4512 15 18 an an DET cana-4512 15 19 essential essential ADJ cana-4512 15 20 role role NOUN cana-4512 15 21 in in ADP cana-4512 15 22 scientific scientific ADJ cana-4512 15 23 study study NOUN cana-4512 15 24 . . PUNCT cana-4512 16 1 integral integral ADJ cana-4512 16 2 equations equation NOUN cana-4512 16 3 that that PRON cana-4512 16 4 are be AUX cana-4512 16 5 closely closely ADV cana-4512 16 6 connected connect VERB cana-4512 16 7 to to PART cana-4512 16 8 differential differential ADJ cana-4512 16 9 equations equation NOUN cana-4512 16 10 represent represent VERB cana-4512 16 11 nearly nearly ADV cana-4512 16 12 all all PRON cana-4512 16 13 of of ADP cana-4512 16 14 fredholm fredholm ADJ cana-4512 16 15 integral integral ADJ cana-4512 16 16 equations equation NOUN cana-4512 16 17 [ [ X cana-4512 16 18 6].thus 6].thus NUM cana-4512 16 19 , , PUNCT cana-4512 16 20 boundary boundary ADJ cana-4512 16 21 value value NOUN cana-4512 16 22 problems problem NOUN cana-4512 16 23 for for ADP cana-4512 16 24 differential differential ADJ cana-4512 16 25 equations equation NOUN cana-4512 16 26 are be AUX cana-4512 16 27 the the DET cana-4512 16 28 source source NOUN cana-4512 16 29 of of ADP cana-4512 16 30 fie fie PROPN cana-4512 16 31 , , PUNCT cana-4512 16 32 which which PRON cana-4512 16 33 are be AUX cana-4512 16 34 subsequently subsequently ADV cana-4512 16 35 resolved resolve VERB cana-4512 16 36 using use VERB cana-4512 16 37 a a DET cana-4512 16 38 variety variety NOUN cana-4512 16 39 of of ADP cana-4512 16 40 simplified simplified ADJ cana-4512 16 41 techniques technique NOUN cana-4512 16 42 . . PUNCT cana-4512 17 1 fredholm fredholm NOUN cana-4512 17 2 was be AUX cana-4512 17 3 highly highly ADV cana-4512 17 4 motivated motivated ADJ cana-4512 17 5 to to PART cana-4512 17 6 find find VERB cana-4512 17 7 this this DET cana-4512 17 8 kind kind NOUN cana-4512 17 9 of of ADP cana-4512 17 10 equation equation NOUN cana-4512 17 11 . . PUNCT cana-4512 18 1 after after ADP cana-4512 18 2 being be AUX cana-4512 18 3 found find VERB cana-4512 18 4 by by ADP cana-4512 18 5 fredholm fredholm NOUN cana-4512 18 6 , , PUNCT cana-4512 18 7 the the DET cana-4512 18 8 equations equation NOUN cana-4512 18 9 were be AUX cana-4512 18 10 given give VERB cana-4512 18 11 the the DET cana-4512 18 12 name name NOUN cana-4512 18 13 fredholm fredholm NOUN cana-4512 18 14 integral integral ADJ cana-4512 18 15 equations equation NOUN cana-4512 18 16 . . PUNCT cana-4512 19 1 it it PRON cana-4512 19 2 served serve VERB cana-4512 19 3 as as ADP cana-4512 19 4 the the DET cana-4512 19 5 foundation foundation NOUN cana-4512 19 6 for for ADP cana-4512 19 7 addressing address VERB cana-4512 19 8 significant significant ADJ cana-4512 19 9 challenges challenge NOUN cana-4512 19 10 preventing prevent VERB cana-4512 19 11 mathematics mathematic NOUN cana-4512 19 12 from from ADP cana-4512 19 13 progressing progress VERB cana-4512 19 14 [ [ X cana-4512 19 15 5 5 NUM cana-4512 19 16 ] ] PUNCT cana-4512 19 17 . . PUNCT cana-4512 20 1 we we PRON cana-4512 20 2 conclude conclude VERB cana-4512 20 3 by by ADP cana-4512 20 4 pointing point VERB cana-4512 20 5 out out ADP cana-4512 20 6 that that SCONJ cana-4512 20 7 fie fie PROPN cana-4512 20 8 can can AUX cana-4512 20 9 also also ADV cana-4512 20 10 be be AUX cana-4512 20 11 found find VERB cana-4512 20 12 in in ADP cana-4512 20 13 both both CCONJ cana-4512 20 14 linear linear ADJ cana-4512 20 15 and and CCONJ cana-4512 20 16 non non ADJ cana-4512 20 17 - - ADJ cana-4512 20 18 linear linear ADJ cana-4512 20 19 forms form NOUN cana-4512 20 20 , , PUNCT cana-4512 20 21 such such ADJ cana-4512 20 22 as as ADP cana-4512 20 23 the the DET cana-4512 20 24 homogeneous homogeneous ADJ cana-4512 20 25 and and CCONJ cana-4512 20 26 nonhomogeneous nonhomogeneous ADJ cana-4512 20 27 varieties variety NOUN cana-4512 20 28 [ [ X cana-4512 20 29 1 1 NUM cana-4512 20 30 , , PUNCT cana-4512 20 31 3 3 NUM cana-4512 20 32 ] ] PUNCT cana-4512 20 33 . . PUNCT cana-4512 21 1 mailto:poonamjagtap2008@gmail.com mailto:poonamjagtap2008@gmail.com X cana-4512 21 2 mailto:avinash.khambayat@sandipuniversity.edu.in mailto:avinash.khambayat@sandipuniversity.edu.in PROPN cana-4512 21 3 communications communication NOUN cana-4512 21 4 on on ADP cana-4512 21 5 applied apply VERB cana-4512 21 6 nonlinear nonlinear ADJ cana-4512 21 7 analysis analysis NOUN cana-4512 21 8 issn issn NOUN cana-4512 21 9 : : PUNCT cana-4512 21 10 1074 1074 NUM cana-4512 21 11 - - PUNCT cana-4512 21 12 133x 133x NUM cana-4512 21 13 vol vol NOUN cana-4512 21 14 32 32 NUM cana-4512 21 15 no no NOUN cana-4512 21 16 . . PUNCT cana-4512 22 1 9s 9s NUM cana-4512 22 2 ( ( PUNCT cana-4512 22 3 2025 2025 NUM cana-4512 22 4 ) ) PUNCT cana-4512 22 5 2272 2272 NUM cana-4512 22 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 22 7 fie fie PROPN cana-4512 22 8 : : PUNCT cana-4512 22 9 a a DET cana-4512 22 10 linear linear NOUN cana-4512 22 11 ie ie X cana-4512 22 12 of of ADP cana-4512 22 13 the the DET cana-4512 22 14 form form NOUN cana-4512 22 15 𝑔(∝ 𝑔(∝ NOUN cana-4512 22 16 ) ) PUNCT cana-4512 22 17 𝑒(∝ 𝑒(∝ PROPN cana-4512 22 18 ) ) PUNCT cana-4512 23 1 = = SYM cana-4512 23 2 𝑓(∝ 𝑓(∝ NOUN cana-4512 23 3 ) ) PUNCT cana-4512 24 1 + + CCONJ cana-4512 24 2 πœ† πœ† DET cana-4512 24 3 ∫ ∫ PROPN cana-4512 24 4 𝐾(∝ 𝐾(∝ PROPN cana-4512 24 5 , , PUNCT cana-4512 24 6 𝜏 𝜏 NOUN cana-4512 24 7 ) ) PUNCT cana-4512 24 8 𝑒(𝜏)𝑑 𝑒(𝜏)𝑑 ADJ cana-4512 24 9 𝑏 𝑏 NOUN cana-4512 24 10 π‘Ž π‘Ž PRON cana-4512 24 11 𝜏 𝜏 NOUN cana-4512 24 12 a a PRON cana-4512 24 13 , , PUNCT cana-4512 24 14 b b NOUN cana-4512 24 15 are be AUX cana-4512 24 16 both both PRON cana-4512 24 17 constants.𝑔(∝ constants.𝑔(∝ ADJ cana-4512 24 18 ) ) PUNCT cana-4512 24 19 , , PUNCT cana-4512 24 20 𝑓(∝)π‘Žπ‘›π‘‘ 𝑓(∝)π‘Žπ‘›π‘‘ NOUN cana-4512 24 21 π‘˜(∝ π‘˜(∝ NOUN cana-4512 24 22 , , PUNCT cana-4512 24 23 𝜏 𝜏 NOUN cana-4512 24 24 ) ) PUNCT cana-4512 24 25 and and CCONJ cana-4512 24 26 are be AUX cana-4512 24 27 known know VERB cana-4512 24 28 functions function NOUN cana-4512 24 29 while while SCONJ cana-4512 24 30 𝑒(∝ 𝑒(∝ ADJ cana-4512 24 31 ) ) PUNCT cana-4512 24 32 are be AUX cana-4512 24 33 unknown unknown ADJ cana-4512 24 34 function.the function.the DET cana-4512 24 35 function function NOUN cana-4512 24 36 π‘˜(∝ π‘˜(∝ NOUN cana-4512 24 37 , , PUNCT cana-4512 25 1 𝜏)is 𝜏)is PROPN cana-4512 25 2 known know VERB cana-4512 25 3 as as ADP cana-4512 25 4 the the DET cana-4512 25 5 kernel kernel NOUN cana-4512 25 6 of of ADP cana-4512 25 7 the the DET cana-4512 25 8 ie ie X cana-4512 25 9 . . PUNCT cana-4512 25 10 fie fie NOUN cana-4512 25 11 of of ADP cana-4512 25 12 the the DET cana-4512 25 13 1st 1st ADJ cana-4512 25 14 kind kind NOUN cana-4512 25 15 : : PUNCT cana-4512 25 16 a a DET cana-4512 25 17 linear linear NOUN cana-4512 25 18 ie ie X cana-4512 25 19 of of ADP cana-4512 25 20 the the DET cana-4512 25 21 form form NOUN cana-4512 25 22 𝑔(∝ 𝑔(∝ NOUN cana-4512 25 23 ) ) PUNCT cana-4512 26 1 = = SYM cana-4512 26 2 0 0 NUM cana-4512 27 1 in in ADP cana-4512 27 2 equation equation NOUN cana-4512 27 3 𝑓(∝ 𝑓(∝ NOUN cana-4512 27 4 ) ) PUNCT cana-4512 27 5 + + CCONJ cana-4512 27 6 πœ† πœ† DET cana-4512 27 7 ∫ ∫ PROPN cana-4512 27 8 𝐾(∝ 𝐾(∝ PROPN cana-4512 27 9 , , PUNCT cana-4512 27 10 𝜏 𝜏 NOUN cana-4512 27 11 ) ) PUNCT cana-4512 27 12 𝑒(𝜏 𝑒(𝜏 PROPN cana-4512 27 13 ) ) PUNCT cana-4512 27 14 𝑑 𝑑 PROPN cana-4512 27 15 𝑏 𝑏 PROPN cana-4512 27 16 π‘Ž π‘Ž PROPN cana-4512 27 17 𝜏= 𝜏= PRON cana-4512 27 18 0 0 NUM cana-4512 27 19 fie fie NOUN cana-4512 27 20 of of ADP cana-4512 27 21 the the DET cana-4512 27 22 2nd 2nd ADJ cana-4512 27 23 kind kind NOUN cana-4512 27 24 : : PUNCT cana-4512 27 25 a a DET cana-4512 27 26 linear linear NOUN cana-4512 27 27 ie ie X cana-4512 27 28 of of ADP cana-4512 27 29 the the DET cana-4512 27 30 form form NOUN cana-4512 27 31 𝑔(∝ 𝑔(∝ NOUN cana-4512 27 32 ) ) PUNCT cana-4512 27 33 = = SYM cana-4512 28 1 1 1 NUM cana-4512 28 2 in in ADP cana-4512 28 3 equation equation NOUN cana-4512 28 4 𝑒(∝ 𝑒(∝ NOUN cana-4512 28 5 ) ) PUNCT cana-4512 29 1 = = SYM cana-4512 29 2 𝑓(∝ 𝑓(∝ NOUN cana-4512 29 3 ) ) PUNCT cana-4512 30 1 + + CCONJ cana-4512 30 2 πœ† πœ† DET cana-4512 30 3 ∫ ∫ PROPN cana-4512 30 4 𝐾(∝ 𝐾(∝ PROPN cana-4512 30 5 , , PUNCT cana-4512 30 6 𝜏 𝜏 NOUN cana-4512 30 7 ) ) PUNCT cana-4512 30 8 𝑒(𝜏)𝑑 𝑒(𝜏)𝑑 ADJ cana-4512 30 9 𝑏 𝑏 PROPN cana-4512 30 10 π‘Ž π‘Ž PRON cana-4512 30 11 𝜏 𝜏 NOUN cana-4512 30 12 methods method NOUN cana-4512 30 13 to to PART cana-4512 30 14 solve solve VERB cana-4512 30 15 fie fie NOUN cana-4512 30 16 of of ADP cana-4512 30 17 the the DET cana-4512 30 18 2nd 2nd ADJ cana-4512 30 19 kind kind NOUN cana-4512 30 20 : : PUNCT cana-4512 30 21 we we PRON cana-4512 30 22 will will AUX cana-4512 30 23 discuss discuss VERB cana-4512 30 24 some some DET cana-4512 30 25 analytical analytical ADJ cana-4512 30 26 and and CCONJ cana-4512 30 27 numerical numerical ADJ cana-4512 30 28 techniques technique NOUN cana-4512 30 29 in in ADP cana-4512 30 30 this this DET cana-4512 30 31 study study NOUN cana-4512 30 32 to to PART cana-4512 30 33 solve solve VERB cana-4512 30 34 second second ADJ cana-4512 30 35 - - PUNCT cana-4512 30 36 kind kind NOUN cana-4512 30 37 fie fie NOUN cana-4512 30 38 . . PUNCT cana-4512 31 1 1.direct 1.direct NUM cana-4512 31 2 computation computation NOUN cana-4512 31 3 method method NOUN cana-4512 31 4 ( ( PUNCT cana-4512 31 5 dcm dcm PROPN cana-4512 31 6 ) ) PUNCT cana-4512 31 7 2.variation 2.variation NUM cana-4512 31 8 iteration iteration NOUN cana-4512 31 9 method method NOUN cana-4512 31 10 ( ( PUNCT cana-4512 31 11 vim vim NOUN cana-4512 31 12 ) ) PUNCT cana-4512 31 13 1.1 1.1 NUM cana-4512 31 14 direct direct ADJ cana-4512 31 15 computation computation NOUN cana-4512 31 16 method method NOUN cana-4512 31 17 ( ( PUNCT cana-4512 31 18 dcm dcm PROPN cana-4512 31 19 ): ): PUNCT cana-4512 31 20 in in ADP cana-4512 31 21 this this DET cana-4512 31 22 part part NOUN cana-4512 31 23 of of ADP cana-4512 31 24 the the DET cana-4512 31 25 study study NOUN cana-4512 31 26 , , PUNCT cana-4512 31 27 the the DET cana-4512 31 28 fies fie NOUN cana-4512 31 29 will will AUX cana-4512 31 30 be be AUX cana-4512 31 31 solved solve VERB cana-4512 31 32 using use VERB cana-4512 31 33 the the DET cana-4512 31 34 dcm dcm NOUN cana-4512 31 35 . . PUNCT cana-4512 32 1 for for ADP cana-4512 32 2 the the DET cana-4512 32 3 provided provide VERB cana-4512 32 4 ie ie NOUN cana-4512 32 5 , , PUNCT cana-4512 32 6 this this DET cana-4512 32 7 approach approach NOUN cana-4512 32 8 provides provide VERB cana-4512 32 9 an an DET cana-4512 32 10 accurate accurate ADJ cana-4512 32 11 answer answer NOUN cana-4512 32 12 in in ADP cana-4512 32 13 closed closed ADJ cana-4512 32 14 form form NOUN cana-4512 32 15 . . PUNCT cana-4512 33 1 it it PRON cana-4512 33 2 is be AUX cana-4512 33 3 important important ADJ cana-4512 33 4 to to PART cana-4512 33 5 keep keep VERB cana-4512 33 6 in in ADP cana-4512 33 7 consideration consideration NOUN cana-4512 33 8 that that SCONJ cana-4512 33 9 this this DET cana-4512 33 10 method method NOUN cana-4512 33 11 can can AUX cana-4512 33 12 be be AUX cana-4512 33 13 applied apply VERB cana-4512 33 14 to to ADP cana-4512 33 15 the the DET cana-4512 33 16 form form NOUN cana-4512 33 17 's 's PART cana-4512 33 18 degenerate degenerate ADJ cana-4512 33 19 and and CCONJ cana-4512 33 20 separable separable ADJ cana-4512 33 21 kernels kernel NOUN cana-4512 33 22 of of ADP cana-4512 33 23 the the DET cana-4512 33 24 form form NOUN cana-4512 33 25 , , PUNCT cana-4512 33 26 𝐾(𝛾 𝐾(𝛾 PRON cana-4512 33 27 , , PUNCT cana-4512 33 28 𝜏 𝜏 NOUN cana-4512 33 29 ) ) PUNCT cana-4512 33 30 = = SYM cana-4512 33 31 βˆ‘ βˆ‘ PUNCT cana-4512 33 32 π‘”π‘˜ π‘”π‘˜ ADP cana-4512 33 33 𝑛 𝑛 PRON cana-4512 33 34 𝑖=1 𝑖=1 PROPN cana-4512 33 35 ( ( PUNCT cana-4512 33 36 𝛾 𝛾 NOUN cana-4512 33 37 ) ) PUNCT cana-4512 33 38 β„Žπ‘˜(𝜏 β„Žπ‘˜(𝜏 NOUN cana-4512 33 39 ) ) PUNCT cana-4512 33 40 ( ( PUNCT cana-4512 33 41 1 1 X cana-4512 33 42 ) ) PUNCT cana-4512 33 43 dcm dcm PROPN cana-4512 33 44 can can AUX cana-4512 33 45 apply apply VERB cana-4512 33 46 in in ADP cana-4512 33 47 following follow VERB cana-4512 33 48 form form NOUN cana-4512 33 49 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 33 50 ) ) PUNCT cana-4512 33 51 = = PUNCT cana-4512 33 52 𝑓(𝛾 𝑓(𝛾 ADV cana-4512 33 53 ) ) PUNCT cana-4512 34 1 + + CCONJ cana-4512 34 2 πœ† πœ† PRON cana-4512 34 3 𝑔(𝛾 𝑔(𝛾 PROPN cana-4512 34 4 ) ) PUNCT cana-4512 34 5 ∫ ∫ PROPN cana-4512 34 6 β„Ž(𝜏 β„Ž(𝜏 NOUN cana-4512 34 7 ) ) PUNCT cana-4512 34 8 𝑒(𝜏)𝑑 𝑒(𝜏)𝑑 ADJ cana-4512 34 9 𝑏 𝑏 PROPN cana-4512 34 10 π‘Ž π‘Ž PRON cana-4512 34 11 𝜏 𝜏 X cana-4512 34 12 ( ( PUNCT cana-4512 34 13 2 2 NUM cana-4512 34 14 ) ) PUNCT cana-4512 34 15 the the DET cana-4512 34 16 r.h.s r.h.s NOUN cana-4512 34 17 of of ADP cana-4512 34 18 equation equation NOUN cana-4512 34 19 ( ( PUNCT cana-4512 34 20 2 2 X cana-4512 34 21 ) ) PUNCT cana-4512 34 22 is be AUX cana-4512 34 23 depends depend VERB cana-4512 34 24 on on ADP cana-4512 34 25 one one NUM cana-4512 34 26 variable variable NOUN cana-4512 34 27 t. t. NOUN cana-4512 34 28 let let VERB cana-4512 34 29 ∫ ∫ PROPN cana-4512 34 30 β„Ž(𝜏 β„Ž(𝜏 NOUN cana-4512 34 31 ) ) PUNCT cana-4512 34 32 𝑒(𝜏)𝑑 𝑒(𝜏)𝑑 ADJ cana-4512 34 33 𝑏 𝑏 PROPN cana-4512 34 34 π‘Ž π‘Ž PROPN cana-4512 34 35 𝜏= 𝜏= ADJ cana-4512 34 36 Ξ± Ξ± NOUN cana-4512 34 37 ( ( PUNCT cana-4512 34 38 3 3 NUM cana-4512 34 39 ) ) PUNCT cana-4512 34 40 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 34 41 ) ) PUNCT cana-4512 34 42 = = PUNCT cana-4512 34 43 𝑓(𝛾 𝑓(𝛾 ADV cana-4512 34 44 ) ) PUNCT cana-4512 35 1 + + CCONJ cana-4512 35 2 πœ† πœ† PRON cana-4512 35 3 𝑔(𝛾 𝑔(𝛾 PROPN cana-4512 35 4 ) ) PUNCT cana-4512 35 5 Ξ± Ξ± NOUN cana-4512 35 6 ( ( PUNCT cana-4512 35 7 4 4 NUM cana-4512 35 8 ) ) PUNCT cana-4512 35 9 substitute substitute NOUN cana-4512 35 10 equation equation NOUN cana-4512 35 11 ( ( PUNCT cana-4512 35 12 4 4 NUM cana-4512 35 13 ) ) PUNCT cana-4512 35 14 into into ADP cana-4512 35 15 ( ( PUNCT cana-4512 35 16 3 3 X cana-4512 35 17 ) ) PUNCT cana-4512 35 18 communications communication NOUN cana-4512 35 19 on on ADP cana-4512 35 20 applied apply VERB cana-4512 35 21 nonlinear nonlinear ADJ cana-4512 35 22 analysis analysis NOUN cana-4512 35 23 issn issn NOUN cana-4512 35 24 : : PUNCT cana-4512 35 25 1074 1074 NUM cana-4512 35 26 - - PUNCT cana-4512 35 27 133x 133x NUM cana-4512 35 28 vol vol NOUN cana-4512 35 29 32 32 NUM cana-4512 35 30 no no NOUN cana-4512 35 31 . . PUNCT cana-4512 36 1 9s 9s NUM cana-4512 36 2 ( ( PUNCT cana-4512 36 3 2025 2025 NUM cana-4512 36 4 ) ) PUNCT cana-4512 36 5 2273 2273 NUM cana-4512 36 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 36 7 1.2 1.2 NUM cana-4512 36 8 variational variational ADJ cana-4512 36 9 iteration iteration NOUN cana-4512 36 10 mehod mehod NOUN cana-4512 36 11 ( ( PUNCT cana-4512 36 12 vim vim NOUN cana-4512 36 13 ): ): PUNCT cana-4512 36 14 if if SCONJ cana-4512 36 15 a a DET cana-4512 36 16 solution solution NOUN cana-4512 36 17 does do AUX cana-4512 36 18 exist exist VERB cana-4512 36 19 , , PUNCT cana-4512 36 20 the the DET cana-4512 36 21 vim vim NOUN cana-4512 36 22 provides provide VERB cana-4512 36 23 consecutive consecutive ADJ cana-4512 36 24 approximations approximation NOUN cana-4512 36 25 of of ADP cana-4512 36 26 it it PRON cana-4512 36 27 that that PRON cana-4512 36 28 may may AUX cana-4512 36 29 converge converge VERB cana-4512 36 30 quickly quickly ADV cana-4512 36 31 to to ADP cana-4512 36 32 the the DET cana-4512 36 33 exact exact ADJ cana-4512 36 34 answer answer NOUN cana-4512 36 35 . . PUNCT cana-4512 37 1 it it PRON cana-4512 37 2 is be AUX cana-4512 37 3 possible possible ADJ cana-4512 37 4 to to PART cana-4512 37 5 use use VERB cana-4512 37 6 the the DET cana-4512 37 7 resulting result VERB cana-4512 37 8 approximation approximation NOUN cana-4512 37 9 for for ADP cana-4512 37 10 numerical numerical ADJ cana-4512 37 11 purposes purpose NOUN cana-4512 37 12 . . PUNCT cana-4512 38 1 it it PRON cana-4512 38 2 is be AUX cana-4512 38 3 necessary necessary ADJ cana-4512 38 4 to to PART cana-4512 38 5 first first ADV cana-4512 38 6 convert convert VERB cana-4512 38 7 the the DET cana-4512 38 8 integral integral ADJ cana-4512 38 9 equation equation NOUN cana-4512 38 10 to to ADP cana-4512 38 11 its its PRON cana-4512 38 12 corresponding corresponding ADJ cana-4512 38 13 integro integro ADJ cana-4512 38 14 differential differential ADJ cana-4512 38 15 equation equation NOUN cana-4512 38 16 in in ADP cana-4512 38 17 order order NOUN cana-4512 38 18 to to PART cana-4512 38 19 solve solve VERB cana-4512 38 20 the the DET cana-4512 38 21 fredholm fredholm ADJ cana-4512 38 22 integral integral ADJ cana-4512 38 23 problem problem NOUN cana-4512 38 24 . . PUNCT cana-4512 39 1 this this DET cana-4512 39 2 method method NOUN cana-4512 39 3 is be AUX cana-4512 39 4 effective effective ADJ cana-4512 39 5 if if SCONJ cana-4512 39 6 the the DET cana-4512 39 7 kernel kernel NOUN cana-4512 39 8 can can AUX cana-4512 39 9 be be AUX cana-4512 39 10 separated separate VERB cana-4512 39 11 of of ADP cana-4512 39 12 the the DET cana-4512 39 13 form form NOUN cana-4512 39 14 , , PUNCT cana-4512 39 15 𝐾(𝛾 𝐾(𝛾 PRON cana-4512 39 16 , , PUNCT cana-4512 39 17 Ο„ Ο„ X cana-4512 39 18 ) ) PUNCT cana-4512 40 1 = = SYM cana-4512 40 2 𝑔(𝛾 𝑔(𝛾 NOUN cana-4512 40 3 ) ) PUNCT cana-4512 40 4 β„Ž(Ο„ β„Ž(Ο„ PROPN cana-4512 40 5 ) ) PUNCT cana-4512 40 6 .. .. PUNCT cana-4512 41 1 ( ( PUNCT cana-4512 41 2 5 5 X cana-4512 41 3 ) ) PUNCT cana-4512 41 4 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 41 5 ) ) PUNCT cana-4512 42 1 = = PUNCT cana-4512 42 2 𝑓(𝛾 𝑓(𝛾 ADV cana-4512 42 3 ) ) PUNCT cana-4512 42 4 + + CCONJ cana-4512 42 5 𝑔(𝛾 𝑔(𝛾 PROPN cana-4512 42 6 ) ) PUNCT cana-4512 42 7 ∫ ∫ PROPN cana-4512 42 8 β„Ž(𝜏 β„Ž(𝜏 NOUN cana-4512 42 9 ) ) PUNCT cana-4512 43 1 𝑒(𝜏)𝑑 𝑒(𝜏)𝑑 ADJ cana-4512 43 2 𝑏 𝑏 NOUN cana-4512 43 3 π‘Ž π‘Ž PRON cana-4512 43 4 𝜏 𝜏 NOUN cana-4512 43 5 differentiate differentiate NOUN cana-4512 43 6 both both DET cana-4512 43 7 sides side NOUN cana-4512 43 8 of of ADP cana-4512 43 9 ( ( PUNCT cana-4512 43 10 5 5 NUM cana-4512 43 11 ) ) PUNCT cana-4512 43 12 with with ADP cana-4512 43 13 respect respect NOUN cana-4512 43 14 to to ADP cana-4512 43 15 𝛾 𝛾 PROPN cana-4512 43 16 we we PRON cana-4512 43 17 get get VERB cana-4512 43 18 , , PUNCT cana-4512 43 19 𝑒′(𝛾 𝑒′(𝛾 PROPN cana-4512 43 20 ) ) PUNCT cana-4512 43 21 = = SYM cana-4512 44 1 𝑓′(𝛾 𝑓′(𝛾 PUNCT cana-4512 44 2 ) ) PUNCT cana-4512 44 3 + + CCONJ cana-4512 44 4 𝑔′(𝛾 𝑔′(𝛾 X cana-4512 44 5 ) ) PUNCT cana-4512 44 6 ∫ ∫ PROPN cana-4512 44 7 β„Ž(𝜏 β„Ž(𝜏 NOUN cana-4512 44 8 ) ) PUNCT cana-4512 44 9 𝑒(𝜏)𝑑 𝑒(𝜏)𝑑 ADJ cana-4512 44 10 𝑏 𝑏 SYM cana-4512 44 11 π‘Ž π‘Ž DET cana-4512 44 12 𝜏 𝜏 X cana-4512 44 13 𝑒𝑛+1 𝑒𝑛+1 NUM cana-4512 44 14 ( ( PUNCT cana-4512 44 15 𝛾 𝛾 NOUN cana-4512 44 16 ) ) PUNCT cana-4512 44 17 = = SYM cana-4512 44 18 𝑒𝑛 𝑒𝑛 NOUN cana-4512 44 19 ( ( PUNCT cana-4512 44 20 𝛾 𝛾 NOUN cana-4512 44 21 ) ) PUNCT cana-4512 45 1 + + CCONJ cana-4512 45 2 ∫ ∫ PROPN cana-4512 45 3 πœ† πœ† ADP cana-4512 45 4 𝛾 𝛾 PROPN cana-4512 45 5 0 0 NUM cana-4512 45 6 ( ( PUNCT cana-4512 45 7 𝜏 𝜏 NOUN cana-4512 45 8 ) ) PUNCT cana-4512 45 9 { { PUNCT cana-4512 45 10 𝑒′𝑛 𝑒′𝑛 NOUN cana-4512 45 11 ( ( PUNCT cana-4512 45 12 𝜏 𝜏 NOUN cana-4512 45 13 ) ) PUNCT cana-4512 45 14 βˆ’ βˆ’ PROPN cana-4512 45 15 𝑓′(𝜏 𝑓′(𝜏 PROPN cana-4512 45 16 ) ) PUNCT cana-4512 45 17 βˆ’ βˆ’ PROPN cana-4512 45 18 𝑔′(𝜏 𝑔′(𝜏 PROPN cana-4512 45 19 ) ) PUNCT cana-4512 45 20 ∫ ∫ PROPN cana-4512 46 1 β„Ž β„Ž PART cana-4512 46 2 𝑏 𝑏 SYM cana-4512 46 3 π‘Ž π‘Ž PRON cana-4512 46 4 ( ( PUNCT cana-4512 46 5 r r NOUN cana-4512 46 6 ) ) PUNCT cana-4512 46 7 𝑒𝑛 𝑒𝑛 NOUN cana-4512 46 8 ( ( PUNCT cana-4512 46 9 π‘Ÿ π‘Ÿ NOUN cana-4512 46 10 ) ) PUNCT cana-4512 46 11 dr dr PROPN cana-4512 46 12 } } PUNCT cana-4512 46 13 d d PROPN cana-4512 46 14 𝜏 𝜏 X cana-4512 46 15 ( ( PUNCT cana-4512 46 16 6 6 NUM cana-4512 46 17 ) ) PUNCT cana-4512 46 18 where where SCONJ cana-4512 46 19 Ξ» Ξ» PROPN cana-4512 46 20 is be AUX cana-4512 46 21 lagrange lagrange NOUN cana-4512 46 22 multiplier multipli ADJ cana-4512 46 23 . . PUNCT cana-4512 47 1 in in ADP cana-4512 47 2 vim vim NOUN cana-4512 47 3 we we PRON cana-4512 47 4 follow follow VERB cana-4512 47 5 two two NUM cana-4512 47 6 steps step NOUN cana-4512 47 7 , , PUNCT cana-4512 47 8 first first ADV cana-4512 47 9 we we PRON cana-4512 47 10 determine determine VERB cana-4512 47 11 the the DET cana-4512 47 12 Ξ» Ξ» PROPN cana-4512 47 13 1.3 1.3 NUM cana-4512 47 14 adomian adomian NOUN cana-4512 47 15 decomposition decomposition NOUN cana-4512 47 16 method method NOUN cana-4512 47 17 ( ( PUNCT cana-4512 47 18 adm adm PROPN cana-4512 47 19 ) ) PUNCT cana-4512 47 20 for for ADP cana-4512 47 21 fie fie NOUN cana-4512 47 22 of of ADP cana-4512 47 23 2nd 2nd ADJ cana-4512 47 24 kind kind NOUN cana-4512 47 25 : : PUNCT cana-4512 47 26 the the DET cana-4512 47 27 adm adm PROPN cana-4512 47 28 was be AUX cana-4512 47 29 developed develop VERB cana-4512 47 30 by by ADP cana-4512 47 31 george george PROPN cana-4512 47 32 adomian adomian PROPN cana-4512 47 33 decomposing decompose VERB cana-4512 47 34 the the DET cana-4512 47 35 unknown unknown ADJ cana-4512 47 36 u(x u(x NOUN cana-4512 47 37 ) ) PUNCT cana-4512 47 38 of of ADP cana-4512 47 39 every every DET cana-4512 47 40 equation equation NOUN cana-4512 47 41 into into ADP cana-4512 47 42 the the DET cana-4512 47 43 sum sum NOUN cana-4512 47 44 of of ADP cana-4512 47 45 an an DET cana-4512 47 46 infinite infinite ADJ cana-4512 47 47 number number NOUN cana-4512 47 48 of of ADP cana-4512 47 49 components component NOUN cana-4512 47 50 described describe VERB cana-4512 47 51 by by ADP cana-4512 47 52 the the DET cana-4512 47 53 decomposition decomposition NOUN cana-4512 47 54 series series NOUN cana-4512 47 55 is be AUX cana-4512 47 56 the the DET cana-4512 47 57 basic basic ADJ cana-4512 47 58 concept concept NOUN cana-4512 47 59 of of ADP cana-4512 47 60 the the DET cana-4512 47 61 adm adm PROPN cana-4512 47 62 . . PUNCT cana-4512 48 1 components component NOUN cana-4512 48 2 defined define VERB cana-4512 48 3 by by ADP cana-4512 48 4 𝑒(∝ 𝑒(∝ ADJ cana-4512 48 5 ) ) PUNCT cana-4512 48 6 = = PUNCT cana-4512 49 1 βˆ‘ βˆ‘ PUNCT cana-4512 49 2 𝑒𝑛 𝑒𝑛 NOUN cana-4512 49 3 𝑛 𝑛 PROPN cana-4512 49 4 𝑖=0 𝑖=0 PROPN cana-4512 49 5 ( ( PUNCT cana-4512 49 6 ∝ ∝ PROPN cana-4512 49 7 ) ) PUNCT cana-4512 49 8 βˆ‘ βˆ‘ PART cana-4512 49 9 𝑒𝑛 𝑒𝑛 NOUN cana-4512 49 10 𝑛 𝑛 PROPN cana-4512 49 11 𝑖=0 𝑖=0 PROPN cana-4512 49 12 ( ( PUNCT cana-4512 49 13 ∝ ∝ NOUN cana-4512 49 14 ) ) PUNCT cana-4512 49 15 = = SYM cana-4512 49 16 𝑓(∝ 𝑓(∝ NOUN cana-4512 49 17 ) ) PUNCT cana-4512 50 1 + + CCONJ cana-4512 50 2 πœ† πœ† DET cana-4512 50 3 ∫ ∫ PROPN cana-4512 50 4 𝐾(∝ 𝐾(∝ PROPN cana-4512 50 5 , , PUNCT cana-4512 50 6 𝜏 𝜏 NOUN cana-4512 50 7 ) ) PUNCT cana-4512 50 8 𝑏 𝑏 NOUN cana-4512 51 1 π‘Ž π‘Ž PRON cana-4512 51 2 𝑒𝑛(𝜏)π‘‘πœ 𝑒𝑛(𝜏)π‘‘πœ ADJ cana-4512 51 3 ( ( PUNCT cana-4512 51 4 7 7 X cana-4512 51 5 ) ) PUNCT cana-4512 51 6 compairing compaire VERB cana-4512 51 7 𝑒0(𝜏 𝑒0(𝜏 PROPN cana-4512 51 8 ) ) PUNCT cana-4512 51 9 = = PUNCT cana-4512 51 10 𝑓(∝ 𝑓(∝ NOUN cana-4512 51 11 ) ) PUNCT cana-4512 51 12 𝑒𝑛+1(𝜏 𝑒𝑛+1(𝜏 NOUN cana-4512 51 13 ) ) PUNCT cana-4512 51 14 = = PUNCT cana-4512 52 1 πœ† πœ† DET cana-4512 52 2 ∫ ∫ PROPN cana-4512 52 3 𝐾(∝ 𝐾(∝ PROPN cana-4512 52 4 , , PUNCT cana-4512 52 5 𝜏 𝜏 NOUN cana-4512 52 6 ) ) PUNCT cana-4512 52 7 𝑏 𝑏 NOUN cana-4512 52 8 π‘Ž π‘Ž NOUN cana-4512 52 9 𝑒𝑛(𝜏)d𝜏 𝑒𝑛(𝜏)d𝜏 ADJ cana-4512 52 10 ( ( PUNCT cana-4512 52 11 8) 8) NUM cana-4512 52 12 which which PRON cana-4512 52 13 is be AUX cana-4512 52 14 equivalent equivalent ADJ cana-4512 52 15 to𝑒0(𝜏 to𝑒0(𝜏 NOUN cana-4512 52 16 ) ) PUNCT cana-4512 52 17 = = PUNCT cana-4512 52 18 𝑓(∝ 𝑓(∝ NOUN cana-4512 52 19 ) ) PUNCT cana-4512 52 20 𝑒𝑛+1(∝ 𝑒𝑛+1(∝ NOUN cana-4512 52 21 ) ) PUNCT cana-4512 52 22 = = SYM cana-4512 53 1 πœ† πœ† DET cana-4512 53 2 ∫ ∫ PROPN cana-4512 53 3 𝐾(∝ 𝐾(∝ PROPN cana-4512 53 4 , , PUNCT cana-4512 53 5 𝜏 𝜏 NOUN cana-4512 53 6 ) ) PUNCT cana-4512 53 7 𝑏 𝑏 NOUN cana-4512 53 8 π‘Ž π‘Ž PRON cana-4512 53 9 𝑒𝑛(𝜏)d𝜏 𝑒𝑛(𝜏)d𝜏 ADJ cana-4512 53 10 𝑒1(∝ 𝑒1(∝ NOUN cana-4512 53 11 ) ) PUNCT cana-4512 53 12 = = SYM cana-4512 53 13 πœ† πœ† PRON cana-4512 53 14 ∫ ∫ PROPN cana-4512 53 15 𝐾(∝ 𝐾(∝ PROPN cana-4512 53 16 , , PUNCT cana-4512 53 17 𝜏 𝜏 NOUN cana-4512 53 18 ) ) PUNCT cana-4512 53 19 𝑏 𝑏 NOUN cana-4512 53 20 π‘Ž π‘Ž NOUN cana-4512 53 21 𝑒0(𝜏)d𝜏 𝑒0(𝜏)d𝜏 NOUN cana-4512 53 22 ( ( PUNCT cana-4512 53 23 9 9 NUM cana-4512 53 24 ) ) PUNCT cana-4512 53 25 𝑒2(∝ 𝑒2(∝ NOUN cana-4512 53 26 ) ) PUNCT cana-4512 53 27 = = SYM cana-4512 54 1 πœ† πœ† DET cana-4512 54 2 ∫ ∫ PROPN cana-4512 54 3 𝐾(∝ 𝐾(∝ PROPN cana-4512 54 4 , , PUNCT cana-4512 54 5 𝜏 𝜏 NOUN cana-4512 54 6 ) ) PUNCT cana-4512 54 7 𝑏 𝑏 NOUN cana-4512 54 8 π‘Ž π‘Ž PRON cana-4512 54 9 𝑒1(𝜏)d𝜏 𝑒1(𝜏)d𝜏 NOUN cana-4512 54 10 ( ( PUNCT cana-4512 54 11 10 10 NUM cana-4512 54 12 ) ) PUNCT cana-4512 54 13 and and CCONJ cana-4512 54 14 so so ADV cana-4512 54 15 on on ADV cana-4512 54 16 . . PUNCT cana-4512 55 1 it it PRON cana-4512 55 2 is be AUX cana-4512 55 3 obvious obvious ADJ cana-4512 55 4 that that SCONJ cana-4512 55 5 the the DET cana-4512 55 6 decomposition decomposition NOUN cana-4512 55 7 technique technique NOUN cana-4512 55 8 transformed transform VERB cana-4512 55 9 the the DET cana-4512 55 10 ie ie NOUN cana-4512 55 11 into into ADP cana-4512 55 12 an an DET cana-4512 55 13 advanced advanced ADJ cana-4512 55 14 computation computation NOUN cana-4512 55 15 of of ADP cana-4512 55 16 the the DET cana-4512 55 17 individual individual ADJ cana-4512 55 18 components component NOUN cana-4512 55 19 . . PUNCT cana-4512 56 1 formally formally ADV cana-4512 56 2 , , PUNCT cana-4512 56 3 it it PRON cana-4512 56 4 has have AUX cana-4512 56 5 been be AUX cana-4512 56 6 demonstrated demonstrate VERB cana-4512 56 7 that that SCONJ cana-4512 56 8 the the DET cana-4512 56 9 derived derive VERB cana-4512 56 10 series series NOUN cana-4512 56 11 rapidly rapidly ADV cana-4512 56 12 communications communication NOUN cana-4512 56 13 on on ADP cana-4512 56 14 applied apply VERB cana-4512 56 15 nonlinear nonlinear ADJ cana-4512 56 16 analysis analysis NOUN cana-4512 56 17 issn issn NOUN cana-4512 56 18 : : PUNCT cana-4512 56 19 1074 1074 NUM cana-4512 56 20 - - PUNCT cana-4512 56 21 133x 133x NUM cana-4512 56 22 vol vol NOUN cana-4512 56 23 32 32 NUM cana-4512 57 1 no no NOUN cana-4512 57 2 . . PUNCT cana-4512 58 1 9s 9s NUM cana-4512 58 2 ( ( PUNCT cana-4512 58 3 2025 2025 NUM cana-4512 58 4 ) ) PUNCT cana-4512 58 5 2274 2274 NUM cana-4512 58 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 58 7 converges converge VERB cana-4512 58 8 to to ADP cana-4512 58 9 the the DET cana-4512 58 10 precise precise ADJ cana-4512 58 11 solution solution NOUN cana-4512 58 12 if if SCONJ cana-4512 58 13 there there PRON cana-4512 58 14 is be VERB cana-4512 58 15 one one NUM cana-4512 58 16 . . PUNCT cana-4512 59 1 in in ADP cana-4512 59 2 order order NOUN cana-4512 59 3 to to PART cana-4512 59 4 confirm confirm VERB cana-4512 59 5 that that SCONJ cana-4512 59 6 the the DET cana-4512 59 7 resulting result VERB cana-4512 59 8 series series NOUN cana-4512 59 9 converges converge VERB cana-4512 59 10 quickly quickly ADV cana-4512 59 11 , , PUNCT cana-4512 59 12 numerous numerous ADJ cana-4512 59 13 researchers researcher NOUN cana-4512 59 14 researched research VERB cana-4512 59 15 deep deep ADV cana-4512 59 16 into into ADP cana-4512 59 17 the the DET cana-4512 59 18 convergence convergence NOUN cana-4512 59 19 idea idea NOUN cana-4512 59 20 of of ADP cana-4512 59 21 the the DET cana-4512 59 22 decomposition decomposition NOUN cana-4512 59 23 series series NOUN cana-4512 59 24 . . PUNCT cana-4512 60 1 however however ADV cana-4512 60 2 , , PUNCT cana-4512 60 3 in in ADP cana-4512 60 4 the the DET cana-4512 60 5 case case NOUN cana-4512 60 6 of of ADP cana-4512 60 7 actual actual ADJ cana-4512 60 8 issues issue NOUN cana-4512 60 9 , , PUNCT cana-4512 60 10 where where SCONJ cana-4512 60 11 a a DET cana-4512 60 12 closed closed ADJ cana-4512 60 13 form form NOUN cana-4512 60 14 solution solution NOUN cana-4512 60 15 is be AUX cana-4512 60 16 unattainable unattainable ADJ cana-4512 60 17 , , PUNCT cana-4512 60 18 numerical numerical ADJ cana-4512 60 19 objectives objective NOUN cana-4512 60 20 usually usually ADV cana-4512 60 21 involve involve VERB cana-4512 60 22 a a DET cana-4512 60 23 simplified simplified ADJ cana-4512 60 24 number number NOUN cana-4512 60 25 of of ADP cana-4512 60 26 terms term NOUN cana-4512 60 27 . . PUNCT cana-4512 61 1 we we PRON cana-4512 61 2 get get VERB cana-4512 61 3 higher high ADJ cana-4512 61 4 accuracy accuracy NOUN cana-4512 61 5 . . PUNCT cana-4512 62 1 1.4 1.4 NUM cana-4512 62 2 m m PROPN cana-4512 62 3 - - PUNCT cana-4512 62 4 adm adm PROPN cana-4512 62 5 for for ADP cana-4512 62 6 fie fie NOUN cana-4512 62 7 of of ADP cana-4512 62 8 2nd 2nd ADJ cana-4512 62 9 kind kind NOUN cana-4512 62 10 : : PUNCT cana-4512 62 11 domain domain NOUN cana-4512 62 12 decomposition decomposition NOUN cana-4512 62 13 methods method NOUN cana-4512 62 14 provide provide VERB cana-4512 62 15 solutions solution NOUN cana-4512 62 16 through through ADP cana-4512 62 17 an an DET cana-4512 62 18 infinite infinite ADJ cana-4512 62 19 series series NOUN cana-4512 62 20 of of ADP cana-4512 62 21 components component NOUN cana-4512 62 22 . . PUNCT cana-4512 63 1 the the DET cana-4512 63 2 fie fie NOUN cana-4512 63 3 ( ( PUNCT cana-4512 63 4 3 3 NUM cana-4512 63 5 ) ) PUNCT cana-4512 63 6 , , PUNCT cana-4512 63 7 𝑒(∝ 𝑒(∝ ADJ cana-4512 63 8 ) ) PUNCT cana-4512 63 9 = = SYM cana-4512 63 10 𝑓(∝ 𝑓(∝ NOUN cana-4512 63 11 ) ) PUNCT cana-4512 63 12 + + CCONJ cana-4512 63 13 πœ† πœ† DET cana-4512 63 14 ∫ ∫ PROPN cana-4512 63 15 𝐾(∝ 𝐾(∝ PROPN cana-4512 63 16 , , PUNCT cana-4512 63 17 𝜏 𝜏 NOUN cana-4512 63 18 ) ) PUNCT cana-4512 63 19 𝑒(𝜏)𝑑 𝑒(𝜏)𝑑 ADJ cana-4512 63 20 𝑏 𝑏 NOUN cana-4512 63 21 π‘Ž π‘Ž PRON cana-4512 63 22 𝜏 𝜏 NOUN cana-4512 63 23 using use VERB cana-4512 63 24 recurrence recurrence NOUN cana-4512 63 25 relation relation PROPN cana-4512 63 26 𝑒0(𝜏 𝑒0(𝜏 PROPN cana-4512 63 27 ) ) PUNCT cana-4512 63 28 = = PUNCT cana-4512 63 29 𝑓(∝ 𝑓(∝ NOUN cana-4512 63 30 ) ) PUNCT cana-4512 63 31 𝑒𝑛+1(𝜏 𝑒𝑛+1(𝜏 NOUN cana-4512 63 32 ) ) PUNCT cana-4512 63 33 = = PUNCT cana-4512 64 1 πœ† πœ† DET cana-4512 64 2 ∫ ∫ PROPN cana-4512 64 3 𝐾(∝ 𝐾(∝ PROPN cana-4512 64 4 , , PUNCT cana-4512 64 5 𝜏 𝜏 NOUN cana-4512 64 6 ) ) PUNCT cana-4512 64 7 𝑏 𝑏 NOUN cana-4512 64 8 π‘Ž π‘Ž NOUN cana-4512 64 9 𝑒𝑛(𝜏)d𝜏 𝑒𝑛(𝜏)d𝜏 ADJ cana-4512 64 10 ( ( PUNCT cana-4512 64 11 11 11 NUM cana-4512 64 12 ) ) PUNCT cana-4512 64 13 consists consist VERB cana-4512 64 14 of of ADP cana-4512 64 15 one one NUM cana-4512 64 16 or or CCONJ cana-4512 64 17 more more ADJ cana-4512 64 18 terms term NOUN cana-4512 64 19 in in ADP cana-4512 64 20 a a DET cana-4512 64 21 polynomial polynomial NOUN cana-4512 64 22 . . PUNCT cana-4512 65 1 in in ADP cana-4512 65 2 the the DET cana-4512 65 3 occurrence occurrence NOUN cana-4512 65 4 that that PRON cana-4512 65 5 the the DET cana-4512 65 6 function function NOUN cana-4512 65 7 f(x f(x PROPN cana-4512 65 8 ) ) PUNCT cana-4512 65 9 is be AUX cana-4512 65 10 formed form VERB cana-4512 65 11 from from ADP cana-4512 65 12 two two NUM cana-4512 65 13 or or CCONJ cana-4512 65 14 more more ADJ cana-4512 65 15 polynomials polynomial NOUN cana-4512 65 16 combined combine VERB cana-4512 65 17 , , PUNCT cana-4512 65 18 hyperbolic hyperbolic ADJ cana-4512 65 19 functions function NOUN cana-4512 65 20 or or CCONJ cana-4512 65 21 trigonometric trigonometric ADJ cana-4512 65 22 functions function NOUN cana-4512 65 23 and and CCONJ cana-4512 65 24 the the DET cana-4512 65 25 evaluation evaluation NOUN cana-4512 65 26 of of ADP cana-4512 65 27 component component NOUN cana-4512 65 28 𝑒𝑗 𝑒𝑗 NOUN cana-4512 65 29 , , PUNCT cana-4512 65 30 𝑗 𝑗 X cana-4512 65 31 β‰₯ β‰₯ NOUN cana-4512 65 32 0 0 NUM cana-4512 65 33 . . PUNCT cana-4512 66 1 the the DET cana-4512 66 2 m m PROPN cana-4512 66 3 - - PUNCT cana-4512 66 4 adm adm PROPN cana-4512 66 5 depends depend VERB cana-4512 66 6 mainly mainly ADV cana-4512 66 7 on on ADP cana-4512 66 8 splitting split VERB cana-4512 66 9 the the DET cana-4512 66 10 function function NOUN cana-4512 66 11 𝑓(∝ 𝑓(∝ NOUN cana-4512 66 12 ) ) PUNCT cana-4512 66 13 into into ADP cana-4512 66 14 two two NUM cana-4512 66 15 parts part NOUN cana-4512 66 16 , , PUNCT cana-4512 66 17 therefore therefore ADV cana-4512 66 18 it it PRON cana-4512 66 19 can can AUX cana-4512 66 20 not not PART cana-4512 66 21 be be AUX cana-4512 66 22 used use VERB cana-4512 66 23 if if SCONJ cana-4512 66 24 the the DET cana-4512 66 25 𝑓(∝)consists 𝑓(∝)consist NOUN cana-4512 66 26 of of ADP cana-4512 66 27 only only ADV cana-4512 66 28 one one NUM cana-4512 66 29 term term NOUN cana-4512 66 30 . . PUNCT cana-4512 67 1 the the DET cana-4512 67 2 m m PROPN cana-4512 67 3 - - PUNCT cana-4512 67 4 adm adm PROPN cana-4512 67 5 presents present VERB cana-4512 67 6 a a DET cana-4512 67 7 small small ADJ cana-4512 67 8 variation variation NOUN cana-4512 67 9 to to ADP cana-4512 67 10 the the DET cana-4512 67 11 recurrence recurrence NOUN cana-4512 67 12 relation relation NOUN cana-4512 67 13 to to PART cana-4512 67 14 determine determine VERB cana-4512 67 15 the the DET cana-4512 67 16 component component NOUN cana-4512 67 17 of of ADP cana-4512 67 18 𝑒(∝ 𝑒(∝ NOUN cana-4512 67 19 ) ) PUNCT cana-4512 67 20 in in ADP cana-4512 67 21 an an DET cana-4512 67 22 faster fast ADJ cana-4512 67 23 and and CCONJ cana-4512 67 24 easier easy ADJ cana-4512 67 25 way.in way.in PUNCT cana-4512 67 26 many many ADJ cana-4512 67 27 examples example NOUN cana-4512 67 28 the the DET cana-4512 67 29 function function NOUN cana-4512 67 30 𝑓(∝ 𝑓(∝ NOUN cana-4512 67 31 ) ) PUNCT cana-4512 67 32 can can AUX cana-4512 67 33 be be AUX cana-4512 67 34 set set VERB cana-4512 67 35 as as ADP cana-4512 67 36 the the DET cana-4512 67 37 sum sum NOUN cana-4512 67 38 of of ADP cana-4512 67 39 two two NUM cana-4512 67 40 partial partial ADJ cana-4512 67 41 functions function NOUN cana-4512 67 42 namely,𝑓1(∝)and namely,𝑓1(∝)and NOUN cana-4512 67 43 𝑓2(∝ 𝑓2(∝ PROPN cana-4512 67 44 ) ) PUNCT cana-4512 67 45 . . PUNCT cana-4512 68 1 the the DET cana-4512 68 2 m m PROPN cana-4512 68 3 - - PUNCT cana-4512 68 4 adm adm PROPN cana-4512 68 5 admits admit VERB cana-4512 68 6 the the DET cana-4512 68 7 use use NOUN cana-4512 68 8 of of ADP cana-4512 68 9 the the DET cana-4512 68 10 modified modify VERB cana-4512 68 11 recurrence recurrence NOUN cana-4512 68 12 relation relation NOUN cana-4512 68 13 . . PUNCT cana-4512 69 1 𝑒0(𝜏 𝑒0(𝜏 PROPN cana-4512 69 2 ) ) PUNCT cana-4512 69 3 = = PUNCT cana-4512 69 4 𝑓1(∝ 𝑓1(∝ X cana-4512 69 5 ) ) PUNCT cana-4512 69 6 ( ( PUNCT cana-4512 69 7 12 12 NUM cana-4512 69 8 ) ) PUNCT cana-4512 69 9 𝑒1(𝜏 𝑒1(𝜏 NOUN cana-4512 69 10 ) ) PUNCT cana-4512 69 11 = = SYM cana-4512 69 12 𝑓2(∝ 𝑓2(∝ NOUN cana-4512 69 13 ) ) PUNCT cana-4512 69 14 + + CCONJ cana-4512 69 15 πœ† πœ† DET cana-4512 69 16 ∫ ∫ PROPN cana-4512 69 17 𝐾(∝ 𝐾(∝ PROPN cana-4512 69 18 , , PUNCT cana-4512 69 19 𝜏 𝜏 NOUN cana-4512 69 20 ) ) PUNCT cana-4512 69 21 𝑏 𝑏 NOUN cana-4512 69 22 π‘Ž π‘Ž PRON cana-4512 69 23 𝑒0(𝜏)d𝜏 𝑒0(𝜏)d𝜏 NOUN cana-4512 69 24 ( ( PUNCT cana-4512 69 25 13 13 NUM cana-4512 69 26 ) ) PUNCT cana-4512 69 27 𝑒𝑛+1(𝜏 𝑒𝑛+1(𝜏 NOUN cana-4512 69 28 ) ) PUNCT cana-4512 69 29 = = PUNCT cana-4512 70 1 πœ† πœ† DET cana-4512 70 2 ∫ ∫ PROPN cana-4512 70 3 𝐾(∝ 𝐾(∝ PROPN cana-4512 70 4 , , PUNCT cana-4512 70 5 𝜏 𝜏 NOUN cana-4512 70 6 ) ) PUNCT cana-4512 70 7 𝑏 𝑏 NOUN cana-4512 70 8 π‘Ž π‘Ž NOUN cana-4512 70 9 𝑒𝑛(𝜏)d𝜏 𝑒𝑛(𝜏)d𝜏 ADJ cana-4512 70 10 , , PUNCT cana-4512 70 11 𝑛 𝑛 DET cana-4512 70 12 β‰₯ β‰₯ NUM cana-4512 70 13 1 1 NUM cana-4512 70 14 ( ( PUNCT cana-4512 70 15 14 14 NUM cana-4512 70 16 ) ) PUNCT cana-4512 70 17 some some DET cana-4512 70 18 example example NOUN cana-4512 70 19 on on ADP cana-4512 70 20 this this DET cana-4512 70 21 method method NOUN cana-4512 70 22 example example NOUN cana-4512 70 23 1 1 NUM cana-4512 70 24 : : PUNCT cana-4512 70 25 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 70 26 ) ) PUNCT cana-4512 70 27 = = SYM cana-4512 70 28 𝛾𝑒𝛾 𝛾𝑒𝛾 VERB cana-4512 70 29 βˆ’ βˆ’ PROPN cana-4512 70 30 𝛾 𝛾 PROPN cana-4512 70 31 + + CCONJ cana-4512 70 32 𝛾 𝛾 ADP cana-4512 70 33 ∫ ∫ PROPN cana-4512 70 34 𝑒 𝑒 NOUN cana-4512 70 35 ( ( PUNCT cana-4512 70 36 1 1 NUM cana-4512 70 37 0 0 NUM cana-4512 70 38 Ο„ Ο„ NOUN cana-4512 70 39 ) ) PUNCT cana-4512 70 40 dΟ„ dΟ„ NOUN cana-4512 70 41 by by ADP cana-4512 70 42 direct direct ADJ cana-4512 70 43 method method NOUN cana-4512 70 44 equation equation NOUN cana-4512 70 45 ( ( PUNCT cana-4512 70 46 i i NOUN cana-4512 70 47 ) ) PUNCT cana-4512 70 48 as as SCONJ cana-4512 70 49 put put VERB cana-4512 70 50 ∫ ∫ PROPN cana-4512 70 51 𝑒 𝑒 PART cana-4512 70 52 ( ( PUNCT cana-4512 70 53 1 1 NUM cana-4512 70 54 0 0 NUM cana-4512 70 55 Ο„ Ο„ NOUN cana-4512 70 56 ) ) PUNCT cana-4512 70 57 dΟ„ dΟ„ NOUN cana-4512 71 1 = = SYM cana-4512 71 2 Ξ± Ξ± PROPN cana-4512 71 3 ( ( PUNCT cana-4512 71 4 ii ii NOUN cana-4512 71 5 ) ) PUNCT cana-4512 71 6 therefore therefore ADV cana-4512 71 7 , , PUNCT cana-4512 71 8 𝑒(𝛾 𝑒(𝛾 PRON cana-4512 71 9 ) ) PUNCT cana-4512 71 10 = = SYM cana-4512 71 11 𝛾𝑒𝛾 𝛾𝑒𝛾 VERB cana-4512 71 12 βˆ’ βˆ’ PROPN cana-4512 71 13 𝛾 𝛾 PROPN cana-4512 71 14 + + CCONJ cana-4512 71 15 𝛾α 𝛾α PROPN cana-4512 71 16 ( ( PUNCT cana-4512 71 17 iii iii NOUN cana-4512 71 18 ) ) PUNCT cana-4512 71 19 put put VERB cana-4512 71 20 these these DET cana-4512 71 21 value value NOUN cana-4512 71 22 of of ADP cana-4512 71 23 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 71 24 ) ) PUNCT cana-4512 71 25 in in ADP cana-4512 71 26 equation equation NOUN cana-4512 71 27 ( ( PUNCT cana-4512 71 28 ii ii NOUN cana-4512 71 29 ) ) PUNCT cana-4512 71 30 Ξ± Ξ± PROPN cana-4512 71 31 = = SYM cana-4512 71 32 ∫ ∫ PROPN cana-4512 71 33 ( ( PUNCT cana-4512 71 34 τ𝑒τ τ𝑒τ NOUN cana-4512 72 1 βˆ’ βˆ’ X cana-4512 72 2 Ο„ Ο„ X cana-4512 72 3 + + NUM cana-4512 72 4 τα τα NOUN cana-4512 72 5 ) ) PUNCT cana-4512 73 1 d d ADP cana-4512 73 2 1 1 NUM cana-4512 73 3 0 0 NUM cana-4512 73 4 Ο„ Ο„ PROPN cana-4512 73 5 𝛼 𝛼 NOUN cana-4512 73 6 = = NOUN cana-4512 73 7 Ξ± Ξ± PROPN cana-4512 73 8 2 2 NUM cana-4512 73 9 + + CCONJ cana-4512 73 10 1 1 NUM cana-4512 73 11 2 2 NUM cana-4512 73 12 communications communication NOUN cana-4512 73 13 on on ADP cana-4512 73 14 applied apply VERB cana-4512 73 15 nonlinear nonlinear ADJ cana-4512 73 16 analysis analysis NOUN cana-4512 73 17 issn issn NOUN cana-4512 73 18 : : PUNCT cana-4512 73 19 1074 1074 NUM cana-4512 73 20 - - PUNCT cana-4512 73 21 133x 133x NUM cana-4512 73 22 vol vol NOUN cana-4512 73 23 32 32 NUM cana-4512 73 24 no no NOUN cana-4512 73 25 . . PUNCT cana-4512 74 1 9s 9s NUM cana-4512 74 2 ( ( PUNCT cana-4512 74 3 2025 2025 NUM cana-4512 74 4 ) ) PUNCT cana-4512 74 5 2275 2275 NUM cana-4512 74 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 74 7 Ξ±= Ξ±= NOUN cana-4512 74 8 1 1 NUM cana-4512 74 9 substitute substitute NOUN cana-4512 74 10 Ξ±= Ξ±= NUM cana-4512 74 11 1 1 NUM cana-4512 74 12 in in ADP cana-4512 74 13 equation equation NOUN cana-4512 74 14 ( ( PUNCT cana-4512 74 15 4 4 X cana-4512 74 16 ) ) PUNCT cana-4512 74 17 we we PRON cana-4512 74 18 get get VERB cana-4512 74 19 exact exact ADJ cana-4512 74 20 solution solution NOUN cana-4512 74 21 , , PUNCT cana-4512 74 22 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 74 23 ) ) PUNCT cana-4512 75 1 = = SYM cana-4512 75 2 𝛾𝑒𝛾 𝛾𝑒𝛾 VERB cana-4512 75 3 βˆ’ βˆ’ PROPN cana-4512 75 4 𝛾 𝛾 PROPN cana-4512 75 5 + + CCONJ cana-4512 75 6 𝛾 𝛾 ADP cana-4512 75 7 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 75 8 ) ) PUNCT cana-4512 76 1 = = SYM cana-4512 76 2 𝛾𝑒𝛾 𝛾𝑒𝛾 NOUN cana-4512 76 3 by by ADP cana-4512 76 4 variation variation NOUN cana-4512 76 5 iteration iteration NOUN cana-4512 76 6 method method NOUN cana-4512 76 7 , , PUNCT cana-4512 76 8 𝑒(𝛾 𝑒(𝛾 PRON cana-4512 76 9 ) ) PUNCT cana-4512 76 10 = = SYM cana-4512 76 11 𝛾𝑒𝛾 𝛾𝑒𝛾 VERB cana-4512 76 12 βˆ’ βˆ’ PROPN cana-4512 76 13 𝛾 𝛾 PROPN cana-4512 76 14 + + CCONJ cana-4512 76 15 𝛾 𝛾 ADP cana-4512 76 16 ∫ ∫ PROPN cana-4512 76 17 𝑒 𝑒 NOUN cana-4512 76 18 ( ( PUNCT cana-4512 76 19 1 1 NUM cana-4512 76 20 0 0 NUM cana-4512 76 21 Ο„ Ο„ NOUN cana-4512 76 22 ) ) PUNCT cana-4512 76 23 dΟ„ dΟ„ NOUN cana-4512 76 24 ( ( PUNCT cana-4512 76 25 i i NOUN cana-4512 76 26 ) ) PUNCT cana-4512 76 27 differentiate differentiate VERB cana-4512 76 28 from from ADP cana-4512 76 29 both both DET cana-4512 76 30 sides side NOUN cana-4512 76 31 of of ADP cana-4512 76 32 equation equation NOUN cana-4512 76 33 ( ( PUNCT cana-4512 76 34 i i NOUN cana-4512 76 35 ) ) PUNCT cana-4512 76 36 with with ADP cana-4512 76 37 respect respect NOUN cana-4512 76 38 to to ADP cana-4512 76 39 𝛾 𝛾 PROPN cana-4512 76 40 𝑒′ 𝑒′ ADJ cana-4512 76 41 = = X cana-4512 76 42 𝛾𝑒𝛾 𝛾𝑒𝛾 PROPN cana-4512 76 43 + + CCONJ cana-4512 76 44 𝛾 𝛾 NOUN cana-4512 76 45 βˆ’ βˆ’ NUM cana-4512 76 46 1 1 NUM cana-4512 76 47 + + NUM cana-4512 76 48 ∫ ∫ PROPN cana-4512 76 49 𝑒 𝑒 PART cana-4512 76 50 ( ( PUNCT cana-4512 76 51 1 1 NUM cana-4512 76 52 0 0 NUM cana-4512 76 53 Ο„ Ο„ NOUN cana-4512 76 54 ) ) PUNCT cana-4512 76 55 dΟ„ dΟ„ NOUN cana-4512 76 56 ( ( PUNCT cana-4512 76 57 iv iv NOUN cana-4512 76 58 ) ) PUNCT cana-4512 76 59 𝑒0 𝑒0 PROPN cana-4512 76 60 ( ( PUNCT cana-4512 76 61 𝛾 𝛾 NOUN cana-4512 76 62 ) ) PUNCT cana-4512 76 63 = = SYM cana-4512 76 64 0 0 NUM cana-4512 76 65 𝑒𝑛+1 𝑒𝑛+1 PROPN cana-4512 76 66 ( ( PUNCT cana-4512 76 67 𝛾 𝛾 NOUN cana-4512 76 68 ) ) PUNCT cana-4512 76 69 = = SYM cana-4512 76 70 𝑒𝑛 𝑒𝑛 NOUN cana-4512 76 71 ( ( PUNCT cana-4512 76 72 𝛾 𝛾 NOUN cana-4512 76 73 ) ) PUNCT cana-4512 77 1 βˆ’ βˆ’ PROPN cana-4512 77 2 ∫ ∫ PROPN cana-4512 77 3 { { PUNCT cana-4512 77 4 𝑒′𝑛 𝑒′𝑛 PROPN cana-4512 77 5 𝛾 𝛾 PROPN cana-4512 77 6 0 0 NUM cana-4512 77 7 ( ( PUNCT cana-4512 77 8 Ο„ Ο„ NOUN cana-4512 77 9 ) ) PUNCT cana-4512 77 10 πœπ‘’πœ πœπ‘’πœ NOUN cana-4512 77 11 βˆ’ βˆ’ PROPN cana-4512 77 12 𝑒τ 𝑒τ ADP cana-4512 77 13 + + CCONJ cana-4512 77 14 1 1 NUM cana-4512 77 15 βˆ’ βˆ’ NUM cana-4512 77 16 ∫ ∫ PROPN cana-4512 77 17 𝑒𝑛 𝑒𝑛 PROPN cana-4512 77 18 ( ( PUNCT cana-4512 77 19 ψ ψ NOUN cana-4512 77 20 ) ) PUNCT cana-4512 77 21 1 1 NUM cana-4512 77 22 0 0 NUM cana-4512 77 23 π‘‘πœ“ π‘‘πœ“ PROPN cana-4512 77 24 } } PUNCT cana-4512 77 25 𝑑τ 𝑑τ PROPN cana-4512 77 26 ( ( PUNCT cana-4512 77 27 v v NOUN cana-4512 77 28 ) ) PUNCT cana-4512 77 29 put put VERB cana-4512 77 30 πœ† πœ† PRON cana-4512 77 31 = = X cana-4512 77 32 βˆ’1 βˆ’1 NOUN cana-4512 77 33 𝑒1(𝛾 𝑒1(𝛾 NOUN cana-4512 77 34 ) ) PUNCT cana-4512 77 35 = = SYM cana-4512 77 36 𝑒0 𝑒0 PROPN cana-4512 77 37 ( ( PUNCT cana-4512 77 38 𝛾 𝛾 NOUN cana-4512 77 39 ) ) PUNCT cana-4512 78 1 βˆ’ βˆ’ PROPN cana-4512 79 1 ∫ ∫ PROPN cana-4512 79 2 { { PUNCT cana-4512 79 3 𝑒′0 𝑒′0 NOUN cana-4512 79 4 𝛾 𝛾 PROPN cana-4512 79 5 0 0 NUM cana-4512 79 6 ( ( PUNCT cana-4512 79 7 Ο„ Ο„ NOUN cana-4512 79 8 ) ) PUNCT cana-4512 79 9 πœπ‘’πœ πœπ‘’πœ NOUN cana-4512 79 10 βˆ’ βˆ’ PROPN cana-4512 79 11 𝑒τ 𝑒τ ADP cana-4512 80 1 + + CCONJ cana-4512 80 2 1 1 NUM cana-4512 80 3 βˆ’ βˆ’ PROPN cana-4512 80 4 ∫ ∫ PROPN cana-4512 80 5 𝑒0 𝑒0 PROPN cana-4512 80 6 ( ( PUNCT cana-4512 80 7 ψ ψ NOUN cana-4512 80 8 ) ) PUNCT cana-4512 80 9 1 1 NUM cana-4512 80 10 0 0 NUM cana-4512 80 11 π‘‘πœ“ π‘‘πœ“ PROPN cana-4512 80 12 } } PUNCT cana-4512 80 13 𝑑τ 𝑑τ ADP cana-4512 80 14 = = NOUN cana-4512 80 15 = = SYM cana-4512 80 16 𝛾𝑒𝛾 𝛾𝑒𝛾 PROPN cana-4512 80 17 βˆ’ βˆ’ PROPN cana-4512 80 18 𝛾 𝛾 PROPN cana-4512 80 19 ( ( PUNCT cana-4512 80 20 vi vi NOUN cana-4512 80 21 ) ) PUNCT cana-4512 80 22 𝑒2(𝛾 𝑒2(𝛾 NOUN cana-4512 80 23 ) ) PUNCT cana-4512 80 24 = = SYM cana-4512 80 25 𝑒1 𝑒1 PROPN cana-4512 80 26 ( ( PUNCT cana-4512 80 27 𝛾 𝛾 NOUN cana-4512 80 28 ) ) PUNCT cana-4512 80 29 βˆ’ βˆ’ PROPN cana-4512 81 1 ∫ ∫ PROPN cana-4512 82 1 { { PUNCT cana-4512 82 2 𝑒′1 𝑒′1 VERB cana-4512 82 3 𝛾 𝛾 PROPN cana-4512 82 4 0 0 NUM cana-4512 82 5 ( ( PUNCT cana-4512 82 6 Ο„ Ο„ NOUN cana-4512 82 7 ) ) PUNCT cana-4512 82 8 πœπ‘’πœ πœπ‘’πœ NOUN cana-4512 82 9 βˆ’ βˆ’ PROPN cana-4512 82 10 𝑒τ 𝑒τ ADP cana-4512 83 1 + + CCONJ cana-4512 83 2 1 1 NUM cana-4512 83 3 βˆ’ βˆ’ NOUN cana-4512 83 4 ∫ ∫ PROPN cana-4512 83 5 𝑒1 𝑒1 PROPN cana-4512 83 6 ( ( PUNCT cana-4512 83 7 ψ ψ NOUN cana-4512 83 8 ) ) PUNCT cana-4512 83 9 1 1 NUM cana-4512 83 10 0 0 NUM cana-4512 83 11 π‘‘πœ“ π‘‘πœ“ PROPN cana-4512 83 12 } } PUNCT cana-4512 83 13 𝑑τ 𝑑τ ADP cana-4512 83 14 = = NOUN cana-4512 83 15 = = SYM cana-4512 83 16 𝛾𝑒𝛾 𝛾𝑒𝛾 NOUN cana-4512 83 17 βˆ’ βˆ’ PROPN cana-4512 83 18 1 1 NUM cana-4512 83 19 2 2 NUM cana-4512 83 20 𝛾 𝛾 NOUN cana-4512 83 21 ( ( PUNCT cana-4512 83 22 vii vii PROPN cana-4512 83 23 ) ) PUNCT cana-4512 83 24 𝑒3(𝛾 𝑒3(𝛾 PROPN cana-4512 83 25 ) ) PUNCT cana-4512 83 26 = = SYM cana-4512 83 27 𝑒2 𝑒2 PROPN cana-4512 83 28 ( ( PUNCT cana-4512 83 29 𝛾 𝛾 PROPN cana-4512 83 30 ) ) PUNCT cana-4512 83 31 βˆ’ βˆ’ PROPN cana-4512 84 1 ∫ ∫ PROPN cana-4512 84 2 { { PUNCT cana-4512 84 3 𝑒′2 𝑒′2 PROPN cana-4512 84 4 𝛾 𝛾 PROPN cana-4512 84 5 0 0 NUM cana-4512 84 6 ( ( PUNCT cana-4512 84 7 Ο„ Ο„ NOUN cana-4512 84 8 ) ) PUNCT cana-4512 84 9 πœπ‘’πœ πœπ‘’πœ NOUN cana-4512 84 10 βˆ’ βˆ’ PROPN cana-4512 84 11 𝑒τ 𝑒τ ADP cana-4512 85 1 + + CCONJ cana-4512 85 2 1 1 NUM cana-4512 85 3 βˆ’ βˆ’ NUM cana-4512 85 4 ∫ ∫ PROPN cana-4512 85 5 𝑒2 𝑒2 PROPN cana-4512 85 6 ( ( PUNCT cana-4512 85 7 ψ ψ NOUN cana-4512 85 8 ) ) PUNCT cana-4512 85 9 1 1 NUM cana-4512 85 10 0 0 NUM cana-4512 85 11 π‘‘πœ“ π‘‘πœ“ PROPN cana-4512 85 12 } } PUNCT cana-4512 85 13 𝑑τ 𝑑τ ADP cana-4512 85 14 ( ( PUNCT cana-4512 85 15 viii viii NOUN cana-4512 85 16 ) ) PUNCT cana-4512 85 17 we we PRON cana-4512 85 18 get get VERB cana-4512 85 19 , , PUNCT cana-4512 85 20 𝑒𝑛 𝑒𝑛 PROPN cana-4512 85 21 ( ( PUNCT cana-4512 85 22 𝛾 𝛾 NOUN cana-4512 85 23 ) ) PUNCT cana-4512 85 24 = = SYM cana-4512 85 25 𝛾𝑒𝛾 𝛾𝑒𝛾 NOUN cana-4512 85 26 βˆ’ βˆ’ PROPN cana-4512 85 27 1 1 NUM cana-4512 85 28 2π‘›βˆ’1 2π‘›βˆ’1 NUM cana-4512 85 29 𝛾 𝛾 ADP cana-4512 85 30 𝑒(𝛾 𝑒(𝛾 NOUN cana-4512 85 31 ) ) PUNCT cana-4512 86 1 = = SYM cana-4512 86 2 lim lim PROPN cana-4512 86 3 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4512 86 4 𝑒𝑛(𝛾 𝑒𝑛(𝛾 NUM cana-4512 86 5 ) ) PUNCT cana-4512 86 6 𝑒(𝛾)= 𝑒(𝛾)= PROPN cana-4512 86 7 𝛾𝑒𝛾 𝛾𝑒𝛾 ADJ cana-4512 86 8 example example NOUN cana-4512 86 9 2 2 NUM cana-4512 86 10 : : PUNCT cana-4512 86 11 fie fie NOUN cana-4512 86 12 by by ADP cana-4512 86 13 adm adm PROPN cana-4512 86 14 method method PROPN cana-4512 86 15 𝑒(∝ 𝑒(∝ PROPN cana-4512 86 16 ) ) PUNCT cana-4512 87 1 = = VERB cana-4512 87 2 βˆ’πœ‹ βˆ’πœ‹ ADJ cana-4512 87 3 ∝ ∝ X cana-4512 88 1 + + ADP cana-4512 88 2 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-4512 88 3 ∝ ∝ PROPN cana-4512 88 4 + + NOUN cana-4512 88 5 ∝ ∝ PROPN cana-4512 88 6 ∫ ∫ PROPN cana-4512 88 7 𝜏 𝜏 X cana-4512 88 8 πœ‹ πœ‹ NOUN cana-4512 88 9 0 0 NUM cana-4512 88 10 u(𝜏)π‘‘πœ u(𝜏)π‘‘πœ PRON cana-4512 88 11 consider consider VERB cana-4512 88 12 , , PUNCT cana-4512 88 13 communications communication NOUN cana-4512 88 14 on on ADP cana-4512 88 15 applied apply VERB cana-4512 88 16 nonlinear nonlinear ADJ cana-4512 88 17 analysis analysis NOUN cana-4512 88 18 issn issn NOUN cana-4512 88 19 : : PUNCT cana-4512 88 20 1074 1074 NUM cana-4512 88 21 - - PUNCT cana-4512 88 22 133x 133x NUM cana-4512 88 23 vol vol NOUN cana-4512 88 24 32 32 NUM cana-4512 88 25 no no NOUN cana-4512 88 26 . . PUNCT cana-4512 89 1 9s 9s NUM cana-4512 89 2 ( ( PUNCT cana-4512 89 3 2025 2025 NUM cana-4512 89 4 ) ) PUNCT cana-4512 89 5 2276 2276 NUM cana-4512 89 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 89 7 𝑒(∝ 𝑒(∝ X cana-4512 89 8 ) ) PUNCT cana-4512 90 1 = = VERB cana-4512 90 2 βˆ’πœ‹ βˆ’πœ‹ ADJ cana-4512 90 3 ∝ ∝ X cana-4512 91 1 + + ADP cana-4512 91 2 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-4512 91 3 ∝ ∝ PROPN cana-4512 91 4 + + NOUN cana-4512 91 5 ∝ ∝ PROPN cana-4512 91 6 ∫ ∫ PROPN cana-4512 91 7 𝜏 𝜏 X cana-4512 91 8 πœ‹ πœ‹ NOUN cana-4512 91 9 0 0 NUM cana-4512 91 10 u(𝜏)π‘‘πœ u(𝜏)π‘‘πœ NUM cana-4512 91 11 𝑒0(∝ 𝑒0(∝ NOUN cana-4512 91 12 ) ) PUNCT cana-4512 91 13 = = PUNCT cana-4512 92 1 βˆ’πœ‹ βˆ’πœ‹ ADJ cana-4512 92 2 ∝ ∝ X cana-4512 92 3 + + ADP cana-4512 92 4 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-4512 92 5 ∝ ∝ PROPN cana-4512 92 6 𝑒𝑛+1(∝ 𝑒𝑛+1(∝ VERB cana-4512 92 7 ) ) PUNCT cana-4512 93 1 = = PUNCT cana-4512 93 2 ∝ ∝ PROPN cana-4512 93 3 ∫ ∫ PROPN cana-4512 93 4 𝜏 𝜏 X cana-4512 93 5 πœ‹ πœ‹ NOUN cana-4512 93 6 0 0 NUM cana-4512 93 7 𝑒𝑛(∝ 𝑒𝑛(∝ NOUN cana-4512 93 8 ) ) PUNCT cana-4512 94 1 = = PUNCT cana-4512 94 2 ∝ ∝ PROPN cana-4512 94 3 ∫ ∫ PROPN cana-4512 95 1 𝜏 𝜏 X cana-4512 95 2 πœ‹ πœ‹ NOUN cana-4512 95 3 0 0 NUM cana-4512 95 4 𝑒0(𝜏)d𝜏 𝑒0(𝜏)d𝜏 NOUN cana-4512 95 5 = = PUNCT cana-4512 95 6 ∝ ∝ PROPN cana-4512 95 7 ∫ ∫ PROPN cana-4512 96 1 𝜏 𝜏 X cana-4512 96 2 πœ‹ πœ‹ NOUN cana-4512 96 3 0 0 NUM cana-4512 97 1 ( ( PUNCT cana-4512 97 2 -πœ‹ -πœ‹ PROPN cana-4512 97 3 𝜏 𝜏 X cana-4512 97 4 + + CCONJ cana-4512 97 5 π‘ π‘–π‘›πœ π‘ π‘–π‘›πœ ADJ cana-4512 97 6 ) ) PUNCT cana-4512 97 7 π‘‘πœ π‘‘πœ NOUN cana-4512 97 8 = = SYM cana-4512 98 1 βˆ’βˆ βˆ’βˆ PROPN cana-4512 98 2 πœ‹4 πœ‹4 NOUN cana-4512 98 3 3 3 NUM cana-4512 98 4 + + NOUN cana-4512 98 5 ∝ ∝ PROPN cana-4512 98 6 πœ‹ πœ‹ NOUN cana-4512 98 7 𝑒2(∝ 𝑒2(∝ NOUN cana-4512 98 8 ) ) PUNCT cana-4512 98 9 = = SYM cana-4512 99 1 ∝ ∝ PROPN cana-4512 99 2 ∫ ∫ PROPN cana-4512 100 1 𝜏 𝜏 X cana-4512 100 2 πœ‹ πœ‹ NOUN cana-4512 100 3 0 0 NUM cana-4512 100 4 𝑒1(𝜏)d𝜏 𝑒1(𝜏)d𝜏 NOUN cana-4512 100 5 = = PUNCT cana-4512 100 6 ∝ ∝ PROPN cana-4512 100 7 ∫ ∫ PROPN cana-4512 100 8 𝜏2 𝜏2 PROPN cana-4512 100 9 πœ‹ πœ‹ NOUN cana-4512 100 10 0 0 NUM cana-4512 100 11 ( ( PUNCT cana-4512 100 12 βˆ’ βˆ’ PROPN cana-4512 100 13 πœ‹4 πœ‹4 NOUN cana-4512 100 14 3 3 NUM cana-4512 100 15 + + CCONJ cana-4512 100 16 πœ‹ πœ‹ NOUN cana-4512 100 17 ) ) PUNCT cana-4512 100 18 π‘‘πœ π‘‘πœ NOUN cana-4512 100 19 = = PUNCT cana-4512 100 20 ∝ ∝ PROPN cana-4512 100 21 πœ‹4 πœ‹4 PROPN cana-4512 100 22 3 3 NUM cana-4512 100 23 βˆ’βˆ βˆ’βˆ PROPN cana-4512 100 24 πœ‹7 πœ‹7 PROPN cana-4512 100 25 3 3 NUM cana-4512 100 26 from from ADP cana-4512 100 27 , , PUNCT cana-4512 100 28 𝑒(∝ 𝑒(∝ ADJ cana-4512 100 29 ) ) PUNCT cana-4512 100 30 = = SYM cana-4512 101 1 βˆ‘ βˆ‘ PUNCT cana-4512 101 2 𝑒𝑛 𝑒𝑛 NOUN cana-4512 101 3 𝑛 𝑛 PROPN cana-4512 101 4 𝑖=0 𝑖=0 PROPN cana-4512 101 5 ( ( PUNCT cana-4512 101 6 ∝ ∝ PROPN cana-4512 101 7 ) ) PUNCT cana-4512 101 8 𝑒(∝ 𝑒(∝ PROPN cana-4512 101 9 ) ) PUNCT cana-4512 102 1 = = VERB cana-4512 102 2 βˆ’πœ‹ βˆ’πœ‹ ADJ cana-4512 102 3 ∝ ∝ X cana-4512 103 1 + + ADP cana-4512 103 2 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-4512 103 3 ∝ ∝ PROPN cana-4512 103 4 βˆ’βˆ βˆ’βˆ PROPN cana-4512 103 5 πœ‹4 πœ‹4 NOUN cana-4512 103 6 3 3 NUM cana-4512 103 7 + + NOUN cana-4512 103 8 ∝ ∝ PROPN cana-4512 103 9 πœ‹+ πœ‹+ PUNCT cana-4512 103 10 ∝ ∝ PROPN cana-4512 103 11 πœ‹4 πœ‹4 PROPN cana-4512 103 12 3 3 NUM cana-4512 103 13 βˆ’βˆ βˆ’βˆ PROPN cana-4512 103 14 πœ‹7 πœ‹7 PROPN cana-4512 103 15 3 3 NUM cana-4512 103 16 . . PUNCT cana-4512 103 17 ... ... PUNCT cana-4512 104 1 cancelling cancel VERB cana-4512 104 2 the the DET cana-4512 104 3 noise noise NOUN cana-4512 104 4 terms term NOUN cana-4512 104 5 we we PRON cana-4512 104 6 get get VERB cana-4512 104 7 , , PUNCT cana-4512 104 8 𝑒(∝ 𝑒(∝ ADJ cana-4512 104 9 ) ) PUNCT cana-4512 105 1 = = SYM cana-4512 105 2 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-4512 105 3 ∝ ∝ PROPN cana-4512 105 4 example example VERB cana-4512 105 5 3 3 NUM cana-4512 105 6 : : PUNCT cana-4512 105 7 the the DET cana-4512 105 8 fie fie NOUN cana-4512 105 9 by by ADP cana-4512 105 10 using use VERB cana-4512 105 11 m m PROPN cana-4512 105 12 - - PUNCT cana-4512 105 13 adm adm PROPN cana-4512 105 14 𝑒(∝ 𝑒(∝ NOUN cana-4512 105 15 ) ) PUNCT cana-4512 105 16 = = SYM cana-4512 105 17 3 3 NUM cana-4512 105 18 ∝ ∝ PROPN cana-4512 105 19 + + ADJ cana-4512 105 20 𝑒4∝ 𝑒4∝ PROPN cana-4512 105 21 βˆ’ βˆ’ PROPN cana-4512 105 22 1 1 NUM cana-4512 105 23 16 16 NUM cana-4512 105 24 ( ( PUNCT cana-4512 105 25 17 17 NUM cana-4512 105 26 + + SYM cana-4512 105 27 3𝑒4∝ 3𝑒4∝ NUM cana-4512 105 28 ) ) PUNCT cana-4512 106 1 + + NUM cana-4512 106 2 ∫ ∫ X cana-4512 106 3 𝜏u(𝜏)π‘‘πœ 𝜏u(𝜏)π‘‘πœ ADJ cana-4512 106 4 1 1 NUM cana-4512 106 5 0 0 NUM cana-4512 106 6 consider consider VERB cana-4512 106 7 , , PUNCT cana-4512 106 8 𝑒(∝ 𝑒(∝ ADJ cana-4512 106 9 ) ) PUNCT cana-4512 106 10 = = SYM cana-4512 106 11 3 3 NUM cana-4512 106 12 ∝ ∝ PROPN cana-4512 106 13 + + ADJ cana-4512 106 14 𝑒4∝ 𝑒4∝ PROPN cana-4512 106 15 βˆ’ βˆ’ PROPN cana-4512 106 16 1 1 NUM cana-4512 106 17 16 16 NUM cana-4512 106 18 ( ( PUNCT cana-4512 106 19 17 17 NUM cana-4512 106 20 + + SYM cana-4512 106 21 3𝑒4∝ 3𝑒4∝ NUM cana-4512 106 22 ) ) PUNCT cana-4512 107 1 + + NUM cana-4512 107 2 ∫ ∫ X cana-4512 107 3 𝜏u(𝜏)π‘‘πœ 𝜏u(𝜏)π‘‘πœ ADJ cana-4512 107 4 1 1 NUM cana-4512 107 5 0 0 NUM cana-4512 107 6 𝑓1(∝ 𝑓1(∝ X cana-4512 107 7 ) ) PUNCT cana-4512 107 8 = = SYM cana-4512 107 9 3 3 NUM cana-4512 107 10 ∝ ∝ PROPN cana-4512 107 11 + + ADJ cana-4512 107 12 𝑒4∝ 𝑒4∝ PROPN cana-4512 107 13 , , PUNCT cana-4512 107 14 𝑓2(∝ 𝑓2(∝ NOUN cana-4512 107 15 ) ) PUNCT cana-4512 107 16 = = SYM cana-4512 107 17 βˆ’ βˆ’ PROPN cana-4512 108 1 1 1 NUM cana-4512 108 2 16 16 NUM cana-4512 108 3 ( ( PUNCT cana-4512 108 4 17 17 NUM cana-4512 108 5 + + SYM cana-4512 108 6 3𝑒4∝ 3𝑒4∝ PROPN cana-4512 108 7 ) ) PUNCT cana-4512 108 8 𝑒0(∝ 𝑒0(∝ NOUN cana-4512 108 9 ) ) PUNCT cana-4512 108 10 = = PUNCT cana-4512 109 1 𝑓1(∝ 𝑓1(∝ X cana-4512 109 2 ) ) PUNCT cana-4512 109 3 = = SYM cana-4512 109 4 3 3 NUM cana-4512 109 5 ∝ ∝ PROPN cana-4512 109 6 + + ADJ cana-4512 109 7 𝑒4∝ 𝑒4∝ PROPN cana-4512 109 8 𝑒0(𝜏 𝑒0(𝜏 PROPN cana-4512 109 9 ) ) PUNCT cana-4512 109 10 = = NOUN cana-4512 109 11 3 3 NUM cana-4512 109 12 𝜏 𝜏 X cana-4512 109 13 + + NUM cana-4512 109 14 𝑒4𝜏 𝑒4𝜏 NUM cana-4512 109 15 ` ` PUNCT cana-4512 109 16 𝑒1(𝜏 𝑒1(𝜏 NOUN cana-4512 109 17 ) ) PUNCT cana-4512 109 18 = = SYM cana-4512 109 19 𝑓2(∝ 𝑓2(∝ NOUN cana-4512 109 20 ) ) PUNCT cana-4512 109 21 + + CCONJ cana-4512 109 22 πœ† πœ† DET cana-4512 109 23 ∫ ∫ PROPN cana-4512 109 24 𝐾(∝ 𝐾(∝ PROPN cana-4512 109 25 , , PUNCT cana-4512 109 26 𝜏 𝜏 NOUN cana-4512 109 27 ) ) PUNCT cana-4512 109 28 𝑏 𝑏 NOUN cana-4512 109 29 π‘Ž π‘Ž DET cana-4512 109 30 𝑒0(𝜏)d𝜏 𝑒0(𝜏)d𝜏 ADJ cana-4512 109 31 𝑒1(𝜏 𝑒1(𝜏 NOUN cana-4512 109 32 ) ) PUNCT cana-4512 109 33 = = SYM cana-4512 110 1 βˆ’ βˆ’ PROPN cana-4512 110 2 1 1 NUM cana-4512 110 3 16 16 NUM cana-4512 110 4 ( ( PUNCT cana-4512 110 5 17 17 NUM cana-4512 110 6 + + CCONJ cana-4512 110 7 3𝑒4𝜏 3𝑒4𝜏 NUM cana-4512 110 8 ) ) PUNCT cana-4512 111 1 + + CCONJ cana-4512 111 2 ∫ ∫ X cana-4512 111 3 𝜏u(𝜏)π‘‘πœ 𝜏u(𝜏)π‘‘πœ ADJ cana-4512 111 4 1 1 NUM cana-4512 111 5 0 0 NUM cana-4512 111 6 = = SYM cana-4512 111 7 0 0 NUM cana-4512 111 8 communications communication NOUN cana-4512 111 9 on on ADP cana-4512 111 10 applied apply VERB cana-4512 111 11 nonlinear nonlinear ADJ cana-4512 111 12 analysis analysis NOUN cana-4512 111 13 issn issn NOUN cana-4512 111 14 : : PUNCT cana-4512 111 15 1074 1074 NUM cana-4512 111 16 - - PUNCT cana-4512 111 17 133x 133x NUM cana-4512 111 18 vol vol NOUN cana-4512 111 19 32 32 NUM cana-4512 111 20 no no NOUN cana-4512 111 21 . . PUNCT cana-4512 112 1 9s 9s NUM cana-4512 112 2 ( ( PUNCT cana-4512 112 3 2025 2025 NUM cana-4512 112 4 ) ) PUNCT cana-4512 112 5 2277 2277 NUM cana-4512 112 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 112 7 𝑒𝑛+1(𝜏 𝑒𝑛+1(𝜏 NOUN cana-4512 112 8 ) ) PUNCT cana-4512 112 9 = = PUNCT cana-4512 113 1 πœ† πœ† DET cana-4512 113 2 ∫ ∫ PROPN cana-4512 113 3 𝐾(∝ 𝐾(∝ PROPN cana-4512 113 4 , , PUNCT cana-4512 113 5 𝜏 𝜏 NOUN cana-4512 113 6 ) ) PUNCT cana-4512 113 7 𝑏 𝑏 NOUN cana-4512 113 8 π‘Ž π‘Ž NOUN cana-4512 113 9 𝑒𝑛(𝜏)d𝜏 𝑒𝑛(𝜏)d𝜏 ADJ cana-4512 113 10 , , PUNCT cana-4512 113 11 𝑛 𝑛 DET cana-4512 113 12 β‰₯ β‰₯ NOUN cana-4512 113 13 1 1 NUM cana-4512 113 14 = = SYM cana-4512 113 15 0 0 NUM cana-4512 113 16 , , PUNCT cana-4512 113 17 𝑛 𝑛 DET cana-4512 113 18 β‰₯ β‰₯ NUM cana-4512 113 19 1 1 NUM cana-4512 113 20 each each DET cana-4512 113 21 component component NOUN cana-4512 113 22 𝑒𝑗 𝑒𝑗 NOUN cana-4512 113 23 , , PUNCT cana-4512 113 24 𝑗 𝑗 X cana-4512 113 25 β‰₯ β‰₯ NOUN cana-4512 113 26 0 0 NUM cana-4512 113 27 is be AUX cana-4512 113 28 zero zero NUM cana-4512 113 29 . . PUNCT cana-4512 113 30 , , PUNCT cana-4512 113 31 𝑒(∝ 𝑒(∝ PROPN cana-4512 113 32 ) ) PUNCT cana-4512 114 1 = = SYM cana-4512 114 2 βˆ‘ βˆ‘ PUNCT cana-4512 114 3 𝑒𝑛 𝑒𝑛 NOUN cana-4512 114 4 𝑛 𝑛 PROPN cana-4512 114 5 𝑖=0 𝑖=0 PROPN cana-4512 114 6 ( ( PUNCT cana-4512 114 7 ∝ ∝ PROPN cana-4512 114 8 ) ) PUNCT cana-4512 114 9 𝑒(∝ 𝑒(∝ PROPN cana-4512 114 10 ) ) PUNCT cana-4512 114 11 = = SYM cana-4512 114 12 3 3 NUM cana-4512 114 13 ∝ ∝ PROPN cana-4512 114 14 + + ADJ cana-4512 114 15 𝑒4∝ 𝑒4∝ PROPN cana-4512 114 16 conclusion conclusion NOUN cana-4512 114 17 : : PUNCT cana-4512 114 18 in in ADP cana-4512 114 19 our our PRON cana-4512 114 20 study study NOUN cana-4512 114 21 we we PRON cana-4512 114 22 discuss discuss VERB cana-4512 114 23 the the DET cana-4512 114 24 various various ADJ cana-4512 114 25 methods method NOUN cana-4512 114 26 to to PART cana-4512 114 27 find find VERB cana-4512 114 28 the the DET cana-4512 114 29 solutions solution NOUN cana-4512 114 30 of of ADP cana-4512 114 31 the the DET cana-4512 114 32 fies fie NOUN cana-4512 114 33 of of ADP cana-4512 114 34 various various ADJ cana-4512 114 35 kind kind NOUN cana-4512 114 36 . . PUNCT cana-4512 115 1 we we PRON cana-4512 115 2 obtain obtain VERB cana-4512 115 3 m m PROPN cana-4512 115 4 - - PUNCT cana-4512 115 5 adm adm PROPN cana-4512 115 6 is be AUX cana-4512 115 7 more more ADV cana-4512 115 8 easy easy ADJ cana-4512 115 9 than than ADP cana-4512 115 10 other other ADJ cana-4512 115 11 method method NOUN cana-4512 115 12 . . PUNCT cana-4512 116 1 references reference NOUN cana-4512 116 2 [ [ X cana-4512 116 3 1 1 NUM cana-4512 116 4 ] ] PUNCT cana-4512 116 5 a.m.wazwaz a.m.wazwaz NOUN cana-4512 116 6 , , PUNCT cana-4512 116 7 β€œ " PUNCT cana-4512 116 8 a a DET cana-4512 116 9 first first ADJ cana-4512 116 10 course course NOUN cana-4512 116 11 in in ADP cana-4512 116 12 integral integral ADJ cana-4512 116 13 equationss equations NOUN cana-4512 116 14 ” " PUNCT cana-4512 116 15 , , PUNCT cana-4512 116 16 world world NOUN cana-4512 116 17 scientific scientific PROPN cana-4512 116 18 , , PUNCT cana-4512 116 19 singapore singapore PROPN cana-4512 116 20 , , PUNCT cana-4512 116 21 ( ( PUNCT cana-4512 116 22 1997 1997 NUM cana-4512 116 23 ) ) PUNCT cana-4512 116 24 . . PUNCT cana-4512 117 1 [ [ X cana-4512 117 2 2 2 NUM cana-4512 117 3 ] ] PUNCT cana-4512 117 4 a.m. a.m. NOUN cana-4512 117 5 wazwaz wazwaz PROPN cana-4512 117 6 , , PUNCT cana-4512 117 7 β€œ " PUNCT cana-4512 117 8 a a DET cana-4512 117 9 reliable reliable ADJ cana-4512 117 10 modification modification NOUN cana-4512 117 11 of of ADP cana-4512 117 12 the the DET cana-4512 117 13 adomian adomian NOUN cana-4512 117 14 decomposition decomposition NOUN cana-4512 117 15 method method NOUN cana-4512 117 16 ” " PUNCT cana-4512 117 17 , , PUNCT cana-4512 117 18 appl.math.comput appl.math.comput PROPN cana-4512 117 19 . . PUNCT cana-4512 117 20 , , PUNCT cana-4512 117 21 102 102 NUM cana-4512 117 22 ( ( PUNCT cana-4512 117 23 1999 1999 NUM cana-4512 117 24 ) ) PUNCT cana-4512 117 25 77–86 77–86 NUM cana-4512 117 26 . . PUNCT cana-4512 118 1 [ [ X cana-4512 118 2 3 3 X cana-4512 118 3 ] ] PUNCT cana-4512 118 4 a.m.wazwaz,β€œnecessary a.m.wazwaz,β€œnecessary ADJ cana-4512 118 5 conditions condition NOUN cana-4512 118 6 for for ADP cana-4512 118 7 the the DET cana-4512 118 8 appearance appearance NOUN cana-4512 118 9 of of ADP cana-4512 118 10 noise noise NOUN cana-4512 118 11 terms term NOUN cana-4512 118 12 in in ADP cana-4512 118 13 decomposition decomposition NOUN cana-4512 118 14 solution solution NOUN cana-4512 118 15 series series NOUN cana-4512 118 16 ” " PUNCT cana-4512 118 17 , , PUNCT cana-4512 118 18 appl appl PROPN cana-4512 118 19 . . PUNCT cana-4512 119 1 math.comput math.comput PROPN cana-4512 119 2 . . PUNCT cana-4512 119 3 , , PUNCT cana-4512 119 4 81 81 NUM cana-4512 119 5 ( ( PUNCT cana-4512 119 6 1997 1997 NUM cana-4512 119 7 ) ) PUNCT cana-4512 120 1 265–274 265–274 NUM cana-4512 120 2 . . PUNCT cana-4512 121 1 [ [ X cana-4512 121 2 4 4 NUM cana-4512 121 3 ] ] PUNCT cana-4512 121 4 a.m. a.m. NOUN cana-4512 121 5 wazwaz wazwaz PROPN cana-4512 121 6 , , PUNCT cana-4512 121 7 β€œ " PUNCT cana-4512 121 8 partial partial ADJ cana-4512 121 9 differential differential NOUN cana-4512 121 10 equations equation NOUN cana-4512 121 11 and and CCONJ cana-4512 121 12 solitary solitary ADJ cana-4512 121 13 waves wave NOUN cana-4512 121 14 theory theory NOUN cana-4512 121 15 ” " PUNCT cana-4512 121 16 , , PUNCT cana-4512 121 17 hep hep NOUN cana-4512 121 18 and and CCONJ cana-4512 121 19 springer springer NOUN cana-4512 121 20 , , PUNCT cana-4512 121 21 beijing beijing PROPN cana-4512 121 22 and and CCONJ cana-4512 121 23 berlin berlin PROPN cana-4512 121 24 , , PUNCT cana-4512 121 25 ( ( PUNCT cana-4512 121 26 2009 2009 NUM cana-4512 121 27 ) ) PUNCT cana-4512 121 28 . . PUNCT cana-4512 122 1 [ [ X cana-4512 122 2 5 5 NUM cana-4512 122 3 ] ] PUNCT cana-4512 122 4 a. a. PROPN cana-4512 122 5 el el PROPN cana-4512 122 6 - - PROPN cana-4512 122 7 sappagh sappagh PROPN cana-4512 122 8 , , PUNCT cana-4512 122 9 ahmed ahmed PROPN cana-4512 122 10 m.a m.a PROPN cana-4512 122 11 . . PROPN cana-4512 122 12 el el PROPN cana-4512 122 13 - - PUNCT cana-4512 122 14 sayed say VERB cana-4512 122 15 , , PUNCT cana-4512 122 16 and and CCONJ cana-4512 122 17 h.m h.m PROPN cana-4512 122 18 . . PROPN cana-4512 122 19 ahmed ahmed PROPN cana-4512 122 20 : : PUNCT cana-4512 122 21 numerical numerical ADJ cana-4512 122 22 , , PUNCT cana-4512 122 23 ” " PUNCT cana-4512 122 24 solution solution NOUN cana-4512 122 25 of of ADP cana-4512 122 26 vies vie NOUN cana-4512 122 27 using use VERB cana-4512 122 28 the the DET cana-4512 122 29 first first ADJ cana-4512 122 30 - - PUNCT cana-4512 122 31 order order NOUN cana-4512 122 32 recursive recursive ADJ cana-4512 122 33 filters filter NOUN cana-4512 122 34 method method NOUN cana-4512 122 35 ” " PUNCT cana-4512 122 36 , , PUNCT cana-4512 122 37 journal journal NOUN cana-4512 122 38 of of ADP cana-4512 122 39 computational computational ADJ cana-4512 122 40 and and CCONJ cana-4512 122 41 appiied appiie VERB cana-4512 122 42 mathematics mathematic NOUN cana-4512 122 43 . . PUNCT cana-4512 123 1 [ [ X cana-4512 123 2 6 6 NUM cana-4512 123 3 ] ] X cana-4512 123 4 c. c. PROPN cana-4512 123 5 constanda,”integral constanda,”integral PROPN cana-4512 123 6 equations equation NOUN cana-4512 123 7 of of ADP cana-4512 123 8 the the DET cana-4512 123 9 first first ADJ cana-4512 123 10 kind kind NOUN cana-4512 123 11 in in ADP cana-4512 123 12 plane plane NOUN cana-4512 123 13 elasticity elasticity NOUN cana-4512 123 14 ” " PUNCT cana-4512 123 15 , , PUNCT cana-4512 123 16 j. j. PROPN cana-4512 123 17 quart quart PROPN cana-4512 123 18 . . PUNCT cana-4512 124 1 appl appl PROPN cana-4512 124 2 . . PROPN cana-4512 124 3 math math NOUN cana-4512 124 4 . . PUNCT cana-4512 125 1 l111 l111 PROPN cana-4512 125 2 ( ( PUNCT cana-4512 125 3 4 4 NUM cana-4512 125 4 ) ) PUNCT cana-4512 125 5 ( ( PUNCT cana-4512 125 6 1995 1995 NUM cana-4512 125 7 ) ) PUNCT cana-4512 126 1 783–793 783–793 NUM cana-4512 126 2 . . PUNCT cana-4512 127 1 [ [ X cana-4512 127 2 7 7 NUM cana-4512 127 3 ] ] SYM cana-4512 127 4 d.bahuguna d.bahuguna NOUN cana-4512 127 5 , , PUNCT cana-4512 127 6 β€œ " PUNCT cana-4512 127 7 a a DET cana-4512 127 8 comparative comparative ADJ cana-4512 127 9 study study NOUN cana-4512 127 10 of of ADP cana-4512 127 11 numerical numerical ADJ cana-4512 127 12 methods method NOUN cana-4512 127 13 for for ADP cana-4512 127 14 solving solve VERB cana-4512 127 15 an an DET cana-4512 127 16 integro integro ADJ cana-4512 127 17 - - PUNCT cana-4512 127 18 differential differential NOUN cana-4512 127 19 equation”,computers equation”,computer NOUN cana-4512 127 20 and and CCONJ cana-4512 127 21 mathematics mathematic NOUN cana-4512 127 22 with with ADP cana-4512 127 23 applications application NOUN cana-4512 127 24 57 57 NUM cana-4512 127 25 , , PUNCT cana-4512 127 26 1485 1485 NUM cana-4512 127 27 – – PUNCT cana-4512 127 28 1493 1493 NUM cana-4512 127 29 ( ( PUNCT cana-4512 127 30 2009 2009 NUM cana-4512 127 31 ) ) PUNCT cana-4512 127 32 . . PUNCT cana-4512 128 1 [ [ X cana-4512 128 2 8 8 NUM cana-4512 128 3 ] ] X cana-4512 128 4 d. d. PROPN cana-4512 128 5 evans evans PROPN cana-4512 128 6 , , PUNCT cana-4512 128 7 ” " PUNCT cana-4512 128 8 the the DET cana-4512 128 9 adomian adomian NOUN cana-4512 128 10 decomposition decomposition NOUN cana-4512 128 11 method method NOUN cana-4512 128 12 for for ADP cana-4512 128 13 solving solve VERB cana-4512 128 14 delay delay NOUN cana-4512 128 15 differential differential ADJ cana-4512 128 16 equation equation NOUN cana-4512 128 17 ” " PUNCT cana-4512 128 18 , , PUNCT cana-4512 128 19 international international ADJ cana-4512 128 20 journal journal NOUN cana-4512 128 21 of of ADP cana-4512 128 22 computer computer NOUN cana-4512 128 23 mathematics mathematic NOUN cana-4512 128 24 00 00 NUM cana-4512 128 25 , , PUNCT cana-4512 128 26 1–6 1–6 NUM cana-4512 128 27 ( ( PUNCT cana-4512 128 28 2004 2004 NUM cana-4512 128 29 ) ) PUNCT cana-4512 128 30 . . PUNCT cana-4512 129 1 [ [ X cana-4512 129 2 9 9 NUM cana-4512 129 3 ] ] X cana-4512 129 4 d.l d.l NOUN cana-4512 129 5 . . PUNCT cana-4512 130 1 phillips,”a phillips,”a NOUN cana-4512 130 2 technique technique NOUN cana-4512 130 3 for for ADP cana-4512 130 4 the the DET cana-4512 130 5 numerical numerical ADJ cana-4512 130 6 solution solution NOUN cana-4512 130 7 of of ADP cana-4512 130 8 certain certain ADJ cana-4512 130 9 ies ie NOUN cana-4512 130 10 of of ADP cana-4512 130 11 the the DET cana-4512 130 12 first first ADJ cana-4512 130 13 kind kind NOUN cana-4512 130 14 ” " PUNCT cana-4512 130 15 , , PUNCT cana-4512 130 16 j. j. PROPN cana-4512 130 17 assoc assoc PROPN cana-4512 130 18 . . PUNCT cana-4512 131 1 comput comput PROPN cana-4512 131 2 . . PUNCT cana-4512 132 1 mach mach PROPN cana-4512 132 2 , , PUNCT cana-4512 132 3 9 9 NUM cana-4512 132 4 ( ( PUNCT cana-4512 132 5 1962 1962 NUM cana-4512 132 6 ) ) PUNCT cana-4512 132 7 84–96 84–96 NOUN cana-4512 132 8 . . PUNCT cana-4512 133 1 theory theory NOUN cana-4512 133 2 to to ADP cana-4512 133 3 applications”,cambridge,(2004 applications”,cambridge,(2004 PROPN cana-4512 133 4 ) ) PUNCT cana-4512 133 5 . . PUNCT cana-4512 134 1 [ [ X cana-4512 134 2 10 10 NUM cana-4512 134 3 ] ] X cana-4512 134 4 e.babolian e.babolian NOUN cana-4512 134 5 , , PUNCT cana-4512 134 6 j.biazar j.biazar NOUN cana-4512 134 7 and and CCONJ cana-4512 134 8 a.vahidi a.vahidi NUM cana-4512 134 9 , , PUNCT cana-4512 134 10 ” " PUNCT cana-4512 134 11 the the DET cana-4512 134 12 decomposition decomposition NOUN cana-4512 134 13 method method NOUN cana-4512 134 14 appiied appiie VERB cana-4512 134 15 to to ADP cana-4512 134 16 systems system NOUN cana-4512 134 17 of of ADP cana-4512 134 18 fredholm fredholm ADJ cana-4512 134 19 integral integral ADJ cana-4512 134 20 equations equation NOUN cana-4512 134 21 of of ADP cana-4512 134 22 the the DET cana-4512 134 23 second second ADJ cana-4512 134 24 kind kind NOUN cana-4512 134 25 , , PUNCT cana-4512 134 26 ” " PUNCT cana-4512 134 27 appiied appiie VERB cana-4512 134 28 mathematics mathematic NOUN cana-4512 134 29 computation computation NOUN cana-4512 134 30 and and CCONJ cana-4512 134 31 , , PUNCT cana-4512 134 32 vol.148,no.2,pp.443452,2004 vol.148,no.2,pp.443452,2004 NOUN cana-4512 134 33 . . PUNCT cana-4512 135 1 [ [ X cana-4512 135 2 11 11 NUM cana-4512 135 3 ] ] PUNCT cana-4512 135 4 e.babolian e.babolian NOUN cana-4512 135 5 , , PUNCT cana-4512 135 6 j.biazar j.biazar NOUN cana-4512 135 7 , , PUNCT cana-4512 135 8 β€œ " PUNCT cana-4512 135 9 solution solution NOUN cana-4512 135 10 of of ADP cana-4512 135 11 a a DET cana-4512 135 12 system system NOUN cana-4512 135 13 of of ADP cana-4512 135 14 nonlinear nonlinear PROPN cana-4512 135 15 volterra volterra PROPN cana-4512 135 16 equations equation NOUN cana-4512 135 17 by by ADP cana-4512 135 18 adomian adomian NOUN cana-4512 135 19 decomposition decomposition NOUN cana-4512 135 20 method method NOUN cana-4512 135 21 , , PUNCT cana-4512 135 22 ” " PUNCT cana-4512 135 23 far far ADV cana-4512 135 24 east east PROPN cana-4512 135 25 j. j. PROPN cana-4512 135 26 math math PROPN cana-4512 135 27 . . PUNCT cana-4512 136 1 sci sci PROPN cana-4512 136 2 . . PROPN cana-4512 136 3 2 2 NUM cana-4512 136 4 ( ( PUNCT cana-4512 136 5 6 6 NUM cana-4512 136 6 ) ) PUNCT cana-4512 136 7 ( ( PUNCT cana-4512 136 8 2000 2000 NUM cana-4512 136 9 ) ) PUNCT cana-4512 137 1 935–945 935–945 NUM cana-4512 137 2 . . PUNCT cana-4512 138 1 [ [ X cana-4512 138 2 12 12 NUM cana-4512 138 3 ] ] X cana-4512 138 4 g g PROPN cana-4512 138 5 .. .. SYM cana-4512 138 6 adomian adomian NOUN cana-4512 138 7 , , PUNCT cana-4512 138 8 β€œ " PUNCT cana-4512 138 9 a a DET cana-4512 138 10 review review NOUN cana-4512 138 11 of of ADP cana-4512 138 12 the the DET cana-4512 138 13 decomposition decomposition NOUN cana-4512 138 14 method method NOUN cana-4512 138 15 in in ADP cana-4512 138 16 appiied appiied ADJ cana-4512 138 17 mathematics mathematic NOUN cana-4512 138 18 ” " PUNCT cana-4512 138 19 . . PUNCT cana-4512 139 1 journal journal PROPN cana-4512 139 2 of of ADP cana-4512 139 3 mathematical mathematical ADJ cana-4512 139 4 analysis analysis NOUN cana-4512 139 5 and and CCONJ cana-4512 139 6 applications application NOUN cana-4512 139 7 135 135 NUM cana-4512 139 8 , , PUNCT cana-4512 139 9 501–544,(1988 501–544,(1988 NUM cana-4512 139 10 ) ) PUNCT cana-4512 139 11 . . PUNCT cana-4512 140 1 communications communication NOUN cana-4512 140 2 on on ADP cana-4512 140 3 applied apply VERB cana-4512 140 4 nonlinear nonlinear ADJ cana-4512 140 5 analysis analysis NOUN cana-4512 140 6 issn issn NOUN cana-4512 140 7 : : PUNCT cana-4512 140 8 1074 1074 NUM cana-4512 140 9 - - PUNCT cana-4512 140 10 133x 133x NUM cana-4512 140 11 vol vol NOUN cana-4512 140 12 32 32 NUM cana-4512 140 13 no no NOUN cana-4512 140 14 . . PUNCT cana-4512 141 1 9s 9s NUM cana-4512 141 2 ( ( PUNCT cana-4512 141 3 2025 2025 NUM cana-4512 141 4 ) ) PUNCT cana-4512 141 5 2278 2278 NUM cana-4512 141 6 https://internationalpubls.com https://internationalpubls.com X cana-4512 142 1 [ [ X cana-4512 142 2 13 13 NUM cana-4512 142 3 ] ] X cana-4512 142 4 g.adomian,”a g.adomian,”a NOUN cana-4512 142 5 review review NOUN cana-4512 142 6 of of ADP cana-4512 142 7 the the DET cana-4512 142 8 decomposition decomposition NOUN cana-4512 142 9 method method NOUN cana-4512 142 10 and and CCONJ cana-4512 142 11 some some DET cana-4512 142 12 recent recent ADJ cana-4512 142 13 results result NOUN cana-4512 142 14 for for ADP cana-4512 142 15 non non ADJ cana-4512 142 16 linear linear PROPN cana-4512 142 17 equations equation NOUN cana-4512 142 18 ” " PUNCT cana-4512 142 19 . . PUNCT cana-4512 143 1 mathematics mathematic NOUN cana-4512 143 2 computer computer NOUN cana-4512 143 3 and and CCONJ cana-4512 143 4 modelling model VERB cana-4512 143 5 13 13 NUM cana-4512 143 6 , , PUNCT cana-4512 143 7 17–43,(1990 17–43,(1990 NUM cana-4512 143 8 ) ) PUNCT cana-4512 144 1 [ [ X cana-4512 144 2 14 14 NUM cana-4512 144 3 ] ] X cana-4512 144 4 g. g. PROPN cana-4512 144 5 adomian adomian PROPN cana-4512 144 6 , , PUNCT cana-4512 144 7 β€œ " PUNCT cana-4512 144 8 modified modified ADJ cana-4512 144 9 adomian adomian NOUN cana-4512 144 10 polynomials polynomial NOUN cana-4512 144 11 ” " PUNCT cana-4512 144 12 . . PUNCT cana-4512 145 1 mathematics mathematic NOUN cana-4512 145 2 and and CCONJ cana-4512 145 3 computer computer NOUN cana-4512 145 4 modelling model VERB cana-4512 145 5 24 24 NUM cana-4512 145 6 , , PUNCT cana-4512 145 7 39–46(1996 39–46(1996 NOUN cana-4512 145 8 ) ) PUNCT cana-4512 145 9 . . PUNCT cana-4512 146 1 [ [ X cana-4512 146 2 15 15 NUM cana-4512 146 3 ] ] X cana-4512 146 4 g g NOUN cana-4512 146 5 adomian adomian NOUN cana-4512 146 6 , , PUNCT cana-4512 146 7 β€œ " PUNCT cana-4512 146 8 noise noise NOUN cana-4512 146 9 terms term NOUN cana-4512 146 10 in in ADP cana-4512 146 11 decomposition decomposition NOUN cana-4512 146 12 series series NOUN cana-4512 146 13 solution solution NOUN cana-4512 146 14 ” " PUNCT cana-4512 146 15 , , PUNCT cana-4512 146 16 comput.math comput.math NOUN cana-4512 146 17 . . PUNCT cana-4512 146 18 appl appl PROPN cana-4512 146 19 . . PROPN cana-4512 146 20 , , PUNCT cana-4512 146 21 24(1992 24(1992 NUM cana-4512 146 22 ) ) PUNCT cana-4512 146 23 61–64 61–64 NUM cana-4512 146 24 . . PUNCT cana-4512 147 1 [ [ X cana-4512 147 2 16 16 NUM cana-4512 147 3 ] ] X cana-4512 147 4 g.mustafa g.mustafa PROPN cana-4512 147 5 and and CCONJ cana-4512 147 6 nasir nasir PROPN cana-4512 147 7 ali ali PROPN cana-4512 147 8 khan”numerical khan”numerical PROPN cana-4512 147 9 solution solution NOUN cana-4512 147 10 of of ADP cana-4512 147 11 vies vie NOUN cana-4512 147 12 with with ADP cana-4512 147 13 delay delay NOUN cana-4512 147 14 using use VERB cana-4512 147 15 block block NOUN cana-4512 147 16 methods method NOUN cana-4512 147 17 ” " PUNCT cana-4512 147 18 , , PUNCT cana-4512 147 19 appiied appiie VERB cana-4512 147 20 mathematics mathematic NOUN cana-4512 147 21 and and CCONJ cana-4512 147 22 computation computation NOUN cana-4512 147 23 [ [ X cana-4512 147 24 17 17 NUM cana-4512 147 25 ] ] X cana-4512 147 26 i.hashim,β€œadomian i.hashim,β€œadomian ADJ cana-4512 147 27 decomposition decomposition NOUN cana-4512 147 28 method method NOUN cana-4512 147 29 for for ADP cana-4512 147 30 solving solve VERB cana-4512 147 31 bvps bvps NOUN cana-4512 147 32 for for ADP cana-4512 147 33 fourth fourth ADJ cana-4512 147 34 - - PUNCT cana-4512 147 35 order order NOUN cana-4512 147 36 integro integro PROPN cana-4512 147 37 differential differential ADJ cana-4512 147 38 equations equation NOUN cana-4512 147 39 ” " PUNCT cana-4512 147 40 , , PUNCT cana-4512 147 41 journal journal NOUN cana-4512 147 42 of of ADP cana-4512 147 43 computational computational ADJ cana-4512 147 44 and and CCONJ cana-4512 147 45 appiied appiie VERB cana-4512 147 46 mathematics mathematic NOUN cana-4512 147 47 193 193 NUM cana-4512 147 48 , , PUNCT cana-4512 147 49 658–664 658–664 NUM cana-4512 147 50 ( ( PUNCT cana-4512 147 51 2006 2006 NUM cana-4512 147 52 ) ) PUNCT cana-4512 147 53 . . PUNCT cana-4512 148 1 [ [ X cana-4512 148 2 18 18 NUM cana-4512 148 3 ] ] X cana-4512 148 4 j. j. PROPN cana-4512 148 5 biazar biazar PROPN cana-4512 148 6 ” " PUNCT cana-4512 148 7 , , PUNCT cana-4512 148 8 solution solution NOUN cana-4512 148 9 of of ADP cana-4512 148 10 the the DET cana-4512 148 11 system system NOUN cana-4512 148 12 of of ADP cana-4512 148 13 ordinary ordinary ADJ cana-4512 148 14 differential differential ADJ cana-4512 148 15 equations equation NOUN cana-4512 148 16 by by ADP cana-4512 148 17 adomian adomian NOUN cana-4512 148 18 decomposition decomposition NOUN cana-4512 148 19 method method NOUN cana-4512 148 20 ” " PUNCT cana-4512 148 21 , , PUNCT cana-4512 148 22 appiied appiie VERB cana-4512 148 23 mathematics mathematic NOUN cana-4512 148 24 and and CCONJ cana-4512 148 25 computation computation NOUN cana-4512 148 26 147 147 NUM cana-4512 148 27 , , PUNCT cana-4512 148 28 713–719 713–719 NUM cana-4512 148 29 ( ( PUNCT cana-4512 148 30 2004 2004 NUM cana-4512 148 31 ) ) PUNCT cana-4512 148 32 . . PUNCT cana-4512 149 1 [ [ X cana-4512 149 2 19 19 NUM cana-4512 149 3 ] ] X cana-4512 149 4 j. j. PROPN cana-4512 149 5 biazar biazar PROPN cana-4512 149 6 and and CCONJ cana-4512 149 7 h. h. PROPN cana-4512 149 8 ebrahimi ebrahimi PROPN cana-4512 149 9 , , PUNCT cana-4512 149 10 β€œ " PUNCT cana-4512 149 11 iteration iteration NOUN cana-4512 149 12 method method NOUN cana-4512 149 13 for for ADP cana-4512 149 14 fies fie NOUN cana-4512 149 15 of of ADP cana-4512 149 16 second second ADJ cana-4512 149 17 kind kind NOUN cana-4512 149 18 , , PUNCT cana-4512 149 19 ” " PUNCT cana-4512 149 20 iranian iranian ADJ cana-4512 149 21 journal journal PROPN cana-4512 149 22 of of ADP cana-4512 149 23 optimization optimization NOUN cana-4512 149 24 , , PUNCT cana-4512 149 25 vol vol NOUN cana-4512 149 26 . . PROPN cana-4512 149 27 1 1 NUM cana-4512 149 28 , , PUNCT cana-4512 149 29 pp pp ADJ cana-4512 149 30 . . PUNCT cana-4512 150 1 13–23 13–23 NUM cana-4512 150 2 , , PUNCT cana-4512 150 3 2009 2009 NUM cana-4512 150 4 . . PUNCT cana-4512 151 1 [ [ X cana-4512 151 2 20 20 NUM cana-4512 151 3 ] ] X cana-4512 151 4 j j PROPN cana-4512 151 5 biazar biazar NOUN cana-4512 151 6 , , PUNCT cana-4512 151 7 ” " PUNCT cana-4512 151 8 extracting extract VERB cana-4512 151 9 a a DET cana-4512 151 10 general general ADJ cana-4512 151 11 iterative iterative NOUN cana-4512 151 12 method method NOUN cana-4512 151 13 from from ADP cana-4512 151 14 an an DET cana-4512 151 15 adomian adomian NOUN cana-4512 151 16 decomposition decomposition NOUN cana-4512 151 17 method method NOUN cana-4512 151 18 and and CCONJ cana-4512 151 19 comparing compare VERB cana-4512 151 20 it it PRON cana-4512 151 21 to to ADP cana-4512 151 22 the the DET cana-4512 151 23 variational variational ADJ cana-4512 151 24 iteration iteration NOUN cana-4512 151 25 method”computers method”computer NOUN cana-4512 151 26 and and CCONJ cana-4512 151 27 mathematics mathematic NOUN cana-4512 151 28 with with ADP cana-4512 151 29 applications application NOUN cana-4512 151 30 59 59 NUM cana-4512 151 31 , , PUNCT cana-4512 151 32 622–628 622–628 NUM cana-4512 151 33 ( ( PUNCT cana-4512 151 34 2010 2010 NUM cana-4512 151 35 ) ) PUNCT cana-4512 151 36 . . PUNCT cana-4512 152 1 [ [ X cana-4512 152 2 21 21 NUM cana-4512 152 3 ] ] SYM cana-4512 152 4 j.bizar j.bizar NOUN cana-4512 152 5 and and CCONJ cana-4512 152 6 ebrahimi ebrahimi PROPN cana-4512 152 7 h. h. PROPN cana-4512 152 8 , , PUNCT cana-4512 152 9 ( ( PUNCT cana-4512 152 10 2009 2009 NUM cana-4512 152 11 ) ) PUNCT cana-4512 152 12 , , PUNCT cana-4512 152 13 β€œ " PUNCT cana-4512 152 14 variational variational ADJ cana-4512 152 15 iteration iteration NOUN cana-4512 152 16 method method NOUN cana-4512 152 17 for for ADP cana-4512 152 18 fredholm fredholm ADJ cana-4512 152 19 integral integral ADJ cana-4512 152 20 equations equation NOUN cana-4512 152 21 of of ADP cana-4512 152 22 the the DET cana-4512 152 23 second second ADJ cana-4512 152 24 kind kind NOUN cana-4512 152 25 ” " PUNCT cana-4512 152 26 , , PUNCT cana-4512 152 27 iranian iranian ADJ cana-4512 152 28 journal journal NOUN cana-4512 152 29 of of ADP cana-4512 152 30 optimization optimization NOUN cana-4512 152 31 , , PUNCT cana-4512 152 32 no.1,pp no.1,pp ADJ cana-4512 152 33 . . PROPN cana-4512 153 1 13 13 NUM cana-4512 153 2 - - SYM cana-4512 153 3 16 16 NUM cana-4512 153 4 . . PUNCT cana-4512 154 1 [ [ X cana-4512 154 2 22 22 NUM cana-4512 154 3 ] ] X cana-4512 154 4 j. j. PROPN cana-4512 154 5 biazar biazar PROPN cana-4512 154 6 and and CCONJ cana-4512 154 7 h. h. PROPN cana-4512 154 8 ebrahimi ebrahimi PROPN cana-4512 154 9 , , PUNCT cana-4512 154 10 β€œ " PUNCT cana-4512 154 11 iteration iteration NOUN cana-4512 154 12 method method NOUN cana-4512 154 13 for for ADP cana-4512 154 14 fies fie NOUN cana-4512 154 15 of of ADP cana-4512 154 16 second second ADJ cana-4512 154 17 kind kind NOUN cana-4512 154 18 ” " PUNCT cana-4512 154 19 , , PUNCT cana-4512 154 20 iranian iranian ADJ cana-4512 154 21 journal journal NOUN cana-4512 154 22 of of ADP cana-4512 154 23 optimization optimization NOUN cana-4512 154 24 , , PUNCT cana-4512 154 25 vol vol NOUN cana-4512 154 26 . . PROPN cana-4512 154 27 1 1 NUM cana-4512 154 28 , , PUNCT cana-4512 154 29 pp pp ADJ cana-4512 154 30 . . PUNCT cana-4512 155 1 13 13 NUM cana-4512 155 2 - - SYM cana-4512 155 3 23 23 NUM cana-4512 155 4 , , PUNCT cana-4512 155 5 2009 2009 NUM cana-4512 155 6 . . PUNCT cana-4512 156 1 [ [ X cana-4512 156 2 23 23 NUM cana-4512 156 3 ] ] X cana-4512 156 4 j. j. PROPN cana-4512 156 5 biazar biazar PROPN cana-4512 156 6 and and CCONJ cana-4512 156 7 h. h. PROPN cana-4512 156 8 ebrahimi ebrahimi PROPN cana-4512 156 9 , , PUNCT cana-4512 156 10 β€œ " PUNCT cana-4512 156 11 iteration iteration NOUN cana-4512 156 12 method method NOUN cana-4512 156 13 for for ADP cana-4512 156 14 fies fie NOUN cana-4512 156 15 of of ADP cana-4512 156 16 second second ADJ cana-4512 156 17 β€œ " PUNCT cana-4512 156 18 ” " PUNCT cana-4512 156 19 , , PUNCT cana-4512 156 20 iranian iranian ADJ cana-4512 156 21 journal journal NOUN cana-4512 156 22 of of ADP cana-4512 156 23 optimization optimization NOUN cana-4512 156 24 . . PUNCT cana-4512 157 1 [ [ X cana-4512 157 2 24 24 NUM cana-4512 157 3 ] ] PUNCT cana-4512 157 4 j. j. PROPN cana-4512 157 5 e. e. PROPN cana-4512 157 6 mamadu mamadu PROPN cana-4512 157 7 and and CCONJ cana-4512 157 8 ignatius ignatius PROPN cana-4512 157 9 n. n. PROPN cana-4512 157 10 njoseh njoseh PROPN cana-4512 157 11 , , PUNCT cana-4512 157 12 β€œ " PUNCT cana-4512 157 13 numerical numerical ADJ cana-4512 157 14 solutions solution NOUN cana-4512 157 15 of of ADP cana-4512 157 16 volterra volterra NOUN cana-4512 157 17 equations equation NOUN cana-4512 157 18 using use VERB cana-4512 157 19 galerkin galerkin ADJ cana-4512 157 20 method method NOUN cana-4512 157 21 with with ADP cana-4512 157 22 certain certain ADJ cana-4512 157 23 orthogonal orthogonal ADJ cana-4512 157 24 polynomials polynomial NOUN cana-4512 157 25 ” " PUNCT cana-4512 157 26 , , PUNCT cana-4512 157 27 journal journal NOUN cana-4512 157 28 of of ADP cana-4512 157 29 appiied appiie VERB cana-4512 157 30 mathematics mathematic NOUN cana-4512 157 31 and and CCONJ cana-4512 157 32 physics physics PROPN cana-4512 157 33 vol.4 vol.4 PROPN cana-4512 157 34 , , PUNCT cana-4512 157 35 pp.376 pp.376 PROPN cana-4512 157 36 - - NOUN cana-4512 157 37 382,2016 382,2016 NUM cana-4512 157 38 . . PUNCT cana-4512 158 1 [ [ X cana-4512 158 2 25 25 NUM cana-4512 158 3 ] ] PUNCT cana-4512 158 4 j. j. PROPN cana-4512 158 5 h. h. PROPN cana-4512 158 6 , , PUNCT cana-4512 158 7 hosseinzadeh hosseinzadeh PROPN cana-4512 158 8 , , PUNCT cana-4512 158 9 h. h. PROPN cana-4512 158 10 and and CCONJ cana-4512 158 11 mohamadzadeh mohamadzadeh PROPN cana-4512 158 12 , , PUNCT cana-4512 158 13 β€œ " PUNCT cana-4512 158 14 numerical numerical ADJ cana-4512 158 15 solution solution NOUN cana-4512 158 16 of of ADP cana-4512 158 17 systemof systemof ADJ cana-4512 158 18 linear linear ADJ cana-4512 158 19 integral integral ADJ cana-4512 158 20 equations equation NOUN cana-4512 158 21 by by ADP cana-4512 158 22 using use VERB cana-4512 158 23 legendre legendre PROPN cana-4512 158 24 wavelets wavelet NOUN cana-4512 158 25 ” " PUNCT cana-4512 158 26 , , PUNCT cana-4512 158 27 int int NOUN cana-4512 158 28 . . PUNCT cana-4512 159 1 j. j. PROPN cana-4512 159 2 open open PROPN cana-4512 159 3 problems problem NOUN cana-4512 159 4 compt compt VERB cana-4512 159 5 . . PUNCT cana-4512 160 1 math math NOUN cana-4512 160 2 . . PUNCT cana-4512 161 1 ,3 ,3 PROPN cana-4512 161 2 , , PUNCT cana-4512 161 3 63 63 NUM cana-4512 161 4 - - SYM cana-4512 161 5 71,2010 71,2010 NUM cana-4512 161 6 . . PUNCT