id sid tid token lemma pos iajs-1918 1 1 microsoft microsoft PROPN iajs-1918 1 2 word word NOUN iajs-1918 1 3 178 178 NUM iajs-1918 1 4 - - SYM iajs-1918 1 5 186 186 NUM iajs-1918 1 6     SPACE iajs-1918 1 7 178 178 NUM iajs-1918 1 8 ibn ibn PROPN iajs-1918 1 9 al al PROPN iajs-1918 1 10 - - PUNCT iajs-1918 1 11 haitham haitham PROPN iajs-1918 1 12 jour.for jour.for PROPN iajs-1918 1 13 pure&appl.sci pure&appl.sci PROPN iajs-1918 1 14 . . PUNCT iajs-1918 1 15 ihjpas ihjpas PROPN iajs-1918 1 16 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 1 17 vol vol NOUN iajs-1918 1 18 . . PUNCT iajs-1918 1 19 32 32 NUM iajs-1918 1 20 ( ( PUNCT iajs-1918 1 21 1 1 NUM iajs-1918 1 22 ) ) PUNCT iajs-1918 1 23 2019 2019 NUM iajs-1918 1 24 a a DET iajs-1918 1 25 parallel parallel ADJ iajs-1918 1 26 numerical numerical ADJ iajs-1918 1 27 algorithm algorithm NOUN iajs-1918 1 28 for for ADP iajs-1918 1 29 solving solve VERB iajs-1918 1 30 some some DET iajs-1918 1 31 fractional fractional ADJ iajs-1918 1 32 integral integral ADJ iajs-1918 1 33 equations equation NOUN iajs-1918 1 34 khalid khalid PROPN iajs-1918 1 35 mindeel mindeel PROPN iajs-1918 1 36 mohammed mohammed PROPN iajs-1918 1 37 al al PROPN iajs-1918 1 38 - - PUNCT iajs-1918 1 39 abrahemee abrahemee PROPN iajs-1918 1 40 khalid.mohammed@qu.edu.iq khalid.mohammed@qu.edu.iq PROPN iajs-1918 1 41 department department PROPN iajs-1918 1 42 of of ADP iajs-1918 1 43 mathematics mathematics PROPN iajs-1918 1 44 , , PUNCT iajs-1918 1 45 college college NOUN iajs-1918 1 46 of of ADP iajs-1918 1 47 education education NOUN iajs-1918 1 48 , , PUNCT iajs-1918 1 49 university university NOUN iajs-1918 1 50 of of ADP iajs-1918 1 51 al al PROPN iajs-1918 1 52 - - PUNCT iajs-1918 1 53 qadisiyhah qadisiyhah NOUN iajs-1918 1 54 , , PUNCT iajs-1918 1 55 qadisiyhah qadisiyhah NOUN iajs-1918 1 56 , , PUNCT iajs-1918 1 57 iraq iraq PROPN iajs-1918 1 58 . . PUNCT iajs-1918 2 1 article article NOUN iajs-1918 2 2 history history NOUN iajs-1918 2 3 : : PUNCT iajs-1918 2 4 received receive VERB iajs-1918 2 5 12 12 NUM iajs-1918 2 6 august august PROPN iajs-1918 2 7 2018 2018 NUM iajs-1918 2 8 , , PUNCT iajs-1918 2 9 accepted accept VERB iajs-1918 2 10 26 26 NUM iajs-1918 2 11 september september PROPN iajs-1918 2 12 2018 2018 NUM iajs-1918 2 13 , , PUNCT iajs-1918 2 14 publish publish VERB iajs-1918 2 15 january january PROPN iajs-1918 2 16 2019 2019 NUM iajs-1918 2 17 abstract abstract ADV iajs-1918 2 18 in in ADP iajs-1918 2 19 this this DET iajs-1918 2 20 study study NOUN iajs-1918 2 21 , , PUNCT iajs-1918 2 22 he he PRON iajs-1918 2 23 's be AUX iajs-1918 2 24 parallel parallel ADJ iajs-1918 2 25 numerical numerical ADJ iajs-1918 2 26 algorithm algorithm NOUN iajs-1918 2 27 by by ADP iajs-1918 2 28 neural neural ADJ iajs-1918 2 29 network network NOUN iajs-1918 2 30 is be AUX iajs-1918 2 31 applied apply VERB iajs-1918 2 32 to to ADP iajs-1918 2 33 type type NOUN iajs-1918 2 34 of of ADP iajs-1918 2 35 integration integration NOUN iajs-1918 2 36 of of ADP iajs-1918 2 37 fractional fractional ADJ iajs-1918 2 38 equations equation NOUN iajs-1918 2 39 is be AUX iajs-1918 2 40 abel abel PROPN iajs-1918 2 41 ’s ’s PART iajs-1918 2 42 integral integral ADJ iajs-1918 2 43 equations equation NOUN iajs-1918 2 44 of of ADP iajs-1918 2 45 the the DET iajs-1918 2 46 1st 1st ADJ iajs-1918 2 47 and and CCONJ iajs-1918 2 48 2nd 2nd ADJ iajs-1918 2 49 kinds kind NOUN iajs-1918 2 50 . . PUNCT iajs-1918 3 1 using use VERB iajs-1918 3 2 a a DET iajs-1918 3 3 levenberge levenberge NOUN iajs-1918 3 4 – – PUNCT iajs-1918 3 5 marquaradt marquaradt ADJ iajs-1918 3 6 training training NOUN iajs-1918 3 7 algorithm algorithm NOUN iajs-1918 3 8 as as ADP iajs-1918 3 9 a a DET iajs-1918 3 10 tool tool NOUN iajs-1918 3 11 to to PART iajs-1918 3 12 train train VERB iajs-1918 3 13 the the DET iajs-1918 3 14 network network NOUN iajs-1918 3 15 . . PUNCT iajs-1918 4 1 to to PART iajs-1918 4 2 show show VERB iajs-1918 4 3 the the DET iajs-1918 4 4 efficiency efficiency NOUN iajs-1918 4 5 of of ADP iajs-1918 4 6 the the DET iajs-1918 4 7 method method NOUN iajs-1918 4 8 , , PUNCT iajs-1918 4 9 some some DET iajs-1918 4 10 type type NOUN iajs-1918 4 11 of of ADP iajs-1918 4 12 abel abel PROPN iajs-1918 4 13 ’s ’s PART iajs-1918 4 14 integral integral ADJ iajs-1918 4 15 equations equation NOUN iajs-1918 4 16 is be AUX iajs-1918 4 17 solved solve VERB iajs-1918 4 18 as as ADP iajs-1918 4 19 numerical numerical ADJ iajs-1918 4 20 examples example NOUN iajs-1918 4 21 . . PUNCT iajs-1918 5 1 numerical numerical PROPN iajs-1918 5 2 results result NOUN iajs-1918 5 3 show show VERB iajs-1918 5 4 that that SCONJ iajs-1918 5 5 the the DET iajs-1918 5 6 new new ADJ iajs-1918 5 7 method method NOUN iajs-1918 5 8 is be AUX iajs-1918 5 9 very very ADV iajs-1918 5 10 efficient efficient ADJ iajs-1918 5 11 problems problem NOUN iajs-1918 5 12 with with ADP iajs-1918 5 13 high high ADJ iajs-1918 5 14 accuracy accuracy NOUN iajs-1918 5 15 . . PUNCT iajs-1918 6 1 keywords keyword NOUN iajs-1918 6 2 : : PUNCT iajs-1918 6 3 neural neural ADJ iajs-1918 6 4 network network NOUN iajs-1918 6 5 , , PUNCT iajs-1918 6 6 abel abel PROPN iajs-1918 6 7 ’s ’s PART iajs-1918 6 8 integral integral ADJ iajs-1918 6 9 equations equation NOUN iajs-1918 6 10 , , PUNCT iajs-1918 6 11 levenberg levenberg PROPN iajs-1918 6 12 – – PUNCT iajs-1918 6 13 marquardt marquardt PROPN iajs-1918 6 14 training training NOUN iajs-1918 6 15 algorithm algorithm NOUN iajs-1918 6 16 . . PUNCT iajs-1918 7 1 1 1 X iajs-1918 7 2 . . X iajs-1918 7 3 introduction introduction NOUN iajs-1918 7 4 abel abel PROPN iajs-1918 7 5 ’s ’s PART iajs-1918 7 6 equations equation NOUN iajs-1918 7 7 are be AUX iajs-1918 7 8 interconnected interconnect VERB iajs-1918 7 9 to to ADP iajs-1918 7 10 a a DET iajs-1918 7 11 wide wide ADV iajs-1918 7 12 - - PUNCT iajs-1918 7 13 ranging ranging NOUN iajs-1918 7 14 of of ADP iajs-1918 7 15 physecal physecal ADJ iajs-1918 7 16 problems problem NOUN iajs-1918 7 17 , , PUNCT iajs-1918 7 18 such such ADJ iajs-1918 7 19 as as ADP iajs-1918 7 20 tempera tempera NOUN iajs-1918 7 21 ture ture NOUN iajs-1918 7 22 transfer transfer NOUN iajs-1918 7 23 [ [ X iajs-1918 7 24 1 1 NUM iajs-1918 7 25 ] ] PUNCT iajs-1918 7 26 , , PUNCT iajs-1918 7 27 nonlinear nonlinear ADJ iajs-1918 7 28 diffusion diffusion NOUN iajs-1918 8 1 [ [ X iajs-1918 8 2 2 2 NUM iajs-1918 8 3 ] ] PUNCT iajs-1918 8 4 , , PUNCT iajs-1918 8 5 the the DET iajs-1918 8 6 propagation propagation NOUN iajs-1918 8 7 of of ADP iajs-1918 8 8 nonlinear nonlinear ADJ iajs-1918 8 9 waves wave NOUN iajs-1918 8 10 [ [ X iajs-1918 8 11 3 3 NUM iajs-1918 8 12 ] ] PUNCT iajs-1918 8 13 , , PUNCT iajs-1918 8 14 and and CCONJ iajs-1918 8 15 were be AUX iajs-1918 8 16 used use VERB iajs-1918 8 17 in in ADP iajs-1918 8 18 the the DET iajs-1918 8 19 theory theory NOUN iajs-1918 8 20 of of ADP iajs-1918 8 21 nautron nautron ADJ iajs-1918 8 22 transportation transportation NOUN iajs-1918 8 23 and and CCONJ iajs-1918 8 24 traffic traffic NOUN iajs-1918 8 25 theory theory NOUN iajs-1918 8 26 . . PUNCT iajs-1918 9 1 there there PRON iajs-1918 9 2 is be VERB iajs-1918 9 3 several several ADJ iajs-1918 9 4 methods method NOUN iajs-1918 9 5 counting count VERB iajs-1918 9 6 numerical numerical ADJ iajs-1918 9 7 analysis analysis NOUN iajs-1918 9 8 thus thus ADV iajs-1918 9 9 far far ADV iajs-1918 9 10 away away ADV iajs-1918 9 11 to to ADP iajs-1918 9 12 reviewing review VERB iajs-1918 9 13 this this DET iajs-1918 9 14 type type NOUN iajs-1918 9 15 of of ADP iajs-1918 9 16 integration integration NOUN iajs-1918 9 17 as as ADV iajs-1918 9 18 well well ADV iajs-1918 9 19 as as ADP iajs-1918 9 20 their their PRON iajs-1918 9 21 variations variation NOUN iajs-1918 9 22 with with ADP iajs-1918 9 23 several several ADJ iajs-1918 9 24 uses use NOUN iajs-1918 9 25 [ [ X iajs-1918 9 26 4–5 4–5 X iajs-1918 9 27 ] ] X iajs-1918 9 28 . . PUNCT iajs-1918 10 1 in in ADP iajs-1918 10 2 [ [ X iajs-1918 10 3 6 6 NUM iajs-1918 10 4 ] ] PUNCT iajs-1918 10 5 he he PRON iajs-1918 10 6 also also ADV iajs-1918 10 7 talked talk VERB iajs-1918 10 8 about about ADP iajs-1918 10 9 solutions solution NOUN iajs-1918 10 10 to to ADP iajs-1918 10 11 this this DET iajs-1918 10 12 type type NOUN iajs-1918 10 13 of of ADP iajs-1918 10 14 integration integration NOUN iajs-1918 10 15 under under ADP iajs-1918 10 16 assured assure VERB iajs-1918 10 17 conditions condition NOUN iajs-1918 10 18 by by ADP iajs-1918 10 19 some some DET iajs-1918 10 20 special special ADJ iajs-1918 10 21 ways way NOUN iajs-1918 10 22 . . PUNCT iajs-1918 11 1 sumner sumner PROPN iajs-1918 11 2 calculated calculate VERB iajs-1918 11 3 this this DET iajs-1918 11 4 type type NOUN iajs-1918 11 5 from from ADP iajs-1918 11 6 the the DET iajs-1918 11 7 point point NOUN iajs-1918 11 8 of of ADP iajs-1918 11 9 observation observation NOUN iajs-1918 11 10 of of ADP iajs-1918 11 11 the the DET iajs-1918 11 12 convolutional convolutional ADJ iajs-1918 11 13 convert convert NOUN iajs-1918 11 14 studied study VERB iajs-1918 11 15 a a DET iajs-1918 11 16 numerical numerical ADJ iajs-1918 11 17 solution solution NOUN iajs-1918 11 18 for for ADP iajs-1918 11 19 this this DET iajs-1918 11 20 integration integration NOUN iajs-1918 11 21 using use VERB iajs-1918 11 22 orthogonal orthogonal ADJ iajs-1918 11 23 polynomials polynomial NOUN iajs-1918 11 24 . . PUNCT iajs-1918 12 1 hilbert hilbert PROPN iajs-1918 12 2 worked work VERB iajs-1918 12 3 on on ADP iajs-1918 12 4 the the DET iajs-1918 12 5 problem problem NOUN iajs-1918 12 6 of of ADP iajs-1918 12 7 nonlinear nonlinear ADJ iajs-1918 12 8 type type NOUN iajs-1918 12 9 of of ADP iajs-1918 12 10 energy energy NOUN iajs-1918 12 11 , , PUNCT iajs-1918 12 12 explained explain VERB iajs-1918 12 13 in in ADP iajs-1918 12 14 locked lock VERB iajs-1918 12 15 shaped shape VERB iajs-1918 12 16 by by ADP iajs-1918 12 17 expressive expressive ADJ iajs-1918 12 18 a a DET iajs-1918 12 19 sectional sectional ADJ iajs-1918 12 20 holomorphic holomorphic ADJ iajs-1918 12 21 function function NOUN iajs-1918 12 22 by by ADP iajs-1918 12 23 earnings earning NOUN iajs-1918 12 24 of of ADP iajs-1918 12 25 an an DET iajs-1918 12 26 integral integral ADJ iajs-1918 12 27 with with ADP iajs-1918 12 28 energy energy NOUN iajs-1918 12 29 nucleus nucleus NOUN iajs-1918 12 30 , , PUNCT iajs-1918 12 31 and and CCONJ iajs-1918 12 32 converted convert VERB iajs-1918 12 33 the the DET iajs-1918 12 34 problem problem NOUN iajs-1918 12 35 to to ADP iajs-1918 12 36 one one NUM iajs-1918 12 37 of of ADP iajs-1918 12 38 resolving resolve VERB iajs-1918 12 39 a a DET iajs-1918 12 40 generalized generalized ADJ iajs-1918 12 41 reviewing reviewing NOUN iajs-1918 12 42 this this DET iajs-1918 12 43 type type NOUN iajs-1918 12 44 of of ADP iajs-1918 12 45 integration integration NOUN iajs-1918 12 46 . . PUNCT iajs-1918 13 1 in in ADP iajs-1918 13 2 [ [ X iajs-1918 13 3 7 7 X iajs-1918 13 4 ] ] PUNCT iajs-1918 13 5 used use VERB iajs-1918 13 6 the the DET iajs-1918 13 7 reverse reverse NOUN iajs-1918 13 8 of of ADP iajs-1918 13 9 several several ADJ iajs-1918 13 10 of of ADP iajs-1918 13 11 fractional fractional ADJ iajs-1918 13 12 order order NOUN iajs-1918 13 13 . . PUNCT iajs-1918 14 1 in in ADP iajs-1918 14 2 [ [ X iajs-1918 14 3 8 8 NUM iajs-1918 14 4 ] ] PUNCT iajs-1918 14 5 resolved resolve VERB iajs-1918 14 6 this this DET iajs-1918 14 7 integration integration NOUN iajs-1918 14 8 numerically numerically ADV iajs-1918 14 9 it it PRON iajs-1918 14 10 is be AUX iajs-1918 14 11 based base VERB iajs-1918 14 12 on on ADP iajs-1918 14 13 the the DET iajs-1918 14 14 process process NOUN iajs-1918 14 15 of of ADP iajs-1918 14 16 approximation approximation NOUN iajs-1918 14 17 of of ADP iajs-1918 14 18 partial partial ADJ iajs-1918 14 19 integrations integration NOUN iajs-1918 14 20 and and CCONJ iajs-1918 14 21 derivatives derivative NOUN iajs-1918 14 22 . . PUNCT iajs-1918 15 1 in in ADP iajs-1918 15 2 [ [ X iajs-1918 15 3 9 9 NUM iajs-1918 15 4 ] ] PUNCT iajs-1918 15 5 the the DET iajs-1918 15 6 researcher researcher NOUN iajs-1918 15 7 used use VERB iajs-1918 15 8 chebyshev chebyshev NOUN iajs-1918 15 9 polynomials polynomial NOUN iajs-1918 15 10 for for AUX iajs-1918 15 11 solve solve VERB iajs-1918 15 12 this this DET iajs-1918 15 13 integration integration NOUN iajs-1918 15 14 . . PUNCT iajs-1918 16 1 in in ADP iajs-1918 16 2 [ [ X iajs-1918 16 3 10 10 NUM iajs-1918 16 4 ] ] PUNCT iajs-1918 16 5 for for ADP iajs-1918 16 6 easy easy ADJ iajs-1918 16 7 calculations calculation NOUN iajs-1918 16 8 the the DET iajs-1918 16 9 researcher researcher NOUN iajs-1918 16 10 used use VERB iajs-1918 16 11 some some DET iajs-1918 16 12 conversions conversion NOUN iajs-1918 16 13 such such ADJ iajs-1918 16 14 homotopy homotopy NOUN iajs-1918 16 15 pertuarbation pertuarbation NOUN iajs-1918 16 16 and and CCONJ iajs-1918 16 17 laplace laplace NOUN iajs-1918 16 18 transforms transform VERB iajs-1918 16 19 algorithm algorithm NOUN iajs-1918 16 20 . . PUNCT iajs-1918 17 1 2 2 X iajs-1918 17 2 . . X iajs-1918 17 3 abel abel PROPN iajs-1918 17 4 ’s ’s PART iajs-1918 17 5 integral integral ADJ iajs-1918 17 6 equations equation NOUN iajs-1918 17 7 this this DET iajs-1918 17 8 type type NOUN iajs-1918 17 9 of of ADP iajs-1918 17 10 integration integration NOUN iajs-1918 17 11 studied study VERB iajs-1918 17 12 by by ADP iajs-1918 17 13 nieal nieal PROPN iajs-1918 17 14 henreik henreik PROPN iajs-1918 17 15 abel abel PROPN iajs-1918 17 16 and and CCONJ iajs-1918 17 17 liouveille liouveille NOUN iajs-1918 17 18 for for ADP iajs-1918 17 19 its its PRON iajs-1918 17 20 real real ADJ iajs-1918 17 21 importance importance NOUN iajs-1918 17 22 in in ADP iajs-1918 17 23 modeling model VERB iajs-1918 17 24 many many ADJ iajs-1918 17 25 phenomena phenomenon NOUN iajs-1918 17 26 in in ADP iajs-1918 17 27 mechanical mechanical ADJ iajs-1918 17 28 , , PUNCT iajs-1918 17 29 electronic electronic ADJ iajs-1918 17 30 , , PUNCT iajs-1918 17 31 engineering engineering NOUN iajs-1918 17 32 , , PUNCT iajs-1918 17 33 chemical chemical NOUN iajs-1918 17 34 and and CCONJ iajs-1918 17 35 basic basic ADJ iajs-1918 17 36 sciences science NOUN iajs-1918 17 37 [ [ X iajs-1918 17 38 1113 1113 NUM iajs-1918 17 39 ] ] PUNCT iajs-1918 17 40 . . PUNCT iajs-1918 18 1 where where SCONJ iajs-1918 18 2 it it PRON iajs-1918 18 3 emerged emerge VERB iajs-1918 18 4 for for ADP iajs-1918 18 5 this this DET iajs-1918 18 6 type type NOUN iajs-1918 18 7 of of ADP iajs-1918 18 8 integration integration NOUN iajs-1918 18 9 two two NUM iajs-1918 18 10 types type NOUN iajs-1918 18 11 first first ADV iajs-1918 18 12 and and CCONJ iajs-1918 18 13 second second ADJ iajs-1918 18 14 as as SCONJ iajs-1918 18 15 follows follow VERB iajs-1918 18 16 respectably respectably ADV iajs-1918 18 17     SPACE iajs-1918 18 18 179 179 NUM iajs-1918 18 19 ibn ibn PROPN iajs-1918 18 20 al al PROPN iajs-1918 18 21 - - PUNCT iajs-1918 18 22 haitham haitham PROPN iajs-1918 18 23 jour.for jour.for PROPN iajs-1918 18 24 pure&appl.sci pure&appl.sci PROPN iajs-1918 18 25 . . PUNCT iajs-1918 19 1 ihjpas ihjpas PROPN iajs-1918 19 2 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 19 3 vol vol NOUN iajs-1918 19 4 . . PUNCT iajs-1918 19 5 32 32 NUM iajs-1918 19 6 ( ( PUNCT iajs-1918 19 7 1 1 NUM iajs-1918 19 8 ) ) PUNCT iajs-1918 19 9 2019 2019 NUM iajs-1918 20 1 𝑓 𝑓 DET iajs-1918 20 2 𝑥 𝑥 X iajs-1918 20 3 𝜆 𝜆 PRON iajs-1918 20 4 𝑦 𝑦 NUM iajs-1918 20 5 𝑡 𝑡 PART iajs-1918 20 6 𝑥 𝑥 PROPN iajs-1918 20 7 𝑡 𝑡 PROPN iajs-1918 20 8 𝑑𝑡 𝑑𝑡 ADP iajs-1918 20 9 1 1 NUM iajs-1918 20 10 and and CCONJ iajs-1918 20 11 𝑦 𝑦 NOUN iajs-1918 20 12 𝑥 𝑥 X iajs-1918 20 13 𝑓 𝑓 DET iajs-1918 20 14 𝑥 𝑥 X iajs-1918 20 15 𝜆 𝜆 ADV iajs-1918 20 16 𝑑𝑡 𝑑𝑡 ADP iajs-1918 20 17 2 2 NUM iajs-1918 20 18 where where SCONJ iajs-1918 20 19 𝜆 𝜆 X iajs-1918 20 20 , , PUNCT iajs-1918 20 21 𝛼 𝛼 NOUN iajs-1918 20 22 𝑎𝑛𝑑 𝑎𝑛𝑑 ADJ iajs-1918 20 23 𝑓 𝑓 DET iajs-1918 20 24 𝑥 𝑥 PROPN iajs-1918 20 25 and and CCONJ iajs-1918 20 26 the the DET iajs-1918 20 27 function function NOUN iajs-1918 20 28 are be AUX iajs-1918 20 29 given give VERB iajs-1918 20 30 where where SCONJ iajs-1918 20 31 0 0 NUM iajs-1918 20 32 𝛼 𝛼 SYM iajs-1918 20 33 1 1 NUM iajs-1918 20 34 , , PUNCT iajs-1918 20 35 𝑓 𝑓 PROPN iajs-1918 20 36 𝑐 𝑐 PROPN iajs-1918 20 37 ∈ ∈ PROPN iajs-1918 20 38 𝑐 𝑐 PROPN iajs-1918 20 39 0,1 0,1 NUM iajs-1918 20 40 , , PUNCT iajs-1918 20 41 0 0 NUM iajs-1918 21 1 𝑥 𝑥 NOUN iajs-1918 21 2 , , PUNCT iajs-1918 21 3 𝑡 𝑡 NOUN iajs-1918 21 4 𝑇 𝑇 PROPN iajs-1918 21 5 and and CCONJ iajs-1918 21 6 t t PROPN iajs-1918 21 7 is be AUX iajs-1918 21 8 constant constant ADJ iajs-1918 21 9 . . PUNCT iajs-1918 22 1 we we PRON iajs-1918 22 2 will will AUX iajs-1918 22 3 solve solve VERB iajs-1918 22 4 this this DET iajs-1918 22 5 type type NOUN iajs-1918 22 6 of of ADP iajs-1918 22 7 integration integration NOUN iajs-1918 22 8 of of ADP iajs-1918 22 9 fractional fractional ADJ iajs-1918 22 10 equations equation NOUN iajs-1918 22 11 by by ADP iajs-1918 22 12 means mean NOUN iajs-1918 22 13 of of ADP iajs-1918 22 14 neural neural ADJ iajs-1918 22 15 networks network NOUN iajs-1918 22 16 using use VERB iajs-1918 22 17 three three NUM iajs-1918 22 18 layers layer NOUN iajs-1918 22 19 : : PUNCT iajs-1918 22 20 the the DET iajs-1918 22 21 input input NOUN iajs-1918 22 22 and and CCONJ iajs-1918 22 23 output output NOUN iajs-1918 22 24 layers layer NOUN iajs-1918 22 25 and and CCONJ iajs-1918 22 26 the the DET iajs-1918 22 27 hidden hide VERB iajs-1918 22 28 layer layer NOUN iajs-1918 22 29 contain contain VERB iajs-1918 22 30 7 7 NUM iajs-1918 22 31 hidden hide VERB iajs-1918 22 32 units unit NOUN iajs-1918 22 33 and and CCONJ iajs-1918 22 34 one one NUM iajs-1918 22 35 linear linear ADJ iajs-1918 22 36 output output NOUN iajs-1918 22 37 unit unit NOUN iajs-1918 22 38 , , PUNCT iajs-1918 22 39 the the DET iajs-1918 22 40 sigmoid sigmoid NOUN iajs-1918 22 41 activation activation NOUN iajs-1918 22 42 function function NOUN iajs-1918 22 43 of of ADP iajs-1918 22 44 each each DET iajs-1918 22 45 unit unit NOUN iajs-1918 22 46 in in ADP iajs-1918 22 47 hidden hide VERB iajs-1918 22 48 layer layer NOUN iajs-1918 22 49 is be AUX iajs-1918 22 50 tanseg tanseg NOUN iajs-1918 22 51 . . PUNCT iajs-1918 23 1 function function NOUN iajs-1918 23 2 by by ADP iajs-1918 23 3 a a DET iajs-1918 23 4 type type NOUN iajs-1918 23 5 of of ADP iajs-1918 23 6 training training NOUN iajs-1918 23 7 , , PUNCT iajs-1918 23 8 the the DET iajs-1918 23 9 levenberge levenberge NOUN iajs-1918 23 10 – – PUNCT iajs-1918 23 11 marqaurdt marqaurdt NOUN iajs-1918 23 12 training training NOUN iajs-1918 23 13 algoreithm algoreithm PROPN iajs-1918 23 14 is be AUX iajs-1918 23 15 used use VERB iajs-1918 23 16 to to PART iajs-1918 23 17 train train VERB iajs-1918 23 18 the the DET iajs-1918 23 19 network network NOUN iajs-1918 23 20 . . PUNCT iajs-1918 24 1 3 3 X iajs-1918 24 2 . . X iajs-1918 24 3 description description NOUN iajs-1918 24 4 of of ADP iajs-1918 24 5 the the DET iajs-1918 24 6 method method NOUN iajs-1918 24 7 in in ADP iajs-1918 24 8 this this DET iajs-1918 24 9 part part NOUN iajs-1918 24 10 we we PRON iajs-1918 24 11 will will AUX iajs-1918 24 12 describe describe VERB iajs-1918 24 13 how how SCONJ iajs-1918 24 14 our our PRON iajs-1918 24 15 method method NOUN iajs-1918 24 16 can can AUX iajs-1918 24 17 be be AUX iajs-1918 24 18 used use VERB iajs-1918 24 19 to to PART iajs-1918 24 20 discovery discovery VERB iajs-1918 24 21 the the DET iajs-1918 24 22 approximate approximate ADJ iajs-1918 24 23 result result NOUN iajs-1918 24 24 of of ADP iajs-1918 24 25 equations equation NOUN iajs-1918 24 26 ( ( PUNCT iajs-1918 24 27 1 1 NUM iajs-1918 24 28 ) ) PUNCT iajs-1918 24 29 and and CCONJ iajs-1918 24 30 ( ( PUNCT iajs-1918 24 31 2 2 X iajs-1918 24 32 ) ) PUNCT iajs-1918 24 33 let let VERB iajs-1918 24 34 𝛾 𝛾 PROPN iajs-1918 24 35 𝜒 𝜒 X iajs-1918 24 36 denotes denote NOUN iajs-1918 24 37 the the DET iajs-1918 24 38 result result NOUN iajs-1918 24 39 to to PART iajs-1918 24 40 be be AUX iajs-1918 24 41 calculated calculate VERB iajs-1918 24 42 , , PUNCT iajs-1918 24 43 𝛾 𝛾 NOUN iajs-1918 24 44 𝜒 𝜒 NUM iajs-1918 24 45 , , PUNCT iajs-1918 24 46 𝜌 𝜌 X iajs-1918 24 47 ) ) PUNCT iajs-1918 24 48 refers refer VERB iajs-1918 24 49 to to ADP iajs-1918 24 50 the the DET iajs-1918 24 51 analytical analytical ADJ iajs-1918 24 52 solution solution NOUN iajs-1918 24 53 . . PUNCT iajs-1918 25 1 in in ADP iajs-1918 25 2 the the DET iajs-1918 25 3 proposed propose VERB iajs-1918 25 4 approach approach NOUN iajs-1918 25 5 , , PUNCT iajs-1918 25 6 the the DET iajs-1918 25 7 ffnn ffnn NOUN iajs-1918 25 8 experimental experimental ADJ iajs-1918 25 9 solution solution NOUN iajs-1918 25 10 is be AUX iajs-1918 25 11 used use VERB iajs-1918 25 12 and and CCONJ iajs-1918 25 13 the the DET iajs-1918 25 14 factors factor NOUN iajs-1918 25 15 𝜌 𝜌 PART iajs-1918 25 16 correspond correspond VERB iajs-1918 25 17 to to ADP iajs-1918 25 18 the the DET iajs-1918 25 19 weights weight NOUN iajs-1918 25 20 and and CCONJ iajs-1918 25 21 bias bias NOUN iajs-1918 25 22 of of ADP iajs-1918 25 23 the the DET iajs-1918 25 24 neural neural ADJ iajs-1918 25 25 architecture architecture NOUN iajs-1918 25 26 . . PUNCT iajs-1918 26 1 we we PRON iajs-1918 26 2 choose choose VERB iajs-1918 26 3 a a DET iajs-1918 26 4 model model NOUN iajs-1918 26 5 for for SCONJ iajs-1918 26 6 a a DET iajs-1918 26 7 pilot pilot NOUN iajs-1918 26 8 function function NOUN iajs-1918 26 9 yt yt NOUN iajs-1918 26 10 ( ( PUNCT iajs-1918 26 11 x x X iajs-1918 26 12 ) ) PUNCT iajs-1918 26 13 to to PART iajs-1918 26 14 meet meet VERB iajs-1918 26 15 bc bc PROPN iajs-1918 26 16 requirements requirement NOUN iajs-1918 26 17 . . PUNCT iajs-1918 27 1 this this PRON iajs-1918 27 2 is be AUX iajs-1918 27 3 attained attain VERB iajs-1918 27 4 by by ADP iajs-1918 27 5 lettering letter VERB iajs-1918 27 6 it it PRON iajs-1918 27 7 as as ADP iajs-1918 27 8 two two NUM iajs-1918 27 9 sets set NOUN iajs-1918 27 10 : : PUNCT iajs-1918 27 11 𝛾 𝛾 NOUN iajs-1918 27 12 𝜒 𝜒 NOUN iajs-1918 27 13 , , PUNCT iajs-1918 27 14 𝜌 𝜌 X iajs-1918 27 15 𝜙 𝜙 X iajs-1918 27 16 𝜒 𝜒 X iajs-1918 27 17 𝜉 𝜉 NOUN iajs-1918 27 18 𝜒 𝜒 NOUN iajs-1918 27 19 , , PUNCT iajs-1918 27 20 ν ν X iajs-1918 27 21 𝜒 𝜒 X iajs-1918 27 22 , , PUNCT iajs-1918 27 23 𝜌 𝜌 X iajs-1918 27 24 ( ( PUNCT iajs-1918 27 25 3 3 NUM iajs-1918 27 26 ) ) PUNCT iajs-1918 27 27 where where SCONJ iajs-1918 27 28 ν ν NOUN iajs-1918 27 29 𝜒 𝜒 X iajs-1918 27 30 , , PUNCT iajs-1918 27 31 𝜌 𝜌 X iajs-1918 27 32 is be AUX iajs-1918 27 33 a a DET iajs-1918 27 34 singleoutput singleoutput VERB iajs-1918 27 35 ffnn ffnn NOUN iajs-1918 27 36 with with ADP iajs-1918 27 37 𝜌 𝜌 ADP iajs-1918 27 38 factors factor NOUN iajs-1918 27 39 and and CCONJ iajs-1918 27 40 one one NUM iajs-1918 27 41 input input NOUN iajs-1918 27 42 element element NOUN iajs-1918 27 43 breast breast NOUN iajs-1918 27 44 - - PUNCT iajs-1918 27 45 feeding feeding NOUN iajs-1918 27 46 on on ADP iajs-1918 27 47 the the DET iajs-1918 27 48 𝜒 𝜒 NOUN iajs-1918 27 49 input input NOUN iajs-1918 27 50 vector vector NOUN iajs-1918 27 51 . . PUNCT iajs-1918 28 1 the the DET iajs-1918 28 2 part part NOUN iajs-1918 28 3 𝜙 𝜙 X iajs-1918 28 4 𝜒 𝜒 NOUN iajs-1918 28 5 does do AUX iajs-1918 28 6 not not PART iajs-1918 28 7 contain contain VERB iajs-1918 28 8 parameters parameter NOUN iajs-1918 28 9 that that PRON iajs-1918 28 10 can can AUX iajs-1918 28 11 be be AUX iajs-1918 28 12 adjusted adjust VERB iajs-1918 28 13 and and CCONJ iajs-1918 28 14 meet meet VERB iajs-1918 28 15 the the DET iajs-1918 28 16 boundary boundary ADJ iajs-1918 28 17 conditions condition NOUN iajs-1918 28 18 . . PUNCT iajs-1918 29 1 the the DET iajs-1918 29 2 second second ADJ iajs-1918 29 3 term term NOUN iajs-1918 29 4 𝜉 𝜉 NOUN iajs-1918 29 5 𝜒 𝜒 NOUN iajs-1918 29 6 , , PUNCT iajs-1918 29 7 ν ν X iajs-1918 29 8 𝜒 𝜒 X iajs-1918 29 9 , , PUNCT iajs-1918 29 10 𝜌 𝜌 PRON iajs-1918 29 11 is be AUX iajs-1918 29 12 created create VERB iajs-1918 29 13 so so SCONJ iajs-1918 29 14 that that SCONJ iajs-1918 29 15 it it PRON iajs-1918 29 16 does do AUX iajs-1918 29 17 not not PART iajs-1918 29 18 contribute contribute VERB iajs-1918 29 19 to to ADP iajs-1918 29 20 bc bc PROPN iajs-1918 29 21 , , PUNCT iajs-1918 29 22 because because SCONJ iajs-1918 29 23 𝛾 𝛾 ADP iajs-1918 29 24 𝜒 𝜒 X iajs-1918 29 25 satisfies satisfie NOUN iajs-1918 29 26 them they PRON iajs-1918 29 27 . . PUNCT iajs-1918 30 1 this this DET iajs-1918 30 2 part part NOUN iajs-1918 30 3 can can AUX iajs-1918 30 4 be be AUX iajs-1918 30 5 molded mold VERB iajs-1918 30 6 using use VERB iajs-1918 30 7 ffnn ffnn NOUN iajs-1918 30 8 whose whose DET iajs-1918 30 9 weights weight NOUN iajs-1918 30 10 and and CCONJ iajs-1918 30 11 biases bias NOUN iajs-1918 30 12 must must AUX iajs-1918 30 13 be be AUX iajs-1918 30 14 adjusted adjust VERB iajs-1918 30 15 to to PART iajs-1918 30 16 address address VERB iajs-1918 30 17 the the DET iajs-1918 30 18 minimization minimization NOUN iajs-1918 30 19 problem problem NOUN iajs-1918 30 20 . . PUNCT iajs-1918 31 1 4 4 X iajs-1918 31 2 . . X iajs-1918 31 3 illustration illustration NOUN iajs-1918 31 4 of of ADP iajs-1918 31 5 the the DET iajs-1918 31 6 method method NOUN iajs-1918 31 7 to to PART iajs-1918 31 8 explain explain VERB iajs-1918 31 9 the the DET iajs-1918 31 10 method method NOUN iajs-1918 31 11 , , PUNCT iajs-1918 31 12 let let VERB iajs-1918 31 13 the the DET iajs-1918 31 14 equations equation NOUN iajs-1918 31 15 ( ( PUNCT iajs-1918 31 16 ( ( PUNCT iajs-1918 31 17 1 1 NUM iajs-1918 31 18 ) ) PUNCT iajs-1918 31 19 and and CCONJ iajs-1918 31 20 ( ( PUNCT iajs-1918 31 21 2 2 NUM iajs-1918 31 22 ) ) PUNCT iajs-1918 31 23 ) ) PUNCT iajs-1918 31 24 , , PUNCT iajs-1918 31 25 where where SCONJ iajs-1918 31 26 x x X iajs-1918 31 27 ∈ ∈ PROPN iajs-1918 31 28 [ [ X iajs-1918 31 29 0 0 NUM iajs-1918 31 30 , , PUNCT iajs-1918 31 31 1 1 NUM iajs-1918 31 32 ] ] PUNCT iajs-1918 31 33 and and CCONJ iajs-1918 31 34 the the DET iajs-1918 31 35 bc bc PROPN iajs-1918 31 36 : : PUNCT iajs-1918 31 37 y(0 y(0 PROPN iajs-1918 31 38 ) ) PUNCT iajs-1918 31 39 = = SYM iajs-1918 31 40 α α PROPN iajs-1918 31 41 and and CCONJ iajs-1918 31 42 y(1 y(1 PROPN iajs-1918 31 43 ) ) PUNCT iajs-1918 31 44 = = SYM iajs-1918 31 45 𝛽 𝛽 NOUN iajs-1918 31 46 , , PUNCT iajs-1918 31 47 the the DET iajs-1918 31 48 approximation approximation NOUN iajs-1918 31 49 of of ADP iajs-1918 31 50 solution solution NOUN iajs-1918 31 51 can can AUX iajs-1918 31 52 be be AUX iajs-1918 31 53 written write VERB iajs-1918 31 54 as as ADP iajs-1918 31 55 : : PUNCT iajs-1918 31 56 𝛾 𝛾 NOUN iajs-1918 31 57 𝜒 𝜒 NOUN iajs-1918 31 58 , , PUNCT iajs-1918 31 59 𝜌 𝜌 ADP iajs-1918 31 60 α α X iajs-1918 31 61 𝛽 𝛽 NOUN iajs-1918 31 62 α α PRON iajs-1918 31 63 𝜒 𝜒 NOUN iajs-1918 31 64 𝜒 𝜒 SYM iajs-1918 31 65 𝜒 𝜒 NUM iajs-1918 31 66 1 1 NUM iajs-1918 31 67 ν ν NOUN iajs-1918 31 68 𝜒 𝜒 NOUN iajs-1918 31 69 , , PUNCT iajs-1918 31 70 𝜌 𝜌 X iajs-1918 31 71 ( ( PUNCT iajs-1918 31 72 4 4 X iajs-1918 31 73 ) ) PUNCT iajs-1918 31 74 the the DET iajs-1918 31 75 error error NOUN iajs-1918 31 76 calculated calculate VERB iajs-1918 31 77 to to PART iajs-1918 31 78 be be AUX iajs-1918 31 79 minimized minimize VERB iajs-1918 31 80 is be AUX iajs-1918 31 81 given give VERB iajs-1918 31 82 by by ADP iajs-1918 31 83 𝐸 𝐸 PROPN iajs-1918 31 84 𝑝 𝑝 PROPN iajs-1918 31 85 ∑ ∑ PUNCT iajs-1918 31 86 𝑓 𝑓 PRON iajs-1918 31 87 𝛾 𝛾 NOUN iajs-1918 31 88 , , PUNCT iajs-1918 31 89 𝛾 𝛾 PROPN iajs-1918 31 90 , , PUNCT iajs-1918 31 91 𝜒 𝜒 X iajs-1918 31 92 𝑑𝑡 𝑑𝑡 ADP iajs-1918 31 93 ( ( PUNCT iajs-1918 31 94 5 5 NUM iajs-1918 31 95 ) ) PUNCT iajs-1918 31 96     SPACE iajs-1918 31 97 180 180 NUM iajs-1918 32 1 ibn ibn PROPN iajs-1918 32 2 al al PROPN iajs-1918 32 3 - - PUNCT iajs-1918 32 4 haitham haitham PROPN iajs-1918 32 5 jour.for jour.for PROPN iajs-1918 32 6 pure&appl.sci pure&appl.sci PROPN iajs-1918 32 7 . . PUNCT iajs-1918 32 8 ihjpas ihjpas PROPN iajs-1918 32 9 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 32 10 vol vol NOUN iajs-1918 32 11 . . PUNCT iajs-1918 32 12 32 32 NUM iajs-1918 32 13 ( ( PUNCT iajs-1918 32 14 1 1 NUM iajs-1918 32 15 ) ) PUNCT iajs-1918 32 16 2019 2019 NUM iajs-1918 32 17 where where SCONJ iajs-1918 32 18 the the DET iajs-1918 32 19 xi xi PROPN iajs-1918 32 20 ’s ’s PART iajs-1918 32 21 are be AUX iajs-1918 32 22 points point NOUN iajs-1918 32 23 in in ADP iajs-1918 32 24 [ [ X iajs-1918 32 25 a a PRON iajs-1918 32 26 , , PUNCT iajs-1918 32 27 b b NOUN iajs-1918 32 28 ] ] X iajs-1918 32 29 . . PUNCT iajs-1918 33 1 5 5 NUM iajs-1918 33 2 . . X iajs-1918 33 3 numerical numerical ADJ iajs-1918 33 4 examples example NOUN iajs-1918 33 5 in in ADP iajs-1918 33 6 this this DET iajs-1918 33 7 section section NOUN iajs-1918 33 8 we we PRON iajs-1918 33 9 will will AUX iajs-1918 33 10 apply apply VERB iajs-1918 33 11 the the DET iajs-1918 33 12 above above ADJ iajs-1918 33 13 algorithm algorithm NOUN iajs-1918 33 14 to to ADP iajs-1918 33 15 three three NUM iajs-1918 33 16 different different ADJ iajs-1918 33 17 examples example NOUN iajs-1918 33 18 of of ADP iajs-1918 33 19 the the DET iajs-1918 33 20 first first ADJ iajs-1918 33 21 kind kind NOUN iajs-1918 33 22 and and CCONJ iajs-1918 33 23 second second ADJ iajs-1918 33 24 kind kind NOUN iajs-1918 33 25 . . PUNCT iajs-1918 34 1 for for ADP iajs-1918 34 2 each each DET iajs-1918 34 3 test test NOUN iajs-1918 34 4 problem problem NOUN iajs-1918 34 5 the the DET iajs-1918 34 6 analytic analytic ADJ iajs-1918 34 7 solution solution NOUN iajs-1918 34 8 𝛾 𝛾 ADP iajs-1918 34 9 𝜒 𝜒 NOUN iajs-1918 34 10 was be AUX iajs-1918 34 11 identified identify VERB iajs-1918 34 12 in in ADP iajs-1918 34 13 improvement improvement NOUN iajs-1918 34 14 . . PUNCT iajs-1918 35 1 so so ADV iajs-1918 35 2 , , PUNCT iajs-1918 35 3 we we PRON iajs-1918 35 4 have have AUX iajs-1918 35 5 tried try VERB iajs-1918 35 6 the the DET iajs-1918 35 7 accuracy accuracy NOUN iajs-1918 35 8 of of ADP iajs-1918 35 9 the the DET iajs-1918 35 10 solutions solution NOUN iajs-1918 35 11 obtained obtain VERB iajs-1918 35 12 through through ADP iajs-1918 35 13 the the DET iajs-1918 35 14 equation equation NOUN iajs-1918 35 15 ; ; PUNCT iajs-1918 35 16 ∆𝑦 ∆𝑦 PROPN iajs-1918 35 17 𝑥 𝑥 PROPN iajs-1918 35 18 |𝛾 |𝛾 NOUN iajs-1918 35 19 𝜒 𝜒 ADP iajs-1918 35 20 𝛾 𝛾 NOUN iajs-1918 35 21 𝜒 𝜒 X iajs-1918 35 22 | | NOUN iajs-1918 35 23 " " PUNCT iajs-1918 35 24 ( ( PUNCT iajs-1918 35 25 6 6 NUM iajs-1918 35 26 ) ) PUNCT iajs-1918 35 27 example example NOUN iajs-1918 35 28 1 1 NUM iajs-1918 35 29 in in ADP iajs-1918 35 30 the the DET iajs-1918 35 31 first first ADJ iajs-1918 35 32 example example NOUN iajs-1918 35 33 we we PRON iajs-1918 35 34 will will AUX iajs-1918 35 35 demonstrate demonstrate VERB iajs-1918 35 36 the the DET iajs-1918 35 37 work work NOUN iajs-1918 35 38 of of ADP iajs-1918 35 39 the the DET iajs-1918 35 40 proposed propose VERB iajs-1918 35 41 algorithm algorithm NOUN iajs-1918 35 42 on on ADP iajs-1918 35 43 the the DET iajs-1918 35 44 first first ADJ iajs-1918 35 45 type type NOUN iajs-1918 35 46 of of ADP iajs-1918 35 47 fractional fractional ADJ iajs-1918 35 48 equations equation NOUN iajs-1918 35 49 for for ADP iajs-1918 35 50 that that DET iajs-1918 35 51 integration integration NOUN iajs-1918 35 52 [ [ X iajs-1918 35 53 14 14 NUM iajs-1918 35 54 ] ] PUNCT iajs-1918 35 55 ∪ ∪ NOUN iajs-1918 35 56 𝑑𝑡 𝑑𝑡 ADP iajs-1918 35 57 , , PUNCT iajs-1918 35 58 0 0 NUM iajs-1918 35 59 1 1 NUM iajs-1918 35 60 , , PUNCT iajs-1918 35 61 where where SCONJ iajs-1918 35 62 the the DET iajs-1918 35 63 exact exact ADJ iajs-1918 35 64 solution solution NOUN iajs-1918 35 65 ∪ ∪ ADP iajs-1918 35 66 𝑡 𝑡 PROPN iajs-1918 35 67 table table NOUN iajs-1918 35 68 1 1 NUM iajs-1918 35 69 . . PUNCT iajs-1918 35 70 analytic analytic ADJ iajs-1918 35 71 and and CCONJ iajs-1918 35 72 neural neural ADJ iajs-1918 35 73 solution solution NOUN iajs-1918 35 74 of of ADP iajs-1918 35 75 example example NOUN iajs-1918 35 76 1 1 NUM iajs-1918 35 77 the the DET iajs-1918 35 78 error error NOUN iajs-1918 35 79 e(x e(x NUM iajs-1918 35 80 ) ) PUNCT iajs-1918 36 1   NUM iajs-1918 37 1 | | ADV iajs-1918 37 2 yt(x yt(x NOUN iajs-1918 37 3 ) ) PUNCT iajs-1918 37 4 ya(x ya(x NOUN iajs-1918 37 5 ) ) PUNCT iajs-1918 38 1 | | ADV iajs-1918 38 2 w w NOUN iajs-1918 38 3 h h NOUN iajs-1918 38 4 e e NOUN iajs-1918 38 5 r r NOUN iajs-1918 38 6 e e X iajs-1918 38 7 yt(x yt(x NOUN iajs-1918 38 8 ) ) PUNCT iajs-1918 38 9 computed compute VERB iajs-1918 38 10 by by ADP iajs-1918 38 11 the the DET iajs-1918 38 12 following follow VERB iajs-1918 38 13 training training NOUN iajs-1918 38 14 algorithm algorithm NOUN iajs-1918 38 15 ( ( PUNCT iajs-1918 38 16 trainlm trainlm PROPN iajs-1918 38 17 ) ) PUNCT iajs-1918 38 18 out out ADP iajs-1918 38 19 of of ADP iajs-1918 38 20 suggested suggest VERB iajs-1918 38 21 ffnn ffnn NOUN iajs-1918 38 22 yt(x yt(x PROPN iajs-1918 38 23 ) ) PUNCT iajs-1918 38 24 for for ADP iajs-1918 38 25 lm lm INTJ iajs-1918 38 26 training training NOUN iajs-1918 38 27 algorithm algorithm NOUN iajs-1918 38 28 analytic analytic ADJ iajs-1918 38 29 solution solution NOUN iajs-1918 38 30 input input NOUN iajs-1918 38 31 trainlmya(x trainlmya(x NOUN iajs-1918 38 32 ) ) PUNCT iajs-1918 38 33 x x SYM iajs-1918 38 34 0.000000000000000.000000000000000.00000000000000 0.000000000000000.000000000000000.00000000000000 PROPN iajs-1918 38 35 0.0 0.0 NUM iajs-1918 38 36 0.000000022894650.1111111340057650.11111111111111 0.000000022894650.1111111340057650.11111111111111 NUM iajs-1918 38 37 0.1 0.1 NUM iajs-1918 38 38 0.000763781650320.222986003872540.22222222222222 0.000763781650320.222986003872540.22222222222222 NUM iajs-1918 38 39 0.2 0.2 NUM iajs-1918 38 40 0.000000000654060.333333333987390.33333333333333 0.000000000654060.333333333987390.33333333333333 NUM iajs-1918 38 41 0.3 0.3 NUM iajs-1918 38 42 0.000000120031940.444444564476380.44444444444444 0.000000120031940.444444564476380.44444444444444 NUM iajs-1918 38 43 0.4 0.4 NUM iajs-1918 38 44 0.000000002082650.555555553472910.55555555555556 0.000000002082650.555555553472910.55555555555556 NUM iajs-1918 38 45 0.5 0.5 NUM iajs-1918 38 46 0.000000000182700.666666666849370.66666666666667 0.000000000182700.666666666849370.66666666666667 NOUN iajs-1918 38 47 0.6 0.6 NUM iajs-1918 38 48 0.000004347869570.777773429908210.77777777777778 0.000004347869570.777773429908210.77777777777778 VERB iajs-1918 38 49 0.7 0.7 NUM iajs-1918 38 50 0.000000003546000.888888885342890.88888888888889 0.000000003546000.888888885342890.88888888888889 PROPN iajs-1918 38 51 0.8 0.8 NUM iajs-1918 38 52 0.0000000000000011.00000000000000 0.0000000000000011.00000000000000 NUM iajs-1918 38 53 0.9 0.9 NUM iajs-1918 38 54 -0.010111456888841.121222567999951.11111111111111 -0.010111456888841.121222567999951.11111111111111 PROPN iajs-1918 39 1 1.0 1.0 NUM iajs-1918 39 2     SPACE iajs-1918 39 3 181 181 NUM iajs-1918 39 4 ibn ibn PROPN iajs-1918 39 5 al al PROPN iajs-1918 39 6 - - PUNCT iajs-1918 39 7 haitham haitham PROPN iajs-1918 39 8 jour.for jour.for PROPN iajs-1918 39 9 pure&appl.sci pure&appl.sci PROPN iajs-1918 39 10 . . PUNCT iajs-1918 39 11 ihjpas ihjpas PROPN iajs-1918 39 12 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 39 13 vol vol NOUN iajs-1918 39 14 . . PUNCT iajs-1918 39 15 32 32 NUM iajs-1918 39 16 ( ( PUNCT iajs-1918 39 17 1 1 NUM iajs-1918 39 18 ) ) SYM iajs-1918 39 19 2019 2019 NUM iajs-1918 39 20 table table NOUN iajs-1918 39 21 2 2 NUM iajs-1918 39 22 . . PUNCT iajs-1918 40 1 the the DET iajs-1918 40 2 performance performance NOUN iajs-1918 40 3 of of ADP iajs-1918 40 4 the the DET iajs-1918 40 5 train train NOUN iajs-1918 40 6 with with ADP iajs-1918 40 7 epoch epoch NOUN iajs-1918 40 8 and and CCONJ iajs-1918 40 9 time time NOUN iajs-1918 40 10 table table NOUN iajs-1918 40 11 3 3 NUM iajs-1918 40 12 . . PUNCT iajs-1918 40 13 initial initial ADJ iajs-1918 40 14 weight weight NOUN iajs-1918 40 15 and and CCONJ iajs-1918 40 16 bias bias NOUN iajs-1918 40 17 of of ADP iajs-1918 40 18 the the DET iajs-1918 40 19 network network NOUN iajs-1918 40 20 for for ADP iajs-1918 40 21 different different ADJ iajs-1918 40 22 training training NOUN iajs-1918 40 23 algorithm algorithm NOUN iajs-1918 40 24 figure figure NOUN iajs-1918 40 25 1 1 NUM iajs-1918 40 26 . . NOUN iajs-1918 40 27 exact exact ADJ iajs-1918 40 28 and and CCONJ iajs-1918 40 29 approximate approximate ADJ iajs-1918 40 30 solution solution NOUN iajs-1918 40 31 of of ADP iajs-1918 40 32 example1 example1 PROPN iajs-1918 40 33 using use VERB iajs-1918 40 34 : : PUNCT iajs-1918 40 35 modify modify VERB iajs-1918 40 36 trainlm trainlm PROPN iajs-1918 40 37 & & CCONJ iajs-1918 40 38 trainlm trainlm PROPN iajs-1918 40 39 algorithms algorithms PROPN iajs-1918 40 40 example example NOUN iajs-1918 40 41 2 2 NUM iajs-1918 40 42 in in ADP iajs-1918 40 43 this this DET iajs-1918 40 44 example example NOUN iajs-1918 40 45 we we PRON iajs-1918 40 46 will will AUX iajs-1918 40 47 demonstrate demonstrate VERB iajs-1918 40 48 the the DET iajs-1918 40 49 work work NOUN iajs-1918 40 50 of of ADP iajs-1918 40 51 the the DET iajs-1918 40 52 offered offer VERB iajs-1918 40 53 algorithm algorithm NOUN iajs-1918 40 54 on on ADP iajs-1918 40 55 the the DET iajs-1918 40 56 first first ADJ iajs-1918 40 57 type type NOUN iajs-1918 40 58 of of ADP iajs-1918 40 59 fractional fractional ADJ iajs-1918 40 60 equations equation NOUN iajs-1918 40 61 for for ADP iajs-1918 40 62 that that DET iajs-1918 40 63 integration integration NOUN iajs-1918 40 64 [ [ X iajs-1918 40 65 15 15 NUM iajs-1918 40 66 ] ] SYM iajs-1918 40 67 ∪ ∪ NOUN iajs-1918 40 68 / / SYM iajs-1918 40 69 𝑑𝑡 𝑑𝑡 ADP iajs-1918 40 70 35 35 NUM iajs-1918 40 71 24 24 NUM iajs-1918 40 72 0 0 NUM iajs-1918 40 73 1 1 NUM iajs-1918 40 74 , , PUNCT iajs-1918 40 75 where where SCONJ iajs-1918 40 76 the the DET iajs-1918 40 77 exact exact ADJ iajs-1918 40 78 solution solution NOUN iajs-1918 40 79 is be AUX iajs-1918 40 80 : : PUNCT iajs-1918 40 81 ∪ ∪ X iajs-1918 40 82 𝑡 𝑡 PROPN iajs-1918 40 83 mse mse PROPN iajs-1918 40 84 time time NOUN iajs-1918 40 85 epoch epoch NOUN iajs-1918 40 86 performance performance NOUN iajs-1918 40 87 of of ADP iajs-1918 40 88 train train NOUN iajs-1918 40 89 train train NOUN iajs-1918 40 90 function function NOUN iajs-1918 40 91 9.34772e‐06 9.34772e‐06 NOUN iajs-1918 40 92     SPACE iajs-1918 41 1 0:00:02 0:00:02 NUM iajs-1918 41 2 50 50 NUM iajs-1918 41 3 2.39 2.39 NUM iajs-1918 41 4 - - SYM iajs-1918 41 5 31 31 NUM iajs-1918 41 6 trainlm trainlm NOUN iajs-1918 41 7 weights weight NOUN iajs-1918 41 8 and and CCONJ iajs-1918 41 9 bias bias NOUN iajs-1918 41 10 for for ADP iajs-1918 41 11 trainlm trainlm NOUN iajs-1918 41 12 net.b{1}net.lw{2,1}net.iu{1,1}net.iw{1,1 net.b{1}net.lw{2,1}net.iu{1,1}net.iw{1,1 NOUN iajs-1918 41 13 } } PUNCT iajs-1918 41 14 0.48760.12390.98760.2436 0.48760.12390.98760.2436 SYM iajs-1918 41 15 0.25420.63980.98430.4563 0.25420.63980.98430.4563 NUM iajs-1918 41 16 0.19840.27600.09870.6534 0.19840.27600.09870.6534 NUM iajs-1918 42 1 0.12950.08340.23410.4237 0.12950.08340.23410.4237 NUM iajs-1918 42 2 0.59340.23140.75230.3122 0.59340.23140.75230.3122 NUM iajs-1918 42 3     SPACE iajs-1918 42 4 182 182 NUM iajs-1918 42 5 ibn ibn PROPN iajs-1918 42 6 al al PROPN iajs-1918 42 7 - - PUNCT iajs-1918 42 8 haitham haitham PROPN iajs-1918 42 9 jour.for jour.for PROPN iajs-1918 42 10 pure&appl.sci pure&appl.sci PROPN iajs-1918 42 11 . . PUNCT iajs-1918 42 12 ihjpas ihjpas PROPN iajs-1918 42 13 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 42 14 vol vol NOUN iajs-1918 42 15 . . PUNCT iajs-1918 42 16 32 32 NUM iajs-1918 42 17 ( ( PUNCT iajs-1918 42 18 1 1 NUM iajs-1918 42 19 ) ) SYM iajs-1918 42 20 2019 2019 NUM iajs-1918 42 21 table table NOUN iajs-1918 42 22 4 4 NUM iajs-1918 42 23 . . PUNCT iajs-1918 42 24 analytic analytic ADJ iajs-1918 42 25 and and CCONJ iajs-1918 42 26 neural neural ADJ iajs-1918 42 27 solution solution NOUN iajs-1918 42 28 of of ADP iajs-1918 42 29 example example NOUN iajs-1918 42 30 2 2 NUM iajs-1918 42 31 table table NOUN iajs-1918 42 32 5 5 NUM iajs-1918 42 33 . . PUNCT iajs-1918 43 1 the the DET iajs-1918 43 2 performance performance NOUN iajs-1918 43 3 of of ADP iajs-1918 43 4 the the DET iajs-1918 43 5 train train NOUN iajs-1918 43 6 with with ADP iajs-1918 43 7 epoch epoch NOUN iajs-1918 43 8 and and CCONJ iajs-1918 43 9 time time NOUN iajs-1918 43 10 mse mse PROPN iajs-1918 43 11 time time PROPN iajs-1918 43 12 epoch epoch PROPN iajs-1918 43 13 performance performance NOUN iajs-1918 43 14 of of ADP iajs-1918 43 15 train train NOUN iajs-1918 43 16 train train NOUN iajs-1918 43 17 function function NOUN iajs-1918 43 18 9.72845e-08 9.72845e-08 NOUN iajs-1918 43 19 0:00:10 0:00:10 NUM iajs-1918 43 20 300 300 NUM iajs-1918 43 21 2.38 2.38 NUM iajs-1918 43 22 - - SYM iajs-1918 43 23 31 31 NUM iajs-1918 43 24 trainlm trainlm NOUN iajs-1918 43 25 table table NOUN iajs-1918 43 26 6 6 NUM iajs-1918 43 27 . . PUNCT iajs-1918 43 28 initial initial ADJ iajs-1918 43 29 weight weight NOUN iajs-1918 43 30 and and CCONJ iajs-1918 43 31 bias bias NOUN iajs-1918 43 32 of of ADP iajs-1918 43 33 the the DET iajs-1918 43 34 network network NOUN iajs-1918 43 35 for for ADP iajs-1918 43 36 different different ADJ iajs-1918 43 37 training training NOUN iajs-1918 43 38 algorithm algorithm NOUN iajs-1918 43 39 the the DET iajs-1918 43 40 error error NOUN iajs-1918 43 41 e(x e(x NUM iajs-1918 43 42 ) ) PUNCT iajs-1918 43 43   NUM iajs-1918 43 44 | | ADV iajs-1918 43 45 yt(x yt(x NOUN iajs-1918 43 46 ) ) PUNCT iajs-1918 43 47 ya(x ya(x NOUN iajs-1918 43 48 ) ) PUNCT iajs-1918 43 49 | | ADV iajs-1918 43 50 w w NOUN iajs-1918 43 51 h h NOUN iajs-1918 43 52 e e NOUN iajs-1918 43 53 r r NOUN iajs-1918 43 54 e e X iajs-1918 43 55 yt(x yt(x NOUN iajs-1918 43 56 ) ) PUNCT iajs-1918 43 57 computed compute VERB iajs-1918 43 58 by by ADP iajs-1918 43 59 the the DET iajs-1918 43 60 following follow VERB iajs-1918 43 61 training training NOUN iajs-1918 43 62 algorithm algorithm NOUN iajs-1918 43 63 ( ( PUNCT iajs-1918 43 64 trainlm trainlm PROPN iajs-1918 43 65 ) ) PUNCT iajs-1918 43 66 out out ADP iajs-1918 43 67 of of ADP iajs-1918 43 68 suggested suggest VERB iajs-1918 43 69 ffnn ffnn NOUN iajs-1918 43 70 yt(x yt(x PROPN iajs-1918 43 71 ) ) PUNCT iajs-1918 43 72 for for ADP iajs-1918 43 73     SPACE iajs-1918 43 74 lm lm INTJ iajs-1918 43 75 training training NOUN iajs-1918 43 76 algorithm algorithm NOUN iajs-1918 43 77 analytic analytic ADJ iajs-1918 43 78 solution solution NOUN iajs-1918 43 79 input input NOUN iajs-1918 43 80 trainlmya(x trainlmya(x NOUN iajs-1918 43 81 ) ) PUNCT iajs-1918 43 82 x x SYM iajs-1918 43 83 0.0000000000000000.000000000000000.0 0.0000000000000000.000000000000000.0 NOUN iajs-1918 43 84 0.000004654342980.099004654342980.099000000000000.1 0.000004654342980.099004654342980.099000000000000.1 NOUN iajs-1918 43 85 0.000045366572450.192045366572450.192000000000000.2 0.000045366572450.192045366572450.192000000000000.2 X iajs-1918 43 86 0.000065872297870.273065872297870.273000000000000.3 0.000065872297870.273065872297870.273000000000000.3 DET iajs-1918 43 87 0.000999435226040.335000564773960.336000000000000.4 0.000999435226040.335000564773960.336000000000000.4 NOUN iajs-1918 43 88 0.000000564773420.375000564773420.375000000000000.5 0.000000564773420.375000564773420.375000000000000.5 NUM iajs-1918 43 89 0.000000354288780.384000354288780.384000000000000.6 0.000000354288780.384000354288780.384000000000000.6 NUM iajs-1918 43 90 0.000006755341980.357006755341980.357000000000000.7 0.000006755341980.357006755341980.357000000000000.7 NUM iajs-1918 43 91 0.000254463901340.287745536098660.288000000000000.8 0.000254463901340.287745536098660.288000000000000.8 NUM iajs-1918 43 92 0.000006453372430.171006453372430.171000000000000.9 0.000006453372430.171006453372430.171000000000000.9 NUM iajs-1918 43 93 0.0000000000000000.000000000000001.0 0.0000000000000000.000000000000001.0 NOUN iajs-1918 43 94 weights weight NOUN iajs-1918 43 95 and and CCONJ iajs-1918 43 96 bias bias NOUN iajs-1918 43 97 for for ADP iajs-1918 43 98 trainlm trainlm NOUN iajs-1918 43 99 net.b{1 net.b{1 PRON iajs-1918 43 100 } } PUNCT iajs-1918 43 101 net.lw{2,1 net.lw{2,1 ADJ iajs-1918 43 102 } } PUNCT iajs-1918 43 103 net.iu{1,1 net.iu{1,1 NOUN iajs-1918 43 104 } } PUNCT iajs-1918 43 105 net.iw{1,1 net.iw{1,1 ADV iajs-1918 43 106 } } PUNCT iajs-1918 43 107 0.12330.93650.09750.6574 0.12330.93650.09750.6574 NUM iajs-1918 43 108 0.75640.92840.43520.9786 0.75640.92840.43520.9786 NOUN iajs-1918 43 109 0.87350.55420.75640.4352 0.87350.55420.75640.4352 NUM iajs-1918 43 110 0.92850.86350.97860.9807 0.92850.86350.97860.9807 NUM iajs-1918 43 111 0.56370.66570.98430.5543 0.56370.66570.98430.5543 NUM iajs-1918 43 112     SPACE iajs-1918 43 113 183 183 NUM iajs-1918 43 114 ibn ibn PROPN iajs-1918 43 115 al al PROPN iajs-1918 43 116 - - PUNCT iajs-1918 43 117 haitham haitham PROPN iajs-1918 43 118 jour.for jour.for PROPN iajs-1918 43 119 pure&appl.sci pure&appl.sci PROPN iajs-1918 43 120 . . PUNCT iajs-1918 43 121 ihjpas ihjpas PROPN iajs-1918 43 122 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 43 123 vol vol NOUN iajs-1918 43 124 . . PUNCT iajs-1918 44 1 32 32 NUM iajs-1918 44 2 ( ( PUNCT iajs-1918 44 3 1 1 NUM iajs-1918 44 4 ) ) PUNCT iajs-1918 44 5 2019 2019 NUM iajs-1918 44 6 figure figure NOUN iajs-1918 44 7 2 2 NUM iajs-1918 44 8 . . NOUN iajs-1918 44 9 exact exact ADJ iajs-1918 44 10 & & CCONJ iajs-1918 44 11 approximate approximate ADJ iajs-1918 44 12 solution solution NOUN iajs-1918 44 13 of of ADP iajs-1918 44 14 example1 example1 NOUN iajs-1918 44 15 by by ADP iajs-1918 44 16 using use VERB iajs-1918 44 17 : : PUNCT iajs-1918 44 18 modify modify VERB iajs-1918 44 19 trainlm trainlm PROPN iajs-1918 44 20 & & CCONJ iajs-1918 44 21 trainlm trainlm PROPN iajs-1918 44 22 algorithms algorithms PROPN iajs-1918 44 23 example example NOUN iajs-1918 44 24 3 3 NUM iajs-1918 44 25 in in ADP iajs-1918 44 26 the the DET iajs-1918 44 27 third third ADJ iajs-1918 44 28 example example NOUN iajs-1918 44 29 we we PRON iajs-1918 44 30 will will AUX iajs-1918 44 31 demonstrate demonstrate VERB iajs-1918 44 32 the the DET iajs-1918 44 33 proposed propose VERB iajs-1918 44 34 technique technique NOUN iajs-1918 44 35 on on ADP iajs-1918 44 36 the the DET iajs-1918 44 37 second second ADJ iajs-1918 44 38 type type NOUN iajs-1918 44 39 of of ADP iajs-1918 44 40 fractional fractional ADJ iajs-1918 44 41 equations equation NOUN iajs-1918 44 42 for for ADP iajs-1918 44 43 that that DET iajs-1918 44 44 integration integration NOUN iajs-1918 45 1 [ [ X iajs-1918 45 2 16 16 NUM iajs-1918 45 3 ] ] X iajs-1918 45 4 𝒖 𝒖 X iajs-1918 45 5 𝟐 𝟐 NUM iajs-1918 45 6 𝟏𝟔 𝟏𝟔 NUM iajs-1918 45 7 𝟏𝟓 𝟏𝟓 NUM iajs-1918 45 8 𝟓 𝟓 NUM iajs-1918 45 9 𝟐 𝟐 NUM iajs-1918 45 10 ∪ ∪ AUX iajs-1918 45 11 𝒕 𝒕 VERB iajs-1918 45 12 𝒕 𝒕 NOUN iajs-1918 45 13 𝟏 𝟏 NUM iajs-1918 45 14 𝟐 𝟐 NUM iajs-1918 45 15 𝒅𝒕 𝒅𝒕 PROPN iajs-1918 45 16 𝒙 𝒙 PROPN iajs-1918 45 17 𝟎 𝟎 NUM iajs-1918 45 18 0 0 NUM iajs-1918 45 19 1 1 NUM iajs-1918 45 20 , , PUNCT iajs-1918 45 21 where where SCONJ iajs-1918 45 22 the the DET iajs-1918 45 23 exact exact ADJ iajs-1918 45 24 solution solution NOUN iajs-1918 45 25 is be AUX iajs-1918 45 26 ∪ ∪ ADJ iajs-1918 45 27 𝑡 𝑡 NOUN iajs-1918 45 28 table table NOUN iajs-1918 45 29 7 7 NUM iajs-1918 45 30 . . PUNCT iajs-1918 45 31 analytic analytic ADJ iajs-1918 45 32 and and CCONJ iajs-1918 45 33 neural neural ADJ iajs-1918 45 34 solution solution NOUN iajs-1918 45 35 of of ADP iajs-1918 45 36 example example NOUN iajs-1918 45 37 3 3 NUM iajs-1918 45 38 the the DET iajs-1918 45 39 error error NOUN iajs-1918 45 40 e(x e(x NUM iajs-1918 45 41 ) ) PUNCT iajs-1918 46 1   NUM iajs-1918 47 1 | | ADV iajs-1918 47 2 yt(x yt(x NOUN iajs-1918 47 3 ) ) PUNCT iajs-1918 47 4 ya(x ya(x NOUN iajs-1918 47 5 ) ) PUNCT iajs-1918 48 1 | | ADV iajs-1918 48 2 w w NOUN iajs-1918 48 3 h h NOUN iajs-1918 48 4 e e NOUN iajs-1918 48 5 r r NOUN iajs-1918 48 6 e e X iajs-1918 48 7 yt(x yt(x NOUN iajs-1918 48 8 ) ) PUNCT iajs-1918 48 9 computed compute VERB iajs-1918 48 10 by by ADP iajs-1918 48 11 the the DET iajs-1918 48 12 following follow VERB iajs-1918 48 13 training training NOUN iajs-1918 48 14 algorithm algorithm NOUN iajs-1918 48 15 ( ( PUNCT iajs-1918 48 16 trainlm trainlm PROPN iajs-1918 48 17 ) ) PUNCT iajs-1918 48 18 out out ADP iajs-1918 48 19 of of ADP iajs-1918 48 20 suggested suggest VERB iajs-1918 48 21 ffnn ffnn NOUN iajs-1918 48 22 yt(x yt(x PROPN iajs-1918 48 23 ) ) PUNCT iajs-1918 48 24 for for ADP iajs-1918 48 25 lm lm INTJ iajs-1918 48 26 training training NOUN iajs-1918 48 27 algorithm algorithm NOUN iajs-1918 48 28 analytic analytic ADJ iajs-1918 48 29 solution solution NOUN iajs-1918 48 30 input input NOUN iajs-1918 48 31 trainlmya(x trainlmya(x NOUN iajs-1918 48 32 ) ) PUNCT iajs-1918 48 33 x x SYM iajs-1918 48 34 0.000000000000000.000000000000000.000000000000000.0 0.000000000000000.000000000000000.000000000000000.0 NUM iajs-1918 48 35 0.000004356208660.010004356208660.010000000000000.1 0.000004356208660.010004356208660.010000000000000.1 NUM iajs-1918 48 36 0.000000653231460.040000653231460.040000000000000.2 0.000000653231460.040000653231460.040000000000000.2 VERB iajs-1918 48 37 0.000000534274650.090000534274650.090000000000000.3 0.000000534274650.090000534274650.090000000000000.3 NUM iajs-1918 48 38 0.000000635277630.160000635277630.160000000000000.4 0.000000635277630.160000635277630.160000000000000.4 NUM iajs-1918 48 39 0.000000006242420.250000006242420.250000000000000.5 0.000000006242420.250000006242420.250000000000000.5 NUM iajs-1918 48 40 0.000003425534270.360003425534270.360000000000000.6 0.000003425534270.360003425534270.360000000000000.6 NOUN iajs-1918 48 41 0.000342776134500.49034277613450.490000000000000.7 0.000342776134500.49034277613450.490000000000000.7 NOUN iajs-1918 48 42 0.001436628759300.64143662875930.640000000000000.8 0.001436628759300.64143662875930.640000000000000.8 PRON iajs-1918 48 43 0.000342876143380.810342876143380.810000000000000.9 0.000342876143380.810342876143380.810000000000000.9 NUM iajs-1918 49 1 0.0000000000000011.000000000000001.0 0.0000000000000011.000000000000001.0 PRON iajs-1918 49 2     SPACE iajs-1918 49 3 184 184 NUM iajs-1918 49 4 ibn ibn PROPN iajs-1918 49 5 al al PROPN iajs-1918 49 6 - - PUNCT iajs-1918 49 7 haitham haitham PROPN iajs-1918 49 8 jour.for jour.for PROPN iajs-1918 49 9 pure&appl.sci pure&appl.sci PROPN iajs-1918 49 10 . . PUNCT iajs-1918 49 11 ihjpas ihjpas PROPN iajs-1918 49 12 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 49 13 vol vol NOUN iajs-1918 49 14 . . PUNCT iajs-1918 49 15 32 32 NUM iajs-1918 49 16 ( ( PUNCT iajs-1918 49 17 1 1 NUM iajs-1918 49 18 ) ) SYM iajs-1918 49 19 2019 2019 NUM iajs-1918 49 20 table table NOUN iajs-1918 49 21 8 8 NUM iajs-1918 49 22 . . PUNCT iajs-1918 50 1 the the DET iajs-1918 50 2 performance performance NOUN iajs-1918 50 3 of of ADP iajs-1918 50 4 the the DET iajs-1918 50 5 train train NOUN iajs-1918 50 6 with with ADP iajs-1918 50 7 epoch epoch NOUN iajs-1918 50 8 and and CCONJ iajs-1918 50 9 time time NOUN iajs-1918 50 10 mse mse PROPN iajs-1918 50 11 time time PROPN iajs-1918 50 12 epoch epoch PROPN iajs-1918 50 13 performance performance NOUN iajs-1918 50 14 of of ADP iajs-1918 50 15 train train NOUN iajs-1918 50 16 train train NOUN iajs-1918 50 17 function function NOUN iajs-1918 50 18 2.08999e-07 2.08999e-07 NUM iajs-1918 50 19 0:00:11 0:00:11 NUM iajs-1918 50 20 700 700 NUM iajs-1918 50 21 3.38 3.38 NUM iajs-1918 50 22 - - SYM iajs-1918 50 23 31 31 NUM iajs-1918 50 24 trainlm trainlm NOUN iajs-1918 50 25 table table NOUN iajs-1918 50 26 9 9 NUM iajs-1918 50 27 . . PUNCT iajs-1918 50 28 initial initial ADJ iajs-1918 50 29 weight weight NOUN iajs-1918 50 30 and and CCONJ iajs-1918 50 31 bias bias NOUN iajs-1918 50 32 of of ADP iajs-1918 50 33 the the DET iajs-1918 50 34 network network NOUN iajs-1918 50 35 for for ADP iajs-1918 50 36 different different ADJ iajs-1918 50 37 training training NOUN iajs-1918 50 38 algorithm algorithm NOUN iajs-1918 50 39 figure figure NOUN iajs-1918 50 40 3 3 NUM iajs-1918 50 41 . . NOUN iajs-1918 50 42 exact exact ADJ iajs-1918 50 43 & & CCONJ iajs-1918 50 44 approximate approximate ADJ iajs-1918 50 45 solution solution NOUN iajs-1918 50 46 of of ADP iajs-1918 50 47 example1 example1 PROPN iajs-1918 50 48 using use VERB iajs-1918 50 49 : : PUNCT iajs-1918 50 50 modify modify VERB iajs-1918 50 51 trainlm trainlm PROPN iajs-1918 50 52 & & CCONJ iajs-1918 50 53 trainlm trainlm PROPN iajs-1918 50 54 algorithms algorithms PROPN iajs-1918 50 55 6 6 NUM iajs-1918 50 56 . . PUNCT iajs-1918 51 1 conclusions conclusion NOUN iajs-1918 51 2 in in ADP iajs-1918 51 3 this this DET iajs-1918 51 4 paper paper NOUN iajs-1918 51 5 , , PUNCT iajs-1918 51 6 a a DET iajs-1918 51 7 new new ADJ iajs-1918 51 8 procedure procedure NOUN iajs-1918 51 9 was be AUX iajs-1918 51 10 used use VERB iajs-1918 51 11 to to PART iajs-1918 51 12 solve solve VERB iajs-1918 51 13 a a DET iajs-1918 51 14 special special ADJ iajs-1918 51 15 type type NOUN iajs-1918 51 16 of of ADP iajs-1918 51 17 fractional fractional ADJ iajs-1918 51 18 integrals integral NOUN iajs-1918 51 19 using use VERB iajs-1918 51 20 parallel parallel ADJ iajs-1918 51 21 processors processor NOUN iajs-1918 51 22 and and CCONJ iajs-1918 51 23 a a DET iajs-1918 51 24 high high ADJ iajs-1918 51 25 - - PUNCT iajs-1918 51 26 level level NOUN iajs-1918 51 27 training training NOUN iajs-1918 51 28 algorithm algorithm NOUN iajs-1918 51 29 . . PUNCT iajs-1918 52 1 then then ADV iajs-1918 52 2 applied apply VERB iajs-1918 52 3 in in ADP iajs-1918 52 4 three three NUM iajs-1918 52 5 examples example NOUN iajs-1918 52 6 of of ADP iajs-1918 52 7 different different ADJ iajs-1918 52 8 types type NOUN iajs-1918 52 9 were be AUX iajs-1918 52 10 the the DET iajs-1918 52 11 results result NOUN iajs-1918 52 12 compared compare VERB iajs-1918 52 13 with with ADP iajs-1918 52 14 analytical analytical ADJ iajs-1918 52 15 solutions solution NOUN iajs-1918 52 16 for for ADP iajs-1918 52 17 these these DET iajs-1918 52 18 integrals integral NOUN iajs-1918 52 19 . . PUNCT iajs-1918 53 1 the the DET iajs-1918 53 2 results result NOUN iajs-1918 53 3 show show VERB iajs-1918 53 4 the the DET iajs-1918 53 5 accuracy accuracy NOUN iajs-1918 53 6 of of ADP iajs-1918 53 7 suggested suggest VERB iajs-1918 53 8 method method NOUN iajs-1918 53 9 is be AUX iajs-1918 53 10 very very ADV iajs-1918 53 11 high high ADJ iajs-1918 53 12 and and CCONJ iajs-1918 53 13 the the DET iajs-1918 53 14 speed speed NOUN iajs-1918 53 15 of of ADP iajs-1918 53 16 convergence convergence NOUN iajs-1918 53 17 is be AUX iajs-1918 53 18 ideal ideal ADJ iajs-1918 53 19 . . PUNCT iajs-1918 54 1 weights weight NOUN iajs-1918 54 2 and and CCONJ iajs-1918 54 3 bias bias NOUN iajs-1918 54 4 for for ADP iajs-1918 54 5 trainlm trainlm NOUN iajs-1918 54 6 net.b{1 net.b{1 PRON iajs-1918 54 7 } } PUNCT iajs-1918 54 8 net.lw{2,1 net.lw{2,1 ADJ iajs-1918 54 9 } } PUNCT iajs-1918 54 10 net.iu{1,1 net.iu{1,1 NOUN iajs-1918 54 11 } } PUNCT iajs-1918 54 12 net.iw{1,1 net.iw{1,1 ADV iajs-1918 54 13 } } PUNCT iajs-1918 54 14 0.92750.95730.64770.7645 0.92750.95730.64770.7645 NUM iajs-1918 54 15 0.95390.36240.62980.8735 0.95390.36240.62980.8735 NUM iajs-1918 54 16 0.23650.87290.64290.1249 0.23650.87290.64290.1249 NOUN iajs-1918 54 17 0.11260.75630.23140.9823 0.11260.75630.23140.9823 NUM iajs-1918 54 18 0.74030.83650.34250.8669 0.74030.83650.34250.8669 NUM iajs-1918 54 19     SPACE iajs-1918 54 20 185 185 NUM iajs-1918 54 21 ibn ibn PROPN iajs-1918 54 22 al al PROPN iajs-1918 54 23 - - PUNCT iajs-1918 54 24 haitham haitham PROPN iajs-1918 54 25 jour.for jour.for PROPN iajs-1918 54 26 pure&appl.sci pure&appl.sci PROPN iajs-1918 54 27 . . PUNCT iajs-1918 54 28 ihjpas ihjpas PROPN iajs-1918 54 29 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 54 30 vol vol NOUN iajs-1918 54 31 . . PUNCT iajs-1918 55 1 32 32 NUM iajs-1918 55 2 ( ( PUNCT iajs-1918 55 3 1 1 NUM iajs-1918 55 4 ) ) PUNCT iajs-1918 55 5 2019 2019 NUM iajs-1918 55 6 references reference NOUN iajs-1918 55 7 1 1 NUM iajs-1918 55 8 . . PUNCT iajs-1918 56 1 mann mann PROPN iajs-1918 56 2 , , PUNCT iajs-1918 56 3 w.r w.r PROPN iajs-1918 56 4 . . PROPN iajs-1918 56 5 ; ; PUNCT iajs-1918 56 6 wolf wolf PROPN iajs-1918 56 7 , , PUNCT iajs-1918 56 8 f. f. PROPN iajs-1918 56 9 heat heat NOUN iajs-1918 56 10 transfer transfer NOUN iajs-1918 56 11 between between ADP iajs-1918 56 12 solids solid NOUN iajs-1918 56 13 and and CCONJ iajs-1918 56 14 gases gas NOUN iajs-1918 56 15 under under ADP iajs-1918 56 16 nonlinear nonlinear ADJ iajs-1918 56 17 boundary boundary ADJ iajs-1918 56 18 conditions condition NOUN iajs-1918 56 19 . . PUNCT iajs-1918 57 1 appl appl PROPN iajs-1918 57 2 math math PROPN iajs-1918 57 3 . . PUNCT iajs-1918 58 1 1951 1951 NUM iajs-1918 58 2 , , PUNCT iajs-1918 58 3 9 9 NUM iajs-1918 58 4 , , PUNCT iajs-1918 58 5 163–184 163–184 NUM iajs-1918 58 6 . . NOUN iajs-1918 59 1 2 2 NUM iajs-1918 59 2 . . X iajs-1918 59 3 goncerzewicz goncerzewicz PROPN iajs-1918 59 4 , , PUNCT iajs-1918 59 5 j. j. PROPN iajs-1918 59 6 ; ; PUNCT iajs-1918 59 7 marcinkowska marcinkowska PROPN iajs-1918 59 8 , , PUNCT iajs-1918 59 9 h. h. PROPN iajs-1918 59 10 ; ; PUNCT iajs-1918 59 11 okrasinski okrasinski PROPN iajs-1918 59 12 , , PUNCT iajs-1918 59 13 w. w. PROPN iajs-1918 59 14 ; ; PUNCT iajs-1918 59 15 tabisz tabisz VERB iajs-1918 59 16 , , PUNCT iajs-1918 59 17 k. k. NOUN iajs-1918 59 18 on on ADP iajs-1918 59 19 percolation percolation NOUN iajs-1918 59 20 of of ADP iajs-1918 59 21 water water NOUN iajs-1918 59 22 from from ADP iajs-1918 59 23 a a DET iajs-1918 59 24 cylindrical cylindrical ADJ iajs-1918 59 25 reservoir reservoir NOUN iajs-1918 59 26 into into ADP iajs-1918 59 27 the the DET iajs-1918 59 28 surrounding surround VERB iajs-1918 59 29 soil soil NOUN iajs-1918 59 30 . . PUNCT iajs-1918 60 1 appl appl PROPN iajs-1918 60 2 . . PROPN iajs-1918 60 3 math math PROPN iajs-1918 60 4 . . PUNCT iajs-1918 61 1 1951 1951 NUM iajs-1918 61 2 , , PUNCT iajs-1918 61 3 16 16 NUM iajs-1918 61 4 , , PUNCT iajs-1918 61 5 249–261 249–261 NUM iajs-1918 61 6 . . PUNCT iajs-1918 62 1 3 3 NUM iajs-1918 62 2 . . X iajs-1918 62 3 keller keller PROPN iajs-1918 62 4 , , PUNCT iajs-1918 62 5 j.j j.j PROPN iajs-1918 62 6 . . PROPN iajs-1918 62 7 propagation propagation NOUN iajs-1918 62 8 of of ADP iajs-1918 62 9 simple simple ADJ iajs-1918 62 10 nonlinear nonlinear ADJ iajs-1918 62 11 waves wave NOUN iajs-1918 62 12 in in ADP iajs-1918 62 13 gas gas NOUN iajs-1918 62 14 filled fill VERB iajs-1918 62 15 tubes tube NOUN iajs-1918 62 16 with with ADP iajs-1918 62 17 friction friction NOUN iajs-1918 62 18 . . PUNCT iajs-1918 63 1 z. z. PROPN iajs-1918 63 2 angew angew PROPN iajs-1918 63 3 . . PUNCT iajs-1918 64 1 math math NOUN iajs-1918 64 2 . . PUNCT iajs-1918 65 1 phys phy NOUN iajs-1918 65 2 . . PUNCT iajs-1918 66 1 1981 1981 NUM iajs-1918 66 2 , , PUNCT iajs-1918 66 3 32 32 NUM iajs-1918 66 4 , , PUNCT iajs-1918 66 5 170–181 170–181 NUM iajs-1918 66 6 . . NOUN iajs-1918 67 1 4 4 NUM iajs-1918 67 2 . . X iajs-1918 67 3 lima lima PROPN iajs-1918 67 4 , , PUNCT iajs-1918 67 5 p. p. NOUN iajs-1918 67 6 ; ; PUNCT iajs-1918 67 7 diogo diogo PROPN iajs-1918 67 8 , , PUNCT iajs-1918 67 9 t. t. PROPN iajs-1918 67 10 numerical numerical ADJ iajs-1918 67 11 solution solution NOUN iajs-1918 67 12 of of ADP iajs-1918 67 13 nonuniquely nonuniquely ADV iajs-1918 67 14 solvable solvable ADJ iajs-1918 67 15 volterra volterra NOUN iajs-1918 67 16 integral integral ADJ iajs-1918 67 17 equation equation NOUN iajs-1918 67 18 using use VERB iajs-1918 67 19 extrapolation extrapolation NOUN iajs-1918 67 20 methods method NOUN iajs-1918 67 21 . . PUNCT iajs-1918 68 1 j. j. PROPN iajs-1918 68 2 comput comput PROPN iajs-1918 68 3 . . PUNCT iajs-1918 69 1 app app PROPN iajs-1918 69 2 . . PROPN iajs-1918 69 3 math math PROPN iajs-1918 69 4 . . PUNCT iajs-1918 70 1 2002 2002 NUM iajs-1918 70 2 , , PUNCT iajs-1918 70 3 140 140 NUM iajs-1918 70 4 , , PUNCT iajs-1918 70 5 537–557 537–557 NUM iajs-1918 70 6 . . NOUN iajs-1918 71 1 5 5 NUM iajs-1918 71 2 . . X iajs-1918 71 3 singh singh PROPN iajs-1918 71 4 , , PUNCT iajs-1918 71 5 o.p o.p PROPN iajs-1918 71 6 . . PROPN iajs-1918 71 7 ; ; PUNCT iajs-1918 71 8 singh singh PROPN iajs-1918 71 9 , , PUNCT iajs-1918 71 10 v.k v.k PROPN iajs-1918 71 11 . . PROPN iajs-1918 71 12 ; ; PUNCT iajs-1918 71 13 pandey pandey PROPN iajs-1918 71 14 , , PUNCT iajs-1918 71 15 r.k r.k PROPN iajs-1918 71 16 . . PROPN iajs-1918 72 1 a a DET iajs-1918 72 2 stable stable ADJ iajs-1918 72 3 numerical numerical ADJ iajs-1918 72 4 inversion inversion NOUN iajs-1918 72 5 of of ADP iajs-1918 72 6 abel abel PROPN iajs-1918 72 7 ’s ’s PART iajs-1918 72 8 integral integral ADJ iajs-1918 72 9 equation equation NOUN iajs-1918 72 10 using use VERB iajs-1918 72 11 almost almost ADV iajs-1918 72 12 bernstein bernstein PROPN iajs-1918 72 13 operational operational ADJ iajs-1918 72 14 matrix matrix NOUN iajs-1918 72 15 . . PUNCT iajs-1918 73 1 j. j. PROPN iajs-1918 73 2 quant quant PROPN iajs-1918 73 3 . . PUNCT iajs-1918 74 1 spectrosc spectrosc PROPN iajs-1918 74 2 . . PUNCT iajs-1918 75 1 radiat radiat PROPN iajs-1918 75 2 . . PUNCT iajs-1918 76 1 transf transf PROPN iajs-1918 76 2 . . PUNCT iajs-1918 77 1 2010 2010 NUM iajs-1918 77 2 , , PUNCT iajs-1918 77 3 111 111 NUM iajs-1918 77 4 , , PUNCT iajs-1918 77 5 245 245 NUM iajs-1918 77 6 – – PUNCT iajs-1918 77 7 252 252 NUM iajs-1918 77 8 . . NOUN iajs-1918 78 1 6 6 NUM iajs-1918 78 2 . . X iajs-1918 78 3 bushell bushell PROPN iajs-1918 78 4 , , PUNCT iajs-1918 78 5 p.j p.j PROPN iajs-1918 78 6 . . PROPN iajs-1918 78 7 ; ; PUNCT iajs-1918 78 8 okrasinski okrasinski PROPN iajs-1918 78 9 , , PUNCT iajs-1918 78 10 w. w. PROPN iajs-1918 78 11 nonlinear nonlinear PROPN iajs-1918 78 12 volterra volterra PROPN iajs-1918 78 13 integral integral ADJ iajs-1918 78 14 equations equation NOUN iajs-1918 78 15 with with ADP iajs-1918 78 16 convolution convolution NOUN iajs-1918 78 17 kernel kernel PROPN iajs-1918 78 18 . . PUNCT iajs-1918 79 1 j. j. PROPN iajs-1918 79 2 lond lond PROPN iajs-1918 79 3 . . PUNCT iajs-1918 80 1 math math PROPN iajs-1918 80 2 . . PUNCT iajs-1918 81 1 soc soc PROPN iajs-1918 81 2 . . PUNCT iajs-1918 82 1 1990 1990 NUM iajs-1918 82 2 , , PUNCT iajs-1918 82 3 41 41 NUM iajs-1918 82 4 , , PUNCT iajs-1918 82 5 503–510 503–510 NUM iajs-1918 82 6 . . PUNCT iajs-1918 83 1 7 7 NUM iajs-1918 83 2 . . X iajs-1918 83 3 li li PROPN iajs-1918 83 4 , , PUNCT iajs-1918 83 5 m. m. NOUN iajs-1918 83 6 ; ; PUNCT iajs-1918 83 7 zhao zhao PROPN iajs-1918 83 8 , , PUNCT iajs-1918 83 9 w. w. PROPN iajs-1918 83 10 solving solve VERB iajs-1918 83 11 abel abel PROPN iajs-1918 83 12 ’s ’s PART iajs-1918 83 13 type type NOUN iajs-1918 83 14 integral integral ADJ iajs-1918 83 15 equation equation NOUN iajs-1918 83 16 with with ADP iajs-1918 83 17 mikusinski mikusinski PROPN iajs-1918 83 18 ’s ’s PART iajs-1918 83 19 operator operator NOUN iajs-1918 83 20 of of ADP iajs-1918 83 21 fractional fractional ADJ iajs-1918 83 22 order order NOUN iajs-1918 83 23 , , PUNCT iajs-1918 83 24 adv adv PROPN iajs-1918 83 25 . . PUNCT iajs-1918 83 26 math math PROPN iajs-1918 83 27 . . PUNCT iajs-1918 84 1 phys phy NOUN iajs-1918 84 2 . . PUNCT iajs-1918 85 1 2013 2013 NUM iajs-1918 85 2 . . PUNCT iajs-1918 86 1 doi:10.1155/2013/806984 doi:10.1155/2013/806984 PROPN iajs-1918 86 2 . . NOUN iajs-1918 86 3 8 8 NUM iajs-1918 86 4 . . PUNCT iajs-1918 87 1 jahanshahi jahanshahi PROPN iajs-1918 87 2 , , PUNCT iajs-1918 87 3 s. s. PROPN iajs-1918 87 4 ; ; PUNCT iajs-1918 87 5 babolian babolian PROPN iajs-1918 87 6 , , PUNCT iajs-1918 87 7 e. e. PROPN iajs-1918 87 8 ; ; PUNCT iajs-1918 87 9 torres torre NOUN iajs-1918 87 10 , , PUNCT iajs-1918 87 11 d. d. PROPN iajs-1918 87 12 ; ; PUNCT iajs-1918 87 13 vahidi vahidi PROPN iajs-1918 87 14 , , PUNCT iajs-1918 87 15 a. a. NOUN iajs-1918 87 16 solving solve VERB iajs-1918 87 17 abel abel PROPN iajs-1918 87 18 integral integral ADJ iajs-1918 87 19 equations equation NOUN iajs-1918 87 20 of of ADP iajs-1918 87 21 first first ADJ iajs-1918 87 22 kind kind NOUN iajs-1918 87 23 via via ADP iajs-1918 87 24 fractional fractional ADJ iajs-1918 87 25 calculus calculus NOUN iajs-1918 87 26 . . PUNCT iajs-1918 88 1 j. j. PROPN iajs-1918 88 2 king king PROPN iajs-1918 88 3 saud saud PROPN iajs-1918 88 4 univ univ PROPN iajs-1918 88 5 . . PUNCT iajs-1918 89 1 sci sci PROPN iajs-1918 89 2 . . PROPN iajs-1918 89 3 2015 2015 NUM iajs-1918 89 4 , , PUNCT iajs-1918 89 5 27 27 NUM iajs-1918 89 6 , , PUNCT iajs-1918 89 7 161–167 161–167 NUM iajs-1918 89 8 . . PUNCT iajs-1918 89 9 9 9 NUM iajs-1918 89 10 . . X iajs-1918 89 11 saleh saleh NOUN iajs-1918 89 12 , , PUNCT iajs-1918 89 13 m.h m.h PROPN iajs-1918 89 14 . . PROPN iajs-1918 89 15 ; ; PUNCT iajs-1918 89 16 amer amer PROPN iajs-1918 89 17 , , PUNCT iajs-1918 89 18 s.m s.m PROPN iajs-1918 89 19 . . PROPN iajs-1918 89 20 ; ; PUNCT iajs-1918 89 21 mohamed mohamed PROPN iajs-1918 89 22 , , PUNCT iajs-1918 89 23 d.s d.s PROPN iajs-1918 89 24 . . PROPN iajs-1918 89 25 ; ; PUNCT iajs-1918 89 26 mahdy mahdy NOUN iajs-1918 89 27 , , PUNCT iajs-1918 89 28 a.e a.e PROPN iajs-1918 89 29 . . PROPN iajs-1918 89 30 fractional fractional ADJ iajs-1918 89 31 calculus calculus NOUN iajs-1918 89 32 for for ADP iajs-1918 89 33 solving solve VERB iajs-1918 89 34 generalized generalized ADJ iajs-1918 89 35 abel abel PROPN iajs-1918 89 36 ’s ’s PART iajs-1918 89 37 integral integral ADJ iajs-1918 89 38 equations equation NOUN iajs-1918 89 39 suing sue VERB iajs-1918 89 40 chebyshev chebyshev NOUN iajs-1918 89 41 polynomials polynomial NOUN iajs-1918 89 42 . . PUNCT iajs-1918 90 1 int int NOUN iajs-1918 90 2 . . PUNCT iajs-1918 91 1 j. j. PROPN iajs-1918 91 2 comput comput PROPN iajs-1918 91 3 . . PUNCT iajs-1918 92 1 appl appl PROPN iajs-1918 92 2 . . PROPN iajs-1918 93 1 2014 2014 NUM iajs-1918 93 2 , , PUNCT iajs-1918 93 3 100 100 NUM iajs-1918 93 4 , , PUNCT iajs-1918 93 5 19–23 19–23 NUM iajs-1918 93 6 . . PUNCT iajs-1918 94 1 10 10 NUM iajs-1918 94 2 . . X iajs-1918 95 1 kumar kumar PROPN iajs-1918 95 2 , , PUNCT iajs-1918 95 3 s. s. PROPN iajs-1918 95 4 ; ; PUNCT iajs-1918 95 5 kumar kumar PROPN iajs-1918 95 6 , , PUNCT iajs-1918 95 7 a. a. PROPN iajs-1918 95 8 ; ; PUNCT iajs-1918 95 9 kumar kumar PROPN iajs-1918 95 10 , , PUNCT iajs-1918 95 11 d. d. PROPN iajs-1918 95 12 ; ; PUNCT iajs-1918 95 13 singh singh PROPN iajs-1918 95 14 , , PUNCT iajs-1918 95 15 j. j. PROPN iajs-1918 95 16 ; ; PUNCT iajs-1918 95 17 singh singh PROPN iajs-1918 95 18 , , PUNCT iajs-1918 95 19 a. a. NOUN iajs-1918 95 20 analytical analytical ADJ iajs-1918 95 21 solution solution NOUN iajs-1918 95 22 of of ADP iajs-1918 95 23 abel abel PROPN iajs-1918 95 24 integral integral ADJ iajs-1918 95 25 equation equation NOUN iajs-1918 95 26 arising arise VERB iajs-1918 95 27 in in ADP iajs-1918 95 28 astrophysics astrophysic NOUN iajs-1918 95 29 via via ADP iajs-1918 95 30 laplace laplace NOUN iajs-1918 95 31 transform transform NOUN iajs-1918 95 32 . . PUNCT iajs-1918 96 1 j. j. PROPN iajs-1918 96 2 egypt egypt PROPN iajs-1918 96 3 . . PUNCT iajs-1918 97 1 math math PROPN iajs-1918 97 2 . . PUNCT iajs-1918 98 1 soc soc PROPN iajs-1918 98 2 . . PUNCT iajs-1918 99 1 2015 2015 NUM iajs-1918 99 2 , , PUNCT iajs-1918 99 3 23 23 NUM iajs-1918 99 4 , , PUNCT iajs-1918 99 5 102–107 102–107 NUM iajs-1918 99 6 . . PUNCT iajs-1918 100 1 11 11 NUM iajs-1918 100 2 . . X iajs-1918 101 1 wazwaz wazwaz NOUN iajs-1918 101 2 , , PUNCT iajs-1918 101 3 a. a. NOUN iajs-1918 101 4 m. m. NOUN iajs-1918 101 5 partial partial ADJ iajs-1918 101 6 differential differential NOUN iajs-1918 101 7 equations equation NOUN iajs-1918 101 8 and and CCONJ iajs-1918 101 9 solitary solitary ADJ iajs-1918 101 10 waves wave NOUN iajs-1918 101 11 theory theory NOUN iajs-1918 101 12 . . PUNCT iajs-1918 102 1 springer springer NOUN iajs-1918 102 2 . . PUNCT iajs-1918 103 1 beijing beijing PROPN iajs-1918 103 2 and and CCONJ iajs-1918 103 3 berlin berlin PROPN iajs-1918 103 4 . . PUNCT iajs-1918 104 1 2009 2009 NUM iajs-1918 104 2 . . PUNCT iajs-1918 105 1 isbn isbn ADJ iajs-1918 105 2 978 978 NUM iajs-1918 105 3 - - SYM iajs-1918 105 4 3 3 NUM iajs-1918 105 5 - - PUNCT iajs-1918 105 6 642 642 NUM iajs-1918 105 7 - - PUNCT iajs-1918 105 8 00250 00250 NUM iajs-1918 105 9 - - PUNCT iajs-1918 105 10 2 2 NUM iajs-1918 105 11 , , PUNCT iajs-1918 105 12 e e NOUN iajs-1918 105 13 - - NOUN iajs-1918 105 14 isbn isbn ADJ iajs-1918 105 15 978 978 NUM iajs-1918 105 16 - - SYM iajs-1918 105 17 3 3 NUM iajs-1918 105 18 - - PUNCT iajs-1918 105 19 642 642 NUM iajs-1918 105 20 - - PUNCT iajs-1918 105 21 00251 00251 NUM iajs-1918 105 22 - - PUNCT iajs-1918 105 23 9 9 NUM iajs-1918 105 24 . . NOUN iajs-1918 105 25 12 12 NUM iajs-1918 105 26 . . PUNCT iajs-1918 105 27 rashidinia rashidinia PROPN iajs-1918 105 28 , , PUNCT iajs-1918 105 29 j. j. PROPN iajs-1918 105 30 ; ; PUNCT iajs-1918 105 31 tahmasebi tahmasebi ADJ iajs-1918 105 32 , , PUNCT iajs-1918 105 33 a. a. NOUN iajs-1918 105 34 approximate approximate ADJ iajs-1918 105 35 solution solution NOUN iajs-1918 105 36 of of ADP iajs-1918 105 37 linear linear PROPN iajs-1918 105 38 integro integro PROPN iajs-1918 105 39 differential differential ADJ iajs-1918 105 40 equations equation NOUN iajs-1918 105 41 by by ADP iajs-1918 105 42 using use VERB iajs-1918 105 43 modified modify VERB iajs-1918 105 44 taylor taylor PROPN iajs-1918 105 45 expansion expansion NOUN iajs-1918 105 46 method method NOUN iajs-1918 105 47 . . PUNCT iajs-1918 106 1 world world PROPN iajs-1918 106 2 journal journal PROPN iajs-1918 106 3 of of ADP iajs-1918 106 4 modelling modelling NOUN iajs-1918 106 5 and and CCONJ iajs-1918 106 6 simulation simulation NOUN iajs-1918 106 7 . . PUNCT iajs-1918 107 1 2013 2013 NUM iajs-1918 107 2 , , PUNCT iajs-1918 107 3 9 9 NUM iajs-1918 107 4 , , PUNCT iajs-1918 107 5 4 4 NUM iajs-1918 107 6 , , PUNCT iajs-1918 107 7 289 289 NUM iajs-1918 107 8 - - SYM iajs-1918 107 9 301 301 NUM iajs-1918 107 10 . . NOUN iajs-1918 108 1 13 13 NUM iajs-1918 108 2 . . PUNCT iajs-1918 109 1 abdulkawi abdulkawi PROPN iajs-1918 109 2 , , PUNCT iajs-1918 109 3 m. m. NOUN iajs-1918 109 4 ; ; PUNCT iajs-1918 109 5 eshkuvatov eshkuvatov PROPN iajs-1918 109 6 , , PUNCT iajs-1918 109 7 z. z. PROPN iajs-1918 109 8 k. k. PROPN iajs-1918 109 9 ; ; PUNCT iajs-1918 109 10 nik nik PROPN iajs-1918 109 11 long long ADV iajs-1918 109 12 , , PUNCT iajs-1918 109 13 n. n. PROPN iajs-1918 109 14 m. m. NOUN iajs-1918 109 15 a. a. PROPN iajs-1918 109 16 a a DET iajs-1918 109 17 note note NOUN iajs-1918 109 18 on on ADP iajs-1918 109 19 the the DET iajs-1918 109 20 numerical numerical ADJ iajs-1918 109 21 solution solution NOUN iajs-1918 109 22 of of ADP iajs-1918 109 23 singular singular ADJ iajs-1918 109 24 integral integral ADJ iajs-1918 109 25 equations equation NOUN iajs-1918 109 26 of of ADP iajs-1918 109 27 cauchy cauchy PROPN iajs-1918 109 28 type type NOUN iajs-1918 109 29 . . PUNCT iajs-1918 110 1 world world PROPN iajs-1918 110 2 academy academy PROPN iajs-1918 110 3 of of ADP iajs-1918 110 4 science science PROPN iajs-1918 110 5 , , PUNCT iajs-1918 110 6 engineering engineering NOUN iajs-1918 110 7 and and CCONJ iajs-1918 110 8 technology technology NOUN iajs-1918 110 9 . . PUNCT iajs-1918 111 1 2009 2009 NUM iajs-1918 111 2 , , PUNCT iajs-1918 111 3 3 3 NUM iajs-1918 111 4 , , PUNCT iajs-1918 111 5 9 9 NUM iajs-1918 111 6 , , PUNCT iajs-1918 111 7 893 893 NUM iajs-1918 111 8 - - SYM iajs-1918 111 9 896 896 NUM iajs-1918 111 10 . . PUNCT iajs-1918 112 1 14 14 NUM iajs-1918 112 2 . . PUNCT iajs-1918 113 1 huang huang PROPN iajs-1918 113 2 , , PUNCT iajs-1918 113 3 l. l. PROPN iajs-1918 113 4 ; ; PUNCT iajs-1918 113 5 huang huang PROPN iajs-1918 113 6 , , PUNCT iajs-1918 113 7 y. y. PROPN iajs-1918 113 8 ; ; PUNCT iajs-1918 113 9 li li PROPN iajs-1918 113 10 , , PUNCT iajs-1918 113 11 x x PROPN iajs-1918 113 12 - - ADJ iajs-1918 113 13 f. f. ADJ iajs-1918 113 14 approximate approximate ADJ iajs-1918 113 15 solution solution NOUN iajs-1918 113 16 of of ADP iajs-1918 113 17 abel abel PROPN iajs-1918 113 18 integral integral ADJ iajs-1918 113 19 equation equation NOUN iajs-1918 113 20 . . PUNCT iajs-1918 114 1 comput comput NOUN iajs-1918 114 2 . . PUNCT iajs-1918 115 1 math math NOUN iajs-1918 115 2 . . PUNCT iajs-1918 116 1 appl appl PROPN iajs-1918 116 2 . . PROPN iajs-1918 116 3 2008 2008 NUM iajs-1918 116 4 , , PUNCT iajs-1918 116 5 56 56 NUM iajs-1918 116 6 , , PUNCT iajs-1918 116 7 1748 1748 NUM iajs-1918 116 8 - - SYM iajs-1918 116 9 1757 1757 NUM iajs-1918 116 10 . . PUNCT iajs-1918 117 1 15 15 NUM iajs-1918 117 2 . . PUNCT iajs-1918 118 1 liu liu PROPN iajs-1918 118 2 , , PUNCT iajs-1918 118 3 y y PROPN iajs-1918 118 4 - - PROPN iajs-1918 118 5 p. p. PROPN iajs-1918 118 6 ; ; PUNCT iajs-1918 118 7 tao tao PROPN iajs-1918 118 8 , , PUNCT iajs-1918 118 9 l. l. PROPN iajs-1918 118 10 high high ADJ iajs-1918 118 11 accuracy accuracy NOUN iajs-1918 118 12 combination combination NOUN iajs-1918 118 13 algorithm algorithm NOUN iajs-1918 118 14 and and CCONJ iajs-1918 118 15 a a DET iajs-1918 118 16 posteriori posteriori NOUN iajs-1918 118 17 error error NOUN iajs-1918 118 18 estimation estimation NOUN iajs-1918 118 19 for for ADP iajs-1918 118 20 solving solve VERB iajs-1918 118 21 the the DET iajs-1918 118 22 first first ADJ iajs-1918 118 23 kind kind NOUN iajs-1918 118 24 abel abel PROPN iajs-1918 118 25 integral integral ADJ iajs-1918 118 26 equations equation NOUN iajs-1918 118 27 . . PUNCT iajs-1918 119 1 appl appl PROPN iajs-1918 119 2 . . PROPN iajs-1918 119 3 math math PROPN iajs-1918 119 4 . . PUNCT iajs-1918 120 1 comput comput NOUN iajs-1918 120 2 . . PUNCT iajs-1918 121 1 2006 2006 NUM iajs-1918 121 2 , , PUNCT iajs-1918 121 3 178 178 NUM iajs-1918 121 4 , , PUNCT iajs-1918 121 5 441451 441451 NUM iajs-1918 121 6 . . PUNCT iajs-1918 122 1 16 16 NUM iajs-1918 122 2 . . PUNCT iajs-1918 123 1 kumar kumar PROPN iajs-1918 123 2 , , PUNCT iajs-1918 123 3 s. s. PROPN iajs-1918 123 4 ; ; PUNCT iajs-1918 123 5 kumar kumar PROPN iajs-1918 123 6 , , PUNCT iajs-1918 123 7 a. a. PROPN iajs-1918 123 8 ; ; PUNCT iajs-1918 123 9 kumar kumar PROPN iajs-1918 123 10 , , PUNCT iajs-1918 123 11 d. d. PROPN iajs-1918 123 12 ; ; PUNCT iajs-1918 123 13 singh singh PROPN iajs-1918 123 14 j. j. PROPN iajs-1918 123 15 ; ; PUNCT iajs-1918 123 16 singh singh PROPN iajs-1918 123 17 , , PUNCT iajs-1918 123 18 a. a. NOUN iajs-1918 123 19 analytical analytical ADJ iajs-1918 123 20 solution solution NOUN iajs-1918 123 21 of of ADP iajs-1918 123 22 abel abel PROPN iajs-1918 123 23 integral integral ADJ iajs-1918 123 24 equation equation NOUN iajs-1918 123 25 arising arise VERB iajs-1918 123 26 in in ADP iajs-1918 123 27 astrophysics astrophysic NOUN iajs-1918 123 28 via via ADP iajs-1918 123 29 laplace laplace NOUN iajs-1918 123 30 transform transform NOUN iajs-1918 123 31 . . PUNCT iajs-1918 124 1 journal journal PROPN iajs-1918 124 2 egypt egypt PROPN iajs-1918 124 3 . . PUNCT iajs-1918 125 1 math math PROPN iajs-1918 125 2 . . PUNCT iajs-1918 126 1 soc soc PROPN iajs-1918 126 2 . . PUNCT iajs-1918 127 1 2014 2014 NUM iajs-1918 127 2 , , PUNCT iajs-1918 127 3 2 2 NUM iajs-1918 127 4 , , PUNCT iajs-1918 127 5 doi doi NOUN iajs-1918 127 6 : : PUNCT iajs-1918 127 7 org/10.1016 org/10.1016 PROPN iajs-1918 127 8 / / SYM iajs-1918 127 9 j.joems.2014.02.004 j.joems.2014.02.004 PROPN iajs-1918 127 10 . . PUNCT iajs-1918 128 1 17 17 NUM iajs-1918 128 2 . . PUNCT iajs-1918 129 1 al al PROPN iajs-1918 129 2 - - PUNCT iajs-1918 129 3 abrahemee abrahemee PROPN iajs-1918 129 4 , , PUNCT iajs-1918 129 5 k.m.m k.m.m PROPN iajs-1918 129 6 . . PROPN iajs-1918 129 7 solution solution NOUN iajs-1918 129 8 of of ADP iajs-1918 129 9 some some DET iajs-1918 129 10 application application NOUN iajs-1918 129 11 of of ADP iajs-1918 129 12 system system NOUN iajs-1918 129 13 of of ADP iajs-1918 129 14 ordinary ordinary ADJ iajs-1918 129 15 initial initial ADJ iajs-1918 129 16 value value NOUN iajs-1918 129 17 problems problem NOUN iajs-1918 129 18 using use VERB iajs-1918 129 19 osculatory osculatory NOUN iajs-1918 129 20 interpolation interpolation NOUN iajs-1918 129 21 technique technique NOUN iajs-1918 129 22 . . PUNCT iajs-1918 130 1 ibn ibn PROPN iajs-1918 130 2 al al PROPN iajs-1918 130 3 - - PUNCT iajs-1918 130 4 haitham haitham PROPN iajs-1918 130 5 journal journal PROPN iajs-1918 130 6 for for ADP iajs-1918 130 7 pure pure ADJ iajs-1918 130 8 and and CCONJ iajs-1918 130 9 applied applied ADJ iajs-1918 130 10 science science NOUN iajs-1918 130 11 . . PUNCT iajs-1918 131 1 2012 2012 NUM iajs-1918 131 2 , , PUNCT iajs-1918 131 3 25 25 NUM iajs-1918 131 4 , , PUNCT iajs-1918 131 5 1 1 NUM iajs-1918 131 6 . . PUNCT iajs-1918 131 7     SPACE iajs-1918 132 1 186 186 NUM iajs-1918 132 2 ibn ibn PROPN iajs-1918 132 3 al al PROPN iajs-1918 132 4 - - PUNCT iajs-1918 132 5 haitham haitham PROPN iajs-1918 132 6 jour.for jour.for PROPN iajs-1918 132 7 pure&appl.sci pure&appl.sci PROPN iajs-1918 132 8 . . PUNCT iajs-1918 132 9 ihjpas ihjpas PROPN iajs-1918 132 10 https://doi.org/10.30526/32.1.1918 https://doi.org/10.30526/32.1.1918 ADJ iajs-1918 132 11 vol vol NOUN iajs-1918 132 12 . . PUNCT iajs-1918 132 13 32 32 NUM iajs-1918 132 14 ( ( PUNCT iajs-1918 132 15 1 1 NUM iajs-1918 132 16 ) ) PUNCT iajs-1918 132 17 2019 2019 NUM iajs-1918 132 18 18 18 NUM iajs-1918 132 19 . . PUNCT iajs-1918 133 1 tawfiq tawfiq PROPN iajs-1918 133 2 , , PUNCT iajs-1918 133 3 l.n.m l.n.m NOUN iajs-1918 133 4 . . PUNCT iajs-1918 134 1 on on ADP iajs-1918 134 2 training training NOUN iajs-1918 134 3 of of ADP iajs-1918 134 4 artificial artificial ADJ iajs-1918 134 5 neural neural ADJ iajs-1918 134 6 networks network NOUN iajs-1918 134 7 . . PUNCT iajs-1918 135 1 al al PROPN iajs-1918 135 2 - - PUNCT iajs-1918 135 3 fatih fatih PROPN iajs-1918 135 4 journal journal NOUN iajs-1918 135 5 . . PUNCT iajs-1918 136 1 2005,1,23,130 2005,1,23,130 NUM iajs-1918 136 2 - - SYM iajs-1918 136 3 139 139 NUM iajs-1918 136 4 . . PUNCT iajs-1918 137 1 19 19 NUM iajs-1918 137 2 . . X iajs-1918 137 3 tawfiq tawfiq PROPN iajs-1918 137 4 , , PUNCT iajs-1918 137 5 l. l. PROPN iajs-1918 137 6 n. n. PROPN iajs-1918 137 7 m. m. PROPN iajs-1918 137 8 ; ; PUNCT iajs-1918 137 9 yassien yassien PROPN iajs-1918 137 10 . . PUNCT iajs-1918 138 1 s. s. PROPN iajs-1918 138 2 m. m. PROPN iajs-1918 138 3 solution solution NOUN iajs-1918 138 4 of of ADP iajs-1918 138 5 high high ADJ iajs-1918 138 6 order order NOUN iajs-1918 138 7 ordinary ordinary ADJ iajs-1918 138 8 boundary boundary ADJ iajs-1918 138 9 value value NOUN iajs-1918 138 10 problems problem NOUN iajs-1918 138 11 using use VERB iajs-1918 138 12 semi semi ADJ iajs-1918 138 13 - - ADJ iajs-1918 138 14 analytic analytic ADJ iajs-1918 138 15 technique technique NOUN iajs-1918 138 16 . . PUNCT iajs-1918 139 1 ibn ibn PROPN iajs-1918 139 2 al al PROPN iajs-1918 139 3 - - PUNCT iajs-1918 139 4 haitham haitham PROPN iajs-1918 139 5 jour jour X iajs-1918 139 6 . . PROPN iajs-1918 139 7 for for ADP iajs-1918 139 8 pure pure ADJ iajs-1918 139 9 & & CCONJ iajs-1918 139 10 appl appl PROPN iajs-1918 139 11 . . PUNCT iajs-1918 140 1 sci sci PROPN iajs-1918 140 2 . . PROPN iajs-1918 140 3 2013 2013 NUM iajs-1918 140 4 , , PUNCT iajs-1918 140 5 26 26 NUM iajs-1918 140 6 , , PUNCT iajs-1918 140 7 1 1 NUM iajs-1918 140 8 . . NOUN iajs-1918 140 9 20 20 NUM iajs-1918 140 10 . . PUNCT iajs-1918 141 1 tawfiq tawfiq PROPN iajs-1918 141 2 , , PUNCT iajs-1918 141 3 l. l. PROPN iajs-1918 141 4 n. n. PROPN iajs-1918 141 5 m. m. PROPN iajs-1918 141 6 ; ; PUNCT iajs-1918 141 7 hussein hussein PROPN iajs-1918 141 8 , , PUNCT iajs-1918 141 9 r. r. PROPN iajs-1918 141 10 w. w. PROPN iajs-1918 141 11 on on ADP iajs-1918 141 12 solution solution NOUN iajs-1918 141 13 of of ADP iajs-1918 141 14 regular regular ADJ iajs-1918 141 15 singular singular ADJ iajs-1918 141 16 initial initial ADJ iajs-1918 141 17 value value NOUN iajs-1918 141 18 problems problem NOUN iajs-1918 141 19 . . PUNCT iajs-1918 142 1 ibn ibn PROPN iajs-1918 142 2 al al PROPN iajs-1918 142 3 - - PUNCT iajs-1918 142 4 haitham haitham PROPN iajs-1918 142 5 jour jour X iajs-1918 142 6 . . PROPN iajs-1918 142 7 for for ADP iajs-1918 142 8 pure pure ADJ iajs-1918 142 9 & & CCONJ iajs-1918 142 10 appl appl PROPN iajs-1918 142 11 . . PUNCT iajs-1918 143 1 sci sci PROPN iajs-1918 143 2 . . PROPN iajs-1918 143 3 2013 2013 NUM iajs-1918 143 4 , , PUNCT iajs-1918 143 5 26 26 NUM iajs-1918 143 6 , , PUNCT iajs-1918 143 7 1 1 NUM iajs-1918 143 8 . . NUM iajs-1918 143 9 21 21 NUM iajs-1918 143 10 . . PUNCT iajs-1918 144 1 tawfiq tawfiq PROPN iajs-1918 144 2 , , PUNCT iajs-1918 144 3 l. l. PROPN iajs-1918 144 4 n. n. PROPN iajs-1918 144 5 m. m. PROPN iajs-1918 144 6 ; ; PUNCT iajs-1918 144 7 rasheed rasheed NOUN iajs-1918 144 8 , , PUNCT iajs-1918 144 9 h. h. PROPN iajs-1918 144 10 w. w. PROPN iajs-1918 144 11 on on ADP iajs-1918 144 12 solution solution NOUN iajs-1918 144 13 of of ADP iajs-1918 144 14 regular regular ADJ iajs-1918 144 15 singular singular ADJ iajs-1918 144 16 ordinary ordinary ADJ iajs-1918 144 17 boundary boundary ADJ iajs-1918 144 18 value value NOUN iajs-1918 144 19 problem problem NOUN iajs-1918 144 20 . . PUNCT iajs-1918 145 1 ibn ibn PROPN iajs-1918 145 2 al al PROPN iajs-1918 145 3 - - PUNCT iajs-1918 145 4 haitham haitham PROPN iajs-1918 145 5 jour jour X iajs-1918 145 6 . . PROPN iajs-1918 145 7 for for ADP iajs-1918 145 8 pure pure ADJ iajs-1918 145 9 & & CCONJ iajs-1918 145 10 appl appl PROPN iajs-1918 145 11 . . PUNCT iajs-1918 146 1 sci sci PROPN iajs-1918 146 2 . . PROPN iajs-1918 146 3 2013 2013 NUM iajs-1918 146 4 , , PUNCT iajs-1918 146 5 26 26 NUM iajs-1918 146 6 , , PUNCT iajs-1918 146 7 2 2 NUM iajs-1918 146 8 . . PUNCT