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PUNCT iajs-2610 6 1 so so ADV iajs-2610 6 2 the the DET iajs-2610 6 3 results result NOUN iajs-2610 6 4 of of ADP iajs-2610 6 5 this this DET iajs-2610 6 6 study study NOUN iajs-2610 6 7 are be AUX iajs-2610 6 8 debated debate VERB iajs-2610 6 9 to to PART iajs-2610 6 10 show show VERB iajs-2610 6 11 how how SCONJ iajs-2610 6 12 useful useful ADJ iajs-2610 6 13 this this DET iajs-2610 6 14 method method NOUN iajs-2610 6 15 is be AUX iajs-2610 6 16 in in ADP iajs-2610 6 17 terms term NOUN iajs-2610 6 18 of of ADP iajs-2610 6 19 being be AUX iajs-2610 6 20 a a DET iajs-2610 6 21 powerful powerful ADJ iajs-2610 6 22 , , PUNCT iajs-2610 6 23 accurate accurate ADJ iajs-2610 6 24 and and CCONJ iajs-2610 6 25 fast fast ADJ iajs-2610 6 26 tool tool NOUN iajs-2610 6 27 with with ADP iajs-2610 6 28 a a DET iajs-2610 6 29 little little ADJ iajs-2610 6 30 effort effort NOUN iajs-2610 6 31 compared compare VERB iajs-2610 6 32 to to ADP iajs-2610 6 33 other other ADJ iajs-2610 6 34 iterative iterative NOUN iajs-2610 6 35 methods method NOUN iajs-2610 6 36 . . 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PUNCT iajs-2610 11 1 with with ADP iajs-2610 11 2 this this DET iajs-2610 11 3 great great ADJ iajs-2610 11 4 expansion expansion NOUN iajs-2610 11 5 of of ADP iajs-2610 11 6 differential differential ADJ iajs-2610 11 7 equations equation NOUN iajs-2610 11 8 in in ADP iajs-2610 11 9 applied applied ADJ iajs-2610 11 10 mathematics mathematic NOUN iajs-2610 11 11 and and CCONJ iajs-2610 11 12 physics physics NOUN iajs-2610 11 13 , , PUNCT iajs-2610 11 14 especially especially ADV iajs-2610 11 15 the the DET iajs-2610 11 16 non non ADJ iajs-2610 11 17 - - ADJ iajs-2610 11 18 linear linear ADJ iajs-2610 11 19 pdes pde NOUN iajs-2610 11 20 which which PRON iajs-2610 11 21 are be AUX iajs-2610 11 22 the the DET iajs-2610 11 23 subject subject NOUN iajs-2610 11 24 of of ADP iajs-2610 11 25 our our PRON iajs-2610 11 26 study study NOUN iajs-2610 11 27 . . PUNCT iajs-2610 12 1 mathematicians mathematician NOUN iajs-2610 12 2 faced face VERB iajs-2610 12 3 some some DET iajs-2610 12 4 problems problem NOUN iajs-2610 12 5 in in ADP iajs-2610 12 6 solving solve VERB iajs-2610 12 7 some some PRON iajs-2610 12 8 of of ADP iajs-2610 12 9 these these DET iajs-2610 12 10 equations equation NOUN iajs-2610 12 11 which which PRON iajs-2610 12 12 required require VERB iajs-2610 12 13 the the DET iajs-2610 12 14 development development NOUN iajs-2610 12 15 of of ADP iajs-2610 12 16 new new ADJ iajs-2610 12 17 methods method NOUN iajs-2610 12 18 to to PART iajs-2610 12 19 find find VERB iajs-2610 12 20 an an DET iajs-2610 12 21 exact exact ADJ iajs-2610 12 22 or or CCONJ iajs-2610 12 23 approximate approximate ADJ iajs-2610 12 24 solution solution NOUN iajs-2610 12 25 to to ADP iajs-2610 12 26 them they PRON iajs-2610 12 27 . . PUNCT iajs-2610 13 1 in in ADP iajs-2610 13 2 recent recent ADJ iajs-2610 13 3 years year NOUN iajs-2610 13 4 , , PUNCT iajs-2610 13 5 most most ADJ iajs-2610 13 6 researchers researcher NOUN iajs-2610 13 7 fundamentally fundamentally ADV iajs-2610 13 8 have have AUX iajs-2610 13 9 studied study VERB iajs-2610 13 10 solutions solution NOUN iajs-2610 13 11 of of ADP iajs-2610 13 12 nonlinear nonlinear ADJ iajs-2610 13 13 pdes pde NOUN iajs-2610 13 14 and and CCONJ iajs-2610 13 15 odes ode NOUN iajs-2610 13 16 via via ADP iajs-2610 13 17 utilizing utilize VERB iajs-2610 13 18 different different ADJ iajs-2610 13 19 methods method NOUN iajs-2610 13 20 , , PUNCT iajs-2610 13 21 like like ADP iajs-2610 13 22 the the DET iajs-2610 13 23 variational variational ADJ iajs-2610 13 24 ibn ibn PROPN iajs-2610 13 25 al al PROPN iajs-2610 13 26 haitham haitham PROPN iajs-2610 13 27 journal journal PROPN iajs-2610 13 28 for for ADP iajs-2610 13 29 pure pure ADJ iajs-2610 13 30 and and CCONJ iajs-2610 13 31 applied apply VERB iajs-2610 13 32 science science NOUN iajs-2610 13 33 journal journal PROPN iajs-2610 13 34 homepage homepage NOUN iajs-2610 13 35 : : PUNCT iajs-2610 13 36 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2610 13 37 doi doi NOUN iajs-2610 13 38 : : PUNCT iajs-2610 13 39 10.30526/34.2.2610 10.30526/34.2.2610 PROPN iajs-2610 13 40 article article NOUN iajs-2610 13 41 history history NOUN iajs-2610 13 42 : : PUNCT iajs-2610 13 43 received,12,february,2020 received,12,february,2020 NOUN iajs-2610 13 44 , , PUNCT iajs-2610 13 45 accepted23 accepted23 NOUN iajs-2610 13 46 june,2020 june,2020 NOUN iajs-2610 13 47 , , PUNCT iajs-2610 13 48 published publish VERB iajs-2610 13 49 in in ADP iajs-2610 13 50 april april PROPN iajs-2610 13 51 2021 2021 NUM iajs-2610 13 52 asmaa asmaa PROPN iajs-2610 13 53 a. a. 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PROPN iajs-2610 19 1 34 34 NUM iajs-2610 19 2 ( ( PUNCT iajs-2610 19 3 2 2 NUM iajs-2610 19 4 ) ) PUNCT iajs-2610 19 5 2021 2021 NUM iajs-2610 19 6 iteration iteration NOUN iajs-2610 19 7 method[1 method[1 NUM iajs-2610 19 8 ] ] PUNCT iajs-2610 19 9 , , PUNCT iajs-2610 19 10 adomian adomian NOUN iajs-2610 19 11 decomposition decomposition NOUN iajs-2610 19 12 method method NOUN iajs-2610 19 13 [ [ X iajs-2610 19 14 2 2 NUM iajs-2610 19 15 - - SYM iajs-2610 19 16 3 3 NUM iajs-2610 19 17 ] ] PUNCT iajs-2610 19 18 , , PUNCT iajs-2610 19 19 homotopy homotopy VERB iajs-2610 19 20 perturbation perturbation NOUN iajs-2610 19 21 method method NOUN iajs-2610 19 22 [ [ X iajs-2610 19 23 4 4 NUM iajs-2610 19 24 ] ] PUNCT iajs-2610 19 25 , , PUNCT iajs-2610 19 26 laplace laplace NOUN iajs-2610 19 27 transform transform NOUN iajs-2610 19 28 and and CCONJ iajs-2610 19 29 modified modify VERB iajs-2610 19 30 variational variational ADJ iajs-2610 19 31 iteration iteration NOUN iajs-2610 20 1 [ [ X iajs-2610 20 2 5 5 NUM iajs-2610 20 3 ] ] PUNCT iajs-2610 20 4 , , PUNCT iajs-2610 20 5 natural natural ADJ iajs-2610 20 6 decomposition decomposition NOUN iajs-2610 20 7 method method NOUN iajs-2610 20 8 [ [ X iajs-2610 20 9 6 6 NUM iajs-2610 20 10 ] ] PUNCT iajs-2610 20 11 , , PUNCT iajs-2610 20 12 the the DET iajs-2610 20 13 reduced reduce VERB iajs-2610 20 14 differential differential NOUN iajs-2610 20 15 transform transform NOUN iajs-2610 20 16 method method NOUN iajs-2610 20 17 as as ADP iajs-2610 20 18 in in ADP iajs-2610 20 19 [ [ PUNCT iajs-2610 20 20 7 7 NUM iajs-2610 20 21 - - PUNCT iajs-2610 20 22 8],the 8],the PRON iajs-2610 20 23 sumudu sumudu NOUN iajs-2610 20 24 transform transform VERB iajs-2610 20 25 [ [ PUNCT iajs-2610 20 26 9 9 NUM iajs-2610 20 27 - - SYM iajs-2610 20 28 10 10 NUM iajs-2610 20 29 ] ] PUNCT iajs-2610 20 30 and and CCONJ iajs-2610 20 31 sumudu sumudu NOUN iajs-2610 20 32 decomposition decomposition NOUN iajs-2610 20 33 method method NOUN iajs-2610 20 34 [ [ X iajs-2610 20 35 11 11 NUM iajs-2610 20 36 ] ] PUNCT iajs-2610 20 37 . . 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PUNCT iajs-2610 27 1 it it PRON iajs-2610 27 2 is be AUX iajs-2610 27 3 a a DET iajs-2610 27 4 qualified qualified ADJ iajs-2610 27 5 method method NOUN iajs-2610 27 6 for for ADP iajs-2610 27 7 reducing reduce VERB iajs-2610 27 8 the the DET iajs-2610 27 9 number number NOUN iajs-2610 27 10 of of ADP iajs-2610 27 11 calculations calculation NOUN iajs-2610 27 12 while while SCONJ iajs-2610 27 13 keeping keep VERB iajs-2610 27 14 the the DET iajs-2610 27 15 solution solution NOUN iajs-2610 27 16 is be AUX iajs-2610 27 17 more more ADV iajs-2610 27 18 accurate accurate ADJ iajs-2610 27 19 and and CCONJ iajs-2610 27 20 efficient efficient ADJ iajs-2610 27 21 . . PUNCT iajs-2610 28 1 in in ADP iajs-2610 28 2 this this DET iajs-2610 28 3 work work NOUN iajs-2610 28 4 , , PUNCT iajs-2610 28 5 the the DET iajs-2610 28 6 examples example NOUN iajs-2610 28 7 of of ADP iajs-2610 28 8 non non ADJ iajs-2610 28 9 - - ADJ iajs-2610 28 10 linear linear ADJ iajs-2610 28 11 pdes pde NOUN iajs-2610 28 12 which which PRON iajs-2610 28 13 are be AUX iajs-2610 28 14 used use VERB iajs-2610 28 15 in in ADP iajs-2610 28 16 [ [ X iajs-2610 28 17 1 1 NUM iajs-2610 28 18 ] ] PUNCT iajs-2610 28 19 will will AUX iajs-2610 28 20 be be AUX iajs-2610 28 21 solved solve VERB iajs-2610 28 22 by by ADP iajs-2610 28 23 using use VERB iajs-2610 28 24 the the DET iajs-2610 28 25 sim sim NOUN iajs-2610 28 26 . . PUNCT iajs-2610 29 1 the the DET iajs-2610 29 2 basic basic ADJ iajs-2610 29 3 concept concept NOUN iajs-2610 29 4 of of ADP iajs-2610 29 5 this this DET iajs-2610 29 6 work work NOUN iajs-2610 29 7 is be AUX iajs-2610 29 8 included include VERB iajs-2610 29 9 in in ADP iajs-2610 29 10 section section NOUN iajs-2610 29 11 2 2 NUM iajs-2610 29 12 and and CCONJ iajs-2610 29 13 3 3 NUM iajs-2610 29 14 as as SCONJ iajs-2610 29 15 we we PRON iajs-2610 29 16 introduce introduce VERB iajs-2610 29 17 the the DET iajs-2610 29 18 definition definition NOUN iajs-2610 29 19 and and CCONJ iajs-2610 29 20 properties property NOUN iajs-2610 29 21 of of ADP iajs-2610 29 22 the the DET iajs-2610 29 23 sumudu sumudu NOUN iajs-2610 29 24 transform transform NOUN iajs-2610 29 25 . . 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PUNCT iajs-2610 32 1 section section NOUN iajs-2610 32 2 6 6 NUM iajs-2610 32 3 is be AUX iajs-2610 32 4 devoted devote VERB iajs-2610 32 5 to to PART iajs-2610 32 6 debate debate VERB iajs-2610 32 7 and and CCONJ iajs-2610 32 8 conclusion conclusion NOUN iajs-2610 32 9 of of ADP iajs-2610 32 10 this this DET iajs-2610 32 11 paper paper NOUN iajs-2610 32 12 . . PUNCT iajs-2610 33 1 2 2 X iajs-2610 33 2 . . X iajs-2610 33 3 sumudu sumudu NOUN iajs-2610 33 4 transform transform VERB iajs-2610 33 5 sumudu sumudu NOUN iajs-2610 33 6 transform transform NOUN iajs-2610 33 7 is be AUX iajs-2610 33 8 one one NUM iajs-2610 33 9 of of ADP iajs-2610 33 10 the the DET iajs-2610 33 11 integral integral ADJ iajs-2610 33 12 transformations transformation NOUN iajs-2610 33 13 that that PRON iajs-2610 33 14 emerges emerge VERB iajs-2610 33 15 after after ADP iajs-2610 33 16 laplace laplace NOUN iajs-2610 33 17 transform transform VERB iajs-2610 33 18 where where SCONJ iajs-2610 33 19 it it PRON iajs-2610 33 20 is be AUX iajs-2610 33 21 derived derive VERB iajs-2610 33 22 from from ADP iajs-2610 33 23 the the DET iajs-2610 33 24 laplace laplace NOUN iajs-2610 33 25 transform transform NOUN iajs-2610 33 26 by by ADP iajs-2610 33 27 setting set VERB iajs-2610 33 28 p p X iajs-2610 33 29 = = PUNCT iajs-2610 33 30 1 1 NUM iajs-2610 33 31 s s NOUN iajs-2610 33 32 to to PART iajs-2610 33 33 resolve resolve VERB iajs-2610 33 34 differential differential ADJ iajs-2610 33 35 equations equation NOUN iajs-2610 33 36 subject subject ADJ iajs-2610 33 37 to to ADP iajs-2610 33 38 the the DET iajs-2610 33 39 initial initial ADJ iajs-2610 33 40 conditions condition NOUN iajs-2610 33 41 . . PUNCT iajs-2610 34 1 this this DET iajs-2610 34 2 transform transform NOUN iajs-2610 34 3 is be AUX iajs-2610 34 4 relatively relatively ADV iajs-2610 34 5 new new ADJ iajs-2610 34 6 but but CCONJ iajs-2610 34 7 has have VERB iajs-2610 34 8 many many ADJ iajs-2610 34 9 good good ADJ iajs-2610 34 10 properties property NOUN iajs-2610 34 11 for for ADP iajs-2610 34 12 resolving resolve VERB iajs-2610 34 13 problems problem NOUN iajs-2610 34 14 in in ADP iajs-2610 34 15 computational computational ADJ iajs-2610 34 16 science science NOUN iajs-2610 34 17 and and CCONJ iajs-2610 34 18 control control PROPN iajs-2610 34 19 engineering engineering NOUN iajs-2610 34 20 . . PUNCT iajs-2610 35 1 it it PRON iajs-2610 35 2 is be AUX iajs-2610 35 3 interesting interesting ADJ iajs-2610 35 4 that that SCONJ iajs-2610 35 5 in in ADP iajs-2610 35 6 sumudu sumudu NOUN iajs-2610 35 7 transform transform VERB iajs-2610 35 8 variable variable ADJ iajs-2610 35 9 u u NOUN iajs-2610 35 10 and and CCONJ iajs-2610 35 11 transformed transform VERB iajs-2610 35 12 function function NOUN iajs-2610 35 13 f(u f(u PROPN iajs-2610 35 14 ) ) PUNCT iajs-2610 35 15 in in ADP iajs-2610 35 16 u u NOUN iajs-2610 35 17 - - NOUN iajs-2610 35 18 domain domain NOUN iajs-2610 35 19 are be AUX iajs-2610 35 20 transacted transact VERB iajs-2610 35 21 as as ADP iajs-2610 35 22 replicas replica NOUN iajs-2610 35 23 of of ADP iajs-2610 35 24 t t PROPN iajs-2610 35 25 and and CCONJ iajs-2610 35 26 f(t f(t NOUN iajs-2610 35 27 ) ) PUNCT iajs-2610 35 28 in in ADP iajs-2610 35 29 the the DET iajs-2610 35 30 t t NOUN iajs-2610 35 31 - - PUNCT iajs-2610 35 32 domain domain NOUN iajs-2610 35 33 so so ADV iajs-2610 35 34 thus thus ADV iajs-2610 35 35 the the DET iajs-2610 35 36 consistency consistency NOUN iajs-2610 35 37 of of ADP iajs-2610 35 38 unit unit NOUN iajs-2610 35 39 in in ADP iajs-2610 35 40 the the DET iajs-2610 35 41 differential differential ADJ iajs-2610 35 42 equations equation NOUN iajs-2610 35 43 describing describe VERB iajs-2610 35 44 a a DET iajs-2610 35 45 physical physical ADJ iajs-2610 35 46 process process NOUN iajs-2610 35 47 can can AUX iajs-2610 35 48 be be AUX iajs-2610 35 49 preserved preserve VERB iajs-2610 35 50 even even ADV iajs-2610 35 51 after after ADP iajs-2610 35 52 transformations transformation NOUN iajs-2610 35 53 , , PUNCT iajs-2610 35 54 as as ADP iajs-2610 35 55 for for ADP iajs-2610 35 56 , , PUNCT iajs-2610 35 57 in in SCONJ iajs-2610 35 58 laplace laplace NOUN iajs-2610 35 59 transform transform VERB iajs-2610 35 60 variable variable NOUN iajs-2610 35 61 s s PART iajs-2610 35 62 and and CCONJ iajs-2610 35 63 transformed transform VERB iajs-2610 35 64 function function NOUN iajs-2610 35 65 f(s f( NOUN iajs-2610 35 66 ) ) PUNCT iajs-2610 35 67 in in ADP iajs-2610 35 68 s s NOUN iajs-2610 35 69 - - NOUN iajs-2610 35 70 domain domain NOUN iajs-2610 35 71 are be AUX iajs-2610 35 72 transacted transact VERB iajs-2610 35 73 as as ADP iajs-2610 35 74 toys toy NOUN iajs-2610 35 75 in in ADP iajs-2610 35 76 this this DET iajs-2610 35 77 operation operation NOUN iajs-2610 36 1 [ [ X iajs-2610 36 2 19 19 NUM iajs-2610 36 3 ] ] PUNCT iajs-2610 36 4 . . 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PUNCT iajs-2610 40 1 then then ADV iajs-2610 40 2 sumudu sumudu VERB iajs-2610 40 3 transform transform VERB iajs-2610 40 4 u u PRON iajs-2610 40 5 ( ( PUNCT iajs-2610 40 6 x x PROPN iajs-2610 40 7 , , PUNCT iajs-2610 40 8 t t PROPN iajs-2610 40 9 ) ) PUNCT iajs-2610 40 10 is be AUX iajs-2610 40 11 written write VERB iajs-2610 40 12 as as ADP iajs-2610 40 13 [ [ X iajs-2610 40 14 20 20 NUM iajs-2610 40 15 ] ] PUNCT iajs-2610 40 16 u(x u(x PROPN iajs-2610 40 17 , , PUNCT iajs-2610 40 18 s s X iajs-2610 40 19 ) ) PUNCT iajs-2610 40 20 = = SYM iajs-2610 40 21 π•Š[u(x π•Š[u(x NOUN iajs-2610 40 22 , , PUNCT iajs-2610 40 23 t t NOUN iajs-2610 40 24 ) ) PUNCT iajs-2610 40 25 ] ] PUNCT iajs-2610 41 1 = = PUNCT iajs-2610 41 2 1 1 NUM iajs-2610 41 3 s s NOUN iajs-2610 41 4 ∫ ∫ PROPN iajs-2610 41 5 u(x u(x PROPN iajs-2610 41 6 , , PUNCT iajs-2610 41 7 t)e t)e NOUN iajs-2610 41 8 βˆ’t βˆ’t PROPN iajs-2610 41 9 s s PART iajs-2610 41 10 dt dt X iajs-2610 41 11 , , PUNCT iajs-2610 41 12 ∞ ∞ PROPN iajs-2610 41 13 0 0 NUM iajs-2610 42 1 βˆ’πœ1 βˆ’πœ1 PROPN iajs-2610 42 2 < < X iajs-2610 42 3 𝑒 𝑒 X iajs-2610 42 4 < < X iajs-2610 42 5 𝜏2,where 𝜏2,where X iajs-2610 42 6 βˆ’πœ1 βˆ’πœ1 NOUN iajs-2610 42 7 and and CCONJ iajs-2610 42 8 𝜏2 𝜏2 VERB iajs-2610 42 9 > > X iajs-2610 42 10 0 0 NUM iajs-2610 42 11 , , PUNCT iajs-2610 42 12 ( ( PUNCT iajs-2610 42 13 2.1 2.1 NUM iajs-2610 42 14 ) ) PUNCT iajs-2610 42 15 3.properties 3.properties PROPN iajs-2610 42 16 of of ADP iajs-2610 42 17 sumudu sumudu NOUN iajs-2610 42 18 transform transform NOUN iajs-2610 42 19 [ [ X iajs-2610 42 20 20 20 NUM iajs-2610 42 21 ] ] PUNCT iajs-2610 42 22 : : PUNCT iajs-2610 42 23 let let VERB iajs-2610 42 24 u u PRON iajs-2610 42 25 ( ( PUNCT iajs-2610 42 26 x x PROPN iajs-2610 42 27 , , PUNCT iajs-2610 42 28 t t PROPN iajs-2610 42 29 ) ) PUNCT iajs-2610 42 30 be be AUX iajs-2610 42 31 a a DET iajs-2610 42 32 piecewise piecewise NOUN iajs-2610 42 33 continuous continuous ADJ iajs-2610 42 34 function function NOUN iajs-2610 42 35 having have VERB iajs-2610 42 36 exponential exponential ADJ iajs-2610 42 37 order order NOUN iajs-2610 42 38 . . PUNCT iajs-2610 43 1 if if SCONJ iajs-2610 43 2 u(x u(x NOUN iajs-2610 43 3 , , PUNCT iajs-2610 43 4 s s PART iajs-2610 43 5 ) ) PUNCT iajs-2610 43 6 is be AUX iajs-2610 43 7 sumudu sumudu NOUN iajs-2610 43 8 transform transform NOUN iajs-2610 43 9 of of ADP iajs-2610 43 10 u u NOUN iajs-2610 43 11 ( ( PUNCT iajs-2610 43 12 x x PROPN iajs-2610 43 13 , , PUNCT iajs-2610 43 14 t t PROPN iajs-2610 43 15 ) ) PUNCT iajs-2610 43 16 , , PUNCT iajs-2610 43 17 then then ADV iajs-2610 43 18 sumudu sumudu NOUN iajs-2610 43 19 transforms transform VERB iajs-2610 43 20 of of ADP iajs-2610 43 21 partial partial ADJ iajs-2610 43 22 derivatives derivative NOUN iajs-2610 43 23 that that PRON iajs-2610 43 24 function function VERB iajs-2610 43 25 as as SCONJ iajs-2610 43 26 follows follow VERB iajs-2610 43 27 , , PUNCT iajs-2610 43 28 ( ( PUNCT iajs-2610 43 29 i i NOUN iajs-2610 43 30 ) ) PUNCT iajs-2610 43 31 π•Š π•Š PROPN iajs-2610 43 32 [ [ PUNCT iajs-2610 43 33 βˆ‚u(x βˆ‚u(x PROPN iajs-2610 43 34 , , PUNCT iajs-2610 43 35 t t PROPN iajs-2610 43 36 ) ) PUNCT iajs-2610 43 37 βˆ‚t βˆ‚t PROPN iajs-2610 43 38 ] ] X iajs-2610 44 1 = = PUNCT iajs-2610 44 2 1 1 NUM iajs-2610 44 3 s s PART iajs-2610 44 4 [ [ X iajs-2610 44 5 u(x u(x X iajs-2610 44 6 , , PUNCT iajs-2610 44 7 s s NOUN iajs-2610 44 8 ) ) PUNCT iajs-2610 44 9 βˆ’ βˆ’ NOUN iajs-2610 45 1 u(x u(x PROPN iajs-2610 45 2 , , PUNCT iajs-2610 45 3 t t PROPN iajs-2610 45 4 ) ) PUNCT iajs-2610 45 5 ] ] PUNCT iajs-2610 45 6 . . PUNCT iajs-2610 46 1 ( ( PUNCT iajs-2610 46 2 3.1 3.1 NUM iajs-2610 46 3 ) ) PUNCT iajs-2610 46 4 ( ( PUNCT iajs-2610 46 5 ii ii NOUN iajs-2610 46 6 ) ) PUNCT iajs-2610 46 7 π•Š π•Š PROPN iajs-2610 46 8 [ [ PUNCT iajs-2610 46 9 βˆ‚u(x βˆ‚u(x PROPN iajs-2610 46 10 , , PUNCT iajs-2610 46 11 t t PROPN iajs-2610 46 12 ) ) PUNCT iajs-2610 46 13 βˆ‚x βˆ‚x NOUN iajs-2610 46 14 ] ] PUNCT iajs-2610 47 1 = = PUNCT iajs-2610 47 2 d[u(x d[u(x NOUN iajs-2610 47 3 , , PUNCT iajs-2610 47 4 s s NOUN iajs-2610 47 5 ) ) PUNCT iajs-2610 47 6 ] ] PUNCT iajs-2610 47 7 dx dx PROPN iajs-2610 47 8 . . PUNCT iajs-2610 48 1 ( ( PUNCT iajs-2610 48 2 3.2 3.2 NUM iajs-2610 48 3 ) ) PUNCT iajs-2610 48 4 ( ( PUNCT iajs-2610 48 5 iii iii X iajs-2610 48 6 ) ) PUNCT iajs-2610 48 7 π•Š π•Š PROPN iajs-2610 48 8 [ [ PUNCT iajs-2610 48 9 βˆ‚2u(x βˆ‚2u(x PROPN iajs-2610 48 10 , , PUNCT iajs-2610 48 11 t t NOUN iajs-2610 48 12 ) ) PUNCT iajs-2610 48 13 βˆ‚t2 βˆ‚t2 NOUN iajs-2610 48 14 ] ] PUNCT iajs-2610 48 15 = = SYM iajs-2610 48 16 1 1 NUM iajs-2610 48 17 𝑠2 𝑠2 NOUN iajs-2610 48 18 u(x u(x NOUN iajs-2610 48 19 , , PUNCT iajs-2610 48 20 s s NOUN iajs-2610 48 21 ) ) PUNCT iajs-2610 48 22 βˆ’ βˆ’ PROPN iajs-2610 48 23 1 1 NUM iajs-2610 48 24 s2 s2 NOUN iajs-2610 48 25 u(x u(x NOUN iajs-2610 48 26 , , PUNCT iajs-2610 48 27 0 0 NUM iajs-2610 48 28 ) ) PUNCT iajs-2610 48 29 βˆ’ βˆ’ ADP iajs-2610 48 30 1 1 NUM iajs-2610 48 31 s s NOUN iajs-2610 48 32 βˆ‚u(x,0 βˆ‚u(x,0 NOUN iajs-2610 48 33 ) ) PUNCT iajs-2610 49 1 βˆ‚t βˆ‚t PROPN iajs-2610 49 2 . . PUNCT iajs-2610 50 1 ( ( PUNCT iajs-2610 50 2 3.3 3.3 NUM iajs-2610 50 3 ) ) PUNCT iajs-2610 50 4 25 25 NUM iajs-2610 50 5 ibn ibn PROPN iajs-2610 50 6 al al PROPN iajs-2610 50 7 - - PUNCT iajs-2610 50 8 haitham haitham PROPN iajs-2610 50 9 jour jour X iajs-2610 50 10 . . PROPN iajs-2610 51 1 for for ADP iajs-2610 51 2 pure pure ADJ iajs-2610 51 3 & & CCONJ iajs-2610 51 4 appl appl PROPN iajs-2610 51 5 . . PUNCT iajs-2610 52 1 sci sci PROPN iajs-2610 52 2 . . PROPN iajs-2610 53 1 34 34 NUM iajs-2610 53 2 ( ( PUNCT iajs-2610 53 3 2 2 NUM iajs-2610 53 4 ) ) PUNCT iajs-2610 53 5 2021 2021 NUM iajs-2610 53 6 ( ( PUNCT iajs-2610 53 7 iv iv X iajs-2610 53 8 ) ) PUNCT iajs-2610 53 9 π•Š π•Š PROPN iajs-2610 53 10 [ [ PUNCT iajs-2610 53 11 βˆ‚2u(x βˆ‚2u(x PROPN iajs-2610 53 12 , , PUNCT iajs-2610 53 13 t t PROPN iajs-2610 53 14 ) ) PUNCT iajs-2610 53 15 βˆ‚x2 βˆ‚x2 NOUN iajs-2610 53 16 ] ] PUNCT iajs-2610 53 17 = = PUNCT iajs-2610 53 18 d2[u(x d2[u(x NOUN iajs-2610 53 19 , , PUNCT iajs-2610 53 20 s s PART iajs-2610 53 21 ) ) PUNCT iajs-2610 53 22 ] ] PUNCT iajs-2610 54 1 dx2 dx2 PROPN iajs-2610 54 2 . . PUNCT iajs-2610 55 1 ( ( PUNCT iajs-2610 55 2 3.4 3.4 NUM iajs-2610 55 3 ) ) PUNCT iajs-2610 55 4 table table NOUN iajs-2610 55 5 1 1 NUM iajs-2610 55 6 . . PUNCT iajs-2610 56 1 s s X iajs-2610 56 2 - - PUNCT iajs-2610 56 3 transform transform NOUN iajs-2610 56 4 of of ADP iajs-2610 56 5 some some DET iajs-2610 56 6 functions function NOUN iajs-2610 56 7 . . PUNCT iajs-2610 57 1 f(t f(t NOUN iajs-2610 57 2 ) ) PUNCT iajs-2610 57 3 g(s g(s NOUN iajs-2610 57 4 ) ) PUNCT iajs-2610 57 5 = = SYM iajs-2610 57 6 π•Š[f(t π•Š[f(t NOUN iajs-2610 57 7 ) ) PUNCT iajs-2610 57 8 ] ] PUNCT iajs-2610 57 9 1 1 NUM iajs-2610 57 10 1 1 NUM iajs-2610 57 11 t t NOUN iajs-2610 57 12 s s X iajs-2610 57 13 tnβˆ’1 tnβˆ’1 PROPN iajs-2610 57 14 ( ( PUNCT iajs-2610 57 15 n n CCONJ iajs-2610 57 16 βˆ’ βˆ’ PROPN iajs-2610 57 17 1 1 NUM iajs-2610 57 18 ) ) PUNCT iajs-2610 57 19 ! ! PUNCT iajs-2610 57 20 , , PUNCT iajs-2610 58 1 n n NOUN iajs-2610 58 2 = = SYM iajs-2610 58 3 1,2 1,2 NUM iajs-2610 58 4 , , PUNCT iajs-2610 58 5 … … PUNCT iajs-2610 58 6 snβˆ’1 snβˆ’1 ADJ iajs-2610 58 7 table table NOUN iajs-2610 58 8 2 2 NUM iajs-2610 58 9 . . PUNCT iajs-2610 58 10 s s X iajs-2610 58 11 - - PUNCT iajs-2610 58 12 transform transform NOUN iajs-2610 58 13 properties property NOUN iajs-2610 58 14 . . PUNCT iajs-2610 59 1 formula formula NOUN iajs-2610 59 2 comment comment NOUN iajs-2610 59 3 π•Š π•Š PROPN iajs-2610 59 4 [ [ PUNCT iajs-2610 59 5 πœ•π‘’(π‘₯ πœ•π‘’(π‘₯ PROPN iajs-2610 59 6 , , PUNCT iajs-2610 59 7 𝑑 𝑑 X iajs-2610 59 8 ) ) PUNCT iajs-2610 59 9 βˆ‚π‘‘ βˆ‚π‘‘ PROPN iajs-2610 59 10 ] ] PUNCT iajs-2610 60 1 = = PUNCT iajs-2610 60 2 g(x g(x X iajs-2610 60 3 , , PUNCT iajs-2610 60 4 s s NOUN iajs-2610 60 5 ) ) PUNCT iajs-2610 60 6 βˆ’ βˆ’ NOUN iajs-2610 60 7 u(x u(x PROPN iajs-2610 60 8 , , PUNCT iajs-2610 60 9 0 0 NUM iajs-2610 60 10 ) ) PUNCT iajs-2610 60 11 s s PART iajs-2610 60 12 sumudu sumudu NOUN iajs-2610 60 13 transforms transform VERB iajs-2610 60 14 of of ADP iajs-2610 60 15 function function NOUN iajs-2610 60 16 derivatives derivative NOUN iajs-2610 60 17 π•Š π•Š PROPN iajs-2610 60 18 [ [ PUNCT iajs-2610 60 19 πœ•2𝑒(π‘₯ πœ•2𝑒(π‘₯ PROPN iajs-2610 60 20 , , PUNCT iajs-2610 60 21 𝑑 𝑑 NOUN iajs-2610 60 22 ) ) PUNCT iajs-2610 60 23 πœ•π‘‘2 πœ•π‘‘2 NOUN iajs-2610 60 24 ] ] PUNCT iajs-2610 60 25 = = SYM iajs-2610 60 26 g(x g(x X iajs-2610 60 27 , , PUNCT iajs-2610 60 28 s s NOUN iajs-2610 60 29 ) ) PUNCT iajs-2610 60 30 βˆ’ βˆ’ NOUN iajs-2610 60 31 u(x u(x PROPN iajs-2610 60 32 , , PUNCT iajs-2610 60 33 0 0 NUM iajs-2610 60 34 ) ) PUNCT iajs-2610 60 35 𝑠2 𝑠2 NOUN iajs-2610 60 36 βˆ’ βˆ’ PROPN iajs-2610 60 37 1 1 NUM iajs-2610 60 38 𝑠 𝑠 PROPN iajs-2610 60 39 πœ•π‘’(π‘₯ πœ•π‘’(π‘₯ PROPN iajs-2610 60 40 , , PUNCT iajs-2610 60 41 0 0 NUM iajs-2610 60 42 ) ) PUNCT iajs-2610 61 1 βˆ‚π‘‘ βˆ‚π‘‘ PROPN iajs-2610 61 2 π•Š π•Š PROPN iajs-2610 61 3 [ [ PUNCT iajs-2610 61 4 πœ•π‘›π‘’(π‘₯ πœ•π‘›π‘’(π‘₯ PROPN iajs-2610 61 5 , , PUNCT iajs-2610 61 6 𝑑 𝑑 NOUN iajs-2610 61 7 ) ) PUNCT iajs-2610 61 8 πœ•π‘‘π‘› πœ•π‘‘π‘› PUNCT iajs-2610 61 9 ] ] PUNCT iajs-2610 62 1 = = PUNCT iajs-2610 62 2 g(x g(x X iajs-2610 62 3 , , PUNCT iajs-2610 62 4 s s NOUN iajs-2610 62 5 ) ) PUNCT iajs-2610 62 6 βˆ’ βˆ’ NOUN iajs-2610 62 7 u(x u(x PROPN iajs-2610 62 8 , , PUNCT iajs-2610 62 9 0 0 NUM iajs-2610 62 10 ) ) PUNCT iajs-2610 62 11 𝑠𝑛 𝑠𝑛 NOUN iajs-2610 62 12 βˆ’ βˆ’ NUM iajs-2610 62 13 1 1 NUM iajs-2610 62 14 π‘ π‘›βˆ’1 π‘ π‘›βˆ’1 PROPN iajs-2610 62 15 πœ•π‘’(π‘₯ πœ•π‘’(π‘₯ PROPN iajs-2610 62 16 , , PUNCT iajs-2610 62 17 0 0 NUM iajs-2610 62 18 ) ) PUNCT iajs-2610 62 19 πœ•π‘‘ πœ•π‘‘ NOUN iajs-2610 62 20 βˆ’ βˆ’ PROPN iajs-2610 63 1 β‹― β‹― NOUN iajs-2610 63 2 βˆ’ βˆ’ PROPN iajs-2610 63 3 1 1 NUM iajs-2610 63 4 𝑠 𝑠 PROPN iajs-2610 63 5 πœ•π‘›βˆ’1𝑒(π‘₯ πœ•π‘›βˆ’1𝑒(π‘₯ PROPN iajs-2610 63 6 , , PUNCT iajs-2610 63 7 0 0 NUM iajs-2610 63 8 ) ) PUNCT iajs-2610 63 9 πœ•π‘‘π‘›βˆ’1 πœ•π‘‘π‘›βˆ’1 NOUN iajs-2610 63 10 π•Š[π‘Žπ‘’1(π‘₯ π•Š[π‘Žπ‘’1(π‘₯ NOUN iajs-2610 63 11 , , PUNCT iajs-2610 63 12 𝑑 𝑑 PROPN iajs-2610 63 13 ) ) PUNCT iajs-2610 63 14 + + CCONJ iajs-2610 63 15 𝑏𝑒2(π‘₯ 𝑏𝑒2(π‘₯ PROPN iajs-2610 63 16 , , PUNCT iajs-2610 63 17 𝑑 𝑑 X iajs-2610 63 18 ) ) PUNCT iajs-2610 63 19 = = SYM iajs-2610 63 20 π‘Žπ•Š[𝑒1(π‘₯ π‘Žπ•Š[𝑒1(π‘₯ NUM iajs-2610 63 21 , , PUNCT iajs-2610 63 22 𝑑)]+ 𝑑)]+ PROPN iajs-2610 63 23 𝑏[𝑒2(π‘₯ 𝑏[𝑒2(π‘₯ NOUN iajs-2610 63 24 , , PUNCT iajs-2610 63 25 𝑑 𝑑 NOUN iajs-2610 63 26 ) ) PUNCT iajs-2610 63 27 ] ] PUNCT iajs-2610 63 28 where where SCONJ iajs-2610 63 29 π‘Ž π‘Ž NOUN iajs-2610 63 30 and and CCONJ iajs-2610 63 31 𝑏 𝑏 PROPN iajs-2610 63 32 are be AUX iajs-2610 63 33 constants constant NOUN iajs-2610 63 34 linearity linearity NOUN iajs-2610 63 35 property property NOUN iajs-2610 63 36 π•Š[∫ π•Š[∫ NOUN iajs-2610 63 37 𝑒(π‘₯ 𝑒(π‘₯ PROPN iajs-2610 63 38 , , PUNCT iajs-2610 63 39 𝜏)π‘‘πœ 𝜏)π‘‘πœ PROPN iajs-2610 63 40 ] ] X iajs-2610 63 41 = = SYM iajs-2610 63 42 𝑠𝐺(π‘₯ 𝑠𝐺(π‘₯ PROPN iajs-2610 63 43 , , PUNCT iajs-2610 63 44 𝑠 𝑠 NOUN iajs-2610 63 45 ) ) PUNCT iajs-2610 63 46 𝑑 𝑑 PROPN iajs-2610 63 47 0 0 NUM iajs-2610 63 48 sumudu sumudu NOUN iajs-2610 63 49 transform transform NOUN iajs-2610 63 50 of of ADP iajs-2610 63 51 an an DET iajs-2610 63 52 integral integral ADJ iajs-2610 63 53 function function NOUN iajs-2610 63 54 π•Š[𝑒(π‘₯ π•Š[𝑒(π‘₯ NOUN iajs-2610 63 55 , , PUNCT iajs-2610 63 56 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2610 63 57 ) ) PUNCT iajs-2610 63 58 ] ] PUNCT iajs-2610 64 1 = = SYM iajs-2610 64 2 𝐺(π‘₯ 𝐺(π‘₯ NOUN iajs-2610 64 3 , , PUNCT iajs-2610 64 4 π‘Žπ‘  π‘Žπ‘  PROPN iajs-2610 64 5 ) ) PUNCT iajs-2610 64 6 where where SCONJ iajs-2610 64 7 π‘Ž π‘Ž NOUN iajs-2610 64 8 is be AUX iajs-2610 64 9 a a DET iajs-2610 64 10 constant constant ADJ iajs-2610 64 11 first first ADJ iajs-2610 64 12 scale scale NOUN iajs-2610 64 13 preserving preserving NOUN iajs-2610 64 14 theorem theorem NOUN iajs-2610 64 15 4 4 NUM iajs-2610 64 16 . . PUNCT iajs-2610 64 17 fundamental fundamental ADJ iajs-2610 64 18 idea idea NOUN iajs-2610 64 19 of of ADP iajs-2610 64 20 the the DET iajs-2610 64 21 iterative iterative NOUN iajs-2610 64 22 method method NOUN iajs-2610 64 23 the the DET iajs-2610 64 24 bacis bacis NOUN iajs-2610 64 25 steps step NOUN iajs-2610 64 26 of of ADP iajs-2610 64 27 i i PRON iajs-2610 64 28 m. m. NOUN iajs-2610 64 29 it it PRON iajs-2610 64 30 is be AUX iajs-2610 64 31 rewritten rewrite VERB iajs-2610 64 32 that that SCONJ iajs-2610 64 33 any any DET iajs-2610 64 34 pde pde NOUN iajs-2610 64 35 can can AUX iajs-2610 64 36 be be AUX iajs-2610 64 37 written write VERB iajs-2610 64 38 as as ADP iajs-2610 64 39 : : PUNCT iajs-2610 64 40 l(u(x l(u(x PROPN iajs-2610 64 41 , , PUNCT iajs-2610 64 42 t t PROPN iajs-2610 64 43 ) ) PUNCT iajs-2610 64 44 ) ) PUNCT iajs-2610 65 1 + + CCONJ iajs-2610 65 2 n(u(x n(u(x NOUN iajs-2610 65 3 , , PUNCT iajs-2610 65 4 t t PROPN iajs-2610 65 5 ) ) PUNCT iajs-2610 65 6 ) ) PUNCT iajs-2610 66 1 + + CCONJ iajs-2610 66 2 h(x h(x PROPN iajs-2610 66 3 , , PUNCT iajs-2610 66 4 t t PROPN iajs-2610 66 5 ) ) PUNCT iajs-2610 66 6 ) ) PUNCT iajs-2610 67 1 = = SYM iajs-2610 67 2 0 0 X iajs-2610 67 3 . . PUNCT iajs-2610 68 1 ( ( PUNCT iajs-2610 68 2 4.1 4.1 NUM iajs-2610 68 3 ) ) PUNCT iajs-2610 68 4 subject subject NOUN iajs-2610 68 5 to to ADP iajs-2610 68 6 the the DET iajs-2610 68 7 conditions condition NOUN iajs-2610 68 8 c(u c(u PROPN iajs-2610 68 9 , , PUNCT iajs-2610 68 10 βˆ‚u βˆ‚u PROPN iajs-2610 68 11 βˆ‚t βˆ‚t PROPN iajs-2610 68 12 ) ) PUNCT iajs-2610 69 1 = = PUNCT iajs-2610 69 2 0 0 X iajs-2610 69 3 . . PUNCT iajs-2610 70 1 ( ( PUNCT iajs-2610 70 2 4.2 4.2 NUM iajs-2610 70 3 ) ) PUNCT iajs-2610 70 4 so so ADV iajs-2610 70 5 , , PUNCT iajs-2610 70 6 thus thus ADV iajs-2610 70 7 u(x u(x PROPN iajs-2610 70 8 , , PUNCT iajs-2610 70 9 t t PROPN iajs-2610 70 10 ) ) PUNCT iajs-2610 70 11 the the DET iajs-2610 70 12 unknown unknown ADJ iajs-2610 70 13 function function NOUN iajs-2610 70 14 , , PUNCT iajs-2610 70 15 x x X iajs-2610 70 16 and and CCONJ iajs-2610 70 17 t t PROPN iajs-2610 70 18 indicates indicate VERB iajs-2610 70 19 the the DET iajs-2610 70 20 independent independent ADJ iajs-2610 70 21 variables variable NOUN iajs-2610 70 22 , , PUNCT iajs-2610 70 23 whereas whereas SCONJ iajs-2610 70 24 l l NOUN iajs-2610 70 25 , , PUNCT iajs-2610 70 26 n n PRON iajs-2610 70 27 represent represent VERB iajs-2610 70 28 linear linear NOUN iajs-2610 70 29 and and CCONJ iajs-2610 70 30 non non ADJ iajs-2610 70 31 - - ADJ iajs-2610 70 32 linear linear ADJ iajs-2610 70 33 operators operator NOUN iajs-2610 70 34 , , PUNCT iajs-2610 70 35 respectively respectively ADV iajs-2610 70 36 , , PUNCT iajs-2610 70 37 h(x h(x PROPN iajs-2610 70 38 , , PUNCT iajs-2610 70 39 t t PROPN iajs-2610 70 40 ) ) PUNCT iajs-2610 70 41 represents represent VERB iajs-2610 70 42 inhomogeneous inhomogeneous ADJ iajs-2610 70 43 term term NOUN iajs-2610 70 44 which which PRON iajs-2610 70 45 is be AUX iajs-2610 70 46 a a DET iajs-2610 70 47 renowned renowned ADJ iajs-2610 70 48 function function NOUN iajs-2610 70 49 and and CCONJ iajs-2610 70 50 c c X iajs-2610 70 51 the the DET iajs-2610 70 52 conditions condition NOUN iajs-2610 70 53 operator operator NOUN iajs-2610 70 54 for for ADP iajs-2610 70 55 a a DET iajs-2610 70 56 problem problem NOUN iajs-2610 70 57 . . PUNCT iajs-2610 71 1 the the DET iajs-2610 71 2 essential essential ADJ iajs-2610 71 3 idea idea NOUN iajs-2610 71 4 of of ADP iajs-2610 71 5 the the DET iajs-2610 71 6 iterative iterative NOUN iajs-2610 71 7 method method NOUN iajs-2610 71 8 to to PART iajs-2610 71 9 solve solve VERB iajs-2610 71 10 eq eq ADP iajs-2610 71 11 . . PUNCT iajs-2610 72 1 ( ( PUNCT iajs-2610 72 2 4.1 4.1 NUM iajs-2610 72 3 ) ) PUNCT iajs-2610 72 4 with with ADP iajs-2610 72 5 initial initial ADJ iajs-2610 72 6 approximation approximation NOUN iajs-2610 72 7 for for ADP iajs-2610 72 8 eq.(4.2 eq.(4.2 NOUN iajs-2610 72 9 ) ) PUNCT iajs-2610 72 10 is be AUX iajs-2610 72 11 a a DET iajs-2610 72 12 primary primary ADJ iajs-2610 72 13 step step NOUN iajs-2610 72 14 , , PUNCT iajs-2610 72 15 by by ADP iajs-2610 72 16 assuming assume VERB iajs-2610 72 17 that that SCONJ iajs-2610 72 18 the the DET iajs-2610 72 19 initial initial ADJ iajs-2610 72 20 guess guess NOUN iajs-2610 72 21 u0(x u0(x SYM iajs-2610 72 22 , , PUNCT iajs-2610 72 23 t t PROPN iajs-2610 72 24 ) ) PUNCT iajs-2610 72 25 is be AUX iajs-2610 72 26 a a DET iajs-2610 72 27 solution solution NOUN iajs-2610 72 28 of of ADP iajs-2610 72 29 the the DET iajs-2610 72 30 problem problem NOUN iajs-2610 72 31 u(x u(x NOUN iajs-2610 72 32 , , PUNCT iajs-2610 72 33 t t PROPN iajs-2610 72 34 ) ) PUNCT iajs-2610 72 35 and and CCONJ iajs-2610 72 36 solution solution NOUN iajs-2610 72 37 of of ADP iajs-2610 72 38 the the DET iajs-2610 72 39 equation equation NOUN iajs-2610 72 40 l(u0(x l(u0(x PROPN iajs-2610 72 41 , , PUNCT iajs-2610 72 42 t t PROPN iajs-2610 72 43 ) ) PUNCT iajs-2610 72 44 ) ) PUNCT iajs-2610 73 1 + + CCONJ iajs-2610 73 2 h(x h(x PROPN iajs-2610 73 3 , , PUNCT iajs-2610 73 4 t t PROPN iajs-2610 73 5 ) ) PUNCT iajs-2610 73 6 = = SYM iajs-2610 73 7 0 0 NUM iajs-2610 73 8 , , PUNCT iajs-2610 73 9 c(u0 c(u0 PROPN iajs-2610 73 10 , , PUNCT iajs-2610 73 11 βˆ‚u0 βˆ‚u0 NOUN iajs-2610 73 12 βˆ‚t βˆ‚t PROPN iajs-2610 73 13 ) ) PUNCT iajs-2610 73 14 = = SYM iajs-2610 73 15 0 0 X iajs-2610 73 16 . . PUNCT iajs-2610 74 1 ( ( PUNCT iajs-2610 74 2 4.3 4.3 NUM iajs-2610 74 3 ) ) PUNCT iajs-2610 74 4 to to PART iajs-2610 74 5 generate generate VERB iajs-2610 74 6 next next ADJ iajs-2610 74 7 iterative iterative NOUN iajs-2610 74 8 of of ADP iajs-2610 74 9 solution solution NOUN iajs-2610 74 10 , , PUNCT iajs-2610 74 11 we we PRON iajs-2610 74 12 put put VERB iajs-2610 74 13 eq.(4.1 eq.(4.1 NOUN iajs-2610 74 14 ) ) PUNCT iajs-2610 74 15 as as SCONJ iajs-2610 74 16 follows follow VERB iajs-2610 74 17 l(u1(x l(u1(x PROPN iajs-2610 74 18 , , PUNCT iajs-2610 74 19 t t PROPN iajs-2610 74 20 ) ) PUNCT iajs-2610 74 21 ) ) PUNCT iajs-2610 75 1 + + CCONJ iajs-2610 75 2 h(x h(x PROPN iajs-2610 75 3 , , PUNCT iajs-2610 75 4 t t PROPN iajs-2610 75 5 ) ) PUNCT iajs-2610 75 6 + + CCONJ iajs-2610 76 1 n(u0(x n(u0(x PROPN iajs-2610 76 2 , , PUNCT iajs-2610 76 3 t t PROPN iajs-2610 76 4 ) ) PUNCT iajs-2610 76 5 ) ) PUNCT iajs-2610 77 1 = = SYM iajs-2610 77 2 0 0 NUM iajs-2610 77 3 , , PUNCT iajs-2610 77 4 c(u1 c(u1 NOUN iajs-2610 77 5 , , PUNCT iajs-2610 77 6 βˆ‚u1 βˆ‚u1 PROPN iajs-2610 77 7 βˆ‚t βˆ‚t PROPN iajs-2610 77 8 ) ) PUNCT iajs-2610 78 1 = = PUNCT iajs-2610 78 2 0 0 X iajs-2610 78 3 . . PUNCT iajs-2610 79 1 ( ( PUNCT iajs-2610 79 2 4.4 4.4 NUM iajs-2610 79 3 ) ) PUNCT iajs-2610 79 4 after after ADP iajs-2610 79 5 several several ADJ iajs-2610 79 6 simple simple ADJ iajs-2610 79 7 iterative iterative NOUN iajs-2610 79 8 steps step NOUN iajs-2610 79 9 of of ADP iajs-2610 79 10 the the DET iajs-2610 79 11 solution solution NOUN iajs-2610 79 12 , , PUNCT iajs-2610 79 13 the the DET iajs-2610 79 14 generic generic ADJ iajs-2610 79 15 form form NOUN iajs-2610 79 16 of of ADP iajs-2610 79 17 this this DET iajs-2610 79 18 equation equation NOUN iajs-2610 79 19 is be AUX iajs-2610 79 20 l(un+1(x l(un+1(x NOUN iajs-2610 79 21 , , PUNCT iajs-2610 79 22 t t PROPN iajs-2610 79 23 ) ) PUNCT iajs-2610 79 24 ) ) PUNCT iajs-2610 80 1 + + CCONJ iajs-2610 80 2 h(x h(x PROPN iajs-2610 80 3 , , PUNCT iajs-2610 80 4 t t PROPN iajs-2610 80 5 ) ) PUNCT iajs-2610 80 6 + + CCONJ iajs-2610 81 1 n(un(x n(un(x PROPN iajs-2610 81 2 , , PUNCT iajs-2610 81 3 t t PROPN iajs-2610 81 4 ) ) PUNCT iajs-2610 81 5 ) ) PUNCT iajs-2610 82 1 = = SYM iajs-2610 82 2 0 0 NUM iajs-2610 82 3 , , PUNCT iajs-2610 82 4 c(un+1 c(un+1 ADJ iajs-2610 82 5 , , PUNCT iajs-2610 82 6 βˆ‚un+1 βˆ‚un+1 ADJ iajs-2610 82 7 βˆ‚t βˆ‚t PROPN iajs-2610 82 8 ) ) PUNCT iajs-2610 83 1 = = SYM iajs-2610 83 2 0 0 NUM iajs-2610 83 3 ( ( PUNCT iajs-2610 83 4 4.5 4.5 NUM iajs-2610 83 5 ) ) PUNCT iajs-2610 83 6 26 26 NUM iajs-2610 83 7 ibn ibn PROPN iajs-2610 83 8 al al PROPN iajs-2610 83 9 - - PUNCT iajs-2610 83 10 haitham haitham PROPN iajs-2610 83 11 jour jour X iajs-2610 83 12 . . PROPN iajs-2610 84 1 for for ADP iajs-2610 84 2 pure pure ADJ iajs-2610 84 3 & & CCONJ iajs-2610 84 4 appl appl PROPN iajs-2610 84 5 . . PUNCT iajs-2610 85 1 sci sci PROPN iajs-2610 85 2 . . PROPN iajs-2610 86 1 34 34 NUM iajs-2610 86 2 ( ( PUNCT iajs-2610 86 3 2 2 NUM iajs-2610 86 4 ) ) PUNCT iajs-2610 86 5 2021 2021 NUM iajs-2610 86 6 evidently evidently ADV iajs-2610 86 7 , , PUNCT iajs-2610 86 8 each each DET iajs-2610 86 9 iteration iteration NOUN iajs-2610 86 10 of of ADP iajs-2610 86 11 the the DET iajs-2610 86 12 function function NOUN iajs-2610 86 13 un(x un(x SYM iajs-2610 86 14 , , PUNCT iajs-2610 86 15 t t PROPN iajs-2610 86 16 ) ) PUNCT iajs-2610 86 17 represents represent VERB iajs-2610 86 18 effectively effectively ADV iajs-2610 86 19 the the DET iajs-2610 86 20 only only ADJ iajs-2610 86 21 solution solution NOUN iajs-2610 86 22 for for ADP iajs-2610 86 23 eq eq NOUN iajs-2610 86 24 . . PUNCT iajs-2610 87 1 ( ( PUNCT iajs-2610 87 2 4.1 4.1 NUM iajs-2610 87 3 ) ) PUNCT iajs-2610 87 4 . . PUNCT iajs-2610 88 1 πŸ“. πŸ“. PUNCT iajs-2610 88 2 sumudu sumudu NOUN iajs-2610 88 3 iterative iterative NOUN iajs-2610 88 4 method method NOUN iajs-2610 88 5 we we PRON iajs-2610 88 6 clarify clarify VERB iajs-2610 88 7 the the DET iajs-2610 88 8 sumudu sumudu NOUN iajs-2610 88 9 iterative iterative NOUN iajs-2610 88 10 method method NOUN iajs-2610 88 11 ( ( PUNCT iajs-2610 88 12 sim sim ADJ iajs-2610 88 13 ) ) PUNCT iajs-2610 88 14 algorithm algorithm NOUN iajs-2610 88 15 via via ADP iajs-2610 88 16 considering consider VERB iajs-2610 88 17 general general ADJ iajs-2610 88 18 nonlinear nonlinear ADJ iajs-2610 88 19 inhomogeneous inhomogeneous ADJ iajs-2610 88 20 pdes pde NOUN iajs-2610 88 21 of of ADP iajs-2610 88 22 the the DET iajs-2610 88 23 form form NOUN iajs-2610 88 24 : : PUNCT iajs-2610 88 25 l(u(x l(u(x NOUN iajs-2610 88 26 , , PUNCT iajs-2610 88 27 t t PROPN iajs-2610 88 28 ) ) PUNCT iajs-2610 88 29 ) ) PUNCT iajs-2610 89 1 + + CCONJ iajs-2610 89 2 r(u(x r(u(x NOUN iajs-2610 89 3 , , PUNCT iajs-2610 89 4 t t PROPN iajs-2610 89 5 ) ) PUNCT iajs-2610 89 6 ) ) PUNCT iajs-2610 90 1 + + CCONJ iajs-2610 90 2 n(u(x n(u(x NOUN iajs-2610 90 3 , , PUNCT iajs-2610 90 4 t t NOUN iajs-2610 90 5 ) ) PUNCT iajs-2610 90 6 ) ) PUNCT iajs-2610 91 1 = = SYM iajs-2610 91 2 h(x h(x PROPN iajs-2610 91 3 , , PUNCT iajs-2610 91 4 t t PROPN iajs-2610 91 5 ) ) PUNCT iajs-2610 91 6 . . PUNCT iajs-2610 92 1 ( ( PUNCT iajs-2610 92 2 5.1 5.1 NUM iajs-2610 92 3 ) ) PUNCT iajs-2610 92 4 submit submit NOUN iajs-2610 92 5 to to ADP iajs-2610 92 6 ics ics NOUN iajs-2610 92 7 u(x u(x NOUN iajs-2610 92 8 , , PUNCT iajs-2610 92 9 0 0 NUM iajs-2610 92 10 ) ) PUNCT iajs-2610 92 11 = = SYM iajs-2610 92 12 f1(x f1(x PROPN iajs-2610 92 13 ) ) PUNCT iajs-2610 92 14 , , PUNCT iajs-2610 92 15 0 0 NUM iajs-2610 92 16 ≀ ≀ NUM iajs-2610 92 17 x x SYM iajs-2610 92 18 ≀ ≀ NUM iajs-2610 92 19 l l NOUN iajs-2610 92 20 and and CCONJ iajs-2610 92 21 ut(x ut(x NOUN iajs-2610 92 22 , , PUNCT iajs-2610 92 23 0 0 NUM iajs-2610 92 24 ) ) PUNCT iajs-2610 92 25 = = SYM iajs-2610 93 1 f2(x f2(x PROPN iajs-2610 93 2 ) ) PUNCT iajs-2610 93 3 , , PUNCT iajs-2610 93 4 t t X iajs-2610 93 5 > > X iajs-2610 93 6 0 0 PROPN iajs-2610 93 7 . . PUNCT iajs-2610 94 1 ( ( PUNCT iajs-2610 94 2 5.2 5.2 NUM iajs-2610 94 3 ) ) PUNCT iajs-2610 94 4 so so ADV iajs-2610 94 5 , , PUNCT iajs-2610 94 6 the the DET iajs-2610 94 7 second second ADJ iajs-2610 94 8 order order NOUN iajs-2610 94 9 linear linear NOUN iajs-2610 94 10 differential differential NOUN iajs-2610 94 11 operator operator NOUN iajs-2610 94 12 is be AUX iajs-2610 94 13 l l NOUN iajs-2610 94 14 about about ADP iajs-2610 94 15 l l NOUN iajs-2610 94 16 = = PUNCT iajs-2610 94 17 πœ•2 πœ•2 NOUN iajs-2610 94 18 πœ•π‘‘2 πœ•π‘‘2 NOUN iajs-2610 94 19 , , PUNCT iajs-2610 94 20 linear linear ADJ iajs-2610 94 21 operator operator NOUN iajs-2610 94 22 for for ADP iajs-2610 94 23 less less ADJ iajs-2610 94 24 order order NOUN iajs-2610 94 25 than than SCONJ iajs-2610 94 26 l l NOUN iajs-2610 94 27 is be AUX iajs-2610 94 28 r r NOUN iajs-2610 94 29 , , PUNCT iajs-2610 94 30 n n PRON iajs-2610 94 31 represents represent VERB iajs-2610 94 32 the the DET iajs-2610 94 33 non non ADJ iajs-2610 94 34 - - ADJ iajs-2610 94 35 linear linear ADJ iajs-2610 94 36 differential differential NOUN iajs-2610 94 37 operator operator NOUN iajs-2610 94 38 and and CCONJ iajs-2610 94 39 the the DET iajs-2610 94 40 source source NOUN iajs-2610 94 41 term term NOUN iajs-2610 94 42 is be AUX iajs-2610 94 43 h(x h(x PROPN iajs-2610 94 44 , , PUNCT iajs-2610 94 45 t t PROPN iajs-2610 94 46 ) ) PUNCT iajs-2610 94 47 . . PUNCT iajs-2610 95 1 we we PRON iajs-2610 95 2 stratify stratify VERB iajs-2610 95 3 s s NOUN iajs-2610 95 4 - - PUNCT iajs-2610 95 5 transform transform NOUN iajs-2610 95 6 of of ADP iajs-2610 95 7 eq.(5.1 eq.(5.1 NOUN iajs-2610 95 8 ) ) PUNCT iajs-2610 95 9 to to PART iajs-2610 95 10 get get VERB iajs-2610 95 11 : : PUNCT iajs-2610 95 12 π•Š[l(u(x π•Š[l(u(x NOUN iajs-2610 95 13 , , PUNCT iajs-2610 95 14 t t PROPN iajs-2610 95 15 ) ) PUNCT iajs-2610 95 16 ) ) PUNCT iajs-2610 95 17 ] ] PUNCT iajs-2610 96 1 + + CCONJ iajs-2610 96 2 π•Š[r(u(x π•Š[r(u(x NOUN iajs-2610 96 3 , , PUNCT iajs-2610 96 4 t t NOUN iajs-2610 96 5 ) ) PUNCT iajs-2610 96 6 ) ) PUNCT iajs-2610 96 7 ] ] PUNCT iajs-2610 97 1 + + CCONJ iajs-2610 97 2 π•Š[n(u(x π•Š[n(u(x NOUN iajs-2610 97 3 , , PUNCT iajs-2610 97 4 t t NOUN iajs-2610 97 5 ) ) PUNCT iajs-2610 97 6 ) ) PUNCT iajs-2610 97 7 ] ] PUNCT iajs-2610 98 1 = = SYM iajs-2610 98 2 π•Š[h(x π•Š[h(x PROPN iajs-2610 98 3 , , PUNCT iajs-2610 98 4 t t PROPN iajs-2610 98 5 ) ) PUNCT iajs-2610 98 6 ] ] PUNCT iajs-2610 98 7 . . PUNCT iajs-2610 99 1 ( ( PUNCT iajs-2610 99 2 5.3 5.3 NUM iajs-2610 99 3 ) ) PUNCT iajs-2610 99 4 utilizing utilize VERB iajs-2610 99 5 properties property NOUN iajs-2610 99 6 in in ADP iajs-2610 99 7 table table NOUN iajs-2610 99 8 1,2 1,2 NUM iajs-2610 99 9 respectively respectively ADV iajs-2610 99 10 above above ADV iajs-2610 99 11 in in ADP iajs-2610 99 12 section section NOUN iajs-2610 99 13 3 3 NUM iajs-2610 99 14 with with ADP iajs-2610 99 15 the the DET iajs-2610 99 16 given give VERB iajs-2610 99 17 initial initial ADJ iajs-2610 99 18 conditions condition NOUN iajs-2610 99 19 and and CCONJ iajs-2610 99 20 for for ADP iajs-2610 99 21 equation equation NOUN iajs-2610 99 22 of of ADP iajs-2610 99 23 order order NOUN iajs-2610 99 24 2 2 NUM iajs-2610 99 25 , , PUNCT iajs-2610 99 26 we we PRON iajs-2610 99 27 get get VERB iajs-2610 99 28 : : PUNCT iajs-2610 100 1 u(x u(x NOUN iajs-2610 100 2 , , PUNCT iajs-2610 100 3 s s NOUN iajs-2610 100 4 ) ) PUNCT iajs-2610 100 5 s2 s2 NOUN iajs-2610 100 6 βˆ’ βˆ’ PROPN iajs-2610 100 7 u(x,0 u(x,0 PROPN iajs-2610 100 8 ) ) PUNCT iajs-2610 100 9 s2 s2 NOUN iajs-2610 100 10 βˆ’ βˆ’ PROPN iajs-2610 100 11 ut(x,0 ut(x,0 PROPN iajs-2610 100 12 ) ) PUNCT iajs-2610 100 13 s s PART iajs-2610 101 1 + + X iajs-2610 101 2 π•Š[r(u(x π•Š[r(u(x NOUN iajs-2610 101 3 , , PUNCT iajs-2610 101 4 t t NOUN iajs-2610 101 5 ) ) PUNCT iajs-2610 101 6 ) ) PUNCT iajs-2610 101 7 ] ] PUNCT iajs-2610 102 1 + + CCONJ iajs-2610 102 2 π•Š[n(u(x π•Š[n(u(x NOUN iajs-2610 102 3 , , PUNCT iajs-2610 102 4 t t NOUN iajs-2610 102 5 ) ) PUNCT iajs-2610 102 6 ) ) PUNCT iajs-2610 102 7 ] ] PUNCT iajs-2610 103 1 = = SYM iajs-2610 103 2 π•Š[h(x π•Š[h(x PROPN iajs-2610 103 3 , , PUNCT iajs-2610 103 4 t t PROPN iajs-2610 103 5 ) ) PUNCT iajs-2610 103 6 ] ] PUNCT iajs-2610 103 7 . . PUNCT iajs-2610 104 1 ( ( PUNCT iajs-2610 104 2 5.4 5.4 NUM iajs-2610 104 3 ) ) PUNCT iajs-2610 104 4 replace replace NOUN iajs-2610 104 5 eq.(5.2 eq.(5.2 NOUN iajs-2610 104 6 ) ) PUNCT iajs-2610 104 7 into into ADP iajs-2610 104 8 eq.(5.4 eq.(5.4 PROPN iajs-2610 104 9 ) ) PUNCT iajs-2610 104 10 to to PART iajs-2610 104 11 get get VERB iajs-2610 104 12 : : PUNCT iajs-2610 104 13 u(x u(x NOUN iajs-2610 104 14 , , PUNCT iajs-2610 104 15 s s NOUN iajs-2610 104 16 ) ) PUNCT iajs-2610 104 17 s2 s2 NOUN iajs-2610 104 18 βˆ’ βˆ’ ADP iajs-2610 104 19 f1(x f1(x NOUN iajs-2610 104 20 ) ) PUNCT iajs-2610 104 21 s2 s2 NOUN iajs-2610 104 22 βˆ’ βˆ’ NOUN iajs-2610 104 23 f2(x f2(x NOUN iajs-2610 104 24 ) ) PUNCT iajs-2610 104 25 s s PART iajs-2610 105 1 + + X iajs-2610 105 2 π•Š[r(u(x π•Š[r(u(x NOUN iajs-2610 105 3 , , PUNCT iajs-2610 105 4 t t NOUN iajs-2610 105 5 ) ) PUNCT iajs-2610 105 6 ) ) PUNCT iajs-2610 105 7 ] ] PUNCT iajs-2610 106 1 + + CCONJ iajs-2610 106 2 π•Š[n(u(x π•Š[n(u(x NOUN iajs-2610 106 3 , , PUNCT iajs-2610 106 4 t t NOUN iajs-2610 106 5 ) ) PUNCT iajs-2610 106 6 ) ) PUNCT iajs-2610 106 7 ] ] PUNCT iajs-2610 107 1 = = SYM iajs-2610 107 2 π•Š[h(x π•Š[h(x PROPN iajs-2610 107 3 , , PUNCT iajs-2610 107 4 t t PROPN iajs-2610 107 5 ) ) PUNCT iajs-2610 107 6 ] ] PUNCT iajs-2610 107 7 . . PUNCT iajs-2610 108 1 ( ( PUNCT iajs-2610 108 2 5.5 5.5 NUM iajs-2610 108 3 ) ) PUNCT iajs-2610 108 4 from from ADP iajs-2610 108 5 eq.(5.5 eq.(5.5 PROPN iajs-2610 108 6 ) ) PUNCT iajs-2610 108 7 , , PUNCT iajs-2610 108 8 we we PRON iajs-2610 108 9 obtain obtain VERB iajs-2610 108 10 : : PUNCT iajs-2610 108 11 u(x u(x NOUN iajs-2610 108 12 , , PUNCT iajs-2610 108 13 s s NOUN iajs-2610 108 14 ) ) PUNCT iajs-2610 108 15 s2 s2 NOUN iajs-2610 108 16 = = SYM iajs-2610 108 17 f1(x f1(x PROPN iajs-2610 108 18 ) ) PUNCT iajs-2610 108 19 s2 s2 NOUN iajs-2610 108 20 + + CCONJ iajs-2610 108 21 f2(x f2(x X iajs-2610 108 22 ) ) PUNCT iajs-2610 108 23 s s PART iajs-2610 109 1 + + X iajs-2610 109 2 π•Š[h(x π•Š[h(x PROPN iajs-2610 109 3 , , PUNCT iajs-2610 109 4 t t PROPN iajs-2610 109 5 ) ) PUNCT iajs-2610 109 6 ] ] PUNCT iajs-2610 110 1 βˆ’ βˆ’ PROPN iajs-2610 110 2 π•Š[r(u(x π•Š[r(u(x NOUN iajs-2610 110 3 , , PUNCT iajs-2610 110 4 t t PROPN iajs-2610 110 5 ) ) PUNCT iajs-2610 110 6 ) ) PUNCT iajs-2610 110 7 ] ] PUNCT iajs-2610 111 1 βˆ’ βˆ’ PROPN iajs-2610 111 2 π•Š[n(u(x π•Š[n(u(x NOUN iajs-2610 111 3 , , PUNCT iajs-2610 111 4 t t PROPN iajs-2610 111 5 ) ) PUNCT iajs-2610 111 6 ) ) PUNCT iajs-2610 111 7 ] ] PUNCT iajs-2610 111 8 . . PUNCT iajs-2610 112 1 ( ( PUNCT iajs-2610 112 2 5.6 5.6 NUM iajs-2610 112 3 ) ) PUNCT iajs-2610 112 4 eq.(5.6 eq.(5.6 NOUN iajs-2610 112 5 ) ) PUNCT iajs-2610 112 6 becomes become VERB iajs-2610 112 7 u(x u(x NOUN iajs-2610 112 8 , , PUNCT iajs-2610 112 9 s s NOUN iajs-2610 112 10 ) ) PUNCT iajs-2610 112 11 = = SYM iajs-2610 112 12 f1(x f1(x PROPN iajs-2610 112 13 ) ) PUNCT iajs-2610 113 1 + + CCONJ iajs-2610 114 1 sf2(x sf2(x PROPN iajs-2610 114 2 ) ) PUNCT iajs-2610 115 1 + + CCONJ iajs-2610 115 2 s2π•Š[h(x s2π•Š[h(x NOUN iajs-2610 115 3 , , PUNCT iajs-2610 115 4 t t PROPN iajs-2610 115 5 ) ) PUNCT iajs-2610 115 6 ] ] PUNCT iajs-2610 116 1 βˆ’ βˆ’ PROPN iajs-2610 116 2 s2π•Š[r(u(x s2π•Š[r(u(x NOUN iajs-2610 116 3 , , PUNCT iajs-2610 116 4 t t PROPN iajs-2610 116 5 ) ) PUNCT iajs-2610 116 6 ) ) PUNCT iajs-2610 116 7 ] ] PUNCT iajs-2610 117 1 βˆ’ βˆ’ PROPN iajs-2610 117 2 s2π•Š[n(u(x s2π•Š[n(u(x NOUN iajs-2610 117 3 , , PUNCT iajs-2610 117 4 t t PROPN iajs-2610 117 5 ) ) PUNCT iajs-2610 117 6 ) ) PUNCT iajs-2610 117 7 ] ] PUNCT iajs-2610 117 8 . . PUNCT iajs-2610 118 1 ( ( PUNCT iajs-2610 118 2 5.7 5.7 NUM iajs-2610 118 3 ) ) PUNCT iajs-2610 118 4 subsequently subsequently ADV iajs-2610 118 5 , , PUNCT iajs-2610 118 6 taking take VERB iajs-2610 118 7 inverse inverse NOUN iajs-2610 118 8 sumudu sumudu NOUN iajs-2610 118 9 transform transform NOUN iajs-2610 118 10 of of ADP iajs-2610 118 11 eq.(5.7 eq.(5.7 NOUN iajs-2610 118 12 ) ) PUNCT iajs-2610 118 13 implies imply VERB iajs-2610 118 14 that that SCONJ iajs-2610 118 15 : : PUNCT iajs-2610 118 16 π•Šβˆ’1{u(x π•Šβˆ’1{u(x PROPN iajs-2610 118 17 , , PUNCT iajs-2610 118 18 s s X iajs-2610 118 19 ) ) PUNCT iajs-2610 118 20 } } PUNCT iajs-2610 118 21 = = PUNCT iajs-2610 118 22 π•Šβˆ’1{f1(x π•Šβˆ’1{f1(x NOUN iajs-2610 118 23 ) ) PUNCT iajs-2610 118 24 + + CCONJ iajs-2610 118 25 sf2(x sf2(x PROPN iajs-2610 118 26 ) ) PUNCT iajs-2610 119 1 + + CCONJ iajs-2610 119 2 s2π•Š[h(x s2π•Š[h(x NOUN iajs-2610 119 3 , , PUNCT iajs-2610 119 4 t t PROPN iajs-2610 119 5 ) ) PUNCT iajs-2610 119 6 ] ] PUNCT iajs-2610 119 7 } } PUNCT iajs-2610 119 8 βˆ’ βˆ’ PUNCT iajs-2610 119 9 π•Šβˆ’1{s2π•Š[r(u(x π•Šβˆ’1{s2π•Š[r(u(x PROPN iajs-2610 119 10 , , PUNCT iajs-2610 119 11 t t PROPN iajs-2610 119 12 ) ) PUNCT iajs-2610 119 13 ) ) PUNCT iajs-2610 119 14 ] ] PUNCT iajs-2610 119 15 } } PUNCT iajs-2610 119 16 βˆ’ βˆ’ PROPN iajs-2610 119 17 π•Šβˆ’1{s2π•Š[n(u(x π•Šβˆ’1{s2π•Š[n(u(x PROPN iajs-2610 119 18 , , PUNCT iajs-2610 119 19 t t PROPN iajs-2610 119 20 ) ) PUNCT iajs-2610 119 21 ) ) PUNCT iajs-2610 119 22 ] ] PUNCT iajs-2610 119 23 } } PUNCT iajs-2610 119 24 . . PUNCT iajs-2610 120 1 ( ( PUNCT iajs-2610 120 2 5.8 5.8 NUM iajs-2610 120 3 ) ) PUNCT iajs-2610 120 4 from from ADP iajs-2610 120 5 eq.(5.8 eq.(5.8 PROPN iajs-2610 120 6 ) ) PUNCT iajs-2610 120 7 , , PUNCT iajs-2610 120 8 we we PRON iajs-2610 120 9 can can AUX iajs-2610 120 10 get get VERB iajs-2610 120 11 u(x u(x NOUN iajs-2610 120 12 , , PUNCT iajs-2610 120 13 t t NOUN iajs-2610 120 14 ) ) PUNCT iajs-2610 121 1 = = SYM iajs-2610 121 2 h(x h(x PROPN iajs-2610 121 3 , , PUNCT iajs-2610 121 4 t t PROPN iajs-2610 121 5 ) ) PUNCT iajs-2610 121 6 βˆ’ βˆ’ PROPN iajs-2610 122 1 π•Šβˆ’1{s2π•Š[r(u(x π•Šβˆ’1{s2π•Š[r(u(x PROPN iajs-2610 122 2 , , PUNCT iajs-2610 122 3 t t PROPN iajs-2610 122 4 ) ) PUNCT iajs-2610 122 5 ) ) PUNCT iajs-2610 123 1 ] ] PUNCT iajs-2610 123 2 } } PUNCT iajs-2610 123 3 βˆ’ βˆ’ PROPN iajs-2610 123 4 π•Šβˆ’1{s2π•Š[n(u(x π•Šβˆ’1{s2π•Š[n(u(x PROPN iajs-2610 123 5 , , PUNCT iajs-2610 123 6 t t PROPN iajs-2610 123 7 ) ) PUNCT iajs-2610 123 8 ) ) PUNCT iajs-2610 123 9 ] ] PUNCT iajs-2610 123 10 } } PUNCT iajs-2610 123 11 . . PUNCT iajs-2610 124 1 ( ( PUNCT iajs-2610 124 2 5.9 5.9 NUM iajs-2610 124 3 ) ) PUNCT iajs-2610 124 4 note note NOUN iajs-2610 124 5 that that SCONJ iajs-2610 124 6 h(x h(x PROPN iajs-2610 124 7 , , PUNCT iajs-2610 124 8 t t PROPN iajs-2610 124 9 ) ) PUNCT iajs-2610 124 10 represents represent VERB iajs-2610 124 11 a a DET iajs-2610 124 12 term term NOUN iajs-2610 124 13 engendering engender VERB iajs-2610 124 14 from from ADP iajs-2610 124 15 source source NOUN iajs-2610 124 16 term term NOUN iajs-2610 124 17 and and CCONJ iajs-2610 124 18 specified specify VERB iajs-2610 124 19 ic ic PROPN iajs-2610 124 20 . . PUNCT iajs-2610 124 21 hence hence ADV iajs-2610 124 22 to to PART iajs-2610 124 23 transact transact VERB iajs-2610 124 24 with with ADP iajs-2610 124 25 the the DET iajs-2610 124 26 new new ADJ iajs-2610 124 27 correction correction NOUN iajs-2610 124 28 function function NOUN iajs-2610 124 29 of of ADP iajs-2610 124 30 the the DET iajs-2610 124 31 sumudu sumudu NOUN iajs-2610 124 32 iterative iterative NOUN iajs-2610 124 33 method method NOUN iajs-2610 124 34 un+1(x un+1(x PROPN iajs-2610 124 35 , , PUNCT iajs-2610 124 36 t t PROPN iajs-2610 124 37 ) ) PUNCT iajs-2610 125 1 = = SYM iajs-2610 125 2 h(x h(x PROPN iajs-2610 125 3 , , PUNCT iajs-2610 125 4 t t PROPN iajs-2610 125 5 ) ) PUNCT iajs-2610 125 6 βˆ’ βˆ’ PROPN iajs-2610 126 1 π•Šβˆ’1{s2π•Š[r(un(x π•Šβˆ’1{s2π•Š[r(un(x PROPN iajs-2610 126 2 , , PUNCT iajs-2610 126 3 t t PROPN iajs-2610 126 4 ) ) PUNCT iajs-2610 126 5 ) ) PUNCT iajs-2610 127 1 ] ] PUNCT iajs-2610 127 2 } } PUNCT iajs-2610 127 3 βˆ’ βˆ’ ADP iajs-2610 127 4 π•Šβˆ’1{s2π•Š[n(un(x π•Šβˆ’1{s2π•Š[n(un(x ADP iajs-2610 127 5 , , PUNCT iajs-2610 127 6 t t PROPN iajs-2610 127 7 ) ) PUNCT iajs-2610 127 8 ) ) PUNCT iajs-2610 128 1 ] ] PUNCT iajs-2610 128 2 } } PUNCT iajs-2610 128 3 , , PUNCT iajs-2610 128 4 n n X iajs-2610 128 5 β‰₯ β‰₯ NOUN iajs-2610 128 6 0 0 NUM iajs-2610 128 7 . . PUNCT iajs-2610 129 1 ( ( PUNCT iajs-2610 129 2 5.10 5.10 NUM iajs-2610 129 3 ) ) PUNCT iajs-2610 129 4 to to PART iajs-2610 129 5 get get VERB iajs-2610 129 6 out out ADP iajs-2610 129 7 how how SCONJ iajs-2610 129 8 this this DET iajs-2610 129 9 method method NOUN iajs-2610 129 10 works work VERB iajs-2610 129 11 , , PUNCT iajs-2610 129 12 we we PRON iajs-2610 129 13 follow follow VERB iajs-2610 129 14 next next ADJ iajs-2610 129 15 steps step NOUN iajs-2610 129 16 as as SCONJ iajs-2610 129 17 follows follow VERB iajs-2610 129 18 : : PUNCT iajs-2610 129 19 step step NOUN iajs-2610 129 20 1 1 NUM iajs-2610 129 21 : : SYM iajs-2610 129 22 27 27 NUM iajs-2610 129 23 ibn ibn PROPN iajs-2610 129 24 al al PROPN iajs-2610 129 25 - - PUNCT iajs-2610 129 26 haitham haitham PROPN iajs-2610 129 27 jour jour X iajs-2610 129 28 . . PROPN iajs-2610 130 1 for for ADP iajs-2610 130 2 pure pure ADJ iajs-2610 130 3 & & CCONJ iajs-2610 130 4 appl appl PROPN iajs-2610 130 5 . . PUNCT iajs-2610 131 1 sci sci PROPN iajs-2610 131 2 . . PROPN iajs-2610 132 1 34 34 NUM iajs-2610 132 2 ( ( PUNCT iajs-2610 132 3 2 2 NUM iajs-2610 132 4 ) ) PUNCT iajs-2610 132 5 2021 2021 NUM iajs-2610 132 6 u0(x u0(x SYM iajs-2610 132 7 , , PUNCT iajs-2610 132 8 t t PROPN iajs-2610 132 9 ) ) PUNCT iajs-2610 132 10 = = SYM iajs-2610 133 1 h(x h(x PROPN iajs-2610 133 2 , , PUNCT iajs-2610 133 3 t t PROPN iajs-2610 133 4 ) ) PUNCT iajs-2610 133 5 step step NOUN iajs-2610 133 6 2 2 NUM iajs-2610 133 7 : : PUNCT iajs-2610 133 8 the the DET iajs-2610 133 9 next next ADJ iajs-2610 133 10 iteration iteration NOUN iajs-2610 133 11 is be AUX iajs-2610 133 12 𝑒1(π‘₯ 𝑒1(π‘₯ NOUN iajs-2610 133 13 , , PUNCT iajs-2610 133 14 𝑑 𝑑 NOUN iajs-2610 133 15 ) ) PUNCT iajs-2610 134 1 = = SYM iajs-2610 134 2 h(x h(x PROPN iajs-2610 134 3 , , PUNCT iajs-2610 134 4 t t PROPN iajs-2610 134 5 ) ) PUNCT iajs-2610 134 6 βˆ’ βˆ’ PROPN iajs-2610 135 1 π•Šβˆ’1{s2π•Š[r(𝑒0(x π•Šβˆ’1{s2π•Š[r(𝑒0(x PROPN iajs-2610 135 2 , , PUNCT iajs-2610 135 3 t t PROPN iajs-2610 135 4 ) ) PUNCT iajs-2610 135 5 ) ) PUNCT iajs-2610 135 6 ] ] PUNCT iajs-2610 135 7 } } PUNCT iajs-2610 135 8 βˆ’ βˆ’ PROPN iajs-2610 135 9 π•Šβˆ’1{s2π•Š[n(𝑒0(x π•Šβˆ’1{s2π•Š[n(𝑒0(x PROPN iajs-2610 135 10 , , PUNCT iajs-2610 135 11 t t PROPN iajs-2610 135 12 ) ) PUNCT iajs-2610 135 13 ) ) PUNCT iajs-2610 135 14 ] ] PUNCT iajs-2610 135 15 } } PUNCT iajs-2610 135 16 . . PUNCT iajs-2610 136 1 ( ( PUNCT iajs-2610 136 2 5.11 5.11 NUM iajs-2610 136 3 ) ) PUNCT iajs-2610 136 4 step step NOUN iajs-2610 136 5 ( ( PUNCT iajs-2610 136 6 3 3 NUM iajs-2610 136 7 ) ) PUNCT iajs-2610 136 8 : : PUNCT iajs-2610 136 9 after after ADP iajs-2610 136 10 several several ADJ iajs-2610 136 11 simple simple ADJ iajs-2610 136 12 sumudu sumudu NOUN iajs-2610 136 13 iterative iterative NOUN iajs-2610 136 14 steps step NOUN iajs-2610 136 15 of of ADP iajs-2610 136 16 the the DET iajs-2610 136 17 solution solution NOUN iajs-2610 136 18 , , PUNCT iajs-2610 136 19 the the DET iajs-2610 136 20 general general ADJ iajs-2610 136 21 form form NOUN iajs-2610 136 22 of of ADP iajs-2610 136 23 this this DET iajs-2610 136 24 equation equation NOUN iajs-2610 136 25 which which PRON iajs-2610 136 26 is be AUX iajs-2610 136 27 𝑒𝑛+1(π‘₯ 𝑒𝑛+1(π‘₯ NUM iajs-2610 136 28 , , PUNCT iajs-2610 136 29 𝑑 𝑑 NOUN iajs-2610 136 30 ) ) PUNCT iajs-2610 136 31 = = SYM iajs-2610 137 1 h(x h(x PROPN iajs-2610 137 2 , , PUNCT iajs-2610 137 3 t t PROPN iajs-2610 137 4 ) ) PUNCT iajs-2610 137 5 βˆ’ βˆ’ PUNCT iajs-2610 137 6 π•Šβˆ’1{s2π•Š[r(𝑒𝑛(x π•Šβˆ’1{s2π•Š[r(𝑒𝑛(x NOUN iajs-2610 137 7 , , PUNCT iajs-2610 137 8 t t PROPN iajs-2610 137 9 ) ) PUNCT iajs-2610 137 10 ) ) PUNCT iajs-2610 137 11 ] ] PUNCT iajs-2610 137 12 } } PUNCT iajs-2610 137 13 βˆ’ βˆ’ ADP iajs-2610 137 14 π•Šβˆ’1{s2π•Š[n(𝑒𝑛(x π•Šβˆ’1{s2π•Š[n(𝑒𝑛(x PROPN iajs-2610 137 15 , , PUNCT iajs-2610 137 16 t t PROPN iajs-2610 137 17 ) ) PUNCT iajs-2610 137 18 ) ) PUNCT iajs-2610 137 19 ] ] PUNCT iajs-2610 137 20 } } PUNCT iajs-2610 137 21 , , PUNCT iajs-2610 137 22 𝑛 𝑛 DET iajs-2610 137 23 β‰₯ β‰₯ NOUN iajs-2610 137 24 0.(5.12 0.(5.12 NOUN iajs-2610 137 25 ) ) PUNCT iajs-2610 137 26 evidently evidently ADV iajs-2610 137 27 , , PUNCT iajs-2610 137 28 each each DET iajs-2610 137 29 iteration iteration NOUN iajs-2610 137 30 of of ADP iajs-2610 137 31 the the DET iajs-2610 137 32 function function NOUN iajs-2610 137 33 ( ( PUNCT iajs-2610 137 34 𝑒𝑛+1(π‘₯ 𝑒𝑛+1(π‘₯ NOUN iajs-2610 137 35 , , PUNCT iajs-2610 137 36 𝑑 𝑑 NOUN iajs-2610 137 37 ) ) PUNCT iajs-2610 137 38 ) ) PUNCT iajs-2610 137 39 represents represent VERB iajs-2610 137 40 effectively effectively ADV iajs-2610 137 41 the the DET iajs-2610 137 42 only only ADJ iajs-2610 137 43 solution solution NOUN iajs-2610 137 44 for for ADP iajs-2610 137 45 eq eq PROPN iajs-2610 137 46 . . PUNCT iajs-2610 138 1 ( ( PUNCT iajs-2610 138 2 5.12 5.12 NUM iajs-2610 138 3 ) ) PUNCT iajs-2610 138 4 . . PUNCT iajs-2610 139 1 in in ADP iajs-2610 139 2 the the DET iajs-2610 139 3 above above ADJ iajs-2610 139 4 steps step NOUN iajs-2610 139 5 , , PUNCT iajs-2610 139 6 this this DET iajs-2610 139 7 method method NOUN iajs-2610 139 8 has have VERB iajs-2610 139 9 the the DET iajs-2610 139 10 merit merit NOUN iajs-2610 139 11 that that PRON iajs-2610 139 12 solution solution NOUN iajs-2610 139 13 can can AUX iajs-2610 139 14 be be AUX iajs-2610 139 15 found find VERB iajs-2610 139 16 in in ADP iajs-2610 139 17 the the DET iajs-2610 139 18 first first ADJ iajs-2610 139 19 steps step NOUN iajs-2610 139 20 manually manually ADV iajs-2610 139 21 and and CCONJ iajs-2610 139 22 easily easily ADV iajs-2610 139 23 . . PUNCT iajs-2610 140 1 6.illustrative 6.illustrative NUM iajs-2610 140 2 examples example NOUN iajs-2610 140 3 in in ADP iajs-2610 140 4 this this DET iajs-2610 140 5 part part NOUN iajs-2610 140 6 , , PUNCT iajs-2610 140 7 we we PRON iajs-2610 140 8 implement implement VERB iajs-2610 140 9 four four NUM iajs-2610 140 10 examples example NOUN iajs-2610 140 11 and and CCONJ iajs-2610 140 12 then then ADV iajs-2610 140 13 compare compare VERB iajs-2610 140 14 our our PRON iajs-2610 140 15 solutions solution NOUN iajs-2610 140 16 to to ADP iajs-2610 140 17 the the DET iajs-2610 140 18 existent existent ADJ iajs-2610 140 19 exact exact ADJ iajs-2610 140 20 solutions solution NOUN iajs-2610 140 21 . . PUNCT iajs-2610 141 1 example example NOUN iajs-2610 141 2 6.1 6.1 NUM iajs-2610 141 3 solve solve NOUN iajs-2610 141 4 the the DET iajs-2610 141 5 inhomogeneous inhomogeneous ADJ iajs-2610 141 6 advection advection NOUN iajs-2610 141 7 problem problem NOUN iajs-2610 141 8 : : PUNCT iajs-2610 141 9 ut ut PROPN iajs-2610 142 1 + + CCONJ iajs-2610 142 2 1 1 NUM iajs-2610 142 3 2 2 NUM iajs-2610 142 4 ( ( PUNCT iajs-2610 142 5 u2)x u2)x NOUN iajs-2610 142 6 = = PUNCT iajs-2610 142 7 x x PROPN iajs-2610 142 8 . . PUNCT iajs-2610 143 1 ( ( PUNCT iajs-2610 143 2 6.1 6.1 NUM iajs-2610 143 3 ) ) PUNCT iajs-2610 143 4 submit submit NOUN iajs-2610 143 5 to to ADP iajs-2610 143 6 the the DET iajs-2610 143 7 ic ic PROPN iajs-2610 143 8 u(x u(x PROPN iajs-2610 143 9 , , PUNCT iajs-2610 143 10 0 0 NUM iajs-2610 143 11 ) ) PUNCT iajs-2610 143 12 = = SYM iajs-2610 144 1 2 2 X iajs-2610 144 2 . . PUNCT iajs-2610 144 3 ( ( PUNCT iajs-2610 144 4 6.2 6.2 NUM iajs-2610 144 5 ) ) PUNCT iajs-2610 144 6 taking take VERB iajs-2610 144 7 sumudu sumudu NOUN iajs-2610 144 8 transform transform NOUN iajs-2610 144 9 of of ADP iajs-2610 144 10 eq eq NOUN iajs-2610 144 11 . . PUNCT iajs-2610 145 1 ( ( PUNCT iajs-2610 145 2 6.1),we 6.1),we NUM iajs-2610 145 3 have have VERB iajs-2610 145 4 : : PUNCT iajs-2610 145 5 π•Š[ut π•Š[ut X iajs-2610 145 6 ] ] X iajs-2610 146 1 + + PROPN iajs-2610 146 2 π•Š π•Š PROPN iajs-2610 146 3 [ [ PUNCT iajs-2610 146 4 1 1 NUM iajs-2610 146 5 2 2 NUM iajs-2610 146 6 ( ( PUNCT iajs-2610 146 7 u2)x]= u2)x]= NOUN iajs-2610 146 8 π•Š π•Š PROPN iajs-2610 146 9 [ [ PUNCT iajs-2610 146 10 x x NOUN iajs-2610 146 11 ] ] X iajs-2610 146 12 . . PUNCT iajs-2610 146 13 utilizing utilize VERB iajs-2610 146 14 the the DET iajs-2610 146 15 properties property NOUN iajs-2610 146 16 in in ADP iajs-2610 146 17 table table NOUN iajs-2610 146 18 1,2 1,2 NUM iajs-2610 146 19 and and CCONJ iajs-2610 146 20 the the DET iajs-2610 146 21 properties property NOUN iajs-2610 146 22 mentioned mention VERB iajs-2610 146 23 above above ADV iajs-2610 146 24 , , PUNCT iajs-2610 146 25 we we PRON iajs-2610 146 26 get get VERB iajs-2610 146 27 : : PUNCT iajs-2610 146 28 u(x u(x NOUN iajs-2610 146 29 , , PUNCT iajs-2610 146 30 s s NOUN iajs-2610 146 31 ) ) PUNCT iajs-2610 146 32 s s PART iajs-2610 146 33 βˆ’ βˆ’ PROPN iajs-2610 146 34 u(x,0 u(x,0 PROPN iajs-2610 146 35 ) ) PUNCT iajs-2610 146 36 s s PART iajs-2610 147 1 = = NOUN iajs-2610 147 2 x x SYM iajs-2610 147 3 βˆ’ βˆ’ NUM iajs-2610 147 4 1 1 NUM iajs-2610 147 5 2 2 NUM iajs-2610 147 6 π•Š[(u2)x π•Š[(u2)x ADJ iajs-2610 147 7 ] ] PUNCT iajs-2610 147 8 . . PUNCT iajs-2610 148 1 ( ( PUNCT iajs-2610 148 2 6.3 6.3 NUM iajs-2610 148 3 ) ) PUNCT iajs-2610 148 4 replace replace NOUN iajs-2610 148 5 eq eq NOUN iajs-2610 148 6 . . PUNCT iajs-2610 149 1 ( ( PUNCT iajs-2610 149 2 6.2 6.2 NUM iajs-2610 149 3 ) ) PUNCT iajs-2610 149 4 into into ADP iajs-2610 149 5 eq eq NOUN iajs-2610 149 6 . . PUNCT iajs-2610 150 1 ( ( PUNCT iajs-2610 150 2 6.3 6.3 NUM iajs-2610 150 3 ) ) PUNCT iajs-2610 150 4 , , PUNCT iajs-2610 150 5 we we PRON iajs-2610 150 6 get get VERB iajs-2610 150 7 : : PUNCT iajs-2610 150 8 u(x u(x NOUN iajs-2610 150 9 , , PUNCT iajs-2610 150 10 s s NOUN iajs-2610 150 11 ) ) PUNCT iajs-2610 150 12 = = SYM iajs-2610 150 13 2 2 NUM iajs-2610 150 14 + + NUM iajs-2610 150 15 xs xs NOUN iajs-2610 150 16 βˆ’ βˆ’ PROPN iajs-2610 150 17 s s PART iajs-2610 150 18 2 2 NUM iajs-2610 150 19 π•Š[(u2)x π•Š[(u2)x NOUN iajs-2610 150 20 ] ] PUNCT iajs-2610 150 21 . . PUNCT iajs-2610 151 1 ( ( PUNCT iajs-2610 151 2 6.4 6.4 NUM iajs-2610 151 3 ) ) PUNCT iajs-2610 151 4 subsequently subsequently ADV iajs-2610 151 5 , , PUNCT iajs-2610 151 6 taking take VERB iajs-2610 151 7 inverse inverse NOUN iajs-2610 151 8 sumudu sumudu NOUN iajs-2610 151 9 transform transform NOUN iajs-2610 151 10 of of ADP iajs-2610 151 11 eq eq NOUN iajs-2610 151 12 . . PUNCT iajs-2610 151 13 ( ( PUNCT iajs-2610 151 14 6.4 6.4 NUM iajs-2610 151 15 ) ) PUNCT iajs-2610 151 16 implies imply VERB iajs-2610 151 17 that that SCONJ iajs-2610 151 18 : : PUNCT iajs-2610 151 19 π•Šβˆ’1[u(x π•Šβˆ’1[u(x NOUN iajs-2610 151 20 , , PUNCT iajs-2610 151 21 s s NOUN iajs-2610 151 22 ) ) PUNCT iajs-2610 151 23 ] ] PUNCT iajs-2610 152 1 = = SYM iajs-2610 152 2 2 2 NUM iajs-2610 152 3 + + CCONJ iajs-2610 152 4 xπ•Šβˆ’1[s xπ•Šβˆ’1[s PROPN iajs-2610 152 5 ] ] X iajs-2610 152 6 βˆ’ βˆ’ PROPN iajs-2610 153 1 π•Šβˆ’1 π•Šβˆ’1 PROPN iajs-2610 153 2 { { PUNCT iajs-2610 153 3 s s NOUN iajs-2610 153 4 2 2 NUM iajs-2610 153 5 π•Š[(u2)x π•Š[(u2)x ADJ iajs-2610 153 6 ] ] PUNCT iajs-2610 153 7 } } PUNCT iajs-2610 153 8 . . PUNCT iajs-2610 154 1 ( ( PUNCT iajs-2610 154 2 6.5 6.5 NUM iajs-2610 154 3 ) ) PUNCT iajs-2610 154 4 table table NOUN iajs-2610 154 5 1and 1and PROPN iajs-2610 154 6 2 2 NUM iajs-2610 154 7 , , PUNCT iajs-2610 154 8 eq.(6.5 eq.(6.5 NOUN iajs-2610 154 9 ) ) PUNCT iajs-2610 154 10 becomes become VERB iajs-2610 154 11 u(x u(x NOUN iajs-2610 154 12 , , PUNCT iajs-2610 154 13 t t NOUN iajs-2610 154 14 ) ) PUNCT iajs-2610 154 15 = = SYM iajs-2610 155 1 2 2 NUM iajs-2610 155 2 + + NUM iajs-2610 155 3 xt xt X iajs-2610 155 4 – – PUNCT iajs-2610 155 5 π•Šβˆ’1 π•Šβˆ’1 NOUN iajs-2610 155 6 { { PUNCT iajs-2610 155 7 s s NOUN iajs-2610 155 8 2 2 NUM iajs-2610 155 9 π•Š[(u2)x π•Š[(u2)x ADJ iajs-2610 155 10 ] ] PUNCT iajs-2610 155 11 } } PUNCT iajs-2610 155 12 . . PUNCT iajs-2610 156 1 ( ( PUNCT iajs-2610 156 2 6.6 6.6 NUM iajs-2610 156 3 ) ) PUNCT iajs-2610 156 4 by by ADP iajs-2610 156 5 the the DET iajs-2610 156 6 new new ADJ iajs-2610 156 7 correction correction NOUN iajs-2610 156 8 function function VERB iajs-2610 156 9 from from ADP iajs-2610 156 10 eq.(6.6 eq.(6.6 PROPN iajs-2610 156 11 ) ) PUNCT iajs-2610 156 12 , , PUNCT iajs-2610 156 13 we we PRON iajs-2610 156 14 can can AUX iajs-2610 156 15 get get VERB iajs-2610 156 16 : : PUNCT iajs-2610 156 17 un+1(x un+1(x ADJ iajs-2610 156 18 , , PUNCT iajs-2610 156 19 t t PROPN iajs-2610 156 20 ) ) PUNCT iajs-2610 156 21 = = SYM iajs-2610 157 1 2 2 NUM iajs-2610 157 2 + + NUM iajs-2610 157 3 xt xt X iajs-2610 157 4 βˆ’π•Šβˆ’1 βˆ’π•Šβˆ’1 PROPN iajs-2610 157 5 { { PUNCT iajs-2610 157 6 s s NOUN iajs-2610 157 7 2 2 NUM iajs-2610 157 8 π•Š[(un π•Š[(un NOUN iajs-2610 157 9 2)x 2)x NUM iajs-2610 157 10 ] ] PUNCT iajs-2610 157 11 } } PUNCT iajs-2610 157 12 , , PUNCT iajs-2610 157 13 n n X iajs-2610 157 14 β‰₯ β‰₯ NOUN iajs-2610 157 15 0 0 NUM iajs-2610 157 16 . . PUNCT iajs-2610 158 1 ( ( PUNCT iajs-2610 158 2 6.7 6.7 NUM iajs-2610 158 3 ) ) PUNCT iajs-2610 158 4 now now ADV iajs-2610 158 5 , , PUNCT iajs-2610 158 6 we we PRON iajs-2610 158 7 apply apply VERB iajs-2610 158 8 the the DET iajs-2610 158 9 new new ADJ iajs-2610 158 10 sumudu sumudu NOUN iajs-2610 158 11 iterative iterative NOUN iajs-2610 158 12 method method NOUN iajs-2610 158 13 . . PUNCT iajs-2610 159 1 28 28 NUM iajs-2610 160 1 ibn ibn PROPN iajs-2610 160 2 al al PROPN iajs-2610 160 3 - - PUNCT iajs-2610 160 4 haitham haitham PROPN iajs-2610 160 5 jour jour X iajs-2610 160 6 . . PROPN iajs-2610 160 7 for for ADP iajs-2610 160 8 pure pure ADJ iajs-2610 160 9 & & CCONJ iajs-2610 160 10 appl appl PROPN iajs-2610 160 11 . . PUNCT iajs-2610 161 1 sci sci PROPN iajs-2610 161 2 . . PROPN iajs-2610 162 1 34 34 NUM iajs-2610 162 2 ( ( PUNCT iajs-2610 162 3 2 2 NUM iajs-2610 162 4 ) ) PUNCT iajs-2610 162 5 2021 2021 NUM iajs-2610 162 6 so so ADV iajs-2610 162 7 from from ADP iajs-2610 162 8 eq.(6.7 eq.(6.7 PROPN iajs-2610 162 9 ) ) PUNCT iajs-2610 162 10 , , PUNCT iajs-2610 162 11 we we PRON iajs-2610 162 12 conclude conclude VERB iajs-2610 162 13 u0(x u0(x PRON iajs-2610 162 14 , , PUNCT iajs-2610 162 15 t t PROPN iajs-2610 162 16 ) ) PUNCT iajs-2610 162 17 = = SYM iajs-2610 162 18 2 2 NUM iajs-2610 162 19 + + CCONJ iajs-2610 162 20 xt xt X iajs-2610 162 21 . . PUNCT iajs-2610 163 1 𝑒1(π‘₯ 𝑒1(π‘₯ PROPN iajs-2610 163 2 , , PUNCT iajs-2610 163 3 𝑑 𝑑 X iajs-2610 163 4 ) ) PUNCT iajs-2610 163 5 = = SYM iajs-2610 163 6 2 2 NUM iajs-2610 163 7 + + CCONJ iajs-2610 163 8 π‘₯𝑑 π‘₯𝑑 ADV iajs-2610 163 9 βˆ’π•Šβˆ’1 βˆ’π•Šβˆ’1 ADJ iajs-2610 163 10 { { PUNCT iajs-2610 163 11 𝑠 𝑠 PROPN iajs-2610 163 12 2 2 NUM iajs-2610 163 13 π•Š[(𝑒0 π•Š[(𝑒0 PROPN iajs-2610 163 14 2)π‘₯ 2)π‘₯ NOUN iajs-2610 163 15 ] ] PUNCT iajs-2610 163 16 } } PUNCT iajs-2610 163 17 . . PUNCT iajs-2610 164 1 𝑒1(π‘₯ 𝑒1(π‘₯ PROPN iajs-2610 164 2 , , PUNCT iajs-2610 164 3 𝑑 𝑑 X iajs-2610 164 4 ) ) PUNCT iajs-2610 164 5 = = SYM iajs-2610 164 6 2 2 NUM iajs-2610 164 7 + + CCONJ iajs-2610 164 8 π‘₯𝑑 π‘₯𝑑 ADP iajs-2610 164 9 βˆ’ βˆ’ PROPN iajs-2610 165 1 𝑑2 𝑑2 ADJ iajs-2610 165 2 βˆ’ βˆ’ PROPN iajs-2610 165 3 1 1 NUM iajs-2610 165 4 3 3 NUM iajs-2610 165 5 π‘₯𝑑3 π‘₯𝑑3 NOUN iajs-2610 165 6 . . PUNCT iajs-2610 166 1 and and CCONJ iajs-2610 166 2 u2(x u2(x PROPN iajs-2610 166 3 , , PUNCT iajs-2610 166 4 t t PROPN iajs-2610 166 5 ) ) PUNCT iajs-2610 166 6 = = SYM iajs-2610 167 1 2 2 NUM iajs-2610 167 2 + + NUM iajs-2610 167 3 xt xt X iajs-2610 167 4 – – PUNCT iajs-2610 167 5 π•Šβˆ’1 π•Šβˆ’1 NOUN iajs-2610 167 6 { { PUNCT iajs-2610 167 7 s s NOUN iajs-2610 167 8 2 2 NUM iajs-2610 167 9 π•Š[(u1 π•Š[(u1 NOUN iajs-2610 167 10 2)x 2)x NUM iajs-2610 167 11 ] ] PUNCT iajs-2610 167 12 } } PUNCT iajs-2610 167 13 , , PUNCT iajs-2610 167 14 n n X iajs-2610 167 15 β‰₯ β‰₯ NOUN iajs-2610 167 16 1 1 NUM iajs-2610 167 17 . . PUNCT iajs-2610 168 1 u2(x u2(x PROPN iajs-2610 168 2 , , PUNCT iajs-2610 168 3 t t PROPN iajs-2610 168 4 ) ) PUNCT iajs-2610 168 5 = = SYM iajs-2610 169 1 2 2 NUM iajs-2610 169 2 + + NUM iajs-2610 169 3 xt xt ADP iajs-2610 169 4 βˆ’ βˆ’ PROPN iajs-2610 169 5 t2 t2 NOUN iajs-2610 169 6 βˆ’ βˆ’ PROPN iajs-2610 169 7 1 1 NUM iajs-2610 169 8 3 3 NUM iajs-2610 169 9 xt3 xt3 NOUN iajs-2610 170 1 + + CCONJ iajs-2610 170 2 5 5 NUM iajs-2610 170 3 12 12 NUM iajs-2610 170 4 t4 t4 PROPN iajs-2610 170 5 + + CCONJ iajs-2610 170 6 2 2 NUM iajs-2610 170 7 15 15 NUM iajs-2610 170 8 xt5 xt5 NOUN iajs-2610 170 9 βˆ’ βˆ’ NUM iajs-2610 170 10 1 1 NUM iajs-2610 170 11 63 63 NUM iajs-2610 170 12 xt7 xt7 NOUN iajs-2610 170 13 ) ) PUNCT iajs-2610 170 14 . . PUNCT iajs-2610 171 1 therefore therefore ADV iajs-2610 171 2 , , PUNCT iajs-2610 171 3 we we PRON iajs-2610 171 4 persist persist VERB iajs-2610 171 5 in in ADP iajs-2610 171 6 this this DET iajs-2610 171 7 way way NOUN iajs-2610 171 8 to to PART iajs-2610 171 9 obtain obtain VERB iajs-2610 171 10 a a DET iajs-2610 171 11 generic generic ADJ iajs-2610 171 12 recursive recursive ADJ iajs-2610 171 13 relation relation NOUN iajs-2610 171 14 via via ADP iajs-2610 171 15 canceling cancel VERB iajs-2610 171 16 a a DET iajs-2610 171 17 noise noise NOUN iajs-2610 171 18 terms term NOUN iajs-2610 171 19 and and CCONJ iajs-2610 171 20 we we PRON iajs-2610 171 21 get get VERB iajs-2610 171 22 : : PUNCT iajs-2610 171 23 un+1(x un+1(x ADJ iajs-2610 171 24 , , PUNCT iajs-2610 171 25 t t PROPN iajs-2610 171 26 ) ) PUNCT iajs-2610 171 27 = = SYM iajs-2610 171 28 2 2 NUM iajs-2610 171 29 ( ( PUNCT iajs-2610 171 30 1 1 NUM iajs-2610 171 31 βˆ’ βˆ’ NUM iajs-2610 171 32 1 1 NUM iajs-2610 171 33 2 2 NUM iajs-2610 171 34 ! ! PUNCT iajs-2610 172 1 t2 t2 NOUN iajs-2610 172 2 + + CCONJ iajs-2610 172 3 5 5 NUM iajs-2610 172 4 4 4 NUM iajs-2610 172 5 ! ! PUNCT iajs-2610 172 6 t4 t4 PROPN iajs-2610 172 7 … … PUNCT iajs-2610 172 8 ) ) PUNCT iajs-2610 173 1 + + CCONJ iajs-2610 173 2 x x X iajs-2610 173 3 ( ( PUNCT iajs-2610 173 4 t t NOUN iajs-2610 173 5 βˆ’ βˆ’ PROPN iajs-2610 173 6 1 1 NUM iajs-2610 173 7 3 3 NUM iajs-2610 173 8 t3 t3 NOUN iajs-2610 173 9 + + CCONJ iajs-2610 173 10 2 2 NUM iajs-2610 173 11 15 15 NUM iajs-2610 173 12 t5 t5 PROPN iajs-2610 173 13 βˆ’ βˆ’ PROPN iajs-2610 173 14 1 1 NUM iajs-2610 173 15 63 63 NUM iajs-2610 173 16 t7 t7 PROPN iajs-2610 173 17 + + CCONJ iajs-2610 173 18 β‹― β‹― PROPN iajs-2610 173 19 … … PUNCT iajs-2610 173 20 ) ) PUNCT iajs-2610 173 21 hence hence ADV iajs-2610 173 22 , , PUNCT iajs-2610 173 23 we we PRON iajs-2610 173 24 can can AUX iajs-2610 173 25 conclude conclude VERB iajs-2610 173 26 the the DET iajs-2610 173 27 exact exact ADJ iajs-2610 173 28 solution solution NOUN iajs-2610 173 29 as as SCONJ iajs-2610 173 30 follows follow VERB iajs-2610 173 31 u(x u(x NOUN iajs-2610 173 32 , , PUNCT iajs-2610 173 33 t t NOUN iajs-2610 173 34 ) ) PUNCT iajs-2610 173 35 = = PUNCT iajs-2610 174 1 2secht 2secht NUM iajs-2610 174 2 + + CCONJ iajs-2610 174 3 x x PUNCT iajs-2610 174 4 tanht tanht NOUN iajs-2610 174 5 . . PUNCT iajs-2610 175 1 example example NOUN iajs-2610 175 2 6.2 6.2 NUM iajs-2610 175 3 . . PUNCT iajs-2610 176 1 solve solve VERB iajs-2610 176 2 the the DET iajs-2610 176 3 burgers burger NOUN iajs-2610 176 4 equation equation NOUN iajs-2610 176 5 of of ADP iajs-2610 176 6 the the DET iajs-2610 176 7 form form NOUN iajs-2610 176 8 : : PUNCT iajs-2610 176 9 ut ut PROPN iajs-2610 177 1 + + PROPN iajs-2610 177 2 uux uux PROPN iajs-2610 177 3 = = PROPN iajs-2610 177 4 uxx uxx PROPN iajs-2610 177 5 , , PUNCT iajs-2610 177 6 with with ADP iajs-2610 177 7 the the DET iajs-2610 177 8 ic ic PROPN iajs-2610 177 9 u(x u(x PROPN iajs-2610 177 10 , , PUNCT iajs-2610 177 11 0 0 NUM iajs-2610 177 12 ) ) PUNCT iajs-2610 177 13 = = SYM iajs-2610 178 1 2tanx 2tanx X iajs-2610 178 2 . . PUNCT iajs-2610 178 3 ( ( PUNCT iajs-2610 178 4 6.8 6.8 X iajs-2610 178 5 ) ) PUNCT iajs-2610 178 6 taking take VERB iajs-2610 178 7 sumudu sumudu NOUN iajs-2610 178 8 transform transform NOUN iajs-2610 178 9 of of ADP iajs-2610 178 10 eq.(6.8 eq.(6.8 NOUN iajs-2610 178 11 ) ) PUNCT iajs-2610 178 12 , , PUNCT iajs-2610 178 13 we we PRON iajs-2610 178 14 have have VERB iajs-2610 178 15 : : PUNCT iajs-2610 178 16 π•Š[𝑒𝑑 π•Š[𝑒𝑑 PROPN iajs-2610 178 17 ] ] PUNCT iajs-2610 179 1 + + CCONJ iajs-2610 179 2 π•Š[𝑒𝑒π‘₯]= π•Š[𝑒𝑒π‘₯]= X iajs-2610 179 3 π•Š[𝑒π‘₯π‘₯ π•Š[𝑒π‘₯π‘₯ NOUN iajs-2610 179 4 ] ] PUNCT iajs-2610 179 5 . . PUNCT iajs-2610 180 1 using use VERB iajs-2610 180 2 the the DET iajs-2610 180 3 properties property NOUN iajs-2610 180 4 in in ADP iajs-2610 180 5 table table NOUN iajs-2610 180 6 1 1 NUM iajs-2610 180 7 , , PUNCT iajs-2610 180 8 2 2 NUM iajs-2610 180 9 and and CCONJ iajs-2610 180 10 submitting submit VERB iajs-2610 180 11 to to ADP iajs-2610 180 12 the the DET iajs-2610 180 13 ic ic PROPN iajs-2610 180 14 above above ADV iajs-2610 180 15 to to PART iajs-2610 180 16 obtain obtain VERB iajs-2610 180 17 : : PUNCT iajs-2610 180 18 π‘ˆ(π‘₯,𝑠 π‘ˆ(π‘₯,𝑠 NUM iajs-2610 180 19 ) ) PUNCT iajs-2610 181 1 𝑠 𝑠 ADP iajs-2610 181 2 βˆ’ βˆ’ NOUN iajs-2610 182 1 2tanx 2tanx NUM iajs-2610 182 2 𝑠 𝑠 PROPN iajs-2610 182 3 = = SYM iajs-2610 182 4 π•Š[𝑒π‘₯π‘₯ π•Š[𝑒π‘₯π‘₯ PROPN iajs-2610 182 5 βˆ’ βˆ’ PROPN iajs-2610 182 6 𝑒𝑒π‘₯ 𝑒𝑒π‘₯ PROPN iajs-2610 182 7 ] ] PUNCT iajs-2610 182 8 . . PUNCT iajs-2610 183 1 ( ( PUNCT iajs-2610 183 2 6.9 6.9 NUM iajs-2610 183 3 ) ) PUNCT iajs-2610 183 4 subsequently subsequently ADV iajs-2610 183 5 , , PUNCT iajs-2610 183 6 taking take VERB iajs-2610 183 7 inverse inverse NOUN iajs-2610 183 8 sumudu sumudu NOUN iajs-2610 183 9 transform transform NOUN iajs-2610 183 10 of of ADP iajs-2610 183 11 eq.(6.9 eq.(6.9 NOUN iajs-2610 183 12 ) ) PUNCT iajs-2610 183 13 to to PART iajs-2610 183 14 get get VERB iajs-2610 183 15 : : PUNCT iajs-2610 183 16 π•Šβˆ’1[u(x π•Šβˆ’1[u(x NOUN iajs-2610 183 17 , , PUNCT iajs-2610 183 18 s s NOUN iajs-2610 183 19 ) ) PUNCT iajs-2610 183 20 ] ] PUNCT iajs-2610 184 1 = = PUNCT iajs-2610 185 1 2tanx 2tanx NUM iajs-2610 185 2 + + CCONJ iajs-2610 185 3 π•Šβˆ’1{s π•Šβˆ’1{s PROPN iajs-2610 185 4 π•Š[uxx π•Š[uxx PROPN iajs-2610 185 5 βˆ’ βˆ’ PROPN iajs-2610 185 6 uux uux PROPN iajs-2610 185 7 ] ] PUNCT iajs-2610 185 8 } } PUNCT iajs-2610 185 9 . . PUNCT iajs-2610 186 1 ( ( PUNCT iajs-2610 186 2 6.10 6.10 NUM iajs-2610 186 3 ) ) PUNCT iajs-2610 186 4 table table NOUN iajs-2610 186 5 1and 1and PROPN iajs-2610 186 6 2 2 NUM iajs-2610 186 7 , , PUNCT iajs-2610 186 8 eq.(6.10 eq.(6.10 PROPN iajs-2610 186 9 ) ) PUNCT iajs-2610 186 10 becomes become VERB iajs-2610 186 11 u(x u(x NOUN iajs-2610 186 12 , , PUNCT iajs-2610 186 13 t t NOUN iajs-2610 186 14 ) ) PUNCT iajs-2610 186 15 = = PUNCT iajs-2610 187 1 2tanx 2tanx NUM iajs-2610 187 2 + + CCONJ iajs-2610 187 3 π•Šβˆ’1{s π•Šβˆ’1{s PROPN iajs-2610 187 4 π•Š[uxx π•Š[uxx PROPN iajs-2610 187 5 βˆ’ βˆ’ PROPN iajs-2610 187 6 uux uux PROPN iajs-2610 187 7 ] ] PUNCT iajs-2610 187 8 } } PUNCT iajs-2610 187 9 . . PUNCT iajs-2610 188 1 ( ( PUNCT iajs-2610 188 2 6.11 6.11 NUM iajs-2610 188 3 ) ) PUNCT iajs-2610 188 4 thus thus ADV iajs-2610 188 5 , , PUNCT iajs-2610 188 6 we we PRON iajs-2610 188 7 persist persist VERB iajs-2610 188 8 in in ADP iajs-2610 188 9 this this DET iajs-2610 188 10 manner manner NOUN iajs-2610 188 11 to to PART iajs-2610 188 12 obtain obtain VERB iajs-2610 188 13 a a DET iajs-2610 188 14 general general ADJ iajs-2610 188 15 recursive recursive ADJ iajs-2610 188 16 relation relation NOUN iajs-2610 188 17 from from ADP iajs-2610 188 18 eq.(6.11 eq.(6.11 PROPN iajs-2610 188 19 ): ): PUNCT iajs-2610 188 20 un+1(x un+1(x PROPN iajs-2610 188 21 , , PUNCT iajs-2610 188 22 t t PROPN iajs-2610 188 23 ) ) PUNCT iajs-2610 188 24 = = PUNCT iajs-2610 189 1 2tanx 2tanx NUM iajs-2610 189 2 + + CCONJ iajs-2610 189 3 π•Šβˆ’1{s π•Šβˆ’1{s NOUN iajs-2610 189 4 π•Š[unxx π•Š[unxx NOUN iajs-2610 189 5 βˆ’ βˆ’ NOUN iajs-2610 189 6 ununx ununx VERB iajs-2610 189 7 ] ] PUNCT iajs-2610 189 8 } } PUNCT iajs-2610 189 9 , , PUNCT iajs-2610 189 10 n n X iajs-2610 189 11 β‰₯ β‰₯ NOUN iajs-2610 189 12 0 0 NUM iajs-2610 189 13 . . PUNCT iajs-2610 190 1 ( ( PUNCT iajs-2610 190 2 6.12 6.12 NUM iajs-2610 190 3 ) ) PUNCT iajs-2610 190 4 now now ADV iajs-2610 190 5 , , PUNCT iajs-2610 190 6 we we PRON iajs-2610 190 7 apply apply VERB iajs-2610 190 8 the the DET iajs-2610 190 9 new new ADJ iajs-2610 190 10 sumudu sumudu NOUN iajs-2610 190 11 iterative iterative NOUN iajs-2610 190 12 method method NOUN iajs-2610 190 13 . . PUNCT iajs-2610 191 1 so so ADV iajs-2610 191 2 from from ADP iajs-2610 191 3 eq.(6.12 eq.(6.12 PROPN iajs-2610 191 4 ) ) PUNCT iajs-2610 191 5 , , PUNCT iajs-2610 191 6 we we PRON iajs-2610 191 7 conclude conclude VERB iajs-2610 191 8 u0(x u0(x PRON iajs-2610 191 9 , , PUNCT iajs-2610 191 10 t t PROPN iajs-2610 191 11 ) ) PUNCT iajs-2610 191 12 = = SYM iajs-2610 192 1 2tanx 2tanx NUM iajs-2610 192 2 note note VERB iajs-2610 192 3 that that SCONJ iajs-2610 192 4 u1(x u1(x PROPN iajs-2610 192 5 , , PUNCT iajs-2610 192 6 t t PROPN iajs-2610 192 7 ) ) PUNCT iajs-2610 192 8 = = PUNCT iajs-2610 193 1 2tanx 2tanx NUM iajs-2610 193 2 + + CCONJ iajs-2610 193 3 π•Šβˆ’1{s π•Šβˆ’1{s ADV iajs-2610 193 4 π•Š[u0xx π•Š[u0xx VERB iajs-2610 193 5 βˆ’ βˆ’ NOUN iajs-2610 193 6 u0u0x u0u0x NUM iajs-2610 193 7 ] ] PUNCT iajs-2610 193 8 } } PUNCT iajs-2610 193 9 . . PUNCT iajs-2610 194 1 29 29 NUM iajs-2610 194 2 ibn ibn PROPN iajs-2610 194 3 al al PROPN iajs-2610 194 4 - - PUNCT iajs-2610 194 5 haitham haitham PROPN iajs-2610 194 6 jour jour X iajs-2610 194 7 . . PROPN iajs-2610 194 8 for for ADP iajs-2610 194 9 pure pure ADJ iajs-2610 194 10 & & CCONJ iajs-2610 194 11 appl appl PROPN iajs-2610 194 12 . . PUNCT iajs-2610 195 1 sci sci PROPN iajs-2610 195 2 . . PROPN iajs-2610 196 1 34 34 NUM iajs-2610 196 2 ( ( PUNCT iajs-2610 196 3 2 2 NUM iajs-2610 196 4 ) ) PUNCT iajs-2610 196 5 2021 2021 NUM iajs-2610 196 6 𝑒1(π‘₯ 𝑒1(π‘₯ ADP iajs-2610 196 7 , , PUNCT iajs-2610 196 8 𝑑 𝑑 X iajs-2610 196 9 ) ) PUNCT iajs-2610 196 10 = = PUNCT iajs-2610 197 1 2π‘‘π‘Žπ‘›π‘₯ 2π‘‘π‘Žπ‘›π‘₯ NUM iajs-2610 197 2 + + CCONJ iajs-2610 197 3 π•Šβˆ’1{𝑠 π•Šβˆ’1{𝑠 NUM iajs-2610 197 4 π•Š[4tan π•Š[4tan PROPN iajs-2610 197 5 ( ( PUNCT iajs-2610 197 6 π‘₯)𝑠𝑒𝑐2(π‘₯ π‘₯)𝑠𝑒𝑐2(π‘₯ NOUN iajs-2610 197 7 ) ) PUNCT iajs-2610 198 1 βˆ’ βˆ’ PROPN iajs-2610 198 2 4tan 4tan PROPN iajs-2610 198 3 ( ( PUNCT iajs-2610 198 4 π‘₯)𝑠𝑒𝑐2(π‘₯ π‘₯)𝑠𝑒𝑐2(π‘₯ NOUN iajs-2610 198 5 ) ) PUNCT iajs-2610 198 6 ] ] PUNCT iajs-2610 198 7 } } PUNCT iajs-2610 198 8 = = SYM iajs-2610 198 9 0 0 X iajs-2610 198 10 . . PUNCT iajs-2610 199 1 so so ADV iajs-2610 199 2 , , PUNCT iajs-2610 199 3 we we PRON iajs-2610 199 4 can can AUX iajs-2610 199 5 conclude conclude VERB iajs-2610 199 6 𝑒𝑛+1(π‘₯ 𝑒𝑛+1(π‘₯ NUM iajs-2610 199 7 , , PUNCT iajs-2610 199 8 𝑑 𝑑 X iajs-2610 199 9 ) ) PUNCT iajs-2610 199 10 = = SYM iajs-2610 199 11 0 0 NUM iajs-2610 199 12 , , PUNCT iajs-2610 199 13 βˆ€π‘› βˆ€π‘› NOUN iajs-2610 199 14 β‰₯ β‰₯ NUM iajs-2610 199 15 0 0 NUM iajs-2610 199 16 . . PUNCT iajs-2610 200 1 hence hence ADV iajs-2610 200 2 the the DET iajs-2610 200 3 exact exact ADJ iajs-2610 200 4 solution solution NOUN iajs-2610 200 5 as as ADP iajs-2610 200 6 u(x u(x NOUN iajs-2610 200 7 , , PUNCT iajs-2610 200 8 t t NOUN iajs-2610 200 9 ) ) PUNCT iajs-2610 200 10 = = PUNCT iajs-2610 201 1 2tanx 2tanx NUM iajs-2610 201 2 . . PUNCT iajs-2610 201 3 example example NOUN iajs-2610 201 4 6.3 6.3 NUM iajs-2610 201 5 . . PUNCT iajs-2610 202 1 consider consider VERB iajs-2610 202 2 the the DET iajs-2610 202 3 non non ADJ iajs-2610 202 4 - - ADJ iajs-2610 202 5 linear linear ADJ iajs-2610 202 6 pde pde NOUN iajs-2610 202 7 ut ut PROPN iajs-2610 203 1 = = SYM iajs-2610 203 2 2u(ux)2 2u(ux)2 PROPN iajs-2610 203 3 + + CCONJ iajs-2610 203 4 u2uxx u2uxx NOUN iajs-2610 203 5 , , PUNCT iajs-2610 203 6 with with ADP iajs-2610 203 7 ic ic PROPN iajs-2610 203 8 u(x u(x NOUN iajs-2610 203 9 , , PUNCT iajs-2610 203 10 0 0 NUM iajs-2610 203 11 ) ) PUNCT iajs-2610 203 12 = = PUNCT iajs-2610 204 1 x+1 x+1 PROPN iajs-2610 204 2 2 2 NUM iajs-2610 204 3 . . PUNCT iajs-2610 205 1 ( ( PUNCT iajs-2610 205 2 6.13 6.13 NUM iajs-2610 205 3 ) ) PUNCT iajs-2610 205 4 taking take VERB iajs-2610 205 5 sumudu sumudu NOUN iajs-2610 205 6 transform transform NOUN iajs-2610 205 7 of of ADP iajs-2610 205 8 eq.(6.13 eq.(6.13 PROPN iajs-2610 205 9 ) ) PUNCT iajs-2610 205 10 , , PUNCT iajs-2610 205 11 we we PRON iajs-2610 205 12 have have VERB iajs-2610 205 13 : : PUNCT iajs-2610 205 14 π•Š[ut π•Š[ut X iajs-2610 205 15 ] ] X iajs-2610 205 16 = = SYM iajs-2610 205 17 π•Š[2u(ux)2 π•Š[2u(ux)2 PROPN iajs-2610 205 18 + + CCONJ iajs-2610 205 19 u2uxx u2uxx NOUN iajs-2610 205 20 ] ] PUNCT iajs-2610 205 21 using use VERB iajs-2610 205 22 the the DET iajs-2610 205 23 properties property NOUN iajs-2610 205 24 in in ADP iajs-2610 205 25 table table NOUN iajs-2610 205 26 1,2 1,2 NUM iajs-2610 205 27 and and CCONJ iajs-2610 205 28 submit submit VERB iajs-2610 205 29 to to ADP iajs-2610 205 30 the the DET iajs-2610 205 31 ic ic PROPN iajs-2610 205 32 above above ADP iajs-2610 205 33 we we PRON iajs-2610 205 34 obtain obtain VERB iajs-2610 205 35 : : PUNCT iajs-2610 205 36 u(x u(x NOUN iajs-2610 205 37 , , PUNCT iajs-2610 205 38 s s NOUN iajs-2610 205 39 ) ) PUNCT iajs-2610 205 40 s s PART iajs-2610 206 1 βˆ’ βˆ’ PROPN iajs-2610 206 2 x+1 x+1 PROPN iajs-2610 206 3 2 2 NUM iajs-2610 206 4 s s NOUN iajs-2610 206 5 = = PUNCT iajs-2610 206 6 π•Š[2u(ux)2 π•Š[2u(ux)2 PROPN iajs-2610 206 7 + + CCONJ iajs-2610 206 8 u2uxx u2uxx NOUN iajs-2610 206 9 ] ] PUNCT iajs-2610 206 10 . . PUNCT iajs-2610 207 1 ( ( PUNCT iajs-2610 207 2 6.14 6.14 NUM iajs-2610 207 3 ) ) PUNCT iajs-2610 207 4 subsequently subsequently ADV iajs-2610 207 5 , , PUNCT iajs-2610 207 6 taking take VERB iajs-2610 207 7 inverse inverse NOUN iajs-2610 207 8 sumudu sumudu NOUN iajs-2610 207 9 transform transform NOUN iajs-2610 207 10 of of ADP iajs-2610 207 11 eq eq NOUN iajs-2610 207 12 . . PUNCT iajs-2610 208 1 ( ( PUNCT iajs-2610 208 2 6.14 6.14 NUM iajs-2610 208 3 ) ) PUNCT iajs-2610 208 4 to to PART iajs-2610 208 5 get get VERB iajs-2610 208 6 : : PUNCT iajs-2610 208 7 π•Šβˆ’1[u(x π•Šβˆ’1[u(x NOUN iajs-2610 208 8 , , PUNCT iajs-2610 208 9 s s NOUN iajs-2610 208 10 ) ) PUNCT iajs-2610 208 11 ] ] PUNCT iajs-2610 209 1 = = PUNCT iajs-2610 209 2 x+1 x+1 NUM iajs-2610 209 3 2 2 NUM iajs-2610 209 4 + + CCONJ iajs-2610 209 5 π•Šβˆ’1{s π•Šβˆ’1{s NUM iajs-2610 209 6 π•Š[2u(ux)2 π•Š[2u(ux)2 NUM iajs-2610 209 7 + + CCONJ iajs-2610 209 8 u2uxx u2uxx NOUN iajs-2610 209 9 ] ] PUNCT iajs-2610 209 10 } } PUNCT iajs-2610 209 11 . . PUNCT iajs-2610 210 1 ( ( PUNCT iajs-2610 210 2 6.15 6.15 NUM iajs-2610 210 3 ) ) PUNCT iajs-2610 210 4 table table NOUN iajs-2610 210 5 1and 1and PROPN iajs-2610 210 6 2 2 NUM iajs-2610 210 7 eq.(6.15 eq.(6.15 NOUN iajs-2610 210 8 ) ) PUNCT iajs-2610 210 9 becomes become VERB iajs-2610 210 10 u(x u(x NOUN iajs-2610 210 11 , , PUNCT iajs-2610 210 12 t t NOUN iajs-2610 210 13 ) ) PUNCT iajs-2610 210 14 = = PUNCT iajs-2610 211 1 x+1 x+1 NUM iajs-2610 211 2 2 2 NUM iajs-2610 211 3 + + CCONJ iajs-2610 211 4 π•Šβˆ’1{s π•Šβˆ’1{s NUM iajs-2610 211 5 π•Š[2u(ux)2 π•Š[2u(ux)2 NUM iajs-2610 211 6 + + CCONJ iajs-2610 211 7 u2uxx u2uxx NOUN iajs-2610 211 8 ] ] PUNCT iajs-2610 211 9 } } PUNCT iajs-2610 211 10 . . PUNCT iajs-2610 212 1 ( ( PUNCT iajs-2610 212 2 6.16 6.16 NUM iajs-2610 212 3 ) ) PUNCT iajs-2610 212 4 thus thus ADV iajs-2610 212 5 , , PUNCT iajs-2610 212 6 we we PRON iajs-2610 212 7 persist persist VERB iajs-2610 212 8 in in ADP iajs-2610 212 9 this this DET iajs-2610 212 10 manner manner NOUN iajs-2610 212 11 to to PART iajs-2610 212 12 obtain obtain VERB iajs-2610 212 13 a a DET iajs-2610 212 14 general general ADJ iajs-2610 212 15 recursive recursive ADJ iajs-2610 212 16 relation relation NOUN iajs-2610 212 17 from from ADP iajs-2610 212 18 eq.(6.16 eq.(6.16 PROPN iajs-2610 212 19 ) ) PUNCT iajs-2610 212 20 : : PUNCT iajs-2610 213 1 𝑒𝑛+1(π‘₯ 𝑒𝑛+1(π‘₯ NUM iajs-2610 213 2 , , PUNCT iajs-2610 213 3 𝑑 𝑑 NOUN iajs-2610 213 4 ) ) PUNCT iajs-2610 213 5 = = SYM iajs-2610 213 6 π‘₯+1 π‘₯+1 SYM iajs-2610 213 7 2 2 NUM iajs-2610 213 8 + + CCONJ iajs-2610 213 9 π•Šβˆ’1{𝑠 π•Šβˆ’1{𝑠 PUNCT iajs-2610 213 10 π•Š[2𝑒𝑛(𝑒𝑛π‘₯)2 π•Š[2𝑒𝑛(𝑒𝑛π‘₯)2 X iajs-2610 214 1 + + CCONJ iajs-2610 214 2 ( ( PUNCT iajs-2610 214 3 𝑒𝑛)2𝑒𝑛π‘₯π‘₯ 𝑒𝑛)2𝑒𝑛π‘₯π‘₯ X iajs-2610 214 4 ] ] X iajs-2610 214 5 } } PUNCT iajs-2610 214 6 , , PUNCT iajs-2610 214 7 𝑛 𝑛 DET iajs-2610 214 8 β‰₯ β‰₯ NOUN iajs-2610 214 9 0 0 NUM iajs-2610 214 10 . . PUNCT iajs-2610 215 1 ( ( PUNCT iajs-2610 215 2 6.17 6.17 NUM iajs-2610 215 3 ) ) PUNCT iajs-2610 215 4 now now ADV iajs-2610 215 5 , , PUNCT iajs-2610 215 6 we we PRON iajs-2610 215 7 apply apply VERB iajs-2610 215 8 the the DET iajs-2610 215 9 new new ADJ iajs-2610 215 10 sumudu sumudu NOUN iajs-2610 215 11 iterative iterative NOUN iajs-2610 215 12 method method NOUN iajs-2610 215 13 . . PUNCT iajs-2610 216 1 so so ADV iajs-2610 216 2 from from ADP iajs-2610 216 3 eq.(6.17 eq.(6.17 PROPN iajs-2610 216 4 ) ) PUNCT iajs-2610 216 5 , , PUNCT iajs-2610 216 6 we we PRON iajs-2610 216 7 conclude conclude VERB iajs-2610 216 8 u0(x u0(x PRON iajs-2610 216 9 , , PUNCT iajs-2610 216 10 t t PROPN iajs-2610 216 11 ) ) PUNCT iajs-2610 216 12 = = PUNCT iajs-2610 217 1 x+1 x+1 PROPN iajs-2610 217 2 2 2 NUM iajs-2610 217 3 , , PUNCT iajs-2610 217 4 u1(x u1(x PROPN iajs-2610 217 5 , , PUNCT iajs-2610 217 6 t t PROPN iajs-2610 217 7 ) ) PUNCT iajs-2610 217 8 = = PUNCT iajs-2610 218 1 x+1 x+1 NUM iajs-2610 218 2 2 2 NUM iajs-2610 218 3 + + CCONJ iajs-2610 218 4 π•Šβˆ’1{s π•Šβˆ’1{s NOUN iajs-2610 218 5 π•Š[2u0(u0x)2 π•Š[2u0(u0x)2 VERB iajs-2610 218 6 + + CCONJ iajs-2610 218 7 ( ( PUNCT iajs-2610 218 8 u0)2u0xx u0)2u0xx PROPN iajs-2610 218 9 ] ] PUNCT iajs-2610 218 10 } } PUNCT iajs-2610 218 11 . . PUNCT iajs-2610 219 1 𝑒1(π‘₯ 𝑒1(π‘₯ PROPN iajs-2610 219 2 , , PUNCT iajs-2610 219 3 𝑑 𝑑 NOUN iajs-2610 219 4 ) ) PUNCT iajs-2610 219 5 = = SYM iajs-2610 219 6 π‘₯+1 π‘₯+1 SYM iajs-2610 219 7 2 2 NUM iajs-2610 219 8 ( ( PUNCT iajs-2610 219 9 1 1 NUM iajs-2610 219 10 + + SYM iajs-2610 219 11 1 1 NUM iajs-2610 219 12 2 2 NUM iajs-2610 219 13 𝑑 𝑑 NOUN iajs-2610 219 14 ) ) PUNCT iajs-2610 219 15 . . PUNCT iajs-2610 220 1 and and CCONJ iajs-2610 220 2 u2(x u2(x PROPN iajs-2610 220 3 , , PUNCT iajs-2610 220 4 t t PROPN iajs-2610 220 5 ) ) PUNCT iajs-2610 220 6 = = PUNCT iajs-2610 221 1 x x PUNCT iajs-2610 221 2 + + NUM iajs-2610 221 3 1 1 NUM iajs-2610 221 4 2 2 NUM iajs-2610 221 5 + + CCONJ iajs-2610 221 6 π•Šβˆ’1{s π•Šβˆ’1{s AUX iajs-2610 221 7 π•Š[2u1(u1x)2 π•Š[2u1(u1x)2 VERB iajs-2610 221 8 + + CCONJ iajs-2610 221 9 ( ( PUNCT iajs-2610 221 10 u1)2u1xx u1)2u1xx PROPN iajs-2610 221 11 ] ] PUNCT iajs-2610 221 12 } } PUNCT iajs-2610 221 13 , , PUNCT iajs-2610 221 14 βˆ€n βˆ€n NUM iajs-2610 221 15 β‰₯ β‰₯ NOUN iajs-2610 221 16 1 1 NUM iajs-2610 221 17 . . PUNCT iajs-2610 222 1 u2(x u2(x PRON iajs-2610 222 2 , , PUNCT iajs-2610 222 3 t t PROPN iajs-2610 222 4 ) ) PUNCT iajs-2610 223 1 = = PUNCT iajs-2610 224 1 x+1 x+1 NUM iajs-2610 224 2 2 2 NUM iajs-2610 224 3 ( ( PUNCT iajs-2610 224 4 1 1 NUM iajs-2610 224 5 + + SYM iajs-2610 224 6 1 1 NUM iajs-2610 224 7 2 2 NUM iajs-2610 224 8 t t NOUN iajs-2610 224 9 + + NOUN iajs-2610 224 10 3 3 NUM iajs-2610 224 11 8 8 NUM iajs-2610 224 12 t2 t2 NOUN iajs-2610 224 13 + + CCONJ iajs-2610 224 14 1 1 NUM iajs-2610 224 15 8 8 NUM iajs-2610 224 16 t3 t3 NOUN iajs-2610 224 17 + + CCONJ iajs-2610 224 18 1 1 NUM iajs-2610 224 19 64 64 NUM iajs-2610 224 20 t4 t4 PROPN iajs-2610 224 21 ) ) PUNCT iajs-2610 224 22 , , PUNCT iajs-2610 224 23 we we PRON iajs-2610 224 24 persist persist VERB iajs-2610 224 25 in in ADP iajs-2610 224 26 this this DET iajs-2610 224 27 technique technique NOUN iajs-2610 224 28 to to PART iajs-2610 224 29 obtain obtain VERB iajs-2610 224 30 : : PUNCT iajs-2610 224 31 : : PUNCT iajs-2610 224 32 u(x u(x PROPN iajs-2610 224 33 , , PUNCT iajs-2610 224 34 t t PROPN iajs-2610 224 35 ) ) PUNCT iajs-2610 224 36 = = PUNCT iajs-2610 225 1 x+1 x+1 NUM iajs-2610 225 2 2 2 NUM iajs-2610 225 3 ( ( PUNCT iajs-2610 225 4 1 1 NUM iajs-2610 225 5 + + SYM iajs-2610 225 6 1 1 NUM iajs-2610 225 7 2 2 NUM iajs-2610 225 8 t t NOUN iajs-2610 225 9 + + NOUN iajs-2610 225 10 3 3 NUM iajs-2610 225 11 8 8 NUM iajs-2610 225 12 t2 t2 NOUN iajs-2610 225 13 + + CCONJ iajs-2610 225 14 1 1 NUM iajs-2610 225 15 8 8 NUM iajs-2610 225 16 t3 t3 NOUN iajs-2610 225 17 + + CCONJ iajs-2610 225 18 1 1 NUM iajs-2610 225 19 64 64 NUM iajs-2610 225 20 t4 t4 PROPN iajs-2610 225 21 + + PROPN iajs-2610 225 22 β‹― β‹― PROPN iajs-2610 225 23 ) ) PUNCT iajs-2610 225 24 . . PUNCT iajs-2610 226 1 which which PRON iajs-2610 226 2 is be AUX iajs-2610 226 3 the the DET iajs-2610 226 4 exact exact ADJ iajs-2610 226 5 solution solution NOUN iajs-2610 226 6 of of ADP iajs-2610 226 7 eq.(6.13 eq.(6.13 PROPN iajs-2610 226 8 ) ) PUNCT iajs-2610 226 9 . . PUNCT iajs-2610 227 1 30 30 NUM iajs-2610 227 2 ibn ibn PROPN iajs-2610 227 3 al al PROPN iajs-2610 227 4 - - PUNCT iajs-2610 227 5 haitham haitham PROPN iajs-2610 227 6 jour jour X iajs-2610 227 7 . . PROPN iajs-2610 227 8 for for ADP iajs-2610 227 9 pure pure ADJ iajs-2610 227 10 & & CCONJ iajs-2610 227 11 appl appl PROPN iajs-2610 227 12 . . PUNCT iajs-2610 228 1 sci sci PROPN iajs-2610 228 2 . . PROPN iajs-2610 229 1 34 34 NUM iajs-2610 229 2 ( ( PUNCT iajs-2610 229 3 2 2 NUM iajs-2610 229 4 ) ) PUNCT iajs-2610 229 5 2021 2021 NUM iajs-2610 229 6 example example NOUN iajs-2610 229 7 6.4 6.4 NUM iajs-2610 229 8 . . PUNCT iajs-2610 230 1 the the DET iajs-2610 230 2 non non ADJ iajs-2610 230 3 - - ADJ iajs-2610 230 4 linear linear ADJ iajs-2610 230 5 inhomogeneous inhomogeneous ADJ iajs-2610 230 6 klein klein PROPN iajs-2610 230 7 gordon gordon PROPN iajs-2610 230 8 equation equation NOUN iajs-2610 230 9 of of ADP iajs-2610 230 10 the the DET iajs-2610 230 11 form form NOUN iajs-2610 230 12 : : PUNCT iajs-2610 230 13 utt utt PROPN iajs-2610 230 14 βˆ’ βˆ’ PROPN iajs-2610 230 15 uxx uxx NOUN iajs-2610 230 16 + + CCONJ iajs-2610 230 17 u2 u2 NOUN iajs-2610 230 18 = = SYM iajs-2610 230 19 x2t2 x2t2 NOUN iajs-2610 230 20 , , PUNCT iajs-2610 230 21 with with ADP iajs-2610 230 22 ics ics NOUN iajs-2610 230 23 u(x u(x NOUN iajs-2610 230 24 , , PUNCT iajs-2610 230 25 0 0 NUM iajs-2610 230 26 ) ) PUNCT iajs-2610 230 27 = = SYM iajs-2610 230 28 0 0 NUM iajs-2610 230 29 , , PUNCT iajs-2610 230 30 ut(x ut(x NOUN iajs-2610 230 31 , , PUNCT iajs-2610 230 32 0 0 NUM iajs-2610 230 33 ) ) PUNCT iajs-2610 230 34 = = PUNCT iajs-2610 230 35 x. x. NOUN iajs-2610 230 36 ( ( PUNCT iajs-2610 230 37 6.18 6.18 NUM iajs-2610 230 38 ) ) PUNCT iajs-2610 230 39 taking take VERB iajs-2610 230 40 sumudu sumudu NOUN iajs-2610 230 41 transform transform NOUN iajs-2610 230 42 of of ADP iajs-2610 230 43 eq.(6.18 eq.(6.18 PROPN iajs-2610 230 44 ) ) PUNCT iajs-2610 230 45 , , PUNCT iajs-2610 230 46 we we PRON iajs-2610 230 47 have have VERB iajs-2610 230 48 : : PUNCT iajs-2610 230 49 π•Š[utt π•Š[utt VERB iajs-2610 230 50 ] ] X iajs-2610 230 51 βˆ’ βˆ’ PROPN iajs-2610 230 52 π•Š[uxx π•Š[uxx X iajs-2610 230 53 ] ] X iajs-2610 230 54 + + NUM iajs-2610 230 55 π•Š[u2 π•Š[u2 NOUN iajs-2610 230 56 ] ] PUNCT iajs-2610 230 57 = = PUNCT iajs-2610 231 1 π•Š[x2t2 π•Š[x2t2 NOUN iajs-2610 231 2 ] ] PUNCT iajs-2610 231 3 utilizing utilize VERB iajs-2610 231 4 properties property NOUN iajs-2610 231 5 in in ADP iajs-2610 231 6 table table NOUN iajs-2610 231 7 1,2 1,2 NUM iajs-2610 231 8 and and CCONJ iajs-2610 231 9 the the DET iajs-2610 231 10 properties property NOUN iajs-2610 231 11 mentioned mention VERB iajs-2610 231 12 above above ADV iajs-2610 231 13 , , PUNCT iajs-2610 231 14 we we PRON iajs-2610 231 15 get get VERB iajs-2610 231 16 : : PUNCT iajs-2610 231 17 u(x u(x NOUN iajs-2610 231 18 , , PUNCT iajs-2610 231 19 s s NOUN iajs-2610 231 20 ) ) PUNCT iajs-2610 231 21 s2 s2 NOUN iajs-2610 231 22 βˆ’ βˆ’ PROPN iajs-2610 231 23 u(x,0 u(x,0 PROPN iajs-2610 231 24 ) ) PUNCT iajs-2610 231 25 s2 s2 NOUN iajs-2610 231 26 βˆ’ βˆ’ PROPN iajs-2610 231 27 ut(x,0 ut(x,0 PROPN iajs-2610 231 28 ) ) PUNCT iajs-2610 231 29 s s PART iajs-2610 232 1 = = PROPN iajs-2610 232 2 2x2s2 2x2s2 PROPN iajs-2610 232 3 + + NUM iajs-2610 232 4 π•Š[uxx π•Š[uxx PROPN iajs-2610 232 5 βˆ’ βˆ’ PROPN iajs-2610 232 6 u2 u2 PROPN iajs-2610 232 7 ] ] PUNCT iajs-2610 232 8 . . PUNCT iajs-2610 233 1 ( ( PUNCT iajs-2610 233 2 6.19 6.19 NUM iajs-2610 233 3 ) ) PUNCT iajs-2610 233 4 substitute substitute NOUN iajs-2610 233 5 the the DET iajs-2610 233 6 initial initial ADJ iajs-2610 233 7 condition condition NOUN iajs-2610 233 8 in in ADP iajs-2610 233 9 eq.(6.19 eq.(6.19 NOUN iajs-2610 233 10 ) ) PUNCT iajs-2610 233 11 to to PART iajs-2610 233 12 obtain obtain VERB iajs-2610 233 13 : : PUNCT iajs-2610 233 14 u(x u(x NOUN iajs-2610 233 15 , , PUNCT iajs-2610 233 16 s s X iajs-2610 233 17 ) ) PUNCT iajs-2610 234 1 = = SYM iajs-2610 234 2 xs xs PROPN iajs-2610 235 1 + + CCONJ iajs-2610 236 1 2x2s4 2x2s4 NUM iajs-2610 236 2 + + CCONJ iajs-2610 236 3 s2π•Š[uxx s2π•Š[uxx PROPN iajs-2610 236 4 βˆ’ βˆ’ PROPN iajs-2610 236 5 u2 u2 NOUN iajs-2610 236 6 ] ] PUNCT iajs-2610 236 7 . . PUNCT iajs-2610 237 1 ( ( PUNCT iajs-2610 237 2 6.20 6.20 NUM iajs-2610 237 3 ) ) PUNCT iajs-2610 237 4 subsequently subsequently ADV iajs-2610 237 5 , , PUNCT iajs-2610 237 6 taking take VERB iajs-2610 237 7 inverse inverse NOUN iajs-2610 237 8 sumudu sumudu NOUN iajs-2610 237 9 transform transform NOUN iajs-2610 237 10 of of ADP iajs-2610 237 11 eq.(6.20 eq.(6.20 NOUN iajs-2610 237 12 ) ) PUNCT iajs-2610 237 13 , , PUNCT iajs-2610 237 14 we we PRON iajs-2610 237 15 get get VERB iajs-2610 237 16 : : PUNCT iajs-2610 237 17 π•Šβˆ’1[u(x π•Šβˆ’1[u(x NOUN iajs-2610 237 18 , , PUNCT iajs-2610 237 19 s s NOUN iajs-2610 237 20 ) ) PUNCT iajs-2610 237 21 ] ] PUNCT iajs-2610 238 1 = = PUNCT iajs-2610 238 2 xπ•Šβˆ’1[s xπ•Šβˆ’1[s PROPN iajs-2610 238 3 ] ] X iajs-2610 238 4 + + CCONJ iajs-2610 238 5 2x2π•Šβˆ’1[s4 2x2π•Šβˆ’1[s4 NOUN iajs-2610 238 6 ] ] X iajs-2610 238 7 + + NUM iajs-2610 238 8 π•Šβˆ’1{s2π•Š[uxx π•Šβˆ’1{s2π•Š[uxx NOUN iajs-2610 238 9 βˆ’ βˆ’ NOUN iajs-2610 238 10 u2 u2 NOUN iajs-2610 238 11 ] ] PUNCT iajs-2610 238 12 } } PUNCT iajs-2610 238 13 . . PUNCT iajs-2610 239 1 ( ( PUNCT iajs-2610 239 2 6.21 6.21 NUM iajs-2610 239 3 ) ) PUNCT iajs-2610 239 4 table table NOUN iajs-2610 239 5 1and 1and NUM iajs-2610 239 6 2 2 NUM iajs-2610 239 7 eq.(6.21 eq.(6.21 NOUN iajs-2610 239 8 ) ) PUNCT iajs-2610 239 9 becomes become VERB iajs-2610 239 10 u(x u(x NOUN iajs-2610 239 11 , , PUNCT iajs-2610 239 12 t t NOUN iajs-2610 239 13 ) ) PUNCT iajs-2610 239 14 = = PUNCT iajs-2610 239 15 xt xt PROPN iajs-2610 240 1 + + CCONJ iajs-2610 240 2 x2t4 x2t4 PROPN iajs-2610 240 3 12 12 NUM iajs-2610 240 4 + + NUM iajs-2610 240 5 π•Šβˆ’1{s2π•Š[uxx π•Šβˆ’1{s2π•Š[uxx NOUN iajs-2610 240 6 βˆ’ βˆ’ NOUN iajs-2610 240 7 u2 u2 NOUN iajs-2610 240 8 ] ] PUNCT iajs-2610 240 9 } } PUNCT iajs-2610 240 10 . . PUNCT iajs-2610 241 1 ( ( PUNCT iajs-2610 241 2 6.22 6.22 NUM iajs-2610 241 3 ) ) PUNCT iajs-2610 241 4 by by ADP iajs-2610 241 5 the the DET iajs-2610 241 6 new new ADJ iajs-2610 241 7 correction correction NOUN iajs-2610 241 8 function function VERB iajs-2610 241 9 from from ADP iajs-2610 241 10 eq.(6.22 eq.(6.22 PROPN iajs-2610 241 11 ) ) PUNCT iajs-2610 241 12 , , PUNCT iajs-2610 241 13 we we PRON iajs-2610 241 14 can can AUX iajs-2610 241 15 get get VERB iajs-2610 241 16 : : PUNCT iajs-2610 241 17 un(x un(x NUM iajs-2610 241 18 , , PUNCT iajs-2610 241 19 t t PROPN iajs-2610 241 20 ) ) PUNCT iajs-2610 241 21 = = PUNCT iajs-2610 241 22 xt xt PROPN iajs-2610 242 1 + + CCONJ iajs-2610 242 2 x2t4 x2t4 PROPN iajs-2610 242 3 12 12 NUM iajs-2610 242 4 + + CCONJ iajs-2610 242 5 π•Šβˆ’1{s2π•Š[unxx π•Šβˆ’1{s2π•Š[unxx ADJ iajs-2610 242 6 βˆ’ βˆ’ NOUN iajs-2610 242 7 ( ( PUNCT iajs-2610 242 8 un)2 un)2 PROPN iajs-2610 242 9 ] ] PUNCT iajs-2610 242 10 } } PUNCT iajs-2610 242 11 , , PUNCT iajs-2610 242 12 n n X iajs-2610 242 13 β‰₯ β‰₯ NOUN iajs-2610 242 14 0 0 NUM iajs-2610 242 15 . . PUNCT iajs-2610 243 1 ( ( PUNCT iajs-2610 243 2 6.23 6.23 NUM iajs-2610 243 3 ) ) PUNCT iajs-2610 243 4 now now ADV iajs-2610 243 5 , , PUNCT iajs-2610 243 6 we we PRON iajs-2610 243 7 apply apply VERB iajs-2610 243 8 the the DET iajs-2610 243 9 new new ADJ iajs-2610 243 10 sumudu sumudu NOUN iajs-2610 243 11 iterative iterative NOUN iajs-2610 243 12 method method NOUN iajs-2610 243 13 . . PUNCT iajs-2610 244 1 so so ADV iajs-2610 244 2 from from ADP iajs-2610 244 3 eq.(6.23 eq.(6.23 NOUN iajs-2610 244 4 ) ) PUNCT iajs-2610 244 5 , , PUNCT iajs-2610 244 6 we we PRON iajs-2610 244 7 conclude conclude VERB iajs-2610 244 8 𝑒0(π‘₯ 𝑒0(π‘₯ NUM iajs-2610 244 9 , , PUNCT iajs-2610 244 10 𝑑 𝑑 NOUN iajs-2610 244 11 ) ) PUNCT iajs-2610 244 12 = = SYM iajs-2610 245 1 π‘₯𝑑 π‘₯𝑑 ADP iajs-2610 245 2 + + CCONJ iajs-2610 245 3 π‘₯2𝑑4 π‘₯2𝑑4 X iajs-2610 245 4 12 12 NUM iajs-2610 245 5 , , PUNCT iajs-2610 245 6 u1(x u1(x PROPN iajs-2610 245 7 , , PUNCT iajs-2610 245 8 t t PROPN iajs-2610 245 9 ) ) PUNCT iajs-2610 245 10 = = PUNCT iajs-2610 245 11 xt xt PROPN iajs-2610 246 1 + + CCONJ iajs-2610 246 2 x2t4 x2t4 PROPN iajs-2610 246 3 12 12 NUM iajs-2610 246 4 + + CCONJ iajs-2610 246 5 π•Šβˆ’1{s2π•Š[u0xx π•Šβˆ’1{s2π•Š[u0xx NUM iajs-2610 246 6 βˆ’ βˆ’ PROPN iajs-2610 246 7 ( ( PUNCT iajs-2610 246 8 u0)2 u0)2 X iajs-2610 246 9 ] ] PUNCT iajs-2610 246 10 } } PUNCT iajs-2610 246 11 . . PUNCT iajs-2610 247 1 u1(x u1(x ADV iajs-2610 247 2 , , PUNCT iajs-2610 247 3 t t PROPN iajs-2610 247 4 ) ) PUNCT iajs-2610 247 5 = = SYM iajs-2610 247 6 xt xt PROPN iajs-2610 248 1 + + CCONJ iajs-2610 248 2 t6 t6 PROPN iajs-2610 248 3 180 180 NUM iajs-2610 248 4 βˆ’ βˆ’ NOUN iajs-2610 248 5 x3t7 x3t7 ADP iajs-2610 248 6 252 252 NUM iajs-2610 248 7 βˆ’ βˆ’ NOUN iajs-2610 248 8 x4t10 x4t10 NUM iajs-2610 248 9 12960 12960 NUM iajs-2610 248 10 . . PUNCT iajs-2610 249 1 we we PRON iajs-2610 249 2 would would AUX iajs-2610 249 3 like like VERB iajs-2610 249 4 to to PART iajs-2610 249 5 mention mention VERB iajs-2610 249 6 here here ADV iajs-2610 249 7 , , PUNCT iajs-2610 249 8 as as SCONJ iajs-2610 249 9 we we PRON iajs-2610 249 10 go go VERB iajs-2610 249 11 forward forward ADV iajs-2610 249 12 with with ADP iajs-2610 249 13 iterations iteration NOUN iajs-2610 249 14 the the DET iajs-2610 249 15 terms term NOUN iajs-2610 249 16 after after SCONJ iajs-2610 249 17 xt xt PROPN iajs-2610 249 18 will will AUX iajs-2610 249 19 begin begin VERB iajs-2610 249 20 to to PART iajs-2610 249 21 decrease decrease VERB iajs-2610 249 22 and and CCONJ iajs-2610 249 23 get get VERB iajs-2610 249 24 close close ADJ iajs-2610 249 25 to to ADP iajs-2610 249 26 zero zero NUM iajs-2610 249 27 , , PUNCT iajs-2610 249 28 hence hence ADV iajs-2610 249 29 we we PRON iajs-2610 249 30 can can AUX iajs-2610 249 31 cancel cancel VERB iajs-2610 249 32 this this DET iajs-2610 249 33 noise noise NOUN iajs-2610 249 34 term term NOUN iajs-2610 249 35 and and CCONJ iajs-2610 249 36 get get VERB iajs-2610 249 37 the the DET iajs-2610 249 38 exact exact ADJ iajs-2610 249 39 solution solution NOUN iajs-2610 249 40 u(x u(x NOUN iajs-2610 249 41 , , PUNCT iajs-2610 249 42 t t PROPN iajs-2610 249 43 ) ) PUNCT iajs-2610 249 44 = = SYM iajs-2610 249 45 xt xt X iajs-2610 249 46 . . PUNCT iajs-2610 250 1 7.conclusion 7.conclusion NUM iajs-2610 250 2 in in ADP iajs-2610 250 3 this this DET iajs-2610 250 4 work work NOUN iajs-2610 250 5 we we PRON iajs-2610 250 6 introduce introduce VERB iajs-2610 250 7 a a DET iajs-2610 250 8 new new ADJ iajs-2610 250 9 efficient efficient ADJ iajs-2610 250 10 method method NOUN iajs-2610 250 11 for for ADP iajs-2610 250 12 resolving resolve VERB iajs-2610 250 13 non non ADJ iajs-2610 250 14 - - ADJ iajs-2610 250 15 linear linear ADJ iajs-2610 250 16 partial partial ADJ iajs-2610 250 17 differential differential NOUN iajs-2610 250 18 equations equation NOUN iajs-2610 250 19 that that PRON iajs-2610 250 20 is be AUX iajs-2610 250 21 the the DET iajs-2610 250 22 sumudu sumudu NOUN iajs-2610 250 23 iterative iterative NOUN iajs-2610 250 24 method method NOUN iajs-2610 250 25 , , PUNCT iajs-2610 250 26 which which PRON iajs-2610 250 27 represents represent VERB iajs-2610 250 28 a a DET iajs-2610 250 29 new new ADJ iajs-2610 250 30 addition addition NOUN iajs-2610 250 31 to to ADP iajs-2610 250 32 the the DET iajs-2610 250 33 previous previous ADJ iajs-2610 250 34 iterative iterative NOUN iajs-2610 250 35 methods method NOUN iajs-2610 250 36 such such ADJ iajs-2610 250 37 as as ADP iajs-2610 250 38 the the DET iajs-2610 250 39 adomain adomain NOUN iajs-2610 250 40 decomposition decomposition NOUN iajs-2610 250 41 method method NOUN iajs-2610 250 42 , , PUNCT iajs-2610 250 43 variation variation NOUN iajs-2610 250 44 iteration iteration NOUN iajs-2610 250 45 method method NOUN iajs-2610 250 46 , , PUNCT iajs-2610 250 47 homotopy homotopy VERB iajs-2610 250 48 perpetration perpetration NOUN iajs-2610 250 49 method method NOUN iajs-2610 250 50 , , PUNCT iajs-2610 250 51 etc etc X iajs-2610 250 52 . . X iajs-2610 251 1 our our PRON iajs-2610 251 2 method method NOUN iajs-2610 251 3 is be AUX iajs-2610 251 4 characterized characterize VERB iajs-2610 251 5 by by ADP iajs-2610 251 6 its its PRON iajs-2610 251 7 speed speed NOUN iajs-2610 251 8 and and CCONJ iajs-2610 251 9 ease ease NOUN iajs-2610 251 10 of of ADP iajs-2610 251 11 use use NOUN iajs-2610 251 12 in in ADP iajs-2610 251 13 addition addition NOUN iajs-2610 251 14 to to ADP iajs-2610 251 15 the the DET iajs-2610 251 16 fact fact NOUN iajs-2610 251 17 that that SCONJ iajs-2610 251 18 it it PRON iajs-2610 251 19 often often ADV iajs-2610 251 20 gives give VERB iajs-2610 251 21 the the DET iajs-2610 251 22 exact exact ADJ iajs-2610 251 23 solution solution NOUN iajs-2610 251 24 after after ADP iajs-2610 251 25 a a DET iajs-2610 251 26 few few ADJ iajs-2610 251 27 iterations iteration NOUN iajs-2610 251 28 . . PUNCT iajs-2610 252 1 our our PRON iajs-2610 252 2 aim aim NOUN iajs-2610 252 3 of of ADP iajs-2610 252 4 this this DET iajs-2610 252 5 work work NOUN iajs-2610 252 6 is be AUX iajs-2610 252 7 a a DET iajs-2610 252 8 step step NOUN iajs-2610 252 9 towards towards ADP iajs-2610 252 10 using use VERB iajs-2610 252 11 applications application NOUN iajs-2610 252 12 of of ADP iajs-2610 252 13 the the DET iajs-2610 252 14 sumudu sumudu NOUN iajs-2610 252 15 iterative iterative NOUN iajs-2610 252 16 method method NOUN iajs-2610 252 17 to to PART iajs-2610 252 18 resolve resolve VERB iajs-2610 252 19 nonlinear nonlinear ADJ iajs-2610 252 20 problems problem NOUN iajs-2610 252 21 arising arise VERB iajs-2610 252 22 in in ADP iajs-2610 252 23 various various ADJ iajs-2610 252 24 field field NOUN iajs-2610 252 25 scopes scope NOUN iajs-2610 252 26 of of ADP iajs-2610 252 27 science science PROPN iajs-2610 252 28 mathematical mathematical ADJ iajs-2610 252 29 physics physics PROPN iajs-2610 252 30 and and CCONJ iajs-2610 252 31 engineering engineering NOUN iajs-2610 252 32 , , PUNCT iajs-2610 252 33 which which PRON iajs-2610 252 34 may may AUX iajs-2610 252 35 be be AUX iajs-2610 252 36 debated debate VERB iajs-2610 252 37 in in ADP iajs-2610 252 38 further further ADJ iajs-2610 252 39 work work NOUN iajs-2610 252 40 . . PUNCT iajs-2610 253 1 31 31 NUM iajs-2610 254 1 ibn ibn PROPN iajs-2610 254 2 al al PROPN iajs-2610 254 3 - - PUNCT iajs-2610 254 4 haitham haitham PROPN iajs-2610 254 5 jour jour X iajs-2610 254 6 . . PROPN iajs-2610 254 7 for for ADP iajs-2610 254 8 pure pure ADJ iajs-2610 254 9 & & CCONJ iajs-2610 254 10 appl appl PROPN iajs-2610 254 11 . . PUNCT iajs-2610 255 1 sci sci PROPN iajs-2610 255 2 . . PROPN iajs-2610 256 1 34 34 NUM iajs-2610 256 2 ( ( PUNCT iajs-2610 256 3 2 2 NUM iajs-2610 256 4 ) ) PUNCT iajs-2610 256 5 2021 2021 NUM iajs-2610 256 6 references reference NOUN iajs-2610 256 7 1 1 NUM iajs-2610 256 8 . . PUNCT iajs-2610 256 9 wazwaz wazwaz NOUN iajs-2610 256 10 , , PUNCT iajs-2610 256 11 a.m. a.m. PROPN iajs-2610 256 12 chapter chapter PROPN iajs-2610 256 13 9 9 NUM iajs-2610 256 14 . . PUNCT iajs-2610 257 1 linear linear ADJ iajs-2610 257 2 and and CCONJ iajs-2610 257 3 nonlinear nonlinear ADJ iajs-2610 257 4 physical physical ADJ iajs-2610 257 5 models model NOUN iajs-2610 257 6 . . PUNCT iajs-2610 258 1 in in ADP iajs-2610 258 2 : : PUNCT iajs-2610 258 3 albert albert PROPN iajs-2610 258 4 cj cj PROPN iajs-2610 258 5 l l PROPN iajs-2610 258 6 , , PUNCT iajs-2610 258 7 nail nail NOUN iajs-2610 258 8 hi hi NOUN iajs-2610 258 9 , , PUNCT iajs-2610 258 10 editors editor NOUN iajs-2610 258 11 . . PUNCT iajs-2610 259 1 partial partial ADJ iajs-2610 259 2 differential differential ADJ iajs-2610 259 3 equations equation NOUN iajs-2610 259 4 and and CCONJ iajs-2610 259 5 solitary solitary ADJ iajs-2610 259 6 waves wave NOUN iajs-2610 259 7 theory theory NOUN iajs-2610 259 8 . . PUNCT iajs-2610 260 1 london london PROPN iajs-2610 260 2 new new PROPN iajs-2610 260 3 york york PROPN iajs-2610 260 4 : : PUNCT iajs-2610 260 5 springer springer NOUN iajs-2610 260 6 dordrecht dordrecht PROPN iajs-2610 260 7 heidelberg heidelberg PROPN iajs-2610 260 8 , , PUNCT iajs-2610 260 9 2009 2009 NUM iajs-2610 260 10 , , PUNCT iajs-2610 260 11 isbn isbn ADJ iajs-2610 260 12 978 978 NUM iajs-2610 260 13 - - SYM iajs-2610 260 14 3 3 NUM iajs-2610 260 15 - - PUNCT iajs-2610 260 16 642 642 NUM iajs-2610 260 17 - - PUNCT iajs-2610 260 18 00250 00250 NUM iajs-2610 260 19 - - SYM iajs-2610 260 20 2 2 NUM iajs-2610 260 21 , , PUNCT iajs-2610 260 22 345 345 NUM iajs-2610 260 23 380 380 NUM iajs-2610 260 24 . . PUNCT iajs-2610 261 1 2 2 X iajs-2610 261 2 . . X iajs-2610 261 3 hashim hashim PROPN iajs-2610 261 4 , , PUNCT iajs-2610 261 5 i. i. PROPN iajs-2610 261 6 ; ; PUNCT iajs-2610 261 7 noorani noorani PROPN iajs-2610 261 8 , , PUNCT iajs-2610 261 9 m.s.m m.s.m PROPN iajs-2610 261 10 . . PROPN iajs-2610 261 11 ; ; PUNCT iajs-2610 261 12 ahmed ahmed PROPN iajs-2610 261 13 , , PUNCT iajs-2610 261 14 r. r. PROPN iajs-2610 261 15 ; ; PUNCT iajs-2610 261 16 bakar bakar PROPN iajs-2610 261 17 , , PUNCT iajs-2610 261 18 s.a s.a PROPN iajs-2610 261 19 . . PROPN iajs-2610 261 20 ; ; PUNCT iajs-2610 261 21 ismail ismail PROPN iajs-2610 261 22 , , PUNCT iajs-2610 261 23 e.s e.s PROPN iajs-2610 261 24 . . PROPN iajs-2610 261 25 ; ; PUNCT iajs-2610 261 26 zakaria zakaria PROPN iajs-2610 261 27 , , PUNCT iajs-2610 261 28 m.accuracy m.accuracy NOUN iajs-2610 261 29 of of ADP iajs-2610 261 30 the the DET iajs-2610 261 31 adomian adomian NOUN iajs-2610 261 32 decomposition decomposition NOUN iajs-2610 261 33 method method NOUN iajs-2610 261 34 applied apply VERB iajs-2610 261 35 to to ADP iajs-2610 261 36 the the DET iajs-2610 261 37 lorenz lorenz PROPN iajs-2610 261 38 system.chaos system.chaos PROPN iajs-2610 261 39 , , PUNCT iajs-2610 261 40 solitons&fractals solitons&fractal NOUN iajs-2610 261 41 . . NOUN iajs-2610 261 42 2006 2006 NUM iajs-2610 261 43 , , PUNCT iajs-2610 261 44 28 28 NUM iajs-2610 261 45 , , PUNCT iajs-2610 261 46 5 5 NUM iajs-2610 261 47 , , PUNCT iajs-2610 261 48 1149–1158,doi:10.1016 1149–1158,doi:10.1016 NUM iajs-2610 261 49 / / SYM iajs-2610 261 50 j.chaos.2005.08.135 j.chaos.2005.08.135 PROPN iajs-2610 261 51 . . PROPN iajs-2610 262 1 3 3 X iajs-2610 262 2 . . PUNCT iajs-2610 262 3 abdul abdul PROPN iajs-2610 262 4 razzaq razzaq PROPN iajs-2610 262 5 , , PUNCT iajs-2610 262 6 e.a.l e.a.l PROPN iajs-2610 262 7 . . PUNCT iajs-2610 262 8 ; ; PUNCT iajs-2610 262 9 yassein yassein PROPN iajs-2610 262 10 , , PUNCT iajs-2610 262 11 s.m s.m PROPN iajs-2610 262 12 . . PROPN iajs-2610 262 13 analytic analytic ADJ iajs-2610 262 14 solutions solution NOUN iajs-2610 262 15 for for ADP iajs-2610 262 16 integro integro ADJ iajs-2610 262 17 - - PUNCT iajs-2610 262 18 differential differential NOUN iajs-2610 262 19 inequalities inequality NOUN iajs-2610 262 20 using use VERB iajs-2610 262 21 modified modify VERB iajs-2610 262 22 adomian adomian NOUN iajs-2610 262 23 decomposition decomposition NOUN iajs-2610 262 24 method method NOUN iajs-2610 262 25 . . PUNCT iajs-2610 263 1 ibn ibn PROPN iajs-2610 263 2 al al PROPN iajs-2610 263 3 - - PUNCT iajs-2610 263 4 haitham haitham PROPN iajs-2610 263 5 jour jour PROPN iajs-2610 263 6 for for ADP iajs-2610 263 7 pure pure ADJ iajs-2610 263 8 & & CCONJ iajs-2610 263 9 appl appl PROPN iajs-2610 263 10 . . PUNCT iajs-2610 264 1 sci sci PROPN iajs-2610 264 2 . . PROPN iajs-2610 264 3 2017 2017 NUM iajs-2610 264 4 , , PUNCT iajs-2610 264 5 30 30 NUM iajs-2610 264 6 , , PUNCT iajs-2610 264 7 1 1 NUM iajs-2610 264 8 , , PUNCT iajs-2610 264 9 177 177 NUM iajs-2610 264 10 - - SYM iajs-2610 264 11 191 191 NUM iajs-2610 264 12 , , PUNCT iajs-2610 264 13 doi doi NOUN iajs-2610 264 14 : : PUNCT iajs-2610 264 15 10.30526/32.2.2139 10.30526/32.2.2139 NUM iajs-2610 264 16 . . PROPN iajs-2610 264 17 4 4 X iajs-2610 264 18 . . X iajs-2610 264 19 singh singh PROPN iajs-2610 264 20 , , PUNCT iajs-2610 264 21 j. j. PROPN iajs-2610 264 22 ; ; PUNCT iajs-2610 264 23 kumar kumar PROPN iajs-2610 264 24 , , PUNCT iajs-2610 264 25 d. d. PROPN iajs-2610 264 26 , , PUNCT iajs-2610 264 27 sushila sushila PROPN iajs-2610 264 28 . . PUNCT iajs-2610 265 1 homotopy homotopy NOUN iajs-2610 265 2 perturbation perturbation NOUN iajs-2610 265 3 sumudu sumudu NOUN iajs-2610 265 4 transform transform VERB iajs-2610 265 5 method method NOUN iajs-2610 265 6 for for ADP iajs-2610 265 7 nonlinear nonlinear ADJ iajs-2610 265 8 equations equation NOUN iajs-2610 265 9 . . PUNCT iajs-2610 266 1 adv adv PROPN iajs-2610 266 2 . . PUNCT iajs-2610 267 1 theor theor PROPN iajs-2610 267 2 . . PUNCT iajs-2610 268 1 appl appl PROPN iajs-2610 268 2 . . PROPN iajs-2610 268 3 mech mech PROPN iajs-2610 268 4 . . PUNCT iajs-2610 269 1 2011 2011 NUM iajs-2610 269 2 , , PUNCT iajs-2610 269 3 4 4 NUM iajs-2610 269 4 , , PUNCT iajs-2610 269 5 4 4 NUM iajs-2610 269 6 , , PUNCT iajs-2610 269 7 165 165 NUM iajs-2610 269 8 - - SYM iajs-2610 269 9 175 175 NUM iajs-2610 269 10 . . PUNCT iajs-2610 270 1 5 5 NUM iajs-2610 270 2 . . X iajs-2610 270 3 elnour elnour PROPN iajs-2610 270 4 , , PUNCT iajs-2610 270 5 e.a e.a PROPN iajs-2610 270 6 . . PROPN iajs-2610 270 7 ; ; PUNCT iajs-2610 271 1 elzaki elzaki PROPN iajs-2610 271 2 , , PUNCT iajs-2610 271 3 t.m t.m PROPN iajs-2610 271 4 . . NOUN iajs-2610 271 5 solution solution NOUN iajs-2610 271 6 of of ADP iajs-2610 271 7 nonlinear nonlinear ADJ iajs-2610 271 8 partial partial ADJ iajs-2610 271 9 differential differential ADJ iajs-2610 271 10 equations equation NOUN iajs-2610 271 11 bythe bythe ADP iajs-2610 271 12 combined combine VERB iajs-2610 271 13 laplace laplace NOUN iajs-2610 271 14 transform transform NOUN iajs-2610 271 15 and and CCONJ iajs-2610 271 16 the the DET iajs-2610 271 17 new new ADJ iajs-2610 271 18 modified modify VERB iajs-2610 271 19 variational variational ADJ iajs-2610 271 20 iteration iteration NOUN iajs-2610 271 21 method method NOUN iajs-2610 271 22 . . PUNCT iajs-2610 272 1 wulfenia wulfenia NOUN iajs-2610 272 2 . . PUNCT iajs-2610 273 1 2013 2013 NUM iajs-2610 273 2 , , PUNCT iajs-2610 273 3 20 20 NUM iajs-2610 273 4 , , PUNCT iajs-2610 273 5 4 4 NUM iajs-2610 273 6 , , PUNCT iajs-2610 273 7 174 174 NUM iajs-2610 273 8 - - SYM iajs-2610 273 9 180 180 NUM iajs-2610 273 10 . . NOUN iajs-2610 274 1 6 6 NUM iajs-2610 274 2 . . X iajs-2610 274 3 rawashdeh rawashdeh PROPN iajs-2610 274 4 , , PUNCT iajs-2610 274 5 m.s m.s PROPN iajs-2610 274 6 . . PROPN iajs-2610 274 7 ; ; PUNCT iajs-2610 274 8 maitama maitama PROPN iajs-2610 274 9 , , PUNCT iajs-2610 274 10 s. s. PROPN iajs-2610 274 11 solving solve VERB iajs-2610 274 12 pdes pde NOUN iajs-2610 274 13 using use VERB iajs-2610 274 14 the the DET iajs-2610 274 15 natural natural ADJ iajs-2610 274 16 decomposition decomposition NOUN iajs-2610 274 17 method method NOUN iajs-2610 274 18 . . PUNCT iajs-2610 275 1 cspcambridge cspcambridge NOUN iajs-2610 275 2 , , PUNCT iajs-2610 275 3 uk;i&s uk;i&s PROPN iajs-2610 275 4 - - PROPN iajs-2610 275 5 florida florida PROPN iajs-2610 275 6 usa usa PROPN iajs-2610 275 7 . . PROPN iajs-2610 275 8 2016 2016 NUM iajs-2610 275 9 , , PUNCT iajs-2610 275 10 23 23 NUM iajs-2610 275 11 , , PUNCT iajs-2610 275 12 1 1 NUM iajs-2610 275 13 , , PUNCT iajs-2610 275 14 1 1 NUM iajs-2610 275 15 - - SYM iajs-2610 275 16 10 10 NUM iajs-2610 275 17 . . PUNCT iajs-2610 276 1 7 7 X iajs-2610 276 2 . . X iajs-2610 276 3 rawashdeh rawashdeh NOUN iajs-2610 276 4 , , PUNCT iajs-2610 276 5 m. m. NOUN iajs-2610 276 6 using use VERB iajs-2610 276 7 the the DET iajs-2610 276 8 reduced reduce VERB iajs-2610 276 9 differential differential NOUN iajs-2610 276 10 transform transform NOUN iajs-2610 276 11 method method NOUN iajs-2610 276 12 to to PART iajs-2610 276 13 solve solve VERB iajs-2610 276 14 nonlinear nonlinear ADJ iajs-2610 276 15 pdes pde NOUN iajs-2610 276 16 arises arise VERB iajs-2610 276 17 in in ADP iajs-2610 276 18 biology biology NOUN iajs-2610 276 19 and and CCONJ iajs-2610 276 20 physics physics NOUN iajs-2610 276 21 . . PUNCT iajs-2610 277 1 world world PROPN iajs-2610 277 2 applied apply VERB iajs-2610 277 3 sciences science NOUN iajs-2610 277 4 journal journal NOUN iajs-2610 277 5 . . PUNCT iajs-2610 278 1 2013 2013 NUM iajs-2610 278 2 , , PUNCT iajs-2610 278 3 23 23 NUM iajs-2610 278 4 , , PUNCT iajs-2610 278 5 8 8 NUM iajs-2610 278 6 , , PUNCT iajs-2610 278 7 10371043 10371043 NUM iajs-2610 278 8 , , PUNCT iajs-2610 278 9 doi doi NOUN iajs-2610 278 10 : : PUNCT iajs-2610 278 11 10.5829 10.5829 NUM iajs-2610 278 12 / / SYM iajs-2610 278 13 idosi.wasj.2013.23.08.899 idosi.wasj.2013.23.08.899 NUM iajs-2610 278 14 . . NOUN iajs-2610 278 15 8 8 NUM iajs-2610 278 16 . . X iajs-2610 278 17 rawashdeh rawashdeh NOUN iajs-2610 278 18 , , PUNCT iajs-2610 278 19 m. m. NOUN iajs-2610 278 20 ; ; PUNCT iajs-2610 278 21 obeidat obeidat NOUN iajs-2610 278 22 , , PUNCT iajs-2610 278 23 n. n. NOUN iajs-2610 278 24 a. a. 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