id sid tid token lemma pos iajs-2678 1 1 60 60 NUM iajs-2678 1 2 new new ADJ iajs-2678 1 3 travelling travel VERB iajs-2678 1 4 wave wave NOUN iajs-2678 1 5 solution solution NOUN iajs-2678 1 6 of of ADP iajs-2678 1 7 burgers burger NOUN iajs-2678 1 8 equations equation NOUN iajs-2678 1 9 abstract abstract ADJ iajs-2678 1 10 in in ADP iajs-2678 1 11 this this DET iajs-2678 1 12 paper paper NOUN iajs-2678 1 13 , , PUNCT iajs-2678 1 14 we we PRON iajs-2678 1 15 studied study VERB iajs-2678 1 16 the the DET iajs-2678 1 17 travelling travel VERB iajs-2678 1 18 wave wave NOUN iajs-2678 1 19 solving solve VERB iajs-2678 1 20 for for ADP iajs-2678 1 21 some some DET iajs-2678 1 22 models model NOUN iajs-2678 1 23 of of ADP iajs-2678 1 24 burger burger NOUN iajs-2678 1 25 's 's PART iajs-2678 1 26 equations equation NOUN iajs-2678 1 27 . . PUNCT iajs-2678 2 1 we we PRON iajs-2678 2 2 used use VERB iajs-2678 2 3 sine sine ADJ iajs-2678 2 4 - - PUNCT iajs-2678 2 5 cosine cosine NOUN iajs-2678 2 6 method method NOUN iajs-2678 2 7 to to AUX iajs-2678 2 8 solution solution NOUN iajs-2678 2 9 nonlinear nonlinear ADJ iajs-2678 2 10 equation equation NOUN iajs-2678 2 11 and and CCONJ iajs-2678 2 12 we we PRON iajs-2678 2 13 used use VERB iajs-2678 2 14 a a DET iajs-2678 2 15 direct direct ADJ iajs-2678 2 16 solution solution NOUN iajs-2678 2 17 after after ADP iajs-2678 2 18 getting get VERB iajs-2678 2 19 travelling travel VERB iajs-2678 2 20 wave wave NOUN iajs-2678 2 21 equation equation NOUN iajs-2678 2 22 . . PUNCT iajs-2678 3 1 keywords keyword NOUN iajs-2678 3 2 : : PUNCT iajs-2678 3 3 travelling travel VERB iajs-2678 3 4 wave wave NOUN iajs-2678 3 5 solution solution NOUN iajs-2678 3 6 , , PUNCT iajs-2678 3 7 cosine cosine NOUN iajs-2678 3 8 - - PUNCT iajs-2678 3 9 sine sine ADJ iajs-2678 3 10 method method NOUN iajs-2678 3 11 , , PUNCT iajs-2678 3 12 nonlinear nonlinear ADJ iajs-2678 3 13 differential differential ADJ iajs-2678 3 14 equations equation NOUN iajs-2678 3 15 , , PUNCT iajs-2678 3 16 wave wave NOUN iajs-2678 3 17 equations equation NOUN iajs-2678 3 18 . . PUNCT iajs-2678 4 1 1.introduction 1.introduction NUM iajs-2678 4 2 a a DET iajs-2678 4 3 large large ADJ iajs-2678 4 4 variety variety NOUN iajs-2678 4 5 of of ADP iajs-2678 4 6 physical physical ADJ iajs-2678 4 7 , , PUNCT iajs-2678 4 8 chemical chemical NOUN iajs-2678 4 9 , , PUNCT iajs-2678 4 10 and and CCONJ iajs-2678 4 11 biological biological ADJ iajs-2678 4 12 phenomena phenomenon NOUN iajs-2678 4 13 are be AUX iajs-2678 4 14 waves wave NOUN iajs-2678 4 15 , , PUNCT iajs-2678 4 16 such such ADJ iajs-2678 4 17 as as ADP iajs-2678 4 18 sound sound ADJ iajs-2678 4 19 waves wave NOUN iajs-2678 4 20 , , PUNCT iajs-2678 4 21 string string NOUN iajs-2678 4 22 waves wave NOUN iajs-2678 4 23 , , PUNCT iajs-2678 4 24 water water NOUN iajs-2678 4 25 waves wave NOUN iajs-2678 4 26 , , PUNCT iajs-2678 4 27 etc etc X iajs-2678 4 28 . . X iajs-2678 4 29 wave wave NOUN iajs-2678 4 30 phenomena phenomenon NOUN iajs-2678 4 31 are be AUX iajs-2678 4 32 very very ADV iajs-2678 4 33 important important ADJ iajs-2678 4 34 in in ADP iajs-2678 4 35 dispersion dispersion NOUN iajs-2678 4 36 , , PUNCT iajs-2678 4 37 diffusion diffusion NOUN iajs-2678 4 38 , , PUNCT iajs-2678 4 39 reaction reaction NOUN iajs-2678 4 40 , , PUNCT iajs-2678 4 41 and and CCONJ iajs-2678 4 42 convection convection NOUN iajs-2678 4 43 , , PUNCT iajs-2678 4 44 and and CCONJ iajs-2678 4 45 such such ADJ iajs-2678 4 46 phenomena phenomenon NOUN iajs-2678 4 47 are be AUX iajs-2678 4 48 represented represent VERB iajs-2678 4 49 by by ADP iajs-2678 4 50 nonlinear nonlinear ADJ iajs-2678 4 51 partial partial ADJ iajs-2678 4 52 differential differential NOUN iajs-2678 4 53 equations equation NOUN iajs-2678 4 54 . . PUNCT iajs-2678 5 1 moving move VERB iajs-2678 5 2 waves wave NOUN iajs-2678 5 3 are be AUX iajs-2678 5 4 visible visible ADJ iajs-2678 5 5 in in ADP iajs-2678 5 6 many many ADJ iajs-2678 5 7 linear linear ADJ iajs-2678 5 8 equations equation NOUN iajs-2678 5 9 and and CCONJ iajs-2678 5 10 nonlinear nonlinear ADJ iajs-2678 5 11 wave wave NOUN iajs-2678 5 12 modeling modeling NOUN iajs-2678 5 13 , , PUNCT iajs-2678 5 14 such such ADJ iajs-2678 5 15 as as ADP iajs-2678 5 16 heat heat NOUN iajs-2678 5 17 waves wave NOUN iajs-2678 5 18 , , PUNCT iajs-2678 5 19 string string NOUN iajs-2678 5 20 waves wave NOUN iajs-2678 5 21 , , PUNCT iajs-2678 5 22 and and CCONJ iajs-2678 5 23 the the DET iajs-2678 5 24 cycle cycle NOUN iajs-2678 5 25 life life NOUN iajs-2678 5 26 of of ADP iajs-2678 5 27 some some DET iajs-2678 5 28 organisms organism NOUN iajs-2678 5 29 one one NUM iajs-2678 5 30 of of ADP iajs-2678 5 31 the the DET iajs-2678 5 32 travelling travel VERB iajs-2678 5 33 waves wave NOUN iajs-2678 5 34 , , PUNCT iajs-2678 5 35 etc etc X iajs-2678 5 36 . . X iajs-2678 5 37 one one NUM iajs-2678 5 38 of of ADP iajs-2678 5 39 the the DET iajs-2678 5 40 most most ADV iajs-2678 5 41 important important ADJ iajs-2678 5 42 methods method NOUN iajs-2678 5 43 for for ADP iajs-2678 5 44 solving solve VERB iajs-2678 5 45 wave wave NOUN iajs-2678 5 46 equations equation NOUN iajs-2678 5 47 is be AUX iajs-2678 5 48 the the DET iajs-2678 5 49 travelling travel VERB iajs-2678 5 50 wave wave NOUN iajs-2678 5 51 solution solution NOUN iajs-2678 5 52 ( ( PUNCT iajs-2678 5 53 tws tws NOUN iajs-2678 5 54 ) ) PUNCT iajs-2678 5 55 , , PUNCT iajs-2678 5 56 which which PRON iajs-2678 5 57 many many ADJ iajs-2678 5 58 researchers researcher NOUN iajs-2678 5 59 have have AUX iajs-2678 5 60 discussed discuss VERB iajs-2678 5 61 ( ( PUNCT iajs-2678 5 62 see see VERB iajs-2678 5 63 [ [ X iajs-2678 5 64 1,2 1,2 NUM iajs-2678 5 65 ] ] PUNCT iajs-2678 5 66 , , PUNCT iajs-2678 5 67 the the DET iajs-2678 5 68 tanh tanh NOUN iajs-2678 5 69 - - PUNCT iajs-2678 5 70 coth coth NOUN iajs-2678 5 71 method method NOUN iajs-2678 5 72 [ [ X iajs-2678 5 73 3 3 NUM iajs-2678 5 74 ] ] PUNCT iajs-2678 5 75 , , PUNCT iajs-2678 5 76 the the DET iajs-2678 5 77 tanh tanh NOUN iajs-2678 5 78 [ [ X iajs-2678 5 79 4,5 4,5 NUM iajs-2678 5 80 ] ] PUNCT iajs-2678 5 81 sine sine ADJ iajs-2678 5 82 - - PUNCT iajs-2678 5 83 cosine cosine NOUN iajs-2678 5 84 method method NOUN iajs-2678 5 85 [ [ X iajs-2678 5 86 6 6 NUM iajs-2678 5 87 ] ] PUNCT iajs-2678 5 88 , , PUNCT iajs-2678 5 89 and and CCONJ iajs-2678 5 90 see see VERB iajs-2678 5 91 [ [ X iajs-2678 5 92 7 7 NUM iajs-2678 5 93 - - SYM iajs-2678 5 94 12 12 NUM iajs-2678 5 95 ] ] PUNCT iajs-2678 5 96 ) ) PUNCT iajs-2678 5 97 . . PUNCT iajs-2678 6 1 2.infinite 2.infinite NUM iajs-2678 6 2 sires sire NOUN iajs-2678 6 3 method method NOUN iajs-2678 6 4 [ [ X iajs-2678 6 5 5,9 5,9 NUM iajs-2678 6 6 ] ] PUNCT iajs-2678 6 7 to to PART iajs-2678 6 8 clarify clarify VERB iajs-2678 6 9 the the DET iajs-2678 6 10 travelling travel VERB iajs-2678 6 11 wave wave NOUN iajs-2678 6 12 solutions solution NOUN iajs-2678 6 13 method method NOUN iajs-2678 6 14 , , PUNCT iajs-2678 6 15 we we PRON iajs-2678 6 16 note note VERB iajs-2678 6 17 the the DET iajs-2678 6 18 following following NOUN iajs-2678 6 19 : : PUNCT iajs-2678 6 20 let let VERB iajs-2678 6 21 the the DET iajs-2678 6 22 nonlinear nonlinear ADJ iajs-2678 6 23 partial partial ADJ iajs-2678 6 24 differential differential NOUN iajs-2678 6 25 equation equation NOUN iajs-2678 6 26 has have VERB iajs-2678 6 27 the the DET iajs-2678 6 28 form form NOUN iajs-2678 6 29 𝑔(𝑒 𝑔(𝑒 PROPN iajs-2678 6 30 , , PUNCT iajs-2678 6 31 𝑒𝑑 𝑒𝑑 NOUN iajs-2678 6 32 , , PUNCT iajs-2678 6 33 𝑒π‘₯ 𝑒π‘₯ ADP iajs-2678 6 34 , , PUNCT iajs-2678 6 35 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ PROPN iajs-2678 6 36 , , PUNCT iajs-2678 6 37 … … PUNCT iajs-2678 6 38 ) ) PUNCT iajs-2678 7 1 = = SYM iajs-2678 7 2 0 0 NUM iajs-2678 7 3 , , PUNCT iajs-2678 7 4 ( ( PUNCT iajs-2678 7 5 1 1 X iajs-2678 7 6 ) ) PUNCT iajs-2678 7 7 ibn ibn PROPN iajs-2678 7 8 al al PROPN iajs-2678 7 9 haitham haitham PROPN iajs-2678 7 10 journal journal PROPN iajs-2678 7 11 for for ADP iajs-2678 7 12 pure pure ADJ iajs-2678 7 13 and and CCONJ iajs-2678 7 14 applied apply VERB iajs-2678 7 15 science science NOUN iajs-2678 7 16 journal journal PROPN iajs-2678 7 17 homepage homepage NOUN iajs-2678 7 18 : : PUNCT iajs-2678 7 19 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2678 7 20 doi doi NOUN iajs-2678 7 21 : : PUNCT iajs-2678 7 22 10.30526/34.3.2678 10.30526/34.3.2678 PROPN iajs-2678 7 23 article article NOUN iajs-2678 7 24 history history NOUN iajs-2678 7 25 : : PUNCT iajs-2678 7 26 received receive VERB iajs-2678 7 27 19 19 NUM iajs-2678 7 28 january january NOUN iajs-2678 7 29 2021 2021 NUM iajs-2678 7 30 , , PUNCT iajs-2678 7 31 accepted accept VERB iajs-2678 7 32 11 11 NUM iajs-2678 7 33 april april PROPN iajs-2678 7 34 2021 2021 NUM iajs-2678 7 35 , , PUNCT iajs-2678 7 36 published publish VERB iajs-2678 7 37 in in ADP iajs-2678 7 38 2021 2021 NUM iajs-2678 7 39 . . PUNCT iajs-2678 8 1 zainab zainab PROPN iajs-2678 8 2 john john PROPN iajs-2678 8 3 department department PROPN iajs-2678 8 4 of of ADP iajs-2678 8 5 mathematical mathematical PROPN iajs-2678 8 6 , , PUNCT iajs-2678 8 7 college college NOUN iajs-2678 8 8 of of ADP iajs-2678 8 9 science science PROPN iajs-2678 8 10 , , PUNCT iajs-2678 8 11 al al PROPN iajs-2678 8 12 - - PUNCT iajs-2678 8 13 mustansiriyah mustansiriyah PROPN iajs-2678 8 14 university university NOUN iajs-2678 8 15 , , PUNCT iajs-2678 8 16 baghdad baghdad PROPN iajs-2678 8 17 , , PUNCT iajs-2678 8 18 iraq iraq PROPN iajs-2678 8 19 . . PUNCT iajs-2678 9 1 zainabjohn22@uomustansiriyah.edu.iq zainabjohn22@uomustansiriyah.edu.iq PROPN iajs-2678 9 2 basim basim PROPN iajs-2678 9 3 akhudir akhudir PROPN iajs-2678 9 4 abbas abbas PROPN iajs-2678 9 5 department department PROPN iajs-2678 9 6 of of ADP iajs-2678 9 7 mathematical mathematical PROPN iajs-2678 9 8 , , PUNCT iajs-2678 9 9 college college NOUN iajs-2678 9 10 of of ADP iajs-2678 9 11 science science PROPN iajs-2678 9 12 , , PUNCT iajs-2678 9 13 al al PROPN iajs-2678 9 14 - - PUNCT iajs-2678 9 15 mustansiriyah mustansiriyah PROPN iajs-2678 9 16 university university NOUN iajs-2678 9 17 , , PUNCT iajs-2678 9 18 baghdad baghdad PROPN iajs-2678 9 19 , , PUNCT iajs-2678 9 20 iraq iraq PROPN iajs-2678 9 21 . . PUNCT iajs-2678 10 1 baasim_math@uomustansiriyah.edu.iq baasim_math@uomustansiriyah.edu.iq PROPN iajs-2678 10 2 ibn ibn PROPN iajs-2678 10 3 al al PROPN iajs-2678 10 4 - - PUNCT iajs-2678 10 5 haitham haitham PROPN iajs-2678 10 6 jour jour X iajs-2678 10 7 . . PROPN iajs-2678 11 1 for for ADP iajs-2678 11 2 pure pure ADJ iajs-2678 11 3 & & CCONJ iajs-2678 11 4 appl appl PROPN iajs-2678 11 5 . . PUNCT iajs-2678 12 1 sci sci PROPN iajs-2678 12 2 . . PROPN iajs-2678 13 1 34(3)2021 34(3)2021 NUM iajs-2678 13 2 61 61 NUM iajs-2678 13 3 where where SCONJ iajs-2678 13 4 𝑒 𝑒 AUX iajs-2678 13 5 = = SYM iajs-2678 13 6 ( ( PUNCT iajs-2678 13 7 π‘₯ π‘₯ PROPN iajs-2678 13 8 , , PUNCT iajs-2678 13 9 𝑑 𝑑 PROPN iajs-2678 13 10 ) ) PUNCT iajs-2678 13 11 is be AUX iajs-2678 13 12 a a DET iajs-2678 13 13 solution solution NOUN iajs-2678 13 14 of of ADP iajs-2678 13 15 equation equation NOUN iajs-2678 13 16 ( ( PUNCT iajs-2678 13 17 1 1 NUM iajs-2678 13 18 ) ) PUNCT iajs-2678 13 19 , , PUNCT iajs-2678 13 20 and and CCONJ iajs-2678 13 21 g g NOUN iajs-2678 13 22 is be AUX iajs-2678 13 23 a a DET iajs-2678 13 24 polynomial polynomial NOUN iajs-2678 13 25 with with ADP iajs-2678 13 26 respect respect NOUN iajs-2678 13 27 to to ADP iajs-2678 13 28 u u NOUN iajs-2678 13 29 and and CCONJ iajs-2678 13 30 with with ADP iajs-2678 13 31 its its PRON iajs-2678 13 32 derivatives derivative NOUN iajs-2678 13 33 . . PUNCT iajs-2678 14 1 to to PART iajs-2678 14 2 solve solve VERB iajs-2678 14 3 eq eq ADP iajs-2678 14 4 . . PUNCT iajs-2678 15 1 ( ( PUNCT iajs-2678 15 2 1 1 NUM iajs-2678 15 3 ) ) PUNCT iajs-2678 15 4 , , PUNCT iajs-2678 15 5 we we PRON iajs-2678 15 6 follow follow VERB iajs-2678 15 7 the the DET iajs-2678 15 8 following follow VERB iajs-2678 15 9 steps step NOUN iajs-2678 15 10 : : PUNCT iajs-2678 15 11 step step NOUN iajs-2678 15 12 1 1 NUM iajs-2678 15 13 : : PUNCT iajs-2678 15 14 first first ADV iajs-2678 15 15 , , PUNCT iajs-2678 15 16 we we PRON iajs-2678 15 17 change change VERB iajs-2678 15 18 the the DET iajs-2678 15 19 independent independent ADJ iajs-2678 15 20 variables variable NOUN iajs-2678 15 21 x x PROPN iajs-2678 15 22 , , PUNCT iajs-2678 15 23 t t PROPN iajs-2678 15 24 , , PUNCT iajs-2678 15 25 by by ADP iajs-2678 15 26 using use VERB iajs-2678 15 27 one one NUM iajs-2678 15 28 independent independent ADJ iajs-2678 15 29 variable variable NOUN iajs-2678 15 30 πœ‰ πœ‰ NOUN iajs-2678 15 31 that that PRON iajs-2678 15 32 combines combine VERB iajs-2678 15 33 the the DET iajs-2678 15 34 two two NUM iajs-2678 15 35 variables variable NOUN iajs-2678 15 36 as as SCONJ iajs-2678 15 37 shown show VERB iajs-2678 15 38 let let VERB iajs-2678 15 39 𝑒(π‘₯ 𝑒(π‘₯ PROPN iajs-2678 15 40 , , PUNCT iajs-2678 15 41 𝑑 𝑑 X iajs-2678 15 42 ) ) PUNCT iajs-2678 15 43 = = SYM iajs-2678 15 44 𝑓(πœ‰ 𝑓(πœ‰ PROPN iajs-2678 15 45 ) ) PUNCT iajs-2678 15 46 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN iajs-2678 15 47 πœ‰ πœ‰ NOUN iajs-2678 15 48 = = PUNCT iajs-2678 15 49 π‘₯ π‘₯ PROPN iajs-2678 15 50 βˆ’ βˆ’ ADV iajs-2678 15 51 𝑐𝑑 𝑐𝑑 INTJ iajs-2678 15 52 , , PUNCT iajs-2678 15 53 ( ( PUNCT iajs-2678 15 54 2 2 X iajs-2678 15 55 ) ) PUNCT iajs-2678 15 56 where where SCONJ iajs-2678 15 57 πœ‰ πœ‰ NOUN iajs-2678 15 58 is be AUX iajs-2678 15 59 a a DET iajs-2678 15 60 wave wave NOUN iajs-2678 15 61 variable variable NOUN iajs-2678 15 62 and and CCONJ iajs-2678 15 63 𝑐 𝑐 PROPN iajs-2678 15 64 is be AUX iajs-2678 15 65 constant constant ADJ iajs-2678 15 66 . . PUNCT iajs-2678 16 1 step2 step2 NOUN iajs-2678 16 2 : : PUNCT iajs-2678 16 3 derivative derivative ADJ iajs-2678 16 4 equation equation NOUN iajs-2678 16 5 ( ( PUNCT iajs-2678 16 6 2 2 NUM iajs-2678 16 7 ) ) PUNCT iajs-2678 16 8 with with ADP iajs-2678 16 9 respect respect NOUN iajs-2678 16 10 to to ADP iajs-2678 16 11 π‘₯ π‘₯ PROPN iajs-2678 16 12 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-2678 16 13 𝑑 𝑑 PROPN iajs-2678 16 14 , , PUNCT iajs-2678 16 15 we we PRON iajs-2678 16 16 have have VERB iajs-2678 16 17 the the DET iajs-2678 16 18 following follow VERB iajs-2678 16 19 ordinary ordinary ADJ iajs-2678 16 20 differential differential ADJ iajs-2678 16 21 equations equation NOUN iajs-2678 16 22 : : PUNCT iajs-2678 16 23 𝑒π‘₯ 𝑒π‘₯ ADP iajs-2678 16 24 = = PUNCT iajs-2678 16 25 𝑒′ 𝑒′ NUM iajs-2678 16 26 = = PUNCT iajs-2678 16 27 𝑑𝑒 𝑑𝑒 PROPN iajs-2678 16 28 𝑑 𝑑 NOUN iajs-2678 16 29 πœ‰ πœ‰ NOUN iajs-2678 16 30 ; ; PUNCT iajs-2678 16 31 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X iajs-2678 16 32 = = SYM iajs-2678 16 33 𝑒′′ 𝑒′′ PROPN iajs-2678 16 34 = = SYM iajs-2678 16 35 𝑑2𝑒 𝑑2𝑒 PROPN iajs-2678 16 36 𝑑 𝑑 PROPN iajs-2678 16 37 πœ‰2 πœ‰2 PROPN iajs-2678 16 38 ; ; PUNCT iajs-2678 16 39 𝑒𝑑 𝑒𝑑 NOUN iajs-2678 16 40 = = PUNCT iajs-2678 16 41 βˆ’π‘π‘’β€² βˆ’π‘π‘’β€² NOUN iajs-2678 16 42 = = PUNCT iajs-2678 17 1 βˆ’π‘ βˆ’π‘ ADV iajs-2678 18 1 𝑑𝑒 𝑑𝑒 X iajs-2678 18 2 𝑑 𝑑 NOUN iajs-2678 18 3 πœ‰ πœ‰ PROPN iajs-2678 18 4 ; ; PUNCT iajs-2678 18 5 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN iajs-2678 18 6 = = SYM iajs-2678 18 7 𝑐2𝑒′′ 𝑐2𝑒′′ PROPN iajs-2678 18 8 = = NOUN iajs-2678 18 9 𝑐2 𝑐2 NOUN iajs-2678 18 10 𝑑2𝑒 𝑑2𝑒 NOUN iajs-2678 18 11 𝑑 𝑑 PROPN iajs-2678 18 12 πœ‰2 πœ‰2 PROPN iajs-2678 18 13 ; ; PUNCT iajs-2678 19 1 𝑒π‘₯𝑑 𝑒π‘₯𝑑 PROPN iajs-2678 19 2 = = SYM iajs-2678 19 3 βˆ’π‘π‘’β€²β€² βˆ’π‘π‘’β€²β€² PROPN iajs-2678 19 4 = = NOUN iajs-2678 19 5 βˆ’π‘ βˆ’π‘ ADJ iajs-2678 19 6 𝑑2𝑒 𝑑2𝑒 PROPN iajs-2678 19 7 𝑑 𝑑 PROPN iajs-2678 19 8 πœ‰2 πœ‰2 PROPN iajs-2678 19 9 ( ( PUNCT iajs-2678 19 10 3 3 NUM iajs-2678 19 11 ) ) PUNCT iajs-2678 19 12 when when SCONJ iajs-2678 19 13 substitute substitute NOUN iajs-2678 19 14 ( ( PUNCT iajs-2678 19 15 3 3 NUM iajs-2678 19 16 ) ) PUNCT iajs-2678 19 17 into into ADP iajs-2678 19 18 ( ( PUNCT iajs-2678 19 19 1 1 NUM iajs-2678 19 20 ) ) PUNCT iajs-2678 19 21 , , PUNCT iajs-2678 19 22 we we PRON iajs-2678 19 23 find find VERB iajs-2678 19 24 that that SCONJ iajs-2678 19 25 equation equation NOUN iajs-2678 19 26 ( ( PUNCT iajs-2678 19 27 1 1 X iajs-2678 19 28 ) ) PUNCT iajs-2678 19 29 is be AUX iajs-2678 19 30 transform transform VERB iajs-2678 19 31 to to ADP iajs-2678 19 32 ( ( PUNCT iajs-2678 19 33 linear linear ADJ iajs-2678 19 34 or or CCONJ iajs-2678 19 35 nonlinear nonlinear ADJ iajs-2678 19 36 ) ) PUNCT iajs-2678 19 37 ordinary ordinary ADJ iajs-2678 19 38 differential differential ADJ iajs-2678 19 39 equation equation NOUN iajs-2678 19 40 transformed transform VERB iajs-2678 19 41 into into ADP iajs-2678 19 42 𝐺 𝐺 PROPN iajs-2678 19 43 ( ( PUNCT iajs-2678 19 44 𝑓(πœ‰ 𝑓(πœ‰ PROPN iajs-2678 19 45 ) ) PUNCT iajs-2678 19 46 , , PUNCT iajs-2678 19 47 𝑑𝑓 𝑑𝑓 PROPN iajs-2678 19 48 π‘‘πœ‰ π‘‘πœ‰ ADP iajs-2678 19 49 , , PUNCT iajs-2678 19 50 𝑑2𝑓 𝑑2𝑓 PROPN iajs-2678 19 51 π‘‘πœ‰2 π‘‘πœ‰2 VERB iajs-2678 19 52 , , PUNCT iajs-2678 19 53 … … PUNCT iajs-2678 19 54 ) ) PUNCT iajs-2678 19 55 ( ( PUNCT iajs-2678 19 56 4 4 X iajs-2678 19 57 ) ) PUNCT iajs-2678 19 58 where where SCONJ iajs-2678 19 59 g g PROPN iajs-2678 19 60 is be AUX iajs-2678 19 61 a a DET iajs-2678 19 62 polynomial polynomial ADJ iajs-2678 19 63 in in ADP iajs-2678 19 64 𝑓(πœ‰ 𝑓(πœ‰ PROPN iajs-2678 19 65 ) ) PUNCT iajs-2678 19 66 and and CCONJ iajs-2678 19 67 with with ADP iajs-2678 19 68 its its PRON iajs-2678 19 69 derivatives derivative NOUN iajs-2678 19 70 , , PUNCT iajs-2678 19 71 step step VERB iajs-2678 19 72 3 3 NUM iajs-2678 19 73 : : PUNCT iajs-2678 19 74 integrate integrate VERB iajs-2678 19 75 ( ( PUNCT iajs-2678 19 76 4 4 NUM iajs-2678 19 77 ) ) PUNCT iajs-2678 19 78 with with ADP iajs-2678 19 79 respect respect NOUN iajs-2678 19 80 to to ADP iajs-2678 19 81 πœ‰ πœ‰ NOUN iajs-2678 19 82 , , PUNCT iajs-2678 19 83 and and CCONJ iajs-2678 19 84 let let VERB iajs-2678 19 85 the the DET iajs-2678 19 86 constant constant NOUN iajs-2678 19 87 of of ADP iajs-2678 19 88 integration integration NOUN iajs-2678 19 89 equal equal ADJ iajs-2678 19 90 to to ADP iajs-2678 19 91 zero zero NUM iajs-2678 19 92 . . PUNCT iajs-2678 20 1 step step NOUN iajs-2678 20 2 4 4 NUM iajs-2678 20 3 : : PUNCT iajs-2678 20 4 ( ( PUNCT iajs-2678 20 5 a a X iajs-2678 20 6 ) ) PUNCT iajs-2678 20 7 we we PRON iajs-2678 20 8 can can AUX iajs-2678 20 9 solve solve VERB iajs-2678 20 10 equation equation NOUN iajs-2678 20 11 ( ( PUNCT iajs-2678 20 12 4 4 NUM iajs-2678 20 13 ) ) PUNCT iajs-2678 20 14 directly directly ADV iajs-2678 20 15 through through ADP iajs-2678 20 16 many many ADJ iajs-2678 20 17 methods method NOUN iajs-2678 20 18 that that PRON iajs-2678 20 19 solve solve VERB iajs-2678 20 20 ordinary ordinary ADJ iajs-2678 20 21 differential differential ADJ iajs-2678 20 22 equations equation NOUN iajs-2678 20 23 . . PUNCT iajs-2678 21 1 or or CCONJ iajs-2678 21 2 ( ( PUNCT iajs-2678 21 3 b b X iajs-2678 21 4 ) ) PUNCT iajs-2678 21 5 use use VERB iajs-2678 21 6 sine sine ADJ iajs-2678 21 7 - - PUNCT iajs-2678 21 8 cosine cosine NOUN iajs-2678 21 9 method method NOUN iajs-2678 21 10 to to PART iajs-2678 21 11 solve solve VERB iajs-2678 21 12 equation equation NOUN iajs-2678 21 13 ( ( PUNCT iajs-2678 21 14 4 4 NUM iajs-2678 21 15 ) ) PUNCT iajs-2678 21 16 . . PUNCT iajs-2678 22 1 3 3 X iajs-2678 22 2 . . X iajs-2678 22 3 applications application NOUN iajs-2678 22 4 of of ADP iajs-2678 22 5 the the DET iajs-2678 22 6 travelling travel VERB iajs-2678 22 7 wave wave NOUN iajs-2678 22 8 solution solution NOUN iajs-2678 22 9 3.1 3.1 NUM iajs-2678 22 10 : : PUNCT iajs-2678 22 11 some some DET iajs-2678 22 12 models model NOUN iajs-2678 22 13 of of ADP iajs-2678 22 14 burgers burger NOUN iajs-2678 22 15 equations equation NOUN iajs-2678 22 16 [ [ X iajs-2678 22 17 13 13 NUM iajs-2678 22 18 ] ] X iajs-2678 22 19 the the DET iajs-2678 22 20 viscid viscid NOUN iajs-2678 22 21 burgers burger NOUN iajs-2678 22 22 equation equation NOUN iajs-2678 22 23 to to PART iajs-2678 22 24 be be AUX iajs-2678 22 25 the the DET iajs-2678 22 26 nonlinear nonlinear ADJ iajs-2678 22 27 parabolic parabolic ADJ iajs-2678 22 28 pde pde NOUN iajs-2678 22 29 has have VERB iajs-2678 22 30 the the DET iajs-2678 22 31 form form NOUN iajs-2678 22 32 𝑣𝑑 𝑣𝑑 ADP iajs-2678 22 33 = = SYM iajs-2678 22 34 𝑣π‘₯π‘₯+∈ 𝑣π‘₯π‘₯+∈ PROPN iajs-2678 22 35 𝑣𝑣π‘₯ 𝑣𝑣π‘₯ NOUN iajs-2678 22 36 . . PUNCT iajs-2678 23 1 , , PUNCT iajs-2678 23 2 0 0 NUM iajs-2678 23 3 < < X iajs-2678 23 4 πœ– πœ– X iajs-2678 23 5 ≀ ≀ NUM iajs-2678 23 6 1 1 NUM iajs-2678 23 7 . . PUNCT iajs-2678 24 1 ( ( PUNCT iajs-2678 24 2 5 5 NUM iajs-2678 24 3 ) ) PUNCT iajs-2678 24 4 3.1.1 3.1.1 NUM iajs-2678 24 5 . . PUNCT iajs-2678 25 1 travelling travel VERB iajs-2678 25 2 wave wave NOUN iajs-2678 25 3 solution solution NOUN iajs-2678 25 4 the the DET iajs-2678 25 5 travelling travel VERB iajs-2678 25 6 wave wave NOUN iajs-2678 25 7 solution solution NOUN iajs-2678 25 8 of of ADP iajs-2678 25 9 eq.(5 eq.(5 NOUN iajs-2678 25 10 ) ) PUNCT iajs-2678 25 11 is be AUX iajs-2678 25 12 πœ‰ πœ‰ X iajs-2678 25 13 = = PUNCT iajs-2678 25 14 π‘₯ π‘₯ PROPN iajs-2678 25 15 βˆ’ βˆ’ ADP iajs-2678 25 16 𝑐𝑑 𝑐𝑑 INTJ iajs-2678 25 17 β‡’ β‡’ PROPN iajs-2678 25 18 𝑣(π‘₯ 𝑣(π‘₯ PROPN iajs-2678 25 19 , , PUNCT iajs-2678 25 20 𝑑 𝑑 X iajs-2678 25 21 ) ) PUNCT iajs-2678 25 22 = = SYM iajs-2678 25 23 𝑣(πœ‰ 𝑣(πœ‰ NOUN iajs-2678 25 24 ) ) PUNCT iajs-2678 25 25 , , PUNCT iajs-2678 25 26 𝑣π‘₯ 𝑣π‘₯ X iajs-2678 25 27 = = PUNCT iajs-2678 25 28 𝑣′ 𝑣′ NUM iajs-2678 25 29 = = PUNCT iajs-2678 26 1 𝑑𝑒 𝑑𝑒 PROPN iajs-2678 26 2 𝑑 𝑑 NOUN iajs-2678 26 3 πœ‰ πœ‰ NOUN iajs-2678 26 4 ; ; PUNCT iajs-2678 26 5 𝑣π‘₯π‘₯ 𝑣π‘₯π‘₯ NOUN iajs-2678 26 6 = = SYM iajs-2678 26 7 𝑣′′ 𝑣′′ PROPN iajs-2678 26 8 = = SYM iajs-2678 26 9 𝑑2𝑣 𝑑2𝑣 PROPN iajs-2678 26 10 𝑑 𝑑 PROPN iajs-2678 26 11 πœ‰2 πœ‰2 PROPN iajs-2678 26 12 ; ; PUNCT iajs-2678 26 13 𝑣𝑑 𝑣𝑑 ADP iajs-2678 26 14 = = SYM iajs-2678 26 15 βˆ’π‘π‘’β€² βˆ’π‘π‘’β€² NOUN iajs-2678 26 16 = = PUNCT iajs-2678 26 17 βˆ’π‘ βˆ’π‘ X iajs-2678 26 18 𝑑𝑣 𝑑𝑣 ADP iajs-2678 26 19 𝑑 𝑑 NOUN iajs-2678 26 20 πœ‰ πœ‰ NOUN iajs-2678 26 21 . . PUNCT iajs-2678 27 1 ( ( PUNCT iajs-2678 27 2 6 6 X iajs-2678 27 3 ) ) PUNCT iajs-2678 27 4 substituting substitute VERB iajs-2678 27 5 eq eq NOUN iajs-2678 27 6 . . PUNCT iajs-2678 28 1 ( ( PUNCT iajs-2678 28 2 6 6 NUM iajs-2678 28 3 ) ) PUNCT iajs-2678 28 4 into into ADP iajs-2678 28 5 eq eq NOUN iajs-2678 28 6 . . PUNCT iajs-2678 29 1 ( ( PUNCT iajs-2678 29 2 5 5 NUM iajs-2678 29 3 ) ) PUNCT iajs-2678 29 4 , , PUNCT iajs-2678 29 5 we we PRON iajs-2678 29 6 have have AUX iajs-2678 29 7 got get VERB iajs-2678 29 8 𝑣′′ 𝑣′′ NOUN iajs-2678 29 9 + + CCONJ iajs-2678 29 10 𝑐𝑣′+∈ 𝑐𝑣′+∈ NUM iajs-2678 29 11 𝑣𝑣′ 𝑣𝑣′ NUM iajs-2678 29 12 = = SYM iajs-2678 29 13 0 0 NUM iajs-2678 29 14 ( ( PUNCT iajs-2678 29 15 7 7 X iajs-2678 29 16 ) ) PUNCT iajs-2678 29 17 𝑣′′ 𝑣′′ NOUN iajs-2678 29 18 + + CCONJ iajs-2678 30 1 𝑐𝑣′+∈ 𝑐𝑣′+∈ X iajs-2678 30 2 ( ( PUNCT iajs-2678 30 3 𝑣2 𝑣2 NOUN iajs-2678 30 4 2 2 NUM iajs-2678 30 5 ) ) PUNCT iajs-2678 30 6 β€² β€² NUM iajs-2678 31 1 = = SYM iajs-2678 31 2 0 0 PUNCT iajs-2678 32 1 ( ( PUNCT iajs-2678 32 2 8) 8) NUM iajs-2678 32 3 integral integral ADJ iajs-2678 32 4 eq eq X iajs-2678 32 5 . . PUNCT iajs-2678 33 1 ( ( PUNCT iajs-2678 33 2 8) 8) NUM iajs-2678 33 3 with with ADP iajs-2678 33 4 respect respect NOUN iajs-2678 33 5 to to ADP iajs-2678 33 6 πœ‰ πœ‰ NOUN iajs-2678 33 7 we we PRON iajs-2678 33 8 have have AUX iajs-2678 33 9 got get VERB iajs-2678 33 10 𝑣′ 𝑣′ NUM iajs-2678 33 11 + + X iajs-2678 33 12 𝑐𝑣 𝑐𝑣 ADP iajs-2678 33 13 + + CCONJ iajs-2678 33 14 πœ– πœ– PROPN iajs-2678 33 15 𝑣2 𝑣2 PROPN iajs-2678 33 16 2 2 NUM iajs-2678 33 17 = = NOUN iajs-2678 33 18 𝑐1 𝑐1 NOUN iajs-2678 33 19 ( ( PUNCT iajs-2678 33 20 9 9 X iajs-2678 33 21 ) ) PUNCT iajs-2678 33 22 ibn ibn PROPN iajs-2678 33 23 al al PROPN iajs-2678 33 24 - - PUNCT iajs-2678 33 25 haitham haitham PROPN iajs-2678 33 26 jour jour X iajs-2678 33 27 . . PROPN iajs-2678 34 1 for for ADP iajs-2678 34 2 pure pure ADJ iajs-2678 34 3 & & CCONJ iajs-2678 34 4 appl appl PROPN iajs-2678 34 5 . . PUNCT iajs-2678 35 1 sci sci PROPN iajs-2678 35 2 . . PROPN iajs-2678 36 1 34(3)2021 34(3)2021 NUM iajs-2678 36 2 62 62 NUM iajs-2678 36 3 let let VERB iajs-2678 36 4 𝑐1 𝑐1 NOUN iajs-2678 36 5 = = NOUN iajs-2678 36 6 0 0 PUNCT iajs-2678 36 7 𝑣′ 𝑣′ NUM iajs-2678 36 8 = = SYM iajs-2678 37 1 βˆ’(𝑐𝑣 βˆ’(𝑐𝑣 PROPN iajs-2678 37 2 + + CCONJ iajs-2678 37 3 πœ– πœ– X iajs-2678 37 4 𝑣2 𝑣2 PROPN iajs-2678 37 5 2 2 NUM iajs-2678 37 6 ) ) PUNCT iajs-2678 37 7 ( ( PUNCT iajs-2678 37 8 10 10 NUM iajs-2678 37 9 ) ) PUNCT iajs-2678 37 10 𝑑𝑣 𝑑𝑣 ADP iajs-2678 37 11 βˆ’(𝑐𝑣+πœ– βˆ’(𝑐𝑣+πœ– PUNCT iajs-2678 37 12 𝑣2 𝑣2 PROPN iajs-2678 37 13 2 2 NUM iajs-2678 37 14 ) ) PUNCT iajs-2678 37 15 = = SYM iajs-2678 37 16 π‘‘πœ‰ π‘‘πœ‰ ADP iajs-2678 37 17 by by ADP iajs-2678 37 18 solving solve VERB iajs-2678 37 19 this this DET iajs-2678 37 20 equation equation NOUN iajs-2678 37 21 , , PUNCT iajs-2678 37 22 we we PRON iajs-2678 37 23 have have AUX iajs-2678 37 24 got get VERB iajs-2678 37 25 𝑣(πœ‰ 𝑣(πœ‰ NOUN iajs-2678 37 26 ) ) PUNCT iajs-2678 38 1 = = SYM iajs-2678 38 2 𝑐 𝑐 PROPN iajs-2678 38 3 ( ( PUNCT iajs-2678 38 4 𝑐𝑒 𝑐𝑒 X iajs-2678 38 5 𝑐(πœ‰+𝑐2)βˆ’πœ–/2 𝑐(πœ‰+𝑐2)βˆ’πœ–/2 PROPN iajs-2678 38 6 ) ) PUNCT iajs-2678 38 7 ( ( PUNCT iajs-2678 38 8 11 11 NUM iajs-2678 38 9 ) ) PUNCT iajs-2678 38 10 where where SCONJ iajs-2678 38 11 = = PUNCT iajs-2678 38 12 π‘₯ π‘₯ X iajs-2678 39 1 βˆ’ βˆ’ ADV iajs-2678 39 2 𝑐𝑑 𝑐𝑑 INTJ iajs-2678 39 3 , , PUNCT iajs-2678 39 4 substituted substitute VERB iajs-2678 39 5 in in ADP iajs-2678 39 6 ( ( PUNCT iajs-2678 39 7 11 11 NUM iajs-2678 39 8 ) ) PUNCT iajs-2678 39 9 , , PUNCT iajs-2678 39 10 we we PRON iajs-2678 39 11 have have VERB iajs-2678 39 12 𝑣(π‘₯ 𝑣(π‘₯ PROPN iajs-2678 39 13 , , PUNCT iajs-2678 39 14 𝑑 𝑑 X iajs-2678 39 15 ) ) PUNCT iajs-2678 39 16 = = SYM iajs-2678 39 17 𝑐 𝑐 PROPN iajs-2678 39 18 ( ( PUNCT iajs-2678 39 19 𝑐𝑒((π‘₯βˆ’π‘π‘‘)+𝑐2)βˆ’πœ–/2 𝑐𝑒((π‘₯βˆ’π‘π‘‘)+𝑐2)βˆ’πœ–/2 NOUN iajs-2678 39 20 ) ) PUNCT iajs-2678 39 21 , , PUNCT iajs-2678 39 22 which which PRON iajs-2678 39 23 is be AUX iajs-2678 39 24 a a DET iajs-2678 39 25 general general ADJ iajs-2678 39 26 solution solution NOUN iajs-2678 39 27 of of ADP iajs-2678 39 28 burgers burger NOUN iajs-2678 39 29 equation equation NOUN iajs-2678 39 30 ( ( PUNCT iajs-2678 39 31 5 5 NUM iajs-2678 39 32 ) ) PUNCT iajs-2678 39 33 . . PUNCT iajs-2678 40 1 by by ADP iajs-2678 40 2 drawing draw VERB iajs-2678 40 3 the the DET iajs-2678 40 4 solution solution NOUN iajs-2678 40 5 function function NOUN iajs-2678 40 6 ( ( PUNCT iajs-2678 40 7 v(x v(x PROPN iajs-2678 40 8 , , PUNCT iajs-2678 40 9 t t PROPN iajs-2678 40 10 ) ) PUNCT iajs-2678 40 11 ) ) PUNCT iajs-2678 40 12 , , PUNCT iajs-2678 40 13 we we PRON iajs-2678 40 14 find find VERB iajs-2678 40 15 the the DET iajs-2678 40 16 following follow VERB iajs-2678 40 17 figure figure NOUN iajs-2678 40 18 singular singular PROPN iajs-2678 40 19 kink kink NOUN iajs-2678 40 20 of of ADP iajs-2678 40 21 u u PRON iajs-2678 40 22 figure figure NOUN iajs-2678 40 23 1 1 NUM iajs-2678 40 24 . . PUNCT iajs-2678 41 1 when when SCONJ iajs-2678 41 2 c=1 c=1 NOUN iajs-2678 41 3 , , PUNCT iajs-2678 41 4 𝛿=1/3 𝛿=1/3 NOUN iajs-2678 41 5 , , PUNCT iajs-2678 41 6 x=[-10,10 x=[-10,10 PROPN iajs-2678 41 7 ] ] PUNCT iajs-2678 41 8 , , PUNCT iajs-2678 41 9 t=[-10,10 t=[-10,10 PROPN iajs-2678 41 10 ] ] PUNCT iajs-2678 41 11 . . PUNCT iajs-2678 41 12 3.2 3.2 NUM iajs-2678 41 13 . . PUNCT iajs-2678 42 1 kdv kdv NOUN iajs-2678 42 2 - - PUNCT iajs-2678 42 3 burgers burger NOUN iajs-2678 42 4 equation equation NOUN iajs-2678 42 5 : : PUNCT iajs-2678 42 6 3.2.1 3.2.1 NUM iajs-2678 42 7 : : PUNCT iajs-2678 42 8 cosine cosine NOUN iajs-2678 42 9 function function NOUN iajs-2678 42 10 method method NOUN iajs-2678 42 11 : : PUNCT iajs-2678 42 12 in in ADP iajs-2678 42 13 this this DET iajs-2678 42 14 part part NOUN iajs-2678 42 15 , , PUNCT iajs-2678 42 16 we we PRON iajs-2678 42 17 want want VERB iajs-2678 42 18 to to PART iajs-2678 42 19 find find VERB iajs-2678 42 20 ( ( PUNCT iajs-2678 42 21 tws tws NOUN iajs-2678 42 22 ) ) PUNCT iajs-2678 42 23 of of ADP iajs-2678 42 24 kdv kdv NOUN iajs-2678 42 25 - - PUNCT iajs-2678 42 26 burgers burger NOUN iajs-2678 42 27 equation equation NOUN iajs-2678 42 28 , , PUNCT iajs-2678 42 29 by by ADP iajs-2678 42 30 using use VERB iajs-2678 42 31 the the DET iajs-2678 42 32 cosine cosine NOUN iajs-2678 42 33 function function NOUN iajs-2678 42 34 method method NOUN iajs-2678 42 35 see see VERB iajs-2678 42 36 [ [ PUNCT iajs-2678 42 37 6,7 6,7 NUM iajs-2678 42 38 ] ] PUNCT iajs-2678 42 39 . . PUNCT iajs-2678 43 1 kdv kdv NOUN iajs-2678 43 2 - - PUNCT iajs-2678 43 3 burgers burger NOUN iajs-2678 43 4 equation equation NOUN iajs-2678 43 5 [ [ X iajs-2678 43 6 13,14 13,14 X iajs-2678 43 7 ] ] X iajs-2678 43 8 has have VERB iajs-2678 43 9 the the DET iajs-2678 43 10 form form NOUN iajs-2678 43 11 𝑒𝑑 𝑒𝑑 NOUN iajs-2678 43 12 + + CCONJ iajs-2678 43 13 ( ( PUNCT iajs-2678 43 14 𝑒2)π‘₯ 𝑒2)π‘₯ ADJ iajs-2678 43 15 βˆ’ βˆ’ NOUN iajs-2678 43 16 𝛾𝑒π‘₯π‘₯ 𝛾𝑒π‘₯π‘₯ ADJ iajs-2678 43 17 + + CCONJ iajs-2678 43 18 𝛿𝑒π‘₯π‘₯π‘₯ 𝛿𝑒π‘₯π‘₯π‘₯ NOUN iajs-2678 43 19 = = NOUN iajs-2678 43 20 0 0 NUM iajs-2678 43 21 , , PUNCT iajs-2678 43 22 ( ( PUNCT iajs-2678 43 23 12 12 NUM iajs-2678 43 24 ) ) PUNCT iajs-2678 43 25 where where SCONJ iajs-2678 43 26 𝛿 𝛿 NOUN iajs-2678 43 27 and and CCONJ iajs-2678 43 28 𝛢>0 𝛢>0 NOUN iajs-2678 43 29 are be AUX iajs-2678 43 30 constants constant NOUN iajs-2678 43 31 and and CCONJ iajs-2678 43 32 𝑒 𝑒 PRON iajs-2678 43 33 is be AUX iajs-2678 43 34 a a DET iajs-2678 43 35 function function NOUN iajs-2678 43 36 of of ADP iajs-2678 43 37 spatial spatial ADJ iajs-2678 43 38 variable variable NOUN iajs-2678 43 39 π‘₯ π‘₯ NOUN iajs-2678 43 40 and and CCONJ iajs-2678 43 41 time time NOUN iajs-2678 43 42 variable variable ADJ iajs-2678 43 43 𝑑 𝑑 PROPN iajs-2678 43 44 let let VERB iajs-2678 43 45 𝑒(π‘₯ 𝑒(π‘₯ PROPN iajs-2678 43 46 , , PUNCT iajs-2678 43 47 𝑑 𝑑 X iajs-2678 43 48 ) ) PUNCT iajs-2678 43 49 = = SYM iajs-2678 43 50 𝑒(πœ‰ 𝑒(πœ‰ PROPN iajs-2678 43 51 ) ) PUNCT iajs-2678 43 52 , , PUNCT iajs-2678 43 53 πœ‰ πœ‰ X iajs-2678 43 54 = = VERB iajs-2678 43 55 π‘₯ π‘₯ PROPN iajs-2678 43 56 βˆ’ βˆ’ PROPN iajs-2678 43 57 𝑐𝑑 𝑐𝑑 INTJ iajs-2678 43 58 ( ( PUNCT iajs-2678 43 59 13 13 NUM iajs-2678 43 60 ) ) PUNCT iajs-2678 43 61 application application NOUN iajs-2678 43 62 ( ( PUNCT iajs-2678 43 63 tws tws NOUN iajs-2678 43 64 ) ) PUNCT iajs-2678 43 65 on on ADP iajs-2678 43 66 eq eq ADP iajs-2678 43 67 . . PUNCT iajs-2678 44 1 ( ( PUNCT iajs-2678 44 2 12 12 NUM iajs-2678 44 3 ) ) PUNCT iajs-2678 44 4 yields yield NOUN iajs-2678 44 5 into into ADP iajs-2678 44 6 the the DET iajs-2678 44 7 ordinary ordinary ADJ iajs-2678 44 8 differential differential ADJ iajs-2678 44 9 equation equation NOUN iajs-2678 44 10 βˆ’π‘π‘’β€² βˆ’π‘π‘’β€² NUM iajs-2678 44 11 + + CCONJ iajs-2678 44 12 ( ( PUNCT iajs-2678 44 13 𝑒2)β€² 𝑒2)β€² INTJ iajs-2678 44 14 βˆ’ βˆ’ PROPN iajs-2678 44 15 𝛾𝑒′′ 𝛾𝑒′′ PROPN iajs-2678 44 16 + + CCONJ iajs-2678 44 17 𝛿𝑒′′′ 𝛿𝑒′′′ NOUN iajs-2678 45 1 = = SYM iajs-2678 45 2 0 0 NUM iajs-2678 45 3 ( ( PUNCT iajs-2678 45 4 14 14 NUM iajs-2678 45 5 ) ) PUNCT iajs-2678 45 6 ibn ibn PROPN iajs-2678 45 7 al al PROPN iajs-2678 45 8 - - PUNCT iajs-2678 45 9 haitham haitham PROPN iajs-2678 45 10 jour jour X iajs-2678 45 11 . . PROPN iajs-2678 45 12 for for ADP iajs-2678 45 13 pure pure ADJ iajs-2678 45 14 & & CCONJ iajs-2678 45 15 appl appl PROPN iajs-2678 45 16 . . PUNCT iajs-2678 46 1 sci sci PROPN iajs-2678 46 2 . . PROPN iajs-2678 47 1 34(3)2021 34(3)2021 NUM iajs-2678 47 2 63 63 NUM iajs-2678 47 3 integrating integrating NOUN iajs-2678 47 4 ( ( PUNCT iajs-2678 47 5 14 14 NUM iajs-2678 47 6 ) ) PUNCT iajs-2678 47 7 we we PRON iajs-2678 47 8 get get VERB iajs-2678 47 9 βˆ’π‘π‘’ βˆ’π‘π‘’ PROPN iajs-2678 47 10 + + CCONJ iajs-2678 47 11 𝑒2 𝑒2 PROPN iajs-2678 47 12 βˆ’ βˆ’ PROPN iajs-2678 47 13 𝛾𝑒′ 𝛾𝑒′ ADJ iajs-2678 47 14 + + CCONJ iajs-2678 47 15 𝛿𝑒′′ 𝛿𝑒′′ NOUN iajs-2678 47 16 = = SYM iajs-2678 47 17 0 0 NUM iajs-2678 47 18 ( ( PUNCT iajs-2678 47 19 15 15 NUM iajs-2678 47 20 ) ) PUNCT iajs-2678 47 21 then then ADV iajs-2678 47 22 we we PRON iajs-2678 47 23 have have VERB iajs-2678 47 24 the the DET iajs-2678 47 25 nonlinear nonlinear ADJ iajs-2678 47 26 travelling travel VERB iajs-2678 47 27 wave wave NOUN iajs-2678 47 28 equation equation NOUN iajs-2678 47 29 ( ( PUNCT iajs-2678 47 30 15 15 NUM iajs-2678 47 31 ) ) PUNCT iajs-2678 47 32 . . PUNCT iajs-2678 48 1 solution solution NOUN iajs-2678 48 2 eq.+(15 eq.+(15 NOUN iajs-2678 48 3 ) ) PUNCT iajs-2678 48 4 by by ADP iajs-2678 48 5 cosine cosine NOUN iajs-2678 48 6 function function NOUN iajs-2678 48 7 method method NOUN iajs-2678 48 8 has have VERB iajs-2678 48 9 the the DET iajs-2678 48 10 form form NOUN iajs-2678 48 11 . . PUNCT iajs-2678 49 1 𝑒(πœ‰ 𝑒(πœ‰ ADJ iajs-2678 49 2 ) ) PUNCT iajs-2678 49 3 = = SYM iajs-2678 49 4 πœ†π‘π‘œπ‘ π›½(πœ‡πœ‰ πœ†π‘π‘œπ‘ π›½(πœ‡πœ‰ NOUN iajs-2678 49 5 ) ) PUNCT iajs-2678 49 6 ( ( PUNCT iajs-2678 49 7 16 16 NUM iajs-2678 49 8 ) ) PUNCT iajs-2678 49 9 where where SCONJ iajs-2678 49 10 π›Œ π›Œ ADP iajs-2678 49 11 , , PUNCT iajs-2678 49 12 𝛃 𝛃 PROPN iajs-2678 49 13 and and CCONJ iajs-2678 49 14 ΞΌ ΞΌ PROPN iajs-2678 49 15 are be AUX iajs-2678 49 16 unknown unknown ADJ iajs-2678 49 17 parameters parameter NOUN iajs-2678 49 18 to to PART iajs-2678 49 19 find find VERB iajs-2678 49 20 its its PRON iajs-2678 49 21 , , PUNCT iajs-2678 49 22 we we PRON iajs-2678 49 23 derivatives derivative NOUN iajs-2678 49 24 ( ( PUNCT iajs-2678 49 25 16 16 NUM iajs-2678 49 26 ) ) PUNCT iajs-2678 49 27 we we PRON iajs-2678 49 28 get get VERB iajs-2678 49 29 𝑒′ 𝑒′ ADJ iajs-2678 49 30 = = SYM iajs-2678 49 31 βˆ’πœ†πœ‡π›½π‘π‘œπ‘ π›½βˆ’1(πœ‡πœ‰)sin βˆ’πœ†πœ‡π›½π‘π‘œπ‘ π›½βˆ’1(πœ‡πœ‰)sin X iajs-2678 49 32 ( ( PUNCT iajs-2678 49 33 πœ‡πœ‰ πœ‡πœ‰ NOUN iajs-2678 49 34 ) ) PUNCT iajs-2678 49 35 ( ( PUNCT iajs-2678 49 36 17 17 NUM iajs-2678 49 37 ) ) PUNCT iajs-2678 49 38 𝑒′′ 𝑒′′ NOUN iajs-2678 49 39 = = SYM iajs-2678 49 40 βˆ’πœ†π›½πœ‡2π‘π‘œπ‘ π›½(πœ‡πœ‰ βˆ’πœ†π›½πœ‡2π‘π‘œπ‘ π›½(πœ‡πœ‰ NOUN iajs-2678 49 41 ) ) PUNCT iajs-2678 50 1 + + SYM iajs-2678 50 2 πœ†πœ‡2𝛽(𝛽 πœ†πœ‡2𝛽(𝛽 NUM iajs-2678 50 3 βˆ’ βˆ’ NUM iajs-2678 50 4 1)π‘π‘œπ‘ π›½βˆ’2(πœ‡πœ‰ 1)π‘π‘œπ‘ π›½βˆ’2(πœ‡πœ‰ NUM iajs-2678 50 5 ) ) PUNCT iajs-2678 50 6 . . PUNCT iajs-2678 51 1 ( ( PUNCT iajs-2678 51 2 18 18 NUM iajs-2678 51 3 ) ) PUNCT iajs-2678 51 4 substituting substitute VERB iajs-2678 51 5 ( ( PUNCT iajs-2678 51 6 16 16 NUM iajs-2678 51 7 - - SYM iajs-2678 51 8 18 18 NUM iajs-2678 51 9 ) ) PUNCT iajs-2678 51 10 into into ADP iajs-2678 51 11 equation equation NOUN iajs-2678 51 12 ( ( PUNCT iajs-2678 51 13 15 15 NUM iajs-2678 51 14 ) ) PUNCT iajs-2678 51 15 gives give VERB iajs-2678 51 16 βˆ’π‘πœ†π‘π‘œπ‘ π›½(πœ‡πœ‰ βˆ’π‘πœ†π‘π‘œπ‘ π›½(πœ‡πœ‰ NOUN iajs-2678 51 17 ) ) PUNCT iajs-2678 52 1 + + CCONJ iajs-2678 52 2 ( ( PUNCT iajs-2678 52 3 πœ†π‘π‘œπ‘ π›½(πœ‡πœ‰ πœ†π‘π‘œπ‘ π›½(πœ‡πœ‰ NOUN iajs-2678 52 4 ) ) PUNCT iajs-2678 52 5 ) ) PUNCT iajs-2678 52 6 2 2 NUM iajs-2678 53 1 + + CCONJ iajs-2678 53 2 π›Ύπœ†πœ‡π›½π‘π‘œπ‘ π›½βˆ’1(πœ‡πœ‰ π›Ύπœ†πœ‡π›½π‘π‘œπ‘ π›½βˆ’1(πœ‡πœ‰ ADJ iajs-2678 53 3 ) ) PUNCT iajs-2678 53 4 sin(πœ‡πœ‰ sin(πœ‡πœ‰ X iajs-2678 53 5 ) ) PUNCT iajs-2678 54 1 + + CCONJ iajs-2678 54 2 𝛿 𝛿 ADJ iajs-2678 54 3 ( ( PUNCT iajs-2678 54 4 βˆ’πœ†π›½πœ‡2π‘π‘œπ‘ π›½(πœ‡πœ‰ βˆ’πœ†π›½πœ‡2π‘π‘œπ‘ π›½(πœ‡πœ‰ NOUN iajs-2678 54 5 ) ) PUNCT iajs-2678 55 1 + + SYM iajs-2678 55 2 πœ†πœ‡2𝛽(𝛽 πœ†πœ‡2𝛽(𝛽 NUM iajs-2678 55 3 βˆ’ βˆ’ NUM iajs-2678 55 4 1)π‘π‘œπ‘ π›½βˆ’2(πœ‡πœ‰ 1)π‘π‘œπ‘ π›½βˆ’2(πœ‡πœ‰ NUM iajs-2678 55 5 ) ) PUNCT iajs-2678 55 6 ) ) PUNCT iajs-2678 56 1 = = SYM iajs-2678 56 2 0 0 PUNCT iajs-2678 57 1 the the DET iajs-2678 57 2 following follow VERB iajs-2678 57 3 algebraic algebraic ADJ iajs-2678 57 4 equation equation NOUN iajs-2678 57 5 system system NOUN iajs-2678 57 6 is be AUX iajs-2678 57 7 obtained obtain VERB iajs-2678 57 8 by by ADP iajs-2678 57 9 equating equate VERB iajs-2678 57 10 the the DET iajs-2678 57 11 exponents exponent NOUN iajs-2678 57 12 and and CCONJ iajs-2678 57 13 the the DET iajs-2678 57 14 coefficient coefficient NOUN iajs-2678 57 15 of of ADP iajs-2678 57 16 each each DET iajs-2678 57 17 pair pair NOUN iajs-2678 57 18 of of ADP iajs-2678 57 19 the the DET iajs-2678 57 20 cosine cosine NOUN iajs-2678 57 21 functions function NOUN iajs-2678 57 22 . . PUNCT iajs-2678 58 1 βˆ’π‘πœ† βˆ’π‘πœ† INTJ iajs-2678 58 2 βˆ’ βˆ’ NOUN iajs-2678 58 3 π›Ώπœ†π›½πœ‡2 π›Ώπœ†π›½πœ‡2 X iajs-2678 59 1 = = SYM iajs-2678 59 2 0 0 NUM iajs-2678 59 3 ( ( PUNCT iajs-2678 59 4 19 19 NUM iajs-2678 59 5 ) ) PUNCT iajs-2678 59 6 πœ†2 πœ†2 NOUN iajs-2678 60 1 + + CCONJ iajs-2678 60 2 π›Ώπœ†πœ‡2𝛽(𝛽 π›Ώπœ†πœ‡2𝛽(𝛽 NOUN iajs-2678 60 3 βˆ’ βˆ’ ADP iajs-2678 60 4 1 1 NUM iajs-2678 60 5 ) ) PUNCT iajs-2678 60 6 = = SYM iajs-2678 60 7 0 0 NUM iajs-2678 60 8 ( ( PUNCT iajs-2678 60 9 20 20 NUM iajs-2678 60 10 ) ) PUNCT iajs-2678 61 1 2𝛽 2𝛽 NOUN iajs-2678 61 2 = = SYM iajs-2678 61 3 𝛽 𝛽 NOUN iajs-2678 61 4 βˆ’ βˆ’ PROPN iajs-2678 61 5 2 2 NUM iajs-2678 61 6 β‡’ β‡’ NOUN iajs-2678 61 7 𝛽 𝛽 NOUN iajs-2678 61 8 = = SYM iajs-2678 61 9 βˆ’2 βˆ’2 NOUN iajs-2678 61 10 , , PUNCT iajs-2678 61 11 substitute substitute NOUN iajs-2678 61 12 in in ADP iajs-2678 61 13 ( ( PUNCT iajs-2678 61 14 19 19 NUM iajs-2678 61 15 - - SYM iajs-2678 61 16 20 20 NUM iajs-2678 61 17 ) ) PUNCT iajs-2678 61 18 we we PRON iajs-2678 61 19 have have VERB iajs-2678 61 20 πœ‡ πœ‡ ADP iajs-2678 61 21 = = X iajs-2678 61 22 βˆ“βˆš βˆ“βˆš PROPN iajs-2678 61 23 𝑐 𝑐 PROPN iajs-2678 61 24 2𝛿 2𝛿 PROPN iajs-2678 61 25 , , PUNCT iajs-2678 61 26 πœ† πœ† PROPN iajs-2678 61 27 = = SYM iajs-2678 61 28 3𝑐 3𝑐 NUM iajs-2678 61 29 , , PUNCT iajs-2678 61 30 substitute substitute NOUN iajs-2678 61 31 in in ADP iajs-2678 61 32 ( ( PUNCT iajs-2678 61 33 16 16 NUM iajs-2678 61 34 ) ) PUNCT iajs-2678 61 35 we we PRON iajs-2678 61 36 get get VERB iajs-2678 61 37 the the DET iajs-2678 61 38 solution solution NOUN iajs-2678 61 39 𝑒(πœ‰ 𝑒(πœ‰ PROPN iajs-2678 61 40 ) ) PUNCT iajs-2678 62 1 = = SYM iajs-2678 62 2 3𝑐 3𝑐 NUM iajs-2678 62 3 π‘π‘œπ‘ βˆ’2(√ π‘π‘œπ‘ βˆ’2(√ PROPN iajs-2678 62 4 𝑐 𝑐 PROPN iajs-2678 62 5 2𝛿 2𝛿 NUM iajs-2678 62 6 πœ‰ πœ‰ NOUN iajs-2678 62 7 ) ) PUNCT iajs-2678 62 8 or or CCONJ iajs-2678 62 9 𝑒(πœ‰ 𝑒(πœ‰ PROPN iajs-2678 62 10 ) ) PUNCT iajs-2678 62 11 = = PUNCT iajs-2678 63 1 βˆ’3𝑐 βˆ’3𝑐 X iajs-2678 63 2 π‘π‘œπ‘ βˆ’2(√ π‘π‘œπ‘ βˆ’2(√ NUM iajs-2678 63 3 𝑐 𝑐 PROPN iajs-2678 63 4 2𝛿 2𝛿 NUM iajs-2678 63 5 πœ‰ πœ‰ NOUN iajs-2678 63 6 ) ) PUNCT iajs-2678 63 7 𝑒(π‘₯ 𝑒(π‘₯ PROPN iajs-2678 63 8 , , PUNCT iajs-2678 63 9 𝑑 𝑑 X iajs-2678 63 10 ) ) PUNCT iajs-2678 63 11 = = SYM iajs-2678 63 12 3𝑐 3𝑐 NUM iajs-2678 63 13 𝑠𝑒𝑐2(√ 𝑠𝑒𝑐2(√ PROPN iajs-2678 63 14 𝑐 𝑐 PROPN iajs-2678 63 15 2𝛿 2𝛿 PROPN iajs-2678 63 16 ( ( PUNCT iajs-2678 63 17 π‘₯ π‘₯ NOUN iajs-2678 63 18 βˆ’ βˆ’ NUM iajs-2678 63 19 𝑐𝑑 𝑐𝑑 NOUN iajs-2678 63 20 ) ) PUNCT iajs-2678 63 21 ) ) PUNCT iajs-2678 63 22 ( ( PUNCT iajs-2678 63 23 21 21 NUM iajs-2678 63 24 ) ) PUNCT iajs-2678 63 25 by by ADP iajs-2678 63 26 drawing draw VERB iajs-2678 63 27 the the DET iajs-2678 63 28 solution solution NOUN iajs-2678 63 29 function function NOUN iajs-2678 63 30 u(x u(x NOUN iajs-2678 63 31 , , PUNCT iajs-2678 63 32 t t PROPN iajs-2678 63 33 ) ) PUNCT iajs-2678 63 34 , , PUNCT iajs-2678 63 35 we we PRON iajs-2678 63 36 find find VERB iajs-2678 63 37 the the DET iajs-2678 63 38 following follow VERB iajs-2678 63 39 figure figure VERB iajs-2678 63 40 the the DET iajs-2678 63 41 periodic periodic ADJ iajs-2678 63 42 solution solution NOUN iajs-2678 63 43 of of ADP iajs-2678 63 44 u u NOUN iajs-2678 63 45 figure figure NOUN iajs-2678 63 46 2 2 NUM iajs-2678 63 47 . . PUNCT iajs-2678 63 48 when when SCONJ iajs-2678 63 49 c=1 c=1 NOUN iajs-2678 63 50 , , PUNCT iajs-2678 63 51 𝛿=1/3 𝛿=1/3 NOUN iajs-2678 63 52 , , PUNCT iajs-2678 63 53 x=[-10,10 x=[-10,10 PROPN iajs-2678 63 54 ] ] PUNCT iajs-2678 63 55 , , PUNCT iajs-2678 63 56 t=[-10,10 t=[-10,10 PROPN iajs-2678 63 57 ] ] PUNCT iajs-2678 63 58 . . PUNCT iajs-2678 64 1 ibn ibn PROPN iajs-2678 64 2 al al PROPN iajs-2678 64 3 - - PUNCT iajs-2678 64 4 haitham haitham PROPN iajs-2678 64 5 jour jour X iajs-2678 64 6 . . PROPN iajs-2678 64 7 for for ADP iajs-2678 64 8 pure pure ADJ iajs-2678 64 9 & & CCONJ iajs-2678 64 10 appl appl PROPN iajs-2678 64 11 . . PUNCT iajs-2678 65 1 sci sci PROPN iajs-2678 65 2 . . PROPN iajs-2678 66 1 34(3)2021 34(3)2021 NUM iajs-2678 66 2 64 64 NUM iajs-2678 66 3 3.2.2.sine 3.2.2.sine NUM iajs-2678 66 4 function function NOUN iajs-2678 66 5 method method NOUN iajs-2678 66 6 : : PUNCT iajs-2678 66 7 solve solve VERB iajs-2678 66 8 equation equation NOUN iajs-2678 66 9 ( ( PUNCT iajs-2678 66 10 15 15 NUM iajs-2678 66 11 ) ) PUNCT iajs-2678 66 12 by by ADP iajs-2678 66 13 use use NOUN iajs-2678 66 14 ( ( PUNCT iajs-2678 66 15 sine sine ADJ iajs-2678 66 16 - - PUNCT iajs-2678 66 17 function function NOUN iajs-2678 66 18 method method NOUN iajs-2678 66 19 ) ) PUNCT iajs-2678 66 20 , , PUNCT iajs-2678 66 21 we we PRON iajs-2678 66 22 have have VERB iajs-2678 66 23 the the DET iajs-2678 66 24 form form NOUN iajs-2678 66 25 , , PUNCT iajs-2678 66 26 𝑒 𝑒 X iajs-2678 66 27 = = SYM iajs-2678 66 28 πœ†π‘ π‘–π‘›π›½(πœ‡πœ‰ πœ†π‘ π‘–π‘›π›½(πœ‡πœ‰ PROPN iajs-2678 66 29 ) ) PUNCT iajs-2678 66 30 ( ( PUNCT iajs-2678 66 31 22 22 NUM iajs-2678 66 32 ) ) PUNCT iajs-2678 66 33 where where SCONJ iajs-2678 66 34 π›Œ π›Œ ADP iajs-2678 66 35 , , PUNCT iajs-2678 66 36 𝛃 𝛃 PROPN iajs-2678 66 37 and and CCONJ iajs-2678 66 38 ΞΌ ΞΌ PROPN iajs-2678 66 39 are be AUX iajs-2678 66 40 unknown unknown ADJ iajs-2678 66 41 parameters parameter NOUN iajs-2678 66 42 , , PUNCT iajs-2678 66 43 to to PART iajs-2678 66 44 find find VERB iajs-2678 66 45 its its PRON iajs-2678 66 46 , , PUNCT iajs-2678 66 47 we we PRON iajs-2678 66 48 derivatives derivative NOUN iajs-2678 66 49 ( ( PUNCT iajs-2678 66 50 22 22 NUM iajs-2678 66 51 ) ) PUNCT iajs-2678 66 52 we we PRON iajs-2678 66 53 get get VERB iajs-2678 66 54 . . PUNCT iajs-2678 67 1 𝑒′ 𝑒′ VERB iajs-2678 67 2 = = PUNCT iajs-2678 67 3 πœ†πœ‡π›½π‘ π‘–π‘›π›½βˆ’1(πœ‡πœ‰)cos(πœ‡πœ‰ πœ†πœ‡π›½π‘ π‘–π‘›π›½βˆ’1(πœ‡πœ‰)cos(πœ‡πœ‰ X iajs-2678 67 4 ) ) PUNCT iajs-2678 67 5 ( ( PUNCT iajs-2678 67 6 23 23 X iajs-2678 67 7 ) ) PUNCT iajs-2678 67 8 𝑒′′ 𝑒′′ NOUN iajs-2678 67 9 = = SYM iajs-2678 67 10 βˆ’πœ†π›½πœ‡2𝑠𝑖𝑛𝛽(πœ‡πœ‰ βˆ’πœ†π›½πœ‡2𝑠𝑖𝑛𝛽(πœ‡πœ‰ PROPN iajs-2678 67 11 ) ) PUNCT iajs-2678 68 1 + + NUM iajs-2678 68 2 πœ†πœ‡2𝛽(𝛽 πœ†πœ‡2𝛽(𝛽 NUM iajs-2678 68 3 βˆ’ βˆ’ ADP iajs-2678 68 4 1)π‘ π‘–π‘›π›½βˆ’2(πœ‡πœ‰ 1)π‘ π‘–π‘›π›½βˆ’2(πœ‡πœ‰ NUM iajs-2678 68 5 ) ) PUNCT iajs-2678 68 6 ( ( PUNCT iajs-2678 68 7 24 24 NUM iajs-2678 68 8 ) ) PUNCT iajs-2678 68 9 substituting substituting NOUN iajs-2678 68 10 ( ( PUNCT iajs-2678 68 11 22 22 NUM iajs-2678 68 12 - - SYM iajs-2678 68 13 24 24 NUM iajs-2678 68 14 ) ) PUNCT iajs-2678 68 15 into into ADP iajs-2678 68 16 equation equation NOUN iajs-2678 68 17 ( ( PUNCT iajs-2678 68 18 15 15 NUM iajs-2678 68 19 ) ) PUNCT iajs-2678 68 20 yield yield NOUN iajs-2678 68 21 in in ADP iajs-2678 68 22 βˆ’π‘πœ†π‘ π‘–π‘›π›½(πœ‡πœ‰ βˆ’π‘πœ†π‘ π‘–π‘›π›½(πœ‡πœ‰ NUM iajs-2678 68 23 ) ) PUNCT iajs-2678 69 1 + + CCONJ iajs-2678 69 2 ( ( PUNCT iajs-2678 69 3 πœ†π‘ π‘–π‘›π›½(πœ‡πœ‰ πœ†π‘ π‘–π‘›π›½(πœ‡πœ‰ NOUN iajs-2678 69 4 ) ) PUNCT iajs-2678 69 5 ) ) PUNCT iajs-2678 69 6 2 2 NUM iajs-2678 69 7 βˆ’ βˆ’ NOUN iajs-2678 69 8 π›Ύπœ†πœ‡π›½π‘ π‘–π‘›π›½βˆ’1(πœ‡πœ‰)cos(πœ‡πœ‰ π›Ύπœ†πœ‡π›½π‘ π‘–π‘›π›½βˆ’1(πœ‡πœ‰)cos(πœ‡πœ‰ NOUN iajs-2678 69 9 ) ) PUNCT iajs-2678 69 10 + + CCONJ iajs-2678 69 11 𝛿(βˆ’πœ†π›½πœ‡2𝑠𝑖𝑛𝛽(πœ‡πœ‰ 𝛿(βˆ’πœ†π›½πœ‡2𝑠𝑖𝑛𝛽(πœ‡πœ‰ X iajs-2678 69 12 ) ) PUNCT iajs-2678 70 1 + + NOUN iajs-2678 70 2 πœ†πœ‡2𝛽(𝛽 πœ†πœ‡2𝛽(𝛽 NUM iajs-2678 70 3 βˆ’ βˆ’ NOUN iajs-2678 70 4 1)π‘ π‘–π‘›π›½βˆ’2(πœ‡πœ‰ 1)π‘ π‘–π‘›π›½βˆ’2(πœ‡πœ‰ NUM iajs-2678 70 5 ) ) PUNCT iajs-2678 70 6 ) ) PUNCT iajs-2678 71 1 = = SYM iajs-2678 71 2 0 0 NUM iajs-2678 72 1 we we PRON iajs-2678 72 2 find find VERB iajs-2678 72 3 the the DET iajs-2678 72 4 following follow VERB iajs-2678 72 5 system system NOUN iajs-2678 72 6 of of ADP iajs-2678 72 7 algebraic algebraic PROPN iajs-2678 72 8 and and CCONJ iajs-2678 72 9 we we PRON iajs-2678 72 10 obtain obtain VERB iajs-2678 72 11 to to ADP iajs-2678 72 12 parameters parameter NOUN iajs-2678 72 13 by by ADP iajs-2678 72 14 equating equate VERB iajs-2678 72 15 the the DET iajs-2678 72 16 exponents exponent NOUN iajs-2678 72 17 and and CCONJ iajs-2678 72 18 the the DET iajs-2678 72 19 coefficient coefficient NOUN iajs-2678 72 20 of of ADP iajs-2678 72 21 each each DET iajs-2678 72 22 pair pair NOUN iajs-2678 72 23 of of ADP iajs-2678 72 24 the the DET iajs-2678 72 25 ( ( PUNCT iajs-2678 72 26 cosine cosine NOUN iajs-2678 72 27 function function NOUN iajs-2678 72 28 ) ) PUNCT iajs-2678 72 29 . . PUNCT iajs-2678 73 1 βˆ’π‘πœ† βˆ’π‘πœ† INTJ iajs-2678 73 2 βˆ’ βˆ’ NOUN iajs-2678 73 3 π›Ώπœ†π›½πœ‡2 π›Ώπœ†π›½πœ‡2 X iajs-2678 74 1 = = SYM iajs-2678 74 2 0 0 NUM iajs-2678 74 3 ( ( PUNCT iajs-2678 74 4 25 25 NUM iajs-2678 74 5 ) ) PUNCT iajs-2678 74 6 πœ†2 πœ†2 NOUN iajs-2678 75 1 + + CCONJ iajs-2678 75 2 πœ†πœ‡2𝛽(𝛽 πœ†πœ‡2𝛽(𝛽 PROPN iajs-2678 75 3 βˆ’ βˆ’ NUM iajs-2678 75 4 1 1 NUM iajs-2678 75 5 ) ) PUNCT iajs-2678 75 6 = = SYM iajs-2678 75 7 0 0 NUM iajs-2678 75 8 , , PUNCT iajs-2678 75 9 ( ( PUNCT iajs-2678 75 10 26 26 NUM iajs-2678 75 11 ) ) PUNCT iajs-2678 75 12 2𝛽 2𝛽 NOUN iajs-2678 75 13 = = SYM iajs-2678 75 14 𝛽 𝛽 NOUN iajs-2678 75 15 βˆ’ βˆ’ PROPN iajs-2678 75 16 2 2 NUM iajs-2678 75 17 β‡’ β‡’ NOUN iajs-2678 75 18 𝛽 𝛽 NOUN iajs-2678 75 19 = = SYM iajs-2678 75 20 βˆ’2 βˆ’2 NOUN iajs-2678 75 21 ( ( PUNCT iajs-2678 75 22 27 27 NUM iajs-2678 75 23 ) ) PUNCT iajs-2678 75 24 substitute substitute NOUN iajs-2678 75 25 ( ( PUNCT iajs-2678 75 26 27 27 NUM iajs-2678 75 27 ) ) PUNCT iajs-2678 75 28 in in ADP iajs-2678 75 29 ( ( PUNCT iajs-2678 75 30 25,26 25,26 X iajs-2678 75 31 ) ) PUNCT iajs-2678 75 32 we we PRON iajs-2678 75 33 have have VERB iajs-2678 75 34 πœ‡ πœ‡ ADP iajs-2678 75 35 = = X iajs-2678 75 36 βˆ“βˆš βˆ“βˆš PROPN iajs-2678 75 37 𝑐 𝑐 PROPN iajs-2678 75 38 2𝛿 2𝛿 PROPN iajs-2678 75 39 , , PUNCT iajs-2678 75 40 πœ† πœ† X iajs-2678 75 41 = = SYM iajs-2678 75 42 3c 3c NUM iajs-2678 75 43 then then ADV iajs-2678 75 44 , , PUNCT iajs-2678 75 45 we we PRON iajs-2678 75 46 have have VERB iajs-2678 75 47 the the DET iajs-2678 75 48 solution solution NOUN iajs-2678 75 49 𝑒 𝑒 NOUN iajs-2678 75 50 = = SYM iajs-2678 75 51 βˆ’3𝑐 βˆ’3𝑐 PROPN iajs-2678 75 52 π‘ π‘–π‘›βˆ’2(√ π‘ π‘–π‘›βˆ’2(√ PROPN iajs-2678 75 53 𝑐 𝑐 PROPN iajs-2678 75 54 2𝛿 2𝛿 NUM iajs-2678 75 55 πœ‰ πœ‰ NOUN iajs-2678 75 56 ) ) PUNCT iajs-2678 75 57 or or CCONJ iajs-2678 75 58 𝑒 𝑒 X iajs-2678 75 59 = = SYM iajs-2678 75 60 3𝑐 3𝑐 NUM iajs-2678 75 61 π‘ π‘–π‘›βˆ’2(√ π‘ π‘–π‘›βˆ’2(√ PROPN iajs-2678 75 62 𝑐 𝑐 PROPN iajs-2678 75 63 2𝛿 2𝛿 NUM iajs-2678 75 64 πœ‰ πœ‰ NOUN iajs-2678 75 65 ) ) PUNCT iajs-2678 75 66 𝑒 𝑒 NOUN iajs-2678 75 67 = = NOUN iajs-2678 75 68 βˆ“3𝑐 βˆ“3𝑐 PROPN iajs-2678 75 69 𝑐𝑠𝑐2(√ 𝑐𝑠𝑐2(√ PROPN iajs-2678 75 70 𝑐 𝑐 PROPN iajs-2678 75 71 2𝛿 2𝛿 PROPN iajs-2678 75 72 ( ( PUNCT iajs-2678 75 73 π‘₯ π‘₯ NOUN iajs-2678 75 74 βˆ’ βˆ’ NUM iajs-2678 75 75 𝑐𝑑 𝑐𝑑 NOUN iajs-2678 75 76 ) ) PUNCT iajs-2678 75 77 ) ) PUNCT iajs-2678 76 1 ibn ibn PROPN iajs-2678 76 2 al al PROPN iajs-2678 76 3 - - PUNCT iajs-2678 76 4 haitham haitham PROPN iajs-2678 76 5 jour jour X iajs-2678 76 6 . . PROPN iajs-2678 76 7 for for ADP iajs-2678 76 8 pure pure ADJ iajs-2678 76 9 & & CCONJ iajs-2678 76 10 appl appl PROPN iajs-2678 76 11 . . PUNCT iajs-2678 77 1 sci sci PROPN iajs-2678 77 2 . . PROPN iajs-2678 78 1 34(3)2021 34(3)2021 NUM iajs-2678 78 2 65 65 NUM iajs-2678 78 3 the the DET iajs-2678 78 4 periodic periodic ADJ iajs-2678 78 5 solutions solution NOUN iajs-2678 78 6 of of ADP iajs-2678 78 7 u u PRON iajs-2678 78 8 figure figure NOUN iajs-2678 78 9 3 3 NUM iajs-2678 78 10 . . PUNCT iajs-2678 79 1 when when SCONJ iajs-2678 79 2 c=1 c=1 NOUN iajs-2678 79 3 , , PUNCT iajs-2678 79 4 𝛿=1/3 𝛿=1/3 NOUN iajs-2678 79 5 , , PUNCT iajs-2678 79 6 x=[-10,10 x=[-10,10 PROPN iajs-2678 79 7 ] ] PUNCT iajs-2678 79 8 , , PUNCT iajs-2678 79 9 t=[-10,10 t=[-10,10 PROPN iajs-2678 79 10 ] ] PUNCT iajs-2678 79 11 . . PUNCT iajs-2678 80 1 conclusion conclusion NOUN iajs-2678 80 2 in in ADP iajs-2678 80 3 this this DET iajs-2678 80 4 work work NOUN iajs-2678 80 5 , , PUNCT iajs-2678 80 6 we we PRON iajs-2678 80 7 conclude conclude VERB iajs-2678 80 8 that that SCONJ iajs-2678 80 9 the the DET iajs-2678 80 10 method method NOUN iajs-2678 80 11 of of ADP iajs-2678 80 12 sine sine NOUN iajs-2678 80 13 and and CCONJ iajs-2678 80 14 cosine cosine NOUN iajs-2678 80 15 function function NOUN iajs-2678 80 16 gives give VERB iajs-2678 80 17 the the DET iajs-2678 80 18 exact exact ADJ iajs-2678 80 19 solution solution NOUN iajs-2678 80 20 to to ADP iajs-2678 80 21 the the DET iajs-2678 80 22 nonlinear nonlinear ADJ iajs-2678 80 23 partial partial ADJ iajs-2678 80 24 differential differential NOUN iajs-2678 80 25 equation equation NOUN iajs-2678 80 26 , , PUNCT iajs-2678 80 27 which which PRON iajs-2678 80 28 is be AUX iajs-2678 80 29 difficult difficult ADJ iajs-2678 80 30 to to PART iajs-2678 80 31 find find VERB iajs-2678 80 32 in in ADP iajs-2678 80 33 other other ADJ iajs-2678 80 34 ways way NOUN iajs-2678 80 35 . . PUNCT iajs-2678 81 1 references reference NOUN iajs-2678 81 2 journal journal NOUN iajs-2678 81 3 of of ADP iajs-2678 81 4 soliton soliton NOUN iajs-2678 81 5 solutions solution NOUN iajs-2678 81 6 of of ADP iajs-2678 81 7 a a DET iajs-2678 81 8 nonlinear nonlinear ADJ iajs-2678 81 9 wave wave NOUN iajs-2678 81 10 equation.‐hirota equation.‐hirota PROPN iajs-2678 81 11 , , PUNCT iajs-2678 81 12 r. r. PROPN iajs-2678 81 13 exact exact PROPN iajs-2678 81 14 envelope envelope PROPN iajs-2678 81 15 .1 .1 NUM iajs-2678 81 16 .809- .809- PUNCT iajs-2678 81 17 ) ) PUNCT iajs-2678 81 18 , , PUNCT iajs-2678 81 19 8057(14 8057(14 NUM iajs-2678 81 20 , , PUNCT iajs-2678 81 21 1973 1973 NUM iajs-2678 81 22 , , PUNCT iajs-2678 81 23 mathematical mathematical ADJ iajs-2678 81 24 physics physics NOUN iajs-2678 81 25 2 2 NUM iajs-2678 81 26 . . PUNCT iajs-2678 82 1 hirota hirota PROPN iajs-2678 82 2 , , PUNCT iajs-2678 82 3 r. r. PROPN iajs-2678 82 4 , , PUNCT iajs-2678 82 5 ; ; PUNCT iajs-2678 82 6 satsuma satsuma PROPN iajs-2678 82 7 , , PUNCT iajs-2678 82 8 j. j. PROPN iajs-2678 82 9 soliton soliton PROPN iajs-2678 82 10 solutions solution NOUN iajs-2678 82 11 of of ADP iajs-2678 82 12 a a DET iajs-2678 82 13 coupled couple VERB iajs-2678 82 14 korteweg korteweg NOUN iajs-2678 82 15 - - PUNCT iajs-2678 82 16 de de NOUN iajs-2678 82 17 vries vries PROPN iajs-2678 82 18 equation equation NOUN iajs-2678 82 19 . . PUNCT iajs-2678 83 1 physics physics NOUN iajs-2678 83 2 letters letter NOUN iajs-2678 83 3 a a PRON iajs-2678 83 4 , , PUNCT iajs-2678 83 5 1981 1981 NUM iajs-2678 83 6 , , PUNCT iajs-2678 83 7 85(8 85(8 NUM iajs-2678 83 8 - - SYM iajs-2678 83 9 9 9 NUM iajs-2678 83 10 ) ) PUNCT iajs-2678 83 11 , , PUNCT iajs-2678 83 12 407 407 NUM iajs-2678 83 13 - - SYM iajs-2678 83 14 408 408 NUM iajs-2678 83 15 . . PUNCT iajs-2678 84 1 3 3 X iajs-2678 84 2 . . X iajs-2678 84 3 kumar kumar PROPN iajs-2678 84 4 , , PUNCT iajs-2678 84 5 a. a. NOUN iajs-2678 84 6 ; ; PUNCT iajs-2678 84 7 pankaj pankaj PROPN iajs-2678 84 8 , , PUNCT iajs-2678 84 9 r. r. PROPN iajs-2678 84 10 d. d. PROPN iajs-2678 84 11 tanh tanh PROPN iajs-2678 84 12 – – PUNCT iajs-2678 84 13 coth coth NOUN iajs-2678 84 14 scheme scheme NOUN iajs-2678 84 15 for for ADP iajs-2678 84 16 traveling travel VERB iajs-2678 84 17 wave wave NOUN iajs-2678 84 18 solutions solution NOUN iajs-2678 84 19 for for ADP iajs-2678 84 20 nonlinear nonlinear ADJ iajs-2678 84 21 wave wave NOUN iajs-2678 84 22 interaction interaction NOUN iajs-2678 84 23 model model NOUN iajs-2678 84 24 . . PUNCT iajs-2678 85 1 journal journal PROPN iajs-2678 85 2 of of ADP iajs-2678 85 3 the the DET iajs-2678 85 4 egyptian egyptian PROPN iajs-2678 85 5 mathematical mathematical PROPN iajs-2678 85 6 society society NOUN iajs-2678 85 7 , , PUNCT iajs-2678 85 8 23(2 23(2 NOUN iajs-2678 85 9 ) ) PUNCT iajs-2678 85 10 , , PUNCT iajs-2678 85 11 282 282 NUM iajs-2678 85 12 - - SYM iajs-2678 85 13 285 285 NUM iajs-2678 85 14 . . PUNCT iajs-2678 86 1 4 4 NUM iajs-2678 86 2 . . X iajs-2678 86 3 taha taha PROPN iajs-2678 86 4 , , PUNCT iajs-2678 86 5 w. w. PROPN iajs-2678 86 6 m. m. PROPN iajs-2678 86 7 , , PUNCT iajs-2678 86 8 hameed hameed PROPN iajs-2678 86 9 , , PUNCT iajs-2678 86 10 r. r. PROPN iajs-2678 86 11 a. a. PROPN iajs-2678 86 12 ; ; PUNCT iajs-2678 86 13 noorani noorani PROPN iajs-2678 86 14 , , PUNCT iajs-2678 86 15 m. m. NOUN iajs-2678 86 16 s. s. PROPN iajs-2678 86 17 m. m. PROPN iajs-2678 86 18 ( ( PUNCT iajs-2678 86 19 2016 2016 NUM iajs-2678 86 20 ) ) PUNCT iajs-2678 86 21 . . PUNCT iajs-2678 87 1 traveling travel VERB iajs-2678 87 2 wave wave NOUN iajs-2678 87 3 solutions solution NOUN iajs-2678 87 4 by by ADP iajs-2678 87 5 using use VERB iajs-2678 87 6 extended extend VERB iajs-2678 87 7 tanh tanh NOUN iajs-2678 87 8 function function NOUN iajs-2678 87 9 method method NOUN iajs-2678 87 10 with with ADP iajs-2678 87 11 special special ADJ iajs-2678 87 12 values value NOUN iajs-2678 87 13 . . PUNCT iajs-2678 88 1 diyala diyala PROPN iajs-2678 88 2 journal journal PROPN iajs-2678 88 3 for for ADP iajs-2678 88 4 pure pure ADJ iajs-2678 88 5 science science NOUN iajs-2678 88 6 , , PUNCT iajs-2678 88 7 2015 2015 NUM iajs-2678 88 8 , , PUNCT iajs-2678 88 9 12 12 NUM iajs-2678 88 10 ( ( PUNCT iajs-2678 88 11 4 4 NUM iajs-2678 88 12 - - PUNCT iajs-2678 88 13 part part NOUN iajs-2678 88 14 2 2 NUM iajs-2678 88 15 ) ) PUNCT iajs-2678 88 16 , , PUNCT iajs-2678 88 17 1 1 NUM iajs-2678 88 18 - - SYM iajs-2678 88 19 13 13 NUM iajs-2678 88 20 . . PUNCT iajs-2678 89 1 5 5 NUM iajs-2678 89 2 . . X iajs-2678 89 3 fan fan PROPN iajs-2678 89 4 , , PUNCT iajs-2678 89 5 e. e. PROPN iajs-2678 89 6 extended extend VERB iajs-2678 89 7 tanh tanh NOUN iajs-2678 89 8 - - PUNCT iajs-2678 89 9 function function NOUN iajs-2678 89 10 method method NOUN iajs-2678 89 11 and and CCONJ iajs-2678 89 12 its its PRON iajs-2678 89 13 applications application NOUN iajs-2678 89 14 to to ADP iajs-2678 89 15 nonlinear nonlinear ADJ iajs-2678 89 16 quations quation NOUN iajs-2678 89 17 . . PUNCT iajs-2678 90 1 physics physics NOUN iajs-2678 90 2 letters letter NOUN iajs-2678 90 3 a a PRON iajs-2678 90 4 , , PUNCT iajs-2678 90 5 2000 2000 NUM iajs-2678 90 6 , , PUNCT iajs-2678 90 7 277(4 277(4 NUM iajs-2678 90 8 - - SYM iajs-2678 90 9 5 5 NUM iajs-2678 90 10 ) ) PUNCT iajs-2678 90 11 , , PUNCT iajs-2678 90 12 212 212 NUM iajs-2678 90 13 - - SYM iajs-2678 90 14 218 218 NUM iajs-2678 90 15 . . PUNCT iajs-2678 91 1 6 6 NUM iajs-2678 91 2 . . X iajs-2678 91 3 bibi bibi NOUN iajs-2678 91 4 , , PUNCT iajs-2678 91 5 s. s. PROPN iajs-2678 91 6 , , PUNCT iajs-2678 91 7 ; ; PUNCT iajs-2678 91 8 mohyud mohyud NOUN iajs-2678 91 9 - - PUNCT iajs-2678 91 10 din din PROPN iajs-2678 91 11 , , PUNCT iajs-2678 91 12 s. s. PROPN iajs-2678 91 13 t. t. PROPN iajs-2678 91 14 traveling traveling PROPN iajs-2678 91 15 wave wave NOUN iajs-2678 91 16 solutions solution NOUN iajs-2678 91 17 of of ADP iajs-2678 91 18 kdvs kdvs NOUN iajs-2678 91 19 using use VERB iajs-2678 91 20 sine sine NOUN iajs-2678 91 21 – – PUNCT iajs-2678 91 22 cosine cosine NOUN iajs-2678 91 23 method method NOUN iajs-2678 91 24 . . PUNCT iajs-2678 92 1 journal journal NOUN iajs-2678 92 2 of of ADP iajs-2678 92 3 the the DET iajs-2678 92 4 association association NOUN iajs-2678 92 5 of of ADP iajs-2678 92 6 arab arab ADJ iajs-2678 92 7 universities university NOUN iajs-2678 92 8 for for ADP iajs-2678 92 9 basic basic ADJ iajs-2678 92 10 and and CCONJ iajs-2678 92 11 applied applied ADJ iajs-2678 92 12 sciences science NOUN iajs-2678 92 13 , , PUNCT iajs-2678 92 14 2014,15(1 2014,15(1 NOUN iajs-2678 92 15 ) ) PUNCT iajs-2678 92 16 , , PUNCT iajs-2678 92 17 90 90 NUM iajs-2678 92 18 - - SYM iajs-2678 92 19 93 93 NUM iajs-2678 92 20 . . PUNCT iajs-2678 93 1 7 7 X iajs-2678 93 2 . . X iajs-2678 93 3 taghizadeh taghizadeh NOUN iajs-2678 93 4 , , PUNCT iajs-2678 93 5 n. n. NOUN iajs-2678 93 6 , , PUNCT iajs-2678 93 7 mirzazadeh mirzazadeh PROPN iajs-2678 93 8 , , PUNCT iajs-2678 93 9 m. m. NOUN iajs-2678 93 10 ; ; PUNCT iajs-2678 93 11 farahrooz farahrooz PROPN iajs-2678 93 12 , , PUNCT iajs-2678 93 13 f. f. 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