id sid tid token lemma pos cana-1224 1 1 communications communication NOUN cana-1224 1 2 on on ADP cana-1224 1 3 applied apply VERB cana-1224 1 4 nonlinear nonlinear ADJ cana-1224 1 5 analysis analysis NOUN cana-1224 1 6 issn issn NOUN cana-1224 1 7 : : PUNCT cana-1224 1 8 1074 1074 NUM cana-1224 1 9 - - PUNCT cana-1224 1 10 133x 133x NUM cana-1224 1 11 vol vol NOUN cana-1224 1 12 31 31 NUM cana-1224 1 13 no no NOUN cana-1224 1 14 . . PUNCT cana-1224 2 1 6s 6s NUM cana-1224 2 2 ( ( PUNCT cana-1224 2 3 2024 2024 NUM cana-1224 2 4 ) ) PUNCT cana-1224 2 5 305 305 NUM cana-1224 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 2 7 elzaki elzaki NOUN cana-1224 2 8 transform transform VERB cana-1224 2 9 homotopy homotopy NOUN cana-1224 2 10 analysis analysis NOUN cana-1224 2 11 techniques technique NOUN cana-1224 2 12 for for ADP cana-1224 2 13 solving solve VERB cana-1224 2 14 fractional fractional ADJ cana-1224 2 15 ( ( PUNCT cana-1224 2 16 2 2 NUM cana-1224 2 17 + + NOUN cana-1224 2 18 1)-d 1)-d NUM cana-1224 2 19 and and CCONJ cana-1224 2 20 ( ( PUNCT cana-1224 2 21 3 3 NUM cana-1224 2 22 + + NOUN cana-1224 2 23 1)-d 1)-d NUM cana-1224 2 24 nonlinear nonlinear ADJ cana-1224 2 25 schrodinger schrodinger PROPN cana-1224 2 26 equations equations PROPN cana-1224 2 27 sandeep sandeep PROPN cana-1224 2 28 sharma1 sharma1 PROPN cana-1224 2 29 , , PUNCT cana-1224 2 30 inderdeep inderdeep VERB cana-1224 2 31 singh2 singh2 VERB cana-1224 3 1 1,2sant 1,2sant NUM cana-1224 3 2 baba baba PROPN cana-1224 3 3 bhag bhag PROPN cana-1224 3 4 singh singh PROPN cana-1224 3 5 university university PROPN cana-1224 3 6 , , PUNCT cana-1224 3 7 jalandhar-144030 jalandhar-144030 NOUN cana-1224 3 8 , , PUNCT cana-1224 3 9 punjab punjab ADJ cana-1224 3 10 , , PUNCT cana-1224 3 11 india india PROPN cana-1224 3 12 email email NOUN cana-1224 3 13 : : PUNCT cana-1224 3 14 1sandeepsharma200@gmail.com 1sandeepsharma200@gmail.com NUM cana-1224 3 15 , , PUNCT cana-1224 3 16 2inderdeeps.ma.12@gmail.com 2inderdeeps.ma.12@gmail.com NUM cana-1224 3 17 article article NOUN cana-1224 3 18 history history NOUN cana-1224 3 19 : : PUNCT cana-1224 3 20 received receive VERB cana-1224 3 21 : : PUNCT cana-1224 3 22 08 08 NUM cana-1224 3 23 - - SYM cana-1224 3 24 06 06 NUM cana-1224 3 25 - - PUNCT cana-1224 3 26 2024 2024 NUM cana-1224 3 27 revised revise VERB cana-1224 3 28 : : PUNCT cana-1224 3 29 09 09 NUM cana-1224 3 30 - - SYM cana-1224 3 31 07 07 NUM cana-1224 3 32 - - PUNCT cana-1224 3 33 2024 2024 NUM cana-1224 3 34 accepted accept VERB cana-1224 3 35 : : PUNCT cana-1224 3 36 29 29 NUM cana-1224 3 37 - - SYM cana-1224 3 38 07 07 NUM cana-1224 3 39 - - PUNCT cana-1224 3 40 2024 2024 NUM cana-1224 3 41 abstract abstract NOUN cana-1224 3 42 : : PUNCT cana-1224 3 43 in in ADP cana-1224 3 44 this this DET cana-1224 3 45 research research NOUN cana-1224 3 46 , , PUNCT cana-1224 3 47 new new ADJ cana-1224 3 48 homotopy homotopy NOUN cana-1224 3 49 analysis analysis NOUN cana-1224 3 50 method method NOUN cana-1224 3 51 for for ADP cana-1224 3 52 solving solve VERB cana-1224 3 53 the the DET cana-1224 3 54 fractional fractional ADJ cana-1224 3 55 ( ( PUNCT cana-1224 3 56 2 2 NUM cana-1224 3 57 + + NOUN cana-1224 3 58 1 1 NUM cana-1224 3 59 ) ) PUNCT cana-1224 3 60 d d NOUN cana-1224 3 61 and and CCONJ cana-1224 3 62 ( ( PUNCT cana-1224 3 63 3 3 NUM cana-1224 3 64 + + NOUN cana-1224 3 65 1 1 NUM cana-1224 3 66 ) ) PUNCT cana-1224 3 67 d d NOUN cana-1224 3 68 non non ADJ cana-1224 3 69 - - ADJ cana-1224 3 70 linear linear ADJ cana-1224 3 71 schrรถdinger schrรถdinger ADJ cana-1224 3 72 equations equation NOUN cana-1224 3 73 by by ADP cana-1224 3 74 elzaki elzaki NOUN cana-1224 3 75 . . PUNCT cana-1224 4 1 to to PART cana-1224 4 2 solve solve VERB cana-1224 4 3 these these DET cana-1224 4 4 equations equation NOUN cana-1224 4 5 , , PUNCT cana-1224 4 6 the the DET cana-1224 4 7 elzaki elzaki NOUN cana-1224 4 8 transform transform NOUN cana-1224 4 9 is be AUX cana-1224 4 10 applied apply VERB cana-1224 4 11 jointly jointly ADV cana-1224 4 12 to to ADP cana-1224 4 13 the the DET cana-1224 4 14 homotopy homotopy NOUN cana-1224 4 15 analysis analysis NOUN cana-1224 4 16 method method NOUN cana-1224 4 17 ( ( PUNCT cana-1224 4 18 ham ham NOUN cana-1224 4 19 ) ) PUNCT cana-1224 4 20 . . PUNCT cana-1224 5 1 this this PRON cana-1224 5 2 has have AUX cana-1224 5 3 proved prove VERB cana-1224 5 4 efficient efficient ADJ cana-1224 5 5 in in ADP cana-1224 5 6 tackling tackle VERB cana-1224 5 7 fractional fractional ADJ cana-1224 5 8 calculus calculus NOUN cana-1224 5 9 and and CCONJ cana-1224 5 10 nonlinear nonlinear ADJ cana-1224 5 11 dynamics dynamic NOUN cana-1224 5 12 since since SCONJ cana-1224 5 13 correct correct ADJ cana-1224 5 14 solutions solution NOUN cana-1224 5 15 are be AUX cana-1224 5 16 offered offer VERB cana-1224 5 17 and and CCONJ cana-1224 5 18 they they PRON cana-1224 5 19 converge converge VERB cana-1224 5 20 at at ADP cana-1224 5 21 a a DET cana-1224 5 22 faster fast ADJ cana-1224 5 23 rate rate NOUN cana-1224 5 24 . . PUNCT cana-1224 6 1 the the DET cana-1224 6 2 accuracy accuracy NOUN cana-1224 6 3 of of ADP cana-1224 6 4 the the DET cana-1224 6 5 proposed propose VERB cana-1224 6 6 technique technique NOUN cana-1224 6 7 has have AUX cana-1224 6 8 been be AUX cana-1224 6 9 corroborated corroborate VERB cana-1224 6 10 by by ADP cana-1224 6 11 analyzing analyze VERB cana-1224 6 12 various various ADJ cana-1224 6 13 examples example NOUN cana-1224 6 14 for for ADP cana-1224 6 15 which which PRON cana-1224 6 16 the the DET cana-1224 6 17 latter latter ADJ cana-1224 6 18 were be AUX cana-1224 6 19 used use VERB cana-1224 6 20 for for ADP cana-1224 6 21 solving solve VERB cana-1224 6 22 highdimensional highdimensional ADJ cana-1224 6 23 non non ADJ cana-1224 6 24 - - ADJ cana-1224 6 25 linear linear ADJ cana-1224 6 26 schrodinger schrodinger NOUN cana-1224 6 27 equation equation NOUN cana-1224 6 28 , , PUNCT cana-1224 6 29 which which PRON cana-1224 6 30 indicates indicate VERB cana-1224 6 31 that that SCONJ cana-1224 6 32 the the DET cana-1224 6 33 technique technique NOUN cana-1224 6 34 is be AUX cana-1224 6 35 quite quite ADV cana-1224 6 36 resilient resilient ADJ cana-1224 6 37 as as ADV cana-1224 6 38 well well ADV cana-1224 6 39 as as ADP cana-1224 6 40 efficient efficient ADJ cana-1224 6 41 ; ; PUNCT cana-1224 6 42 thus thus ADV cana-1224 6 43 , , PUNCT cana-1224 6 44 making make VERB cana-1224 6 45 it it PRON cana-1224 6 46 an an DET cana-1224 6 47 effective effective ADJ cana-1224 6 48 tool tool NOUN cana-1224 6 49 in in ADP cana-1224 6 50 theoretical theoretical ADJ cana-1224 6 51 physics physics NOUN cana-1224 6 52 and and CCONJ cana-1224 6 53 other other ADJ cana-1224 6 54 applied apply VERB cana-1224 6 55 sciences science NOUN cana-1224 6 56 . . PUNCT cana-1224 7 1 keywords keyword NOUN cana-1224 7 2 : : PUNCT cana-1224 7 3 elzaki elzaki NOUN cana-1224 7 4 transform transform NOUN cana-1224 7 5 , , PUNCT cana-1224 7 6 homotopy homotopy VERB cana-1224 7 7 analysis analysis NOUN cana-1224 7 8 method method NOUN cana-1224 7 9 , , PUNCT cana-1224 7 10 ( ( PUNCT cana-1224 7 11 2 2 NUM cana-1224 7 12 + + NOUN cana-1224 7 13 1)-dand 1)-dand NUM cana-1224 7 14 ( ( PUNCT cana-1224 7 15 3 3 NUM cana-1224 7 16 + + SYM cana-1224 7 17 1)-d 1)-d NUM cana-1224 7 18 nonlinear nonlinear ADJ cana-1224 7 19 fractional fractional ADJ cana-1224 7 20 schrodinger schrodinger PROPN cana-1224 7 21 equations equation NOUN cana-1224 7 22 . . PUNCT cana-1224 8 1 1 1 X cana-1224 8 2 . . X cana-1224 8 3 introduction introduction NOUN cana-1224 8 4 this this DET cana-1224 8 5 research research NOUN cana-1224 8 6 paper paper NOUN cana-1224 8 7 is be AUX cana-1224 8 8 devoted devote VERB cana-1224 8 9 to to ADP cana-1224 8 10 finding find VERB cana-1224 8 11 the the DET cana-1224 8 12 semi semi ADJ cana-1224 8 13 - - ADJ cana-1224 8 14 analytical analytical ADJ cana-1224 8 15 solutions solution NOUN cana-1224 8 16 of of ADP cana-1224 8 17 ( ( PUNCT cana-1224 8 18 2 2 NUM cana-1224 8 19 + + NOUN cana-1224 8 20 1)-d 1)-d NUM cana-1224 8 21 and and CCONJ cana-1224 8 22 ( ( PUNCT cana-1224 8 23 3 3 NUM cana-1224 8 24 + + SYM cana-1224 8 25 1)-d 1)-d NUM cana-1224 8 26 nonlinear nonlinear ADJ cana-1224 8 27 fractional fractional ADJ cana-1224 8 28 schrodinger schrodinger PROPN cana-1224 8 29 equations equation NOUN cana-1224 8 30 of of ADP cana-1224 8 31 the the DET cana-1224 8 32 form form NOUN cana-1224 8 33 : : PUNCT cana-1224 8 34 ๐‘–๐‘ค๐‘ก ๐‘–๐‘ค๐‘ก NOUN cana-1224 8 35 ๐›ผ(ฯ‰ ๐›ผ(ฯ‰ NOUN cana-1224 8 36 ) ) PUNCT cana-1224 9 1 + + CCONJ cana-1224 9 2 ๐‘Žโˆ†2๐‘ค(ฯ‰ ๐‘Žโˆ†2๐‘ค(ฯ‰ PROPN cana-1224 9 3 ) ) PUNCT cana-1224 10 1 + + NUM cana-1224 10 2 ๐›ผ(ฯ‰)๐‘ค(ฯ‰ ๐›ผ(ฯ‰)๐‘ค(ฯ‰ NOUN cana-1224 10 3 ) ) PUNCT cana-1224 10 4 โˆ’ โˆ’ NOUN cana-1224 10 5 ๐›ฝ|๐‘ค|2๐‘ค(ฯ‰ ๐›ฝ|๐‘ค|2๐‘ค(ฯ‰ NOUN cana-1224 10 6 ) ) PUNCT cana-1224 10 7 = = SYM cana-1224 10 8 0 0 NUM cana-1224 10 9 , , PUNCT cana-1224 10 10 ( ( PUNCT cana-1224 10 11 1 1 X cana-1224 10 12 ) ) PUNCT cana-1224 10 13 with with ADP cana-1224 10 14 initial initial ADJ cana-1224 10 15 condition condition NOUN cana-1224 10 16 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ PROPN cana-1224 10 17 , , PUNCT cana-1224 10 18 0 0 NUM cana-1224 10 19 ) ) PUNCT cana-1224 10 20 = = SYM cana-1224 10 21 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ NOUN cana-1224 10 22 ) ) PUNCT cana-1224 10 23 and and CCONJ cana-1224 10 24 ๐‘–2 ๐‘–2 PROPN cana-1224 11 1 = = SYM cana-1224 11 2 โˆ’1 โˆ’1 NOUN cana-1224 11 3 . . PUNCT cana-1224 12 1 here here ADV cana-1224 12 2 , , PUNCT cana-1224 12 3 ฯ‰ ฯ‰ PROPN cana-1224 12 4 is be AUX cana-1224 12 5 either either CCONJ cana-1224 12 6 ( ( PUNCT cana-1224 12 7 ๐‘ฅ ๐‘ฅ NUM cana-1224 12 8 , , PUNCT cana-1224 12 9 ๐‘ฆ ๐‘ฆ NOUN cana-1224 12 10 ) ) PUNCT cana-1224 12 11 or or CCONJ cana-1224 12 12 ( ( PUNCT cana-1224 12 13 ๐‘ฅ ๐‘ฅ NOUN cana-1224 12 14 , , PUNCT cana-1224 12 15 ๐‘ฆ ๐‘ฆ NOUN cana-1224 12 16 , , PUNCT cana-1224 12 17 ๐‘ง ๐‘ง NOUN cana-1224 12 18 ) ) PUNCT cana-1224 12 19 , , PUNCT cana-1224 12 20 ๐‘Ž ๐‘Ž PROPN cana-1224 12 21 , , PUNCT cana-1224 12 22 ๐›ฝ ๐›ฝ NOUN cana-1224 12 23 are be AUX cana-1224 12 24 constants constant NOUN cana-1224 12 25 and and CCONJ cana-1224 12 26 ๐›ผ ๐›ผ PRON cana-1224 12 27 is be AUX cana-1224 12 28 a a DET cana-1224 12 29 function function NOUN cana-1224 12 30 of of ADP cana-1224 12 31 variables variable NOUN cana-1224 12 32 ๐‘ฅ ๐‘ฅ PROPN cana-1224 12 33 , , PUNCT cana-1224 12 34 ๐‘ฆ ๐‘ฆ NOUN cana-1224 12 35 and and CCONJ cana-1224 12 36 ๐‘ง. ๐‘ง. NOUN cana-1224 12 37 elzaki elzaki NOUN cana-1224 12 38 integral integral ADJ cana-1224 12 39 transform transform NOUN cana-1224 12 40 has have AUX cana-1224 12 41 been be AUX cana-1224 12 42 introduced introduce VERB cana-1224 12 43 in in ADP cana-1224 12 44 [ [ X cana-1224 12 45 1 1 NUM cana-1224 12 46 ] ] PUNCT cana-1224 12 47 for for ADP cana-1224 12 48 solving solve VERB cana-1224 12 49 differential differential ADJ cana-1224 12 50 equations equation NOUN cana-1224 12 51 . . PUNCT cana-1224 13 1 in in ADP cana-1224 13 2 [ [ X cana-1224 13 3 2 2 NUM cana-1224 13 4 ] ] PUNCT cana-1224 13 5 , , PUNCT cana-1224 13 6 various various ADJ cana-1224 13 7 applications application NOUN cana-1224 13 8 of of ADP cana-1224 13 9 elzaki elzaki NOUN cana-1224 13 10 transform transform NOUN cana-1224 13 11 had have AUX cana-1224 13 12 been be AUX cana-1224 13 13 used use VERB cana-1224 13 14 for for ADP cana-1224 13 15 solving solve VERB cana-1224 13 16 several several ADJ cana-1224 13 17 mathematical mathematical ADJ cana-1224 13 18 models model NOUN cana-1224 13 19 in in ADP cana-1224 13 20 the the DET cana-1224 13 21 pde pde NOUN cana-1224 13 22 ( ( PUNCT cana-1224 13 23 partial partial ADJ cana-1224 13 24 differential differential NOUN cana-1224 13 25 equations equation NOUN cana-1224 13 26 ) ) PUNCT cana-1224 13 27 form form NOUN cana-1224 13 28 . . PUNCT cana-1224 14 1 in in ADP cana-1224 14 2 [ [ X cana-1224 14 3 3 3 NUM cana-1224 14 4 ] ] PUNCT cana-1224 14 5 , , PUNCT cana-1224 14 6 the the DET cana-1224 14 7 authors author NOUN cana-1224 14 8 have have AUX cana-1224 14 9 presented present VERB cana-1224 14 10 a a DET cana-1224 14 11 comparison comparison NOUN cana-1224 14 12 study study NOUN cana-1224 14 13 between between ADP cana-1224 14 14 laplace laplace NOUN cana-1224 14 15 and and CCONJ cana-1224 14 16 elzaki elzaki NOUN cana-1224 14 17 transforms transform VERB cana-1224 14 18 . . PUNCT cana-1224 15 1 for for ADP cana-1224 15 2 solving solve VERB cana-1224 15 3 various various ADJ cana-1224 15 4 differential differential ADJ cana-1224 15 5 equations equation NOUN cana-1224 15 6 , , PUNCT cana-1224 15 7 a a DET cana-1224 15 8 new new ADJ cana-1224 15 9 transform transform NOUN cana-1224 15 10 called call VERB cana-1224 15 11 sumudu sumudu NOUN cana-1224 15 12 transform transform NOUN cana-1224 15 13 - - PUNCT cana-1224 15 14 based base VERB cana-1224 15 15 technique technique NOUN cana-1224 15 16 had have AUX cana-1224 15 17 been be AUX cana-1224 15 18 utilized utilize VERB cana-1224 15 19 in in ADP cana-1224 15 20 [ [ X cana-1224 15 21 4 4 NUM cana-1224 15 22 ] ] PUNCT cana-1224 15 23 . . PUNCT cana-1224 16 1 a a DET cana-1224 16 2 brief brief ADJ cana-1224 16 3 discussion discussion NOUN cana-1224 16 4 about about ADP cana-1224 16 5 integral integral ADJ cana-1224 16 6 transform transform NOUN cana-1224 16 7 for for ADP cana-1224 16 8 solving solve VERB cana-1224 16 9 differential differential ADJ cana-1224 16 10 equations equation NOUN cana-1224 16 11 had have AUX cana-1224 16 12 been be AUX cana-1224 16 13 represented represent VERB cana-1224 16 14 in in ADP cana-1224 16 15 [ [ X cana-1224 16 16 5 5 NUM cana-1224 16 17 ] ] PUNCT cana-1224 16 18 . . PUNCT cana-1224 17 1 homotopy homotopy VERB cana-1224 17 2 analysis analysis NOUN cana-1224 17 3 approaches approach NOUN cana-1224 17 4 had have AUX cana-1224 17 5 been be AUX cana-1224 17 6 applied apply VERB cana-1224 17 7 for for ADP cana-1224 17 8 solving solve VERB cana-1224 17 9 generalized generalize VERB cana-1224 17 10 benjamin benjamin NOUN cana-1224 17 11 - - PUNCT cana-1224 17 12 bona bona ADJ cana-1224 17 13 - - PUNCT cana-1224 17 14 mahony mahony NOUN cana-1224 17 15 equation equation NOUN cana-1224 17 16 and and CCONJ cana-1224 17 17 fifth fifth ADJ cana-1224 17 18 - - PUNCT cana-1224 17 19 order order NOUN cana-1224 17 20 kdv kdv NOUN cana-1224 17 21 eqns eqns X cana-1224 17 22 in in ADP cana-1224 17 23 [ [ PUNCT cana-1224 17 24 6 6 NUM cana-1224 17 25 - - SYM cana-1224 17 26 7 7 NUM cana-1224 17 27 ] ] PUNCT cana-1224 17 28 . . PUNCT cana-1224 18 1 homotopy homotopy VERB cana-1224 18 2 analysis analysis NOUN cana-1224 18 3 addressed address VERB cana-1224 18 4 nonlinear nonlinear ADJ cana-1224 18 5 issues issue NOUN cana-1224 18 6 in in ADP cana-1224 18 7 science science NOUN cana-1224 18 8 and and CCONJ cana-1224 18 9 engineering engineering NOUN cana-1224 18 10 [ [ X cana-1224 18 11 8 8 NUM cana-1224 18 12 ] ] PUNCT cana-1224 18 13 . . PUNCT cana-1224 19 1 a a DET cana-1224 19 2 comparison comparison NOUN cana-1224 19 3 study study NOUN cana-1224 19 4 has have AUX cana-1224 19 5 been be AUX cana-1224 19 6 presented present VERB cana-1224 19 7 in in ADP cana-1224 19 8 [ [ X cana-1224 19 9 9 9 NUM cana-1224 19 10 ] ] PUNCT cana-1224 19 11 for for ADP cana-1224 19 12 solving solve VERB cana-1224 19 13 various various ADJ cana-1224 19 14 differential differential ADJ cana-1224 19 15 equations equation NOUN cana-1224 19 16 . . PUNCT cana-1224 20 1 for for ADP cana-1224 20 2 this this DET cana-1224 20 3 purpose purpose NOUN cana-1224 20 4 , , PUNCT cana-1224 20 5 homotopy homotopy VERB cana-1224 20 6 analysis analysis NOUN cana-1224 20 7 method method NOUN cana-1224 20 8 and and CCONJ cana-1224 20 9 homotoy homotoy VERB cana-1224 20 10 perturbation perturbation NOUN cana-1224 20 11 method method NOUN cana-1224 20 12 have have AUX cana-1224 20 13 been be AUX cana-1224 20 14 used use VERB cana-1224 20 15 . . PUNCT cana-1224 21 1 in in ADP cana-1224 21 2 [ [ X cana-1224 21 3 10 10 NUM cana-1224 21 4 ] ] PUNCT cana-1224 21 5 , , PUNCT cana-1224 21 6 fractional fractional ADJ cana-1224 21 7 kdvburgers kdvburger NOUN cana-1224 21 8 - - PUNCT cana-1224 21 9 kuramoto kuramoto NOUN cana-1224 21 10 eqn eqn PROPN cana-1224 21 11 had have AUX cana-1224 21 12 been be AUX cana-1224 21 13 solved solve VERB cana-1224 21 14 with with ADP cana-1224 21 15 the the DET cana-1224 21 16 help help NOUN cana-1224 21 17 of of ADP cana-1224 21 18 homotopy homotopy VERB cana-1224 21 19 analysis analysis NOUN cana-1224 21 20 approach approach NOUN cana-1224 21 21 . . PUNCT cana-1224 22 1 homotopy homotopy VERB cana-1224 22 2 analysis analysis NOUN cana-1224 22 3 [ [ X cana-1224 22 4 11 11 NUM cana-1224 22 5 ] ] PUNCT cana-1224 22 6 finds find VERB cana-1224 22 7 semi semi ADJ cana-1224 22 8 - - ADJ cana-1224 22 9 analytical analytical ADJ cana-1224 22 10 solutions solution NOUN cana-1224 22 11 for for ADP cana-1224 22 12 nonlinear nonlinear ADJ cana-1224 22 13 fractional fractional ADJ cana-1224 22 14 differential differential ADJ cana-1224 22 15 equations equation NOUN cana-1224 22 16 . . PUNCT cana-1224 23 1 linear linear ADJ cana-1224 23 2 along along ADP cana-1224 23 3 with with ADP cana-1224 23 4 nonlinear nonlinear ADJ cana-1224 23 5 fractional fractional ADJ cana-1224 23 6 diffusionwave diffusionwave NOUN cana-1224 23 7 eqns eqns PROPN cana-1224 23 8 had have AUX cana-1224 23 9 been be AUX cana-1224 23 10 solved solve VERB cana-1224 23 11 by by ADP cana-1224 23 12 using use VERB cana-1224 23 13 homotopy homotopy NOUN cana-1224 23 14 analysis analysis NOUN cana-1224 23 15 approach approach NOUN cana-1224 23 16 in in ADP cana-1224 23 17 [ [ X cana-1224 23 18 12 12 NUM cana-1224 23 19 ] ] PUNCT cana-1224 23 20 . . PUNCT cana-1224 24 1 in in ADP cana-1224 24 2 [ [ X cana-1224 24 3 13 13 NUM cana-1224 24 4 ] ] PUNCT cana-1224 24 5 , , PUNCT cana-1224 24 6 homotopy homotopy VERB cana-1224 24 7 analysis analysis NOUN cana-1224 24 8 solves solve NOUN cana-1224 24 9 linear linear ADJ cana-1224 24 10 and and CCONJ cana-1224 24 11 nonlinear nonlinear ADJ cana-1224 24 12 schrodinger schrodinger PROPN cana-1224 24 13 equations equation NOUN cana-1224 24 14 . . PUNCT cana-1224 25 1 to to ADP cana-1224 25 2 communications communication NOUN cana-1224 25 3 on on ADP cana-1224 25 4 applied apply VERB cana-1224 25 5 nonlinear nonlinear ADJ cana-1224 25 6 analysis analysis NOUN cana-1224 25 7 issn issn NOUN cana-1224 25 8 : : PUNCT cana-1224 25 9 1074 1074 NUM cana-1224 25 10 - - PUNCT cana-1224 25 11 133x 133x NUM cana-1224 25 12 vol vol NOUN cana-1224 25 13 31 31 NUM cana-1224 25 14 no no NOUN cana-1224 25 15 . . PUNCT cana-1224 26 1 6s 6s NUM cana-1224 26 2 ( ( PUNCT cana-1224 26 3 2024 2024 NUM cana-1224 26 4 ) ) PUNCT cana-1224 26 5 306 306 NUM cana-1224 26 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 26 7 solve solve NOUN cana-1224 26 8 heat heat NOUN cana-1224 26 9 radiation radiation NOUN cana-1224 26 10 equations equation NOUN cana-1224 26 11 , , PUNCT cana-1224 26 12 homotopy homotopy VERB cana-1224 26 13 analysis analysis NOUN cana-1224 26 14 was be AUX cana-1224 26 15 devised devise VERB cana-1224 26 16 [ [ X cana-1224 26 17 14 14 NUM cana-1224 26 18 ] ] PUNCT cana-1224 26 19 . . PUNCT cana-1224 27 1 2d 2d PROPN cana-1224 27 2 schrodinger schrodinger PROPN cana-1224 27 3 eqns eqns PROPN cana-1224 27 4 were be AUX cana-1224 27 5 solved solve VERB cana-1224 27 6 utilizing utilize VERB cana-1224 27 7 a a DET cana-1224 27 8 compact compact ADJ cana-1224 27 9 boundary boundary ADJ cana-1224 27 10 value value NOUN cana-1224 27 11 approach approach NOUN cana-1224 27 12 [ [ X cana-1224 27 13 18 18 NUM cana-1224 27 14 ] ] PUNCT cana-1224 27 15 . . PUNCT cana-1224 28 1 in in ADP cana-1224 28 2 [ [ X cana-1224 28 3 19 19 NUM cana-1224 28 4 ] ] PUNCT cana-1224 28 5 , , PUNCT cana-1224 28 6 decomposition decomposition NOUN cana-1224 28 7 solves solve VERB cana-1224 28 8 cubic cubic ADJ cana-1224 28 9 schrodinger schrodinger PROPN cana-1224 28 10 equations equation NOUN cana-1224 28 11 . . PUNCT cana-1224 29 1 this this DET cana-1224 29 2 research research NOUN cana-1224 29 3 paper paper NOUN cana-1224 29 4 is be AUX cana-1224 29 5 constituted constitute VERB cana-1224 29 6 as as SCONJ cana-1224 29 7 follows follow VERB cana-1224 29 8 : : PUNCT cana-1224 29 9 section section NOUN cana-1224 29 10 2 2 NUM cana-1224 29 11 consists consist VERB cana-1224 29 12 of of ADP cana-1224 29 13 the the DET cana-1224 29 14 basic basic ADJ cana-1224 29 15 definitions definition NOUN cana-1224 29 16 of of ADP cana-1224 29 17 fractional fractional ADJ cana-1224 29 18 calculus calculus NOUN cana-1224 29 19 and and CCONJ cana-1224 29 20 the the DET cana-1224 29 21 basic basic ADJ cana-1224 29 22 properties property NOUN cana-1224 29 23 of of ADP cana-1224 29 24 the the DET cana-1224 29 25 elzaki elzaki NOUN cana-1224 29 26 transform transform NOUN cana-1224 29 27 . . PUNCT cana-1224 30 1 โ€œ " PUNCT cana-1224 30 2 homotopy homotopy VERB cana-1224 30 3 analysis analysis NOUN cana-1224 30 4 approach approach NOUN cana-1224 30 5 has have AUX cana-1224 30 6 been be AUX cana-1224 30 7 discussed discuss VERB cana-1224 30 8 in in ADP cana-1224 30 9 section section NOUN cana-1224 30 10 3 3 NUM cana-1224 30 11 . . PUNCT cana-1224 31 1 the the DET cana-1224 31 2 suggested suggest VERB cana-1224 31 3 scheme scheme NOUN cana-1224 31 4 on on ADP cana-1224 31 5 the the DET cana-1224 31 6 basis basis NOUN cana-1224 31 7 of of ADP cana-1224 31 8 the the DET cana-1224 31 9 combination combination NOUN cana-1224 31 10 of of ADP cana-1224 31 11 homotopy homotopy NOUN cana-1224 31 12 analysis analysis NOUN cana-1224 31 13 as as ADV cana-1224 31 14 well well ADV cana-1224 31 15 as as ADP cana-1224 31 16 the the DET cana-1224 31 17 elzaki elzaki NOUN cana-1224 31 18 transform transform VERB cana-1224 31 19 approach approach NOUN cana-1224 31 20 for for ADP cana-1224 31 21 solving solve VERB cana-1224 31 22 ( ( PUNCT cana-1224 31 23 2 2 NUM cana-1224 31 24 + + NOUN cana-1224 31 25 1)-d 1)-d NUM cana-1224 31 26 and and CCONJ cana-1224 31 27 ( ( PUNCT cana-1224 31 28 3 3 NUM cana-1224 31 29 + + SYM cana-1224 31 30 1)-d 1)-d NUM cana-1224 31 31 nonlinear nonlinear ADJ cana-1224 31 32 fractional fractional ADJ cana-1224 31 33 schrodinger schrodinger PROPN cana-1224 31 34 equations equation NOUN cana-1224 31 35 in in ADP cana-1224 31 36 section section NOUN cana-1224 31 37 4 4 NUM cana-1224 31 38 . . PUNCT cana-1224 31 39 test test NOUN cana-1224 31 40 experiments experiment NOUN cana-1224 31 41 have have AUX cana-1224 31 42 been be AUX cana-1224 31 43 performed perform VERB cana-1224 31 44 to to PART cana-1224 31 45 solve solve VERB cana-1224 31 46 nonlinear nonlinear ADJ cana-1224 31 47 ( ( PUNCT cana-1224 31 48 2 2 NUM cana-1224 31 49 + + NUM cana-1224 31 50 1)d 1)d NUM cana-1224 31 51 and and CCONJ cana-1224 31 52 ( ( PUNCT cana-1224 32 1 3 3 NUM cana-1224 32 2 + + NOUN cana-1224 32 3 1)-d 1)-d NUM cana-1224 32 4 fractional fractional ADJ cana-1224 32 5 โ€ " PUNCT cana-1224 32 6 schrodinger schrodinger PROPN cana-1224 32 7 equations equation NOUN cana-1224 32 8 in in ADP cana-1224 32 9 section section NOUN cana-1224 32 10 5 5 NUM cana-1224 32 11 . . PUNCT cana-1224 33 1 the the DET cana-1224 33 2 conclusion conclusion NOUN cana-1224 33 3 has have AUX cana-1224 33 4 been be AUX cana-1224 33 5 discussed discuss VERB cana-1224 33 6 in in ADP cana-1224 33 7 section section NOUN cana-1224 33 8 6 6 NUM cana-1224 33 9 . . NOUN cana-1224 34 1 2 2 NUM cana-1224 34 2 . . NUM cana-1224 34 3 basic basic ADJ cana-1224 34 4 of of ADP cana-1224 34 5 fractional fractional ADJ cana-1224 34 6 calculus calculus NOUN cana-1224 34 7 and and CCONJ cana-1224 34 8 elzaki elzaki NOUN cana-1224 34 9 transform transform VERB cana-1224 34 10 this this DET cana-1224 34 11 section section NOUN cana-1224 34 12 covers cover VERB cana-1224 34 13 fractional fractional ADJ cana-1224 34 14 calculus calculus NOUN cana-1224 34 15 basics basic NOUN cana-1224 34 16 . . PUNCT cana-1224 35 1 definition definition NOUN cana-1224 35 2 2.1 2.1 NUM cana-1224 35 3 . . PUNCT cana-1224 36 1 a a DET cana-1224 36 2 real real ADJ cana-1224 36 3 function function NOUN cana-1224 36 4 โ„Ž(๐‘ก โ„Ž(๐‘ก PROPN cana-1224 36 5 ) ) PUNCT cana-1224 36 6 โˆˆ โˆˆ PROPN cana-1224 37 1 ๐ถ๐œ‡ ๐ถ๐œ‡ PROPN cana-1224 37 2 , , PUNCT cana-1224 37 3 ๐‘ก ๐‘ก X cana-1224 37 4 > > X cana-1224 37 5 0 0 NUM cana-1224 37 6 , , PUNCT cana-1224 37 7 ๐œ‡ ๐œ‡ ADP cana-1224 37 8 โˆˆ โˆˆ PROPN cana-1224 37 9 โ„› โ„› NOUN cana-1224 37 10 if if SCONJ cana-1224 37 11 โˆƒ โˆƒ PROPN cana-1224 37 12 ๐‘ž ๐‘ž X cana-1224 37 13 โˆˆ โˆˆ PROPN cana-1224 37 14 โ„› โ„› PROPN cana-1224 37 15 ; ; PUNCT cana-1224 37 16 ( ( PUNCT cana-1224 37 17 ๐‘ž ๐‘ž X cana-1224 37 18 > > X cana-1224 37 19 ๐œ‡ ๐œ‡ PROPN cana-1224 37 20 ) ) PUNCT cana-1224 37 21 , , PUNCT cana-1224 37 22 s.t s.t PROPN cana-1224 37 23 โ„Ž(๐‘ก โ„Ž(๐‘ก PROPN cana-1224 37 24 ) ) PUNCT cana-1224 37 25 = = SYM cana-1224 37 26 ๐‘ก๐‘ž๐‘š(๐‘ก ๐‘ก๐‘ž๐‘š(๐‘ก NUM cana-1224 37 27 ) ) PUNCT cana-1224 37 28 , , PUNCT cana-1224 37 29 where where SCONJ cana-1224 37 30 ๐‘š(๐‘ก ๐‘š(๐‘ก X cana-1224 37 31 ) ) PUNCT cana-1224 37 32 โˆˆ โˆˆ PROPN cana-1224 37 33 ๐ถ[0,โˆž ๐ถ[0,โˆž NUM cana-1224 37 34 ) ) PUNCT cana-1224 37 35 & & CCONJ cana-1224 37 36 โ„Ž(๐‘ก โ„Ž(๐‘ก PROPN cana-1224 37 37 ) ) PUNCT cana-1224 37 38 โˆˆ โˆˆ PROPN cana-1224 38 1 ๐ถ๐œ‡ ๐ถ๐œ‡ PROPN cana-1224 38 2 ๐‘› ๐‘› ADP cana-1224 38 3 if if SCONJ cana-1224 38 4 โ„Ž(๐‘› โ„Ž(๐‘› NOUN cana-1224 38 5 ) ) PUNCT cana-1224 38 6 โˆˆ โˆˆ PROPN cana-1224 38 7 ๐ถ๐œ‡ ๐ถ๐œ‡ PROPN cana-1224 38 8 , , PUNCT cana-1224 38 9 ๐‘› ๐‘› PRON cana-1224 38 10 โˆˆ โˆˆ NOUN cana-1224 38 11 ๐‘. ๐‘. PROPN cana-1224 38 12 definition definition NOUN cana-1224 38 13 2.2 2.2 NUM cana-1224 38 14 . . PUNCT cana-1224 39 1 the the DET cana-1224 39 2 caputo caputo PROPN cana-1224 39 3 fractional fractional PROPN cana-1224 39 4 derivative derivative NOUN cana-1224 39 5 of of ADP cana-1224 39 6 โ„Ž(๐œ โ„Ž(๐œ NOUN cana-1224 39 7 ) ) PUNCT cana-1224 39 8 is be AUX cana-1224 39 9 written write VERB cana-1224 39 10 as as ADP cana-1224 39 11 : : PUNCT cana-1224 39 12 ๐œ•๐›ผ ๐œ•๐›ผ NOUN cana-1224 39 13 ๐œ•๐œ๐›ผ ๐œ•๐œ๐›ผ NOUN cana-1224 39 14 โ„Ž(๐œ โ„Ž(๐œ NOUN cana-1224 39 15 ) ) PUNCT cana-1224 39 16 = = SYM cana-1224 39 17 ๐ฝ(๐‘›โˆ’๐›ผ ๐ฝ(๐‘›โˆ’๐›ผ NOUN cana-1224 39 18 ) ) PUNCT cana-1224 39 19 ๐œ•๐‘› ๐œ•๐‘› NOUN cana-1224 39 20 ๐œ•๐œ๐‘› ๐œ•๐œ๐‘› PUNCT cana-1224 40 1 โ„Ž(๐œ โ„Ž(๐œ NOUN cana-1224 40 2 ) ) PUNCT cana-1224 40 3 = = SYM cana-1224 40 4 1 1 NUM cana-1224 40 5 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-1224 40 6 โˆ’ โˆ’ NUM cana-1224 40 7 ๐›ผ ๐›ผ NOUN cana-1224 40 8 ) ) PUNCT cana-1224 40 9 โˆซ(๐œ โˆซ(๐œ PROPN cana-1224 40 10 โˆ’ โˆ’ PROPN cana-1224 40 11 ฯ‰)๐‘›โˆ’๐›ผโˆ’1 ฯ‰)๐‘›โˆ’๐›ผโˆ’1 PROPN cana-1224 40 12 ๐œ ๐œ NOUN cana-1224 40 13 0 0 NUM cana-1224 40 14 โ„Ž๐‘›(ฯ‰)๐‘‘ฯ‰ โ„Ž๐‘›(ฯ‰)๐‘‘ฯ‰ NUM cana-1224 40 15 , , PUNCT cana-1224 40 16 where where SCONJ cana-1224 40 17 โ„Ž โ„Ž X cana-1224 40 18 โˆˆ โˆˆ VERB cana-1224 40 19 ๐ถโˆ’1 ๐ถโˆ’1 ADP cana-1224 40 20 ๐‘› ๐‘› PROPN cana-1224 40 21 , , PUNCT cana-1224 40 22 ๐‘› ๐‘› DET cana-1224 40 23 โˆ’ โˆ’ PROPN cana-1224 40 24 1 1 NUM cana-1224 40 25 < < X cana-1224 40 26 ๐›ผ ๐›ผ PROPN cana-1224 40 27 โ‰ค โ‰ค NUM cana-1224 40 28 ๐‘› ๐‘› PROPN cana-1224 40 29 , , PUNCT cana-1224 40 30 ๐‘› ๐‘› PRON cana-1224 40 31 โˆˆ โˆˆ PROPN cana-1224 40 32 โ„• โ„• PROPN cana-1224 40 33 , , PUNCT cana-1224 40 34 ๐œ ๐œ X cana-1224 40 35 > > X cana-1224 40 36 0 0 X cana-1224 40 37 . . PUNCT cana-1224 41 1 here here ADV cana-1224 41 2 , , PUNCT cana-1224 41 3 ๐œ•๐›ผ ๐œ•๐›ผ NOUN cana-1224 41 4 ๐œ•๐œ๐›ผ ๐œ•๐œ๐›ผ PROPN cana-1224 41 5 is be AUX cana-1224 41 6 caputo caputo PROPN cana-1224 41 7 derivative derivative ADJ cana-1224 41 8 operator operator NOUN cana-1224 41 9 & & CCONJ cana-1224 41 10 ฮณ ฮณ PROPN cana-1224 41 11 as as ADP cana-1224 41 12 gamma gamma PROPN cana-1224 41 13 function function NOUN cana-1224 41 14 . . PUNCT cana-1224 42 1 definition definition NOUN cana-1224 42 2 2.3.the 2.3.the NUM cana-1224 42 3 function function NOUN cana-1224 42 4 ๐‘”1(๐‘ก ๐‘”1(๐‘ก NOUN cana-1224 42 5 ) ) PUNCT cana-1224 42 6 elzaki elzaki NOUN cana-1224 42 7 transform transform NOUN cana-1224 42 8 has have AUX cana-1224 42 9 been be AUX cana-1224 42 10 expressed express VERB cana-1224 42 11 as as ADP cana-1224 42 12 : : PUNCT cana-1224 42 13 ๐ธ{๐‘”1(๐‘ก ๐ธ{๐‘”1(๐‘ก NOUN cana-1224 42 14 ) ) PUNCT cana-1224 42 15 } } PUNCT cana-1224 43 1 = = PUNCT cana-1224 43 2 ๐‘ฃ ๐‘ฃ PRON cana-1224 43 3 โˆซ โˆซ PROPN cana-1224 43 4 ๐‘”1(๐‘ก ๐‘”1(๐‘ก NOUN cana-1224 43 5 ) ) PUNCT cana-1224 43 6 . . PUNCT cana-1224 44 1 ๐‘’ ๐‘’ PROPN cana-1224 44 2 โˆ’ โˆ’ PROPN cana-1224 44 3 ๐‘ก ๐‘ก PROPN cana-1224 44 4 ๐‘ฃ๐‘‘๐‘ก ๐‘ฃ๐‘‘๐‘ก NOUN cana-1224 44 5 โˆž โˆž PROPN cana-1224 44 6 0 0 NUM cana-1224 44 7 , , PUNCT cana-1224 44 8 ๐‘ก ๐‘ก X cana-1224 44 9 > > X cana-1224 44 10 0 0 PUNCT cana-1224 45 1 definition definition NOUN cana-1224 45 2 2.4 2.4 NUM cana-1224 45 3 . . PUNCT cana-1224 46 1 for for ADP cana-1224 46 2 2 2 NUM cana-1224 46 3 parameters parameter NOUN cana-1224 46 4 ๐‘Ž ๐‘Ž PROPN cana-1224 46 5 & & CCONJ cana-1224 46 6 ๐‘ ๐‘ NOUN cana-1224 46 7 , , PUNCT cana-1224 46 8 the the DET cana-1224 46 9 mittag mittag ADJ cana-1224 46 10 - - PUNCT cana-1224 46 11 leffler leffler NOUN cana-1224 46 12 function function NOUN cana-1224 46 13 is be AUX cana-1224 46 14 defined define VERB cana-1224 46 15 as as ADP cana-1224 46 16 : : PUNCT cana-1224 46 17 ๐ธ๐‘Ž,๐‘(๐œ ๐ธ๐‘Ž,๐‘(๐œ NUM cana-1224 46 18 ) ) PUNCT cana-1224 47 1 = = PUNCT cana-1224 47 2 โˆ‘ โˆ‘ PUNCT cana-1224 47 3 ๐œ๐‘› ๐œ๐‘› ADP cana-1224 47 4 ฮณ(๐‘Ž๐‘› ฮณ(๐‘Ž๐‘› NOUN cana-1224 47 5 + + CCONJ cana-1224 47 6 ๐‘ ๐‘ NOUN cana-1224 47 7 ) ) PUNCT cana-1224 47 8 , , PUNCT cana-1224 47 9 ๐‘Ž ๐‘Ž X cana-1224 47 10 , , PUNCT cana-1224 47 11 ๐‘ ๐‘ NOUN cana-1224 47 12 > > X cana-1224 47 13 0 0 NUM cana-1224 47 14 โˆž โˆž NUM cana-1224 47 15 ๐‘›=0 ๐‘›=0 VERB cana-1224 47 16 some some DET cana-1224 47 17 basic basic ADJ cana-1224 47 18 properties property NOUN cana-1224 47 19 โ€ข โ€ข ADP cana-1224 47 20 the the DET cana-1224 47 21 caputo caputo PROPN cana-1224 47 22 fractional fractional PROPN cana-1224 47 23 derivative derivative ADJ cana-1224 47 24 elzaki elzaki NOUN cana-1224 47 25 transform transform NOUN cana-1224 47 26 is be AUX cana-1224 47 27 : : PUNCT cana-1224 47 28 ๐ธ ๐ธ PROPN cana-1224 47 29 { { PUNCT cana-1224 47 30 ๐œ•๐›ผ ๐œ•๐›ผ NOUN cana-1224 47 31 ๐œ•๐œ๐›ผ ๐œ•๐œ๐›ผ NOUN cana-1224 47 32 โ„Ž(๐œ โ„Ž(๐œ NOUN cana-1224 47 33 ) ) PUNCT cana-1224 47 34 } } PUNCT cana-1224 47 35 = = SYM cana-1224 47 36 ๐ธ{โ„Ž(๐œ ๐ธ{โ„Ž(๐œ NOUN cana-1224 47 37 ) ) PUNCT cana-1224 47 38 } } PUNCT cana-1224 47 39 ๐‘ฃ๐›ผ ๐‘ฃ๐›ผ ADP cana-1224 47 40 โˆ’ โˆ’ NOUN cana-1224 47 41 โˆ‘ โˆ‘ PUNCT cana-1224 47 42 ๐‘ฃ๐‘˜โˆ’๐›ผ+2โ„Ž๐‘˜(0 ๐‘ฃ๐‘˜โˆ’๐›ผ+2โ„Ž๐‘˜(0 PROPN cana-1224 47 43 ) ) PUNCT cana-1224 47 44 , , PUNCT cana-1224 47 45 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-1224 47 46 ๐‘˜=0 ๐‘˜=0 VERB cana-1224 47 47 ๐‘› ๐‘› PRON cana-1224 47 48 โˆ’ โˆ’ NUM cana-1224 47 49 1 1 NUM cana-1224 47 50 < < X cana-1224 47 51 ๐‘˜ ๐‘˜ DET cana-1224 47 52 โ‰ค โ‰ค NUM cana-1224 47 53 ๐‘› ๐‘› DET cana-1224 47 54 โ€ข โ€ข ADV cana-1224 47 55 below below ADV cana-1224 47 56 are be AUX cana-1224 47 57 the the DET cana-1224 47 58 elzaki elzaki NOUN cana-1224 47 59 transformations transformation NOUN cana-1224 47 60 of of ADP cana-1224 47 61 certain certain ADJ cana-1224 47 62 partial partial ADJ cana-1224 47 63 derivatives derivative NOUN cana-1224 47 64 : : PUNCT cana-1224 47 65 a a X cana-1224 47 66 ) ) PUNCT cana-1224 47 67 ๐ธ ๐ธ PROPN cana-1224 47 68 [ [ PUNCT cana-1224 47 69 ๐œ• ๐œ• NOUN cana-1224 47 70 ๐œ•๐‘ก ๐œ•๐‘ก NOUN cana-1224 47 71 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-1224 47 72 , , PUNCT cana-1224 47 73 ๐‘ก ๐‘ก NOUN cana-1224 47 74 ) ) PUNCT cana-1224 47 75 ] ] PUNCT cana-1224 48 1 = = PUNCT cana-1224 48 2 ๐ธ[๐‘“(๐‘ฅ,๐‘ก ๐ธ[๐‘“(๐‘ฅ,๐‘ก NUM cana-1224 48 3 ) ) PUNCT cana-1224 48 4 ] ] PUNCT cana-1224 49 1 ๐‘ฃ ๐‘ฃ ADP cana-1224 49 2 โˆ’ โˆ’ PROPN cana-1224 49 3 ๐‘ฃ. ๐‘ฃ. NOUN cana-1224 49 4 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-1224 49 5 , , PUNCT cana-1224 49 6 0 0 NUM cana-1224 49 7 ) ) PUNCT cana-1224 49 8 , , PUNCT cana-1224 49 9 communications communication NOUN cana-1224 49 10 on on ADP cana-1224 49 11 applied apply VERB cana-1224 49 12 nonlinear nonlinear ADJ cana-1224 49 13 analysis analysis NOUN cana-1224 49 14 issn issn NOUN cana-1224 49 15 : : PUNCT cana-1224 49 16 1074 1074 NUM cana-1224 49 17 - - PUNCT cana-1224 49 18 133x 133x NUM cana-1224 49 19 vol vol NOUN cana-1224 49 20 31 31 NUM cana-1224 49 21 no no NOUN cana-1224 49 22 . . PUNCT cana-1224 50 1 6s 6s NUM cana-1224 50 2 ( ( PUNCT cana-1224 50 3 2024 2024 NUM cana-1224 50 4 ) ) PUNCT cana-1224 50 5 307 307 NUM cana-1224 50 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 50 7 b b X cana-1224 50 8 ) ) PUNCT cana-1224 50 9 ๐ธ ๐ธ PROPN cana-1224 50 10 [ [ PUNCT cana-1224 50 11 ๐œ•2 ๐œ•2 NOUN cana-1224 50 12 ๐œ•๐‘ก2 ๐œ•๐‘ก2 NOUN cana-1224 50 13 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-1224 50 14 , , PUNCT cana-1224 50 15 ๐‘ก ๐‘ก NOUN cana-1224 50 16 ) ) PUNCT cana-1224 50 17 ] ] PUNCT cana-1224 51 1 = = SYM cana-1224 51 2 1 1 NUM cana-1224 51 3 ๐‘ฃ2 ๐‘ฃ2 NUM cana-1224 51 4 ๐ธ[๐‘“(๐‘ฅ ๐ธ[๐‘“(๐‘ฅ NOUN cana-1224 51 5 , , PUNCT cana-1224 51 6 ๐‘ก ๐‘ก NOUN cana-1224 51 7 ) ) PUNCT cana-1224 51 8 ] ] PUNCT cana-1224 51 9 โˆ’ โˆ’ PUNCT cana-1224 51 10 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-1224 51 11 , , PUNCT cana-1224 51 12 0 0 NUM cana-1224 51 13 ) ) PUNCT cana-1224 51 14 โˆ’ โˆ’ PROPN cana-1224 51 15 ๐‘ฃ. ๐‘ฃ. NOUN cana-1224 51 16 ๐œ•๐‘“ ๐œ•๐‘“ PROPN cana-1224 51 17 ๐œ•๐‘ก ๐œ•๐‘ก PROPN cana-1224 51 18 ( ( PUNCT cana-1224 51 19 ๐‘ฅ ๐‘ฅ PROPN cana-1224 51 20 , , PUNCT cana-1224 51 21 0 0 NUM cana-1224 51 22 ) ) PUNCT cana-1224 51 23 , , PUNCT cana-1224 51 24 c c X cana-1224 51 25 ) ) PUNCT cana-1224 51 26 ๐ธ ๐ธ PROPN cana-1224 51 27 [ [ PUNCT cana-1224 51 28 ๐œ• ๐œ• NOUN cana-1224 51 29 ๐œ•๐‘ฅ ๐œ•๐‘ฅ NOUN cana-1224 51 30 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-1224 51 31 , , PUNCT cana-1224 51 32 ๐‘ก ๐‘ก NOUN cana-1224 51 33 ) ) PUNCT cana-1224 51 34 ] ] PUNCT cana-1224 52 1 = = PUNCT cana-1224 52 2 ๐‘‘ ๐‘‘ NOUN cana-1224 52 3 ๐‘‘๐‘ฅ ๐‘‘๐‘ฅ VERB cana-1224 52 4 ๐ธ[๐‘“(๐‘ฅ ๐ธ[๐‘“(๐‘ฅ NOUN cana-1224 52 5 , , PUNCT cana-1224 52 6 ๐‘ก ๐‘ก NOUN cana-1224 52 7 ) ) PUNCT cana-1224 52 8 ] ] PUNCT cana-1224 52 9 , , PUNCT cana-1224 52 10 d d X cana-1224 52 11 ) ) PUNCT cana-1224 52 12 ๐ธ ๐ธ PROPN cana-1224 52 13 [ [ PUNCT cana-1224 52 14 ๐œ•2 ๐œ•2 NOUN cana-1224 52 15 ๐œ•๐‘ฅ2 ๐œ•๐‘ฅ2 NOUN cana-1224 52 16 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-1224 52 17 , , PUNCT cana-1224 52 18 ๐‘ก ๐‘ก NOUN cana-1224 52 19 ) ) PUNCT cana-1224 52 20 ] ] PUNCT cana-1224 53 1 = = PUNCT cana-1224 53 2 ๐‘‘2 ๐‘‘2 NOUN cana-1224 53 3 ๐‘‘๐‘ฅ2 ๐‘‘๐‘ฅ2 NOUN cana-1224 53 4 ๐ธ[๐‘“(๐‘ฅ ๐ธ[๐‘“(๐‘ฅ NOUN cana-1224 53 5 , , PUNCT cana-1224 53 6 ๐‘ก ๐‘ก NOUN cana-1224 53 7 ) ) PUNCT cana-1224 53 8 ] ] PUNCT cana-1224 53 9 . . PUNCT cana-1224 54 1 โ€ข โ€ข NUM cana-1224 54 2 the the DET cana-1224 54 3 elzaki elzaki NOUN cana-1224 54 4 transform transform NOUN cana-1224 54 5 of of ADP cana-1224 54 6 certain certain ADJ cana-1224 54 7 functions function NOUN cana-1224 54 8 is be AUX cana-1224 54 9 provided provide VERB cana-1224 54 10 in in ADP cana-1224 54 11 the the DET cana-1224 54 12 list list NOUN cana-1224 54 13 : : PUNCT cana-1224 54 14 ๐ธ(1 ๐ธ(1 NUM cana-1224 54 15 ) ) PUNCT cana-1224 54 16 = = SYM cana-1224 54 17 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 54 18 , , PUNCT cana-1224 54 19 ๐ธ(๐‘ก ๐ธ(๐‘ก NUM cana-1224 54 20 ) ) PUNCT cana-1224 54 21 = = SYM cana-1224 54 22 ๐‘ฃ3 ๐‘ฃ3 ADJ cana-1224 54 23 , , PUNCT cana-1224 54 24 ๐ธ(๐‘ก๐‘› ๐ธ(๐‘ก๐‘› X cana-1224 54 25 ) ) PUNCT cana-1224 54 26 = = SYM cana-1224 54 27 ๐‘› ๐‘› PROPN cana-1224 54 28 ! ! NOUN cana-1224 54 29 ๐‘ฃ๐‘›+2 ๐‘ฃ๐‘›+2 NUM cana-1224 54 30 , , PUNCT cana-1224 54 31 ๐ธ(๐‘’๐‘Ž๐‘ก ๐ธ(๐‘’๐‘Ž๐‘ก VERB cana-1224 54 32 ) ) PUNCT cana-1224 54 33 = = SYM cana-1224 55 1 ๐‘ฃ2 ๐‘ฃ2 NOUN cana-1224 55 2 1 1 NUM cana-1224 55 3 โˆ’ โˆ’ PROPN cana-1224 55 4 ๐‘Ž๐‘ฃ ๐‘Ž๐‘ฃ PROPN cana-1224 55 5 , , PUNCT cana-1224 55 6 ๐ธ(sin ๐ธ(sin NOUN cana-1224 55 7 ๐‘Ž๐‘ก ๐‘Ž๐‘ก ADP cana-1224 55 8 ) ) PUNCT cana-1224 55 9 = = PUNCT cana-1224 55 10 ๐‘Ž๐‘ฃ3 ๐‘Ž๐‘ฃ3 NOUN cana-1224 55 11 1 1 NUM cana-1224 55 12 + + CCONJ cana-1224 55 13 ๐‘Ž2๐‘ฃ2 ๐‘Ž2๐‘ฃ2 PROPN cana-1224 55 14 3 3 NUM cana-1224 55 15 . . PUNCT cana-1224 55 16 homotopy homotopy VERB cana-1224 55 17 analysis analysis NOUN cana-1224 55 18 method method NOUN cana-1224 55 19 [ [ X cana-1224 55 20 15 15 NUM cana-1224 55 21 - - SYM cana-1224 55 22 17 17 NUM cana-1224 55 23 ] ] PUNCT cana-1224 55 24 take take VERB cana-1224 55 25 into into ADP cana-1224 55 26 account account NOUN cana-1224 55 27 the the DET cana-1224 55 28 subsequent subsequent ADJ cana-1224 55 29 nonlinear nonlinear ADJ cana-1224 55 30 differential differential ADJ cana-1224 55 31 equation equation NOUN cana-1224 55 32 ๐‘[๐‘ค(ฯ‰ ๐‘[๐‘ค(ฯ‰ NOUN cana-1224 55 33 , , PUNCT cana-1224 55 34 ๐‘ก ๐‘ก NOUN cana-1224 55 35 ) ) PUNCT cana-1224 55 36 ] ] PUNCT cana-1224 56 1 = = SYM cana-1224 56 2 0 0 PUNCT cana-1224 56 3 ( ( PUNCT cana-1224 56 4 2 2 NUM cana-1224 56 5 ) ) PUNCT cana-1224 56 6 where where SCONJ cana-1224 56 7 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ PROPN cana-1224 56 8 , , PUNCT cana-1224 56 9 ๐‘ก ๐‘ก PROPN cana-1224 56 10 ) ) PUNCT cana-1224 56 11 as as ADP cana-1224 56 12 an an DET cana-1224 56 13 unknown unknown ADJ cana-1224 56 14 function function NOUN cana-1224 56 15 , , PUNCT cana-1224 56 16 ๐‘ ๐‘ PROPN cana-1224 56 17 as as ADP cana-1224 56 18 a a DET cana-1224 56 19 nonlinear nonlinear ADJ cana-1224 56 20 operator operator NOUN cana-1224 56 21 , , PUNCT cana-1224 56 22 and and CCONJ cana-1224 56 23 ฯ‰ ฯ‰ NOUN cana-1224 56 24 may may AUX cana-1224 56 25 be be AUX cana-1224 56 26 { { PUNCT cana-1224 56 27 ๐‘ฅ ๐‘ฅ NOUN cana-1224 56 28 , , PUNCT cana-1224 56 29 ๐‘ฆ ๐‘ฆ NOUN cana-1224 56 30 } } PUNCT cana-1224 56 31 or or CCONJ cana-1224 56 32 { { PUNCT cana-1224 56 33 ๐‘ฅ ๐‘ฅ NOUN cana-1224 56 34 , , PUNCT cana-1224 56 35 ๐‘ฆ ๐‘ฆ NOUN cana-1224 56 36 , , PUNCT cana-1224 56 37 ๐‘ง ๐‘ง NOUN cana-1224 56 38 } } PUNCT cana-1224 56 39 . . PUNCT cana-1224 57 1 the the DET cana-1224 57 2 variables variable NOUN cana-1224 57 3 ๐‘ฅ ๐‘ฅ PROPN cana-1224 57 4 , , PUNCT cana-1224 57 5 ๐‘ฆ ๐‘ฆ NOUN cana-1224 57 6 , , PUNCT cana-1224 57 7 ๐‘ง ๐‘ง NOUN cana-1224 57 8 , , PUNCT cana-1224 57 9 and and CCONJ cana-1224 57 10 ๐‘ก ๐‘ก PROPN cana-1224 57 11 as as ADP cana-1224 57 12 the the DET cana-1224 57 13 temporal temporal ADJ cana-1224 57 14 and and CCONJ cana-1224 57 15 spatial spatial ADJ cana-1224 57 16 independent independent ADJ cana-1224 57 17 variables variable NOUN cana-1224 57 18 , , PUNCT cana-1224 57 19 correspondingly correspondingly ADV cana-1224 57 20 . . PUNCT cana-1224 58 1 utilizing utilize VERB cana-1224 58 2 the the DET cana-1224 58 3 classical classical ADJ cana-1224 58 4 homotopy homotopy NOUN cana-1224 58 5 method method NOUN cana-1224 58 6 โ€œ " PUNCT cana-1224 58 7 ( ( PUNCT cana-1224 58 8 invented invent VERB cana-1224 58 9 by by ADP cana-1224 58 10 liao liao PROPN cana-1224 58 11 ) ) PUNCT cana-1224 58 12 ( ( PUNCT cana-1224 58 13 1 1 NUM cana-1224 58 14 โˆ’ โˆ’ NOUN cana-1224 58 15 ๐‘)๐ฟ ๐‘)๐ฟ NOUN cana-1224 58 16 [ [ PUNCT cana-1224 58 17 ๐œ‘(ฯ‰ ๐œ‘(ฯ‰ PROPN cana-1224 58 18 , , PUNCT cana-1224 58 19 ๐‘ก ๐‘ก X cana-1224 58 20 ; ; PUNCT cana-1224 58 21 ๐‘ ๐‘ NOUN cana-1224 58 22 ) ) PUNCT cana-1224 58 23 โˆ’ โˆ’ PROPN cana-1224 59 1 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ PROPN cana-1224 59 2 , , PUNCT cana-1224 59 3 ๐‘ก ๐‘ก NOUN cana-1224 59 4 ) ) PUNCT cana-1224 59 5 ] ] PUNCT cana-1224 60 1 = = PUNCT cana-1224 60 2 ๐‘โ„Ž๐‘[๐œ‘(ฯ‰ ๐‘โ„Ž๐‘[๐œ‘(ฯ‰ PROPN cana-1224 60 3 , , PUNCT cana-1224 60 4 ๐‘ก ๐‘ก X cana-1224 60 5 ; ; PUNCT cana-1224 60 6 ๐‘ ๐‘ NOUN cana-1224 60 7 ) ) PUNCT cana-1224 60 8 ] ] PUNCT cana-1224 60 9 ( ( PUNCT cana-1224 60 10 3 3 X cana-1224 60 11 ) ) PUNCT cana-1224 60 12 where where SCONJ cana-1224 60 13 โ„Ž โ„Ž PROPN cana-1224 60 14 is be AUX cana-1224 60 15 a a DET cana-1224 60 16 nonzero nonzero ADJ cana-1224 60 17 auxiliary auxiliary ADJ cana-1224 60 18 parameter parameter NOUN cana-1224 60 19 , , PUNCT cana-1224 60 20 ๐‘ ๐‘ PROPN cana-1224 60 21 โˆˆ โˆˆ PROPN cana-1224 61 1 [ [ X cana-1224 61 2 0,1 0,1 NUM cana-1224 61 3 ] ] PUNCT cana-1224 61 4 is be AUX cana-1224 61 5 an an DET cana-1224 61 6 embedding embed VERB cana-1224 61 7 parameter parameter NOUN cana-1224 61 8 , , PUNCT cana-1224 61 9 ๐ฟ ๐ฟ PROPN cana-1224 61 10 is be AUX cana-1224 61 11 an an DET cana-1224 61 12 auxiliary auxiliary ADJ cana-1224 61 13 linear linear NOUN cana-1224 61 14 operator operator NOUN cana-1224 61 15 , , PUNCT cana-1224 61 16 ๐œ‘ ๐œ‘ PROPN cana-1224 61 17 ( ( PUNCT cana-1224 61 18 ฯ‰ ฯ‰ PROPN cana-1224 61 19 , , PUNCT cana-1224 61 20 ๐‘ก ๐‘ก PROPN cana-1224 61 21 ; ; PUNCT cana-1224 61 22 ๐‘ ๐‘ NOUN cana-1224 61 23 ) ) PUNCT cana-1224 61 24 as as ADP cana-1224 61 25 an an DET cana-1224 61 26 unknown unknown ADJ cana-1224 61 27 function function NOUN cana-1224 61 28 and and CCONJ cana-1224 61 29 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ NOUN cana-1224 61 30 , , PUNCT cana-1224 61 31 ๐‘ก ๐‘ก X cana-1224 61 32 ) ) PUNCT cana-1224 61 33 is be AUX cana-1224 61 34 as as ADP cana-1224 61 35 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ NOUN cana-1224 61 36 , , PUNCT cana-1224 61 37 ๐‘ก ๐‘ก NOUN cana-1224 61 38 ) ) PUNCT cana-1224 61 39 initial initial ADJ cana-1224 61 40 guess guess NOUN cana-1224 61 41 . . PUNCT cana-1224 62 1 if if SCONJ cana-1224 62 2 ๐‘ ๐‘ NOUN cana-1224 62 3 = = SYM cana-1224 62 4 0 0 PROPN cana-1224 62 5 & & CCONJ cana-1224 62 6 ๐‘ ๐‘ NOUN cana-1224 63 1 = = SYM cana-1224 63 2 1 1 NUM cana-1224 63 3 , , PUNCT cana-1224 63 4 it it PRON cana-1224 63 5 holds hold VERB cana-1224 63 6 ๐œ‘ ๐œ‘ PROPN cana-1224 63 7 ( ( PUNCT cana-1224 63 8 ฯ‰ ฯ‰ PROPN cana-1224 63 9 , , PUNCT cana-1224 63 10 ๐‘ก ๐‘ก X cana-1224 63 11 ; ; PUNCT cana-1224 63 12 0 0 NUM cana-1224 63 13 ) ) PUNCT cana-1224 63 14 = = SYM cana-1224 64 1 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ PROPN cana-1224 64 2 , , PUNCT cana-1224 64 3 ๐‘ก ๐‘ก NOUN cana-1224 64 4 ) ) PUNCT cana-1224 64 5 , , PUNCT cana-1224 64 6 and and CCONJ cana-1224 64 7 ๐œ‘ ๐œ‘ PROPN cana-1224 64 8 ( ( PUNCT cana-1224 64 9 ฯ‰ ฯ‰ PROPN cana-1224 64 10 , , PUNCT cana-1224 64 11 ๐‘ก ๐‘ก X cana-1224 64 12 ; ; PUNCT cana-1224 64 13 1 1 X cana-1224 64 14 ) ) PUNCT cana-1224 64 15 = = SYM cana-1224 64 16 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ PROPN cana-1224 64 17 , , PUNCT cana-1224 64 18 ๐‘ก ๐‘ก NOUN cana-1224 64 19 ) ) PUNCT cana-1224 64 20 therefore therefore ADV cana-1224 64 21 as as SCONJ cana-1224 64 22 ๐‘ ๐‘ PROPN cana-1224 64 23 rises rise VERB cana-1224 64 24 from from ADP cana-1224 64 25 0 0 NUM cana-1224 64 26 - - SYM cana-1224 64 27 1 1 NUM cana-1224 64 28 , , PUNCT cana-1224 64 29 solution solution NOUN cana-1224 64 30 ๐œ‘ ๐œ‘ PROPN cana-1224 64 31 ( ( PUNCT cana-1224 64 32 ฯ‰ ฯ‰ PROPN cana-1224 64 33 , , PUNCT cana-1224 64 34 ๐‘ก ๐‘ก PROPN cana-1224 64 35 ; ; PUNCT cana-1224 64 36 ๐‘ ๐‘ NOUN cana-1224 64 37 ) ) PUNCT cana-1224 64 38 which which PRON cana-1224 64 39 has have AUX cana-1224 64 40 been be AUX cana-1224 64 41 differs differ NOUN cana-1224 64 42 from from ADP cana-1224 64 43 the the DET cana-1224 64 44 initial initial ADJ cana-1224 64 45 guess guess NOUN cana-1224 64 46 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ PROPN cana-1224 64 47 , , PUNCT cana-1224 64 48 ๐‘ก ๐‘ก NOUN cana-1224 64 49 ) ) PUNCT cana-1224 64 50 to to ADP cana-1224 64 51 solution solution NOUN cana-1224 64 52 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ PROPN cana-1224 64 53 , , PUNCT cana-1224 64 54 ๐‘ก ๐‘ก NOUN cana-1224 64 55 ) ) PUNCT cana-1224 64 56 . . PUNCT cana-1224 65 1 expanding expand VERB cana-1224 65 2 ๐œ‘ ๐œ‘ PRON cana-1224 65 3 ( ( PUNCT cana-1224 65 4 ฯ‰ ฯ‰ PROPN cana-1224 65 5 , , PUNCT cana-1224 65 6 ๐‘ก ๐‘ก PROPN cana-1224 65 7 ; ; PUNCT cana-1224 65 8 ๐‘ ๐‘ NOUN cana-1224 65 9 ) ) PUNCT cana-1224 65 10 n n PROPN cana-1224 66 1 taylor taylor PROPN cana-1224 66 2 series series PROPN cana-1224 66 3 regarding regard VERB cana-1224 66 4 ๐‘ ๐‘ PROPN cana-1224 66 5 , , PUNCT cana-1224 66 6 then then ADV cana-1224 66 7 we we PRON cana-1224 66 8 have have VERB cana-1224 66 9 ๐œ‘ ๐œ‘ PROPN cana-1224 66 10 ( ( PUNCT cana-1224 66 11 ฯ‰ ฯ‰ PROPN cana-1224 66 12 , , PUNCT cana-1224 66 13 ๐‘ก ๐‘ก PROPN cana-1224 66 14 ; ; PUNCT cana-1224 66 15 ๐‘ ๐‘ NOUN cana-1224 66 16 ) ) PUNCT cana-1224 66 17 = = SYM cana-1224 67 1 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ PROPN cana-1224 67 2 , , PUNCT cana-1224 67 3 ๐‘ก ๐‘ก X cana-1224 67 4 ) ) PUNCT cana-1224 67 5 + + CCONJ cana-1224 67 6 โˆ‘ โˆ‘ PROPN cana-1224 67 7 ๐‘ค๐‘š ๐‘ค๐‘š PROPN cana-1224 67 8 ( ( PUNCT cana-1224 67 9 ฯ‰ ฯ‰ PROPN cana-1224 67 10 , , PUNCT cana-1224 67 11 ๐‘ก)๐‘๐‘š ๐‘ก)๐‘๐‘š PROPN cana-1224 67 12 โˆž โˆž ADJ cana-1224 67 13 ๐‘š=1 ๐‘š=1 X cana-1224 67 14 ( ( PUNCT cana-1224 67 15 4 4 NUM cana-1224 67 16 ) ) PUNCT cana-1224 67 17 where where SCONJ cana-1224 67 18 , , PUNCT cana-1224 67 19 ๐‘ค๐‘š(ฯ‰ ๐‘ค๐‘š(ฯ‰ X cana-1224 67 20 , , PUNCT cana-1224 67 21 ๐‘ก ๐‘ก NOUN cana-1224 67 22 ) ) PUNCT cana-1224 67 23 = = SYM cana-1224 67 24 1 1 NUM cana-1224 67 25 ๐‘š ๐‘š NOUN cana-1224 67 26 ! ! PROPN cana-1224 67 27 ๐œ•๐‘š๐œ‘(ฯ‰ ๐œ•๐‘š๐œ‘(ฯ‰ NOUN cana-1224 67 28 , , PUNCT cana-1224 67 29 ๐‘ก ๐‘ก X cana-1224 67 30 ; ; PUNCT cana-1224 67 31 ๐‘ ๐‘ NOUN cana-1224 67 32 ) ) PUNCT cana-1224 67 33 ๐œ•๐‘๐‘š ๐œ•๐‘๐‘š PUNCT cana-1224 68 1 | | ADV cana-1224 68 2 ๐‘=0 ๐‘=0 ADP cana-1224 68 3 if if SCONJ cana-1224 68 4 the the DET cana-1224 68 5 auxiliary auxiliary ADJ cana-1224 68 6 linear linear NOUN cana-1224 68 7 operator operator NOUN cana-1224 68 8 , , PUNCT cana-1224 68 9 auxiliary auxiliary ADJ cana-1224 68 10 parameter parameter NOUN cana-1224 68 11 โ„Ž โ„Ž PROPN cana-1224 68 12 , , PUNCT cana-1224 68 13 initial initial ADJ cana-1224 68 14 guess guess NOUN cana-1224 68 15 , , PUNCT cana-1224 68 16 and and CCONJ cana-1224 68 17 auxiliary auxiliary ADJ cana-1224 68 18 function function NOUN cana-1224 68 19 which which PRON cana-1224 68 20 have have AUX cana-1224 68 21 been be AUX cana-1224 68 22 appropriately appropriately ADV cana-1224 68 23 selected select VERB cana-1224 68 24 , , PUNCT cana-1224 68 25 then then ADV cana-1224 68 26 the the DET cana-1224 68 27 series series NOUN cana-1224 68 28 ( ( PUNCT cana-1224 68 29 4 4 X cana-1224 68 30 ) ) PUNCT cana-1224 68 31 converges converge NOUN cana-1224 68 32 at at ADP cana-1224 68 33 ๐‘ ๐‘ NOUN cana-1224 68 34 = = SYM cana-1224 68 35 1 1 NUM cana-1224 68 36 and and CCONJ cana-1224 68 37 we we PRON cana-1224 68 38 get get VERB cana-1224 68 39 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ PROPN cana-1224 68 40 , , PUNCT cana-1224 68 41 ๐‘ก ๐‘ก X cana-1224 68 42 ) ) PUNCT cana-1224 68 43 = = PUNCT cana-1224 68 44 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ PROPN cana-1224 68 45 , , PUNCT cana-1224 68 46 ๐‘ก ๐‘ก X cana-1224 68 47 ) ) PUNCT cana-1224 68 48 + + CCONJ cana-1224 68 49 โˆ‘ โˆ‘ PROPN cana-1224 68 50 ๐‘ค๐‘š ๐‘ค๐‘š NOUN cana-1224 68 51 โˆž โˆž NUM cana-1224 68 52 ๐‘š=1 ๐‘š=1 X cana-1224 68 53 ( ( PUNCT cana-1224 68 54 ฯ‰ ฯ‰ NOUN cana-1224 68 55 , , PUNCT cana-1224 68 56 ๐‘ก ๐‘ก PROPN cana-1224 68 57 ) ) PUNCT cana-1224 68 58 , , PUNCT cana-1224 68 59 ( ( PUNCT cana-1224 68 60 5 5 X cana-1224 68 61 ) ) PUNCT cana-1224 68 62 communications communication NOUN cana-1224 68 63 on on ADP cana-1224 68 64 applied apply VERB cana-1224 68 65 nonlinear nonlinear ADJ cana-1224 68 66 analysis analysis NOUN cana-1224 68 67 issn issn NOUN cana-1224 68 68 : : PUNCT cana-1224 68 69 1074 1074 NUM cana-1224 68 70 - - PUNCT cana-1224 68 71 133x 133x NUM cana-1224 68 72 vol vol NOUN cana-1224 68 73 31 31 NUM cana-1224 68 74 no no NOUN cana-1224 68 75 . . PUNCT cana-1224 69 1 6s 6s NUM cana-1224 69 2 ( ( PUNCT cana-1224 69 3 2024 2024 NUM cana-1224 69 4 ) ) PUNCT cana-1224 69 5 308 308 NUM cana-1224 69 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 70 1 this this PRON cana-1224 70 2 should should AUX cana-1224 70 3 be be AUX cana-1224 70 4 a a DET cana-1224 70 5 valid valid ADJ cana-1224 70 6 solution solution NOUN cana-1224 70 7 to to ADP cana-1224 70 8 the the DET cana-1224 70 9 original original ADJ cana-1224 70 10 nonlinear nonlinear NOUN cana-1224 70 11 eqn eqn NOUN cana-1224 70 12 . . PUNCT cana-1224 71 1 the the DET cana-1224 71 2 governing govern VERB cana-1224 71 3 eqn eqn NOUN cana-1224 71 4 could could AUX cana-1224 71 5 be be AUX cana-1224 71 6 derived derive VERB cana-1224 71 7 from from ADP cana-1224 71 8 the the DET cana-1224 71 9 0 0 NUM cana-1224 71 10 - - PUNCT cana-1224 71 11 order order NOUN cana-1224 71 12 deformation deformation NOUN cana-1224 71 13 eqn eqn NOUN cana-1224 71 14 ( ( PUNCT cana-1224 71 15 3 3 NUM cana-1224 71 16 ) ) PUNCT cana-1224 71 17 based base VERB cana-1224 71 18 on on ADP cana-1224 71 19 definition definition NOUN cana-1224 71 20 ( ( PUNCT cana-1224 71 21 5 5 NUM cana-1224 71 22 ) ) PUNCT cana-1224 71 23 . . PUNCT cana-1224 72 1 define define VERB cana-1224 72 2 the the DET cana-1224 72 3 vector vector NOUN cana-1224 72 4 ๐‘ค๐‘›โƒ—โƒ—โƒ—โƒ— ๐‘ค๐‘›โƒ—โƒ—โƒ—โƒ— PROPN cana-1224 72 5 โƒ— โƒ— PROPN cana-1224 72 6 = = SYM cana-1224 72 7 { { PUNCT cana-1224 72 8 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ NOUN cana-1224 72 9 , , PUNCT cana-1224 72 10 ๐‘ก ๐‘ก NOUN cana-1224 72 11 ) ) PUNCT cana-1224 72 12 , , PUNCT cana-1224 72 13 ๐‘ค1(ฯ‰ ๐‘ค1(ฯ‰ NOUN cana-1224 72 14 , , PUNCT cana-1224 72 15 ๐‘ก ๐‘ก NOUN cana-1224 72 16 ) ) PUNCT cana-1224 72 17 , , PUNCT cana-1224 72 18 ๐‘ค2(ฯ‰ ๐‘ค2(ฯ‰ NOUN cana-1224 72 19 , , PUNCT cana-1224 72 20 ๐‘ก ๐‘ก NOUN cana-1224 72 21 ) ) PUNCT cana-1224 72 22 โ€ฆ โ€ฆ PUNCT cana-1224 72 23 โ€ฆ โ€ฆ PUNCT cana-1224 72 24 . . PUNCT cana-1224 72 25 . . PUNCT cana-1224 73 1 ๐‘ค๐‘›(ฯ‰ ๐‘ค๐‘›(ฯ‰ X cana-1224 73 2 , , PUNCT cana-1224 73 3 ๐‘ก ๐‘ก NOUN cana-1224 73 4 ) ) PUNCT cana-1224 73 5 } } PUNCT cana-1224 73 6 differentiating differentiate VERB cana-1224 73 7 the the DET cana-1224 73 8 zeroorder zeroorder NOUN cana-1224 73 9 deformation deformation NOUN cana-1224 73 10 eqn eqn NOUN cana-1224 73 11 ( ( PUNCT cana-1224 73 12 3 3 NUM cana-1224 73 13 ) ) PUNCT cana-1224 73 14 , , PUNCT cana-1224 73 15 ๐‘š ๐‘š ADP cana-1224 73 16 โˆ’times โˆ’time NOUN cana-1224 73 17 regarding regard VERB cana-1224 73 18 embedding embed VERB cana-1224 73 19 parameter parameter NOUN cana-1224 73 20 ๐‘. ๐‘. NOUN cana-1224 73 21 after after ADP cana-1224 73 22 that that PRON cana-1224 73 23 putting put VERB cana-1224 73 24 ๐‘ ๐‘ NOUN cana-1224 73 25 = = PUNCT cana-1224 73 26 0 0 PUNCT cana-1224 74 1 and and CCONJ cana-1224 74 2 then then ADV cana-1224 74 3 dividing divide VERB cana-1224 74 4 it it PRON cana-1224 74 5 with with ADP cana-1224 74 6 ๐‘š ๐‘š PROPN cana-1224 74 7 ! ! PUNCT cana-1224 74 8 , , PUNCT cana-1224 74 9 then then ADV cana-1224 74 10 the the DET cana-1224 74 11 ๐‘šth ๐‘šth NOUN cana-1224 74 12 - - PUNCT cana-1224 74 13 order order NOUN cana-1224 74 14 deformation deformation NOUN cana-1224 74 15 eqn eqn NOUN cana-1224 74 16 is be AUX cana-1224 74 17 : : PUNCT cana-1224 74 18 ๐ฟ ๐ฟ PROPN cana-1224 74 19 [ [ NOUN cana-1224 74 20 ๐‘ค๐‘š(ฯ‰ ๐‘ค๐‘š(ฯ‰ NUM cana-1224 74 21 , , PUNCT cana-1224 74 22 ๐‘ก ๐‘ก NOUN cana-1224 74 23 ) ) PUNCT cana-1224 74 24 โˆ’ โˆ’ PROPN cana-1224 74 25 ๐œ’๐‘š ๐œ’๐‘š PROPN cana-1224 74 26 ๐‘ค๐‘šโˆ’1 ๐‘ค๐‘šโˆ’1 PROPN cana-1224 74 27 ( ( PUNCT cana-1224 74 28 ฯ‰ ฯ‰ PROPN cana-1224 74 29 , , PUNCT cana-1224 74 30 ๐‘ก ๐‘ก NOUN cana-1224 74 31 ) ) PUNCT cana-1224 74 32 ] ] PUNCT cana-1224 75 1 = = PUNCT cana-1224 75 2 โ„Ž โ„Ž X cana-1224 75 3 ๐‘…๐‘š[๐‘ค๐‘šโˆ’1 ๐‘…๐‘š[๐‘ค๐‘šโˆ’1 PROPN cana-1224 75 4 ( ( PUNCT cana-1224 75 5 ฯ‰ ฯ‰ PROPN cana-1224 75 6 , , PUNCT cana-1224 75 7 ๐‘ก ๐‘ก PROPN cana-1224 75 8 ) ) PUNCT cana-1224 75 9 ] ] PUNCT cana-1224 75 10 where where SCONJ cana-1224 75 11 ๐‘…๐‘š(๐‘ค๐‘šโˆ’1)โƒ—โƒ— ๐‘…๐‘š(๐‘ค๐‘šโˆ’1)โƒ—โƒ— PROPN cana-1224 75 12 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 75 13 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 75 14 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 75 15 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 75 16 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 75 17 โƒ— โƒ— PROPN cana-1224 75 18 = = SYM cana-1224 75 19 1 1 NUM cana-1224 75 20 ๐‘šโˆ’1 ๐‘šโˆ’1 PROPN cana-1224 75 21 ! ! PUNCT cana-1224 76 1 ๐œ•๐‘šโˆ’1 ๐œ•๐‘šโˆ’1 PROPN cana-1224 76 2 ๐‘[๐œ‘(ฯ‰,๐‘ก;๐‘ ๐‘[๐œ‘(ฯ‰,๐‘ก;๐‘ PROPN cana-1224 76 3 ) ) PUNCT cana-1224 76 4 ๐œ•๐‘๐‘šโˆ’1 ๐œ•๐‘๐‘šโˆ’1 NOUN cana-1224 76 5 | | CCONJ cana-1224 76 6 ๐‘=0 ๐‘=0 PROPN cana-1224 76 7 and and CCONJ cana-1224 76 8 ๐œ’๐‘š ๐œ’๐‘š NOUN cana-1224 76 9 = = SYM cana-1224 76 10 { { PUNCT cana-1224 76 11 0 0 NUM cana-1224 76 12 , , PUNCT cana-1224 76 13 ๐‘š ๐‘š PROPN cana-1224 76 14 โ‰ค โ‰ค NUM cana-1224 76 15 1 1 NUM cana-1224 76 16 1 1 NUM cana-1224 76 17 , , PUNCT cana-1224 76 18 ๐‘š ๐‘š X cana-1224 76 19 > > X cana-1224 76 20 1 1 NUM cana-1224 76 21 . . PUNCT cana-1224 77 1 4 4 X cana-1224 77 2 . . X cana-1224 77 3 elzaki elzaki AUX cana-1224 77 4 transform transform VERB cana-1224 77 5 homotopy homotopy NOUN cana-1224 77 6 analysis analysis NOUN cana-1224 77 7 method method NOUN cana-1224 77 8 rewrite rewrite NOUN cana-1224 77 9 โ€ " PUNCT cana-1224 77 10 equation equation NOUN cana-1224 77 11 ( ( PUNCT cana-1224 77 12 1 1 NUM cana-1224 77 13 ) ) PUNCT cana-1224 77 14 as as ADP cana-1224 77 15 : : PUNCT cana-1224 77 16 ๐‘ค๐‘ก ๐‘ค๐‘ก PRON cana-1224 77 17 ๐›ผ(ฯ‰ ๐›ผ(ฯ‰ NOUN cana-1224 77 18 ) ) PUNCT cana-1224 77 19 = = SYM cana-1224 77 20 ๐‘–{๐‘Žโˆ†2๐‘ค(ฯ‰ ๐‘–{๐‘Žโˆ†2๐‘ค(ฯ‰ NOUN cana-1224 77 21 ) ) PUNCT cana-1224 78 1 + + NUM cana-1224 78 2 ๐›ผ(ฯ‰)๐‘ค(ฯ‰ ๐›ผ(ฯ‰)๐‘ค(ฯ‰ NOUN cana-1224 78 3 ) ) PUNCT cana-1224 78 4 โˆ’ โˆ’ ADP cana-1224 78 5 ๐›ฝ๐‘ค2 ๐›ฝ๐‘ค2 NOUN cana-1224 78 6 ๏ฟฝ ๏ฟฝ NOUN cana-1224 78 7 ฬ… ฬ… NOUN cana-1224 78 8 ๏ฟฝ ๏ฟฝ NOUN cana-1224 78 9 } } PUNCT cana-1224 78 10 . . PUNCT cana-1224 79 1 taking take VERB cana-1224 79 2 elzaki elzaki NOUN cana-1224 79 3 transform transform VERB cana-1224 79 4 both both DET cana-1224 79 5 sides side NOUN cana-1224 79 6 , , PUNCT cana-1224 79 7 we we PRON cana-1224 79 8 obtain obtain VERB cana-1224 79 9 ๐ธ{๐‘ค๐‘ก ๐ธ{๐‘ค๐‘ก PROPN cana-1224 79 10 ๐›ผ(ฯ‰ ๐›ผ(ฯ‰ NOUN cana-1224 79 11 ) ) PUNCT cana-1224 79 12 } } PUNCT cana-1224 79 13 = = SYM cana-1224 79 14 ๐‘–๐ธ{๐‘Žโˆ†2๐‘ค(ฯ‰ ๐‘–๐ธ{๐‘Žโˆ†2๐‘ค(ฯ‰ NOUN cana-1224 79 15 ) ) PUNCT cana-1224 80 1 + + CCONJ cana-1224 80 2 ๐›ผ(ฯ‰)๐‘ค(ฯ‰ ๐›ผ(ฯ‰)๐‘ค(ฯ‰ NOUN cana-1224 80 3 ) ) PUNCT cana-1224 80 4 โˆ’ โˆ’ ADP cana-1224 80 5 ๐›ฝ๐‘ค2 ๐›ฝ๐‘ค2 NOUN cana-1224 80 6 ๏ฟฝ ๏ฟฝ NOUN cana-1224 80 7 ฬ… ฬ… NOUN cana-1224 80 8 ๏ฟฝ ๏ฟฝ NOUN cana-1224 80 9 } } PUNCT cana-1224 80 10 . . PUNCT cana-1224 81 1 using use VERB cana-1224 81 2 applications application NOUN cana-1224 81 3 of of ADP cana-1224 81 4 elzaki elzaki NOUN cana-1224 81 5 transform transform NOUN cana-1224 81 6 as as ADV cana-1224 81 7 well well ADV cana-1224 81 8 as as ADP cana-1224 81 9 an an DET cana-1224 81 10 initial initial ADJ cana-1224 81 11 condition condition NOUN cana-1224 81 12 , , PUNCT cana-1224 81 13 we we PRON cana-1224 81 14 obtain obtain VERB cana-1224 81 15 ๐ธ{๐‘ค(ฯ‰ ๐ธ{๐‘ค(ฯ‰ NUM cana-1224 81 16 , , PUNCT cana-1224 81 17 ๐‘ก ๐‘ก NOUN cana-1224 81 18 ) ) PUNCT cana-1224 81 19 } } PUNCT cana-1224 81 20 = = PUNCT cana-1224 81 21 ๐‘ฃ2๐‘ค0(ฯ‰ ๐‘ฃ2๐‘ค0(ฯ‰ NOUN cana-1224 81 22 ) ) PUNCT cana-1224 81 23 + + NUM cana-1224 81 24 ๐‘ฃ๐›ผ๐‘–๐ธ{๐‘Žโˆ†2๐‘ค(ฯ‰ ๐‘ฃ๐›ผ๐‘–๐ธ{๐‘Žโˆ†2๐‘ค(ฯ‰ NOUN cana-1224 81 25 ) ) PUNCT cana-1224 82 1 + + NUM cana-1224 82 2 ๐œ“(ฯ‰)๐‘ค(ฯ‰ ๐œ“(ฯ‰)๐‘ค(ฯ‰ NOUN cana-1224 82 3 ) ) PUNCT cana-1224 82 4 โˆ’ โˆ’ ADP cana-1224 82 5 ๐›ฝ๐‘ค2 ๐›ฝ๐‘ค2 NOUN cana-1224 82 6 ๏ฟฝ ๏ฟฝ NOUN cana-1224 82 7 ฬ… ฬ… NOUN cana-1224 82 8 ๏ฟฝ ๏ฟฝ NOUN cana-1224 82 9 } } PUNCT cana-1224 82 10 . . PUNCT cana-1224 83 1 taking take VERB cana-1224 83 2 the the DET cana-1224 83 3 nonlinear nonlinear ADJ cana-1224 83 4 part part NOUN cana-1224 83 5 as as ADP cana-1224 83 6 : : PUNCT cana-1224 83 7 ๐‘…[๐œ‘(ฯ‰ ๐‘…[๐œ‘(ฯ‰ NOUN cana-1224 83 8 , , PUNCT cana-1224 83 9 ๐‘ก ๐‘ก X cana-1224 83 10 ; ; PUNCT cana-1224 83 11 ๐‘ ๐‘ NOUN cana-1224 83 12 ) ) PUNCT cana-1224 83 13 ] ] PUNCT cana-1224 84 1 = = PUNCT cana-1224 84 2 ๐ธ(๐œ‘ ๐ธ(๐œ‘ NUM cana-1224 84 3 ) ) PUNCT cana-1224 84 4 โˆ’ โˆ’ PRON cana-1224 84 5 ๐‘ฃ2๐‘ค0(ฯ‰ ๐‘ฃ2๐‘ค0(ฯ‰ NOUN cana-1224 84 6 ) ) PUNCT cana-1224 84 7 โˆ’ โˆ’ NOUN cana-1224 84 8 ๐‘ฃ๐›ผ๐‘–๐ธ{๐‘Žโˆ†2๐œ‘(ฯ‰ ๐‘ฃ๐›ผ๐‘–๐ธ{๐‘Žโˆ†2๐œ‘(ฯ‰ NUM cana-1224 84 9 ) ) PUNCT cana-1224 84 10 + + NUM cana-1224 84 11 ๐œ“(ฯ‰)๐œ‘(ฯ‰ ๐œ“(ฯ‰)๐œ‘(ฯ‰ NOUN cana-1224 84 12 ) ) PUNCT cana-1224 84 13 โˆ’ โˆ’ ADP cana-1224 84 14 ๐›ฝ๐œ‘2 ๐›ฝ๐œ‘2 PROPN cana-1224 84 15 ๏ฟฝ ๏ฟฝ PROPN cana-1224 84 16 ฬ… ฬ… NOUN cana-1224 84 17 ๏ฟฝ ๏ฟฝ NOUN cana-1224 84 18 } } PUNCT cana-1224 84 19 . . PUNCT cana-1224 85 1 we we PRON cana-1224 85 2 formulate formulate VERB cana-1224 85 3 the the DET cana-1224 85 4 zero zero NUM cana-1224 85 5 - - PUNCT cana-1224 85 6 order order NOUN cana-1224 85 7 deformation deformation NOUN cana-1224 85 8 eqn eqn NOUN cana-1224 85 9 under under ADP cana-1224 85 10 the the DET cana-1224 85 11 given give VERB cana-1224 85 12 assumption assumption NOUN cana-1224 85 13 . . PUNCT cana-1224 86 1 ๐ป(๐‘ฅ ๐ป(๐‘ฅ PROPN cana-1224 86 2 , , PUNCT cana-1224 86 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 86 4 , , PUNCT cana-1224 86 5 ๐‘ง ๐‘ง NOUN cana-1224 86 6 , , PUNCT cana-1224 86 7 ๐‘ก ๐‘ก NOUN cana-1224 86 8 ) ) PUNCT cana-1224 86 9 = = SYM cana-1224 86 10 1 1 NUM cana-1224 86 11 , , PUNCT cana-1224 86 12 we we PRON cana-1224 86 13 have have VERB cana-1224 86 14 ( ( PUNCT cana-1224 86 15 1 1 NUM cana-1224 86 16 โˆ’ โˆ’ PROPN cana-1224 86 17 ๐‘)๐ธ{๐œ‘(ฯ‰ ๐‘)๐ธ{๐œ‘(ฯ‰ PROPN cana-1224 86 18 , , PUNCT cana-1224 86 19 ๐‘ก ๐‘ก X cana-1224 86 20 ) ) PUNCT cana-1224 86 21 โˆ’ โˆ’ ADP cana-1224 87 1 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ PROPN cana-1224 87 2 , , PUNCT cana-1224 87 3 ๐‘ก ๐‘ก NOUN cana-1224 87 4 ) ) PUNCT cana-1224 87 5 } } PUNCT cana-1224 87 6 = = SYM cana-1224 87 7 ๐‘โ„Ž๐‘…[๐œ‘(ฯ‰ ๐‘โ„Ž๐‘…[๐œ‘(ฯ‰ PROPN cana-1224 87 8 , , PUNCT cana-1224 87 9 ๐‘ก ๐‘ก X cana-1224 87 10 ; ; PUNCT cana-1224 87 11 ๐‘ ๐‘ NOUN cana-1224 87 12 ) ) PUNCT cana-1224 87 13 ] ] PUNCT cana-1224 87 14 . . PUNCT cana-1224 88 1 when when SCONJ cana-1224 88 2 ๐‘ ๐‘ X cana-1224 88 3 = = SYM cana-1224 88 4 0 0 PROPN cana-1224 88 5 & & CCONJ cana-1224 88 6 ๐‘ ๐‘ NOUN cana-1224 88 7 = = SYM cana-1224 88 8 1 1 NUM cana-1224 88 9 , , PUNCT cana-1224 88 10 we we PRON cana-1224 88 11 get get VERB cana-1224 88 12 , , PUNCT cana-1224 88 13 { { PUNCT cana-1224 88 14 ๐œ‘(ฯ‰ ๐œ‘(ฯ‰ ADV cana-1224 88 15 , , PUNCT cana-1224 88 16 ๐‘ก ๐‘ก X cana-1224 88 17 ; ; PUNCT cana-1224 88 18 0 0 NUM cana-1224 88 19 ) ) PUNCT cana-1224 89 1 = = SYM cana-1224 89 2 ๐‘ค0(ฯ‰ ๐‘ค0(ฯ‰ PROPN cana-1224 89 3 , , PUNCT cana-1224 89 4 0 0 NUM cana-1224 89 5 ) ) PUNCT cana-1224 89 6 ๐œ‘(ฯ‰ ๐œ‘(ฯ‰ PROPN cana-1224 89 7 , , PUNCT cana-1224 89 8 ๐‘ก ๐‘ก X cana-1224 89 9 ; ; PUNCT cana-1224 89 10 1 1 X cana-1224 89 11 ) ) PUNCT cana-1224 89 12 = = SYM cana-1224 89 13 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ PROPN cana-1224 89 14 , , PUNCT cana-1224 89 15 ๐‘ก ๐‘ก NOUN cana-1224 89 16 ) ) PUNCT cana-1224 89 17 . . PUNCT cana-1224 90 1 hence hence ADV cana-1224 90 2 , , PUNCT cana-1224 90 3 we we PRON cana-1224 90 4 obtain obtain VERB cana-1224 90 5 the the DET cana-1224 90 6 eqn eqn NOUN cana-1224 90 7 of of ADP cana-1224 90 8 deformation deformation NOUN cana-1224 90 9 of of ADP cana-1224 90 10 order order NOUN cana-1224 90 11 m. m. NOUN cana-1224 90 12 ๐ธ{๐‘ค๐‘š(ฯ‰ ๐ธ{๐‘ค๐‘š(ฯ‰ PROPN cana-1224 90 13 , , PUNCT cana-1224 90 14 ๐‘ก ๐‘ก NOUN cana-1224 90 15 ) ) PUNCT cana-1224 90 16 โˆ’ โˆ’ PRON cana-1224 90 17 ๐œ’๐‘š๐‘ค๐‘šโˆ’1(ฯ‰ ๐œ’๐‘š๐‘ค๐‘šโˆ’1(ฯ‰ NOUN cana-1224 90 18 , , PUNCT cana-1224 90 19 ๐‘ก ๐‘ก NOUN cana-1224 90 20 ) ) PUNCT cana-1224 90 21 } } PUNCT cana-1224 90 22 = = SYM cana-1224 90 23 โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— PROPN cana-1224 90 24 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 90 25 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 90 26 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 90 27 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 90 28 โƒ—(ฯ‰ โƒ—(ฯ‰ PROPN cana-1224 90 29 , , PUNCT cana-1224 90 30 ๐‘ก ๐‘ก NOUN cana-1224 90 31 ) ) PUNCT cana-1224 90 32 ) ) PUNCT cana-1224 90 33 . . PUNCT cana-1224 91 1 inverse inverse PROPN cana-1224 91 2 elzaki elzaki NOUN cana-1224 91 3 transforms transform VERB cana-1224 91 4 both both DET cana-1224 91 5 sides side NOUN cana-1224 91 6 , , PUNCT cana-1224 91 7 we we PRON cana-1224 91 8 obtain obtain VERB cana-1224 91 9 , , PUNCT cana-1224 91 10 ๐‘ค๐‘š(ฯ‰ ๐‘ค๐‘š(ฯ‰ ADV cana-1224 91 11 , , PUNCT cana-1224 91 12 ๐‘ก ๐‘ก NOUN cana-1224 91 13 ) ) PUNCT cana-1224 91 14 โˆ’ โˆ’ PRON cana-1224 91 15 ๐œ’๐‘š๐‘ค๐‘šโˆ’1(ฯ‰ ๐œ’๐‘š๐‘ค๐‘šโˆ’1(ฯ‰ NOUN cana-1224 91 16 , , PUNCT cana-1224 91 17 ๐‘ก ๐‘ก ADJ cana-1224 91 18 ) ) PUNCT cana-1224 91 19 = = SYM cana-1224 91 20 ๐ธโˆ’1{โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— ๐ธโˆ’1{โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— PROPN cana-1224 91 21 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 91 22 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 91 23 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 91 24 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 91 25 โƒ—(ฯ‰ โƒ—(ฯ‰ PROPN cana-1224 91 26 , , PUNCT cana-1224 91 27 ๐‘ก ๐‘ก NOUN cana-1224 91 28 ) ) PUNCT cana-1224 91 29 ) ) PUNCT cana-1224 91 30 } } PUNCT cana-1224 91 31 . . PUNCT cana-1224 92 1 from from ADP cana-1224 92 2 the the DET cana-1224 92 3 above above ADJ cana-1224 92 4 eqon eqon NOUN cana-1224 92 5 , , PUNCT cana-1224 92 6 we we PRON cana-1224 92 7 get get VERB cana-1224 92 8 communications communication NOUN cana-1224 92 9 on on ADP cana-1224 92 10 applied apply VERB cana-1224 92 11 nonlinear nonlinear ADJ cana-1224 92 12 analysis analysis NOUN cana-1224 92 13 issn issn NOUN cana-1224 92 14 : : PUNCT cana-1224 92 15 1074 1074 NUM cana-1224 92 16 - - PUNCT cana-1224 92 17 133x 133x NUM cana-1224 92 18 vol vol NOUN cana-1224 92 19 31 31 NUM cana-1224 92 20 no no NOUN cana-1224 92 21 . . PUNCT cana-1224 93 1 6s 6s NUM cana-1224 93 2 ( ( PUNCT cana-1224 93 3 2024 2024 NUM cana-1224 93 4 ) ) PUNCT cana-1224 93 5 309 309 NUM cana-1224 93 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 93 7 ๐‘ค1(ฯ‰ ๐‘ค1(ฯ‰ PROPN cana-1224 93 8 , , PUNCT cana-1224 93 9 ๐‘ก ๐‘ก X cana-1224 93 10 ) ) PUNCT cana-1224 93 11 = = SYM cana-1224 93 12 โˆ’๐ธโˆ’1{๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— โˆ’๐ธโˆ’1{๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— X cana-1224 93 13 โƒ—(ฯ‰ โƒ—(ฯ‰ PROPN cana-1224 93 14 , , PUNCT cana-1224 93 15 ๐‘ก ๐‘ก NOUN cana-1224 93 16 ) ) PUNCT cana-1224 93 17 ) ) PUNCT cana-1224 93 18 } } PUNCT cana-1224 93 19 , , PUNCT cana-1224 93 20 ๐‘ค2(ฯ‰ ๐‘ค2(ฯ‰ NOUN cana-1224 93 21 , , PUNCT cana-1224 93 22 ๐‘ก ๐‘ก NOUN cana-1224 93 23 ) ) PUNCT cana-1224 93 24 = = SYM cana-1224 93 25 ๐‘ค1(ฯ‰ ๐‘ค1(ฯ‰ PROPN cana-1224 93 26 , , PUNCT cana-1224 93 27 ๐‘ก ๐‘ก NOUN cana-1224 93 28 ) ) PUNCT cana-1224 93 29 โˆ’ โˆ’ PROPN cana-1224 93 30 ๐ธโˆ’1{๐‘…2(๐‘ค1โƒ—โƒ— ๐ธโˆ’1{๐‘…2(๐‘ค1โƒ—โƒ— NOUN cana-1224 93 31 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 93 32 ( ( PUNCT cana-1224 93 33 ฯ‰ ฯ‰ PROPN cana-1224 93 34 , , PUNCT cana-1224 93 35 ๐‘ก ๐‘ก NOUN cana-1224 93 36 ) ) PUNCT cana-1224 93 37 ) ) PUNCT cana-1224 93 38 } } PUNCT cana-1224 93 39 , , PUNCT cana-1224 93 40 ๐‘ค3(ฯ‰ ๐‘ค3(ฯ‰ PROPN cana-1224 93 41 , , PUNCT cana-1224 93 42 ๐‘ก ๐‘ก NOUN cana-1224 93 43 ) ) PUNCT cana-1224 93 44 = = SYM cana-1224 94 1 ๐‘ค2(ฯ‰ ๐‘ค2(ฯ‰ NOUN cana-1224 94 2 , , PUNCT cana-1224 94 3 ๐‘ก ๐‘ก NOUN cana-1224 94 4 ) ) PUNCT cana-1224 94 5 โˆ’ โˆ’ NOUN cana-1224 94 6 ๐ธโˆ’1{๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— ๐ธโˆ’1{๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— NOUN cana-1224 94 7 โƒ—(ฯ‰ โƒ—(ฯ‰ NUM cana-1224 94 8 , , PUNCT cana-1224 94 9 ๐‘ก ๐‘ก NOUN cana-1224 94 10 ) ) PUNCT cana-1224 94 11 ) ) PUNCT cana-1224 94 12 } } PUNCT cana-1224 94 13 , , PUNCT cana-1224 94 14 โ‹ฎ โ‹ฎ NOUN cana-1224 94 15 therefore therefore ADV cana-1224 94 16 , , PUNCT cana-1224 94 17 the the DET cana-1224 94 18 solution solution NOUN cana-1224 94 19 is be AUX cana-1224 94 20 : : PUNCT cana-1224 94 21 ๐‘ค(ฯ‰ ๐‘ค(ฯ‰ PROPN cana-1224 94 22 , , PUNCT cana-1224 94 23 ๐‘ก ๐‘ก NOUN cana-1224 94 24 ) ) PUNCT cana-1224 94 25 = = SYM cana-1224 94 26 ๐‘ค0 ๐‘ค0 PROPN cana-1224 94 27 + + CCONJ cana-1224 94 28 ๐‘ค1 ๐‘ค1 VERB cana-1224 94 29 + + CCONJ cana-1224 94 30 ๐‘ค2 ๐‘ค2 NOUN cana-1224 94 31 + + CCONJ cana-1224 94 32 โ‹ฏ โ‹ฏ PROPN cana-1224 94 33 5 5 NUM cana-1224 94 34 . . PUNCT cana-1224 94 35 test test NOUN cana-1224 94 36 examples example NOUN cana-1224 94 37 : : PUNCT cana-1224 94 38 in in ADP cana-1224 94 39 โ€œ " PUNCT cana-1224 94 40 this this DET cana-1224 94 41 section section NOUN cana-1224 94 42 , , PUNCT cana-1224 94 43 we we PRON cana-1224 94 44 will will AUX cana-1224 94 45 perform perform VERB cana-1224 94 46 some some DET cana-1224 94 47 test test NOUN cana-1224 94 48 examples example NOUN cana-1224 94 49 to to PART cana-1224 94 50 find find VERB cana-1224 94 51 semi semi ADJ cana-1224 94 52 - - ADJ cana-1224 94 53 analytical analytical ADJ cana-1224 94 54 solutions solution NOUN cana-1224 94 55 of of ADP cana-1224 94 56 nonlinear nonlinear ADJ cana-1224 94 57 fractional fractional ADJ cana-1224 94 58 ( ( PUNCT cana-1224 94 59 2 2 NUM cana-1224 94 60 + + NOUN cana-1224 94 61 1)-d 1)-d NUM cana-1224 94 62 and and CCONJ cana-1224 94 63 ( ( PUNCT cana-1224 94 64 3 3 NUM cana-1224 94 65 + + SYM cana-1224 94 66 1)-d 1)-d NUM cana-1224 94 67 nonlinear nonlinear ADJ cana-1224 94 68 fractional fractional ADJ cana-1224 94 69 โ€ " PUNCT cana-1224 94 70 schrodinger schrodinger PROPN cana-1224 94 71 equations equation NOUN cana-1224 94 72 . . PUNCT cana-1224 95 1 example example NOUN cana-1224 95 2 1 1 NUM cana-1224 95 3 : : PUNCT cana-1224 95 4 consider consider VERB cana-1224 95 5 the the DET cana-1224 95 6 ( ( PUNCT cana-1224 95 7 3 3 NUM cana-1224 95 8 + + NOUN cana-1224 95 9 1)-d 1)-d NUM cana-1224 95 10 fractional fractional ADJ cana-1224 95 11 nonlinear nonlinear PROPN cana-1224 95 12 schrodinger schrodinger PROPN cana-1224 95 13 eqn eqn PROPN cana-1224 95 14 of of ADP cana-1224 95 15 form form NOUN cana-1224 95 16 ๐‘–๐‘ค๐‘ก ๐‘–๐‘ค๐‘ก NOUN cana-1224 95 17 ๐›ผ ๐›ผ NOUN cana-1224 95 18 + + CCONJ cana-1224 95 19 ๐‘ค๐‘ฅ๐‘ฅ ๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 95 20 + + CCONJ cana-1224 95 21 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 95 22 + + CCONJ cana-1224 95 23 ๐‘ค๐‘ง๐‘ง ๐‘ค๐‘ง๐‘ง NOUN cana-1224 95 24 + + ADJ cana-1224 95 25 4|๐‘ค|2๐‘ค 4|๐‘ค|2๐‘ค NUM cana-1224 95 26 = = SYM cana-1224 95 27 0 0 NUM cana-1224 95 28 , , PUNCT cana-1224 95 29 ( ( PUNCT cana-1224 95 30 6 6 NUM cana-1224 95 31 ) ) PUNCT cana-1224 95 32 with with ADP cana-1224 95 33 initial initial ADJ cana-1224 95 34 condition condition NOUN cana-1224 95 35 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 95 36 , , PUNCT cana-1224 95 37 ๐‘ฆ ๐‘ฆ NOUN cana-1224 95 38 , , PUNCT cana-1224 95 39 ๐‘ง ๐‘ง PROPN cana-1224 95 40 , , PUNCT cana-1224 95 41 0 0 NUM cana-1224 95 42 ) ) PUNCT cana-1224 95 43 = = SYM cana-1224 95 44 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 95 45 ) ) PUNCT cana-1224 95 46 . . PUNCT cana-1224 96 1 the the DET cana-1224 96 2 problem problem NOUN cana-1224 96 3 ( ( PUNCT cana-1224 96 4 ๐›ผ ๐›ผ NOUN cana-1224 96 5 = = SYM cana-1224 96 6 1)๐‘กโ„Ž๐‘’ 1)๐‘กโ„Ž๐‘’ NUM cana-1224 96 7 exact exact ADJ cana-1224 96 8 solution solution NOUN cana-1224 96 9 is be AUX cana-1224 96 10 : : PUNCT cana-1224 96 11 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 96 12 , , PUNCT cana-1224 96 13 ๐‘ฆ ๐‘ฆ NOUN cana-1224 96 14 , , PUNCT cana-1224 96 15 ๐‘ง ๐‘ง NOUN cana-1224 96 16 , , PUNCT cana-1224 96 17 ๐‘ก ๐‘ก NOUN cana-1224 96 18 ) ) PUNCT cana-1224 96 19 = = SYM cana-1224 96 20 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง+๐‘ก ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง+๐‘ก PROPN cana-1224 96 21 ) ) PUNCT cana-1224 96 22 rewrite rewrite VERB cana-1224 96 23 the the DET cana-1224 96 24 given give VERB cana-1224 96 25 problem problem NOUN cana-1224 96 26 as as ADP cana-1224 96 27 : : PUNCT cana-1224 96 28 ๐‘–๐‘ค๐‘ก ๐‘–๐‘ค๐‘ก NOUN cana-1224 96 29 ๐›ผ ๐›ผ NOUN cana-1224 96 30 = = PUNCT cana-1224 96 31 โˆ’(๐‘ค๐‘ฅ๐‘ฅ โˆ’(๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 96 32 + + NOUN cana-1224 96 33 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 96 34 + + CCONJ cana-1224 96 35 ๐‘ค๐‘ง๐‘ง ๐‘ค๐‘ง๐‘ง NOUN cana-1224 96 36 + + ADJ cana-1224 96 37 4๐‘ค2 4๐‘ค2 NUM cana-1224 96 38 ๏ฟฝ ๏ฟฝ NOUN cana-1224 96 39 ฬ… ฬ… NOUN cana-1224 96 40 ๏ฟฝ ๏ฟฝ NOUN cana-1224 96 41 ) ) PUNCT cana-1224 96 42 , , PUNCT cana-1224 96 43 it it PRON cana-1224 96 44 implies imply VERB cana-1224 96 45 ๐‘ค๐‘ก ๐‘ค๐‘ก NOUN cana-1224 96 46 ๐›ผ ๐›ผ NOUN cana-1224 96 47 = = PUNCT cana-1224 96 48 ๐‘–(๐‘ค๐‘ฅ๐‘ฅ ๐‘–(๐‘ค๐‘ฅ๐‘ฅ PROPN cana-1224 96 49 + + PUNCT cana-1224 96 50 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 96 51 + + CCONJ cana-1224 96 52 ๐‘ค๐‘ง๐‘ง ๐‘ค๐‘ง๐‘ง NOUN cana-1224 96 53 + + ADJ cana-1224 96 54 4๐‘ค2 4๐‘ค2 NUM cana-1224 96 55 ๏ฟฝ ๏ฟฝ NOUN cana-1224 96 56 ฬ… ฬ… NOUN cana-1224 96 57 ๏ฟฝ ๏ฟฝ NOUN cana-1224 96 58 ) ) PUNCT cana-1224 96 59 ( ( PUNCT cana-1224 96 60 7 7 X cana-1224 96 61 ) ) PUNCT cana-1224 96 62 taking take VERB cana-1224 96 63 elzaki elzaki NOUN cana-1224 96 64 transform transform NOUN cana-1224 96 65 to to ADP cana-1224 96 66 both both DET cana-1224 96 67 the the DET cana-1224 96 68 sides side NOUN cana-1224 96 69 of of ADP cana-1224 96 70 eqn eqn NOUN cana-1224 96 71 ( ( PUNCT cana-1224 96 72 7 7 NUM cana-1224 96 73 ) ) PUNCT cana-1224 96 74 , , PUNCT cana-1224 96 75 we we PRON cana-1224 96 76 get get VERB cana-1224 96 77 , , PUNCT cana-1224 96 78 ๐ธ[๐‘ค๐‘ก ๐ธ[๐‘ค๐‘ก PUNCT cana-1224 97 1 ๐›ผ ๐›ผ VERB cana-1224 97 2 ] ] X cana-1224 97 3 = = SYM cana-1224 97 4 ๐ธ[๐‘–(๐‘ค๐‘ฅ๐‘ฅ ๐ธ[๐‘–(๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 97 5 + + CCONJ cana-1224 97 6 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 97 7 + + CCONJ cana-1224 97 8 ๐‘ค๐‘ง๐‘ง ๐‘ค๐‘ง๐‘ง NOUN cana-1224 97 9 + + ADJ cana-1224 97 10 4๐‘ค2 4๐‘ค2 NUM cana-1224 97 11 ๏ฟฝ ๏ฟฝ NOUN cana-1224 97 12 ฬ… ฬ… NOUN cana-1224 97 13 ๏ฟฝ ๏ฟฝ NOUN cana-1224 97 14 ) ) PUNCT cana-1224 97 15 ] ] PUNCT cana-1224 98 1 this this PRON cana-1224 98 2 implies imply VERB cana-1224 98 3 ๐ธ[๐‘ค(๐‘ฅ ๐ธ[๐‘ค(๐‘ฅ PROPN cana-1224 98 4 , , PUNCT cana-1224 98 5 ๐‘ฆ ๐‘ฆ NOUN cana-1224 98 6 , , PUNCT cana-1224 98 7 ๐‘ง ๐‘ง NOUN cana-1224 98 8 , , PUNCT cana-1224 98 9 ๐‘ก ๐‘ก NOUN cana-1224 98 10 ) ) PUNCT cana-1224 98 11 ] ] PUNCT cana-1224 99 1 = = PUNCT cana-1224 99 2 โˆ‘ โˆ‘ PUNCT cana-1224 99 3 ๐‘ฃ๐‘–+2 ๐‘ฃ๐‘–+2 PROPN cana-1224 99 4 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-1224 99 5 ๐‘–=0 ๐‘–=0 SYM cana-1224 99 6 ๐‘ค(๐‘–)(๐‘ฅ ๐‘ค(๐‘–)(๐‘ฅ PROPN cana-1224 99 7 , , PUNCT cana-1224 99 8 ๐‘ฆ ๐‘ฆ NOUN cana-1224 99 9 , , PUNCT cana-1224 99 10 ๐‘ง ๐‘ง PROPN cana-1224 99 11 , , PUNCT cana-1224 99 12 0 0 NUM cana-1224 99 13 ) ) PUNCT cana-1224 100 1 + + CCONJ cana-1224 100 2 ๐‘ฃ๐›ผ๐‘–๐ธ[๐‘ค๐‘ฅ๐‘ฅ ๐‘ฃ๐›ผ๐‘–๐ธ[๐‘ค๐‘ฅ๐‘ฅ PUNCT cana-1224 100 3 + + CCONJ cana-1224 100 4 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 100 5 + + CCONJ cana-1224 100 6 ๐‘ค๐‘ง๐‘ง ๐‘ค๐‘ง๐‘ง NOUN cana-1224 100 7 + + ADJ cana-1224 100 8 4๐‘ค2 4๐‘ค2 NUM cana-1224 100 9 ๏ฟฝ ๏ฟฝ NOUN cana-1224 100 10 ฬ… ฬ… NOUN cana-1224 100 11 ๏ฟฝ ๏ฟฝ NOUN cana-1224 100 12 ] ] PUNCT cana-1224 100 13 after after ADP cana-1224 100 14 applying apply VERB cana-1224 100 15 initial initial ADJ cana-1224 100 16 conditions condition NOUN cana-1224 100 17 , , PUNCT cana-1224 100 18 we we PRON cana-1224 100 19 get get VERB cana-1224 100 20 ๐ธ[๐‘ค(๐‘ฅ ๐ธ[๐‘ค(๐‘ฅ PRON cana-1224 100 21 , , PUNCT cana-1224 100 22 ๐‘ฆ ๐‘ฆ NOUN cana-1224 100 23 , , PUNCT cana-1224 100 24 ๐‘ง ๐‘ง NOUN cana-1224 100 25 , , PUNCT cana-1224 100 26 ๐‘ก ๐‘ก NOUN cana-1224 100 27 ) ) PUNCT cana-1224 100 28 ] ] PUNCT cana-1224 101 1 = = SYM cana-1224 101 2 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 101 3 . . PUNCT cana-1224 102 1 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 102 2 ) ) PUNCT cana-1224 103 1 + + NUM cana-1224 103 2 ๐‘ฃ๐›ผ๐‘–๐ธ[๐‘ค๐‘ฅ๐‘ฅ ๐‘ฃ๐›ผ๐‘–๐ธ[๐‘ค๐‘ฅ๐‘ฅ PUNCT cana-1224 104 1 + + CCONJ cana-1224 104 2 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 104 3 + + CCONJ cana-1224 104 4 ๐‘ค๐‘ง๐‘ง ๐‘ค๐‘ง๐‘ง NOUN cana-1224 104 5 + + ADJ cana-1224 104 6 4๐‘ค2 4๐‘ค2 NUM cana-1224 104 7 ๏ฟฝ ๏ฟฝ NOUN cana-1224 104 8 ฬ… ฬ… NOUN cana-1224 104 9 ๏ฟฝ ๏ฟฝ NOUN cana-1224 104 10 ] ] PUNCT cana-1224 104 11 or or CCONJ cana-1224 104 12 ๐ธ[๐‘ค(๐‘ฅ ๐ธ[๐‘ค(๐‘ฅ PROPN cana-1224 104 13 , , PUNCT cana-1224 104 14 ๐‘ฆ ๐‘ฆ NOUN cana-1224 104 15 , , PUNCT cana-1224 104 16 ๐‘ง ๐‘ง NOUN cana-1224 104 17 , , PUNCT cana-1224 104 18 ๐‘ก ๐‘ก NOUN cana-1224 104 19 ) ) PUNCT cana-1224 104 20 ] ] PUNCT cana-1224 105 1 โˆ’ โˆ’ PROPN cana-1224 105 2 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 105 3 . . PUNCT cana-1224 106 1 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 106 2 ) ) PUNCT cana-1224 107 1 โˆ’ โˆ’ PROPN cana-1224 107 2 ๐‘ฃ๐›ผ๐‘–๐ธ[๐‘ค๐‘ฅ๐‘ฅ ๐‘ฃ๐›ผ๐‘–๐ธ[๐‘ค๐‘ฅ๐‘ฅ PUNCT cana-1224 108 1 + + NUM cana-1224 108 2 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 108 3 + + CCONJ cana-1224 108 4 ๐‘ค๐‘ง๐‘ง ๐‘ค๐‘ง๐‘ง NOUN cana-1224 108 5 + + ADJ cana-1224 108 6 4๐‘ค2 4๐‘ค2 NUM cana-1224 108 7 ๏ฟฝ ๏ฟฝ NOUN cana-1224 108 8 ฬ… ฬ… NOUN cana-1224 108 9 ๏ฟฝ ๏ฟฝ NOUN cana-1224 108 10 ] ] X cana-1224 108 11 = = SYM cana-1224 108 12 0 0 NUM cana-1224 109 1 the the DET cana-1224 109 2 nonlinear nonlinear ADJ cana-1224 109 3 component component NOUN cana-1224 109 4 is be AUX cana-1224 109 5 defined define VERB cana-1224 109 6 as as ADP cana-1224 109 7 : : PUNCT cana-1224 109 8 ๐‘…[๐œ‘(๐‘ฅ ๐‘…[๐œ‘(๐‘ฅ PROPN cana-1224 109 9 , , PUNCT cana-1224 109 10 ๐‘ฆ ๐‘ฆ NOUN cana-1224 109 11 , , PUNCT cana-1224 109 12 ๐‘ง ๐‘ง PROPN cana-1224 109 13 , , PUNCT cana-1224 109 14 ๐‘ก ๐‘ก NOUN cana-1224 109 15 ; ; PUNCT cana-1224 109 16 ๐‘ ๐‘ NOUN cana-1224 109 17 ) ) PUNCT cana-1224 109 18 ] ] PUNCT cana-1224 110 1 = = PUNCT cana-1224 110 2 ๐ธ[๐œ‘ ๐ธ[๐œ‘ VERB cana-1224 110 3 ] ] PUNCT cana-1224 110 4 โˆ’ โˆ’ PROPN cana-1224 110 5 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 110 6 . . PUNCT cana-1224 111 1 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 111 2 ) ) PUNCT cana-1224 112 1 โˆ’ โˆ’ PROPN cana-1224 112 2 ๐‘ฃ๐›ผ๐‘–๐ธ[๐œ‘๐‘ฅ๐‘ฅ ๐‘ฃ๐›ผ๐‘–๐ธ[๐œ‘๐‘ฅ๐‘ฅ PROPN cana-1224 112 3 + + CCONJ cana-1224 112 4 ๐œ‘๐‘ฆ๐‘ฆ ๐œ‘๐‘ฆ๐‘ฆ ADP cana-1224 112 5 + + NUM cana-1224 112 6 ๐œ‘๐‘ง๐‘ง ๐œ‘๐‘ง๐‘ง NOUN cana-1224 112 7 + + CCONJ cana-1224 112 8 4๐œ‘2 4๐œ‘2 NUM cana-1224 112 9 ๏ฟฝ ๏ฟฝ NOUN cana-1224 112 10 ฬ… ฬ… NOUN cana-1224 112 11 ๏ฟฝ ๏ฟฝ NOUN cana-1224 112 12 ] ] X cana-1224 112 13 ( ( PUNCT cana-1224 112 14 8) 8) NUM cana-1224 112 15 we we PRON cana-1224 112 16 formulate formulate VERB cana-1224 112 17 the the DET cana-1224 112 18 zero zero NUM cana-1224 112 19 - - PUNCT cana-1224 112 20 order order NOUN cana-1224 112 21 deformation deformation NOUN cana-1224 112 22 eqn eqn NOUN cana-1224 112 23 under under ADP cana-1224 112 24 the the DET cana-1224 112 25 given give VERB cana-1224 112 26 assumption assumption NOUN cana-1224 112 27 ๐ป(๐‘ฅ ๐ป(๐‘ฅ NOUN cana-1224 112 28 , , PUNCT cana-1224 112 29 ๐‘ฆ ๐‘ฆ NOUN cana-1224 112 30 , , PUNCT cana-1224 112 31 ๐‘ง ๐‘ง NOUN cana-1224 112 32 , , PUNCT cana-1224 112 33 ๐‘ก ๐‘ก NOUN cana-1224 112 34 ) ) PUNCT cana-1224 112 35 = = SYM cana-1224 112 36 1 1 NUM cana-1224 112 37 , , PUNCT cana-1224 112 38 we we PRON cana-1224 112 39 โ€œ " PUNCT cana-1224 112 40 have have VERB cana-1224 112 41 communications communication NOUN cana-1224 112 42 on on ADP cana-1224 112 43 applied apply VERB cana-1224 112 44 nonlinear nonlinear ADJ cana-1224 112 45 analysis analysis NOUN cana-1224 112 46 issn issn NOUN cana-1224 112 47 : : PUNCT cana-1224 112 48 1074 1074 NUM cana-1224 112 49 - - PUNCT cana-1224 112 50 133x 133x NUM cana-1224 112 51 vol vol NOUN cana-1224 112 52 31 31 NUM cana-1224 112 53 no no NOUN cana-1224 112 54 . . PUNCT cana-1224 113 1 6s 6s NUM cana-1224 113 2 ( ( PUNCT cana-1224 113 3 2024 2024 NUM cana-1224 113 4 ) ) PUNCT cana-1224 113 5 310 310 NUM cana-1224 113 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 113 7 ( ( PUNCT cana-1224 113 8 1 1 NUM cana-1224 113 9 โˆ’ โˆ’ PROPN cana-1224 113 10 ๐‘)๐ธ{๐œ‘(๐‘ฅ ๐‘)๐ธ{๐œ‘(๐‘ฅ NOUN cana-1224 113 11 , , PUNCT cana-1224 113 12 ๐‘ฆ ๐‘ฆ NOUN cana-1224 113 13 , , PUNCT cana-1224 113 14 ๐‘ง ๐‘ง NOUN cana-1224 113 15 , , PUNCT cana-1224 113 16 ๐‘ก ๐‘ก NOUN cana-1224 113 17 ) ) PUNCT cana-1224 113 18 โˆ’ โˆ’ ADP cana-1224 113 19 ๐‘ค0(๐‘ฅ ๐‘ค0(๐‘ฅ PROPN cana-1224 113 20 , , PUNCT cana-1224 113 21 ๐‘ฆ ๐‘ฆ NOUN cana-1224 113 22 , , PUNCT cana-1224 113 23 ๐‘ง ๐‘ง NOUN cana-1224 113 24 , , PUNCT cana-1224 113 25 ๐‘ก ๐‘ก NOUN cana-1224 113 26 ) ) PUNCT cana-1224 113 27 } } PUNCT cana-1224 113 28 = = SYM cana-1224 114 1 ๐‘โ„Ž๐‘…[๐œ‘(๐‘ฅ ๐‘โ„Ž๐‘…[๐œ‘(๐‘ฅ ADJ cana-1224 114 2 , , PUNCT cana-1224 114 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 4 , , PUNCT cana-1224 114 5 ๐‘ง ๐‘ง PROPN cana-1224 114 6 , , PUNCT cana-1224 114 7 ๐‘ก ๐‘ก NOUN cana-1224 114 8 ; ; PUNCT cana-1224 114 9 ๐‘ ๐‘ NOUN cana-1224 114 10 ) ) PUNCT cana-1224 114 11 ] ] PUNCT cana-1224 114 12 when when SCONJ cana-1224 114 13 ๐‘ ๐‘ X cana-1224 114 14 = = SYM cana-1224 114 15 0 0 PROPN cana-1224 114 16 & & CCONJ cana-1224 114 17 ๐‘ ๐‘ NOUN cana-1224 114 18 = = SYM cana-1224 114 19 1 1 NUM cana-1224 114 20 , , PUNCT cana-1224 114 21 we we PRON cana-1224 114 22 have have VERB cana-1224 114 23 { { PUNCT cana-1224 114 24 ๐œ‘(๐‘ฅ ๐œ‘(๐‘ฅ NOUN cana-1224 114 25 , , PUNCT cana-1224 114 26 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 27 , , PUNCT cana-1224 114 28 ๐‘ง ๐‘ง PROPN cana-1224 114 29 , , PUNCT cana-1224 114 30 ๐‘ก ๐‘ก X cana-1224 114 31 ; ; PUNCT cana-1224 114 32 0 0 NUM cana-1224 114 33 ) ) PUNCT cana-1224 114 34 = = SYM cana-1224 114 35 ๐‘ค0(๐‘ฅ ๐‘ค0(๐‘ฅ PROPN cana-1224 114 36 , , PUNCT cana-1224 114 37 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 38 , , PUNCT cana-1224 114 39 ๐‘ง ๐‘ง PROPN cana-1224 114 40 , , PUNCT cana-1224 114 41 0 0 NUM cana-1224 114 42 ) ) PUNCT cana-1224 114 43 ๐œ‘(๐‘ฅ ๐œ‘(๐‘ฅ NOUN cana-1224 114 44 , , PUNCT cana-1224 114 45 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 46 , , PUNCT cana-1224 114 47 ๐‘ง ๐‘ง PRON cana-1224 114 48 , , PUNCT cana-1224 114 49 ๐‘ก ๐‘ก X cana-1224 114 50 ; ; PUNCT cana-1224 114 51 1 1 X cana-1224 114 52 ) ) PUNCT cana-1224 114 53 = = SYM cana-1224 114 54 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 114 55 , , PUNCT cana-1224 114 56 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 57 , , PUNCT cana-1224 114 58 ๐‘ง ๐‘ง NOUN cana-1224 114 59 , , PUNCT cana-1224 114 60 ๐‘ก ๐‘ก NOUN cana-1224 114 61 ) ) PUNCT cana-1224 114 62 so so ADV cana-1224 114 63 , , PUNCT cana-1224 114 64 the the DET cana-1224 114 65 mth mth NOUN cana-1224 114 66 - - PUNCT cana-1224 114 67 order order NOUN cana-1224 114 68 deformation deformation NOUN cana-1224 114 69 eqn eqn PROPN cana-1224 114 70 ๐ธ{๐‘ค๐‘š(๐‘ฅ ๐ธ{๐‘ค๐‘š(๐‘ฅ PROPN cana-1224 114 71 , , PUNCT cana-1224 114 72 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 73 , , PUNCT cana-1224 114 74 ๐‘ง ๐‘ง NOUN cana-1224 114 75 , , PUNCT cana-1224 114 76 ๐‘ก ๐‘ก NOUN cana-1224 114 77 ) ) PUNCT cana-1224 114 78 โˆ’ โˆ’ NOUN cana-1224 114 79 ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ NOUN cana-1224 114 80 , , PUNCT cana-1224 114 81 ๐‘ฆ ๐‘ฆ PRON cana-1224 114 82 , , PUNCT cana-1224 114 83 ๐‘ง ๐‘ง NOUN cana-1224 114 84 , , PUNCT cana-1224 114 85 ๐‘ก ๐‘ก NOUN cana-1224 114 86 ) ) PUNCT cana-1224 114 87 } } PUNCT cana-1224 114 88 = = SYM cana-1224 114 89 โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— PROPN cana-1224 114 90 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 91 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 92 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 93 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 94 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 114 95 , , PUNCT cana-1224 114 96 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 97 , , PUNCT cana-1224 114 98 ๐‘ง ๐‘ง NOUN cana-1224 114 99 , , PUNCT cana-1224 114 100 ๐‘ก ๐‘ก NOUN cana-1224 114 101 ) ) PUNCT cana-1224 114 102 ) ) PUNCT cana-1224 114 103 ( ( PUNCT cana-1224 114 104 9 9 X cana-1224 114 105 ) ) PUNCT cana-1224 114 106 by by ADP cana-1224 114 107 inverse inverse NOUN cana-1224 114 108 elzaki elzaki NOUN cana-1224 114 109 transform transform VERB cana-1224 114 110 both both DET cana-1224 114 111 sides side NOUN cana-1224 114 112 , , PUNCT cana-1224 114 113 we we PRON cana-1224 114 114 obtain obtain VERB cana-1224 114 115 ๐‘ค๐‘š(๐‘ฅ ๐‘ค๐‘š(๐‘ฅ NUM cana-1224 114 116 , , PUNCT cana-1224 114 117 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 118 , , PUNCT cana-1224 114 119 ๐‘ง ๐‘ง NOUN cana-1224 114 120 , , PUNCT cana-1224 114 121 ๐‘ก ๐‘ก NOUN cana-1224 114 122 ) ) PUNCT cana-1224 114 123 โˆ’ โˆ’ NOUN cana-1224 114 124 ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ NOUN cana-1224 114 125 , , PUNCT cana-1224 114 126 ๐‘ฆ ๐‘ฆ PRON cana-1224 114 127 , , PUNCT cana-1224 114 128 ๐‘ง ๐‘ง NOUN cana-1224 114 129 , , PUNCT cana-1224 114 130 ๐‘ก ๐‘ก NOUN cana-1224 114 131 ) ) PUNCT cana-1224 114 132 = = SYM cana-1224 114 133 ๐ธโˆ’1{โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— ๐ธโˆ’1{โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— PROPN cana-1224 114 134 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 135 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 136 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 137 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 114 138 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 114 139 , , PUNCT cana-1224 114 140 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 141 , , PUNCT cana-1224 114 142 ๐‘ง ๐‘ง NOUN cana-1224 114 143 , , PUNCT cana-1224 114 144 ๐‘ก ๐‘ก NOUN cana-1224 114 145 ) ) PUNCT cana-1224 114 146 ) ) PUNCT cana-1224 114 147 } } PUNCT cana-1224 114 148 ( ( PUNCT cana-1224 114 149 10 10 NUM cana-1224 114 150 ) ) PUNCT cana-1224 114 151 from from ADP cana-1224 114 152 equation equation NOUN cana-1224 114 153 ( ( PUNCT cana-1224 114 154 10 10 NUM cana-1224 114 155 ) ) PUNCT cana-1224 114 156 ( ( PUNCT cana-1224 114 157 taking take VERB cana-1224 114 158 โ„Ž โ„Ž PROPN cana-1224 114 159 = = SYM cana-1224 114 160 โˆ’1 โˆ’1 NOUN cana-1224 114 161 ) ) PUNCT cana-1224 114 162 , , PUNCT cana-1224 114 163 we we PRON cana-1224 114 164 obtain obtain VERB cana-1224 114 165 ๐‘ค1(๐‘ฅ ๐‘ค1(๐‘ฅ NUM cana-1224 114 166 , , PUNCT cana-1224 114 167 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 168 , , PUNCT cana-1224 114 169 ๐‘ง ๐‘ง NOUN cana-1224 114 170 , , PUNCT cana-1224 114 171 ๐‘ก ๐‘ก NOUN cana-1224 114 172 ) ) PUNCT cana-1224 114 173 = = SYM cana-1224 114 174 โˆ’๐ธโˆ’1{๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— โˆ’๐ธโˆ’1{๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— X cana-1224 114 175 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 114 176 , , PUNCT cana-1224 114 177 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 178 , , PUNCT cana-1224 114 179 ๐‘ง ๐‘ง NOUN cana-1224 114 180 , , PUNCT cana-1224 114 181 ๐‘ก ๐‘ก NOUN cana-1224 114 182 ) ) PUNCT cana-1224 114 183 ) ) PUNCT cana-1224 114 184 } } PUNCT cana-1224 114 185 , , PUNCT cana-1224 114 186 ๐‘ค2(๐‘ฅ ๐‘ค2(๐‘ฅ NOUN cana-1224 114 187 , , PUNCT cana-1224 114 188 ๐‘ฆ ๐‘ฆ NOUN cana-1224 114 189 , , PUNCT cana-1224 114 190 ๐‘ง ๐‘ง NOUN cana-1224 114 191 , , PUNCT cana-1224 114 192 ๐‘ก ๐‘ก NOUN cana-1224 114 193 ) ) PUNCT cana-1224 114 194 = = PUNCT cana-1224 115 1 ๐‘ค1(๐‘ฅ ๐‘ค1(๐‘ฅ PROPN cana-1224 115 2 , , PUNCT cana-1224 115 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 115 4 , , PUNCT cana-1224 115 5 ๐‘ง ๐‘ง NOUN cana-1224 115 6 , , PUNCT cana-1224 115 7 ๐‘ก ๐‘ก NOUN cana-1224 115 8 ) ) PUNCT cana-1224 115 9 โˆ’ โˆ’ PROPN cana-1224 115 10 ๐ธโˆ’1{๐‘…2(๐‘ค1โƒ—โƒ— ๐ธโˆ’1{๐‘…2(๐‘ค1โƒ—โƒ— NOUN cana-1224 115 11 โƒ—โƒ— โƒ—โƒ— NOUN cana-1224 115 12 ( ( PUNCT cana-1224 115 13 ๐‘ฅ ๐‘ฅ PROPN cana-1224 115 14 , , PUNCT cana-1224 115 15 ๐‘ฆ ๐‘ฆ NOUN cana-1224 115 16 , , PUNCT cana-1224 115 17 ๐‘ง ๐‘ง NOUN cana-1224 115 18 , , PUNCT cana-1224 115 19 ๐‘ก ๐‘ก NOUN cana-1224 115 20 ) ) PUNCT cana-1224 115 21 ) ) PUNCT cana-1224 115 22 } } PUNCT cana-1224 115 23 , , PUNCT cana-1224 115 24 ๐‘ค3(๐‘ฅ ๐‘ค3(๐‘ฅ NOUN cana-1224 115 25 , , PUNCT cana-1224 115 26 ๐‘ฆ ๐‘ฆ NOUN cana-1224 115 27 , , PUNCT cana-1224 115 28 ๐‘ง ๐‘ง NOUN cana-1224 115 29 , , PUNCT cana-1224 115 30 ๐‘ก ๐‘ก NOUN cana-1224 115 31 ) ) PUNCT cana-1224 115 32 = = SYM cana-1224 116 1 ๐‘ค2(๐‘ฅ ๐‘ค2(๐‘ฅ PROPN cana-1224 116 2 , , PUNCT cana-1224 116 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 116 4 , , PUNCT cana-1224 116 5 ๐‘ง ๐‘ง NOUN cana-1224 116 6 , , PUNCT cana-1224 116 7 ๐‘ก ๐‘ก NOUN cana-1224 116 8 ) ) PUNCT cana-1224 116 9 โˆ’ โˆ’ NUM cana-1224 116 10 ๐ธโˆ’1{๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— ๐ธโˆ’1{๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— NOUN cana-1224 116 11 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 116 12 , , PUNCT cana-1224 116 13 ๐‘ฆ ๐‘ฆ NOUN cana-1224 116 14 , , PUNCT cana-1224 116 15 ๐‘ง ๐‘ง NOUN cana-1224 116 16 , , PUNCT cana-1224 116 17 ๐‘ก ๐‘ก NOUN cana-1224 116 18 ) ) PUNCT cana-1224 116 19 ) ) PUNCT cana-1224 116 20 } } PUNCT cana-1224 116 21 , , PUNCT cana-1224 116 22 โ‹ฎ โ‹ฎ NOUN cana-1224 116 23 where where SCONJ cana-1224 116 24 , , PUNCT cana-1224 116 25 โ€ " PUNCT cana-1224 116 26 ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— X cana-1224 116 27 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 116 28 , , PUNCT cana-1224 116 29 ๐‘ฆ ๐‘ฆ NOUN cana-1224 116 30 , , PUNCT cana-1224 116 31 ๐‘ง ๐‘ง NOUN cana-1224 116 32 , , PUNCT cana-1224 116 33 ๐‘ก ๐‘ก NOUN cana-1224 116 34 ) ) PUNCT cana-1224 116 35 ) ) PUNCT cana-1224 116 36 = = PUNCT cana-1224 117 1 ๐ธ[๐‘ค0 ๐ธ[๐‘ค0 PROPN cana-1224 117 2 ] ] PUNCT cana-1224 117 3 โˆ’ โˆ’ PROPN cana-1224 117 4 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 117 5 . . PUNCT cana-1224 118 1 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 118 2 ) ) PUNCT cana-1224 119 1 โˆ’ โˆ’ PROPN cana-1224 119 2 ๐‘ฃ๐›ผ๐‘–๐ธ[(๐‘ค0)๐‘ฅ๐‘ฅ ๐‘ฃ๐›ผ๐‘–๐ธ[(๐‘ค0)๐‘ฅ๐‘ฅ ADJ cana-1224 120 1 + + CCONJ cana-1224 120 2 ( ( PUNCT cana-1224 120 3 ๐‘ค0)๐‘ฆ๐‘ฆ ๐‘ค0)๐‘ฆ๐‘ฆ NOUN cana-1224 120 4 + + CCONJ cana-1224 120 5 ( ( PUNCT cana-1224 120 6 ๐‘ค0)๐‘ง๐‘ง ๐‘ค0)๐‘ง๐‘ง PROPN cana-1224 120 7 + + CCONJ cana-1224 120 8 4๐‘ค0 4๐‘ค0 NUM cana-1224 120 9 2๐‘ค0ฬ…ฬ…ฬ…ฬ… 2๐‘ค0ฬ…ฬ…ฬ…ฬ… NUM cana-1224 120 10 ] ] PUNCT cana-1224 120 11 , , PUNCT cana-1224 120 12 ๐‘…2(๐‘ค1โƒ—โƒ— ๐‘…2(๐‘ค1โƒ—โƒ— PROPN cana-1224 120 13 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 120 14 ( ( PUNCT cana-1224 120 15 ๐‘ฅ ๐‘ฅ PROPN cana-1224 120 16 , , PUNCT cana-1224 120 17 ๐‘ฆ ๐‘ฆ NOUN cana-1224 120 18 , , PUNCT cana-1224 120 19 ๐‘ง ๐‘ง NOUN cana-1224 120 20 , , PUNCT cana-1224 120 21 ๐‘ก ๐‘ก NOUN cana-1224 120 22 ) ) PUNCT cana-1224 120 23 ) ) PUNCT cana-1224 121 1 = = SYM cana-1224 121 2 ๐ธ[๐‘ค1 ๐ธ[๐‘ค1 NOUN cana-1224 121 3 ] ] X cana-1224 121 4 โˆ’ โˆ’ PROPN cana-1224 121 5 ๐‘ฃ๐›ผ๐‘–๐ธ[(๐‘ค1)๐‘ฅ๐‘ฅ ๐‘ฃ๐›ผ๐‘–๐ธ[(๐‘ค1)๐‘ฅ๐‘ฅ PROPN cana-1224 121 6 + + CCONJ cana-1224 121 7 ( ( PUNCT cana-1224 121 8 ๐‘ค1)๐‘ฆ๐‘ฆ ๐‘ค1)๐‘ฆ๐‘ฆ ADJ cana-1224 121 9 + + CCONJ cana-1224 121 10 ( ( PUNCT cana-1224 121 11 ๐‘ค1)๐‘ง๐‘ง ๐‘ค1)๐‘ง๐‘ง PROPN cana-1224 121 12 + + CCONJ cana-1224 121 13 4๐‘ค0 4๐‘ค0 NUM cana-1224 121 14 2๐‘ค1ฬ…ฬ…ฬ…ฬ… 2๐‘ค1ฬ…ฬ…ฬ…ฬ… NUM cana-1224 121 15 + + CCONJ cana-1224 121 16 8๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… 8๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… NUM cana-1224 121 17 ๐‘ค1 ๐‘ค1 VERB cana-1224 121 18 ] ] PUNCT cana-1224 121 19 , , PUNCT cana-1224 121 20 ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— X cana-1224 121 21 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 121 22 , , PUNCT cana-1224 121 23 ๐‘ฆ ๐‘ฆ NOUN cana-1224 121 24 , , PUNCT cana-1224 121 25 ๐‘ง ๐‘ง NOUN cana-1224 121 26 , , PUNCT cana-1224 121 27 ๐‘ก ๐‘ก NOUN cana-1224 121 28 ) ) PUNCT cana-1224 121 29 ) ) PUNCT cana-1224 121 30 = = SYM cana-1224 121 31 ๐ธ[๐‘ค2 ๐ธ[๐‘ค2 PROPN cana-1224 121 32 ] ] X cana-1224 122 1 โˆ’๐‘ฃ๐›ผ๐‘–๐ธ[(๐‘ค2)๐‘ฅ๐‘ฅ โˆ’๐‘ฃ๐›ผ๐‘–๐ธ[(๐‘ค2)๐‘ฅ๐‘ฅ X cana-1224 122 2 + + CCONJ cana-1224 122 3 ( ( PUNCT cana-1224 122 4 ๐‘ค2)๐‘ฆ๐‘ฆ ๐‘ค2)๐‘ฆ๐‘ฆ PROPN cana-1224 122 5 + + CCONJ cana-1224 122 6 ( ( PUNCT cana-1224 122 7 ๐‘ค2)๐‘ง๐‘ง ๐‘ค2)๐‘ง๐‘ง PROPN cana-1224 122 8 + + CCONJ cana-1224 122 9 4๐‘ค0 4๐‘ค0 NUM cana-1224 122 10 2๐‘ค2ฬ…ฬ…ฬ…ฬ… 2๐‘ค2ฬ…ฬ…ฬ…ฬ… NUM cana-1224 122 11 + + CCONJ cana-1224 122 12 8๐‘ค0๐‘ค1ฬ…ฬ…ฬ…ฬ… 8๐‘ค0๐‘ค1ฬ…ฬ…ฬ…ฬ… NUM cana-1224 122 13 ๐‘ค1 ๐‘ค1 VERB cana-1224 122 14 + + CCONJ cana-1224 122 15 8๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… 8๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… NUM cana-1224 122 16 ๐‘ค2 ๐‘ค2 NOUN cana-1224 122 17 + + CCONJ cana-1224 122 18 4๐‘ค0ฬ…ฬ…ฬ…ฬ… 4๐‘ค0ฬ…ฬ…ฬ…ฬ… NUM cana-1224 122 19 ๐‘ค1 ๐‘ค1 VERB cana-1224 122 20 2 2 NUM cana-1224 122 21 ] ] PUNCT cana-1224 122 22 , , PUNCT cana-1224 122 23 โ‹ฎ โ‹ฎ NOUN cana-1224 122 24 after after ADP cana-1224 122 25 simplifications simplification NOUN cana-1224 122 26 , , PUNCT cana-1224 122 27 we we PRON cana-1224 122 28 obtain obtain VERB cana-1224 122 29 ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— X cana-1224 122 30 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 122 31 , , PUNCT cana-1224 122 32 ๐‘ฆ ๐‘ฆ NOUN cana-1224 122 33 , , PUNCT cana-1224 122 34 ๐‘ง ๐‘ง NOUN cana-1224 122 35 , , PUNCT cana-1224 122 36 ๐‘ก ๐‘ก NOUN cana-1224 122 37 ) ) PUNCT cana-1224 122 38 ) ) PUNCT cana-1224 123 1 = = PUNCT cana-1224 123 2 โˆ’๐‘–. โˆ’๐‘–. PROPN cana-1224 123 3 ๐‘ฃ๐›ผ+2๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘ฃ๐›ผ+2๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 123 4 ) ) PUNCT cana-1224 123 5 , , PUNCT cana-1224 123 6 ๐‘…2(๐‘ค1โƒ—โƒ— ๐‘…2(๐‘ค1โƒ—โƒ— PROPN cana-1224 123 7 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 123 8 ( ( PUNCT cana-1224 123 9 ๐‘ฅ ๐‘ฅ PROPN cana-1224 123 10 , , PUNCT cana-1224 123 11 ๐‘ฆ ๐‘ฆ NOUN cana-1224 123 12 , , PUNCT cana-1224 123 13 ๐‘ง ๐‘ง NOUN cana-1224 123 14 , , PUNCT cana-1224 123 15 ๐‘ก ๐‘ก NOUN cana-1224 123 16 ) ) PUNCT cana-1224 123 17 ) ) PUNCT cana-1224 124 1 = = SYM cana-1224 124 2 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง)(๐‘–๐‘ฃ๐›ผ+2 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง)(๐‘–๐‘ฃ๐›ผ+2 NOUN cana-1224 125 1 + + CCONJ cana-1224 125 2 ๐‘ฃ๐›ผ+3 ๐‘ฃ๐›ผ+3 NOUN cana-1224 125 3 ) ) PUNCT cana-1224 125 4 , , PUNCT cana-1224 125 5 ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— X cana-1224 126 1 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 126 2 , , PUNCT cana-1224 126 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 126 4 , , PUNCT cana-1224 126 5 ๐‘ง ๐‘ง NOUN cana-1224 126 6 , , PUNCT cana-1224 126 7 ๐‘ก ๐‘ก NOUN cana-1224 126 8 ) ) PUNCT cana-1224 126 9 ) ) PUNCT cana-1224 127 1 = = PUNCT cana-1224 127 2 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง){โˆ’๐‘ฃ๐›ผ+3 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง){โˆ’๐‘ฃ๐›ผ+3 NOUN cana-1224 127 3 โˆ’ โˆ’ PROPN cana-1224 127 4 ๐‘–๐‘ฃ๐›ผ+4 ๐‘–๐‘ฃ๐›ผ+4 PROPN cana-1224 127 5 } } PUNCT cana-1224 127 6 , , PUNCT cana-1224 127 7 โ‹ฎ โ‹ฎ NOUN cana-1224 127 8 therefore therefore ADV cana-1224 127 9 , , PUNCT cana-1224 127 10 ๐‘ค1(๐‘ฅ ๐‘ค1(๐‘ฅ PROPN cana-1224 127 11 , , PUNCT cana-1224 127 12 ๐‘ฆ ๐‘ฆ NOUN cana-1224 127 13 , , PUNCT cana-1224 127 14 ๐‘ง ๐‘ง NOUN cana-1224 127 15 , , PUNCT cana-1224 127 16 ๐‘ก ๐‘ก NOUN cana-1224 127 17 ) ) PUNCT cana-1224 127 18 = = PUNCT cana-1224 128 1 ๐‘– ๐‘– PROPN cana-1224 128 2 ๐‘ก๐›ผ ๐‘ก๐›ผ X cana-1224 128 3 ( ( PUNCT cana-1224 128 4 ๐›ผ ๐›ผ NOUN cana-1224 128 5 ) ) PUNCT cana-1224 128 6 ! ! PUNCT cana-1224 129 1 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 129 2 ) ) PUNCT cana-1224 129 3 , , PUNCT cana-1224 129 4 ๐‘ค2(๐‘ฅ ๐‘ค2(๐‘ฅ PROPN cana-1224 129 5 , , PUNCT cana-1224 129 6 ๐‘ฆ ๐‘ฆ NOUN cana-1224 129 7 , , PUNCT cana-1224 129 8 ๐‘ง ๐‘ง NOUN cana-1224 129 9 , , PUNCT cana-1224 129 10 ๐‘ก ๐‘ก NOUN cana-1224 129 11 ) ) PUNCT cana-1224 129 12 = = SYM cana-1224 129 13 ๐‘–2 ๐‘–2 PROPN cana-1224 129 14 ๐‘ก๐›ผ+1 ๐‘ก๐›ผ+1 NUM cana-1224 129 15 ( ( PUNCT cana-1224 129 16 ๐›ผ ๐›ผ X cana-1224 129 17 + + NOUN cana-1224 129 18 1 1 NUM cana-1224 129 19 ) ) PUNCT cana-1224 129 20 ! ! PUNCT cana-1224 130 1 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 130 2 ) ) PUNCT cana-1224 130 3 , , PUNCT cana-1224 130 4 communications communication NOUN cana-1224 130 5 on on ADP cana-1224 130 6 applied apply VERB cana-1224 130 7 nonlinear nonlinear ADJ cana-1224 130 8 analysis analysis NOUN cana-1224 130 9 issn issn NOUN cana-1224 130 10 : : PUNCT cana-1224 130 11 1074 1074 NUM cana-1224 130 12 - - PUNCT cana-1224 130 13 133x 133x NUM cana-1224 130 14 vol vol NOUN cana-1224 130 15 31 31 NUM cana-1224 130 16 no no NOUN cana-1224 130 17 . . PUNCT cana-1224 131 1 6s 6s NUM cana-1224 131 2 ( ( PUNCT cana-1224 131 3 2024 2024 NUM cana-1224 131 4 ) ) PUNCT cana-1224 131 5 311 311 NUM cana-1224 131 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 131 7 ๐‘ค3(๐‘ฅ ๐‘ค3(๐‘ฅ NUM cana-1224 131 8 , , PUNCT cana-1224 131 9 ๐‘ฆ ๐‘ฆ NOUN cana-1224 131 10 , , PUNCT cana-1224 131 11 ๐‘ง ๐‘ง NOUN cana-1224 131 12 , , PUNCT cana-1224 131 13 ๐‘ก ๐‘ก NOUN cana-1224 131 14 ) ) PUNCT cana-1224 131 15 = = SYM cana-1224 131 16 ๐‘–3 ๐‘–3 ADJ cana-1224 131 17 ๐‘ก๐›ผ+2 ๐‘ก๐›ผ+2 NUM cana-1224 131 18 ( ( PUNCT cana-1224 131 19 ๐›ผ ๐›ผ NOUN cana-1224 131 20 + + NOUN cana-1224 131 21 2 2 NUM cana-1224 131 22 ) ) PUNCT cana-1224 131 23 ! ! PUNCT cana-1224 132 1 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 132 2 ) ) PUNCT cana-1224 132 3 , , PUNCT cana-1224 132 4 โ‹ฎ โ‹ฎ NOUN cana-1224 132 5 for for ADP cana-1224 132 6 ๐›ผ ๐›ผ NOUN cana-1224 132 7 = = SYM cana-1224 132 8 1 1 NUM cana-1224 132 9 , , PUNCT cana-1224 132 10 the the DET cana-1224 132 11 solution solution NOUN cana-1224 132 12 is be AUX cana-1224 132 13 : : PUNCT cana-1224 132 14 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 132 15 , , PUNCT cana-1224 132 16 ๐‘ฆ ๐‘ฆ NOUN cana-1224 132 17 , , PUNCT cana-1224 132 18 ๐‘ง ๐‘ง NOUN cana-1224 132 19 , , PUNCT cana-1224 132 20 ๐‘ก ๐‘ก NOUN cana-1224 132 21 ) ) PUNCT cana-1224 132 22 = = SYM cana-1224 132 23 ๐‘ค0 ๐‘ค0 PROPN cana-1224 132 24 + + CCONJ cana-1224 132 25 ๐‘ค1 ๐‘ค1 VERB cana-1224 132 26 + + CCONJ cana-1224 132 27 ๐‘ค2 ๐‘ค2 NOUN cana-1224 132 28 + + CCONJ cana-1224 132 29 ๐‘ค3 ๐‘ค3 PROPN cana-1224 132 30 + + CCONJ cana-1224 132 31 โ‹ฏ โ‹ฏ NOUN cana-1224 132 32 or or CCONJ cana-1224 132 33 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ X cana-1224 132 34 , , PUNCT cana-1224 132 35 ๐‘ฆ ๐‘ฆ NOUN cana-1224 132 36 , , PUNCT cana-1224 132 37 ๐‘ง ๐‘ง NOUN cana-1224 132 38 , , PUNCT cana-1224 132 39 ๐‘ก ๐‘ก NOUN cana-1224 132 40 ) ) PUNCT cana-1224 132 41 = = SYM cana-1224 132 42 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง PROPN cana-1224 132 43 ) ) PUNCT cana-1224 132 44 { { PUNCT cana-1224 132 45 1 1 NUM cana-1224 132 46 + + CCONJ cana-1224 132 47 ( ( PUNCT cana-1224 132 48 ๐‘–๐‘ก ๐‘–๐‘ก NOUN cana-1224 132 49 ) ) PUNCT cana-1224 132 50 + + CCONJ cana-1224 132 51 ( ( PUNCT cana-1224 132 52 ๐‘–๐‘ก)2 ๐‘–๐‘ก)2 PROPN cana-1224 132 53 2 2 NUM cana-1224 132 54 ! ! PUNCT cana-1224 133 1 + + PUNCT cana-1224 133 2 โ‹ฏ โ‹ฏ NOUN cana-1224 133 3 } } PUNCT cana-1224 133 4 = = SYM cana-1224 133 5 ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง+๐‘ก ๐‘’๐‘–(๐‘ฅ+๐‘ฆ+๐‘ง+๐‘ก NOUN cana-1224 133 6 ) ) PUNCT cana-1224 133 7 figure figure NOUN cana-1224 133 8 1 1 NUM cana-1224 133 9 : : PUNCT cana-1224 133 10 physical physical ADJ cana-1224 133 11 behavior behavior NOUN cana-1224 133 12 of of ADP cana-1224 133 13 solutions solution NOUN cana-1224 133 14 of of ADP cana-1224 133 15 real real ADJ cana-1224 133 16 part part NOUN cana-1224 133 17 for for ADP cana-1224 133 18 ๐‘ง ๐‘ง NOUN cana-1224 133 19 = = SYM cana-1224 133 20 2 2 NUM cana-1224 133 21 and and CCONJ cana-1224 133 22 ๐‘ก ๐‘ก PROPN cana-1224 133 23 = = SYM cana-1224 133 24 0.5 0.5 NUM cana-1224 133 25 figure figure NOUN cana-1224 133 26 2 2 NUM cana-1224 133 27 : : PUNCT cana-1224 133 28 physical physical ADJ cana-1224 133 29 behavior behavior NOUN cana-1224 133 30 of of ADP cana-1224 133 31 solutions solution NOUN cana-1224 133 32 of of ADP cana-1224 133 33 imaginary imaginary ADJ cana-1224 133 34 part part NOUN cana-1224 133 35 for for ADP cana-1224 133 36 ๐‘ง ๐‘ง NOUN cana-1224 133 37 = = SYM cana-1224 133 38 2 2 NUM cana-1224 133 39 and and CCONJ cana-1224 133 40 ๐‘ก ๐‘ก X cana-1224 133 41 = = SYM cana-1224 133 42 0.5 0.5 NUM cana-1224 133 43 -5 -5 NOUN cana-1224 133 44 -4 -4 INTJ cana-1224 133 45 -3 -3 INTJ cana-1224 134 1 -2 -2 INTJ cana-1224 135 1 -1 -1 SYM cana-1224 135 2 0 0 NUM cana-1224 135 3 1 1 NUM cana-1224 135 4 2 2 NUM cana-1224 135 5 3 3 NUM cana-1224 135 6 4 4 NUM cana-1224 135 7 5 5 NUM cana-1224 135 8 -5 -5 NOUN cana-1224 135 9 0 0 NUM cana-1224 135 10 5 5 NUM cana-1224 135 11 -1 -1 SYM cana-1224 135 12 -0.5 -0.5 X cana-1224 135 13 0 0 NUM cana-1224 136 1 0.5 0.5 NUM cana-1224 136 2 1 1 NUM cana-1224 136 3 y y PROPN cana-1224 136 4 w w PROPN cana-1224 136 5 ( ( PUNCT cana-1224 136 6 real real ADJ cana-1224 136 7 part part NOUN cana-1224 136 8 ) ) PUNCT cana-1224 137 1 x x SYM cana-1224 137 2 w w X cana-1224 137 3 ( ( PUNCT cana-1224 137 4 r r NOUN cana-1224 137 5 e e PROPN cana-1224 137 6 a a DET cana-1224 137 7 l l NOUN cana-1224 137 8 p p NOUN cana-1224 137 9 a a DET cana-1224 137 10 rt rt NOUN cana-1224 137 11 ) ) PUNCT cana-1224 137 12 -5 -5 INTJ cana-1224 137 13 -4 -4 INTJ cana-1224 138 1 -3 -3 INTJ cana-1224 138 2 -2 -2 INTJ cana-1224 139 1 -1 -1 SYM cana-1224 139 2 0 0 NUM cana-1224 139 3 1 1 NUM cana-1224 139 4 2 2 NUM cana-1224 139 5 3 3 NUM cana-1224 139 6 4 4 NUM cana-1224 139 7 5 5 NUM cana-1224 139 8 -5 -5 NOUN cana-1224 139 9 0 0 NUM cana-1224 139 10 5 5 NUM cana-1224 139 11 -1 -1 SYM cana-1224 139 12 0 0 NUM cana-1224 139 13 1 1 NUM cana-1224 139 14 y y PROPN cana-1224 139 15 w(imaginary w(imaginary NOUN cana-1224 139 16 part part NOUN cana-1224 139 17 ) ) PUNCT cana-1224 140 1 x x SYM cana-1224 140 2 w w X cana-1224 140 3 ( ( PUNCT cana-1224 140 4 i i PRON cana-1224 140 5 m m VERB cana-1224 140 6 a a DET cana-1224 140 7 g g NOUN cana-1224 140 8 in in ADP cana-1224 140 9 a a DET cana-1224 140 10 ry ry NOUN cana-1224 140 11 p p X cana-1224 140 12 a a DET cana-1224 140 13 rt rt PROPN cana-1224 140 14 ) ) PUNCT cana-1224 140 15 communications communication NOUN cana-1224 140 16 on on ADP cana-1224 140 17 applied apply VERB cana-1224 140 18 nonlinear nonlinear ADJ cana-1224 140 19 analysis analysis NOUN cana-1224 140 20 issn issn NOUN cana-1224 140 21 : : PUNCT cana-1224 140 22 1074 1074 NUM cana-1224 140 23 - - PUNCT cana-1224 140 24 133x 133x NUM cana-1224 140 25 vol vol NOUN cana-1224 140 26 31 31 NUM cana-1224 140 27 no no NOUN cana-1224 140 28 . . PUNCT cana-1224 141 1 6s 6s NUM cana-1224 141 2 ( ( PUNCT cana-1224 141 3 2024 2024 NUM cana-1224 141 4 ) ) PUNCT cana-1224 141 5 312 312 NUM cana-1224 141 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 141 7 figure figure NOUN cana-1224 141 8 3 3 NUM cana-1224 141 9 : : PUNCT cana-1224 141 10 physical physical ADJ cana-1224 141 11 behavior behavior NOUN cana-1224 141 12 of of ADP cana-1224 141 13 solutions solution NOUN cana-1224 141 14 of of ADP cana-1224 141 15 real real ADJ cana-1224 141 16 part part NOUN cana-1224 141 17 for for ADP cana-1224 141 18 ๐‘ง ๐‘ง NOUN cana-1224 141 19 = = SYM cana-1224 141 20 10 10 NUM cana-1224 141 21 and and CCONJ cana-1224 141 22 ๐‘ก ๐‘ก X cana-1224 141 23 = = SYM cana-1224 141 24 2 2 NUM cana-1224 141 25 figure figure NOUN cana-1224 141 26 4 4 NUM cana-1224 141 27 : : PUNCT cana-1224 141 28 physical physical ADJ cana-1224 141 29 behavior behavior NOUN cana-1224 141 30 of of ADP cana-1224 141 31 solutions solution NOUN cana-1224 141 32 of of ADP cana-1224 141 33 imaginary imaginary ADJ cana-1224 141 34 part part NOUN cana-1224 141 35 for for ADP cana-1224 141 36 ๐‘ง ๐‘ง NOUN cana-1224 141 37 = = SYM cana-1224 141 38 10 10 NUM cana-1224 141 39 and and CCONJ cana-1224 141 40 ๐‘ก ๐‘ก PROPN cana-1224 141 41 = = SYM cana-1224 141 42 2 2 NUM cana-1224 141 43 figures figure NOUN cana-1224 141 44 1 1 NUM cana-1224 141 45 & & CCONJ cana-1224 141 46 2 2 NUM cana-1224 141 47 show show VERB cana-1224 141 48 the the DET cana-1224 141 49 real real NOUN cana-1224 141 50 & & CCONJ cana-1224 141 51 the the DET cana-1224 141 52 imaginary imaginary ADJ cana-1224 141 53 part part NOUN cana-1224 141 54 solutions solution NOUN cana-1224 141 55 ' ' PART cana-1224 141 56 physical physical ADJ cana-1224 141 57 behavior behavior NOUN cana-1224 141 58 of of ADP cana-1224 141 59 example example NOUN cana-1224 141 60 1 1 NUM cana-1224 141 61 at at ADP cana-1224 141 62 ๐‘ง ๐‘ง NOUN cana-1224 141 63 = = ADJ cana-1224 141 64 2 2 NUM cana-1224 141 65 , , PUNCT cana-1224 141 66 ๐‘ก ๐‘ก X cana-1224 141 67 = = NOUN cana-1224 141 68 0.5 0.5 NUM cana-1224 141 69 respectively respectively ADV cana-1224 141 70 . . PUNCT cana-1224 142 1 figures figure NOUN cana-1224 142 2 3 3 NUM cana-1224 142 3 & & CCONJ cana-1224 142 4 4 4 NUM cana-1224 142 5 show show VERB cana-1224 142 6 the the DET cana-1224 142 7 real real NOUN cana-1224 142 8 & & CCONJ cana-1224 142 9 the the DET cana-1224 142 10 imaginary imaginary ADJ cana-1224 142 11 part part NOUN cana-1224 142 12 solution solution NOUN cana-1224 142 13 's 's PART cana-1224 142 14 physical physical ADJ cana-1224 142 15 behavior behavior NOUN cana-1224 142 16 of of ADP cana-1224 142 17 example example NOUN cana-1224 142 18 1 1 NUM cana-1224 142 19 at at ADP cana-1224 142 20 ๐‘ง ๐‘ง NOUN cana-1224 142 21 = = SYM cana-1224 142 22 10 10 NUM cana-1224 142 23 , , PUNCT cana-1224 142 24 ๐‘ก ๐‘ก X cana-1224 142 25 = = NOUN cana-1224 142 26 2 2 NUM cana-1224 142 27 respectively respectively ADV cana-1224 142 28 . . PUNCT cana-1224 142 29 example example NOUN cana-1224 142 30 2 2 NUM cana-1224 142 31 : : PUNCT cana-1224 142 32 consider consider VERB cana-1224 142 33 the the DET cana-1224 142 34 ( ( PUNCT cana-1224 142 35 2 2 NUM cana-1224 142 36 + + NOUN cana-1224 142 37 1)-d 1)-d NUM cana-1224 142 38 fractional fractional ADJ cana-1224 142 39 nonlinear nonlinear PROPN cana-1224 142 40 schrodinger schrodinger PROPN cana-1224 142 41 eqn eqn PROPN cana-1224 142 42 of of ADP cana-1224 142 43 the the DET cana-1224 142 44 form form NOUN cana-1224 142 45 ๐‘–๐‘ค๐‘ก ๐‘–๐‘ค๐‘ก NOUN cana-1224 142 46 ๐›ผ ๐›ผ NOUN cana-1224 142 47 = = NOUN cana-1224 142 48 โˆ’ โˆ’ PROPN cana-1224 142 49 1 1 NUM cana-1224 142 50 4 4 NUM cana-1224 142 51 ๐‘ค๐‘ฅ๐‘ฅ ๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 142 52 โˆ’ โˆ’ NOUN cana-1224 142 53 1 1 NUM cana-1224 142 54 4 4 NUM cana-1224 142 55 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 142 56 โˆ’ โˆ’ NOUN cana-1224 142 57 ๐‘คsin2๐‘ฅ ๐‘คsin2๐‘ฅ NOUN cana-1224 142 58 sin2๐‘ฆ sin2๐‘ฆ PROPN cana-1224 143 1 + + CCONJ cana-1224 143 2 |๐‘ค|2๐‘ค |๐‘ค|2๐‘ค X cana-1224 143 3 , , PUNCT cana-1224 143 4 ( ( PUNCT cana-1224 143 5 11 11 NUM cana-1224 143 6 ) ) PUNCT cana-1224 143 7 with with ADP cana-1224 143 8 initial initial ADJ cana-1224 143 9 โ€œ " PUNCT cana-1224 143 10 condition condition NOUN cana-1224 143 11 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 143 12 , , PUNCT cana-1224 143 13 ๐‘ฆ ๐‘ฆ NOUN cana-1224 143 14 , , PUNCT cana-1224 143 15 ๐‘ง ๐‘ง PROPN cana-1224 143 16 , , PUNCT cana-1224 143 17 0 0 NUM cana-1224 143 18 ) ) PUNCT cana-1224 143 19 = = VERB cana-1224 143 20 sin sin NOUN cana-1224 143 21 ๐‘ฅ ๐‘ฅ PRON cana-1224 143 22 sin sin NOUN cana-1224 143 23 ๐‘ฆ. ๐‘ฆ. VERB cana-1224 143 24 the the DET cana-1224 143 25 exact exact ADJ cana-1224 143 26 solution solution NOUN cana-1224 143 27 to to ADP cana-1224 143 28 the the DET cana-1224 143 29 problem problem NOUN cana-1224 143 30 ( ( PUNCT cana-1224 143 31 ๐›ผ ๐›ผ X cana-1224 143 32 = = SYM cana-1224 143 33 1 1 NUM cana-1224 143 34 ) ) PUNCT cana-1224 143 35 is be AUX cana-1224 143 36 : : PUNCT cana-1224 143 37 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 143 38 , , PUNCT cana-1224 143 39 ๐‘ฆ ๐‘ฆ NOUN cana-1224 143 40 , , PUNCT cana-1224 143 41 ๐‘ก ๐‘ก NOUN cana-1224 143 42 ) ) PUNCT cana-1224 143 43 = = PUNCT cana-1224 143 44 ๐‘’โˆ’๐‘–๐‘ก/2 ๐‘’โˆ’๐‘–๐‘ก/2 PUNCT cana-1224 143 45 sin sin VERB cana-1224 143 46 ๐‘ฅ ๐‘ฅ DET cana-1224 143 47 sin sin NOUN cana-1224 143 48 ๐‘ฆ ๐‘ฆ PRON cana-1224 143 49 rewrite rewrite VERB cana-1224 143 50 the the DET cana-1224 143 51 given give VERB cana-1224 143 52 problem problem NOUN cana-1224 143 53 as as ADP cana-1224 143 54 : : PUNCT cana-1224 143 55 ๐‘ค๐‘ก ๐‘ค๐‘ก NOUN cana-1224 143 56 ๐›ผ ๐›ผ NOUN cana-1224 143 57 = = SYM cana-1224 143 58 ๐‘– ๐‘– X cana-1224 143 59 ( ( PUNCT cana-1224 143 60 1 1 NUM cana-1224 143 61 4 4 NUM cana-1224 143 62 ๐‘ค๐‘ฅ๐‘ฅ ๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 143 63 + + CCONJ cana-1224 143 64 1 1 NUM cana-1224 143 65 4 4 NUM cana-1224 143 66 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 143 67 + + CCONJ cana-1224 143 68 ๐‘คsin2๐‘ฅ ๐‘คsin2๐‘ฅ ADJ cana-1224 143 69 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 143 70 โˆ’ โˆ’ NOUN cana-1224 143 71 ๐‘ค2 ๐‘ค2 NOUN cana-1224 143 72 ๏ฟฝ ๏ฟฝ NOUN cana-1224 143 73 ฬ… ฬ… NOUN cana-1224 143 74 ๏ฟฝ ๏ฟฝ NOUN cana-1224 143 75 ) ) PUNCT cana-1224 143 76 ( ( PUNCT cana-1224 143 77 12 12 X cana-1224 143 78 ) ) PUNCT cana-1224 143 79 taking take VERB cana-1224 143 80 elzaki elzaki NOUN cana-1224 143 81 transform transform NOUN cana-1224 143 82 to to ADP cana-1224 143 83 both both DET cana-1224 143 84 sides side NOUN cana-1224 143 85 of of ADP cana-1224 143 86 eqn eqn NOUN cana-1224 143 87 ( ( PUNCT cana-1224 143 88 12 12 NUM cana-1224 143 89 ) ) PUNCT cana-1224 143 90 , , PUNCT cana-1224 143 91 we we PRON cana-1224 143 92 obtain obtain VERB cana-1224 143 93 ๐ธ[๐‘ค๐‘ก ๐ธ[๐‘ค๐‘ก NOUN cana-1224 143 94 ๐›ผ ๐›ผ NOUN cana-1224 143 95 ] ] X cana-1224 143 96 = = PUNCT cana-1224 143 97 ๐ธ ๐ธ PROPN cana-1224 144 1 [ [ X cana-1224 144 2 ๐‘– ๐‘– X cana-1224 144 3 ( ( PUNCT cana-1224 144 4 1 1 NUM cana-1224 144 5 4 4 NUM cana-1224 144 6 ๐‘ค๐‘ฅ๐‘ฅ ๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 144 7 + + CCONJ cana-1224 144 8 1 1 NUM cana-1224 144 9 4 4 NUM cana-1224 144 10 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 144 11 + + CCONJ cana-1224 144 12 ๐‘คsin2๐‘ฅ ๐‘คsin2๐‘ฅ ADJ cana-1224 144 13 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 144 14 โˆ’ โˆ’ NOUN cana-1224 144 15 ๐‘ค2 ๐‘ค2 NOUN cana-1224 144 16 ๏ฟฝ ๏ฟฝ NOUN cana-1224 144 17 ฬ… ฬ… NOUN cana-1224 144 18 ๏ฟฝ ๏ฟฝ NOUN cana-1224 144 19 ) ) PUNCT cana-1224 144 20 ] ] PUNCT cana-1224 145 1 this this PRON cana-1224 145 2 implies imply VERB cana-1224 145 3 -2 -2 INTJ cana-1224 145 4 -1 -1 SYM cana-1224 145 5 0 0 NUM cana-1224 145 6 1 1 NUM cana-1224 145 7 2 2 NUM cana-1224 145 8 -2 -2 NOUN cana-1224 145 9 0 0 NUM cana-1224 145 10 2 2 NUM cana-1224 145 11 - - SYM cana-1224 145 12 1 1 NUM cana-1224 145 13 -0.5 -0.5 NUM cana-1224 145 14 0 0 NUM cana-1224 145 15 0.5 0.5 NUM cana-1224 145 16 1 1 NUM cana-1224 145 17 y y PROPN cana-1224 145 18 w w PROPN cana-1224 145 19 ( ( PUNCT cana-1224 145 20 real real ADJ cana-1224 145 21 part part NOUN cana-1224 145 22 ) ) PUNCT cana-1224 145 23 x x SYM cana-1224 145 24 w w X cana-1224 145 25 ( ( PUNCT cana-1224 145 26 re re ADP cana-1224 145 27 al al PROPN cana-1224 145 28 p p PROPN cana-1224 145 29 ar ar PROPN cana-1224 145 30 t t PROPN cana-1224 145 31 ) ) PUNCT cana-1224 145 32 -2 -2 NOUN cana-1224 146 1 -1 -1 NOUN cana-1224 146 2 0 0 NUM cana-1224 146 3 1 1 NUM cana-1224 146 4 2 2 NUM cana-1224 146 5 -2 -2 NOUN cana-1224 146 6 0 0 NUM cana-1224 146 7 2 2 NUM cana-1224 146 8 -1 -1 SYM cana-1224 146 9 -0.5 -0.5 X cana-1224 146 10 0 0 NUM cana-1224 146 11 0.5 0.5 NUM cana-1224 146 12 1 1 NUM cana-1224 146 13 y y PROPN cana-1224 146 14 x x SYM cana-1224 146 15 w w PROPN cana-1224 146 16 ( ( PUNCT cana-1224 146 17 imaginary imaginary ADJ cana-1224 146 18 part part NOUN cana-1224 146 19 ) ) PUNCT cana-1224 147 1 w w PROPN cana-1224 147 2 ( ( PUNCT cana-1224 148 1 i i PRON cana-1224 148 2 m m VERB cana-1224 148 3 ag ag PROPN cana-1224 148 4 in in ADP cana-1224 148 5 ar ar PROPN cana-1224 148 6 y y PROPN cana-1224 148 7 pa pa PROPN cana-1224 148 8 rt rt PROPN cana-1224 148 9 ) ) PUNCT cana-1224 148 10 communications communication NOUN cana-1224 148 11 on on ADP cana-1224 148 12 applied apply VERB cana-1224 148 13 nonlinear nonlinear ADJ cana-1224 148 14 analysis analysis NOUN cana-1224 148 15 issn issn NOUN cana-1224 148 16 : : PUNCT cana-1224 148 17 1074 1074 NUM cana-1224 148 18 - - PUNCT cana-1224 148 19 133x 133x NUM cana-1224 148 20 vol vol NOUN cana-1224 148 21 31 31 NUM cana-1224 148 22 no no NOUN cana-1224 148 23 . . PUNCT cana-1224 149 1 6s 6s NUM cana-1224 149 2 ( ( PUNCT cana-1224 149 3 2024 2024 NUM cana-1224 149 4 ) ) PUNCT cana-1224 149 5 313 313 NUM cana-1224 149 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 149 7 ๐ธ[๐‘ค(๐‘ฅ ๐ธ[๐‘ค(๐‘ฅ PROPN cana-1224 149 8 , , PUNCT cana-1224 149 9 ๐‘ฆ ๐‘ฆ NOUN cana-1224 149 10 , , PUNCT cana-1224 149 11 ๐‘ก ๐‘ก NOUN cana-1224 149 12 ) ) PUNCT cana-1224 149 13 ] ] PUNCT cana-1224 150 1 = = PUNCT cana-1224 150 2 โˆ‘ โˆ‘ PUNCT cana-1224 150 3 ๐‘ฃ๐‘–+2 ๐‘ฃ๐‘–+2 PROPN cana-1224 150 4 ๐‘›โˆ’1 ๐‘›โˆ’1 PROPN cana-1224 150 5 ๐‘–=0 ๐‘–=0 SYM cana-1224 150 6 ๐‘ค(๐‘–)(๐‘ฅ ๐‘ค(๐‘–)(๐‘ฅ PROPN cana-1224 150 7 , , PUNCT cana-1224 150 8 ๐‘ฆ ๐‘ฆ NOUN cana-1224 150 9 , , PUNCT cana-1224 150 10 0 0 NUM cana-1224 150 11 ) ) PUNCT cana-1224 151 1 + + CCONJ cana-1224 151 2 ๐‘ฃ๐›ผ๐‘–๐ธ ๐‘ฃ๐›ผ๐‘–๐ธ ADJ cana-1224 151 3 [ [ PUNCT cana-1224 151 4 1 1 NUM cana-1224 151 5 4 4 NUM cana-1224 151 6 ๐‘ค๐‘ฅ๐‘ฅ ๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 151 7 + + CCONJ cana-1224 151 8 1 1 NUM cana-1224 151 9 4 4 NUM cana-1224 151 10 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 151 11 + + CCONJ cana-1224 151 12 ๐‘คsin2๐‘ฅ ๐‘คsin2๐‘ฅ ADJ cana-1224 151 13 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 151 14 โˆ’ โˆ’ NOUN cana-1224 151 15 ๐‘ค2 ๐‘ค2 NOUN cana-1224 151 16 ๏ฟฝ ๏ฟฝ NOUN cana-1224 151 17 ฬ… ฬ… NOUN cana-1224 151 18 ๏ฟฝ ๏ฟฝ NOUN cana-1224 151 19 ] ] PUNCT cana-1224 151 20 after after ADP cana-1224 151 21 applying apply VERB cana-1224 151 22 initial initial ADJ cana-1224 151 23 conditions condition NOUN cana-1224 151 24 , , PUNCT cana-1224 151 25 we we PRON cana-1224 151 26 obtain obtain VERB cana-1224 151 27 ๐ธ[๐‘ค(๐‘ฅ ๐ธ[๐‘ค(๐‘ฅ NOUN cana-1224 151 28 , , PUNCT cana-1224 151 29 ๐‘ฆ ๐‘ฆ NOUN cana-1224 151 30 , , PUNCT cana-1224 151 31 ๐‘ก ๐‘ก NOUN cana-1224 151 32 ) ) PUNCT cana-1224 151 33 ] ] PUNCT cana-1224 151 34 = = SYM cana-1224 151 35 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 151 36 . . PUNCT cana-1224 151 37 sin sin PROPN cana-1224 151 38 ๐‘ฅ ๐‘ฅ DET cana-1224 151 39 sin sin VERB cana-1224 151 40 ๐‘ฆ ๐‘ฆ NOUN cana-1224 151 41 + + CCONJ cana-1224 151 42 ๐‘ฃ๐›ผ๐‘–๐ธ ๐‘ฃ๐›ผ๐‘–๐ธ PROPN cana-1224 151 43 [ [ PUNCT cana-1224 151 44 1 1 NUM cana-1224 151 45 4 4 NUM cana-1224 151 46 ๐‘ค๐‘ฅ๐‘ฅ ๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 151 47 + + CCONJ cana-1224 151 48 1 1 NUM cana-1224 151 49 4 4 NUM cana-1224 151 50 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 151 51 + + CCONJ cana-1224 151 52 ๐‘คsin2๐‘ฅ ๐‘คsin2๐‘ฅ ADJ cana-1224 151 53 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 151 54 โˆ’ โˆ’ NOUN cana-1224 151 55 ๐‘ค2 ๐‘ค2 NOUN cana-1224 151 56 ๏ฟฝ ๏ฟฝ NOUN cana-1224 151 57 ฬ… ฬ… NOUN cana-1224 151 58 ๏ฟฝ ๏ฟฝ NOUN cana-1224 151 59 ] ] PUNCT cana-1224 151 60 or or CCONJ cana-1224 151 61 ๐ธ[๐‘ค(๐‘ฅ ๐ธ[๐‘ค(๐‘ฅ PROPN cana-1224 151 62 , , PUNCT cana-1224 151 63 ๐‘ฆ ๐‘ฆ NOUN cana-1224 151 64 , , PUNCT cana-1224 151 65 ๐‘ก ๐‘ก NOUN cana-1224 151 66 ) ) PUNCT cana-1224 151 67 ] ] PUNCT cana-1224 152 1 โˆ’ โˆ’ PROPN cana-1224 152 2 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 152 3 . . PUNCT cana-1224 152 4 sin sin PROPN cana-1224 152 5 ๐‘ฅ ๐‘ฅ DET cana-1224 152 6 sin sin VERB cana-1224 152 7 ๐‘ฆ ๐‘ฆ NOUN cana-1224 152 8 โˆ’ โˆ’ NOUN cana-1224 152 9 ๐‘ฃ๐›ผ๐‘–๐ธ ๐‘ฃ๐›ผ๐‘–๐ธ PROPN cana-1224 152 10 [ [ PUNCT cana-1224 152 11 1 1 NUM cana-1224 152 12 4 4 NUM cana-1224 152 13 ๐‘ค๐‘ฅ๐‘ฅ ๐‘ค๐‘ฅ๐‘ฅ NOUN cana-1224 152 14 + + CCONJ cana-1224 152 15 1 1 NUM cana-1224 152 16 4 4 NUM cana-1224 152 17 ๐‘ค๐‘ฆ๐‘ฆ ๐‘ค๐‘ฆ๐‘ฆ NOUN cana-1224 152 18 + + CCONJ cana-1224 152 19 ๐‘คsin2๐‘ฅ ๐‘คsin2๐‘ฅ ADJ cana-1224 152 20 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 152 21 โˆ’ โˆ’ NOUN cana-1224 152 22 ๐‘ค2 ๐‘ค2 NOUN cana-1224 152 23 ๏ฟฝ ๏ฟฝ NOUN cana-1224 152 24 ฬ… ฬ… NOUN cana-1224 152 25 ๏ฟฝ ๏ฟฝ NOUN cana-1224 152 26 ] ] X cana-1224 152 27 = = SYM cana-1224 152 28 0 0 NUM cana-1224 153 1 the the DET cana-1224 153 2 nonlinear nonlinear ADJ cana-1224 153 3 component component NOUN cana-1224 153 4 is be AUX cana-1224 153 5 : : PUNCT cana-1224 153 6 ๐‘…[๐œ‘(๐‘ฅ ๐‘…[๐œ‘(๐‘ฅ PROPN cana-1224 153 7 , , PUNCT cana-1224 153 8 ๐‘ฆ ๐‘ฆ NOUN cana-1224 153 9 , , PUNCT cana-1224 153 10 ๐‘ก ๐‘ก NOUN cana-1224 153 11 ; ; PUNCT cana-1224 153 12 ๐‘ ๐‘ NOUN cana-1224 153 13 ) ) PUNCT cana-1224 153 14 ] ] PUNCT cana-1224 154 1 = = PUNCT cana-1224 154 2 ๐ธ[๐œ‘ ๐ธ[๐œ‘ VERB cana-1224 154 3 ] ] PUNCT cana-1224 154 4 โˆ’ โˆ’ PROPN cana-1224 154 5 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 154 6 . . PUNCT cana-1224 154 7 sin sin PROPN cana-1224 154 8 ๐‘ฅ ๐‘ฅ DET cana-1224 154 9 sin sin VERB cana-1224 154 10 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 11 โˆ’ โˆ’ NOUN cana-1224 154 12 ๐‘ฃ๐›ผ๐‘–๐ธ ๐‘ฃ๐›ผ๐‘–๐ธ PROPN cana-1224 154 13 [ [ PUNCT cana-1224 154 14 1 1 NUM cana-1224 154 15 4 4 NUM cana-1224 154 16 ๐œ‘๐‘ฅ๐‘ฅ ๐œ‘๐‘ฅ๐‘ฅ NOUN cana-1224 154 17 + + CCONJ cana-1224 154 18 1 1 NUM cana-1224 154 19 4 4 NUM cana-1224 154 20 ๐œ‘๐‘ฆ๐‘ฆ ๐œ‘๐‘ฆ๐‘ฆ ADP cana-1224 154 21 + + CCONJ cana-1224 154 22 ๐œ‘sin2๐‘ฅ ๐œ‘sin2๐‘ฅ PROPN cana-1224 154 23 sin2๐‘ฆ sin2๐‘ฆ PROPN cana-1224 154 24 โˆ’ โˆ’ PROPN cana-1224 154 25 ๐œ‘2 ๐œ‘2 PROPN cana-1224 154 26 ๏ฟฝ ๏ฟฝ PROPN cana-1224 154 27 ฬ… ฬ… NOUN cana-1224 154 28 ๏ฟฝ ๏ฟฝ NOUN cana-1224 154 29 ] ] X cana-1224 154 30 ( ( PUNCT cana-1224 154 31 13 13 NUM cana-1224 154 32 ) ) PUNCT cana-1224 154 33 we we PRON cana-1224 154 34 build build VERB cana-1224 154 35 the the DET cana-1224 154 36 zero zero NUM cana-1224 154 37 - - PUNCT cana-1224 154 38 order order NOUN cana-1224 154 39 deformation deformation NOUN cana-1224 154 40 eqn eqn NOUN cana-1224 154 41 with with ADP cana-1224 154 42 the the DET cana-1224 154 43 assumption assumption NOUN cana-1224 154 44 ๐ป(๐‘ฅ ๐ป(๐‘ฅ NOUN cana-1224 154 45 , , PUNCT cana-1224 154 46 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 47 , , PUNCT cana-1224 154 48 ๐‘ก ๐‘ก NOUN cana-1224 154 49 ) ) PUNCT cana-1224 154 50 = = SYM cana-1224 154 51 1 1 NUM cana-1224 154 52 , , PUNCT cana-1224 154 53 ( ( PUNCT cana-1224 154 54 1 1 NUM cana-1224 154 55 โˆ’ โˆ’ NOUN cana-1224 154 56 ๐‘)๐ธ{๐œ‘(๐‘ฅ ๐‘)๐ธ{๐œ‘(๐‘ฅ NOUN cana-1224 154 57 , , PUNCT cana-1224 154 58 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 59 , , PUNCT cana-1224 154 60 ๐‘ก ๐‘ก NOUN cana-1224 154 61 ) ) PUNCT cana-1224 154 62 โˆ’ โˆ’ ADP cana-1224 154 63 ๐‘ค0(๐‘ฅ ๐‘ค0(๐‘ฅ PROPN cana-1224 154 64 , , PUNCT cana-1224 154 65 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 66 , , PUNCT cana-1224 154 67 ๐‘ก ๐‘ก NOUN cana-1224 154 68 ) ) PUNCT cana-1224 154 69 } } PUNCT cana-1224 154 70 = = SYM cana-1224 154 71 ๐‘โ„Ž๐‘…[๐œ‘(๐‘ฅ ๐‘โ„Ž๐‘…[๐œ‘(๐‘ฅ ADJ cana-1224 154 72 , , PUNCT cana-1224 154 73 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 74 , , PUNCT cana-1224 154 75 ๐‘ก ๐‘ก NOUN cana-1224 154 76 ; ; PUNCT cana-1224 154 77 ๐‘ ๐‘ NOUN cana-1224 154 78 ) ) PUNCT cana-1224 154 79 ] ] PUNCT cana-1224 154 80 when when SCONJ cana-1224 154 81 ๐‘ ๐‘ X cana-1224 154 82 = = SYM cana-1224 154 83 0 0 PROPN cana-1224 154 84 & & CCONJ cana-1224 154 85 ๐‘ ๐‘ NOUN cana-1224 154 86 = = SYM cana-1224 154 87 1 1 NUM cana-1224 154 88 , , PUNCT cana-1224 154 89 we we PRON cana-1224 154 90 get get VERB cana-1224 154 91 { { PUNCT cana-1224 154 92 ๐œ‘(๐‘ฅ ๐œ‘(๐‘ฅ PROPN cana-1224 154 93 , , PUNCT cana-1224 154 94 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 95 , , PUNCT cana-1224 154 96 ๐‘ก ๐‘ก X cana-1224 154 97 ; ; PUNCT cana-1224 154 98 0 0 NUM cana-1224 154 99 ) ) PUNCT cana-1224 154 100 = = SYM cana-1224 154 101 ๐‘ค0(๐‘ฅ ๐‘ค0(๐‘ฅ PROPN cana-1224 154 102 , , PUNCT cana-1224 154 103 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 104 , , PUNCT cana-1224 154 105 0 0 NUM cana-1224 154 106 ) ) PUNCT cana-1224 154 107 ๐œ‘(๐‘ฅ ๐œ‘(๐‘ฅ NOUN cana-1224 154 108 , , PUNCT cana-1224 154 109 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 110 , , PUNCT cana-1224 154 111 ๐‘ก ๐‘ก X cana-1224 154 112 ; ; PUNCT cana-1224 154 113 1 1 X cana-1224 154 114 ) ) PUNCT cana-1224 154 115 = = SYM cana-1224 154 116 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 154 117 , , PUNCT cana-1224 154 118 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 119 , , PUNCT cana-1224 154 120 ๐‘ก ๐‘ก NOUN cana-1224 154 121 ) ) PUNCT cana-1224 154 122 therefore therefore ADV cana-1224 154 123 , , PUNCT cana-1224 154 124 the the DET cana-1224 154 125 mth mth NOUN cana-1224 154 126 - - PUNCT cana-1224 154 127 order order NOUN cana-1224 154 128 deformation deformation NOUN cana-1224 154 129 eqn eqn PROPN cana-1224 154 130 ๐ธ{๐‘ค๐‘š(๐‘ฅ ๐ธ{๐‘ค๐‘š(๐‘ฅ PROPN cana-1224 154 131 , , PUNCT cana-1224 154 132 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 133 , , PUNCT cana-1224 154 134 ๐‘ก ๐‘ก NOUN cana-1224 154 135 ) ) PUNCT cana-1224 154 136 โˆ’ โˆ’ NOUN cana-1224 154 137 ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ NOUN cana-1224 154 138 , , PUNCT cana-1224 154 139 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 140 , , PUNCT cana-1224 154 141 ๐‘ก ๐‘ก NOUN cana-1224 154 142 ) ) PUNCT cana-1224 154 143 } } PUNCT cana-1224 154 144 = = SYM cana-1224 154 145 โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— PROPN cana-1224 154 146 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 147 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 148 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 149 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 150 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 154 151 , , PUNCT cana-1224 154 152 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 153 , , PUNCT cana-1224 154 154 ๐‘ก ๐‘ก NOUN cana-1224 154 155 ) ) PUNCT cana-1224 154 156 ) ) PUNCT cana-1224 154 157 ( ( PUNCT cana-1224 154 158 14 14 X cana-1224 154 159 ) ) PUNCT cana-1224 154 160 inverse inverse NOUN cana-1224 154 161 elzaki elzaki NOUN cana-1224 154 162 transforms transform VERB cana-1224 154 163 both both DET cana-1224 154 164 sides side NOUN cana-1224 154 165 and and CCONJ cana-1224 154 166 gives give VERB cana-1224 154 167 ๐‘ค๐‘š(๐‘ฅ ๐‘ค๐‘š(๐‘ฅ NUM cana-1224 154 168 , , PUNCT cana-1224 154 169 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 170 , , PUNCT cana-1224 154 171 ๐‘ก ๐‘ก NOUN cana-1224 154 172 ) ) PUNCT cana-1224 154 173 โˆ’ โˆ’ NOUN cana-1224 154 174 ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ ๐œ’๐‘š๐‘ค๐‘šโˆ’1(๐‘ฅ NOUN cana-1224 154 175 , , PUNCT cana-1224 154 176 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 177 , , PUNCT cana-1224 154 178 ๐‘ก ๐‘ก NOUN cana-1224 154 179 ) ) PUNCT cana-1224 154 180 = = SYM cana-1224 154 181 ๐ธโˆ’1{โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— ๐ธโˆ’1{โ„Ž๐‘…๐‘š(๐‘ค๐‘šโˆ’1โƒ—โƒ— PROPN cana-1224 154 182 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 183 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 184 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 185 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 154 186 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 154 187 , , PUNCT cana-1224 154 188 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 189 , , PUNCT cana-1224 154 190 ๐‘ก ๐‘ก NOUN cana-1224 154 191 ) ) PUNCT cana-1224 154 192 ) ) PUNCT cana-1224 154 193 } } PUNCT cana-1224 154 194 ( ( PUNCT cana-1224 154 195 15 15 NUM cana-1224 154 196 ) ) PUNCT cana-1224 154 197 from from ADP cana-1224 154 198 equation equation NOUN cana-1224 154 199 ( ( PUNCT cana-1224 154 200 15 15 NUM cana-1224 154 201 ) ) PUNCT cana-1224 154 202 ( ( PUNCT cana-1224 154 203 taking take VERB cana-1224 154 204 โ„Ž โ„Ž PROPN cana-1224 154 205 = = SYM cana-1224 154 206 โˆ’1 โˆ’1 NOUN cana-1224 154 207 ) ) PUNCT cana-1224 154 208 , , PUNCT cana-1224 154 209 we we PRON cana-1224 154 210 obtain obtain VERB cana-1224 154 211 ๐‘ค1(๐‘ฅ ๐‘ค1(๐‘ฅ NUM cana-1224 154 212 , , PUNCT cana-1224 154 213 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 214 , , PUNCT cana-1224 154 215 ๐‘ก ๐‘ก NOUN cana-1224 154 216 ) ) PUNCT cana-1224 154 217 = = SYM cana-1224 154 218 โˆ’๐ธโˆ’1{๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— โˆ’๐ธโˆ’1{๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— X cana-1224 154 219 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 154 220 , , PUNCT cana-1224 154 221 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 222 , , PUNCT cana-1224 154 223 ๐‘ก ๐‘ก NOUN cana-1224 154 224 ) ) PUNCT cana-1224 154 225 ) ) PUNCT cana-1224 154 226 } } PUNCT cana-1224 154 227 , , PUNCT cana-1224 154 228 ๐‘ค2(๐‘ฅ ๐‘ค2(๐‘ฅ NOUN cana-1224 154 229 , , PUNCT cana-1224 154 230 ๐‘ฆ ๐‘ฆ NOUN cana-1224 154 231 , , PUNCT cana-1224 154 232 ๐‘ก ๐‘ก NOUN cana-1224 154 233 ) ) PUNCT cana-1224 154 234 = = PUNCT cana-1224 155 1 ๐‘ค1(๐‘ฅ ๐‘ค1(๐‘ฅ PROPN cana-1224 155 2 , , PUNCT cana-1224 155 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 155 4 , , PUNCT cana-1224 155 5 ๐‘ก ๐‘ก NOUN cana-1224 155 6 ) ) PUNCT cana-1224 155 7 โˆ’ โˆ’ PROPN cana-1224 155 8 ๐ธโˆ’1{๐‘…2(๐‘ค1โƒ—โƒ— ๐ธโˆ’1{๐‘…2(๐‘ค1โƒ—โƒ— NOUN cana-1224 155 9 โƒ—โƒ— โƒ—โƒ— NOUN cana-1224 155 10 ( ( PUNCT cana-1224 155 11 ๐‘ฅ ๐‘ฅ PROPN cana-1224 155 12 , , PUNCT cana-1224 155 13 ๐‘ฆ ๐‘ฆ NOUN cana-1224 155 14 , , PUNCT cana-1224 155 15 ๐‘ก ๐‘ก NOUN cana-1224 155 16 ) ) PUNCT cana-1224 155 17 ) ) PUNCT cana-1224 155 18 } } PUNCT cana-1224 155 19 , , PUNCT cana-1224 155 20 ๐‘ค3(๐‘ฅ ๐‘ค3(๐‘ฅ NOUN cana-1224 155 21 , , PUNCT cana-1224 155 22 ๐‘ฆ ๐‘ฆ NOUN cana-1224 155 23 , , PUNCT cana-1224 155 24 ๐‘ก ๐‘ก NOUN cana-1224 155 25 ) ) PUNCT cana-1224 155 26 = = SYM cana-1224 156 1 ๐‘ค2(๐‘ฅ ๐‘ค2(๐‘ฅ PROPN cana-1224 156 2 , , PUNCT cana-1224 156 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 156 4 , , PUNCT cana-1224 156 5 ๐‘ก ๐‘ก NOUN cana-1224 156 6 ) ) PUNCT cana-1224 156 7 โˆ’ โˆ’ NUM cana-1224 156 8 ๐ธโˆ’1{๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— ๐ธโˆ’1{๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— NOUN cana-1224 156 9 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 156 10 , , PUNCT cana-1224 156 11 ๐‘ฆ ๐‘ฆ NOUN cana-1224 156 12 , , PUNCT cana-1224 156 13 ๐‘ก ๐‘ก NOUN cana-1224 156 14 ) ) PUNCT cana-1224 156 15 ) ) PUNCT cana-1224 156 16 } } PUNCT cana-1224 156 17 , , PUNCT cana-1224 156 18 โ‹ฎ โ‹ฎ NOUN cana-1224 156 19 where where SCONJ cana-1224 156 20 ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— X cana-1224 156 21 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 156 22 , , PUNCT cana-1224 156 23 ๐‘ฆ ๐‘ฆ NOUN cana-1224 156 24 , , PUNCT cana-1224 156 25 ๐‘ก ๐‘ก NOUN cana-1224 156 26 ) ) PUNCT cana-1224 156 27 ) ) PUNCT cana-1224 156 28 = = PUNCT cana-1224 157 1 ๐ธ[๐‘ค0 ๐ธ[๐‘ค0 PROPN cana-1224 157 2 ] ] PUNCT cana-1224 157 3 โˆ’ โˆ’ PROPN cana-1224 157 4 ๐‘ฃ2 ๐‘ฃ2 PROPN cana-1224 157 5 . . PUNCT cana-1224 158 1 sin sin PROPN cana-1224 159 1 ๐‘ฅ ๐‘ฅ DET cana-1224 159 2 sin sin VERB cana-1224 159 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 159 4 โˆ’๐‘ฃ๐›ผ๐‘–๐ธ โˆ’๐‘ฃ๐›ผ๐‘–๐ธ NOUN cana-1224 159 5 [ [ PUNCT cana-1224 159 6 1 1 NUM cana-1224 159 7 4 4 NUM cana-1224 159 8 ( ( PUNCT cana-1224 159 9 ๐‘ค0)๐‘ฅ๐‘ฅ ๐‘ค0)๐‘ฅ๐‘ฅ PROPN cana-1224 159 10 + + NUM cana-1224 159 11 1 1 NUM cana-1224 159 12 4 4 NUM cana-1224 159 13 ( ( PUNCT cana-1224 159 14 ๐‘ค0)๐‘ฆ๐‘ฆ ๐‘ค0)๐‘ฆ๐‘ฆ NOUN cana-1224 159 15 + + CCONJ cana-1224 159 16 ๐‘ค0sin ๐‘ค0sin PROPN cana-1224 159 17 2๐‘ฅ 2๐‘ฅ NOUN cana-1224 159 18 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 159 19 โˆ’ โˆ’ PROPN cana-1224 159 20 ๐‘ค0 ๐‘ค0 PROPN cana-1224 159 21 2๐‘ค0ฬ…ฬ…ฬ…ฬ… 2๐‘ค0ฬ…ฬ…ฬ…ฬ… PROPN cana-1224 159 22 ] ] PUNCT cana-1224 159 23 , , PUNCT cana-1224 159 24 ๐‘…2(๐‘ค1โƒ—โƒ— ๐‘…2(๐‘ค1โƒ—โƒ— PROPN cana-1224 159 25 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 159 26 ( ( PUNCT cana-1224 159 27 ๐‘ฅ ๐‘ฅ PROPN cana-1224 159 28 , , PUNCT cana-1224 159 29 ๐‘ฆ ๐‘ฆ NOUN cana-1224 159 30 , , PUNCT cana-1224 159 31 ๐‘ก ๐‘ก NOUN cana-1224 159 32 ) ) PUNCT cana-1224 159 33 ) ) PUNCT cana-1224 160 1 = = SYM cana-1224 160 2 ๐ธ[๐‘ค1 ๐ธ[๐‘ค1 NOUN cana-1224 160 3 ] ] X cana-1224 160 4 โˆ’ โˆ’ X cana-1224 160 5 ๐‘ฃ๐›ผ๐‘–๐ธ ๐‘ฃ๐›ผ๐‘–๐ธ X cana-1224 160 6 [ [ PUNCT cana-1224 160 7 1 1 NUM cana-1224 160 8 4 4 NUM cana-1224 160 9 ( ( PUNCT cana-1224 160 10 ๐‘ค1)๐‘ฅ๐‘ฅ ๐‘ค1)๐‘ฅ๐‘ฅ NOUN cana-1224 160 11 + + CCONJ cana-1224 160 12 1 1 NUM cana-1224 160 13 4 4 NUM cana-1224 160 14 ( ( PUNCT cana-1224 160 15 ๐‘ค1)๐‘ฆ๐‘ฆ ๐‘ค1)๐‘ฆ๐‘ฆ PROPN cana-1224 160 16 + + CCONJ cana-1224 160 17 ๐‘ค1sin ๐‘ค1sin PROPN cana-1224 160 18 2๐‘ฅ 2๐‘ฅ NOUN cana-1224 160 19 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 161 1 โˆ’ โˆ’ PROPN cana-1224 162 1 ๐‘ค0 ๐‘ค0 PROPN cana-1224 162 2 2๐‘ค1ฬ…ฬ…ฬ…ฬ… 2๐‘ค1ฬ…ฬ…ฬ…ฬ… PROPN cana-1224 162 3 โˆ’ โˆ’ PROPN cana-1224 162 4 2๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… 2๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… PROPN cana-1224 162 5 ๐‘ค1 ๐‘ค1 PROPN cana-1224 162 6 ] ] PUNCT cana-1224 162 7 , , PUNCT cana-1224 162 8 ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— X cana-1224 163 1 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 163 2 , , PUNCT cana-1224 163 3 ๐‘ฆ ๐‘ฆ NOUN cana-1224 163 4 , , PUNCT cana-1224 163 5 ๐‘ก ๐‘ก NOUN cana-1224 163 6 ) ) PUNCT cana-1224 163 7 ) ) PUNCT cana-1224 164 1 = = SYM cana-1224 164 2 ๐ธ[๐‘ค2 ๐ธ[๐‘ค2 PROPN cana-1224 164 3 ] ] PUNCT cana-1224 164 4 communications communication NOUN cana-1224 164 5 on on ADP cana-1224 164 6 applied apply VERB cana-1224 164 7 nonlinear nonlinear ADJ cana-1224 164 8 analysis analysis NOUN cana-1224 164 9 issn issn NOUN cana-1224 164 10 : : PUNCT cana-1224 164 11 1074 1074 NUM cana-1224 164 12 - - PUNCT cana-1224 164 13 133x 133x NUM cana-1224 164 14 vol vol NOUN cana-1224 164 15 31 31 NUM cana-1224 164 16 no no NOUN cana-1224 164 17 . . PUNCT cana-1224 165 1 6s 6s NUM cana-1224 165 2 ( ( PUNCT cana-1224 165 3 2024 2024 NUM cana-1224 165 4 ) ) PUNCT cana-1224 165 5 314 314 NUM cana-1224 165 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 165 7 โˆ’๐‘ฃ๐›ผ๐‘–๐ธ โˆ’๐‘ฃ๐›ผ๐‘–๐ธ PRON cana-1224 165 8 [ [ PUNCT cana-1224 165 9 1 1 NUM cana-1224 165 10 4 4 NUM cana-1224 165 11 ( ( PUNCT cana-1224 165 12 ๐‘ค2)๐‘ฅ๐‘ฅ ๐‘ค2)๐‘ฅ๐‘ฅ PROPN cana-1224 165 13 + + CCONJ cana-1224 165 14 1 1 NUM cana-1224 165 15 4 4 NUM cana-1224 165 16 ( ( PUNCT cana-1224 165 17 ๐‘ค2)๐‘ฆ๐‘ฆ ๐‘ค2)๐‘ฆ๐‘ฆ NOUN cana-1224 165 18 + + CCONJ cana-1224 165 19 ๐‘ค2sin ๐‘ค2sin ADP cana-1224 165 20 2๐‘ฅ 2๐‘ฅ NOUN cana-1224 165 21 sin2๐‘ฆ sin2๐‘ฆ NUM cana-1224 165 22 โˆ’ โˆ’ PROPN cana-1224 165 23 ๐‘ค0 ๐‘ค0 PROPN cana-1224 165 24 2๐‘ค2ฬ…ฬ…ฬ…ฬ… 2๐‘ค2ฬ…ฬ…ฬ…ฬ… NUM cana-1224 165 25 โˆ’ โˆ’ PROPN cana-1224 165 26 2๐‘ค0๐‘ค1ฬ…ฬ…ฬ…ฬ… 2๐‘ค0๐‘ค1ฬ…ฬ…ฬ…ฬ… NUM cana-1224 165 27 ๐‘ค1 ๐‘ค1 VERB cana-1224 165 28 โˆ’ โˆ’ PROPN cana-1224 165 29 2๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… 2๐‘ค0๐‘ค0ฬ…ฬ…ฬ…ฬ… PROPN cana-1224 165 30 ๐‘ค2 ๐‘ค2 PROPN cana-1224 165 31 โˆ’ โˆ’ PROPN cana-1224 165 32 ๐‘ค0ฬ…ฬ…ฬ…ฬ… ๐‘ค0ฬ…ฬ…ฬ…ฬ… PROPN cana-1224 165 33 ๐‘ค1 ๐‘ค1 VERB cana-1224 165 34 2 2 NUM cana-1224 165 35 ] ] PUNCT cana-1224 165 36 , , PUNCT cana-1224 165 37 โ‹ฎ โ‹ฎ NOUN cana-1224 165 38 after after ADP cana-1224 165 39 simplifications simplification NOUN cana-1224 165 40 , , PUNCT cana-1224 165 41 we we PRON cana-1224 165 42 obtain obtain VERB cana-1224 165 43 ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— ๐‘…1(๐‘ค0โƒ—โƒ—โƒ—โƒ— X cana-1224 165 44 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 165 45 , , PUNCT cana-1224 165 46 ๐‘ฆ ๐‘ฆ NOUN cana-1224 165 47 , , PUNCT cana-1224 165 48 ๐‘ก ๐‘ก NOUN cana-1224 165 49 ) ) PUNCT cana-1224 165 50 ) ) PUNCT cana-1224 166 1 = = PUNCT cana-1224 166 2 ๐‘–. ๐‘–. ADJ cana-1224 166 3 ๐‘ฃ๐›ผ+2 ๐‘ฃ๐›ผ+2 NUM cana-1224 166 4 sin sin NOUN cana-1224 166 5 ๐‘ฅ ๐‘ฅ PRON cana-1224 166 6 sin sin NOUN cana-1224 166 7 ๐‘ฆ ๐‘ฆ NOUN cana-1224 166 8 , , PUNCT cana-1224 166 9 ๐‘…2(๐‘ค1โƒ—โƒ— ๐‘…2(๐‘ค1โƒ—โƒ— PROPN cana-1224 166 10 โƒ—โƒ— โƒ—โƒ— PROPN cana-1224 166 11 ( ( PUNCT cana-1224 166 12 ๐‘ฅ ๐‘ฅ PROPN cana-1224 166 13 , , PUNCT cana-1224 166 14 ๐‘ฆ ๐‘ฆ NOUN cana-1224 166 15 , , PUNCT cana-1224 166 16 ๐‘ก ๐‘ก NOUN cana-1224 166 17 ) ) PUNCT cana-1224 166 18 ) ) PUNCT cana-1224 167 1 = = PUNCT cana-1224 167 2 sin sin NOUN cana-1224 167 3 ๐‘ฅ ๐‘ฅ PRON cana-1224 167 4 sin sin NOUN cana-1224 167 5 ๐‘ฆ ๐‘ฆ NOUN cana-1224 167 6 ( ( PUNCT cana-1224 167 7 โˆ’ โˆ’ PROPN cana-1224 167 8 ๐‘– ๐‘– SYM cana-1224 167 9 2 2 NUM cana-1224 167 10 ๐‘ฃ๐›ผ+2 ๐‘ฃ๐›ผ+2 NUM cana-1224 167 11 + + NUM cana-1224 167 12 1 1 NUM cana-1224 167 13 4 4 NUM cana-1224 167 14 ๐‘ฃ๐›ผ+3 ๐‘ฃ๐›ผ+3 NOUN cana-1224 167 15 ) ) PUNCT cana-1224 167 16 , , PUNCT cana-1224 167 17 ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— ๐‘…3(๐‘ค2โƒ—โƒ—โƒ—โƒ— X cana-1224 167 18 โƒ—(๐‘ฅ โƒ—(๐‘ฅ NUM cana-1224 167 19 , , PUNCT cana-1224 167 20 ๐‘ฆ ๐‘ฆ NOUN cana-1224 167 21 , , PUNCT cana-1224 167 22 ๐‘ก ๐‘ก NOUN cana-1224 167 23 ) ) PUNCT cana-1224 167 24 ) ) PUNCT cana-1224 168 1 = = PUNCT cana-1224 168 2 sin sin NOUN cana-1224 168 3 ๐‘ฅ ๐‘ฅ DET cana-1224 168 4 sin sin NOUN cana-1224 168 5 ๐‘ฆ ๐‘ฆ NOUN cana-1224 168 6 { { PUNCT cana-1224 168 7 โˆ’ โˆ’ PROPN cana-1224 168 8 1 1 NUM cana-1224 168 9 4 4 NUM cana-1224 168 10 ๐‘ฃ๐›ผ+3 ๐‘ฃ๐›ผ+3 NOUN cana-1224 168 11 โˆ’ โˆ’ NOUN cana-1224 168 12 ๐‘– ๐‘– SYM cana-1224 168 13 8 8 NUM cana-1224 168 14 ๐‘ฃ๐›ผ+4 ๐‘ฃ๐›ผ+4 NUM cana-1224 168 15 } } PUNCT cana-1224 168 16 , , PUNCT cana-1224 168 17 โ‹ฎ โ‹ฎ NOUN cana-1224 168 18 therefore therefore ADV cana-1224 168 19 , , PUNCT cana-1224 168 20 ๐‘ค1(๐‘ฅ ๐‘ค1(๐‘ฅ PROPN cana-1224 168 21 , , PUNCT cana-1224 168 22 ๐‘ฆ ๐‘ฆ NOUN cana-1224 168 23 , , PUNCT cana-1224 168 24 ๐‘ง ๐‘ง NOUN cana-1224 168 25 , , PUNCT cana-1224 168 26 ๐‘ก ๐‘ก NOUN cana-1224 168 27 ) ) PUNCT cana-1224 168 28 = = SYM cana-1224 169 1 โˆ’๐‘– โˆ’๐‘– ADJ cana-1224 169 2 ( ( PUNCT cana-1224 169 3 ๐‘ก ๐‘ก PROPN cana-1224 169 4 2โ„ 2โ„ NUM cana-1224 169 5 ) ) PUNCT cana-1224 169 6 ๐›ผ ๐›ผ PRON cana-1224 169 7 ๐›ผ ๐›ผ NOUN cana-1224 169 8 ! ! PUNCT cana-1224 169 9 sin sin NOUN cana-1224 169 10 ๐‘ฅ ๐‘ฅ DET cana-1224 169 11 sin sin NOUN cana-1224 169 12 ๐‘ฆ ๐‘ฆ NOUN cana-1224 169 13 , , PUNCT cana-1224 169 14 ๐‘ค2(๐‘ฅ ๐‘ค2(๐‘ฅ NOUN cana-1224 169 15 , , PUNCT cana-1224 169 16 ๐‘ฆ ๐‘ฆ NOUN cana-1224 169 17 , , PUNCT cana-1224 169 18 ๐‘ง ๐‘ง NOUN cana-1224 169 19 , , PUNCT cana-1224 169 20 ๐‘ก ๐‘ก NOUN cana-1224 169 21 ) ) PUNCT cana-1224 169 22 = = SYM cana-1224 169 23 ๐‘–2 ๐‘–2 PROPN cana-1224 169 24 ( ( PUNCT cana-1224 169 25 ๐‘ก ๐‘ก PROPN cana-1224 169 26 2โ„ 2โ„ NUM cana-1224 169 27 ) ) PUNCT cana-1224 169 28 ๐›ผ+1 ๐›ผ+1 NUM cana-1224 169 29 ( ( PUNCT cana-1224 169 30 ๐›ผ ๐›ผ X cana-1224 169 31 + + NOUN cana-1224 169 32 1 1 NUM cana-1224 169 33 ) ) PUNCT cana-1224 169 34 ! ! PUNCT cana-1224 170 1 sin sin VERB cana-1224 170 2 ๐‘ฅ ๐‘ฅ DET cana-1224 170 3 sin sin NOUN cana-1224 170 4 ๐‘ฆ ๐‘ฆ NOUN cana-1224 170 5 , , PUNCT cana-1224 170 6 ๐‘ค3(๐‘ฅ ๐‘ค3(๐‘ฅ NOUN cana-1224 170 7 , , PUNCT cana-1224 170 8 ๐‘ฆ ๐‘ฆ NOUN cana-1224 170 9 , , PUNCT cana-1224 170 10 ๐‘ง ๐‘ง NOUN cana-1224 170 11 , , PUNCT cana-1224 170 12 ๐‘ก ๐‘ก NOUN cana-1224 170 13 ) ) PUNCT cana-1224 170 14 = = SYM cana-1224 170 15 โˆ’๐‘–3 โˆ’๐‘–3 NOUN cana-1224 170 16 ( ( PUNCT cana-1224 170 17 ๐‘ก ๐‘ก PROPN cana-1224 170 18 2โ„ 2โ„ NUM cana-1224 170 19 ) ) PUNCT cana-1224 170 20 ๐›ผ+2 ๐›ผ+2 NUM cana-1224 170 21 ( ( PUNCT cana-1224 170 22 ๐›ผ ๐›ผ NOUN cana-1224 170 23 + + NOUN cana-1224 170 24 2 2 NUM cana-1224 170 25 ) ) PUNCT cana-1224 170 26 ! ! PUNCT cana-1224 171 1 sin sin VERB cana-1224 171 2 ๐‘ฅ ๐‘ฅ PRON cana-1224 171 3 sin sin NOUN cana-1224 171 4 ๐‘ฆ ๐‘ฆ NOUN cana-1224 171 5 , , PUNCT cana-1224 171 6 โ‹ฎ โ‹ฎ NOUN cana-1224 171 7 for for ADP cana-1224 171 8 ๐›ผ ๐›ผ NOUN cana-1224 171 9 = = SYM cana-1224 171 10 1 1 NUM cana-1224 171 11 , , PUNCT cana-1224 171 12 the the DET cana-1224 171 13 solution solution NOUN cana-1224 171 14 โ€ " PUNCT cana-1224 171 15 is be AUX cana-1224 171 16 : : PUNCT cana-1224 171 17 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-1224 171 18 , , PUNCT cana-1224 171 19 ๐‘ฆ ๐‘ฆ NOUN cana-1224 171 20 , , PUNCT cana-1224 171 21 ๐‘ง ๐‘ง NOUN cana-1224 171 22 , , PUNCT cana-1224 171 23 ๐‘ก ๐‘ก NOUN cana-1224 171 24 ) ) PUNCT cana-1224 171 25 = = SYM cana-1224 171 26 ๐‘ค0 ๐‘ค0 PROPN cana-1224 171 27 + + CCONJ cana-1224 171 28 ๐‘ค1 ๐‘ค1 VERB cana-1224 171 29 + + CCONJ cana-1224 171 30 ๐‘ค2 ๐‘ค2 NOUN cana-1224 171 31 + + CCONJ cana-1224 171 32 ๐‘ค3 ๐‘ค3 PROPN cana-1224 171 33 + + CCONJ cana-1224 171 34 โ‹ฏ โ‹ฏ NOUN cana-1224 171 35 or or CCONJ cana-1224 171 36 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ X cana-1224 171 37 , , PUNCT cana-1224 171 38 ๐‘ฆ ๐‘ฆ NOUN cana-1224 171 39 , , PUNCT cana-1224 171 40 ๐‘ง ๐‘ง NOUN cana-1224 171 41 , , PUNCT cana-1224 171 42 ๐‘ก ๐‘ก NOUN cana-1224 171 43 ) ) PUNCT cana-1224 171 44 = = PUNCT cana-1224 171 45 sin sin NOUN cana-1224 171 46 ๐‘ฅ ๐‘ฅ DET cana-1224 171 47 sin sin NOUN cana-1224 171 48 ๐‘ฆ ๐‘ฆ NOUN cana-1224 171 49 { { PUNCT cana-1224 171 50 1 1 NUM cana-1224 171 51 + + CCONJ cana-1224 171 52 ( ( PUNCT cana-1224 171 53 ๐‘–๐‘ก ๐‘–๐‘ก INTJ cana-1224 171 54 2 2 NUM cana-1224 171 55 ) ) PUNCT cana-1224 172 1 + + CCONJ cana-1224 172 2 ( ( PUNCT cana-1224 172 3 ๐‘–๐‘ก ๐‘–๐‘ก NOUN cana-1224 172 4 2 2 NUM cana-1224 172 5 ) ) PUNCT cana-1224 172 6 2 2 NUM cana-1224 172 7 2 2 NUM cana-1224 172 8 ! ! PUNCT cana-1224 173 1 + + PUNCT cana-1224 173 2 โ‹ฏ โ‹ฏ NOUN cana-1224 173 3 } } PUNCT cana-1224 173 4 = = NOUN cana-1224 173 5 ๐‘’โˆ’๐‘–๐‘ก/2 ๐‘’โˆ’๐‘–๐‘ก/2 PUNCT cana-1224 173 6 sin sin NOUN cana-1224 173 7 ๐‘ฅ ๐‘ฅ DET cana-1224 173 8 sin sin NOUN cana-1224 173 9 ๐‘ฆ ๐‘ฆ PRON cana-1224 173 10 which which PRON cana-1224 173 11 is be AUX cana-1224 173 12 totally totally ADV cana-1224 173 13 equal equal ADJ cana-1224 173 14 to to ADP cana-1224 173 15 the the DET cana-1224 173 16 exact exact ADJ cana-1224 173 17 solution solution NOUN cana-1224 173 18 . . PUNCT cana-1224 174 1 communications communication NOUN cana-1224 174 2 on on ADP cana-1224 174 3 applied apply VERB cana-1224 174 4 nonlinear nonlinear ADJ cana-1224 174 5 analysis analysis NOUN cana-1224 174 6 issn issn NOUN cana-1224 174 7 : : PUNCT cana-1224 174 8 1074 1074 NUM cana-1224 174 9 - - PUNCT cana-1224 174 10 133x 133x NUM cana-1224 174 11 vol vol NOUN cana-1224 174 12 31 31 NUM cana-1224 174 13 no no NOUN cana-1224 174 14 . . PUNCT cana-1224 175 1 6s 6s NUM cana-1224 175 2 ( ( PUNCT cana-1224 175 3 2024 2024 NUM cana-1224 175 4 ) ) PUNCT cana-1224 175 5 315 315 NUM cana-1224 175 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 175 7 figure figure NOUN cana-1224 175 8 5 5 NUM cana-1224 175 9 : : PUNCT cana-1224 175 10 physical physical ADJ cana-1224 175 11 behavior behavior NOUN cana-1224 175 12 of of ADP cana-1224 175 13 real real ADJ cana-1224 175 14 part part NOUN cana-1224 175 15 solution solution NOUN cana-1224 175 16 at at ADP cana-1224 175 17 ๐‘ก ๐‘ก PROPN cana-1224 175 18 = = SYM cana-1224 175 19 0.5 0.5 NUM cana-1224 175 20 figure figure NOUN cana-1224 175 21 6 6 NUM cana-1224 175 22 : : PUNCT cana-1224 175 23 physical physical ADJ cana-1224 175 24 behavior behavior NOUN cana-1224 175 25 of of ADP cana-1224 175 26 imaginary imaginary ADJ cana-1224 175 27 part part NOUN cana-1224 175 28 solution solution NOUN cana-1224 175 29 at at ADP cana-1224 175 30 ๐‘ก ๐‘ก PROPN cana-1224 175 31 = = SYM cana-1224 175 32 0.5 0.5 NUM cana-1224 175 33 figure figure NOUN cana-1224 175 34 7 7 NUM cana-1224 175 35 : : PUNCT cana-1224 175 36 physical physical ADJ cana-1224 175 37 behavior behavior NOUN cana-1224 175 38 of of ADP cana-1224 175 39 the the DET cana-1224 175 40 solution solution NOUN cana-1224 175 41 of of ADP cana-1224 175 42 real real ADJ cana-1224 175 43 part part NOUN cana-1224 175 44 at at ADP cana-1224 175 45 ๐‘ก ๐‘ก NOUN cana-1224 175 46 = = SYM cana-1224 175 47 2 2 NUM cana-1224 175 48 -5 -5 NOUN cana-1224 175 49 0 0 NUM cana-1224 175 50 5 5 NUM cana-1224 175 51 -5 -5 NOUN cana-1224 175 52 0 0 NUM cana-1224 175 53 5 5 NUM cana-1224 175 54 -1 -1 SYM cana-1224 175 55 -0.5 -0.5 X cana-1224 175 56 0 0 NUM cana-1224 175 57 0.5 0.5 NUM cana-1224 175 58 1 1 NUM cana-1224 175 59 x x SYM cana-1224 175 60 w w NOUN cana-1224 175 61 ( ( PUNCT cana-1224 175 62 real real ADJ cana-1224 175 63 part part NOUN cana-1224 175 64 ) ) PUNCT cana-1224 176 1 y y PROPN cana-1224 176 2 w w PROPN cana-1224 176 3 ( ( PUNCT cana-1224 176 4 r r NOUN cana-1224 176 5 ea ea NUM cana-1224 176 6 l l NOUN cana-1224 176 7 p p PROPN cana-1224 176 8 ar ar PROPN cana-1224 176 9 t t PROPN cana-1224 176 10 ) ) PUNCT cana-1224 176 11 -5 -5 PUNCT cana-1224 176 12 0 0 NUM cana-1224 176 13 5 5 NUM cana-1224 176 14 -5 -5 NOUN cana-1224 176 15 0 0 NUM cana-1224 176 16 5 5 NUM cana-1224 176 17 -0.4 -0.4 NUM cana-1224 176 18 -0.2 -0.2 PROPN cana-1224 176 19 0 0 NUM cana-1224 176 20 0.2 0.2 NUM cana-1224 176 21 0.4 0.4 NUM cana-1224 176 22 x x SYM cana-1224 176 23 w w ADJ cana-1224 176 24 ( ( PUNCT cana-1224 176 25 imaginary imaginary ADJ cana-1224 176 26 part part NOUN cana-1224 176 27 ) ) PUNCT cana-1224 177 1 y y PROPN cana-1224 177 2 w w PROPN cana-1224 177 3 ( ( PUNCT cana-1224 177 4 i i PRON cana-1224 177 5 m m VERB cana-1224 177 6 ag ag PROPN cana-1224 177 7 in in ADP cana-1224 177 8 ar ar PROPN cana-1224 177 9 y y PROPN cana-1224 177 10 pa pa PROPN cana-1224 177 11 rt rt PROPN cana-1224 177 12 ) ) PUNCT cana-1224 177 13 -2 -2 INTJ cana-1224 178 1 -1 -1 SYM cana-1224 178 2 0 0 NUM cana-1224 178 3 1 1 NUM cana-1224 178 4 2 2 NUM cana-1224 178 5 - - SYM cana-1224 178 6 2 2 NUM cana-1224 178 7 0 0 NUM cana-1224 178 8 2 2 NUM cana-1224 178 9 -0.8 -0.8 PROPN cana-1224 178 10 -0.6 -0.6 X cana-1224 178 11 -0.4 -0.4 X cana-1224 178 12 -0.2 -0.2 PROPN cana-1224 178 13 0 0 NUM cana-1224 179 1 0.2 0.2 NUM cana-1224 179 2 0.4 0.4 NUM cana-1224 179 3 0.6 0.6 NUM cana-1224 179 4 x x SYM cana-1224 179 5 w w NOUN cana-1224 179 6 ( ( PUNCT cana-1224 179 7 real real ADJ cana-1224 179 8 part part NOUN cana-1224 179 9 ) ) PUNCT cana-1224 180 1 y y PROPN cana-1224 180 2 w w PROPN cana-1224 180 3 ( ( PUNCT cana-1224 180 4 re re PROPN cana-1224 180 5 al al PROPN cana-1224 180 6 p p PROPN cana-1224 180 7 ar ar PROPN cana-1224 180 8 t t PROPN cana-1224 180 9 ) ) PUNCT cana-1224 180 10 communications communication NOUN cana-1224 180 11 on on ADP cana-1224 180 12 applied apply VERB cana-1224 180 13 nonlinear nonlinear ADJ cana-1224 180 14 analysis analysis NOUN cana-1224 180 15 issn issn NOUN cana-1224 180 16 : : PUNCT cana-1224 180 17 1074 1074 NUM cana-1224 180 18 - - PUNCT cana-1224 180 19 133x 133x NUM cana-1224 180 20 vol vol NOUN cana-1224 180 21 31 31 NUM cana-1224 180 22 no no NOUN cana-1224 180 23 . . PUNCT cana-1224 181 1 6s 6s NUM cana-1224 181 2 ( ( PUNCT cana-1224 181 3 2024 2024 NUM cana-1224 181 4 ) ) PUNCT cana-1224 181 5 316 316 NUM cana-1224 181 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 181 7 figure figure NOUN cana-1224 181 8 8 8 NUM cana-1224 181 9 : : PUNCT cana-1224 181 10 physical physical ADJ cana-1224 181 11 behavior behavior NOUN cana-1224 181 12 of of ADP cana-1224 181 13 the the DET cana-1224 181 14 solution solution NOUN cana-1224 181 15 of of ADP cana-1224 181 16 imaginary imaginary ADJ cana-1224 181 17 part part NOUN cana-1224 181 18 at at ADP cana-1224 181 19 ๐‘ก ๐‘ก PROPN cana-1224 181 20 = = SYM cana-1224 181 21 2 2 NUM cana-1224 181 22 figures figure NOUN cana-1224 181 23 5 5 NUM cana-1224 181 24 & & CCONJ cana-1224 181 25 6 6 NUM cana-1224 181 26 show show VERB cana-1224 181 27 the the DET cana-1224 181 28 real real ADJ cana-1224 181 29 & & CCONJ cana-1224 181 30 imaginary imaginary ADJ cana-1224 181 31 part part NOUN cana-1224 181 32 solutions solution NOUN cana-1224 181 33 physical physical ADJ cana-1224 181 34 behavior behavior NOUN cana-1224 181 35 of of ADP cana-1224 181 36 example example NOUN cana-1224 181 37 2 2 NUM cana-1224 181 38 at at ADP cana-1224 181 39 ๐‘ก ๐‘ก NOUN cana-1224 181 40 = = SYM cana-1224 181 41 0.5 0.5 NUM cana-1224 181 42 respectively respectively ADV cana-1224 181 43 . . PUNCT cana-1224 182 1 figures figure NOUN cana-1224 182 2 7 7 NUM cana-1224 182 3 & & CCONJ cana-1224 182 4 8 8 NUM cana-1224 182 5 show show VERB cana-1224 182 6 the the DET cana-1224 182 7 real real ADJ cana-1224 182 8 & & CCONJ cana-1224 182 9 imaginary imaginary ADJ cana-1224 182 10 part part NOUN cana-1224 182 11 solutions solution NOUN cana-1224 182 12 physical physical ADJ cana-1224 182 13 behavior behavior NOUN cana-1224 182 14 of of ADP cana-1224 182 15 example example NOUN cana-1224 182 16 2 2 NUM cana-1224 182 17 at at ADP cana-1224 182 18 ๐‘ก ๐‘ก NOUN cana-1224 182 19 = = NOUN cana-1224 182 20 2 2 NUM cana-1224 182 21 respectively respectively ADV cana-1224 182 22 . . PUNCT cana-1224 183 1 6 6 X cana-1224 183 2 . . X cana-1224 183 3 conclusion conclusion NOUN cana-1224 183 4 the the DET cana-1224 183 5 numerical numerical ADJ cana-1224 183 6 data data PROPN cana-1224 183 7 suggests suggest VERB cana-1224 183 8 that that SCONJ cana-1224 183 9 the the DET cana-1224 183 10 elzaki elzaki NOUN cana-1224 183 11 transform transform VERB cana-1224 183 12 - - PUNCT cana-1224 183 13 based base VERB cana-1224 183 14 homotopy homotopy NOUN cana-1224 183 15 analysis analysis NOUN cana-1224 183 16 technique technique NOUN cana-1224 183 17 provides provide VERB cana-1224 183 18 accurate accurate ADJ cana-1224 183 19 solutions solution NOUN cana-1224 183 20 for for ADP cana-1224 183 21 solving solve VERB cana-1224 183 22 ( ( PUNCT cana-1224 183 23 2 2 NUM cana-1224 183 24 + + NOUN cana-1224 183 25 1)-d 1)-d NUM cana-1224 183 26 and and CCONJ cana-1224 183 27 ( ( PUNCT cana-1224 184 1 3 3 NUM cana-1224 184 2 + + SYM cana-1224 184 3 1)-d 1)-d NUM cana-1224 184 4 nonlinear nonlinear ADJ cana-1224 184 5 fractional fractional ADJ cana-1224 184 6 schrodinger schrodinger PROPN cana-1224 184 7 eqns eqns PROPN cana-1224 184 8 . . PUNCT cana-1224 185 1 in in ADP cana-1224 185 2 the the DET cana-1224 185 3 future future NOUN cana-1224 185 4 , , PUNCT cana-1224 185 5 this this DET cana-1224 185 6 approach approach NOUN cana-1224 185 7 will will AUX cana-1224 185 8 be be AUX cana-1224 185 9 valid valid ADJ cana-1224 185 10 for for ADP cana-1224 185 11 various various ADJ cana-1224 185 12 applications application NOUN cana-1224 185 13 of of ADP cana-1224 185 14 sciences science NOUN cana-1224 185 15 and and CCONJ cana-1224 185 16 engineering engineering NOUN cana-1224 185 17 . . PUNCT cana-1224 186 1 references reference NOUN cana-1224 186 2 [ [ X cana-1224 186 3 1 1 NUM cana-1224 186 4 ] ] X cana-1224 186 5 elzaki elzaki NOUN cana-1224 186 6 , , PUNCT cana-1224 186 7 t.m t.m PROPN cana-1224 186 8 . . PUNCT cana-1224 187 1 the the DET cana-1224 187 2 new new ADJ cana-1224 187 3 integral integral ADJ cana-1224 187 4 transform transform NOUN cana-1224 187 5 โ€œ " PUNCT cana-1224 187 6 elzaki elzaki NOUN cana-1224 187 7 transform transform NOUN cana-1224 187 8 โ€ " PUNCT cana-1224 187 9 global global ADJ cana-1224 187 10 journal journal NOUN cana-1224 187 11 of of ADP cana-1224 187 12 pure pure ADJ cana-1224 187 13 and and CCONJ cana-1224 187 14 applied applied ADJ cana-1224 187 15 mathematics mathematic NOUN cana-1224 187 16 . . PUNCT cana-1224 188 1 2011 2011 NUM cana-1224 188 2 , , PUNCT cana-1224 188 3 1 1 NUM cana-1224 188 4 : : SYM cana-1224 188 5 57 57 NUM cana-1224 188 6 - - SYM cana-1224 188 7 64 64 NUM cana-1224 188 8 . . PUNCT cana-1224 189 1 [ [ X cana-1224 189 2 2 2 NUM cana-1224 189 3 ] ] X cana-1224 189 4 elzaki elzaki NOUN cana-1224 189 5 , , PUNCT cana-1224 189 6 t.m t.m PROPN cana-1224 189 7 . . PROPN cana-1224 189 8 and and CCONJ cana-1224 189 9 elzaki elzaki PROPN cana-1224 189 10 , , PUNCT cana-1224 189 11 s. s. PROPN cana-1224 189 12 m. m. PROPN cana-1224 189 13 application application NOUN cana-1224 189 14 of of ADP cana-1224 189 15 new new ADJ cana-1224 189 16 transform transform NOUN cana-1224 189 17 โ€œ " PUNCT cana-1224 189 18 elzaki elzaki NOUN cana-1224 189 19 transform transform NOUN cana-1224 189 20 โ€ " PUNCT cana-1224 189 21 to to ADP cana-1224 189 22 partial partial ADJ cana-1224 189 23 differential differential NOUN cana-1224 189 24 equations equation NOUN cana-1224 189 25 , , PUNCT cana-1224 189 26 global global ADJ cana-1224 189 27 journal journal NOUN cana-1224 189 28 of of ADP cana-1224 189 29 pure pure ADJ cana-1224 189 30 and and CCONJ cana-1224 189 31 applied applied ADJ cana-1224 189 32 mathematics mathematic NOUN cana-1224 189 33 . . PUNCT cana-1224 190 1 2011 2011 NUM cana-1224 190 2 , , PUNCT cana-1224 190 3 1 1 NUM cana-1224 190 4 : : SYM cana-1224 190 5 65 65 NUM cana-1224 190 6 - - SYM cana-1224 190 7 70 70 NUM cana-1224 190 8 . . PUNCT cana-1224 191 1 [ [ X cana-1224 191 2 3 3 NUM cana-1224 191 3 ] ] X cana-1224 191 4 elzaki elzaki NOUN cana-1224 191 5 , , PUNCT cana-1224 191 6 t.m t.m PROPN cana-1224 191 7 . . PROPN cana-1224 191 8 and and CCONJ cana-1224 191 9 elzaki elzaki PROPN cana-1224 191 10 , , PUNCT cana-1224 191 11 s. s. PROPN cana-1224 191 12 m. m. PROPN cana-1224 191 13 on on ADP cana-1224 191 14 the the DET cana-1224 191 15 connections connection NOUN cana-1224 191 16 between between ADP cana-1224 191 17 laplace laplace NOUN cana-1224 191 18 and and CCONJ cana-1224 191 19 elzaki elzaki NOUN cana-1224 191 20 transforms transform VERB cana-1224 191 21 , , PUNCT cana-1224 191 22 advances advance NOUN cana-1224 191 23 in in ADP cana-1224 191 24 theoretical theoretical ADJ cana-1224 191 25 and and CCONJ cana-1224 191 26 applied applied ADJ cana-1224 191 27 mathematics mathematic NOUN cana-1224 191 28 . . PUNCT cana-1224 192 1 2011 2011 NUM cana-1224 192 2 , , PUNCT cana-1224 192 3 6(1 6(1 NUM cana-1224 192 4 ): ): PUNCT cana-1224 192 5 1 1 NUM cana-1224 192 6 - - SYM cana-1224 192 7 11 11 NUM cana-1224 192 8 . . PUNCT cana-1224 193 1 [ [ X cana-1224 193 2 4 4 NUM cana-1224 193 3 ] ] X cana-1224 193 4 eltayeb eltayeb PROPN cana-1224 193 5 , , PUNCT cana-1224 193 6 h. h. PROPN cana-1224 193 7 and and CCONJ cana-1224 193 8 kilicman kilicman PROPN cana-1224 193 9 , , PUNCT cana-1224 193 10 a. a. NOUN cana-1224 193 11 a a DET cana-1224 193 12 note note NOUN cana-1224 193 13 on on ADP cana-1224 193 14 the the DET cana-1224 193 15 sumudu sumudu NOUN cana-1224 193 16 transforms transform VERB cana-1224 193 17 and and CCONJ cana-1224 193 18 differential differential ADJ cana-1224 193 19 equations equation NOUN cana-1224 193 20 , , PUNCT cana-1224 193 21 applied apply VERB cana-1224 193 22 mathematical mathematical ADJ cana-1224 193 23 sciences science NOUN cana-1224 193 24 . . PUNCT cana-1224 194 1 2010 2010 NUM cana-1224 194 2 , , PUNCT cana-1224 194 3 4(22 4(22 NUM cana-1224 194 4 ): ): PUNCT cana-1224 194 5 1089 1089 NUM cana-1224 194 6 - - SYM cana-1224 194 7 1098 1098 NUM cana-1224 194 8 [ [ X cana-1224 194 9 5 5 NUM cana-1224 194 10 ] ] PUNCT cana-1224 194 11 kilicman kilicman NOUN cana-1224 194 12 a. a. PROPN cana-1224 194 13 and and CCONJ cana-1224 194 14 eltayeb eltayeb PROPN cana-1224 194 15 , , PUNCT cana-1224 194 16 h. h. PROPN cana-1224 194 17 a a DET cana-1224 194 18 note note NOUN cana-1224 194 19 on on ADP cana-1224 194 20 integral integral ADJ cana-1224 194 21 transform transform NOUN cana-1224 194 22 and and CCONJ cana-1224 194 23 partial partial ADJ cana-1224 194 24 differential differential NOUN cana-1224 194 25 equation equation NOUN cana-1224 194 26 , , PUNCT cana-1224 194 27 applied apply VERB cana-1224 194 28 mathematical mathematical ADJ cana-1224 194 29 sciences science NOUN cana-1224 194 30 . . PUNCT cana-1224 195 1 2010 2010 NUM cana-1224 195 2 , , PUNCT cana-1224 195 3 4(3):109 4(3):109 NUM cana-1224 195 4 - - SYM cana-1224 195 5 118 118 NUM cana-1224 195 6 . . PUNCT cana-1224 196 1 [ [ X cana-1224 196 2 6 6 NUM cana-1224 196 3 ] ] PUNCT cana-1224 196 4 abbasbandy abbasbandy PROPN cana-1224 196 5 , , PUNCT cana-1224 196 6 s. s. PROPN cana-1224 196 7 homotopy homotopy VERB cana-1224 196 8 analysis analysis NOUN cana-1224 196 9 method method NOUN cana-1224 196 10 for for ADP cana-1224 196 11 generalized generalized ADJ cana-1224 196 12 benjamin benjamin NOUN cana-1224 196 13 - - PUNCT cana-1224 196 14 bona bona ADJ cana-1224 196 15 - - PUNCT cana-1224 196 16 mahony mahony NOUN cana-1224 196 17 equation equation NOUN cana-1224 196 18 , , PUNCT cana-1224 196 19 z. z. PROPN cana-1224 196 20 angew angew PROPN cana-1224 196 21 . . PUNCT cana-1224 197 1 math math NOUN cana-1224 197 2 . . PUNCT cana-1224 198 1 phys phy NOUN cana-1224 198 2 . . PUNCT cana-1224 199 1 2008 2008 NUM cana-1224 199 2 , , PUNCT cana-1224 199 3 59 59 NUM cana-1224 199 4 : : SYM cana-1224 199 5 51 51 NUM cana-1224 199 6 - - SYM cana-1224 199 7 62 62 NUM cana-1224 199 8 . . PUNCT cana-1224 200 1 [ [ X cana-1224 200 2 7 7 NUM cana-1224 200 3 ] ] X cana-1224 200 4 abbasbandy abbasbandy PROPN cana-1224 200 5 , , PUNCT cana-1224 200 6 s. s. PROPN cana-1224 200 7 and and CCONJ cana-1224 200 8 zakaria zakaria PROPN cana-1224 200 9 , , PUNCT cana-1224 200 10 f.s f.s PROPN cana-1224 200 11 . . PROPN cana-1224 200 12 soliton soliton NOUN cana-1224 200 13 solutions solution NOUN cana-1224 200 14 for for ADP cana-1224 200 15 the the DET cana-1224 200 16 fifth fifth ADJ cana-1224 200 17 - - PUNCT cana-1224 200 18 order order NOUN cana-1224 200 19 k k PROPN cana-1224 200 20 - - PUNCT cana-1224 200 21 dv dv PROPN cana-1224 200 22 equation equation NOUN cana-1224 200 23 with with ADP cana-1224 200 24 the the DET cana-1224 200 25 homotopy homotopy NOUN cana-1224 200 26 analysis analysis NOUN cana-1224 200 27 method method NOUN cana-1224 200 28 , , PUNCT cana-1224 200 29 nonlinear nonlinear ADJ cana-1224 200 30 dyn dyn NOUN cana-1224 200 31 . . PUNCT cana-1224 201 1 2008 2008 NUM cana-1224 201 2 , , PUNCT cana-1224 201 3 51 51 NUM cana-1224 201 4 : : SYM cana-1224 201 5 83 83 NUM cana-1224 201 6 - - SYM cana-1224 201 7 87 87 NUM cana-1224 201 8 . . PUNCT cana-1224 202 1 [ [ X cana-1224 202 2 8 8 NUM cana-1224 202 3 ] ] X cana-1224 202 4 liao liao PROPN cana-1224 202 5 , , PUNCT cana-1224 202 6 s.j s.j PROPN cana-1224 202 7 . . PROPN cana-1224 202 8 on on ADP cana-1224 202 9 the the DET cana-1224 202 10 homotopy homotopy NOUN cana-1224 202 11 analysis analysis NOUN cana-1224 202 12 method method NOUN cana-1224 202 13 for for ADP cana-1224 202 14 nonlinear nonlinear ADJ cana-1224 202 15 problems problem NOUN cana-1224 202 16 , , PUNCT cana-1224 202 17 appl appl PROPN cana-1224 202 18 . . PROPN cana-1224 202 19 math math PROPN cana-1224 202 20 . . PUNCT cana-1224 203 1 comput comput NOUN cana-1224 203 2 . . PUNCT cana-1224 204 1 2004 2004 NUM cana-1224 204 2 , , PUNCT cana-1224 204 3 147 147 NUM cana-1224 204 4 : : PUNCT cana-1224 205 1 499โ€“513 499โ€“513 NUM cana-1224 205 2 . . PUNCT cana-1224 206 1 [ [ X cana-1224 206 2 9 9 NUM cana-1224 206 3 ] ] X cana-1224 206 4 liao liao PROPN cana-1224 206 5 , , PUNCT cana-1224 206 6 s.j s.j PROPN cana-1224 206 7 . . PROPN cana-1224 206 8 comparison comparison NOUN cana-1224 206 9 between between ADP cana-1224 206 10 the the DET cana-1224 206 11 homotopy homotopy NOUN cana-1224 206 12 analysis analysis NOUN cana-1224 206 13 method method NOUN cana-1224 206 14 and and CCONJ cana-1224 206 15 homotopy homotopy VERB cana-1224 206 16 perturbation perturbation NOUN cana-1224 206 17 method method NOUN cana-1224 206 18 , , PUNCT cana-1224 206 19 appl appl PROPN cana-1224 206 20 . . PROPN cana-1224 206 21 math math NOUN cana-1224 206 22 . . PUNCT cana-1224 207 1 comput comput NOUN cana-1224 207 2 . . PUNCT cana-1224 208 1 2005 2005 NUM cana-1224 208 2 , , PUNCT cana-1224 208 3 169 169 NUM cana-1224 208 4 : : SYM cana-1224 208 5 1186โ€“1194 1186โ€“1194 NUM cana-1224 208 6 . . PUNCT cana-1224 209 1 [ [ X cana-1224 209 2 10 10 NUM cana-1224 209 3 ] ] X cana-1224 209 4 song song NOUN cana-1224 209 5 , , PUNCT cana-1224 209 6 l. l. PROPN cana-1224 209 7 and and CCONJ cana-1224 209 8 zhang zhang PROPN cana-1224 209 9 , , PUNCT cana-1224 209 10 h. h. PROPN cana-1224 209 11 application application NOUN cana-1224 209 12 of of ADP cana-1224 209 13 homotopy homotopy NOUN cana-1224 209 14 analysis analysis NOUN cana-1224 209 15 method method NOUN cana-1224 209 16 to to PART cana-1224 209 17 fractional fractional VERB cana-1224 209 18 kdv kdv NOUN cana-1224 209 19 - - PUNCT cana-1224 209 20 burgers burger NOUN cana-1224 209 21 - - PUNCT cana-1224 209 22 kuramoto kuramoto NOUN cana-1224 209 23 equation equation NOUN cana-1224 209 24 , , PUNCT cana-1224 209 25 physics physics NOUN cana-1224 209 26 letters letter NOUN cana-1224 209 27 a. a. PROPN cana-1224 209 28 2007 2007 NUM cana-1224 209 29 , , PUNCT cana-1224 209 30 367 367 NUM cana-1224 209 31 : : PUNCT cana-1224 209 32 88 88 NUM cana-1224 209 33 - - SYM cana-1224 209 34 94 94 NUM cana-1224 209 35 . . PUNCT cana-1224 210 1 [ [ X cana-1224 210 2 11 11 NUM cana-1224 210 3 ] ] X cana-1224 210 4 ganjiani ganjiani ADJ cana-1224 210 5 , , PUNCT cana-1224 210 6 m. m. NOUN cana-1224 210 7 solution solution NOUN cana-1224 210 8 of of ADP cana-1224 210 9 nonlinear nonlinear ADJ cana-1224 210 10 fractional fractional ADJ cana-1224 210 11 differential differential NOUN cana-1224 210 12 equation equation NOUN cana-1224 210 13 using use VERB cana-1224 210 14 homotopy homotopy NOUN cana-1224 210 15 analysis analysis NOUN cana-1224 210 16 method method NOUN cana-1224 210 17 , , PUNCT cana-1224 210 18 applied apply VERB cana-1224 210 19 mathematical mathematical ADJ cana-1224 210 20 modeling modeling NOUN cana-1224 210 21 . . PUNCT cana-1224 211 1 2010 2010 NUM cana-1224 211 2 , , PUNCT cana-1224 211 3 34 34 NUM cana-1224 211 4 : : SYM cana-1224 211 5 1634 1634 NUM cana-1224 211 6 - - SYM cana-1224 211 7 1641 1641 NUM cana-1224 211 8 . . PUNCT cana-1224 212 1 [ [ X cana-1224 212 2 12 12 NUM cana-1224 212 3 ] ] X cana-1224 212 4 jafari jafari PROPN cana-1224 212 5 , , PUNCT cana-1224 212 6 h. h. PROPN cana-1224 212 7 and and CCONJ cana-1224 212 8 seifi seifi NOUN cana-1224 212 9 , , PUNCT cana-1224 212 10 s. s. PROPN cana-1224 212 11 homotopy homotopy VERB cana-1224 212 12 analysis analysis NOUN cana-1224 212 13 method method NOUN cana-1224 212 14 for for ADP cana-1224 212 15 solving solve VERB cana-1224 212 16 linear linear NOUN cana-1224 212 17 and and CCONJ cana-1224 212 18 nonlinear nonlinear ADJ cana-1224 212 19 fractional fractional ADJ cana-1224 212 20 diffusion diffusion NOUN cana-1224 212 21 - - PUNCT cana-1224 212 22 wave wave NOUN cana-1224 212 23 equation equation NOUN cana-1224 212 24 . . PUNCT cana-1224 213 1 comun comun PROPN cana-1224 213 2 . . PROPN cana-1224 214 1 nonlin nonlin PROPN cana-1224 214 2 . . PUNCT cana-1224 215 1 sci sci PROPN cana-1224 215 2 . . PUNCT cana-1224 216 1 num num PROPN cana-1224 216 2 . . PUNCT cana-1224 217 1 sim sim PROPN cana-1224 217 2 . . PUNCT cana-1224 218 1 2009 2009 NUM cana-1224 218 2 , , PUNCT cana-1224 218 3 14(5 14(5 NUM cana-1224 218 4 ): ): PUNCT cana-1224 218 5 2006 2006 NUM cana-1224 218 6 - - SYM cana-1224 218 7 2012 2012 NUM cana-1224 218 8 . . PUNCT cana-1224 219 1 -2 -2 NOUN cana-1224 219 2 0 0 NUM cana-1224 220 1 2 2 NUM cana-1224 220 2 - - SYM cana-1224 220 3 2 2 NUM cana-1224 220 4 - - PUNCT cana-1224 220 5 1012 1012 NUM cana-1224 220 6 -1 -1 SYM cana-1224 220 7 -0.5 -0.5 X cana-1224 220 8 0 0 NUM cana-1224 220 9 0.5 0.5 NUM cana-1224 220 10 1 1 NUM cana-1224 220 11 x x SYM cana-1224 220 12 w w ADJ cana-1224 220 13 ( ( PUNCT cana-1224 220 14 imaginary imaginary ADJ cana-1224 220 15 part part NOUN cana-1224 220 16 ) ) PUNCT cana-1224 221 1 y y PROPN cana-1224 221 2 w w PROPN cana-1224 221 3 ( ( PUNCT cana-1224 221 4 i i NOUN cana-1224 221 5 m m PROPN cana-1224 221 6 ag ag PROPN cana-1224 221 7 in in ADP cana-1224 221 8 ar ar PROPN cana-1224 221 9 y y PROPN cana-1224 221 10 pa pa PROPN cana-1224 221 11 rt rt PROPN cana-1224 221 12 ) ) PUNCT cana-1224 221 13 communications communication NOUN cana-1224 221 14 on on ADP cana-1224 221 15 applied apply VERB cana-1224 221 16 nonlinear nonlinear ADJ cana-1224 221 17 analysis analysis NOUN cana-1224 221 18 issn issn NOUN cana-1224 221 19 : : PUNCT cana-1224 221 20 1074 1074 NUM cana-1224 221 21 - - PUNCT cana-1224 221 22 133x 133x NUM cana-1224 221 23 vol vol NOUN cana-1224 221 24 31 31 NUM cana-1224 221 25 no no NOUN cana-1224 221 26 . . PUNCT cana-1224 222 1 6s 6s NUM cana-1224 222 2 ( ( PUNCT cana-1224 222 3 2024 2024 NUM cana-1224 222 4 ) ) PUNCT cana-1224 222 5 317 317 NUM cana-1224 222 6 https://internationalpubls.com https://internationalpubls.com X cana-1224 223 1 [ [ X cana-1224 223 2 13 13 NUM cana-1224 223 3 ] ] X cana-1224 223 4 alomari alomari X cana-1224 223 5 , , PUNCT cana-1224 223 6 a.k a.k PROPN cana-1224 223 7 . . PROPN cana-1224 223 8 , , PUNCT cana-1224 223 9 noorani noorani PROPN cana-1224 223 10 , , PUNCT cana-1224 223 11 m.s.m m.s.m PROPN cana-1224 223 12 . . PROPN cana-1224 223 13 and and CCONJ cana-1224 223 14 nazar nazar PROPN cana-1224 223 15 , , PUNCT cana-1224 223 16 r. r. PROPN cana-1224 223 17 explicit explicit ADJ cana-1224 223 18 series series NOUN cana-1224 223 19 solutions solution NOUN cana-1224 223 20 of of ADP cana-1224 223 21 some some DET cana-1224 223 22 linear linear ADJ cana-1224 223 23 and and CCONJ cana-1224 223 24 nonlinear nonlinear ADJ cana-1224 223 25 schrodinger schrodinger PROPN cana-1224 223 26 equations equation NOUN cana-1224 223 27 via via ADP cana-1224 223 28 the the DET cana-1224 223 29 homotopy homotopy NOUN cana-1224 223 30 analysis analysis NOUN cana-1224 223 31 method method NOUN cana-1224 223 32 , , PUNCT cana-1224 223 33 communications communication NOUN cana-1224 223 34 in in ADP cana-1224 223 35 nonlinear nonlinear ADJ cana-1224 223 36 science science NOUN cana-1224 223 37 and and CCONJ cana-1224 223 38 numerical numerical PROPN cana-1224 223 39 simulation simulation PROPN cana-1224 223 40 . . PUNCT cana-1224 224 1 2009 2009 NUM cana-1224 224 2 , , PUNCT cana-1224 224 3 14(4 14(4 NUM cana-1224 224 4 ): ): PUNCT cana-1224 224 5 1196โ€“1207 1196โ€“1207 NUM cana-1224 224 6 . . PUNCT cana-1224 225 1 [ [ X cana-1224 225 2 14 14 NUM cana-1224 225 3 ] ] PUNCT cana-1224 225 4 biazar biazar NOUN cana-1224 225 5 , , PUNCT cana-1224 225 6 j. j. PROPN cana-1224 225 7 and and CCONJ cana-1224 225 8 ghanbari ghanbari PROPN cana-1224 225 9 , , PUNCT cana-1224 225 10 b. b. PROPN cana-1224 225 11 ham ham PROPN cana-1224 225 12 solution solution NOUN cana-1224 225 13 of of ADP cana-1224 225 14 some some DET cana-1224 225 15 initial initial ADJ cana-1224 225 16 value value NOUN cana-1224 225 17 problems problem NOUN cana-1224 225 18 arising arise VERB cana-1224 225 19 in in ADP cana-1224 225 20 heat heat NOUN cana-1224 225 21 radiation radiation NOUN cana-1224 225 22 equations equation NOUN cana-1224 225 23 , , PUNCT cana-1224 225 24 journal journal NOUN cana-1224 225 25 of of ADP cana-1224 225 26 king king PROPN cana-1224 225 27 saud saud PROPN cana-1224 225 28 university university PROPN cana-1224 225 29 - - PUNCT cana-1224 225 30 science science NOUN cana-1224 225 31 . . PUNCT cana-1224 225 32 2012 2012 NUM cana-1224 225 33 , , PUNCT cana-1224 225 34 24(2 24(2 NUM cana-1224 225 35 ): ): PUNCT cana-1224 225 36 161โ€“165 161โ€“165 NUM cana-1224 225 37 . . PUNCT cana-1224 226 1 [ [ X cana-1224 226 2 15 15 NUM cana-1224 226 3 ] ] X cana-1224 226 4 liao liao PROPN cana-1224 226 5 , , PUNCT cana-1224 226 6 s.j s.j PROPN cana-1224 226 7 . . PROPN cana-1224 226 8 beyond beyond ADP cana-1224 226 9 perturbation perturbation NOUN cana-1224 226 10 : : PUNCT cana-1224 226 11 introduction introduction NOUN cana-1224 226 12 to to ADP cana-1224 226 13 the the DET cana-1224 226 14 homotopy homotopy NOUN cana-1224 226 15 analysis analysis NOUN cana-1224 226 16 method method NOUN cana-1224 226 17 , , PUNCT cana-1224 226 18 chapman chapman PROPN cana-1224 226 19 & & CCONJ cana-1224 226 20 hall hall PROPN cana-1224 226 21 , , PUNCT cana-1224 226 22 crc crc PROPN cana-1224 226 23 press press PROPN cana-1224 226 24 , , PUNCT cana-1224 226 25 boca boca PROPN cana-1224 226 26 raton raton PROPN cana-1224 226 27 , , PUNCT cana-1224 226 28 fla fla PROPN cana-1224 226 29 , , PUNCT cana-1224 226 30 usa usa PROPN cana-1224 226 31 , , PUNCT cana-1224 226 32 2003 2003 NUM cana-1224 226 33 . . PUNCT cana-1224 227 1 [ [ X cana-1224 227 2 16 16 NUM cana-1224 227 3 ] ] X cana-1224 227 4 liao liao PROPN cana-1224 227 5 , , PUNCT cana-1224 227 6 s.j s.j PROPN cana-1224 227 7 . . PROPN cana-1224 227 8 notes note NOUN cana-1224 227 9 on on ADP cana-1224 227 10 the the DET cana-1224 227 11 homotopy homotopy NOUN cana-1224 227 12 analysis analysis NOUN cana-1224 227 13 method method NOUN cana-1224 227 14 : : PUNCT cana-1224 227 15 some some DET cana-1224 227 16 definitions definition NOUN cana-1224 227 17 and and CCONJ cana-1224 227 18 theorems theorem NOUN cana-1224 227 19 , , PUNCT cana-1224 227 20 communications communication NOUN cana-1224 227 21 in in ADP cana-1224 227 22 nonlinear nonlinear ADJ cana-1224 227 23 science science NOUN cana-1224 227 24 and and CCONJ cana-1224 227 25 numerical numerical PROPN cana-1224 227 26 simulation simulation PROPN cana-1224 227 27 . . PUNCT cana-1224 227 28 2009 2009 NUM cana-1224 227 29 , , PUNCT cana-1224 227 30 14(4 14(4 NUM cana-1224 227 31 ): ): PUNCT cana-1224 227 32 983โ€“997 983โ€“997 NUM cana-1224 227 33 . . PUNCT cana-1224 228 1 [ [ X cana-1224 228 2 17 17 NUM cana-1224 228 3 ] ] X cana-1224 228 4 liao liao PROPN cana-1224 228 5 , , PUNCT cana-1224 228 6 s.j s.j PROPN cana-1224 228 7 . . PROPN cana-1224 228 8 a a DET cana-1224 228 9 new new ADJ cana-1224 228 10 branch branch NOUN cana-1224 228 11 of of ADP cana-1224 228 12 solutions solution NOUN cana-1224 228 13 of of ADP cana-1224 228 14 boundary boundary ADJ cana-1224 228 15 - - PUNCT cana-1224 228 16 layer layer NOUN cana-1224 228 17 flows flow NOUN cana-1224 228 18 over over ADP cana-1224 228 19 an an DET cana-1224 228 20 impermeable impermeable ADJ cana-1224 228 21 stretched stretch VERB cana-1224 228 22 plate plate NOUN cana-1224 228 23 , , PUNCT cana-1224 228 24 international international ADJ cana-1224 228 25 journal journal NOUN cana-1224 228 26 of of ADP cana-1224 228 27 heat heat NOUN cana-1224 228 28 and and CCONJ cana-1224 228 29 mass mass NOUN cana-1224 228 30 transfer transfer NOUN cana-1224 228 31 . . PUNCT cana-1224 229 1 2005 2005 NUM cana-1224 229 2 , , PUNCT cana-1224 229 3 48(12 48(12 NUM cana-1224 229 4 ): ): PUNCT cana-1224 229 5 2529โ€“2539 2529โ€“2539 NUM cana-1224 229 6 . . PUNCT cana-1224 230 1 [ [ X cana-1224 230 2 18 18 NUM cana-1224 230 3 ] ] X cana-1224 230 4 mohebbi mohebbi PROPN cana-1224 230 5 , , PUNCT cana-1224 230 6 a. a. NOUN cana-1224 230 7 and and CCONJ cana-1224 230 8 dehghan dehghan PROPN cana-1224 230 9 , , PUNCT cana-1224 230 10 m. m. NOUN cana-1224 230 11 the the DET cana-1224 230 12 use use NOUN cana-1224 230 13 of of ADP cana-1224 230 14 compact compact ADJ cana-1224 230 15 boundary boundary ADJ cana-1224 230 16 value value NOUN cana-1224 230 17 method method NOUN cana-1224 230 18 for for ADP cana-1224 230 19 the the DET cana-1224 230 20 solution solution NOUN cana-1224 230 21 of of ADP cana-1224 230 22 two two NUM cana-1224 230 23 - - PUNCT cana-1224 230 24 dimensional dimensional ADJ cana-1224 230 25 schrodinger schrodinger NOUN cana-1224 230 26 equation equation NOUN cana-1224 230 27 , , PUNCT cana-1224 230 28 journal journal NOUN cana-1224 230 29 of of ADP cana-1224 230 30 computational computational ADJ cana-1224 230 31 and and CCONJ cana-1224 230 32 applied applied ADJ cana-1224 230 33 mathematics mathematic NOUN cana-1224 230 34 , , PUNCT cana-1224 230 35 2009 2009 NUM cana-1224 230 36 , , PUNCT cana-1224 230 37 225(1 225(1 NUM cana-1224 230 38 ): ): PUNCT cana-1224 230 39 124โ€“134 124โ€“134 NUM cana-1224 230 40 . . PUNCT cana-1224 231 1 [ [ X cana-1224 231 2 19 19 NUM cana-1224 231 3 ] ] X cana-1224 231 4 khuri khuri PROPN cana-1224 231 5 , , PUNCT cana-1224 231 6 s.a s.a PROPN cana-1224 231 7 . . PROPN cana-1224 231 8 a a DET cana-1224 231 9 new new ADJ cana-1224 231 10 approach approach NOUN cana-1224 231 11 to to ADP cana-1224 231 12 the the DET cana-1224 231 13 cubic cubic ADJ cana-1224 231 14 schrodinger schrodinger PROPN cana-1224 231 15 equation equation NOUN cana-1224 231 16 : : PUNCT cana-1224 231 17 an an DET cana-1224 231 18 application application NOUN cana-1224 231 19 of of ADP cana-1224 231 20 the the DET cana-1224 231 21 decomposition decomposition NOUN cana-1224 231 22 technique technique NOUN cana-1224 231 23 , , PUNCT cana-1224 231 24 applied apply VERB cana-1224 231 25 mathematics mathematic NOUN cana-1224 231 26 and and CCONJ cana-1224 231 27 computation computation NOUN cana-1224 231 28 . . PUNCT cana-1224 232 1 1998 1998 NUM cana-1224 232 2 , , PUNCT cana-1224 232 3 97 97 NUM cana-1224 232 4 : : PUNCT cana-1224 232 5 251โ€“254 251โ€“254 NUM cana-1224 232 6 . . PUNCT