id sid tid token lemma pos cana-2815 1 1 type type NOUN cana-2815 1 2 of of ADP cana-2815 1 3 the the DET cana-2815 1 4 paper paper NOUN cana-2815 1 5 ( ( PUNCT cana-2815 1 6 article article NOUN cana-2815 1 7 communications communication NOUN cana-2815 1 8 on on ADP cana-2815 1 9 applied apply VERB cana-2815 1 10 nonlinear nonlinear ADJ cana-2815 1 11 analysis analysis NOUN cana-2815 1 12 issn issn NOUN cana-2815 1 13 : : PUNCT cana-2815 1 14 1074 1074 NUM cana-2815 1 15 - - PUNCT cana-2815 1 16 133x 133x NUM cana-2815 1 17 vol vol NOUN cana-2815 1 18 32 32 NUM cana-2815 1 19 no no NOUN cana-2815 1 20 . . PUNCT cana-2815 2 1 4s 4s NUM cana-2815 2 2 ( ( PUNCT cana-2815 2 3 2025 2025 NUM cana-2815 2 4 ) ) PUNCT cana-2815 2 5 309 309 NUM cana-2815 2 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 2 7 time time NOUN cana-2815 2 8 - - PUNCT cana-2815 2 9 fractional fractional ADJ cana-2815 2 10 hyperbolic hyperbolic ADJ cana-2815 2 11 telegraph telegraph NOUN cana-2815 2 12 equation equation NOUN cana-2815 2 13 : : PUNCT cana-2815 2 14 a a DET cana-2815 2 15 semi semi ADJ cana-2815 2 16 - - ADJ cana-2815 2 17 analytic analytic ADJ cana-2815 2 18 approach approach NOUN cana-2815 2 19 using use VERB cana-2815 2 20 modified modify VERB cana-2815 2 21 adomian adomian NOUN cana-2815 2 22 decomposition decomposition NOUN cana-2815 2 23 elzaki elzaki NOUN cana-2815 2 24 transform transform VERB cana-2815 2 25 method method NOUN cana-2815 2 26 parmeshwari parmeshwari PROPN cana-2815 2 27 aland aland PROPN cana-2815 2 28 1,2 1,2 NUM cana-2815 2 29 , , PUNCT cana-2815 2 30 prince prince NOUN cana-2815 2 31 singh3 singh3 NOUN cana-2815 2 32 1 1 NUM cana-2815 2 33 research research NOUN cana-2815 2 34 scholar scholar NOUN cana-2815 2 35 , , PUNCT cana-2815 2 36 department department NOUN cana-2815 2 37 of of ADP cana-2815 2 38 mathematics mathematic NOUN cana-2815 2 39 , , PUNCT cana-2815 2 40 lovely lovely ADJ cana-2815 2 41 professional professional ADJ cana-2815 2 42 university university NOUN cana-2815 2 43 , , PUNCT cana-2815 2 44 phagwara phagwara ADJ cana-2815 2 45 , , PUNCT cana-2815 2 46 punjab-144411 punjab-144411 ADJ cana-2815 2 47 , , PUNCT cana-2815 2 48 india india PROPN cana-2815 2 49 ; ; PUNCT cana-2815 2 50 parmeshwarialand20@gmail.com parmeshwarialand20@gmail.com ADP cana-2815 2 51 2 2 NUM cana-2815 2 52 assistant assistant NOUN cana-2815 2 53 professor professor NOUN cana-2815 2 54 , , PUNCT cana-2815 2 55 school school NOUN cana-2815 2 56 of of ADP cana-2815 2 57 engineering engineering NOUN cana-2815 2 58 , , PUNCT cana-2815 2 59 ajeenkya ajeenkya PROPN cana-2815 2 60 dy dy PROPN cana-2815 2 61 patil patil PROPN cana-2815 2 62 university university PROPN cana-2815 2 63 , , PUNCT cana-2815 2 64 lohegaon lohegaon NOUN cana-2815 2 65 , , PUNCT cana-2815 2 66 pune-412105 pune-412105 NOUN cana-2815 2 67 , , PUNCT cana-2815 2 68 india india PROPN cana-2815 2 69 ; ; PUNCT cana-2815 2 70 facultyit415@adypu.edu.in facultyit415@adypu.edu.in NUM cana-2815 2 71 3 3 NUM cana-2815 2 72 department department NOUN cana-2815 2 73 of of ADP cana-2815 2 74 mathematics mathematic NOUN cana-2815 2 75 , , PUNCT cana-2815 2 76 school school NOUN cana-2815 2 77 of of ADP cana-2815 2 78 chemical chemical NOUN cana-2815 2 79 engineering engineering NOUN cana-2815 2 80 and and CCONJ cana-2815 2 81 physical physical ADJ cana-2815 2 82 science science NOUN cana-2815 2 83 . . PUNCT cana-2815 2 84 , , PUNCT cana-2815 2 85 lovely lovely ADJ cana-2815 2 86 professional professional ADJ cana-2815 2 87 university university NOUN cana-2815 2 88 , , PUNCT cana-2815 2 89 phagwara phagwara ADJ cana-2815 2 90 , , PUNCT cana-2815 2 91 punjab-144411 punjab-144411 ADJ cana-2815 2 92 , , PUNCT cana-2815 2 93 india india PROPN cana-2815 2 94 ; ; PUNCT cana-2815 2 95 princesingh16092@gmail.com princesingh16092@gmail.com X cana-2815 2 96 article article NOUN cana-2815 2 97 history history NOUN cana-2815 2 98 : : PUNCT cana-2815 2 99 received receive VERB cana-2815 2 100 : : PUNCT cana-2815 2 101 25 25 NUM cana-2815 2 102 - - PUNCT cana-2815 2 103 09 09 NUM cana-2815 2 104 - - PUNCT cana-2815 2 105 2024 2024 NUM cana-2815 2 106 revised revise VERB cana-2815 2 107 : : PUNCT cana-2815 2 108 27 27 NUM cana-2815 2 109 - - SYM cana-2815 2 110 11 11 NUM cana-2815 2 111 - - PUNCT cana-2815 2 112 2024 2024 NUM cana-2815 2 113 accepted accept VERB cana-2815 2 114 : : PUNCT cana-2815 2 115 07 07 NUM cana-2815 2 116 - - SYM cana-2815 2 117 12 12 NUM cana-2815 2 118 - - PUNCT cana-2815 2 119 2024 2024 NUM cana-2815 2 120 abstract abstract NOUN cana-2815 2 121 : : PUNCT cana-2815 2 122 in in ADP cana-2815 2 123 this this DET cana-2815 2 124 research research NOUN cana-2815 2 125 paper paper NOUN cana-2815 2 126 , , PUNCT cana-2815 2 127 an an DET cana-2815 2 128 approximate approximate ADJ cana-2815 2 129 analytical analytical ADJ cana-2815 2 130 solution solution NOUN cana-2815 2 131 approach approach NOUN cana-2815 2 132 known know VERB cana-2815 2 133 as as ADP cana-2815 2 134 the the DET cana-2815 2 135 modified modify VERB cana-2815 2 136 adomian adomian NOUN cana-2815 2 137 decomposition decomposition NOUN cana-2815 2 138 method method NOUN cana-2815 2 139 with with ADP cana-2815 2 140 the the DET cana-2815 2 141 coupling coupling NOUN cana-2815 2 142 of of ADP cana-2815 2 143 elzaki elzaki NOUN cana-2815 2 144 transform transform NOUN cana-2815 2 145 ( ( PUNCT cana-2815 2 146 madetm madetm X cana-2815 2 147 ) ) PUNCT cana-2815 2 148 is be AUX cana-2815 2 149 deployed deploy VERB cana-2815 2 150 for for ADP cana-2815 2 151 addressing address VERB cana-2815 2 152 one one NUM cana-2815 2 153 - - PUNCT cana-2815 2 154 dimensional dimensional ADJ cana-2815 2 155 , , PUNCT cana-2815 2 156 twodimensional twodimensional ADJ cana-2815 2 157 , , PUNCT cana-2815 2 158 and and CCONJ cana-2815 2 159 three three NUM cana-2815 2 160 - - PUNCT cana-2815 2 161 dimensional dimensional ADJ cana-2815 2 162 time time NOUN cana-2815 2 163 - - PUNCT cana-2815 2 164 fractional fractional ADJ cana-2815 2 165 hyperbolic hyperbolic ADJ cana-2815 2 166 telegraph telegraph NOUN cana-2815 2 167 equations equation NOUN cana-2815 2 168 . . PUNCT cana-2815 3 1 the the DET cana-2815 3 2 caputo caputo PROPN cana-2815 3 3 derivative derivative ADJ cana-2815 3 4 operator operator NOUN cana-2815 3 5 yields yield VERB cana-2815 3 6 the the DET cana-2815 3 7 approximate approximate ADJ cana-2815 3 8 analytical analytical ADJ cana-2815 3 9 solution solution NOUN cana-2815 3 10 . . PUNCT cana-2815 4 1 the the DET cana-2815 4 2 impact impact NOUN cana-2815 4 3 ness ness NOUN cana-2815 4 4 and and CCONJ cana-2815 4 5 its its PRON cana-2815 4 6 accuracy accuracy NOUN cana-2815 4 7 of of ADP cana-2815 4 8 the the DET cana-2815 4 9 adopted adopt VERB cana-2815 4 10 method method NOUN cana-2815 4 11 are be AUX cana-2815 4 12 demonstrated demonstrate VERB cana-2815 4 13 through through ADP cana-2815 4 14 comparison comparison NOUN cana-2815 4 15 of of ADP cana-2815 4 16 the the DET cana-2815 4 17 approximate approximate ADJ cana-2815 4 18 results result NOUN cana-2815 4 19 with with ADP cana-2815 4 20 the the DET cana-2815 4 21 exact exact ADJ cana-2815 4 22 solutions solution NOUN cana-2815 4 23 , , PUNCT cana-2815 4 24 both both PRON cana-2815 4 25 presented present VERB cana-2815 4 26 graphically graphically ADV cana-2815 4 27 by by ADP cana-2815 4 28 plotting plot VERB cana-2815 4 29 its its PRON cana-2815 4 30 surface surface NOUN cana-2815 4 31 graph graph NOUN cana-2815 4 32 , , PUNCT cana-2815 4 33 line line NOUN cana-2815 4 34 graph graph NOUN cana-2815 4 35 through through ADP cana-2815 4 36 analyzing analyze VERB cana-2815 4 37 its its PRON cana-2815 4 38 error error NOUN cana-2815 4 39 . . PUNCT cana-2815 5 1 the the DET cana-2815 5 2 madetm madetm NOUN cana-2815 5 3 proves prove VERB cana-2815 5 4 to to PART cana-2815 5 5 be be AUX cana-2815 5 6 a a DET cana-2815 5 7 reliable reliable ADJ cana-2815 5 8 and and CCONJ cana-2815 5 9 efficient efficient ADJ cana-2815 5 10 tool tool NOUN cana-2815 5 11 for for ADP cana-2815 5 12 deriving derive VERB cana-2815 5 13 approximate approximate ADJ cana-2815 5 14 and and CCONJ cana-2815 5 15 exact exact ADJ cana-2815 5 16 solutions solution NOUN cana-2815 5 17 for for ADP cana-2815 5 18 a a DET cana-2815 5 19 large large ADJ cana-2815 5 20 class class NOUN cana-2815 5 21 of of ADP cana-2815 5 22 partial partial ADJ cana-2815 5 23 differential differential ADJ cana-2815 5 24 equations equation NOUN cana-2815 5 25 ( ( PUNCT cana-2815 5 26 pdes pde NOUN cana-2815 5 27 ) ) PUNCT cana-2815 5 28 , , PUNCT cana-2815 5 29 fractional fractional ADJ cana-2815 5 30 pdes pde NOUN cana-2815 5 31 , , PUNCT cana-2815 5 32 and and CCONJ cana-2815 5 33 ordinary ordinary ADJ cana-2815 5 34 differential differential ADJ cana-2815 5 35 equations equation NOUN cana-2815 5 36 ( ( PUNCT cana-2815 5 37 odes ode NOUN cana-2815 5 38 ) ) PUNCT cana-2815 5 39 . . PUNCT cana-2815 6 1 the the DET cana-2815 6 2 considered consider VERB cana-2815 6 3 method method NOUN cana-2815 6 4 yields yield VERB cana-2815 6 5 a a DET cana-2815 6 6 solution solution NOUN cana-2815 6 7 in in ADP cana-2815 6 8 series series NOUN cana-2815 6 9 form form NOUN cana-2815 6 10 with with ADP cana-2815 6 11 low low ADJ cana-2815 6 12 computational computational ADJ cana-2815 6 13 complexity complexity NOUN cana-2815 6 14 and and CCONJ cana-2815 6 15 swiftly swiftly ADV cana-2815 6 16 converges converge VERB cana-2815 6 17 towards towards ADP cana-2815 6 18 precise precise ADJ cana-2815 6 19 solutions solution NOUN cana-2815 6 20 . . PUNCT cana-2815 7 1 the the DET cana-2815 7 2 outcomes outcome NOUN cana-2815 7 3 showcase showcase VERB cana-2815 7 4 an an DET cana-2815 7 5 effective effective ADJ cana-2815 7 6 and and CCONJ cana-2815 7 7 uncomplicated uncomplicated ADJ cana-2815 7 8 approach approach NOUN cana-2815 7 9 for for ADP cana-2815 7 10 examining examine VERB cana-2815 7 11 issues issue NOUN cana-2815 7 12 across across ADP cana-2815 7 13 diverse diverse ADJ cana-2815 7 14 scientific scientific ADJ cana-2815 7 15 and and CCONJ cana-2815 7 16 technological technological ADJ cana-2815 7 17 domains domain NOUN cana-2815 7 18 . . PUNCT cana-2815 8 1 keywords keyword NOUN cana-2815 8 2 : : PUNCT cana-2815 8 3 modified modified ADJ cana-2815 8 4 adomian adomian NOUN cana-2815 8 5 decomposition decomposition NOUN cana-2815 8 6 technique technique NOUN cana-2815 8 7 , , PUNCT cana-2815 8 8 hyperbolic hyperbolic ADJ cana-2815 8 9 time time NOUN cana-2815 8 10 fractional fractional PROPN cana-2815 8 11 telegraph telegraph NOUN cana-2815 8 12 equations equation NOUN cana-2815 8 13 , , PUNCT cana-2815 8 14 elzaki elzaki AUX cana-2815 8 15 transform transform VERB cana-2815 8 16 operator operator NOUN cana-2815 8 17 communications communication NOUN cana-2815 8 18 on on ADP cana-2815 8 19 applied apply VERB cana-2815 8 20 nonlinear nonlinear ADJ cana-2815 8 21 analysis analysis NOUN cana-2815 8 22 issn issn NOUN cana-2815 8 23 : : PUNCT cana-2815 8 24 1074 1074 NUM cana-2815 8 25 - - PUNCT cana-2815 8 26 133x 133x NUM cana-2815 8 27 vol vol NOUN cana-2815 8 28 32 32 NUM cana-2815 8 29 no no NOUN cana-2815 8 30 . . PUNCT cana-2815 9 1 4s 4s NUM cana-2815 9 2 ( ( PUNCT cana-2815 9 3 2025 2025 NUM cana-2815 9 4 ) ) PUNCT cana-2815 9 5 310 310 NUM cana-2815 9 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 9 7 1 1 NUM cana-2815 9 8 . . PUNCT cana-2815 9 9 introduction introduction NOUN cana-2815 9 10 fractional fractional ADJ cana-2815 9 11 order order NOUN cana-2815 9 12 differential differential ADJ cana-2815 9 13 equations equation NOUN cana-2815 9 14 ( ( PUNCT cana-2815 9 15 fdes fde NOUN cana-2815 9 16 ) ) PUNCT cana-2815 9 17 indeed indeed ADV cana-2815 9 18 have have AUX cana-2815 9 19 gained gain VERB cana-2815 9 20 significance significance NOUN cana-2815 9 21 in in ADP cana-2815 9 22 applied applied ADJ cana-2815 9 23 mathematics mathematic NOUN cana-2815 9 24 , , PUNCT cana-2815 9 25 being be AUX cana-2815 9 26 applied apply VERB cana-2815 9 27 in in ADP cana-2815 9 28 various various ADJ cana-2815 9 29 systems system NOUN cana-2815 9 30 within within ADP cana-2815 9 31 applied applied ADJ cana-2815 9 32 science science NOUN cana-2815 9 33 . . PUNCT cana-2815 10 1 these these DET cana-2815 10 2 equations equation NOUN cana-2815 10 3 provide provide VERB cana-2815 10 4 a a DET cana-2815 10 5 substitute substitute ADJ cana-2815 10 6 approach approach NOUN cana-2815 10 7 to to ADP cana-2815 10 8 non non ADJ cana-2815 10 9 - - ADJ cana-2815 10 10 linear linear ADJ cana-2815 10 11 equations equation NOUN cana-2815 10 12 and and CCONJ cana-2815 10 13 have have AUX cana-2815 10 14 proven prove VERB cana-2815 10 15 essential essential ADJ cana-2815 10 16 in in ADP cana-2815 10 17 mathematical mathematical ADJ cana-2815 10 18 modelling modelling NOUN cana-2815 10 19 in in ADP cana-2815 10 20 fields field NOUN cana-2815 10 21 such such ADJ cana-2815 10 22 as as ADP cana-2815 10 23 mechanics mechanic NOUN cana-2815 10 24 , , PUNCT cana-2815 10 25 process process NOUN cana-2815 10 26 control control NOUN cana-2815 10 27 , , PUNCT cana-2815 10 28 complex complex ADJ cana-2815 10 29 systems system NOUN cana-2815 10 30 , , PUNCT cana-2815 10 31 and and CCONJ cana-2815 10 32 technology technology NOUN cana-2815 10 33 . . PUNCT cana-2815 11 1 integral integral ADJ cana-2815 11 2 equations equation NOUN cana-2815 11 3 work work VERB cana-2815 11 4 as as ADP cana-2815 11 5 crucial crucial ADJ cana-2815 11 6 role role NOUN cana-2815 11 7 in in ADP cana-2815 11 8 efficiently efficiently ADV cana-2815 11 9 elaborate elaborate ADJ cana-2815 11 10 mathematical mathematical ADJ cana-2815 11 11 problems problem NOUN cana-2815 11 12 associated associate VERB cana-2815 11 13 with with ADP cana-2815 11 14 fdes fde NOUN cana-2815 11 15 and and CCONJ cana-2815 11 16 pdes pde NOUN cana-2815 11 17 . . PUNCT cana-2815 12 1 by by ADP cana-2815 12 2 selecting select VERB cana-2815 12 3 appropriate appropriate ADJ cana-2815 12 4 integral integral ADJ cana-2815 12 5 transformations transformation NOUN cana-2815 12 6 , , PUNCT cana-2815 12 7 one one PRON cana-2815 12 8 can can AUX cana-2815 12 9 convert convert VERB cana-2815 12 10 fdes fde NOUN cana-2815 12 11 and and CCONJ cana-2815 12 12 pdes pde NOUN cana-2815 12 13 into into ADP cana-2815 12 14 algebraic algebraic ADJ cana-2815 12 15 equations equation NOUN cana-2815 12 16 , , PUNCT cana-2815 12 17 simplifying simplify VERB cana-2815 12 18 the the DET cana-2815 12 19 problem problem NOUN cana-2815 12 20 - - PUNCT cana-2815 12 21 solving solve VERB cana-2815 12 22 process process NOUN cana-2815 12 23 . . PUNCT cana-2815 13 1 integral integral ADJ cana-2815 13 2 transforms transform NOUN cana-2815 13 3 offer offer VERB cana-2815 13 4 a a DET cana-2815 13 5 convenient convenient ADJ cana-2815 13 6 method method NOUN cana-2815 13 7 to to PART cana-2815 13 8 address address VERB cana-2815 13 9 the the DET cana-2815 13 10 complexity complexity NOUN cana-2815 13 11 of of ADP cana-2815 13 12 various various ADJ cana-2815 13 13 types type NOUN cana-2815 13 14 of of ADP cana-2815 13 15 differential differential ADJ cana-2815 13 16 equations equation NOUN cana-2815 13 17 . . PUNCT cana-2815 14 1 the the DET cana-2815 14 2 development development NOUN cana-2815 14 3 and and CCONJ cana-2815 14 4 implementation implementation NOUN cana-2815 14 5 of of ADP cana-2815 14 6 integral integral ADJ cana-2815 14 7 transforms transform NOUN cana-2815 14 8 , , PUNCT cana-2815 14 9 like like ADP cana-2815 14 10 laplace laplace NOUN cana-2815 14 11 transform transform NOUN cana-2815 14 12 , , PUNCT cana-2815 14 13 elzaki elzaki NOUN cana-2815 14 14 transformation transformation NOUN cana-2815 14 15 , , PUNCT cana-2815 14 16 elzaki elzaki NOUN cana-2815 14 17 - - PUNCT cana-2815 14 18 laplace laplace NOUN cana-2815 14 19 transform transform NOUN cana-2815 14 20 , , PUNCT cana-2815 14 21 shehu shehu NOUN cana-2815 14 22 transform transform NOUN cana-2815 14 23 , , PUNCT cana-2815 14 24 and and CCONJ cana-2815 14 25 sawi sawi ADJ cana-2815 14 26 transform transform NOUN cana-2815 14 27 , , PUNCT cana-2815 14 28 natural natural ADJ cana-2815 14 29 transforms transform NOUN cana-2815 14 30 have have AUX cana-2815 14 31 been be AUX cana-2815 14 32 instrumental instrumental ADJ cana-2815 14 33 in in ADP cana-2815 14 34 advancing advance VERB cana-2815 14 35 research research NOUN cana-2815 14 36 in in ADP cana-2815 14 37 this this DET cana-2815 14 38 area area NOUN cana-2815 15 1 [ [ X cana-2815 15 2 [ [ X cana-2815 15 3 1],[2],[3],[4],[5],[9 1],[2],[3],[4],[5],[9 X cana-2815 15 4 ] ] X cana-2815 15 5 ] ] X cana-2815 15 6 moreover moreover ADV cana-2815 15 7 , , PUNCT cana-2815 15 8 research research NOUN cana-2815 15 9 efforts effort NOUN cana-2815 15 10 over over ADP cana-2815 15 11 the the DET cana-2815 15 12 past past ADJ cana-2815 15 13 few few ADJ cana-2815 15 14 decades decade NOUN cana-2815 15 15 have have AUX cana-2815 15 16 extensively extensively ADV cana-2815 15 17 explored explore VERB cana-2815 15 18 the the DET cana-2815 15 19 application application NOUN cana-2815 15 20 of of ADP cana-2815 15 21 integral integral ADJ cana-2815 15 22 transformations transformation NOUN cana-2815 15 23 to to PART cana-2815 15 24 solve solve VERB cana-2815 15 25 fractional fractional ADJ cana-2815 15 26 and and CCONJ cana-2815 15 27 fdes fde NOUN cana-2815 15 28 . . PUNCT cana-2815 16 1 these these DET cana-2815 16 2 studies study NOUN cana-2815 16 3 have have AUX cana-2815 16 4 involved involve VERB cana-2815 16 5 various various ADJ cana-2815 16 6 operators operator NOUN cana-2815 16 7 like like ADP cana-2815 16 8 caputo caputo PROPN cana-2815 16 9 , , PUNCT cana-2815 16 10 atanga atanga ADV cana-2815 16 11 baleanu baleanu NOUN cana-2815 16 12 , , PUNCT cana-2815 16 13 erdelyi erdelyi NOUN cana-2815 16 14 - - PUNCT cana-2815 16 15 kober kober PROPN cana-2815 16 16 , , PUNCT cana-2815 16 17 hepolynomials hepolynomial NOUN cana-2815 16 18 and and CCONJ cana-2815 16 19 grunwald grunwald NOUN cana-2815 16 20 - - PUNCT cana-2815 16 21 letnikov letnikov NOUN cana-2815 16 22 operation operation NOUN cana-2815 16 23 , , PUNCT cana-2815 16 24 riemann riemann PROPN cana-2815 16 25 - - PUNCT cana-2815 16 26 liouville liouville VERB cana-2815 16 27 types type NOUN cana-2815 16 28 are be AUX cana-2815 16 29 leading lead VERB cana-2815 16 30 to to ADP cana-2815 16 31 applications application NOUN cana-2815 16 32 in in ADP cana-2815 16 33 diverse diverse ADJ cana-2815 16 34 fields field NOUN cana-2815 16 35 beyond beyond ADP cana-2815 16 36 mathematics mathematic NOUN cana-2815 17 1 [ [ X cana-2815 17 2 [ [ X cana-2815 17 3 11],[12],[13 11],[12],[13 NUM cana-2815 17 4 ] ] X cana-2815 17 5 ] ] PUNCT cana-2815 17 6 . . PUNCT cana-2815 18 1 multiple multiple ADJ cana-2815 18 2 integral integral ADJ cana-2815 18 3 equations equation NOUN cana-2815 18 4 , , PUNCT cana-2815 18 5 odes ode NOUN cana-2815 18 6 , , PUNCT cana-2815 18 7 pdes pde NOUN cana-2815 18 8 , , PUNCT cana-2815 18 9 and and CCONJ cana-2815 18 10 fractional fractional ADJ cana-2815 18 11 pdes pde NOUN cana-2815 18 12 are be AUX cana-2815 18 13 solved solve VERB cana-2815 18 14 using use VERB cana-2815 18 15 these these DET cana-2815 18 16 transformations transformation NOUN cana-2815 18 17 that that PRON cana-2815 18 18 are be AUX cana-2815 18 19 offered offer VERB cana-2815 18 20 in in ADP cana-2815 18 21 the the DET cana-2815 18 22 literature literature NOUN cana-2815 18 23 . . PUNCT cana-2815 19 1 a a DET cana-2815 19 2 range range NOUN cana-2815 19 3 of of ADP cana-2815 19 4 analytical analytical ADJ cana-2815 19 5 and and CCONJ cana-2815 19 6 numeration numeration NOUN cana-2815 19 7 techniques technique NOUN cana-2815 19 8 will will AUX cana-2815 19 9 be be AUX cana-2815 19 10 utilized utilize VERB cana-2815 19 11 to to PART cana-2815 19 12 solve solve VERB cana-2815 19 13 hyperbolic hyperbolic ADJ cana-2815 19 14 time time NOUN cana-2815 19 15 . . PUNCT cana-2815 20 1 fractional fractional ADJ cana-2815 20 2 telegraph telegraph NOUN cana-2815 20 3 equations equation NOUN cana-2815 20 4 . . PUNCT cana-2815 21 1 these these DET cana-2815 21 2 methods method NOUN cana-2815 21 3 encompass encompass VERB cana-2815 21 4 the the DET cana-2815 21 5 homotopy homotopy NOUN cana-2815 21 6 perturbation perturbation NOUN cana-2815 21 7 transform transform VERB cana-2815 21 8 [ [ X cana-2815 21 9 [ [ X cana-2815 21 10 14 14 NUM cana-2815 21 11 ] ] X cana-2815 21 12 ] ] X cana-2815 21 13 technique technique NOUN cana-2815 21 14 , , PUNCT cana-2815 21 15 sinc sinc ADJ cana-2815 21 16 - - PUNCT cana-2815 21 17 collocation collocation NOUN cana-2815 21 18 technique technique NOUN cana-2815 21 19 [ [ X cana-2815 21 20 [ [ X cana-2815 21 21 15 15 NUM cana-2815 21 22 ] ] X cana-2815 21 23 ] ] PUNCT cana-2815 21 24 , , PUNCT cana-2815 21 25 adomian adomian NOUN cana-2815 21 26 decomposition decomposition NOUN cana-2815 21 27 technique technique NOUN cana-2815 22 1 [ [ X cana-2815 22 2 [ [ X cana-2815 22 3 16 16 NUM cana-2815 22 4 ] ] X cana-2815 22 5 ] ] X cana-2815 22 6 , , PUNCT cana-2815 22 7 q q X cana-2815 22 8 - - PUNCT cana-2815 22 9 homotopy homotopy NOUN cana-2815 22 10 analysis analysis NOUN cana-2815 22 11 transform transform NOUN cana-2815 22 12 technique technique NOUN cana-2815 22 13 [ [ X cana-2815 22 14 [ [ X cana-2815 22 15 17 17 NUM cana-2815 22 16 ] ] SYM cana-2815 22 17 ] ] PUNCT cana-2815 22 18 , , PUNCT cana-2815 22 19 reduction reduction NOUN cana-2815 22 20 differential differential NOUN cana-2815 22 21 transform transform NOUN cana-2815 22 22 technique technique NOUN cana-2815 23 1 [ [ X cana-2815 23 2 [ [ X cana-2815 23 3 18 18 NUM cana-2815 23 4 ] ] X cana-2815 23 5 ] ] X cana-2815 23 6 , , PUNCT cana-2815 23 7 reproducing reproduce VERB cana-2815 23 8 kernel kernel PROPN cana-2815 23 9 method method NOUN cana-2815 23 10 [ [ X cana-2815 23 11 [ [ X cana-2815 23 12 19 19 NUM cana-2815 23 13 ] ] X cana-2815 23 14 ] ] X cana-2815 23 15 variational variational ADJ cana-2815 23 16 iteration iteration NOUN cana-2815 23 17 method method NOUN cana-2815 23 18 [ [ X cana-2815 23 19 [ [ X cana-2815 23 20 20 20 NUM cana-2815 23 21 ] ] SYM cana-2815 23 22 ] ] PUNCT cana-2815 23 23 , , PUNCT cana-2815 23 24 and and CCONJ cana-2815 23 25 haar haar PROPN cana-2815 23 26 wavelet wavelet NOUN cana-2815 23 27 technique technique NOUN cana-2815 24 1 [ [ X cana-2815 24 2 [ [ X cana-2815 24 3 21 21 NUM cana-2815 24 4 ] ] X cana-2815 24 5 ] ] X cana-2815 24 6 the the DET cana-2815 24 7 communication communication NOUN cana-2815 24 8 process process NOUN cana-2815 24 9 is be AUX cana-2815 24 10 essential essential ADJ cana-2815 24 11 in in ADP cana-2815 24 12 the the DET cana-2815 24 13 modern modern ADJ cana-2815 24 14 global global ADJ cana-2815 24 15 community community NOUN cana-2815 24 16 . . PUNCT cana-2815 25 1 due due ADP cana-2815 25 2 to to ADP cana-2815 25 3 the the DET cana-2815 25 4 extensive extensive ADJ cana-2815 25 5 usage usage NOUN cana-2815 25 6 of of ADP cana-2815 25 7 radio radio NOUN cana-2815 25 8 frequency frequency NOUN cana-2815 25 9 systems system NOUN cana-2815 25 10 and and CCONJ cana-2815 25 11 microwave microwave NOUN cana-2815 25 12 communication communication NOUN cana-2815 25 13 , , PUNCT cana-2815 25 14 technologies technology NOUN cana-2815 25 15 continue continue VERB cana-2815 25 16 to to PART cana-2815 25 17 get get VERB cana-2815 25 18 substantial substantial ADJ cana-2815 25 19 industrial industrial ADJ cana-2815 25 20 attention attention NOUN cana-2815 25 21 . . PUNCT cana-2815 26 1 importantly importantly ADV cana-2815 26 2 , , PUNCT cana-2815 26 3 all all DET cana-2815 26 4 transmission transmission NOUN cana-2815 26 5 media medium NOUN cana-2815 26 6 experience experience VERB cana-2815 26 7 the the DET cana-2815 26 8 signal signal ADJ cana-2815 26 9 deficit deficit NOUN cana-2815 26 10 , , PUNCT cana-2815 26 11 which which PRON cana-2815 26 12 must must AUX cana-2815 26 13 be be AUX cana-2815 26 14 measured measure VERB cana-2815 26 15 for for ADP cana-2815 26 16 each each DET cana-2815 26 17 medium medium NOUN cana-2815 26 18 . . PUNCT cana-2815 27 1 telegraph telegraph NOUN cana-2815 27 2 equations equation NOUN cana-2815 27 3 are be AUX cana-2815 27 4 employed employ VERB cana-2815 27 5 to to AUX cana-2815 27 6 analyses analyse VERB cana-2815 27 7 electrical electrical ADJ cana-2815 27 8 signal signal NOUN cana-2815 27 9 propagation propagation NOUN cana-2815 27 10 , , PUNCT cana-2815 27 11 random random ADJ cana-2815 27 12 walks walk NOUN cana-2815 27 13 , , PUNCT cana-2815 27 14 wave wave NOUN cana-2815 27 15 propagation propagation NOUN cana-2815 27 16 , , PUNCT cana-2815 27 17 and and CCONJ cana-2815 27 18 transmission transmission NOUN cana-2815 27 19 line line NOUN cana-2815 27 20 cables cable NOUN cana-2815 27 21 and and CCONJ cana-2815 27 22 similar similar ADJ cana-2815 27 23 phenomena phenomenon NOUN cana-2815 27 24 . . PUNCT cana-2815 28 1 heaviside heaviside PROPN cana-2815 28 2 introduced introduce VERB cana-2815 28 3 the the DET cana-2815 28 4 concept concept NOUN cana-2815 28 5 of of ADP cana-2815 28 6 the the DET cana-2815 28 7 transmission transmission NOUN cana-2815 28 8 line line NOUN cana-2815 28 9 , , PUNCT cana-2815 28 10 which which PRON cana-2815 28 11 will will AUX cana-2815 28 12 be be AUX cana-2815 28 13 divided divide VERB cana-2815 28 14 into into ADP cana-2815 28 15 two two NUM cana-2815 28 16 types type NOUN cana-2815 28 17 : : PUNCT cana-2815 28 18 guided guide VERB cana-2815 28 19 and and CCONJ cana-2815 28 20 unguided unguided ADJ cana-2815 28 21 . . PUNCT cana-2815 29 1 in in ADP cana-2815 29 2 guided guide VERB cana-2815 29 3 media medium NOUN cana-2815 29 4 , , PUNCT cana-2815 29 5 signals signal NOUN cana-2815 29 6 are be AUX cana-2815 29 7 transmitted transmit VERB cana-2815 29 8 through through ADP cana-2815 29 9 physical physical ADJ cana-2815 29 10 systems system NOUN cana-2815 29 11 such such ADJ cana-2815 29 12 as as ADP cana-2815 29 13 copper copper NOUN cana-2815 29 14 wires wire NOUN cana-2815 29 15 , , PUNCT cana-2815 29 16 which which PRON cana-2815 29 17 carry carry VERB cana-2815 29 18 voltage voltage NOUN cana-2815 29 19 waves wave NOUN cana-2815 29 20 and and CCONJ cana-2815 29 21 higher high ADJ cana-2815 29 22 frequency frequency NOUN cana-2815 29 23 currents current NOUN cana-2815 29 24 . . PUNCT cana-2815 30 1 conversely conversely ADV cana-2815 30 2 , , PUNCT cana-2815 30 3 the the DET cana-2815 30 4 unguided unguided ADJ cana-2815 30 5 media medium NOUN cana-2815 30 6 use use VERB cana-2815 30 7 magnetic magnetic ADJ cana-2815 30 8 fields field NOUN cana-2815 30 9 to to PART cana-2815 30 10 transmit transmit VERB cana-2815 30 11 signals signal NOUN cana-2815 30 12 across across ADP cana-2815 30 13 communication communication NOUN cana-2815 30 14 channels channel NOUN cana-2815 30 15 , , PUNCT cana-2815 30 16 employing employ VERB cana-2815 30 17 microwave microwave NOUN cana-2815 30 18 communication communication NOUN cana-2815 30 19 and and CCONJ cana-2815 30 20 radio radio NOUN cana-2815 30 21 frequency frequency NOUN cana-2815 30 22 systems system NOUN cana-2815 30 23 , , PUNCT cana-2815 30 24 with with ADP cana-2815 30 25 antennas antenna NOUN cana-2815 30 26 facilitating facilitate VERB cana-2815 30 27 the the DET cana-2815 30 28 broadcasting broadcasting NOUN cana-2815 30 29 and and CCONJ cana-2815 30 30 reception reception NOUN cana-2815 30 31 of of ADP cana-2815 30 32 these these DET cana-2815 30 33 electromagnetic electromagnetic ADJ cana-2815 30 34 waves wave NOUN cana-2815 30 35 . . PUNCT cana-2815 31 1 to to PART cana-2815 31 2 increase increase VERB cana-2815 31 3 the the DET cana-2815 31 4 efficiency efficiency NOUN cana-2815 31 5 of of ADP cana-2815 31 6 telegraph telegraph NOUN cana-2815 31 7 transmission transmission NOUN cana-2815 31 8 , , PUNCT cana-2815 31 9 cable cable NOUN cana-2815 31 10 transmission transmission NOUN cana-2815 31 11 media medium NOUN cana-2815 31 12 are be AUX cana-2815 31 13 communications communication NOUN cana-2815 31 14 on on ADP cana-2815 31 15 applied apply VERB cana-2815 31 16 nonlinear nonlinear ADJ cana-2815 31 17 analysis analysis NOUN cana-2815 31 18 issn issn NOUN cana-2815 31 19 : : PUNCT cana-2815 31 20 1074 1074 NUM cana-2815 31 21 - - PUNCT cana-2815 31 22 133x 133x NUM cana-2815 31 23 vol vol NOUN cana-2815 31 24 32 32 NUM cana-2815 31 25 no no NOUN cana-2815 31 26 . . PUNCT cana-2815 32 1 4s 4s NUM cana-2815 32 2 ( ( PUNCT cana-2815 32 3 2025 2025 NUM cana-2815 32 4 ) ) PUNCT cana-2815 32 5 311 311 NUM cana-2815 32 6 https://internationalpubls.com https://internationalpubls.com NUM cana-2815 32 7 researched research VERB cana-2815 32 8 in in ADP cana-2815 32 9 controlled control VERB cana-2815 32 10 transmission transmission NOUN cana-2815 32 11 environments environment NOUN cana-2815 32 12 . . PUNCT cana-2815 33 1 direct direct ADJ cana-2815 33 2 information information NOUN cana-2815 33 3 propagation propagation NOUN cana-2815 33 4 between between ADP cana-2815 33 5 two two NUM cana-2815 33 6 or or CCONJ cana-2815 33 7 more more ADJ cana-2815 33 8 sites site NOUN cana-2815 33 9 is be AUX cana-2815 33 10 represented represent VERB cana-2815 33 11 by by ADP cana-2815 33 12 a a DET cana-2815 33 13 physical physical ADJ cana-2815 33 14 system system NOUN cana-2815 33 15 using use VERB cana-2815 33 16 a a DET cana-2815 33 17 directed direct VERB cana-2815 33 18 transmission transmission NOUN cana-2815 33 19 medium medium NOUN cana-2815 33 20 . . PUNCT cana-2815 34 1 power power NOUN cana-2815 34 2 and and CCONJ cana-2815 34 3 signal signal NOUN cana-2815 34 4 losses loss NOUN cana-2815 34 5 must must AUX cana-2815 34 6 be be AUX cana-2815 34 7 predicted predict VERB cana-2815 34 8 or or CCONJ cana-2815 34 9 calculated calculate VERB cana-2815 34 10 because because SCONJ cana-2815 34 11 they they PRON cana-2815 34 12 are be AUX cana-2815 34 13 a a DET cana-2815 34 14 necessary necessary ADJ cana-2815 34 15 part part NOUN cana-2815 34 16 of of ADP cana-2815 34 17 any any DET cana-2815 34 18 system system NOUN cana-2815 34 19 to to PART cana-2815 34 20 enhance enhance VERB cana-2815 34 21 controlled control VERB cana-2815 34 22 communication communication NOUN cana-2815 35 1 [ [ X cana-2815 35 2 [ [ X cana-2815 35 3 22 22 NUM cana-2815 35 4 ] ] X cana-2815 35 5 ] ] PUNCT cana-2815 35 6 . . PUNCT cana-2815 36 1 in in ADP cana-2815 36 2 the the DET cana-2815 36 3 last last ADJ cana-2815 36 4 few few ADJ cana-2815 36 5 years year NOUN cana-2815 36 6 , , PUNCT cana-2815 36 7 fractional fractional ADJ cana-2815 36 8 order order NOUN cana-2815 36 9 partial partial ADJ cana-2815 36 10 differentiation differentiation NOUN cana-2815 36 11 equations equation NOUN cana-2815 36 12 ( ( PUNCT cana-2815 36 13 pdes pde NOUN cana-2815 36 14 ) ) PUNCT cana-2815 36 15 gained gain VERB cana-2815 36 16 significant significant ADJ cana-2815 36 17 attention attention NOUN cana-2815 36 18 due due ADP cana-2815 36 19 to to ADP cana-2815 36 20 their their PRON cana-2815 36 21 extensive extensive ADJ cana-2815 36 22 application application NOUN cana-2815 36 23 in in ADP cana-2815 36 24 a a DET cana-2815 36 25 variety variety NOUN cana-2815 36 26 of of ADP cana-2815 36 27 technical technical ADJ cana-2815 36 28 and and CCONJ cana-2815 36 29 scientific scientific ADJ cana-2815 36 30 domains domain NOUN cana-2815 36 31 , , PUNCT cana-2815 36 32 from from ADP cana-2815 36 33 scientists scientist NOUN cana-2815 36 34 and and CCONJ cana-2815 36 35 researchers researcher NOUN cana-2815 36 36 . . PUNCT cana-2815 37 1 the the DET cana-2815 37 2 fractional fractional ADJ cana-2815 37 3 derivative derivative NOUN cana-2815 37 4 within within ADP cana-2815 37 5 these these DET cana-2815 37 6 models model NOUN cana-2815 37 7 offers offer VERB cana-2815 37 8 a a DET cana-2815 37 9 high high ADJ cana-2815 37 10 degree degree NOUN cana-2815 37 11 of of ADP cana-2815 37 12 flexibility flexibility NOUN cana-2815 37 13 , , PUNCT cana-2815 37 14 offering offer VERB cana-2815 37 15 a a DET cana-2815 37 16 great great ADJ cana-2815 37 17 tool tool NOUN cana-2815 37 18 for for ADP cana-2815 37 19 characterizing characterize VERB cana-2815 37 20 the the DET cana-2815 37 21 inherited inherit VERB cana-2815 37 22 traits trait NOUN cana-2815 37 23 of of ADP cana-2815 37 24 various various ADJ cana-2815 37 25 prototypes prototype NOUN cana-2815 37 26 and and CCONJ cana-2815 37 27 their their PRON cana-2815 37 28 varying vary VERB cana-2815 37 29 histories history NOUN cana-2815 37 30 . . PUNCT cana-2815 38 1 extensive extensive ADJ cana-2815 38 2 research research NOUN cana-2815 38 3 has have AUX cana-2815 38 4 been be AUX cana-2815 38 5 conducted conduct VERB cana-2815 38 6 to to PART cana-2815 38 7 develop develop VERB cana-2815 38 8 analytical analytical ADJ cana-2815 38 9 , , PUNCT cana-2815 38 10 semi semi ADJ cana-2815 38 11 - - ADJ cana-2815 38 12 analytical analytical ADJ cana-2815 38 13 and and CCONJ cana-2815 38 14 numerical numerical ADJ cana-2815 38 15 solutions solution NOUN cana-2815 38 16 for for ADP cana-2815 38 17 solving solve VERB cana-2815 38 18 both both CCONJ cana-2815 38 19 nonlinear nonlinear ADJ cana-2815 38 20 and and CCONJ cana-2815 38 21 linear linear ADJ cana-2815 38 22 fractional fractional ADJ cana-2815 38 23 pdes pde NOUN cana-2815 39 1 [ [ X cana-2815 39 2 [ [ X cana-2815 39 3 23 23 NUM cana-2815 39 4 ] ] X cana-2815 39 5 ] ] PUNCT cana-2815 39 6 . . PUNCT cana-2815 40 1 the the DET cana-2815 40 2 time time NOUN cana-2815 40 3 - - PUNCT cana-2815 40 4 fractional fractional ADJ cana-2815 40 5 telegraph telegraph NOUN cana-2815 40 6 equations equation NOUN cana-2815 40 7 are be AUX cana-2815 40 8 deployed deploy VERB cana-2815 40 9 in in ADP cana-2815 40 10 this this DET cana-2815 40 11 article article NOUN cana-2815 40 12 with with ADP cana-2815 40 13 the the DET cana-2815 40 14 use use NOUN cana-2815 40 15 of of ADP cana-2815 40 16 madetm madetm NOUN cana-2815 40 17 . . PUNCT cana-2815 41 1 hyperbolic hyperbolic ADJ cana-2815 41 2 time time NOUN cana-2815 41 3 fractional fractional PROPN cana-2815 41 4 telegraph telegraph NOUN cana-2815 41 5 equations equation NOUN cana-2815 41 6 are be AUX cana-2815 41 7 deployed deploy VERB cana-2815 41 8 in in ADP cana-2815 41 9 this this DET cana-2815 41 10 study study NOUN cana-2815 41 11 through through ADP cana-2815 41 12 demonstration demonstration NOUN cana-2815 41 13 of of ADP cana-2815 41 14 the the DET cana-2815 41 15 modified modify VERB cana-2815 41 16 adomian adomian NOUN cana-2815 41 17 decomposition decomposition NOUN cana-2815 41 18 with with ADP cana-2815 41 19 coupling couple VERB cana-2815 41 20 elzaki elzaki NOUN cana-2815 41 21 transformation transformation NOUN cana-2815 41 22 method method NOUN cana-2815 41 23 ( ( PUNCT cana-2815 41 24 madetm madetm NOUN cana-2815 41 25 ) ) PUNCT cana-2815 41 26 . . PUNCT cana-2815 42 1 certain certain ADJ cana-2815 42 2 fractional fractional ADJ cana-2815 42 3 - - PUNCT cana-2815 42 4 order order NOUN cana-2815 42 5 telegraph telegraph NOUN cana-2815 42 6 equation equation NOUN cana-2815 42 7 models model NOUN cana-2815 42 8 are be AUX cana-2815 42 9 used use VERB cana-2815 42 10 to to PART cana-2815 42 11 determine determine VERB cana-2815 42 12 the the DET cana-2815 42 13 madetm madetm NOUN cana-2815 42 14 solutions solution NOUN cana-2815 42 15 . . PUNCT cana-2815 43 1 the the DET cana-2815 43 2 method method NOUN cana-2815 43 3 demonstrates demonstrate VERB cana-2815 43 4 increased increase VERB cana-2815 43 5 precision precision NOUN cana-2815 43 6 and and CCONJ cana-2815 43 7 efficiency efficiency NOUN cana-2815 43 8 , , PUNCT cana-2815 43 9 as as SCONJ cana-2815 43 10 evidenced evidence VERB cana-2815 43 11 by by ADP cana-2815 43 12 graphical graphical ADJ cana-2815 43 13 comparisons comparison NOUN cana-2815 43 14 with with ADP cana-2815 43 15 the the DET cana-2815 43 16 exact exact ADJ cana-2815 43 17 solution solution NOUN cana-2815 43 18 . . PUNCT cana-2815 44 1 the the DET cana-2815 44 2 edm edm PROPN cana-2815 44 3 solutions solution NOUN cana-2815 44 4 for for ADP cana-2815 44 5 fractional fractional ADJ cana-2815 44 6 - - PUNCT cana-2815 44 7 order order NOUN cana-2815 44 8 telegraph telegraph NOUN cana-2815 44 9 equations equation NOUN cana-2815 44 10 exhibit exhibit VERB cana-2815 44 11 a a DET cana-2815 44 12 high high ADJ cana-2815 44 13 rate rate NOUN cana-2815 44 14 of of ADP cana-2815 44 15 convergence convergence NOUN cana-2815 44 16 . . PUNCT cana-2815 45 1 consequently consequently ADV cana-2815 45 2 , , PUNCT cana-2815 45 3 this this DET cana-2815 45 4 technique technique NOUN cana-2815 45 5 is be AUX cana-2815 45 6 promising promise VERB cana-2815 45 7 for for ADP cana-2815 45 8 simplifying simplify VERB cana-2815 45 9 other other ADJ cana-2815 45 10 fractional fractional ADJ cana-2815 45 11 forms form NOUN cana-2815 45 12 linearly linearly ADV cana-2815 45 13 and and CCONJ cana-2815 45 14 nonlinearly nonlinearly ADV cana-2815 45 15 partial partial ADJ cana-2815 45 16 differentiation differentiation NOUN cana-2815 45 17 equations equation NOUN cana-2815 45 18 . . PUNCT cana-2815 46 1 β€’ β€’ NUM cana-2815 46 2 one one NUM cana-2815 46 3 - - PUNCT cana-2815 46 4 dimensional dimensional ADJ cana-2815 46 5 fractional fractional ADJ cana-2815 46 6 order order NOUN cana-2815 46 7 telegraph telegraph NOUN cana-2815 46 8 equation equation NOUN cana-2815 46 9 : : PUNCT cana-2815 47 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 47 2 2𝛼 2𝛼 NUM cana-2815 47 3 [ [ X cana-2815 47 4 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 47 5 , , PUNCT cana-2815 47 6 𝑑 𝑑 PROPN cana-2815 47 7 ) ) PUNCT cana-2815 47 8 ] ] PUNCT cana-2815 48 1 + + CCONJ cana-2815 49 1 2𝛼𝐷𝑑 2𝛼𝐷𝑑 NUM cana-2815 49 2 𝛼 𝛼 PRON cana-2815 49 3 [ [ X cana-2815 49 4 βˆ…(π‘₯ βˆ…(π‘₯ INTJ cana-2815 49 5 , , PUNCT cana-2815 49 6 𝑑 𝑑 PROPN cana-2815 49 7 ) ) PUNCT cana-2815 49 8 ] ] PUNCT cana-2815 50 1 + + NUM cana-2815 50 2 𝛽2 𝛽2 NOUN cana-2815 50 3 = = SYM cana-2815 50 4 βˆ…π‘₯π‘₯(π‘₯ βˆ…π‘₯π‘₯(π‘₯ X cana-2815 50 5 , , PUNCT cana-2815 50 6 𝑑 𝑑 X cana-2815 50 7 ) ) PUNCT cana-2815 50 8 + + CCONJ cana-2815 50 9 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-2815 50 10 , , PUNCT cana-2815 50 11 𝑑 𝑑 X cana-2815 50 12 ) ) PUNCT cana-2815 50 13 0 0 PUNCT cana-2815 51 1 < < X cana-2815 51 2 ∝ ∝ X cana-2815 51 3 , , PUNCT cana-2815 51 4 π‘₯ π‘₯ PROPN cana-2815 51 5 ≀ ≀ NUM cana-2815 51 6 1 1 NUM cana-2815 51 7 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 51 8 , , PUNCT cana-2815 51 9 0 0 NUM cana-2815 51 10 ) ) PUNCT cana-2815 51 11 = = SYM cana-2815 51 12 𝑓1(π‘₯ 𝑓1(π‘₯ NOUN cana-2815 51 13 ) ) PUNCT cana-2815 51 14 , , PUNCT cana-2815 51 15 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 51 16 , , PUNCT cana-2815 51 17 0 0 NUM cana-2815 51 18 ) ) PUNCT cana-2815 51 19 = = SYM cana-2815 51 20 𝑓2(π‘₯ 𝑓2(π‘₯ X cana-2815 51 21 ) ) PUNCT cana-2815 51 22 βˆ…(0 βˆ…(0 PROPN cana-2815 51 23 , , PUNCT cana-2815 51 24 𝑑 𝑑 X cana-2815 51 25 ) ) PUNCT cana-2815 51 26 = = SYM cana-2815 51 27 𝑓1(𝑑 𝑓1(𝑑 PROPN cana-2815 51 28 ) ) PUNCT cana-2815 51 29 , , PUNCT cana-2815 51 30 βˆ…π‘₯(π‘₯ βˆ…π‘₯(π‘₯ NOUN cana-2815 51 31 , , PUNCT cana-2815 51 32 𝑑 𝑑 X cana-2815 51 33 ) ) PUNCT cana-2815 51 34 = = SYM cana-2815 52 1 𝑓2(𝑑 𝑓2(𝑑 NOUN cana-2815 52 2 ) ) PUNCT cana-2815 52 3 where where SCONJ cana-2815 52 4 𝛽 𝛽 NOUN cana-2815 52 5 β†’ β†’ SYM cana-2815 52 6 arbitrary arbitrary ADJ cana-2815 52 7 constants constant NOUN cana-2815 52 8 and and CCONJ cana-2815 52 9 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 52 10 , , PUNCT cana-2815 52 11 𝑑 𝑑 PROPN cana-2815 52 12 ) ) PUNCT cana-2815 52 13 is be AUX cana-2815 52 14 an an DET cana-2815 52 15 unknown unknown ADJ cana-2815 52 16 function function NOUN cana-2815 52 17 . . PUNCT cana-2815 53 1 β€’ β€’ NUM cana-2815 53 2 two two NUM cana-2815 53 3 - - PUNCT cana-2815 53 4 dimensional dimensional ADJ cana-2815 53 5 fractional fractional ADJ cana-2815 53 6 order order NOUN cana-2815 53 7 telegraph telegraph NOUN cana-2815 53 8 equation equation NOUN cana-2815 53 9 : : PUNCT cana-2815 54 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 54 2 2𝛼 2𝛼 NUM cana-2815 55 1 [ [ X cana-2815 55 2 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 55 3 , , PUNCT cana-2815 55 4 𝑦 𝑦 NOUN cana-2815 55 5 , , PUNCT cana-2815 55 6 𝑑 𝑑 PROPN cana-2815 55 7 ) ) PUNCT cana-2815 55 8 ] ] PUNCT cana-2815 56 1 + + CCONJ cana-2815 57 1 2𝛼𝐷𝑑 2𝛼𝐷𝑑 NUM cana-2815 57 2 𝛼 𝛼 PRON cana-2815 57 3 [ [ X cana-2815 57 4 βˆ…(π‘₯ βˆ…(π‘₯ INTJ cana-2815 57 5 , , PUNCT cana-2815 57 6 𝑦 𝑦 NOUN cana-2815 57 7 , , PUNCT cana-2815 57 8 𝑑 𝑑 PROPN cana-2815 57 9 ) ) PUNCT cana-2815 57 10 ] ] PUNCT cana-2815 58 1 + + CCONJ cana-2815 58 2 𝛽2 𝛽2 NOUN cana-2815 58 3 [ [ X cana-2815 58 4 βˆ…(π‘₯ βˆ…(π‘₯ INTJ cana-2815 58 5 , , PUNCT cana-2815 58 6 𝑦 𝑦 NOUN cana-2815 58 7 , , PUNCT cana-2815 58 8 𝑑 𝑑 PROPN cana-2815 58 9 ) ) PUNCT cana-2815 58 10 ] ] PUNCT cana-2815 58 11 = = SYM cana-2815 58 12 βˆ…π‘₯π‘₯(π‘₯ βˆ…π‘₯π‘₯(π‘₯ X cana-2815 58 13 , , PUNCT cana-2815 58 14 𝑦 𝑦 PRON cana-2815 58 15 , , PUNCT cana-2815 58 16 𝑑 𝑑 PROPN cana-2815 58 17 ) ) PUNCT cana-2815 58 18 + + CCONJ cana-2815 58 19 βˆ…π‘¦π‘¦(π‘₯ βˆ…π‘¦π‘¦(π‘₯ X cana-2815 58 20 , , PUNCT cana-2815 58 21 𝑦 𝑦 NOUN cana-2815 58 22 , , PUNCT cana-2815 58 23 𝑑 𝑑 PROPN cana-2815 58 24 ) ) PUNCT cana-2815 58 25 + + CCONJ cana-2815 58 26 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-2815 58 27 , , PUNCT cana-2815 58 28 𝑦 𝑦 PRON cana-2815 58 29 , , PUNCT cana-2815 58 30 𝑑 𝑑 PROPN cana-2815 58 31 ) ) PUNCT cana-2815 58 32 0 0 PUNCT cana-2815 58 33 < < X cana-2815 58 34 βˆβ‰€ βˆβ‰€ X cana-2815 58 35 1 1 NUM cana-2815 58 36 , , PUNCT cana-2815 58 37 π‘₯ π‘₯ NOUN cana-2815 58 38 = = SYM cana-2815 58 39 1 1 NUM cana-2815 58 40 with with ADP cana-2815 58 41 initial initial ADJ cana-2815 58 42 conditions condition NOUN cana-2815 58 43 : : PUNCT cana-2815 58 44 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 58 45 , , PUNCT cana-2815 58 46 𝑦 𝑦 NOUN cana-2815 58 47 , , PUNCT cana-2815 58 48 0 0 NUM cana-2815 58 49 ) ) PUNCT cana-2815 58 50 = = PUNCT cana-2815 58 51 𝑔1(π‘₯ 𝑔1(π‘₯ NOUN cana-2815 58 52 , , PUNCT cana-2815 58 53 𝑦 𝑦 NOUN cana-2815 58 54 ) ) PUNCT cana-2815 58 55 , , PUNCT cana-2815 58 56 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 58 57 , , PUNCT cana-2815 58 58 𝑦 𝑦 NOUN cana-2815 58 59 , , PUNCT cana-2815 58 60 0 0 NUM cana-2815 58 61 ) ) PUNCT cana-2815 58 62 = = SYM cana-2815 58 63 𝑔2(π‘₯ 𝑔2(π‘₯ PROPN cana-2815 58 64 , , PUNCT cana-2815 58 65 𝑦 𝑦 NOUN cana-2815 58 66 ) ) PUNCT cana-2815 58 67 with with ADP cana-2815 58 68 initial initial ADJ cana-2815 58 69 conditions condition NOUN cana-2815 58 70 : : PUNCT cana-2815 58 71 communications communication NOUN cana-2815 58 72 on on ADP cana-2815 58 73 applied apply VERB cana-2815 58 74 nonlinear nonlinear ADJ cana-2815 58 75 analysis analysis NOUN cana-2815 58 76 issn issn NOUN cana-2815 58 77 : : PUNCT cana-2815 58 78 1074 1074 NUM cana-2815 58 79 - - PUNCT cana-2815 58 80 133x 133x NUM cana-2815 58 81 vol vol NOUN cana-2815 58 82 32 32 NUM cana-2815 58 83 no no NOUN cana-2815 58 84 . . PUNCT cana-2815 59 1 4s 4s NUM cana-2815 59 2 ( ( PUNCT cana-2815 59 3 2025 2025 NUM cana-2815 59 4 ) ) PUNCT cana-2815 59 5 312 312 NUM cana-2815 59 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 59 7 β€’ β€’ NUM cana-2815 59 8 𝐷𝑑 𝐷𝑑 PROPN cana-2815 59 9 2𝛼 2𝛼 NUM cana-2815 60 1 [ [ X cana-2815 60 2 βˆ…(π‘₯. βˆ…(π‘₯. X cana-2815 60 3 , , PUNCT cana-2815 60 4 𝑦 𝑦 PRON cana-2815 60 5 , , PUNCT cana-2815 60 6 𝑧 𝑧 X cana-2815 60 7 , , PUNCT cana-2815 60 8 𝑑 𝑑 NOUN cana-2815 60 9 ) ) PUNCT cana-2815 60 10 ] ] PUNCT cana-2815 61 1 + + CCONJ cana-2815 62 1 2𝛼𝐷𝑑 2𝛼𝐷𝑑 NUM cana-2815 62 2 𝛼 𝛼 PRON cana-2815 62 3 [ [ X cana-2815 62 4 βˆ…(π‘₯ βˆ…(π‘₯ INTJ cana-2815 62 5 , , PUNCT cana-2815 62 6 𝑦 𝑦 NOUN cana-2815 62 7 , , PUNCT cana-2815 62 8 𝑧 𝑧 X cana-2815 62 9 , , PUNCT cana-2815 62 10 𝑑 𝑑 NOUN cana-2815 62 11 ) ) PUNCT cana-2815 62 12 ] ] PUNCT cana-2815 63 1 + + CCONJ cana-2815 63 2 𝛽2 𝛽2 NOUN cana-2815 63 3 [ [ X cana-2815 63 4 βˆ…(π‘₯. βˆ…(π‘₯. X cana-2815 63 5 , , PUNCT cana-2815 63 6 𝑦 𝑦 PRON cana-2815 63 7 , , PUNCT cana-2815 63 8 𝑧 𝑧 X cana-2815 63 9 , , PUNCT cana-2815 63 10 𝑑 𝑑 NOUN cana-2815 63 11 ) ) PUNCT cana-2815 63 12 ] ] PUNCT cana-2815 64 1 = = SYM cana-2815 64 2 βˆ…π‘₯π‘₯(π‘₯ βˆ…π‘₯π‘₯(π‘₯ X cana-2815 64 3 , , PUNCT cana-2815 64 4 𝑦 𝑦 PRON cana-2815 64 5 , , PUNCT cana-2815 64 6 𝑧 𝑧 X cana-2815 64 7 , , PUNCT cana-2815 64 8 𝑑 𝑑 NOUN cana-2815 64 9 ) ) PUNCT cana-2815 64 10 + + CCONJ cana-2815 64 11 βˆ…π‘¦π‘¦(π‘₯ βˆ…π‘¦π‘¦(π‘₯ X cana-2815 64 12 , , PUNCT cana-2815 64 13 𝑦 𝑦 NOUN cana-2815 64 14 , , PUNCT cana-2815 64 15 𝑧 𝑧 X cana-2815 64 16 , , PUNCT cana-2815 64 17 𝑑 𝑑 X cana-2815 64 18 ) ) PUNCT cana-2815 64 19 + + CCONJ cana-2815 64 20 βˆ…π‘§π‘§(π‘₯ βˆ…π‘§π‘§(π‘₯ ADJ cana-2815 64 21 , , PUNCT cana-2815 64 22 𝑦 𝑦 PRON cana-2815 64 23 , , PUNCT cana-2815 64 24 𝑧 𝑧 X cana-2815 64 25 , , PUNCT cana-2815 64 26 𝑑 𝑑 X cana-2815 64 27 ) ) PUNCT cana-2815 64 28 + + CCONJ cana-2815 64 29 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-2815 64 30 , , PUNCT cana-2815 64 31 𝑦 𝑦 PRON cana-2815 64 32 , , PUNCT cana-2815 64 33 𝑧 𝑧 X cana-2815 64 34 , , PUNCT cana-2815 64 35 𝑑 𝑑 PROPN cana-2815 64 36 ) ) PUNCT cana-2815 64 37 0 0 PUNCT cana-2815 64 38 < < X cana-2815 64 39 βˆβ‰€ βˆβ‰€ X cana-2815 64 40 1 1 NUM cana-2815 64 41 , , PUNCT cana-2815 64 42 π‘₯ π‘₯ NOUN cana-2815 64 43 = = SYM cana-2815 64 44 1 1 NUM cana-2815 64 45 with with ADP cana-2815 64 46 initial initial ADJ cana-2815 64 47 conditions condition NOUN cana-2815 64 48 : : PUNCT cana-2815 65 1 βˆ…(π‘₯. βˆ…(π‘₯. NOUN cana-2815 65 2 , , PUNCT cana-2815 65 3 𝑦 𝑦 PRON cana-2815 65 4 , , PUNCT cana-2815 65 5 𝑧 𝑧 X cana-2815 65 6 , , PUNCT cana-2815 65 7 𝑑 𝑑 NOUN cana-2815 65 8 ) ) PUNCT cana-2815 65 9 = = SYM cana-2815 65 10 β„Ž1(π‘₯ β„Ž1(π‘₯ X cana-2815 65 11 , , PUNCT cana-2815 65 12 𝑦 𝑦 NOUN cana-2815 65 13 , , PUNCT cana-2815 65 14 𝑧 𝑧 PROPN cana-2815 65 15 ) ) PUNCT cana-2815 65 16 , , PUNCT cana-2815 65 17 βˆ…π‘‘(π‘₯. βˆ…π‘‘(π‘₯. PROPN cana-2815 65 18 , , PUNCT cana-2815 65 19 𝑦 𝑦 PRON cana-2815 65 20 , , PUNCT cana-2815 65 21 𝑧 𝑧 X cana-2815 65 22 , , PUNCT cana-2815 65 23 𝑑 𝑑 NOUN cana-2815 65 24 ) ) PUNCT cana-2815 65 25 = = SYM cana-2815 65 26 β„Ž2(π‘₯ β„Ž2(π‘₯ PROPN cana-2815 65 27 , , PUNCT cana-2815 65 28 𝑦 𝑦 NOUN cana-2815 65 29 , , PUNCT cana-2815 65 30 𝑧 𝑧 PROPN cana-2815 65 31 ) ) PUNCT cana-2815 65 32 the the DET cana-2815 65 33 hyperbolic hyperbolic ADJ cana-2815 65 34 telegraph telegraph NOUN cana-2815 65 35 equation equation NOUN cana-2815 65 36 is be AUX cana-2815 65 37 broadly broadly ADV cana-2815 65 38 applied apply VERB cana-2815 65 39 in in ADP cana-2815 65 40 the the DET cana-2815 65 41 signal signal NOUN cana-2815 65 42 processing processing NOUN cana-2815 65 43 for for ADP cana-2815 65 44 transmitting transmit VERB cana-2815 65 45 wave wave NOUN cana-2815 65 46 theory theory NOUN cana-2815 65 47 and and CCONJ cana-2815 65 48 electric electric ADJ cana-2815 65 49 impulses impulse NOUN cana-2815 65 50 . . PUNCT cana-2815 66 1 it it PRON cana-2815 66 2 has have AUX cana-2815 66 3 found find VERB cana-2815 66 4 various various ADJ cana-2815 66 5 applications application NOUN cana-2815 66 6 in in ADP cana-2815 66 7 biomedical biomedical ADJ cana-2815 66 8 sciences science NOUN cana-2815 66 9 and and CCONJ cana-2815 66 10 aerospace aerospace NOUN cana-2815 66 11 . . PUNCT cana-2815 67 1 researchers researcher NOUN cana-2815 67 2 are be AUX cana-2815 67 3 particularly particularly ADV cana-2815 67 4 interested interested ADJ cana-2815 67 5 in in ADP cana-2815 67 6 solving solve VERB cana-2815 67 7 problems problem NOUN cana-2815 67 8 involving involve VERB cana-2815 67 9 fractional fractional ADJ cana-2815 67 10 derivatives derivative NOUN cana-2815 67 11 . . PUNCT cana-2815 68 1 fractionalorder fractionalorder ADJ cana-2815 68 2 partial partial ADJ cana-2815 68 3 differential differential NOUN cana-2815 68 4 equations equation NOUN cana-2815 68 5 ( ( PUNCT cana-2815 68 6 pdes pde NOUN cana-2815 68 7 ) ) PUNCT cana-2815 68 8 are be AUX cana-2815 68 9 essentially essentially ADV cana-2815 68 10 a a DET cana-2815 68 11 type type NOUN cana-2815 68 12 of of ADP cana-2815 68 13 integer integer NOUN cana-2815 68 14 - - PUNCT cana-2815 68 15 order order NOUN cana-2815 68 16 pdes pde NOUN cana-2815 68 17 . . PUNCT cana-2815 69 1 fractionalorder fractionalorder ADJ cana-2815 69 2 methods method NOUN cana-2815 69 3 yield yield VERB cana-2815 69 4 results result NOUN cana-2815 69 5 that that PRON cana-2815 69 6 converge converge VERB cana-2815 69 7 to to ADP cana-2815 69 8 those those PRON cana-2815 69 9 of of ADP cana-2815 69 10 integer integer NOUN cana-2815 69 11 - - PUNCT cana-2815 69 12 order order NOUN cana-2815 69 13 methods method NOUN cana-2815 69 14 . . PUNCT cana-2815 70 1 2 2 X cana-2815 70 2 . . NUM cana-2815 70 3 preliminaries preliminary NOUN cana-2815 70 4 and and CCONJ cana-2815 70 5 notations notation NOUN cana-2815 70 6 elzaki elzaki NOUN cana-2815 70 7 transformations transformation NOUN cana-2815 70 8 , , PUNCT cana-2815 70 9 also also ADV cana-2815 70 10 called call VERB cana-2815 70 11 elzaki elzaki ADJ cana-2815 70 12 integral integral ADJ cana-2815 70 13 transformations transformation NOUN cana-2815 70 14 , , PUNCT cana-2815 70 15 are be AUX cana-2815 70 16 algebraic algebraic ADJ cana-2815 70 17 equation equation NOUN cana-2815 70 18 transformations transformation NOUN cana-2815 70 19 that that PRON cana-2815 70 20 are be AUX cana-2815 70 21 used use VERB cana-2815 70 22 to to PART cana-2815 70 23 solve solve VERB cana-2815 70 24 differential differential ADJ cana-2815 70 25 equations equation NOUN cana-2815 70 26 in in ADP cana-2815 70 27 ordinary ordinary ADJ cana-2815 70 28 form form NOUN cana-2815 70 29 ( ( PUNCT cana-2815 70 30 odes ode NOUN cana-2815 70 31 ) ) PUNCT cana-2815 70 32 . . PUNCT cana-2815 71 1 elzaki elzaki PROPN cana-2815 71 2 ali ali PROPN cana-2815 71 3 elzaki elzaki PROPN cana-2815 71 4 , , PUNCT cana-2815 71 5 a a DET cana-2815 71 6 mathematician mathematician NOUN cana-2815 71 7 from from ADP cana-2815 71 8 sudan sudan PROPN cana-2815 71 9 , , PUNCT cana-2815 71 10 introduced introduce VERB cana-2815 71 11 it it PRON cana-2815 71 12 in in ADP cana-2815 71 13 the the DET cana-2815 71 14 1960s 1960s NUM cana-2815 71 15 . . PUNCT cana-2815 72 1 heat heat NOUN cana-2815 72 2 conduction conduction NOUN cana-2815 72 3 , , PUNCT cana-2815 72 4 fluid fluid ADJ cana-2815 72 5 dynamical dynamical ADJ cana-2815 72 6 mechanics mechanic NOUN cana-2815 72 7 , , PUNCT cana-2815 72 8 and and CCONJ cana-2815 72 9 electrical electrical ADJ cana-2815 72 10 circuits circuit NOUN cana-2815 72 11 are be AUX cana-2815 72 12 only only ADV cana-2815 72 13 a a DET cana-2815 72 14 few few ADJ cana-2815 72 15 of of ADP cana-2815 72 16 the the DET cana-2815 72 17 applied apply VERB cana-2815 72 18 scientific scientific ADJ cana-2815 72 19 and and CCONJ cana-2815 72 20 engineering engineering NOUN cana-2815 72 21 domains domain NOUN cana-2815 72 22 where where SCONJ cana-2815 72 23 the the DET cana-2815 72 24 elzaki elzaki NOUN cana-2815 72 25 transformation transformation NOUN cana-2815 72 26 has have AUX cana-2815 72 27 been be AUX cana-2815 72 28 effectively effectively ADV cana-2815 72 29 used use VERB cana-2815 72 30 . . PUNCT cana-2815 73 1 it it PRON cana-2815 73 2 offers offer VERB cana-2815 73 3 an an DET cana-2815 73 4 alternate alternate ADJ cana-2815 73 5 strategy strategy NOUN cana-2815 73 6 for for ADP cana-2815 73 7 resolving resolve VERB cana-2815 73 8 odes ode NOUN cana-2815 73 9 , , PUNCT cana-2815 73 10 especially especially ADV cana-2815 73 11 in in ADP cana-2815 73 12 situations situation NOUN cana-2815 73 13 when when SCONJ cana-2815 73 14 existing existing ADJ cana-2815 73 15 methods method NOUN cana-2815 73 16 are be AUX cana-2815 73 17 not not PART cana-2815 73 18 easily easily ADV cana-2815 73 19 able able ADJ cana-2815 73 20 to to PART cana-2815 73 21 produce produce VERB cana-2815 73 22 analytical analytical ADJ cana-2815 73 23 solutions solution NOUN cana-2815 73 24 . . PUNCT cana-2815 74 1 when when SCONJ cana-2815 74 2 analytical analytical ADJ cana-2815 74 3 solutions solution NOUN cana-2815 74 4 are be AUX cana-2815 74 5 hard hard ADJ cana-2815 74 6 to to PART cana-2815 74 7 come come VERB cana-2815 74 8 by by ADP cana-2815 74 9 with with ADP cana-2815 74 10 other other ADJ cana-2815 74 11 approaches approach NOUN cana-2815 74 12 , , PUNCT cana-2815 74 13 applying apply VERB cana-2815 74 14 the the DET cana-2815 74 15 elzaki elzaki NOUN cana-2815 74 16 transform transform NOUN cana-2815 74 17 to to PART cana-2815 74 18 solve solve VERB cana-2815 74 19 fpdes fpde NOUN cana-2815 74 20 can can AUX cana-2815 74 21 be be AUX cana-2815 74 22 quite quite ADV cana-2815 74 23 helpful helpful ADJ cana-2815 74 24 . . PUNCT cana-2815 75 1 this this DET cana-2815 75 2 section section NOUN cana-2815 75 3 presents present VERB cana-2815 75 4 a a DET cana-2815 75 5 basic basic ADJ cana-2815 75 6 explanation explanation NOUN cana-2815 75 7 of of ADP cana-2815 75 8 elzaki elzaki NOUN cana-2815 75 9 transformation transformation NOUN cana-2815 75 10 and and CCONJ cana-2815 75 11 offers offer VERB cana-2815 75 12 a a DET cana-2815 75 13 framework framework NOUN cana-2815 75 14 for for ADP cana-2815 75 15 transforming transform VERB cana-2815 75 16 the the DET cana-2815 75 17 problem problem NOUN cana-2815 75 18 into into ADP cana-2815 75 19 an an DET cana-2815 75 20 algebraic algebraic ADJ cana-2815 75 21 form form NOUN cana-2815 75 22 that that PRON cana-2815 75 23 can can AUX cana-2815 75 24 be be AUX cana-2815 75 25 solved solve VERB cana-2815 75 26 using use VERB cana-2815 75 27 well well ADV cana-2815 75 28 - - PUNCT cana-2815 75 29 established establish VERB cana-2815 75 30 algebraic algebraic ADJ cana-2815 75 31 techniques technique NOUN cana-2815 75 32 . . PUNCT cana-2815 76 1 definition definition NOUN cana-2815 76 2 1 1 NUM cana-2815 76 3 : : PUNCT cana-2815 76 4 fundamental fundamental ADJ cana-2815 76 5 principle principle NOUN cana-2815 76 6 of of ADP cana-2815 76 7 elzaki elzaki NOUN cana-2815 76 8 transformation transformation NOUN cana-2815 76 9 the the DET cana-2815 76 10 exponential exponential ADJ cana-2815 76 11 form form NOUN cana-2815 76 12 of of ADP cana-2815 76 13 function function NOUN cana-2815 76 14 in in ADP cana-2815 76 15 a a DET cana-2815 76 16 series series NOUN cana-2815 76 17 , , PUNCT cana-2815 76 18 defined define VERB cana-2815 76 19 by by ADP cana-2815 76 20 set set VERB cana-2815 76 21 a a DET cana-2815 76 22 expressed express VERB cana-2815 76 23 as as ADP cana-2815 76 24 new new ADJ cana-2815 76 25 transformation transformation NOUN cana-2815 76 26 which which PRON cana-2815 76 27 renamed rename VERB cana-2815 76 28 as as ADP cana-2815 76 29 an an DET cana-2815 76 30 elzaki elzaki NOUN cana-2815 76 31 transformation transformation NOUN cana-2815 77 1 [ [ X cana-2815 77 2 [ [ X cana-2815 77 3 24 24 NUM cana-2815 77 4 ] ] X cana-2815 77 5 ] ] X cana-2815 77 6 𝐴 𝐴 PROPN cana-2815 77 7 = = SYM cana-2815 77 8 { { PUNCT cana-2815 77 9 𝑓(𝑑 𝑓(𝑑 PROPN cana-2815 77 10 ): ): PUNCT cana-2815 77 11 βˆƒ βˆƒ PROPN cana-2815 77 12 𝑀 𝑀 PROPN cana-2815 77 13 , , PUNCT cana-2815 77 14 π‘˜1 π‘˜1 PROPN cana-2815 77 15 , , PUNCT cana-2815 77 16 π‘˜2 π‘˜2 PROPN cana-2815 77 17 > > X cana-2815 77 18 0 0 PROPN cana-2815 77 19 , , PUNCT cana-2815 77 20 |𝑓(𝑑)| |𝑓(𝑑)| PROPN cana-2815 77 21 < < X cana-2815 77 22 𝑀𝑒 𝑀𝑒 PROPN cana-2815 77 23 |𝑑| |𝑑| AUX cana-2815 77 24 π‘˜π‘— π‘˜π‘— PROPN cana-2815 77 25 , , PUNCT cana-2815 77 26 𝑖𝑓 𝑖𝑓 NUM cana-2815 77 27 𝑑 𝑑 PROPN cana-2815 77 28 ∈ ∈ PROPN cana-2815 77 29 ( ( PUNCT cana-2815 77 30 βˆ’1)𝑗 βˆ’1)𝑗 NOUN cana-2815 77 31 𝑋 𝑋 PROPN cana-2815 77 32 [ [ PUNCT cana-2815 77 33 0,∞ 0,∞ NUM cana-2815 77 34 ) ) PUNCT cana-2815 77 35 } } PUNCT cana-2815 77 36 regarding regard VERB cana-2815 77 37 the the DET cana-2815 77 38 specified specify VERB cana-2815 77 39 function function NOUN cana-2815 77 40 in in ADP cana-2815 77 41 the the DET cana-2815 77 42 set set NOUN cana-2815 77 43 , , PUNCT cana-2815 77 44 m m VERB cana-2815 77 45 is be AUX cana-2815 77 46 defined define VERB cana-2815 77 47 as as ADP cana-2815 77 48 finite finite NOUN cana-2815 77 49 or or CCONJ cana-2815 77 50 infinite infinite ADJ cana-2815 77 51 number number NOUN cana-2815 77 52 , , PUNCT cana-2815 77 53 as as ADP cana-2815 77 54 π‘˜1 π‘˜1 PROPN cana-2815 77 55 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-2815 77 56 π‘˜2 π‘˜2 PROPN cana-2815 77 57 . . PUNCT cana-2815 78 1 the the DET cana-2815 78 2 elzaki elzaki NOUN cana-2815 78 3 transform transform NOUN cana-2815 78 4 defined define VERB cana-2815 78 5 as as ADP cana-2815 78 6 operator operator NOUN cana-2815 78 7 𝐸(g(𝜏 𝐸(g(𝜏 NOUN cana-2815 78 8 ) ) PUNCT cana-2815 78 9 ) ) PUNCT cana-2815 78 10 in in ADP cana-2815 78 11 the the DET cana-2815 78 12 integral integral ADJ cana-2815 78 13 form form NOUN cana-2815 78 14 as as SCONJ cana-2815 78 15 follows follow VERB cana-2815 78 16 . . PUNCT cana-2815 79 1 𝐸 𝐸 PRON cana-2815 79 2 [ [ PUNCT cana-2815 79 3 𝑓(𝜏 𝑓(𝜏 PROPN cana-2815 79 4 ) ) PUNCT cana-2815 79 5 ] ] PUNCT cana-2815 80 1 = = PUNCT cana-2815 80 2 𝑣 𝑣 DET cana-2815 80 3 ∫ ∫ PROPN cana-2815 80 4 𝑓(𝜏)βˆ’ 𝑓(𝜏)βˆ’ PROPN cana-2815 80 5 𝜏 𝜏 PROPN cana-2815 80 6 𝑣 𝑣 X cana-2815 80 7 π‘‘πœ π‘‘πœ NOUN cana-2815 80 8 = = SYM cana-2815 80 9 𝑇 𝑇 PROPN cana-2815 80 10 ( ( PUNCT cana-2815 80 11 𝑣 𝑣 NOUN cana-2815 80 12 ) ) PUNCT cana-2815 80 13 , , PUNCT cana-2815 80 14 𝜏 𝜏 PRON cana-2815 80 15 β‰₯ β‰₯ NOUN cana-2815 80 16 0 0 NUM cana-2815 80 17 , , PUNCT cana-2815 80 18 ∞ ∞ PROPN cana-2815 80 19 0 0 NUM cana-2815 80 20 π‘˜1 π‘˜1 PROPN cana-2815 80 21 ≀ ≀ NUM cana-2815 80 22 𝑣 𝑣 DET cana-2815 80 23 ≀ ≀ ADJ cana-2815 80 24 π‘˜2 π‘˜2 PROPN cana-2815 80 25 three three NUM cana-2815 80 26 - - PUNCT cana-2815 80 27 dimensional dimensional ADJ cana-2815 80 28 fractional fractional ADJ cana-2815 80 29 order order NOUN cana-2815 80 30 telegraph telegraph NOUN cana-2815 80 31 equation equation NOUN cana-2815 80 32 : : PUNCT cana-2815 80 33 communications communication NOUN cana-2815 80 34 on on ADP cana-2815 80 35 applied apply VERB cana-2815 80 36 nonlinear nonlinear ADJ cana-2815 80 37 analysis analysis NOUN cana-2815 80 38 issn issn NOUN cana-2815 80 39 : : PUNCT cana-2815 80 40 1074 1074 NUM cana-2815 80 41 - - PUNCT cana-2815 80 42 133x 133x NUM cana-2815 80 43 vol vol NOUN cana-2815 80 44 32 32 NUM cana-2815 80 45 no no NOUN cana-2815 80 46 . . PUNCT cana-2815 81 1 4s 4s NUM cana-2815 81 2 ( ( PUNCT cana-2815 81 3 2025 2025 NUM cana-2815 81 4 ) ) PUNCT cana-2815 81 5 313 313 NUM cana-2815 81 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 81 7 the the DET cana-2815 81 8 elzaki elzaki NOUN cana-2815 81 9 transformation transformation NOUN cana-2815 81 10 for for ADP cana-2815 81 11 some some DET cana-2815 81 12 functions function NOUN cana-2815 81 13 is be AUX cana-2815 81 14 defined define VERB cana-2815 81 15 below below ADP cana-2815 81 16 [ [ X cana-2815 81 17 [ [ X cana-2815 81 18 24 24 NUM cana-2815 81 19 ] ] X cana-2815 81 20 ] ] PUNCT cana-2815 81 21 . . PUNCT cana-2815 82 1 𝒇(𝒕 𝒇(𝒕 NOUN cana-2815 82 2 ) ) PUNCT cana-2815 82 3 𝑬[𝒇(𝒕 𝑬[𝒇(𝒕 PROPN cana-2815 82 4 ) ) PUNCT cana-2815 82 5 ] ] PUNCT cana-2815 83 1 = = SYM cana-2815 83 2 𝑻(𝑣 𝑻(𝑣 X cana-2815 83 3 ) ) PUNCT cana-2815 83 4 1 1 NUM cana-2815 83 5 𝟏 𝟏 NUM cana-2815 83 6 π‘£πŸ π‘£πŸ NOUN cana-2815 83 7 2 2 NUM cana-2815 83 8 𝒕 𝒕 NOUN cana-2815 83 9 π‘£πŸ‘ π‘£πŸ‘ NOUN cana-2815 83 10 3 3 NUM cana-2815 83 11 𝒕𝒏 𝒕𝒏 INTJ cana-2815 83 12 𝒏 𝒏 PROPN cana-2815 83 13 ! ! PROPN cana-2815 83 14 𝑣𝒏+𝟐 𝑣𝒏+𝟐 VERB cana-2815 84 1 the the DET cana-2815 84 2 following follow VERB cana-2815 84 3 conclusion conclusion NOUN cana-2815 84 4 was be AUX cana-2815 84 5 established establish VERB cana-2815 84 6 based base VERB cana-2815 84 7 on on ADP cana-2815 84 8 the the DET cana-2815 84 9 description description NOUN cana-2815 84 10 and and CCONJ cana-2815 84 11 fundamental fundamental ADJ cana-2815 84 12 analyses analysis NOUN cana-2815 84 13 . . PUNCT cana-2815 85 1 𝐸[𝑑𝑛 𝐸[𝑑𝑛 X cana-2815 85 2 ] ] PUNCT cana-2815 85 3 = = SYM cana-2815 85 4 𝑛 𝑛 PROPN cana-2815 85 5 ! ! NOUN cana-2815 85 6 𝑣𝑛+2 𝑣𝑛+2 NUM cana-2815 85 7 𝐸[𝑓 𝐸[𝑓 PROPN cana-2815 85 8 β€²(𝑑 β€²(𝑑 PROPN cana-2815 85 9 ) ) PUNCT cana-2815 85 10 ] ] PUNCT cana-2815 86 1 = = SYM cana-2815 86 2 𝐹 𝐹 PROPN cana-2815 86 3 ( ( PUNCT cana-2815 86 4 𝑣 𝑣 NOUN cana-2815 86 5 ) ) PUNCT cana-2815 86 6 𝑣 𝑣 ADP cana-2815 86 7 βˆ’ βˆ’ PROPN cana-2815 86 8 𝑣𝑓(0 𝑣𝑓(0 NOUN cana-2815 86 9 ) ) PUNCT cana-2815 86 10 𝐸[𝑓 𝐸[𝑓 NOUN cana-2815 86 11 β€²β€²(𝑑 β€²β€²(𝑑 PROPN cana-2815 86 12 ) ) PUNCT cana-2815 86 13 ] ] PUNCT cana-2815 86 14 = = PUNCT cana-2815 86 15 𝐹(𝜸 𝐹(𝜸 NUM cana-2815 86 16 ) ) PUNCT cana-2815 86 17 𝑣2 𝑣2 PROPN cana-2815 86 18 βˆ’ βˆ’ PROPN cana-2815 86 19 𝑓(0 𝑓(0 PROPN cana-2815 86 20 ) ) PUNCT cana-2815 86 21 βˆ’ βˆ’ NOUN cana-2815 86 22 𝑣𝑓 𝑣𝑓 ADP cana-2815 86 23 β€²(0 β€²(0 NOUN cana-2815 86 24 ) ) PUNCT cana-2815 86 25 𝐸[𝑓(𝑛)(𝑑 𝐸[𝑓(𝑛)(𝑑 PROPN cana-2815 86 26 ) ) PUNCT cana-2815 86 27 ] ] PUNCT cana-2815 86 28 = = PUNCT cana-2815 86 29 𝐹(𝑣 𝐹(𝑣 NUM cana-2815 86 30 ) ) PUNCT cana-2815 86 31 𝑣𝑛 𝑣𝑛 ADP cana-2815 86 32 βˆ’ βˆ’ PROPN cana-2815 86 33 βˆ‘ βˆ‘ PUNCT cana-2815 86 34 𝑣2βˆ’π‘›+π‘˜π‘“(π‘˜)(0 𝑣2βˆ’π‘›+π‘˜π‘“(π‘˜)(0 PROPN cana-2815 86 35 ) ) PUNCT cana-2815 86 36 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-2815 86 37 π‘˜=0 π‘˜=0 PRON cana-2815 86 38 definition definition NOUN cana-2815 86 39 2 2 NUM cana-2815 86 40 : : PUNCT cana-2815 86 41 caputo caputo PROPN cana-2815 86 42 fractional fractional PROPN cana-2815 86 43 elzaki elzaki PROPN cana-2815 86 44 transform transform NOUN cana-2815 86 45 operator operator NOUN cana-2815 86 46 the the DET cana-2815 86 47 caputo caputo PROPN cana-2815 86 48 fractional fractional PROPN cana-2815 86 49 operator operator NOUN cana-2815 86 50 's 's PART cana-2815 86 51 elzaki elzaki NOUN cana-2815 86 52 transformation transformation NOUN cana-2815 86 53 is be AUX cana-2815 86 54 as as SCONJ cana-2815 86 55 follows follow VERB cana-2815 86 56 : : PUNCT cana-2815 86 57 𝐸 𝐸 PROPN cana-2815 86 58 [ [ PUNCT cana-2815 86 59 πœ•π›Ό πœ•π›Ό NOUN cana-2815 86 60 πœ•πœπ›Ό πœ•πœπ›Ό NOUN cana-2815 86 61 𝑓(𝜏 𝑓(𝜏 PROPN cana-2815 86 62 ) ) PUNCT cana-2815 86 63 ] ] PUNCT cana-2815 87 1 = = PUNCT cana-2815 87 2 𝐸 𝐸 PROPN cana-2815 87 3 [ [ X cana-2815 87 4 𝑓 𝑓 X cana-2815 87 5 ( ( PUNCT cana-2815 87 6 𝜏 𝜏 NOUN cana-2815 87 7 ) ) PUNCT cana-2815 87 8 ] ] PUNCT cana-2815 88 1 𝑣𝛼 𝑣𝛼 ADP cana-2815 88 2 βˆ’ βˆ’ PROPN cana-2815 88 3 βˆ‘ βˆ‘ PUNCT cana-2815 88 4 π‘£π‘˜βˆ’π›Ό+2 π‘£π‘˜βˆ’π›Ό+2 VERB cana-2815 88 5 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-2815 88 6 π‘˜=0 π‘˜=0 ADP cana-2815 88 7 𝑓(π‘˜)(0 𝑓(π‘˜)(0 PROPN cana-2815 88 8 ) ) PUNCT cana-2815 88 9 , , PUNCT cana-2815 89 1 𝑛 𝑛 DET cana-2815 89 2 βˆ’ βˆ’ PROPN cana-2815 89 3 1 1 NUM cana-2815 89 4 < < X cana-2815 89 5 𝛼 𝛼 PROPN cana-2815 89 6 ≀ ≀ NUM cana-2815 89 7 𝑛 𝑛 DET cana-2815 89 8 3 3 NUM cana-2815 89 9 . . PUNCT cana-2815 90 1 the the DET cana-2815 90 2 methodology methodology NOUN cana-2815 90 3 for for ADP cana-2815 90 4 modified modified ADJ cana-2815 90 5 adomian adomian NOUN cana-2815 90 6 decomposition decomposition NOUN cana-2815 90 7 elzaki elzaki NOUN cana-2815 90 8 transform transform NOUN cana-2815 90 9 ( ( PUNCT cana-2815 90 10 madetm madetm ADJ cana-2815 90 11 ): ): PUNCT cana-2815 90 12 consider consider VERB cana-2815 90 13 the the DET cana-2815 90 14 partial partial ADJ cana-2815 90 15 differentiation differentiation NOUN cana-2815 90 16 equation equation NOUN cana-2815 90 17 of of ADP cana-2815 90 18 fractional fractional ADJ cana-2815 90 19 order order NOUN cana-2815 90 20 non non ADJ cana-2815 90 21 - - NOUN cana-2815 90 22 linearity linearity ADJ cana-2815 90 23 . . PUNCT cana-2815 91 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 91 2 𝛼 𝛼 PROPN cana-2815 91 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 91 4 , , PUNCT cana-2815 91 5 𝑑 𝑑 PROPN cana-2815 91 6 ) ) PUNCT cana-2815 91 7 + + CCONJ cana-2815 91 8 𝑅 𝑅 PROPN cana-2815 91 9 [ [ X cana-2815 91 10 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 91 11 , , PUNCT cana-2815 91 12 𝑑 𝑑 PROPN cana-2815 91 13 ) ) PUNCT cana-2815 91 14 ] ] PUNCT cana-2815 92 1 + + CCONJ cana-2815 92 2 𝑁[βˆ…(π‘₯ 𝑁[βˆ…(π‘₯ NOUN cana-2815 92 3 , , PUNCT cana-2815 92 4 𝑑 𝑑 NOUN cana-2815 92 5 ) ) PUNCT cana-2815 92 6 ] ] PUNCT cana-2815 93 1 = = PUNCT cana-2815 93 2 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-2815 93 3 , , PUNCT cana-2815 93 4 𝑑 𝑑 NOUN cana-2815 93 5 ) ) PUNCT cana-2815 93 6 π‘₯ π‘₯ NOUN cana-2815 93 7 , , PUNCT cana-2815 93 8 𝑑 𝑑 X cana-2815 93 9 β‰₯ β‰₯ NOUN cana-2815 93 10 0 0 NUM cana-2815 93 11 π‘š π‘š ADP cana-2815 93 12 βˆ’ βˆ’ PROPN cana-2815 93 13 1 1 NUM cana-2815 93 14 < < X cana-2815 93 15 𝛼 𝛼 X cana-2815 93 16 < < X cana-2815 93 17 π‘š π‘š X cana-2815 93 18 with with ADP cana-2815 93 19 initial initial ADJ cana-2815 93 20 condition condition NOUN cana-2815 93 21 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 93 22 , , PUNCT cana-2815 93 23 0 0 NUM cana-2815 93 24 ) ) PUNCT cana-2815 93 25 = = SYM cana-2815 93 26 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-2815 93 27 ) ) PUNCT cana-2815 93 28 the the DET cana-2815 93 29 caputo caputo PROPN cana-2815 93 30 fractional fractional PROPN cana-2815 93 31 function function PROPN cana-2815 93 32 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 93 33 , , PUNCT cana-2815 93 34 𝑑 𝑑 PROPN cana-2815 93 35 ) ) PUNCT cana-2815 93 36 defined define VERB cana-2815 93 37 as as ADP cana-2815 93 38 : : PUNCT cana-2815 93 39 𝐷𝑑 𝐷𝑑 PROPN cana-2815 93 40 𝛼 𝛼 PROPN cana-2815 93 41 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 93 42 , , PUNCT cana-2815 93 43 𝑑 𝑑 PROPN cana-2815 93 44 ) ) PUNCT cana-2815 93 45 = = SYM cana-2815 93 46 πœ•π›Όβˆ…(π‘₯ πœ•π›Όβˆ…(π‘₯ PROPN cana-2815 93 47 , , PUNCT cana-2815 93 48 𝑑 𝑑 NOUN cana-2815 93 49 ) ) PUNCT cana-2815 93 50 πœ•π‘‘π›Ό πœ•π‘‘π›Ό PUNCT cana-2815 94 1 = = PRON cana-2815 94 2 { { PUNCT cana-2815 94 3 1 1 NUM cana-2815 94 4 ⌈(𝑛 ⌈(𝑛 NUM cana-2815 94 5 βˆ’ βˆ’ NOUN cana-2815 94 6 𝛼 𝛼 NOUN cana-2815 94 7 ) ) PUNCT cana-2815 94 8 ∫(π‘‘βˆ’π‘₯)π‘›βˆ’π›Όβˆ’1 ∫(π‘‘βˆ’π‘₯)π‘›βˆ’π›Όβˆ’1 NOUN cana-2815 94 9 πœ•π‘›βˆ…(π‘₯ πœ•π‘›βˆ…(π‘₯ NOUN cana-2815 94 10 , , PUNCT cana-2815 94 11 𝑑 𝑑 PROPN cana-2815 94 12 ) ) PUNCT cana-2815 94 13 πœ•π‘‘π‘› πœ•π‘‘π‘› PUNCT cana-2815 94 14 𝑑𝑑 𝑑𝑑 ADP cana-2815 94 15 , , PUNCT cana-2815 94 16 𝑛 𝑛 PRON cana-2815 94 17 βˆ’ βˆ’ PROPN cana-2815 94 18 1 1 NUM cana-2815 94 19 < < X cana-2815 94 20 𝛼 𝛼 X cana-2815 94 21 < < X cana-2815 94 22 𝑛 𝑛 PRON cana-2815 94 23 𝑑 𝑑 PROPN cana-2815 94 24 π‘Ž π‘Ž PROPN cana-2815 94 25 πœ•π‘›βˆ…(π‘₯ πœ•π‘›βˆ…(π‘₯ PROPN cana-2815 94 26 , , PUNCT cana-2815 94 27 𝑑 𝑑 PROPN cana-2815 94 28 ) ) PUNCT cana-2815 94 29 πœ•π‘‘π‘› πœ•π‘‘π‘› PUNCT cana-2815 94 30 𝛼 𝛼 X cana-2815 94 31 = = NOUN cana-2815 94 32 𝑛 𝑛 PRON cana-2815 94 33 ∈ ∈ NOUN cana-2815 94 34 𝑁 𝑁 NOUN cana-2815 94 35 where where SCONJ cana-2815 94 36 𝐷𝑑 𝐷𝑑 PROPN cana-2815 94 37 𝛼 𝛼 NOUN cana-2815 94 38 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 94 39 , , PUNCT cana-2815 94 40 𝑑 𝑑 PROPN cana-2815 94 41 ) ) PUNCT cana-2815 94 42 is be AUX cana-2815 94 43 caputo caputo PROPN cana-2815 94 44 fractional fractional ADJ cana-2815 94 45 order order NOUN cana-2815 94 46 derivative derivative ADJ cana-2815 94 47 𝛼 𝛼 NOUN cana-2815 94 48 , , PUNCT cana-2815 94 49 n n PROPN cana-2815 94 50 and and CCONJ cana-2815 94 51 are be AUX cana-2815 94 52 r r NOUN cana-2815 94 53 nonlinear nonlinear ADJ cana-2815 94 54 and and CCONJ cana-2815 94 55 linear linear ADJ cana-2815 94 56 terms term NOUN cana-2815 94 57 respectively respectively ADV cana-2815 94 58 , , PUNCT cana-2815 94 59 and and CCONJ cana-2815 94 60 𝑔 𝑔 PROPN cana-2815 94 61 is be AUX cana-2815 94 62 source source NOUN cana-2815 94 63 term term NOUN cana-2815 94 64 . . PUNCT cana-2815 95 1 taking take VERB cana-2815 95 2 the the DET cana-2815 95 3 elzaki elzaki NOUN cana-2815 95 4 transform transform NOUN cana-2815 95 5 on on ADP cana-2815 95 6 both both DET cana-2815 95 7 sides side NOUN cana-2815 95 8 of of ADP cana-2815 95 9 equation equation NOUN cana-2815 95 10 ( ( PUNCT cana-2815 95 11 2 2 NUM cana-2815 95 12 ) ) PUNCT cana-2815 95 13 e e NOUN cana-2815 95 14 [ [ PUNCT cana-2815 95 15 𝐷𝑑 𝐷𝑑 PROPN cana-2815 95 16 𝛼 𝛼 NOUN cana-2815 95 17 βˆ… βˆ… NOUN cana-2815 95 18 ( ( PUNCT cana-2815 95 19 π‘₯. π‘₯. ADV cana-2815 95 20 , , PUNCT cana-2815 95 21 𝑑 𝑑 NOUN cana-2815 95 22 ) ) PUNCT cana-2815 95 23 ] ] PUNCT cana-2815 96 1 + + CCONJ cana-2815 96 2 𝐸 𝐸 PROPN cana-2815 96 3 [ [ X cana-2815 96 4 𝑅 𝑅 PROPN cana-2815 97 1 [ [ X cana-2815 97 2 βˆ…(π‘₯. βˆ…(π‘₯. NOUN cana-2815 97 3 , , PUNCT cana-2815 97 4 𝑑 𝑑 NOUN cana-2815 97 5 ) ) PUNCT cana-2815 97 6 ] ] PUNCT cana-2815 98 1 + + CCONJ cana-2815 98 2 𝑁[βˆ…(π‘₯ 𝑁[βˆ…(π‘₯ NOUN cana-2815 98 3 , , PUNCT cana-2815 98 4 𝑑 𝑑 PROPN cana-2815 98 5 ) ) PUNCT cana-2815 98 6 ] ] PUNCT cana-2815 98 7 ] ] PUNCT cana-2815 99 1 = = SYM cana-2815 99 2 𝐸[𝑔(π‘₯. 𝐸[𝑔(π‘₯. X cana-2815 99 3 , , PUNCT cana-2815 99 4 𝑑 𝑑 PROPN cana-2815 99 5 ) ) PUNCT cana-2815 99 6 ] ] PUNCT cana-2815 99 7 1 1 NUM cana-2815 99 8 𝑣𝛼 𝑣𝛼 NOUN cana-2815 99 9 𝐸[βˆ… 𝐸[βˆ… PROPN cana-2815 99 10 ( ( PUNCT cana-2815 99 11 π‘₯ π‘₯ PROPN cana-2815 99 12 , , PUNCT cana-2815 99 13 𝑑 𝑑 PROPN cana-2815 99 14 ) ) PUNCT cana-2815 99 15 ] ] PUNCT cana-2815 100 1 βˆ’ βˆ’ PROPN cana-2815 100 2 𝑣2βˆ’π›Ό 𝑣2βˆ’π›Ό PROPN cana-2815 100 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 100 4 , , PUNCT cana-2815 100 5 0 0 NUM cana-2815 100 6 ) ) PUNCT cana-2815 100 7 = = SYM cana-2815 100 8 𝐸[𝑔(π‘₯ 𝐸[𝑔(π‘₯ ADJ cana-2815 100 9 , , PUNCT cana-2815 100 10 𝑑 𝑑 PROPN cana-2815 100 11 ) ) PUNCT cana-2815 100 12 ] ] PUNCT cana-2815 100 13 βˆ’ βˆ’ PROPN cana-2815 100 14 𝐸 𝐸 PROPN cana-2815 100 15 [ [ X cana-2815 100 16 𝑅 𝑅 PROPN cana-2815 100 17 [ [ X cana-2815 100 18 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 100 19 , , PUNCT cana-2815 100 20 𝑑 𝑑 PROPN cana-2815 100 21 ) ) PUNCT cana-2815 100 22 ] ] PUNCT cana-2815 100 23 + + CCONJ cana-2815 100 24 𝑁[βˆ…(π‘₯ 𝑁[βˆ…(π‘₯ NOUN cana-2815 100 25 , , PUNCT cana-2815 100 26 𝑑 𝑑 PROPN cana-2815 100 27 ) ) PUNCT cana-2815 100 28 ] ] X cana-2815 100 29 ] ] X cana-2815 100 30 𝐸 𝐸 PROPN cana-2815 100 31 [ [ X cana-2815 100 32 βˆ… βˆ… NOUN cana-2815 100 33 ( ( PUNCT cana-2815 100 34 π‘₯ π‘₯ INTJ cana-2815 100 35 , , PUNCT cana-2815 100 36 𝑑 𝑑 PROPN cana-2815 100 37 ) ) PUNCT cana-2815 100 38 ] ] PUNCT cana-2815 101 1 = = SYM cana-2815 101 2 𝑣2 𝑣2 NUM cana-2815 101 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 101 4 , , PUNCT cana-2815 101 5 0 0 NUM cana-2815 101 6 ) ) PUNCT cana-2815 102 1 + + CCONJ cana-2815 102 2 𝑣𝛼 𝑣𝛼 ADP cana-2815 102 3 𝐸[𝑔(π‘₯ 𝐸[𝑔(π‘₯ ADJ cana-2815 102 4 , , PUNCT cana-2815 102 5 𝑑 𝑑 PROPN cana-2815 102 6 ) ) PUNCT cana-2815 102 7 ] ] PUNCT cana-2815 102 8 βˆ’ βˆ’ PROPN cana-2815 103 1 𝑣𝛼 𝑣𝛼 NOUN cana-2815 103 2 𝐸 𝐸 PROPN cana-2815 104 1 [ [ X cana-2815 104 2 𝑅 𝑅 PROPN cana-2815 104 3 [ [ X cana-2815 104 4 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 104 5 , , PUNCT cana-2815 104 6 𝑑 𝑑 PROPN cana-2815 104 7 ) ) PUNCT cana-2815 104 8 ] ] PUNCT cana-2815 104 9 + + CCONJ cana-2815 104 10 𝑁[βˆ…(π‘₯ 𝑁[βˆ…(π‘₯ NOUN cana-2815 104 11 , , PUNCT cana-2815 104 12 𝑑 𝑑 PROPN cana-2815 104 13 ) ) PUNCT cana-2815 104 14 ] ] PUNCT cana-2815 104 15 ] ] PUNCT cana-2815 104 16 from from ADP cana-2815 104 17 equation equation NOUN cana-2815 104 18 ( ( PUNCT cana-2815 104 19 3 3 X cana-2815 104 20 ) ) PUNCT cana-2815 104 21 its its PRON cana-2815 104 22 initial initial ADJ cana-2815 104 23 conditions condition NOUN cana-2815 104 24 is be AUX cana-2815 104 25 : : PUNCT cana-2815 104 26 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 104 27 , , PUNCT cana-2815 104 28 0 0 NUM cana-2815 104 29 ) ) PUNCT cana-2815 104 30 = = SYM cana-2815 105 1 𝑓 𝑓 PROPN cana-2815 105 2 ( ( PUNCT cana-2815 105 3 π‘₯ π‘₯ NOUN cana-2815 105 4 ) ) PUNCT cana-2815 105 5 𝐸 𝐸 PROPN cana-2815 106 1 [ [ PUNCT cana-2815 106 2 βˆ… βˆ… NOUN cana-2815 106 3 ( ( PUNCT cana-2815 106 4 π‘₯ π‘₯ INTJ cana-2815 106 5 , , PUNCT cana-2815 106 6 𝑑 𝑑 PROPN cana-2815 106 7 ) ) PUNCT cana-2815 106 8 ] ] PUNCT cana-2815 106 9 = = PUNCT cana-2815 106 10 𝑣2 𝑣2 NUM cana-2815 106 11 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-2815 106 12 ) ) PUNCT cana-2815 107 1 + + CCONJ cana-2815 107 2 𝑣𝛼 𝑣𝛼 ADP cana-2815 107 3 𝐸[𝑔(π‘₯ 𝐸[𝑔(π‘₯ ADJ cana-2815 107 4 , , PUNCT cana-2815 107 5 𝑑 𝑑 PROPN cana-2815 107 6 ) ) PUNCT cana-2815 107 7 ] ] PUNCT cana-2815 108 1 βˆ’ βˆ’ PROPN cana-2815 108 2 𝑣𝛼 𝑣𝛼 NOUN cana-2815 108 3 𝐸 𝐸 PROPN cana-2815 109 1 [ [ X cana-2815 109 2 𝑅 𝑅 PROPN cana-2815 109 3 [ [ X cana-2815 109 4 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 109 5 , , PUNCT cana-2815 109 6 𝑑 𝑑 PROPN cana-2815 109 7 ) ) PUNCT cana-2815 109 8 ] ] PUNCT cana-2815 109 9 + + CCONJ cana-2815 109 10 𝑁[βˆ…(π‘₯ 𝑁[βˆ…(π‘₯ NOUN cana-2815 109 11 , , PUNCT cana-2815 109 12 𝑑 𝑑 PROPN cana-2815 109 13 ) ) PUNCT cana-2815 109 14 ] ] X cana-2815 109 15 ] ] X cana-2815 109 16 ( ( PUNCT cana-2815 109 17 2 2 NUM cana-2815 109 18 ) ) PUNCT cana-2815 109 19 ( ( PUNCT cana-2815 109 20 5 5 NUM cana-2815 109 21 ) ) PUNCT cana-2815 109 22 ( ( PUNCT cana-2815 109 23 1 1 X cana-2815 109 24 ) ) PUNCT cana-2815 109 25 ( ( PUNCT cana-2815 109 26 4 4 NUM cana-2815 109 27 ) ) PUNCT cana-2815 109 28 ( ( PUNCT cana-2815 109 29 3 3 X cana-2815 109 30 ) ) PUNCT cana-2815 109 31 communications communication NOUN cana-2815 109 32 on on ADP cana-2815 109 33 applied apply VERB cana-2815 109 34 nonlinear nonlinear ADJ cana-2815 109 35 analysis analysis NOUN cana-2815 109 36 issn issn NOUN cana-2815 109 37 : : PUNCT cana-2815 109 38 1074 1074 NUM cana-2815 109 39 - - PUNCT cana-2815 109 40 133x 133x NUM cana-2815 109 41 vol vol NOUN cana-2815 109 42 32 32 NUM cana-2815 109 43 no no NOUN cana-2815 109 44 . . PUNCT cana-2815 110 1 4s 4s NUM cana-2815 110 2 ( ( PUNCT cana-2815 110 3 2025 2025 NUM cana-2815 110 4 ) ) PUNCT cana-2815 110 5 314 314 NUM cana-2815 110 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 110 7 applying apply VERB cana-2815 110 8 inverse inverse NOUN cana-2815 110 9 elzaki elzaki NOUN cana-2815 110 10 transform transform VERB cana-2815 110 11 on on ADP cana-2815 110 12 equation equation NOUN cana-2815 110 13 ( ( PUNCT cana-2815 110 14 5 5 NUM cana-2815 110 15 ) ) PUNCT cana-2815 110 16 πΈβˆ’1[𝐸 πΈβˆ’1[𝐸 NOUN cana-2815 110 17 [ [ PUNCT cana-2815 110 18 βˆ… βˆ… NOUN cana-2815 110 19 ( ( PUNCT cana-2815 110 20 π‘₯ π‘₯ INTJ cana-2815 110 21 , , PUNCT cana-2815 110 22 𝑑 𝑑 PROPN cana-2815 110 23 ) ) PUNCT cana-2815 110 24 ] ] PUNCT cana-2815 110 25 ] ] X cana-2815 111 1 = = PUNCT cana-2815 111 2 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 112 1 [ [ X cana-2815 112 2 𝑣2 𝑣2 NOUN cana-2815 112 3 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-2815 112 4 ) ) PUNCT cana-2815 113 1 + + CCONJ cana-2815 113 2 𝑣𝛼 𝑣𝛼 ADP cana-2815 113 3 𝐸[𝑔(π‘₯ 𝐸[𝑔(π‘₯ ADJ cana-2815 113 4 , , PUNCT cana-2815 113 5 𝑑 𝑑 PROPN cana-2815 113 6 ) ) PUNCT cana-2815 113 7 ] ] PUNCT cana-2815 114 1 βˆ’ βˆ’ PROPN cana-2815 114 2 𝑣𝛼𝐸 𝑣𝛼𝐸 PROPN cana-2815 114 3 [ [ X cana-2815 114 4 𝑅 𝑅 PROPN cana-2815 114 5 [ [ X cana-2815 114 6 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 114 7 , , PUNCT cana-2815 114 8 𝑑 𝑑 PROPN cana-2815 114 9 ) ) PUNCT cana-2815 114 10 ] ] PUNCT cana-2815 115 1 + + CCONJ cana-2815 115 2 𝑁[βˆ…(π‘₯ 𝑁[βˆ…(π‘₯ NOUN cana-2815 115 3 , , PUNCT cana-2815 115 4 𝑑 𝑑 PROPN cana-2815 115 5 ) ) PUNCT cana-2815 115 6 ] ] PUNCT cana-2815 115 7 ] ] X cana-2815 115 8 ] ] X cana-2815 115 9 βˆ… βˆ… NOUN cana-2815 115 10 ( ( PUNCT cana-2815 115 11 π‘₯ π‘₯ INTJ cana-2815 115 12 , , PUNCT cana-2815 115 13 𝑑 𝑑 PROPN cana-2815 115 14 ) ) PUNCT cana-2815 115 15 = = PUNCT cana-2815 115 16 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 116 1 [ [ X cana-2815 116 2 𝑣2 𝑣2 NUM cana-2815 116 3 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-2815 116 4 ) ) PUNCT cana-2815 116 5 ] ] PUNCT cana-2815 117 1 + + CCONJ cana-2815 117 2 πΈβˆ’1[𝑣𝛼 πΈβˆ’1[𝑣𝛼 PROPN cana-2815 117 3 𝐸 𝐸 PROPN cana-2815 118 1 [ [ X cana-2815 118 2 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-2815 118 3 , , PUNCT cana-2815 118 4 𝑑 𝑑 PROPN cana-2815 118 5 ) ) PUNCT cana-2815 118 6 ] ] PUNCT cana-2815 118 7 ] ] PUNCT cana-2815 118 8 βˆ’ βˆ’ PUNCT cana-2815 118 9 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 119 1 [ [ X cana-2815 119 2 𝑣𝛼𝐸 𝑣𝛼𝐸 PROPN cana-2815 119 3 [ [ X cana-2815 119 4 𝑅 𝑅 PROPN cana-2815 119 5 [ [ X cana-2815 119 6 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 119 7 , , PUNCT cana-2815 119 8 𝑑 𝑑 PROPN cana-2815 119 9 ) ) PUNCT cana-2815 119 10 ] ] PUNCT cana-2815 120 1 + + CCONJ cana-2815 120 2 𝑁[βˆ…(π‘₯ 𝑁[βˆ…(π‘₯ NOUN cana-2815 120 3 , , PUNCT cana-2815 120 4 𝑑 𝑑 NOUN cana-2815 120 5 ) ) PUNCT cana-2815 120 6 ] ] PUNCT cana-2815 120 7 ] ] X cana-2815 120 8 ] ] PUNCT cana-2815 120 9 by by ADP cana-2815 120 10 applying apply VERB cana-2815 120 11 madm madm NOUN cana-2815 120 12 on on ADP cana-2815 120 13 right right ADJ cana-2815 120 14 hand hand NOUN cana-2815 120 15 side side NOUN cana-2815 120 16 of of ADP cana-2815 120 17 equation equation NOUN cana-2815 120 18 occurs occur VERB cana-2815 120 19 the the DET cana-2815 120 20 solution solution NOUN cana-2815 120 21 in in ADP cana-2815 120 22 infinite infinite ADJ cana-2815 120 23 series series NOUN cana-2815 120 24 given give VERB cana-2815 120 25 below below ADP cana-2815 120 26 . . PUNCT cana-2815 121 1 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 121 2 , , PUNCT cana-2815 121 3 𝑑 𝑑 PROPN cana-2815 121 4 ) ) PUNCT cana-2815 121 5 = = PUNCT cana-2815 121 6 βˆ‘ βˆ‘ PUNCT cana-2815 121 7 βˆ…π‘›(π‘₯ βˆ…π‘›(π‘₯ NOUN cana-2815 121 8 , , PUNCT cana-2815 121 9 𝑑 𝑑 PROPN cana-2815 121 10 ) ) PUNCT cana-2815 121 11 ∞ ∞ NUM cana-2815 122 1 𝑛=0 𝑛=0 PROPN cana-2815 122 2 the the DET cana-2815 122 3 non non ADJ cana-2815 122 4 - - ADJ cana-2815 122 5 linear linear ADJ cana-2815 122 6 terms term NOUN cana-2815 122 7 n n CCONJ cana-2815 122 8 in in ADP cana-2815 122 9 an an DET cana-2815 122 10 adomian adomian NOUN cana-2815 122 11 polynomial polynomial NOUN cana-2815 122 12 represented represent VERB cana-2815 122 13 as as SCONJ cana-2815 122 14 follows follow VERB cana-2815 122 15 . . PUNCT cana-2815 123 1 𝑁 𝑁 PROPN cana-2815 123 2 [ [ X cana-2815 123 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 123 4 , , PUNCT cana-2815 123 5 𝑑 𝑑 PROPN cana-2815 123 6 ) ) PUNCT cana-2815 123 7 ] ] PUNCT cana-2815 124 1 = = PUNCT cana-2815 124 2 βˆ‘ βˆ‘ PUNCT cana-2815 124 3 𝐴𝑛 𝐴𝑛 NOUN cana-2815 124 4 ∞ ∞ NUM cana-2815 124 5 𝑛=0 𝑛=0 NOUN cana-2815 124 6 where where SCONJ cana-2815 124 7 𝐴𝑛 𝐴𝑛 NOUN cana-2815 124 8 = = SYM cana-2815 124 9 1 1 NUM cana-2815 124 10 𝑛 𝑛 NOUN cana-2815 124 11 ! ! PUNCT cana-2815 125 1 [ [ PUNCT cana-2815 125 2 𝑑𝑛 𝑑𝑛 PUNCT cana-2815 125 3 π‘‘πœ†π‘› π‘‘πœ†π‘› VERB cana-2815 126 1 [ [ X cana-2815 126 2 𝑁 𝑁 PROPN cana-2815 126 3 βˆ‘ βˆ‘ DET cana-2815 126 4 πœ†π‘–βˆ…π‘– πœ†π‘–βˆ…π‘– PROPN cana-2815 126 5 ∞ ∞ NUM cana-2815 126 6 𝑖=0 𝑖=0 PROPN cana-2815 127 1 ] ] X cana-2815 127 2 ] ] X cana-2815 127 3 πœ†=0 πœ†=0 PUNCT cana-2815 127 4 ; ; PUNCT cana-2815 127 5 𝑖 𝑖 X cana-2815 127 6 = = PUNCT cana-2815 127 7 0,1,2 0,1,2 NUM cana-2815 127 8 , , PUNCT cana-2815 127 9 3 3 NUM cana-2815 127 10 , , PUNCT cana-2815 127 11 … … PUNCT cana-2815 127 12 … … PUNCT cana-2815 127 13 . . PUNCT cana-2815 128 1 the the DET cana-2815 128 2 nonlinear nonlinear ADJ cana-2815 128 3 terms term NOUN cana-2815 128 4 denoted denote VERB cana-2815 128 5 by by ADP cana-2815 128 6 𝑁 𝑁 PROPN cana-2815 128 7 are be AUX cana-2815 128 8 explained explain VERB cana-2815 128 9 with with ADP cana-2815 128 10 adapted adapt VERB cana-2815 128 11 modified modify VERB cana-2815 128 12 adomian adomian NOUN cana-2815 128 13 decomposition decomposition NOUN cana-2815 128 14 technique technique NOUN cana-2815 128 15 for for ADP cana-2815 128 16 handling handle VERB cana-2815 128 17 nonlinear nonlinear ADJ cana-2815 128 18 polynomial polynomial ADJ cana-2815 128 19 system system NOUN cana-2815 128 20 solution solution NOUN cana-2815 128 21 . . PUNCT cana-2815 129 1 following follow VERB cana-2815 129 2 the the DET cana-2815 129 3 utilization utilization NOUN cana-2815 129 4 of of ADP cana-2815 129 5 the the DET cana-2815 129 6 elzaki elzaki NOUN cana-2815 129 7 transformation transformation NOUN cana-2815 129 8 as as SCONJ cana-2815 129 9 specified specify VERB cana-2815 129 10 below below ADV cana-2815 129 11 : : PUNCT cana-2815 129 12 { { PUNCT cana-2815 129 13 𝐴𝑛 𝐴𝑛 NOUN cana-2815 129 14 } } PUNCT cana-2815 129 15 = = SYM cana-2815 129 16 { { PUNCT cana-2815 129 17 𝑁1 𝑁1 PROPN cana-2815 129 18 ( ( PUNCT cana-2815 129 19 𝑠𝑛 𝑠𝑛 NOUN cana-2815 129 20 ) ) PUNCT cana-2815 129 21 βˆ’ βˆ’ PROPN cana-2815 129 22 𝑁1 𝑁1 PROPN cana-2815 129 23 ( ( PUNCT cana-2815 129 24 π‘ π‘›βˆ’1 π‘ π‘›βˆ’1 NOUN cana-2815 129 25 ) ) PUNCT cana-2815 129 26 } } PUNCT cana-2815 129 27 equation equation NOUN cana-2815 129 28 ( ( PUNCT cana-2815 129 29 6 6 NUM cana-2815 129 30 ) ) PUNCT cana-2815 129 31 is be AUX cana-2815 129 32 obtained obtain VERB cana-2815 129 33 by by ADP cana-2815 129 34 substituting substitute VERB cana-2815 129 35 equations equation NOUN cana-2815 129 36 ( ( PUNCT cana-2815 129 37 7 7 NUM cana-2815 129 38 ) ) PUNCT cana-2815 129 39 and and CCONJ cana-2815 129 40 ( ( PUNCT cana-2815 129 41 8) 8) NUM cana-2815 129 42 βˆ‘ βˆ‘ NOUN cana-2815 129 43 βˆ…π‘›(π‘₯ βˆ…π‘›(π‘₯ NOUN cana-2815 129 44 , , PUNCT cana-2815 129 45 𝑑 𝑑 NOUN cana-2815 129 46 ) ) PUNCT cana-2815 129 47 ∞ ∞ NUM cana-2815 129 48 𝑛=0 𝑛=0 NOUN cana-2815 129 49 = = PUNCT cana-2815 129 50 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 ADP cana-2815 129 51 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-2815 129 52 ) ) PUNCT cana-2815 129 53 ] ] PUNCT cana-2815 130 1 + + CCONJ cana-2815 130 2 πΈβˆ’1[𝑣𝛼 πΈβˆ’1[𝑣𝛼 PROPN cana-2815 130 3 𝐸 𝐸 PROPN cana-2815 131 1 [ [ X cana-2815 131 2 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-2815 131 3 , , PUNCT cana-2815 131 4 𝑑 𝑑 NOUN cana-2815 131 5 ) ) PUNCT cana-2815 131 6 ] ] PUNCT cana-2815 131 7 ] ] PUNCT cana-2815 131 8 βˆ’ βˆ’ PUNCT cana-2815 131 9 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 132 1 [ [ X cana-2815 132 2 𝑣𝛼𝐸 𝑣𝛼𝐸 NOUN cana-2815 132 3 [ [ PUNCT cana-2815 132 4 𝑅[βˆ‘ 𝑅[βˆ‘ ADJ cana-2815 132 5 βˆ…π‘›(π‘₯ βˆ…π‘›(π‘₯ NOUN cana-2815 132 6 , , PUNCT cana-2815 132 7 𝑑 𝑑 NOUN cana-2815 132 8 ) ) PUNCT cana-2815 132 9 ∞ ∞ NUM cana-2815 132 10 𝑛=0 𝑛=0 PROPN cana-2815 132 11 ] ] PUNCT cana-2815 133 1 + + CCONJ cana-2815 133 2 [ [ X cana-2815 133 3 βˆ‘ βˆ‘ X cana-2815 133 4 𝐴𝑛 𝐴𝑛 NOUN cana-2815 133 5 ∞ ∞ NUM cana-2815 133 6 𝑛=0 𝑛=0 X cana-2815 133 7 ] ] PUNCT cana-2815 133 8 ] ] X cana-2815 133 9 ] ] PUNCT cana-2815 133 10 since since SCONJ cana-2815 133 11 , , PUNCT cana-2815 133 12 πΈβˆ’1(𝑣2 πΈβˆ’1(𝑣2 PROPN cana-2815 133 13 ) ) PUNCT cana-2815 133 14 = = SYM cana-2815 133 15 1 1 NUM cana-2815 133 16 βˆ‘βˆ…π‘›(π‘₯ βˆ‘βˆ…π‘›(π‘₯ NOUN cana-2815 133 17 , , PUNCT cana-2815 133 18 𝑑 𝑑 NOUN cana-2815 133 19 ) ) PUNCT cana-2815 133 20 ∞ ∞ NUM cana-2815 133 21 𝑛=0 𝑛=0 NOUN cana-2815 133 22 = = PUNCT cana-2815 133 23 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-2815 133 24 ) ) PUNCT cana-2815 134 1 + + CCONJ cana-2815 134 2 πΈβˆ’1[𝑣𝛼 πΈβˆ’1[𝑣𝛼 PROPN cana-2815 134 3 𝐸[𝑔(π‘₯ 𝐸[𝑔(π‘₯ PROPN cana-2815 134 4 , , PUNCT cana-2815 134 5 𝑑 𝑑 X cana-2815 134 6 ) ) PUNCT cana-2815 134 7 ] ] PUNCT cana-2815 134 8 ] ] PUNCT cana-2815 134 9 βˆ’ βˆ’ PUNCT cana-2815 134 10 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 135 1 [ [ X cana-2815 135 2 𝑣𝛼𝐸 𝑣𝛼𝐸 PROPN cana-2815 135 3 [ [ X cana-2815 135 4 𝑅 𝑅 PROPN cana-2815 135 5 [ [ X cana-2815 135 6 βˆ‘βˆ…π‘›(π‘₯ βˆ‘βˆ…π‘›(π‘₯ PROPN cana-2815 135 7 , , PUNCT cana-2815 135 8 𝑑 𝑑 NOUN cana-2815 135 9 ) ) PUNCT cana-2815 135 10 ∞ ∞ NUM cana-2815 135 11 𝑛=0 𝑛=0 PROPN cana-2815 135 12 ] ] PUNCT cana-2815 136 1 + + CCONJ cana-2815 136 2 [ [ X cana-2815 136 3 βˆ‘π΄π‘› βˆ‘π΄π‘› NOUN cana-2815 136 4 ∞ ∞ NUM cana-2815 136 5 𝑛=0 𝑛=0 X cana-2815 136 6 ] ] PUNCT cana-2815 136 7 ] ] X cana-2815 136 8 ] ] X cana-2815 136 9 analysing analyse VERB cana-2815 136 10 both both DET cana-2815 136 11 sides side NOUN cana-2815 136 12 of of ADP cana-2815 136 13 the the DET cana-2815 136 14 equation equation NOUN cana-2815 136 15 ( ( PUNCT cana-2815 136 16 9 9 NUM cana-2815 136 17 ) ) PUNCT cana-2815 136 18 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 136 19 , , PUNCT cana-2815 136 20 𝑑 𝑑 NOUN cana-2815 136 21 ) ) PUNCT cana-2815 136 22 = = SYM cana-2815 136 23 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-2815 136 24 ) ) PUNCT cana-2815 137 1 + + CCONJ cana-2815 137 2 𝐸 𝐸 PROPN cana-2815 137 3 βˆ’1[𝑣𝛼 βˆ’1[𝑣𝛼 PROPN cana-2815 137 4 𝐸[𝑔(π‘₯ 𝐸[𝑔(π‘₯ PROPN cana-2815 137 5 , , PUNCT cana-2815 137 6 𝑑 𝑑 PROPN cana-2815 137 7 ) ) PUNCT cana-2815 137 8 ] ] PUNCT cana-2815 137 9 ] ] PUNCT cana-2815 137 10 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 137 11 , , PUNCT cana-2815 137 12 𝑑 𝑑 PROPN cana-2815 137 13 ) ) PUNCT cana-2815 137 14 = = SYM cana-2815 137 15 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 138 1 [ [ X cana-2815 138 2 𝑅[βˆ…0(π‘₯ 𝑅[βˆ…0(π‘₯ PROPN cana-2815 138 3 , , PUNCT cana-2815 138 4 𝑑 𝑑 NOUN cana-2815 138 5 ) ) PUNCT cana-2815 138 6 ] ] PUNCT cana-2815 139 1 + + CCONJ cana-2815 139 2 𝐴0 𝐴0 PROPN cana-2815 139 3 ] ] X cana-2815 139 4 ] ] PUNCT cana-2815 139 5 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 139 6 , , PUNCT cana-2815 139 7 𝑑 𝑑 X cana-2815 139 8 ) ) PUNCT cana-2815 139 9 = = SYM cana-2815 139 10 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 140 1 [ [ X cana-2815 140 2 𝑅[βˆ…1(π‘₯ 𝑅[βˆ…1(π‘₯ PROPN cana-2815 140 3 , , PUNCT cana-2815 140 4 𝑑 𝑑 NOUN cana-2815 140 5 ) ) PUNCT cana-2815 140 6 ] ] PUNCT cana-2815 141 1 + + CCONJ cana-2815 141 2 𝐴1 𝐴1 PROPN cana-2815 141 3 ] ] X cana-2815 141 4 ] ] PUNCT cana-2815 141 5 . . PUNCT cana-2815 142 1 : : PUNCT cana-2815 142 2 : : PUNCT cana-2815 142 3 βˆ…π‘›+1(π‘₯ βˆ…π‘›+1(π‘₯ NUM cana-2815 142 4 , , PUNCT cana-2815 142 5 𝑑 𝑑 NOUN cana-2815 142 6 ) ) PUNCT cana-2815 142 7 = = SYM cana-2815 142 8 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 143 1 [ [ X cana-2815 143 2 𝑅[βˆ…n(π‘₯ 𝑅[βˆ…n(π‘₯ ADV cana-2815 143 3 , , PUNCT cana-2815 143 4 𝑑 𝑑 NOUN cana-2815 143 5 ) ) PUNCT cana-2815 143 6 ] ] PUNCT cana-2815 144 1 + + CCONJ cana-2815 144 2 𝐴n 𝐴n PROPN cana-2815 144 3 ] ] X cana-2815 144 4 ] ] X cana-2815 144 5 ( ( PUNCT cana-2815 144 6 6 6 NUM cana-2815 144 7 ) ) PUNCT cana-2815 144 8 ( ( PUNCT cana-2815 144 9 8) 8) NUM cana-2815 144 10 ( ( PUNCT cana-2815 144 11 10 10 NUM cana-2815 144 12 ) ) PUNCT cana-2815 144 13 ( ( PUNCT cana-2815 144 14 9 9 NUM cana-2815 144 15 ) ) PUNCT cana-2815 144 16 ( ( PUNCT cana-2815 144 17 7 7 X cana-2815 144 18 ) ) PUNCT cana-2815 144 19 communications communication NOUN cana-2815 144 20 on on ADP cana-2815 144 21 applied apply VERB cana-2815 144 22 nonlinear nonlinear ADJ cana-2815 144 23 analysis analysis NOUN cana-2815 144 24 issn issn NOUN cana-2815 144 25 : : PUNCT cana-2815 144 26 1074 1074 NUM cana-2815 144 27 - - PUNCT cana-2815 144 28 133x 133x NUM cana-2815 144 29 vol vol NOUN cana-2815 144 30 32 32 NUM cana-2815 144 31 no no NOUN cana-2815 144 32 . . PUNCT cana-2815 145 1 4s 4s NUM cana-2815 145 2 ( ( PUNCT cana-2815 145 3 2025 2025 NUM cana-2815 145 4 ) ) PUNCT cana-2815 145 5 315 315 NUM cana-2815 145 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 145 7 the the DET cana-2815 145 8 analytic analytic ADJ cana-2815 145 9 solution solution NOUN cana-2815 145 10 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 145 11 , , PUNCT cana-2815 145 12 𝑑 𝑑 PROPN cana-2815 145 13 ) ) PUNCT cana-2815 145 14 is be AUX cana-2815 145 15 finally finally ADV cana-2815 145 16 approximated approximate VERB cana-2815 145 17 using use VERB cana-2815 145 18 truncated truncated ADJ cana-2815 145 19 series series NOUN cana-2815 145 20 : : PUNCT cana-2815 145 21 the the DET cana-2815 145 22 following follow VERB cana-2815 145 23 different different ADJ cana-2815 145 24 forms form NOUN cana-2815 145 25 of of ADP cana-2815 145 26 nonlinear nonlinear ADJ cana-2815 145 27 one one NUM cana-2815 145 28 - - PUNCT cana-2815 145 29 dimensional dimensional ADJ cana-2815 145 30 , , PUNCT cana-2815 145 31 two two NUM cana-2815 145 32 - - PUNCT cana-2815 145 33 dimensional dimensional ADJ cana-2815 145 34 and and CCONJ cana-2815 145 35 three three NUM cana-2815 145 36 - - PUNCT cana-2815 145 37 dimensional dimensional ADJ cana-2815 145 38 fraction fraction NOUN cana-2815 145 39 form form NOUN cana-2815 145 40 telegraph telegraph NOUN cana-2815 145 41 equations equation NOUN cana-2815 145 42 are be AUX cana-2815 145 43 demonstrated demonstrate VERB cana-2815 145 44 with with ADP cana-2815 145 45 adopted adopt VERB cana-2815 145 46 technique technique NOUN cana-2815 145 47 for for ADP cana-2815 145 48 validation validation NOUN cana-2815 145 49 of of ADP cana-2815 145 50 results result NOUN cana-2815 145 51 in in ADP cana-2815 145 52 the the DET cana-2815 145 53 following follow VERB cana-2815 145 54 applications application NOUN cana-2815 145 55 . . PUNCT cana-2815 146 1 4 4 X cana-2815 146 2 . . X cana-2815 146 3 application application NOUN cana-2815 146 4 one one NUM cana-2815 146 5 - - PUNCT cana-2815 146 6 dimensional dimensional ADJ cana-2815 146 7 non non ADJ cana-2815 146 8 - - ADJ cana-2815 146 9 linear linear ADJ cana-2815 146 10 telegraph telegraph NOUN cana-2815 146 11 equation equation NOUN cana-2815 146 12 : : PUNCT cana-2815 146 13 example example NOUN cana-2815 146 14 1 1 X cana-2815 146 15 . . X cana-2815 146 16 consider consider VERB cana-2815 146 17 one one NUM cana-2815 146 18 - - PUNCT cana-2815 146 19 dimensional dimensional ADJ cana-2815 146 20 nonlinear nonlinear ADJ cana-2815 146 21 telegraph telegraph NOUN cana-2815 146 22 equation equation NOUN cana-2815 146 23 [ [ X cana-2815 146 24 330 330 NUM cana-2815 146 25 ] ] PUNCT cana-2815 146 26 : : PUNCT cana-2815 147 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 147 2 π›Όβˆ…(π‘₯ π›Όβˆ…(π‘₯ PROPN cana-2815 147 3 , , PUNCT cana-2815 147 4 𝑑 𝑑 NOUN cana-2815 147 5 ) ) PUNCT cana-2815 147 6 = = SYM cana-2815 147 7 βˆ…π‘₯π‘₯(π‘₯ βˆ…π‘₯π‘₯(π‘₯ NOUN cana-2815 147 8 , , PUNCT cana-2815 147 9 𝑑 𝑑 PROPN cana-2815 147 10 ) ) PUNCT cana-2815 147 11 + + CCONJ cana-2815 147 12 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 147 13 , , PUNCT cana-2815 147 14 𝑑 𝑑 PROPN cana-2815 147 15 ) ) PUNCT cana-2815 147 16 βˆ’ βˆ’ NOUN cana-2815 147 17 βˆ… βˆ… NOUN cana-2815 147 18 2 2 NUM cana-2815 147 19 + + CCONJ cana-2815 147 20 π‘₯βˆ…βˆ…π‘₯(π‘₯ π‘₯βˆ…βˆ…π‘₯(π‘₯ PROPN cana-2815 147 21 , , PUNCT cana-2815 147 22 𝑑)where 𝑑)where ADV cana-2815 147 23 0 0 NUM cana-2815 147 24 < < X cana-2815 147 25 βˆβ‰€ βˆβ‰€ X cana-2815 147 26 2 2 NUM cana-2815 147 27 ( ( PUNCT cana-2815 147 28 12 12 NUM cana-2815 147 29 ) ) PUNCT cana-2815 147 30 initial initial ADJ cana-2815 147 31 conditions condition NOUN cana-2815 147 32 : : PUNCT cana-2815 147 33 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 147 34 , , PUNCT cana-2815 147 35 0 0 NUM cana-2815 147 36 ) ) PUNCT cana-2815 147 37 = = SYM cana-2815 148 1 π‘₯ π‘₯ PROPN cana-2815 148 2 , , PUNCT cana-2815 148 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 148 4 , , PUNCT cana-2815 148 5 0 0 NUM cana-2815 148 6 ) ) PUNCT cana-2815 148 7 = = SYM cana-2815 148 8 π‘₯ π‘₯ PROPN cana-2815 148 9 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 148 10 , , PUNCT cana-2815 148 11 𝑑 𝑑 PROPN cana-2815 148 12 ) ) PUNCT cana-2815 148 13 = = NOUN cana-2815 148 14 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 148 15 , , PUNCT cana-2815 148 16 0 0 NUM cana-2815 148 17 ) ) PUNCT cana-2815 149 1 + + CCONJ cana-2815 149 2 𝑑 𝑑 PROPN cana-2815 149 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 149 4 , , PUNCT cana-2815 149 5 0 0 NUM cana-2815 149 6 ) ) PUNCT cana-2815 149 7 = = SYM cana-2815 150 1 π‘₯ π‘₯ PROPN cana-2815 151 1 + + NUM cana-2815 151 2 π‘₯𝑑 π‘₯𝑑 ADV cana-2815 151 3 = = SYM cana-2815 151 4 π‘₯(1 π‘₯(1 NOUN cana-2815 151 5 + + SYM cana-2815 151 6 𝑑 𝑑 NOUN cana-2815 151 7 ) ) PUNCT cana-2815 151 8 apply apply VERB cana-2815 151 9 the the DET cana-2815 151 10 elzaki elzaki NOUN cana-2815 151 11 transformation transformation NOUN cana-2815 151 12 on on ADP cana-2815 151 13 equation equation NOUN cana-2815 151 14 ( ( PUNCT cana-2815 151 15 12 12 NUM cana-2815 151 16 ) ) PUNCT cana-2815 151 17 , , PUNCT cana-2815 151 18 𝐸 𝐸 PROPN cana-2815 151 19 [ [ X cana-2815 151 20 βˆ… βˆ… NOUN cana-2815 151 21 ( ( PUNCT cana-2815 151 22 π‘₯ π‘₯ INTJ cana-2815 151 23 , , PUNCT cana-2815 151 24 𝑑 𝑑 PROPN cana-2815 151 25 ) ) PUNCT cana-2815 151 26 ] ] PUNCT cana-2815 152 1 = = SYM cana-2815 152 2 𝑣2 𝑣2 NUM cana-2815 152 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 152 4 , , PUNCT cana-2815 152 5 0 0 NUM cana-2815 152 6 ) ) PUNCT cana-2815 153 1 + + CCONJ cana-2815 153 2 𝑣𝛼 𝑣𝛼 NOUN cana-2815 154 1 [ [ X cana-2815 154 2 𝐸[βˆ…π‘₯π‘₯ 𝐸[βˆ…π‘₯π‘₯ NOUN cana-2815 154 3 + + X cana-2815 154 4 βˆ…π‘‘ βˆ…π‘‘ ADP cana-2815 154 5 βˆ’ βˆ’ ADP cana-2815 154 6 βˆ… βˆ… NOUN cana-2815 154 7 2 2 NUM cana-2815 154 8 + + CCONJ cana-2815 154 9 π‘₯ π‘₯ PROPN cana-2815 154 10 βˆ…βˆ…π‘₯ βˆ…βˆ…π‘₯ PROPN cana-2815 154 11 ] ] X cana-2815 154 12 ] ] X cana-2815 154 13 applying apply VERB cana-2815 154 14 inverse inverse NOUN cana-2815 154 15 elzaki elzaki NOUN cana-2815 154 16 transform transform VERB cana-2815 154 17 on on ADP cana-2815 154 18 above above ADP cana-2815 154 19 equation equation NOUN cana-2815 154 20 βˆ… βˆ… NOUN cana-2815 154 21 ( ( PUNCT cana-2815 154 22 π‘₯ π‘₯ NOUN cana-2815 154 23 , , PUNCT cana-2815 154 24 𝑑 𝑑 NOUN cana-2815 154 25 ) ) PUNCT cana-2815 154 26 = = SYM cana-2815 154 27 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 X cana-2815 154 28 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 154 29 , , PUNCT cana-2815 154 30 0 0 NUM cana-2815 154 31 ) ) PUNCT cana-2815 154 32 ] ] PUNCT cana-2815 155 1 + + CCONJ cana-2815 155 2 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 156 1 [ [ X cana-2815 156 2 𝑣𝛼 𝑣𝛼 NOUN cana-2815 156 3 𝐸 𝐸 PROPN cana-2815 157 1 [ [ X cana-2815 157 2 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ X cana-2815 157 3 + + X cana-2815 157 4 βˆ…π‘‘ βˆ…π‘‘ ADP cana-2815 157 5 βˆ’ βˆ’ ADP cana-2815 157 6 βˆ… βˆ… NOUN cana-2815 157 7 2 2 NUM cana-2815 157 8 + + CCONJ cana-2815 157 9 π‘₯ π‘₯ PROPN cana-2815 157 10 βˆ…βˆ…π‘₯ βˆ…βˆ…π‘₯ PROPN cana-2815 157 11 ] ] X cana-2815 157 12 ] ] X cana-2815 157 13 βˆ… βˆ… NOUN cana-2815 157 14 ( ( PUNCT cana-2815 157 15 π‘₯ π‘₯ NOUN cana-2815 157 16 , , PUNCT cana-2815 157 17 𝑑 𝑑 NOUN cana-2815 157 18 ) ) PUNCT cana-2815 157 19 = = SYM cana-2815 157 20 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 157 21 , , PUNCT cana-2815 157 22 0 0 NUM cana-2815 157 23 ) ) PUNCT cana-2815 157 24 + + CCONJ cana-2815 157 25 πΈβˆ’1[𝑣𝛼 πΈβˆ’1[𝑣𝛼 PROPN cana-2815 157 26 𝐸 𝐸 PROPN cana-2815 158 1 [ [ X cana-2815 158 2 𝑅[βˆ… 𝑅[βˆ… NOUN cana-2815 158 3 ] ] X cana-2815 158 4 + + CCONJ cana-2815 158 5 𝑁[βˆ… 𝑁[βˆ… NOUN cana-2815 158 6 ] ] X cana-2815 158 7 ] ] X cana-2815 158 8 ] ] X cana-2815 158 9 here here ADV cana-2815 158 10 , , PUNCT cana-2815 158 11 πΈβˆ’1(𝑣2 πΈβˆ’1(𝑣2 PROPN cana-2815 158 12 ) ) PUNCT cana-2815 158 13 = = SYM cana-2815 158 14 1 1 NUM cana-2815 158 15 ; ; PUNCT cana-2815 158 16 𝑅[βˆ… 𝑅[βˆ… PROPN cana-2815 158 17 ] ] X cana-2815 158 18 = = PUNCT cana-2815 158 19 ( ( PUNCT cana-2815 158 20 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ X cana-2815 158 21 + + CCONJ cana-2815 158 22 βˆ…π‘‘ βˆ…π‘‘ ADJ cana-2815 158 23 ) ) PUNCT cana-2815 158 24 and and CCONJ cana-2815 158 25 𝑁[βˆ… 𝑁[βˆ… NOUN cana-2815 158 26 ] ] X cana-2815 158 27 = = SYM cana-2815 158 28 ( ( PUNCT cana-2815 158 29 π‘₯ π‘₯ INTJ cana-2815 158 30 βˆ…βˆ…π‘₯ βˆ…βˆ…π‘₯ PROPN cana-2815 158 31 βˆ’ βˆ’ PROPN cana-2815 158 32 βˆ… βˆ… NOUN cana-2815 158 33 2 2 NUM cana-2815 158 34 ) ) PUNCT cana-2815 158 35 appling apple VERB cana-2815 158 36 the the DET cana-2815 158 37 madetm madetm NOUN cana-2815 158 38 process process NOUN cana-2815 158 39 on on ADP cana-2815 158 40 equation equation NOUN cana-2815 158 41 ( ( PUNCT cana-2815 158 42 14 14 NUM cana-2815 158 43 ) ) PUNCT cana-2815 158 44 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 158 45 , , PUNCT cana-2815 158 46 𝑑 𝑑 NOUN cana-2815 158 47 ) ) PUNCT cana-2815 158 48 = = SYM cana-2815 158 49 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 158 50 , , PUNCT cana-2815 158 51 0 0 NUM cana-2815 158 52 ) ) PUNCT cana-2815 158 53 = = PUNCT cana-2815 159 1 π‘₯(1 π‘₯(1 VERB cana-2815 159 2 + + SYM cana-2815 159 3 𝑑 𝑑 NOUN cana-2815 159 4 ) ) PUNCT cana-2815 159 5 appling apple VERB cana-2815 159 6 the the DET cana-2815 159 7 recursive recursive ADJ cana-2815 159 8 series series NOUN cana-2815 159 9 as as SCONJ cana-2815 159 10 shown show VERB cana-2815 159 11 in in ADP cana-2815 159 12 equation equation NOUN cana-2815 159 13 ( ( PUNCT cana-2815 159 14 10 10 NUM cana-2815 159 15 ) ) PUNCT cana-2815 159 16 , , PUNCT cana-2815 159 17 βˆ…π‘›+1(π‘₯ βˆ…π‘›+1(π‘₯ NUM cana-2815 159 18 , , PUNCT cana-2815 159 19 𝑑 𝑑 NOUN cana-2815 159 20 ) ) PUNCT cana-2815 159 21 = = PUNCT cana-2815 159 22 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 159 23 [ [ X cana-2815 159 24 𝑅 𝑅 NOUN cana-2815 159 25 ( ( PUNCT cana-2815 159 26 βˆ…π‘› βˆ…π‘› NOUN cana-2815 159 27 ) ) PUNCT cana-2815 160 1 + + CCONJ cana-2815 160 2 𝑁 𝑁 NOUN cana-2815 160 3 ( ( PUNCT cana-2815 160 4 βˆ…π‘› βˆ…π‘› NOUN cana-2815 160 5 ) ) PUNCT cana-2815 160 6 ] ] PUNCT cana-2815 160 7 ] ] PUNCT cana-2815 160 8 for for ADP cana-2815 160 9 𝑛 𝑛 PROPN cana-2815 160 10 = = SYM cana-2815 160 11 0 0 NUM cana-2815 160 12 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 160 13 , , PUNCT cana-2815 160 14 𝑑 𝑑 X cana-2815 160 15 ) ) PUNCT cana-2815 160 16 = = SYM cana-2815 160 17 𝐸 𝐸 PROPN cana-2815 160 18 βˆ’1[𝑣𝛼𝐸 βˆ’1[𝑣𝛼𝐸 PROPN cana-2815 160 19 [ [ X cana-2815 160 20 𝑅 𝑅 PROPN cana-2815 160 21 ( ( PUNCT cana-2815 160 22 βˆ…0 βˆ…0 NOUN cana-2815 160 23 ) ) PUNCT cana-2815 161 1 + + CCONJ cana-2815 161 2 𝑁 𝑁 PROPN cana-2815 161 3 [ [ PUNCT cana-2815 161 4 βˆ…0 βˆ…0 NOUN cana-2815 161 5 ] ] X cana-2815 161 6 ] ] X cana-2815 161 7 ] ] X cana-2815 161 8 here here ADV cana-2815 161 9 , , PUNCT cana-2815 161 10 𝑅 𝑅 PROPN cana-2815 161 11 ( ( PUNCT cana-2815 161 12 βˆ…0 βˆ…0 NOUN cana-2815 161 13 ) ) PUNCT cana-2815 161 14 = = PUNCT cana-2815 161 15 βˆ…0π‘₯π‘₯ βˆ…0π‘₯π‘₯ NOUN cana-2815 161 16 + + CCONJ cana-2815 161 17 βˆ…0𝑑 βˆ…0𝑑 NOUN cana-2815 162 1 = = PUNCT cana-2815 162 2 π‘₯ π‘₯ X cana-2815 162 3 𝑁 𝑁 PROPN cana-2815 162 4 ( ( PUNCT cana-2815 162 5 βˆ…0 βˆ…0 NOUN cana-2815 162 6 ) ) PUNCT cana-2815 162 7 = = SYM cana-2815 163 1 π‘₯ π‘₯ NOUN cana-2815 163 2 βˆ…0βˆ…0π‘₯ βˆ…0βˆ…0π‘₯ ADP cana-2815 163 3 βˆ’ βˆ’ PROPN cana-2815 163 4 βˆ…0 βˆ…0 NOUN cana-2815 163 5 2 2 NUM cana-2815 163 6 = = SYM cana-2815 163 7 0 0 PUNCT cana-2815 163 8 therefore therefore ADV cana-2815 163 9 , , PUNCT cana-2815 163 10 above above ADP cana-2815 163 11 equation equation NOUN cana-2815 163 12 implies imply VERB cana-2815 163 13 , , PUNCT cana-2815 163 14 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 163 15 , , PUNCT cana-2815 163 16 𝑑 𝑑 X cana-2815 163 17 ) ) PUNCT cana-2815 163 18 = = PUNCT cana-2815 163 19 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 163 20 [ [ X cana-2815 163 21 𝑅 𝑅 NOUN cana-2815 163 22 ( ( PUNCT cana-2815 163 23 βˆ…0 βˆ…0 NOUN cana-2815 163 24 ) ) PUNCT cana-2815 163 25 + + CCONJ cana-2815 164 1 𝑁 𝑁 PROPN cana-2815 164 2 [ [ PUNCT cana-2815 164 3 βˆ…0 βˆ…0 NOUN cana-2815 164 4 ] ] X cana-2815 164 5 ] ] X cana-2815 164 6 ] ] X cana-2815 164 7 βˆ…1 βˆ…1 X cana-2815 164 8 ( ( PUNCT cana-2815 164 9 π‘₯ π‘₯ PROPN cana-2815 164 10 , , PUNCT cana-2815 164 11 𝑑 𝑑 NOUN cana-2815 164 12 ) ) PUNCT cana-2815 164 13 = = PUNCT cana-2815 164 14 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 165 1 [ [ X cana-2815 165 2 π‘₯ π‘₯ X cana-2815 165 3 + + NOUN cana-2815 165 4 0 0 NUM cana-2815 165 5 ] ] X cana-2815 165 6 ] ] X cana-2815 166 1 = = PUNCT cana-2815 166 2 4𝑒π‘₯πΈβˆ’1[𝑣𝛼𝐸 4𝑒π‘₯πΈβˆ’1[𝑣𝛼𝐸 PUNCT cana-2815 167 1 [ [ X cana-2815 167 2 π‘₯ π‘₯ X cana-2815 167 3 ] ] X cana-2815 167 4 ] ] X cana-2815 167 5 = = PUNCT cana-2815 167 6 π‘₯πΈβˆ’1[𝑣𝛼𝐸(1 π‘₯πΈβˆ’1[𝑣𝛼𝐸(1 PROPN cana-2815 167 7 ) ) PUNCT cana-2815 167 8 ] ] PUNCT cana-2815 168 1 βˆ…1 βˆ…1 X cana-2815 168 2 ( ( PUNCT cana-2815 168 3 π‘₯ π‘₯ PROPN cana-2815 168 4 , , PUNCT cana-2815 168 5 𝑑 𝑑 PROPN cana-2815 168 6 ) ) PUNCT cana-2815 168 7 = = SYM cana-2815 168 8 π‘₯ π‘₯ NOUN cana-2815 168 9 πΈβˆ’1(𝑣𝛼+2 πΈβˆ’1(𝑣𝛼+2 PROPN cana-2815 168 10 ) ) PUNCT cana-2815 168 11 = = PUNCT cana-2815 169 1 π‘₯ π‘₯ PROPN cana-2815 169 2 𝑑𝛼 𝑑𝛼 X cana-2815 169 3 ⌈(𝛼 ⌈(𝛼 PROPN cana-2815 170 1 + + CCONJ cana-2815 170 2 1 1 X cana-2815 170 3 ) ) PUNCT cana-2815 170 4 ( ( PUNCT cana-2815 170 5 13 13 NUM cana-2815 170 6 ) ) PUNCT cana-2815 170 7 ( ( PUNCT cana-2815 170 8 14 14 NUM cana-2815 170 9 ) ) PUNCT cana-2815 170 10 ( ( PUNCT cana-2815 170 11 15 15 NUM cana-2815 170 12 ) ) PUNCT cana-2815 171 1 ( ( PUNCT cana-2815 171 2 11 11 NUM cana-2815 171 3 ) ) PUNCT cana-2815 171 4 ( ( PUNCT cana-2815 171 5 12 12 NUM cana-2815 171 6 ) ) PUNCT cana-2815 171 7 communications communication NOUN cana-2815 171 8 on on ADP cana-2815 171 9 applied apply VERB cana-2815 171 10 nonlinear nonlinear ADJ cana-2815 171 11 analysis analysis NOUN cana-2815 171 12 issn issn NOUN cana-2815 171 13 : : PUNCT cana-2815 171 14 1074 1074 NUM cana-2815 171 15 - - PUNCT cana-2815 171 16 133x 133x NUM cana-2815 171 17 vol vol NOUN cana-2815 171 18 32 32 NUM cana-2815 171 19 no no NOUN cana-2815 171 20 . . PUNCT cana-2815 172 1 4s 4s NUM cana-2815 172 2 ( ( PUNCT cana-2815 172 3 2025 2025 NUM cana-2815 172 4 ) ) PUNCT cana-2815 172 5 316 316 NUM cana-2815 172 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 172 7 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 172 8 , , PUNCT cana-2815 172 9 𝑑 𝑑 X cana-2815 172 10 ) ) PUNCT cana-2815 172 11 = = SYM cana-2815 173 1 π‘₯ π‘₯ PROPN cana-2815 173 2 𝑑𝛼 𝑑𝛼 PROPN cana-2815 173 3 ⌈(𝛼+1 ⌈(𝛼+1 PROPN cana-2815 173 4 ) ) PUNCT cana-2815 174 1 for for ADP cana-2815 174 2 𝑛 𝑛 NOUN cana-2815 174 3 = = SYM cana-2815 174 4 2 2 NUM cana-2815 174 5 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 174 6 , , PUNCT cana-2815 174 7 𝑑 𝑑 X cana-2815 174 8 ) ) PUNCT cana-2815 174 9 = = PUNCT cana-2815 174 10 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 175 1 [ [ X cana-2815 175 2 𝑅 𝑅 PROPN cana-2815 175 3 ( ( PUNCT cana-2815 175 4 βˆ…1 βˆ…1 NOUN cana-2815 175 5 ) ) PUNCT cana-2815 176 1 + + CCONJ cana-2815 177 1 𝑁 𝑁 PROPN cana-2815 177 2 [ [ PUNCT cana-2815 177 3 βˆ…1 βˆ…1 NOUN cana-2815 177 4 ] ] X cana-2815 177 5 ] ] X cana-2815 177 6 ] ] X cana-2815 177 7 here here ADV cana-2815 177 8 , , PUNCT cana-2815 177 9 𝑅 𝑅 PROPN cana-2815 177 10 ( ( PUNCT cana-2815 177 11 βˆ…1 βˆ…1 NOUN cana-2815 177 12 ) ) PUNCT cana-2815 177 13 = = PUNCT cana-2815 177 14 βˆ…1π‘₯π‘₯ βˆ…1π‘₯π‘₯ NOUN cana-2815 178 1 + + CCONJ cana-2815 178 2 βˆ…1𝑑 βˆ…1𝑑 NOUN cana-2815 178 3 = = SYM cana-2815 178 4 π‘₯ π‘₯ X cana-2815 178 5 𝛼 𝛼 NOUN cana-2815 178 6 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 178 7 ) ) PUNCT cana-2815 178 8 π‘‘π›Όβˆ’1 π‘‘π›Όβˆ’1 VERB cana-2815 178 9 𝑁 𝑁 PROPN cana-2815 178 10 ( ( PUNCT cana-2815 178 11 βˆ…1 βˆ…1 NOUN cana-2815 178 12 ) ) PUNCT cana-2815 178 13 = = SYM cana-2815 178 14 π‘₯ π‘₯ NOUN cana-2815 178 15 βˆ…0βˆ…0π‘₯ βˆ…0βˆ…0π‘₯ ADP cana-2815 179 1 βˆ’ βˆ’ PROPN cana-2815 179 2 βˆ…0 βˆ…0 NOUN cana-2815 179 3 2 2 NUM cana-2815 179 4 = = SYM cana-2815 179 5 0 0 NUM cana-2815 179 6 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 179 7 , , PUNCT cana-2815 179 8 𝑑 𝑑 X cana-2815 179 9 ) ) PUNCT cana-2815 180 1 = = PUNCT cana-2815 180 2 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 181 1 [ [ X cana-2815 181 2 𝑅 𝑅 PROPN cana-2815 181 3 ( ( PUNCT cana-2815 181 4 βˆ…1 βˆ…1 NOUN cana-2815 181 5 ) ) PUNCT cana-2815 182 1 + + CCONJ cana-2815 183 1 𝑁 𝑁 PROPN cana-2815 183 2 [ [ PUNCT cana-2815 183 3 βˆ…1 βˆ…1 NOUN cana-2815 183 4 ] ] X cana-2815 183 5 ] ] X cana-2815 183 6 ] ] X cana-2815 183 7 = = PUNCT cana-2815 183 8 𝐸 𝐸 PROPN cana-2815 183 9 βˆ’1 βˆ’1 NOUN cana-2815 184 1 [ [ X cana-2815 184 2 𝑣𝛼𝐸 𝑣𝛼𝐸 PUNCT cana-2815 184 3 [ [ X cana-2815 184 4 π‘₯ π‘₯ X cana-2815 184 5 𝛼 𝛼 PRON cana-2815 184 6 ⌈(𝛼 ⌈(𝛼 PROPN cana-2815 184 7 + + CCONJ cana-2815 184 8 1 1 X cana-2815 184 9 ) ) PUNCT cana-2815 184 10 π‘‘π›Όβˆ’1 π‘‘π›Όβˆ’1 NOUN cana-2815 184 11 ] ] X cana-2815 184 12 ] ] X cana-2815 184 13 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 184 14 , , PUNCT cana-2815 184 15 𝑑 𝑑 X cana-2815 184 16 ) ) PUNCT cana-2815 184 17 = = PUNCT cana-2815 185 1 π‘₯ π‘₯ DET cana-2815 185 2 𝛼 𝛼 NOUN cana-2815 185 3 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 185 4 ⌈(𝛼 ⌈(𝛼 PROPN cana-2815 186 1 + + CCONJ cana-2815 186 2 1 1 X cana-2815 186 3 ) ) PUNCT cana-2815 186 4 𝑑2π›Όβˆ’1 𝑑2π›Όβˆ’1 VERB cana-2815 186 5 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 186 6 ) ) PUNCT cana-2815 186 7 considering consider VERB cana-2815 186 8 𝑛 𝑛 PROPN cana-2815 186 9 = = SYM cana-2815 186 10 3 3 NUM cana-2815 186 11 , , PUNCT cana-2815 186 12 4 4 NUM cana-2815 186 13 … … PUNCT cana-2815 186 14 .. .. PUNCT cana-2815 187 1 βˆ…3(π‘₯ βˆ…3(π‘₯ VERB cana-2815 187 2 , , PUNCT cana-2815 187 3 𝑑 𝑑 X cana-2815 187 4 ) ) PUNCT cana-2815 187 5 = = PUNCT cana-2815 187 6 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 187 7 [ [ X cana-2815 187 8 𝑅 𝑅 NOUN cana-2815 187 9 ( ( PUNCT cana-2815 187 10 βˆ…2 βˆ…2 NOUN cana-2815 187 11 ) ) PUNCT cana-2815 187 12 + + CCONJ cana-2815 188 1 𝑁 𝑁 PROPN cana-2815 188 2 [ [ PUNCT cana-2815 188 3 βˆ…2 βˆ…2 NOUN cana-2815 188 4 ] ] PUNCT cana-2815 188 5 ] ] X cana-2815 188 6 ] ] X cana-2815 189 1 = = PUNCT cana-2815 189 2 π‘₯ π‘₯ X cana-2815 189 3 𝛼 𝛼 NOUN cana-2815 189 4 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 189 5 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 189 6 ) ) PUNCT cana-2815 189 7 ( ( PUNCT cana-2815 189 8 2π›Όβˆ’1 2π›Όβˆ’1 NOUN cana-2815 189 9 ) ) PUNCT cana-2815 189 10 ⌈(2π›Όβˆ’1 ⌈(2π›Όβˆ’1 ADJ cana-2815 189 11 ) ) PUNCT cana-2815 189 12 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 189 13 ) ) PUNCT cana-2815 189 14 𝑑3π›Όβˆ’1 𝑑3π›Όβˆ’1 NUM cana-2815 189 15 ⌈(3π›Όβˆ’1 ⌈(3π›Όβˆ’1 NOUN cana-2815 189 16 ) ) PUNCT cana-2815 189 17 βˆ…4(π‘₯ βˆ…4(π‘₯ NOUN cana-2815 189 18 , , PUNCT cana-2815 189 19 𝑑 𝑑 NOUN cana-2815 189 20 ) ) PUNCT cana-2815 189 21 = = PUNCT cana-2815 189 22 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 189 23 [ [ X cana-2815 189 24 𝑅 𝑅 PROPN cana-2815 189 25 ( ( PUNCT cana-2815 189 26 βˆ…3 βˆ…3 NOUN cana-2815 189 27 ) ) PUNCT cana-2815 190 1 + + CCONJ cana-2815 190 2 𝑁 𝑁 PROPN cana-2815 190 3 [ [ PUNCT cana-2815 190 4 βˆ…3 βˆ…3 NOUN cana-2815 190 5 ] ] X cana-2815 190 6 ] ] X cana-2815 190 7 ] ] X cana-2815 191 1 = = PUNCT cana-2815 191 2 π‘₯ π‘₯ X cana-2815 191 3 𝛼 𝛼 NOUN cana-2815 191 4 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 191 5 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 191 6 ) ) PUNCT cana-2815 191 7 ( ( PUNCT cana-2815 191 8 2π›Όβˆ’1 2π›Όβˆ’1 NOUN cana-2815 191 9 ) ) PUNCT cana-2815 191 10 ⌈(2π›Όβˆ’1 ⌈(2π›Όβˆ’1 ADJ cana-2815 191 11 ) ) PUNCT cana-2815 191 12 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 191 13 ) ) PUNCT cana-2815 191 14 ( ( PUNCT cana-2815 191 15 3π›Όβˆ’1 3π›Όβˆ’1 NUM cana-2815 191 16 ) ) PUNCT cana-2815 191 17 ⌈(3π›Όβˆ’1 ⌈(3π›Όβˆ’1 NOUN cana-2815 191 18 ) ) PUNCT cana-2815 191 19 ⌈(3𝛼 ⌈(3𝛼 PROPN cana-2815 191 20 ) ) PUNCT cana-2815 191 21 𝑑4π›Όβˆ’1 𝑑4π›Όβˆ’1 PROPN cana-2815 191 22 ⌈(4π›Όβˆ’1 ⌈(4π›Όβˆ’1 NOUN cana-2815 191 23 ) ) PUNCT cana-2815 191 24 therefore therefore ADV cana-2815 191 25 , , PUNCT cana-2815 191 26 series series NOUN cana-2815 191 27 representation representation NOUN cana-2815 191 28 of of ADP cana-2815 191 29 the the DET cana-2815 191 30 solution solution NOUN cana-2815 191 31 βˆ… βˆ… NOUN cana-2815 191 32 ( ( PUNCT cana-2815 191 33 π‘₯ π‘₯ NOUN cana-2815 191 34 , , PUNCT cana-2815 191 35 𝑑 𝑑 PROPN cana-2815 191 36 ) ) PUNCT cana-2815 191 37 is be AUX cana-2815 191 38 as as SCONJ cana-2815 191 39 follows follow VERB cana-2815 191 40 : : PUNCT cana-2815 191 41 βˆ… βˆ… NOUN cana-2815 191 42 ( ( PUNCT cana-2815 191 43 π‘₯ π‘₯ NOUN cana-2815 191 44 , , PUNCT cana-2815 191 45 𝑑 𝑑 NOUN cana-2815 191 46 ) ) PUNCT cana-2815 191 47 = = SYM cana-2815 191 48 βˆ…0 βˆ…0 NOUN cana-2815 191 49 ( ( PUNCT cana-2815 191 50 π‘₯ π‘₯ NOUN cana-2815 191 51 , , PUNCT cana-2815 191 52 𝑑)+βˆ…1 𝑑)+βˆ…1 PROPN cana-2815 191 53 ( ( PUNCT cana-2815 191 54 π‘₯ π‘₯ PROPN cana-2815 191 55 , , PUNCT cana-2815 191 56 𝑑)+βˆ…2 𝑑)+βˆ…2 NUM cana-2815 191 57 ( ( PUNCT cana-2815 191 58 π‘₯ π‘₯ PROPN cana-2815 191 59 , , PUNCT cana-2815 191 60 𝑑 𝑑 NOUN cana-2815 191 61 ) ) PUNCT cana-2815 191 62 + + SYM cana-2815 191 63 βˆ…3 βˆ…3 PROPN cana-2815 191 64 ( ( PUNCT cana-2815 191 65 π‘₯ π‘₯ PROPN cana-2815 191 66 , , PUNCT cana-2815 191 67 𝑑 𝑑 NOUN cana-2815 191 68 ) ) PUNCT cana-2815 191 69 + + NUM cana-2815 191 70 βˆ…4 βˆ…4 NOUN cana-2815 191 71 ( ( PUNCT cana-2815 191 72 π‘₯ π‘₯ NOUN cana-2815 191 73 , , PUNCT cana-2815 191 74 𝑑 𝑑 NOUN cana-2815 191 75 ) ) PUNCT cana-2815 191 76 + + X cana-2815 191 77 β‹― β‹― NOUN cana-2815 191 78 …… …… NOUN cana-2815 191 79 βˆ… βˆ… NOUN cana-2815 191 80 ( ( PUNCT cana-2815 191 81 π‘₯ π‘₯ INTJ cana-2815 191 82 , , PUNCT cana-2815 191 83 𝑑 𝑑 PROPN cana-2815 191 84 ) ) PUNCT cana-2815 191 85 = = PUNCT cana-2815 192 1 π‘₯(1 π‘₯(1 PROPN cana-2815 192 2 + + SYM cana-2815 192 3 𝑑 𝑑 NOUN cana-2815 192 4 ) ) PUNCT cana-2815 192 5 + + NUM cana-2815 192 6 π‘₯ π‘₯ VERB cana-2815 192 7 𝑑𝛼 𝑑𝛼 PROPN cana-2815 192 8 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 192 9 ) ) PUNCT cana-2815 193 1 + + CCONJ cana-2815 193 2 π‘₯ π‘₯ PRON cana-2815 193 3 𝛼 𝛼 PRON cana-2815 193 4 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 193 5 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 193 6 ) ) PUNCT cana-2815 193 7 𝑑2π›Όβˆ’1 𝑑2π›Όβˆ’1 VERB cana-2815 193 8 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 193 9 ) ) PUNCT cana-2815 194 1 + + CCONJ cana-2815 194 2 π‘₯ π‘₯ PRON cana-2815 194 3 𝛼 𝛼 PRON cana-2815 194 4 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 194 5 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 194 6 ) ) PUNCT cana-2815 194 7 ( ( PUNCT cana-2815 194 8 2π›Όβˆ’1 2π›Όβˆ’1 NOUN cana-2815 194 9 ) ) PUNCT cana-2815 194 10 ⌈(2π›Όβˆ’1 ⌈(2π›Όβˆ’1 ADJ cana-2815 194 11 ) ) PUNCT cana-2815 194 12 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 194 13 ) ) PUNCT cana-2815 194 14 𝑑3π›Όβˆ’1 𝑑3π›Όβˆ’1 NUM cana-2815 194 15 ⌈(3π›Όβˆ’1 ⌈(3π›Όβˆ’1 NOUN cana-2815 194 16 ) ) PUNCT cana-2815 195 1 + + CCONJ cana-2815 195 2 π‘₯ π‘₯ DET cana-2815 195 3 𝛼 𝛼 PRON cana-2815 195 4 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 195 5 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 195 6 ) ) PUNCT cana-2815 195 7 ( ( PUNCT cana-2815 195 8 2π›Όβˆ’1 2π›Όβˆ’1 NOUN cana-2815 195 9 ) ) PUNCT cana-2815 195 10 ⌈(2π›Όβˆ’1 ⌈(2π›Όβˆ’1 ADJ cana-2815 195 11 ) ) PUNCT cana-2815 195 12 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 195 13 ) ) PUNCT cana-2815 195 14 ( ( PUNCT cana-2815 195 15 3π›Όβˆ’1 3π›Όβˆ’1 NUM cana-2815 195 16 ) ) PUNCT cana-2815 195 17 ⌈(3π›Όβˆ’1 ⌈(3π›Όβˆ’1 NOUN cana-2815 195 18 ) ) PUNCT cana-2815 195 19 ⌈(3𝛼 ⌈(3𝛼 PROPN cana-2815 195 20 ) ) PUNCT cana-2815 195 21 𝑑4π›Όβˆ’1 𝑑4π›Όβˆ’1 PROPN cana-2815 195 22 ⌈(4π›Όβˆ’1 ⌈(4π›Όβˆ’1 NOUN cana-2815 195 23 ) ) PUNCT cana-2815 196 1 + + NOUN cana-2815 196 2 β‹― β‹― NOUN cana-2815 196 3 …… …… NOUN cana-2815 196 4 . . PUNCT cana-2815 197 1 βˆ… βˆ… NOUN cana-2815 197 2 ( ( PUNCT cana-2815 197 3 π‘₯ π‘₯ NOUN cana-2815 197 4 , , PUNCT cana-2815 197 5 𝑑 𝑑 NOUN cana-2815 197 6 ) ) PUNCT cana-2815 197 7 = = SYM cana-2815 198 1 π‘₯ π‘₯ PROPN cana-2815 198 2 [ [ X cana-2815 198 3 ( ( PUNCT cana-2815 198 4 1 1 NUM cana-2815 198 5 + + NUM cana-2815 198 6 𝑑 𝑑 NOUN cana-2815 198 7 ) ) PUNCT cana-2815 199 1 + + CCONJ cana-2815 199 2 𝑑𝛼 𝑑𝛼 PROPN cana-2815 199 3 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 199 4 ) ) PUNCT cana-2815 200 1 + + NUM cana-2815 200 2 𝛼 𝛼 PRON cana-2815 200 3 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 200 4 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 200 5 ) ) PUNCT cana-2815 200 6 𝑑2π›Όβˆ’1 𝑑2π›Όβˆ’1 VERB cana-2815 200 7 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 200 8 ) ) PUNCT cana-2815 201 1 + + NUM cana-2815 201 2 𝛼 𝛼 PRON cana-2815 201 3 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 201 4 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 201 5 ) ) PUNCT cana-2815 201 6 ( ( PUNCT cana-2815 201 7 2π›Όβˆ’1 2π›Όβˆ’1 NOUN cana-2815 201 8 ) ) PUNCT cana-2815 201 9 ⌈(2π›Όβˆ’1 ⌈(2π›Όβˆ’1 ADJ cana-2815 201 10 ) ) PUNCT cana-2815 201 11 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 201 12 ) ) PUNCT cana-2815 201 13 𝑑3π›Όβˆ’1 𝑑3π›Όβˆ’1 NUM cana-2815 201 14 ⌈(3π›Όβˆ’1 ⌈(3π›Όβˆ’1 NOUN cana-2815 201 15 ) ) PUNCT cana-2815 201 16 + + NUM cana-2815 201 17 𝛼 𝛼 PRON cana-2815 201 18 βŒˆπ›Ό βŒˆπ›Ό NOUN cana-2815 201 19 ⌈(𝛼+1 ⌈(𝛼+1 NOUN cana-2815 201 20 ) ) PUNCT cana-2815 201 21 ( ( PUNCT cana-2815 201 22 2π›Όβˆ’1 2π›Όβˆ’1 NOUN cana-2815 201 23 ) ) PUNCT cana-2815 201 24 ⌈(2π›Όβˆ’1 ⌈(2π›Όβˆ’1 ADJ cana-2815 201 25 ) ) PUNCT cana-2815 201 26 ⌈(2𝛼 ⌈(2𝛼 NUM cana-2815 201 27 ) ) PUNCT cana-2815 201 28 ( ( PUNCT cana-2815 201 29 3π›Όβˆ’1 3π›Όβˆ’1 NUM cana-2815 201 30 ) ) PUNCT cana-2815 201 31 ⌈(3π›Όβˆ’1 ⌈(3π›Όβˆ’1 NOUN cana-2815 201 32 ) ) PUNCT cana-2815 201 33 ⌈(3𝛼 ⌈(3𝛼 PROPN cana-2815 201 34 ) ) PUNCT cana-2815 201 35 𝑑4π›Όβˆ’1 𝑑4π›Όβˆ’1 PROPN cana-2815 201 36 ⌈(4π›Όβˆ’1 ⌈(4π›Όβˆ’1 NOUN cana-2815 201 37 ) ) PUNCT cana-2815 201 38 … … PUNCT cana-2815 201 39 …… …… X cana-2815 201 40 . . PUNCT cana-2815 201 41 ] ] PUNCT cana-2815 202 1 in in ADP cana-2815 202 2 particular particular ADJ cana-2815 202 3 when when SCONJ cana-2815 202 4 𝛼 𝛼 X cana-2815 202 5 = = SYM cana-2815 202 6 2 2 NUM cana-2815 202 7 , , PUNCT cana-2815 202 8 the the DET cana-2815 202 9 solution solution NOUN cana-2815 202 10 is be AUX cana-2815 202 11 of of ADP cana-2815 202 12 the the DET cana-2815 202 13 form form NOUN cana-2815 202 14 : : PUNCT cana-2815 202 15 βˆ… βˆ… NOUN cana-2815 202 16 ( ( PUNCT cana-2815 202 17 π‘₯ π‘₯ NOUN cana-2815 202 18 , , PUNCT cana-2815 202 19 𝑑 𝑑 NOUN cana-2815 202 20 ) ) PUNCT cana-2815 202 21 = = SYM cana-2815 203 1 π‘₯ π‘₯ PROPN cana-2815 204 1 [ [ X cana-2815 204 2 1 1 NUM cana-2815 204 3 + + NUM cana-2815 204 4 𝑑 𝑑 PROPN cana-2815 204 5 1 1 NUM cana-2815 204 6 ! ! PUNCT cana-2815 205 1 + + CCONJ cana-2815 205 2 t2 t2 PROPN cana-2815 205 3 2 2 NUM cana-2815 205 4 ! ! PUNCT cana-2815 206 1 + + CCONJ cana-2815 206 2 t3 t3 PROPN cana-2815 206 3 3 3 NUM cana-2815 206 4 ! ! PUNCT cana-2815 207 1 + + CCONJ cana-2815 207 2 t4 t4 PROPN cana-2815 207 3 4 4 NUM cana-2815 207 4 ! ! PUNCT cana-2815 208 1 + + CCONJ cana-2815 208 2 t5 t5 PROPN cana-2815 208 3 5 5 NUM cana-2815 208 4 ! ! NUM cana-2815 208 5 … … PUNCT cana-2815 208 6 . . PUNCT cana-2815 208 7 . . PUNCT cana-2815 209 1 ] ] PUNCT cana-2815 210 1 the the DET cana-2815 210 2 exact exact ADJ cana-2815 210 3 solution solution NOUN cana-2815 210 4 for for ADP cana-2815 210 5 equation equation NOUN cana-2815 210 6 ( ( PUNCT cana-2815 210 7 12 12 NUM cana-2815 210 8 ) ) PUNCT cana-2815 210 9 is be AUX cana-2815 210 10 : : PUNCT cana-2815 210 11 βˆ… βˆ… NOUN cana-2815 210 12 ( ( PUNCT cana-2815 210 13 π‘₯ π‘₯ NOUN cana-2815 210 14 , , PUNCT cana-2815 210 15 𝑑 𝑑 NOUN cana-2815 210 16 ) ) PUNCT cana-2815 210 17 = = SYM cana-2815 211 1 π‘₯𝑒𝑑 π‘₯𝑒𝑑 NOUN cana-2815 211 2 example example NOUN cana-2815 211 3 2 2 NUM cana-2815 211 4 . . X cana-2815 211 5 consider consider VERB cana-2815 211 6 the the DET cana-2815 211 7 following follow VERB cana-2815 211 8 one one NUM cana-2815 211 9 - - PUNCT cana-2815 211 10 dimensional dimensional ADJ cana-2815 211 11 nonlinear nonlinear ADJ cana-2815 211 12 telegraph telegraph NOUN cana-2815 211 13 equation equation NOUN cana-2815 211 14 [ [ X cana-2815 211 15 25 25 NUM cana-2815 211 16 ] ] X cana-2815 211 17 𝐷𝑑 𝐷𝑑 PROPN cana-2815 211 18 π›Όβˆ…(π‘₯ π›Όβˆ…(π‘₯ PROPN cana-2815 211 19 , , PUNCT cana-2815 211 20 𝑑 𝑑 X cana-2815 211 21 ) ) PUNCT cana-2815 211 22 = = PUNCT cana-2815 211 23 βˆ…π‘₯(βˆ… βˆ…π‘₯(βˆ… NOUN cana-2815 211 24 2(π‘₯ 2(π‘₯ NUM cana-2815 211 25 , , PUNCT cana-2815 211 26 𝑑 𝑑 NOUN cana-2815 211 27 ) ) PUNCT cana-2815 211 28 . . PUNCT cana-2815 212 1 βˆ…π‘₯(π‘₯ βˆ…π‘₯(π‘₯ NOUN cana-2815 212 2 , , PUNCT cana-2815 212 3 𝑑 𝑑 PROPN cana-2815 212 4 ) ) PUNCT cana-2815 212 5 ) ) PUNCT cana-2815 212 6 ( ( PUNCT cana-2815 212 7 19 19 NUM cana-2815 212 8 ) ) PUNCT cana-2815 212 9 initial initial ADJ cana-2815 212 10 condition condition NOUN cana-2815 212 11 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 212 12 , , PUNCT cana-2815 212 13 0 0 NUM cana-2815 212 14 ) ) PUNCT cana-2815 212 15 = = PUNCT cana-2815 213 1 π‘₯+𝑏 π‘₯+𝑏 X cana-2815 213 2 2𝑐 2𝑐 NOUN cana-2815 213 3 ; ; PUNCT cana-2815 213 4 where where SCONJ cana-2815 213 5 𝑐 𝑐 PROPN cana-2815 213 6 > > X cana-2815 213 7 0 0 PROPN cana-2815 213 8 , , PUNCT cana-2815 213 9 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-2815 213 10 𝑏 𝑏 PROPN cana-2815 213 11 is be AUX cana-2815 213 12 arbitrary arbitrary ADJ cana-2815 213 13 constant constant ADJ cana-2815 213 14 . . PUNCT cana-2815 214 1 ( ( PUNCT cana-2815 214 2 20 20 NUM cana-2815 214 3 ) ) PUNCT cana-2815 214 4 apply apply VERB cana-2815 214 5 the the DET cana-2815 214 6 elzaki elzaki NOUN cana-2815 214 7 transformation transformation NOUN cana-2815 214 8 on on ADP cana-2815 214 9 equation equation NOUN cana-2815 214 10 ( ( PUNCT cana-2815 214 11 19 19 NUM cana-2815 214 12 ) ) PUNCT cana-2815 214 13 𝐸 𝐸 PROPN cana-2815 214 14 [ [ PUNCT cana-2815 214 15 𝐷𝑑 𝐷𝑑 PROPN cana-2815 214 16 π›Όβˆ…(π‘₯ π›Όβˆ…(π‘₯ PROPN cana-2815 214 17 , , PUNCT cana-2815 214 18 𝑑 𝑑 NOUN cana-2815 214 19 ) ) PUNCT cana-2815 214 20 ] ] PUNCT cana-2815 215 1 = = SYM cana-2815 215 2 𝐸[2βˆ…(π‘₯ 𝐸[2βˆ…(π‘₯ X cana-2815 215 3 , , PUNCT cana-2815 215 4 𝑑 𝑑 NOUN cana-2815 215 5 ) ) PUNCT cana-2815 215 6 βˆ…π‘₯ βˆ…π‘₯ ADJ cana-2815 215 7 2 2 NUM cana-2815 215 8 ( ( PUNCT cana-2815 215 9 π‘₯ π‘₯ NOUN cana-2815 215 10 , , PUNCT cana-2815 215 11 𝑑 𝑑 NOUN cana-2815 215 12 ) ) PUNCT cana-2815 215 13 + + NUM cana-2815 215 14 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 215 15 , , PUNCT cana-2815 215 16 𝑑)βˆ…π‘₯π‘₯(π‘₯ 𝑑)βˆ…π‘₯π‘₯(π‘₯ ADJ cana-2815 215 17 , , PUNCT cana-2815 215 18 𝑑 𝑑 PROPN cana-2815 215 19 ) ) PUNCT cana-2815 215 20 ] ] PUNCT cana-2815 215 21 ( ( PUNCT cana-2815 215 22 16 16 NUM cana-2815 215 23 ) ) PUNCT cana-2815 215 24 ( ( PUNCT cana-2815 215 25 18 18 NUM cana-2815 215 26 ) ) PUNCT cana-2815 215 27 ( ( PUNCT cana-2815 215 28 17 17 NUM cana-2815 215 29 ) ) PUNCT cana-2815 215 30 ( ( PUNCT cana-2815 215 31 19 19 NUM cana-2815 215 32 ) ) PUNCT cana-2815 215 33 ( ( PUNCT cana-2815 215 34 20 20 NUM cana-2815 215 35 ) ) PUNCT cana-2815 215 36 communications communication NOUN cana-2815 215 37 on on ADP cana-2815 215 38 applied apply VERB cana-2815 215 39 nonlinear nonlinear ADJ cana-2815 215 40 analysis analysis NOUN cana-2815 215 41 issn issn NOUN cana-2815 215 42 : : PUNCT cana-2815 215 43 1074 1074 NUM cana-2815 215 44 - - PUNCT cana-2815 215 45 133x 133x NUM cana-2815 215 46 vol vol NOUN cana-2815 215 47 32 32 NUM cana-2815 215 48 no no NOUN cana-2815 215 49 . . PUNCT cana-2815 216 1 4s 4s NUM cana-2815 216 2 ( ( PUNCT cana-2815 216 3 2025 2025 NUM cana-2815 216 4 ) ) PUNCT cana-2815 216 5 317 317 NUM cana-2815 216 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 216 7 𝐸 𝐸 PROPN cana-2815 217 1 [ [ X cana-2815 217 2 βˆ… βˆ… NOUN cana-2815 217 3 ( ( PUNCT cana-2815 217 4 π‘₯ π‘₯ INTJ cana-2815 217 5 , , PUNCT cana-2815 217 6 𝑑 𝑑 PROPN cana-2815 217 7 ) ) PUNCT cana-2815 217 8 ] ] PUNCT cana-2815 217 9 = = SYM cana-2815 217 10 𝑣2 𝑣2 NUM cana-2815 217 11 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 217 12 , , PUNCT cana-2815 217 13 0 0 NUM cana-2815 217 14 ) ) PUNCT cana-2815 218 1 + + NUM cana-2815 218 2 𝑣𝛼 𝑣𝛼 PRON cana-2815 219 1 [ [ X cana-2815 219 2 𝐸[2βˆ…. 𝐸[2βˆ…. NOUN cana-2815 219 3 βˆ…π‘₯ βˆ…π‘₯ ADJ cana-2815 219 4 2 2 NUM cana-2815 219 5 + + SYM cana-2815 219 6 βˆ…2 βˆ…2 NOUN cana-2815 219 7 . . PUNCT cana-2815 220 1 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ VERB cana-2815 220 2 ] ] X cana-2815 220 3 ] ] X cana-2815 220 4 applying apply VERB cana-2815 220 5 inverse inverse NOUN cana-2815 220 6 elzaki elzaki NOUN cana-2815 220 7 transform transform VERB cana-2815 220 8 on on ADP cana-2815 220 9 equation equation NOUN cana-2815 220 10 ( ( PUNCT cana-2815 220 11 21 21 NUM cana-2815 220 12 ) ) PUNCT cana-2815 220 13 βˆ… βˆ… NOUN cana-2815 220 14 ( ( PUNCT cana-2815 220 15 π‘₯ π‘₯ NOUN cana-2815 220 16 , , PUNCT cana-2815 220 17 𝑑 𝑑 NOUN cana-2815 220 18 ) ) PUNCT cana-2815 220 19 = = SYM cana-2815 220 20 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 X cana-2815 220 21 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 220 22 , , PUNCT cana-2815 220 23 0 0 NUM cana-2815 220 24 ) ) PUNCT cana-2815 220 25 ] ] PUNCT cana-2815 221 1 + + CCONJ cana-2815 221 2 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 222 1 [ [ X cana-2815 222 2 𝑣𝛼 𝑣𝛼 NOUN cana-2815 222 3 𝐸 𝐸 PROPN cana-2815 222 4 [ [ X cana-2815 222 5 2βˆ…. 2βˆ…. NUM cana-2815 222 6 βˆ…π‘₯ βˆ…π‘₯ ADJ cana-2815 222 7 2 2 NUM cana-2815 222 8 + + CCONJ cana-2815 222 9 βˆ…2 βˆ…2 NOUN cana-2815 222 10 . . PUNCT cana-2815 223 1 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ VERB cana-2815 223 2 ] ] PUNCT cana-2815 223 3 ] ] X cana-2815 223 4 βˆ… βˆ… NOUN cana-2815 223 5 ( ( PUNCT cana-2815 223 6 π‘₯ π‘₯ NOUN cana-2815 223 7 , , PUNCT cana-2815 223 8 𝑑 𝑑 NOUN cana-2815 223 9 ) ) PUNCT cana-2815 223 10 = = SYM cana-2815 224 1 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 224 2 , , PUNCT cana-2815 224 3 0 0 NUM cana-2815 224 4 ) ) PUNCT cana-2815 225 1 + + CCONJ cana-2815 225 2 πΈβˆ’1[𝑣𝛼 πΈβˆ’1[𝑣𝛼 PROPN cana-2815 225 3 𝐸 𝐸 PROPN cana-2815 226 1 [ [ X cana-2815 226 2 𝑁1(βˆ… 𝑁1(βˆ… X cana-2815 226 3 ) ) PUNCT cana-2815 227 1 + + NUM cana-2815 227 2 𝑁2(βˆ… 𝑁2(βˆ… NOUN cana-2815 227 3 ) ) PUNCT cana-2815 227 4 ] ] PUNCT cana-2815 227 5 ] ] PUNCT cana-2815 227 6 here here ADV cana-2815 227 7 , , PUNCT cana-2815 227 8 πΈβˆ’1(𝑣2 πΈβˆ’1(𝑣2 PROPN cana-2815 227 9 ) ) PUNCT cana-2815 227 10 = = SYM cana-2815 227 11 1 1 NUM cana-2815 227 12 ; ; PUNCT cana-2815 227 13 𝑁1(βˆ… 𝑁1(βˆ… PROPN cana-2815 227 14 ) ) PUNCT cana-2815 228 1 = = PUNCT cana-2815 228 2 ( ( PUNCT cana-2815 228 3 2βˆ…. 2βˆ…. NUM cana-2815 228 4 βˆ…π‘₯ βˆ…π‘₯ ADJ cana-2815 228 5 2 2 NUM cana-2815 228 6 ) ) PUNCT cana-2815 228 7 and and CCONJ cana-2815 228 8 𝑁2(βˆ… 𝑁2(βˆ… NOUN cana-2815 228 9 ) ) PUNCT cana-2815 229 1 = = PRON cana-2815 229 2 ( ( PUNCT cana-2815 229 3 βˆ…2 βˆ…2 NOUN cana-2815 229 4 . . PUNCT cana-2815 230 1 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ NOUN cana-2815 230 2 ) ) PUNCT cana-2815 230 3 applying apply VERB cana-2815 230 4 the the DET cana-2815 230 5 madetm madetm NOUN cana-2815 230 6 process process NOUN cana-2815 230 7 on on ADP cana-2815 230 8 equation equation NOUN cana-2815 230 9 ( ( PUNCT cana-2815 230 10 22 22 NUM cana-2815 230 11 ) ) PUNCT cana-2815 230 12 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 230 13 , , PUNCT cana-2815 230 14 𝑑 𝑑 NOUN cana-2815 230 15 ) ) PUNCT cana-2815 230 16 = = SYM cana-2815 230 17 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 230 18 , , PUNCT cana-2815 230 19 0 0 NUM cana-2815 230 20 ) ) PUNCT cana-2815 230 21 = = SYM cana-2815 230 22 π‘₯+𝑏 π‘₯+𝑏 X cana-2815 230 23 2𝑐 2𝑐 NUM cana-2815 230 24 ( ( PUNCT cana-2815 230 25 23 23 NUM cana-2815 230 26 ) ) PUNCT cana-2815 230 27 applying apply VERB cana-2815 230 28 the the DET cana-2815 230 29 recursive recursive ADJ cana-2815 230 30 series series NOUN cana-2815 230 31 as as SCONJ cana-2815 230 32 shown show VERB cana-2815 230 33 in in ADP cana-2815 230 34 equation equation NOUN cana-2815 230 35 ( ( PUNCT cana-2815 230 36 10 10 NUM cana-2815 230 37 ) ) PUNCT cana-2815 230 38 , , PUNCT cana-2815 230 39 βˆ…π‘›+1(π‘₯ βˆ…π‘›+1(π‘₯ NUM cana-2815 230 40 , , PUNCT cana-2815 230 41 𝑑 𝑑 NOUN cana-2815 230 42 ) ) PUNCT cana-2815 230 43 = = PUNCT cana-2815 230 44 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 230 45 [ [ X cana-2815 230 46 𝑁1 𝑁1 X cana-2815 230 47 ( ( PUNCT cana-2815 230 48 βˆ…π‘› βˆ…π‘› NOUN cana-2815 230 49 ) ) PUNCT cana-2815 231 1 + + CCONJ cana-2815 231 2 𝑁2 𝑁2 NOUN cana-2815 231 3 ( ( PUNCT cana-2815 231 4 βˆ…π‘› βˆ…π‘› NOUN cana-2815 231 5 ) ) PUNCT cana-2815 231 6 ] ] PUNCT cana-2815 231 7 ] ] PUNCT cana-2815 231 8 for for ADP cana-2815 231 9 𝑛 𝑛 PROPN cana-2815 231 10 = = SYM cana-2815 231 11 0 0 NUM cana-2815 231 12 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 231 13 , , PUNCT cana-2815 231 14 𝑑 𝑑 X cana-2815 231 15 ) ) PUNCT cana-2815 231 16 = = PUNCT cana-2815 231 17 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 232 1 [ [ X cana-2815 232 2 𝑁1 𝑁1 X cana-2815 232 3 ( ( PUNCT cana-2815 232 4 βˆ…0 βˆ…0 NOUN cana-2815 232 5 ) ) PUNCT cana-2815 233 1 + + CCONJ cana-2815 233 2 𝑁2 𝑁2 NOUN cana-2815 233 3 ( ( PUNCT cana-2815 233 4 βˆ…0 βˆ…0 NOUN cana-2815 233 5 ) ) PUNCT cana-2815 233 6 ] ] PUNCT cana-2815 233 7 ] ] X cana-2815 234 1 βˆ…1 βˆ…1 X cana-2815 234 2 ( ( PUNCT cana-2815 234 3 π‘₯ π‘₯ PROPN cana-2815 234 4 , , PUNCT cana-2815 234 5 𝑑 𝑑 NOUN cana-2815 234 6 ) ) PUNCT cana-2815 234 7 = = PUNCT cana-2815 234 8 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 235 1 [ [ X cana-2815 235 2 𝑣𝛼𝐸 𝑣𝛼𝐸 NOUN cana-2815 235 3 [ [ X cana-2815 235 4 2βˆ…0 2βˆ…0 NOUN cana-2815 235 5 . . PUNCT cana-2815 235 6 βˆ…0π‘₯ βˆ…0π‘₯ PROPN cana-2815 235 7 2 2 NUM cana-2815 236 1 + + NUM cana-2815 236 2 βˆ…0 βˆ…0 NOUN cana-2815 236 3 2 2 NUM cana-2815 236 4 . . PUNCT cana-2815 236 5 βˆ…0π‘₯π‘₯ βˆ…0π‘₯π‘₯ ADP cana-2815 236 6 ] ] PUNCT cana-2815 236 7 ] ] X cana-2815 236 8 βˆ…1 βˆ…1 X cana-2815 236 9 ( ( PUNCT cana-2815 236 10 π‘₯ π‘₯ PROPN cana-2815 236 11 , , PUNCT cana-2815 236 12 𝑑 𝑑 NOUN cana-2815 236 13 ) ) PUNCT cana-2815 236 14 = = SYM cana-2815 237 1 π‘₯ π‘₯ PROPN cana-2815 238 1 + + CCONJ cana-2815 238 2 𝑏 𝑏 PROPN cana-2815 238 3 4𝑐3 4𝑐3 NUM cana-2815 238 4 πΈβˆ’1(𝑣𝛼+2 πΈβˆ’1(𝑣𝛼+2 PROPN cana-2815 238 5 ) ) PUNCT cana-2815 239 1 = = PUNCT cana-2815 240 1 π‘₯ π‘₯ PROPN cana-2815 241 1 + + CCONJ cana-2815 241 2 𝑏 𝑏 PROPN cana-2815 241 3 4𝑐3 4𝑐3 NUM cana-2815 241 4 𝑑𝛼 𝑑𝛼 PROPN cana-2815 241 5 ⌈(𝛼 ⌈(𝛼 PROPN cana-2815 242 1 + + CCONJ cana-2815 242 2 1 1 NUM cana-2815 242 3 ) ) PUNCT cana-2815 242 4 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 242 5 , , PUNCT cana-2815 242 6 𝑑 𝑑 X cana-2815 242 7 ) ) PUNCT cana-2815 242 8 = = SYM cana-2815 242 9 π‘₯+𝑏 π‘₯+𝑏 X cana-2815 242 10 4𝑐3 4𝑐3 NUM cana-2815 242 11 𝑑𝛼 𝑑𝛼 PROPN cana-2815 242 12 ⌈(𝛼+1 ⌈(𝛼+1 PROPN cana-2815 242 13 ) ) PUNCT cana-2815 242 14 for for ADP cana-2815 242 15 𝑛 𝑛 NOUN cana-2815 242 16 = = SYM cana-2815 242 17 2 2 NUM cana-2815 242 18 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 242 19 , , PUNCT cana-2815 242 20 𝑑 𝑑 X cana-2815 242 21 ) ) PUNCT cana-2815 243 1 = = SYM cana-2815 243 2 𝐸 𝐸 PROPN cana-2815 243 3 βˆ’1[𝑣𝛼𝐸 βˆ’1[𝑣𝛼𝐸 PROPN cana-2815 244 1 [ [ X cana-2815 244 2 𝑁1 𝑁1 PROPN cana-2815 244 3 ( ( PUNCT cana-2815 244 4 βˆ…1 βˆ…1 NUM cana-2815 244 5 ) ) PUNCT cana-2815 244 6 + + CCONJ cana-2815 244 7 𝑁2 𝑁2 NOUN cana-2815 244 8 ( ( PUNCT cana-2815 244 9 βˆ…1 βˆ…1 NOUN cana-2815 244 10 ) ) PUNCT cana-2815 244 11 ] ] X cana-2815 244 12 ] ] PUNCT cana-2815 244 13 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 244 14 , , PUNCT cana-2815 244 15 𝑑 𝑑 X cana-2815 244 16 ) ) PUNCT cana-2815 244 17 = = PUNCT cana-2815 244 18 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 245 1 [ [ X cana-2815 245 2 𝑣𝛼𝐸 𝑣𝛼𝐸 NOUN cana-2815 245 3 [ [ X cana-2815 245 4 2βˆ…1 2βˆ…1 NUM cana-2815 245 5 . . PUNCT cana-2815 245 6 βˆ…1π‘₯ βˆ…1π‘₯ ADJ cana-2815 245 7 2 2 NUM cana-2815 245 8 + + CCONJ cana-2815 245 9 βˆ…1 βˆ…1 VERB cana-2815 245 10 2 2 NUM cana-2815 245 11 . . PUNCT cana-2815 245 12 βˆ…1π‘₯π‘₯ βˆ…1π‘₯π‘₯ NOUN cana-2815 246 1 ] ] X cana-2815 246 2 ] ] X cana-2815 246 3 = = PUNCT cana-2815 246 4 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 247 1 [ [ X cana-2815 247 2 𝑣𝛼𝐸 𝑣𝛼𝐸 PROPN cana-2815 247 3 [ [ PUNCT cana-2815 247 4 ( ( PUNCT cana-2815 247 5 π‘₯ π‘₯ PROPN cana-2815 247 6 + + CCONJ cana-2815 247 7 𝑏 𝑏 NOUN cana-2815 247 8 ) ) PUNCT cana-2815 247 9 4𝑐5 4𝑐5 NUM cana-2815 247 10 3 3 NUM cana-2815 247 11 𝑑𝛼 𝑑𝛼 ADP cana-2815 247 12 2⌈(𝛼 2⌈(𝛼 NUM cana-2815 247 13 + + CCONJ cana-2815 247 14 1 1 NUM cana-2815 247 15 ) ) PUNCT cana-2815 247 16 ] ] PUNCT cana-2815 247 17 ] ] PUNCT cana-2815 247 18 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 247 19 , , PUNCT cana-2815 247 20 𝑑 𝑑 X cana-2815 247 21 ) ) PUNCT cana-2815 247 22 = = PUNCT cana-2815 248 1 3(π‘₯ 3(π‘₯ NUM cana-2815 248 2 + + CCONJ cana-2815 248 3 𝑏 𝑏 NOUN cana-2815 248 4 ) ) PUNCT cana-2815 248 5 8𝑐5 8𝑐5 NUM cana-2815 248 6 𝑑2𝛼 𝑑2𝛼 ADP cana-2815 248 7 ⌈(2𝛼 ⌈(2𝛼 NOUN cana-2815 248 8 + + NOUN cana-2815 248 9 1 1 X cana-2815 248 10 ) ) PUNCT cana-2815 248 11 considering consider VERB cana-2815 248 12 𝑛 𝑛 PROPN cana-2815 248 13 = = SYM cana-2815 248 14 3 3 NUM cana-2815 248 15 , , PUNCT cana-2815 248 16 4 4 NUM cana-2815 248 17 … … PUNCT cana-2815 248 18 .. .. PUNCT cana-2815 249 1 βˆ…3(π‘₯ βˆ…3(π‘₯ VERB cana-2815 249 2 , , PUNCT cana-2815 249 3 𝑑 𝑑 X cana-2815 249 4 ) ) PUNCT cana-2815 249 5 = = PUNCT cana-2815 249 6 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 249 7 [ [ X cana-2815 249 8 𝑁1 𝑁1 X cana-2815 249 9 ( ( PUNCT cana-2815 249 10 βˆ…2 βˆ…2 NOUN cana-2815 249 11 ) ) PUNCT cana-2815 250 1 + + CCONJ cana-2815 250 2 𝑁2 𝑁2 NOUN cana-2815 250 3 [ [ PUNCT cana-2815 250 4 βˆ…2 βˆ…2 NOUN cana-2815 250 5 ] ] PUNCT cana-2815 250 6 ] ] X cana-2815 250 7 ] ] X cana-2815 251 1 = = SYM cana-2815 251 2 4 4 NUM cana-2815 251 3 ( ( PUNCT cana-2815 251 4 π‘₯+𝑏 π‘₯+𝑏 NOUN cana-2815 251 5 ) ) PUNCT cana-2815 251 6 16 16 NUM cana-2815 251 7 𝑐5 𝑐5 ADJ cana-2815 251 8 𝑑3𝛼 𝑑3𝛼 NOUN cana-2815 251 9 ⌈(3𝛼+1 ⌈(3𝛼+1 NOUN cana-2815 251 10 ) ) PUNCT cana-2815 251 11 . . PUNCT cana-2815 252 1 : : PUNCT cana-2815 252 2 therefore therefore ADV cana-2815 252 3 , , PUNCT cana-2815 252 4 series series NOUN cana-2815 252 5 representation representation NOUN cana-2815 252 6 of of ADP cana-2815 252 7 the the DET cana-2815 252 8 solution solution NOUN cana-2815 252 9 βˆ… βˆ… NOUN cana-2815 252 10 ( ( PUNCT cana-2815 252 11 π‘₯ π‘₯ NOUN cana-2815 252 12 , , PUNCT cana-2815 252 13 𝑑 𝑑 PROPN cana-2815 252 14 ) ) PUNCT cana-2815 252 15 is be AUX cana-2815 252 16 as as SCONJ cana-2815 252 17 follows follow VERB cana-2815 252 18 : : PUNCT cana-2815 252 19 βˆ… βˆ… NOUN cana-2815 252 20 ( ( PUNCT cana-2815 252 21 π‘₯ π‘₯ NOUN cana-2815 252 22 , , PUNCT cana-2815 252 23 𝑑 𝑑 NOUN cana-2815 252 24 ) ) PUNCT cana-2815 252 25 = = SYM cana-2815 252 26 βˆ…0 βˆ…0 NOUN cana-2815 252 27 ( ( PUNCT cana-2815 252 28 π‘₯ π‘₯ NOUN cana-2815 252 29 , , PUNCT cana-2815 252 30 𝑑)+βˆ…1 𝑑)+βˆ…1 PROPN cana-2815 252 31 ( ( PUNCT cana-2815 252 32 π‘₯ π‘₯ PROPN cana-2815 252 33 , , PUNCT cana-2815 252 34 𝑑)+βˆ…2 𝑑)+βˆ…2 NUM cana-2815 252 35 ( ( PUNCT cana-2815 252 36 π‘₯ π‘₯ PROPN cana-2815 252 37 , , PUNCT cana-2815 252 38 𝑑 𝑑 NOUN cana-2815 252 39 ) ) PUNCT cana-2815 252 40 + + SYM cana-2815 252 41 βˆ…3 βˆ…3 PROPN cana-2815 252 42 ( ( PUNCT cana-2815 252 43 π‘₯ π‘₯ PROPN cana-2815 252 44 , , PUNCT cana-2815 252 45 𝑑 𝑑 NOUN cana-2815 252 46 ) ) PUNCT cana-2815 252 47 + + NUM cana-2815 252 48 βˆ…4 βˆ…4 NOUN cana-2815 252 49 ( ( PUNCT cana-2815 252 50 π‘₯ π‘₯ NOUN cana-2815 252 51 , , PUNCT cana-2815 252 52 𝑑 𝑑 NOUN cana-2815 252 53 ) ) PUNCT cana-2815 253 1 + + X cana-2815 253 2 β‹― β‹― NOUN cana-2815 253 3 …… …… NOUN cana-2815 253 4 ( ( PUNCT cana-2815 253 5 21 21 NUM cana-2815 253 6 ) ) PUNCT cana-2815 253 7 ( ( PUNCT cana-2815 253 8 24 24 NUM cana-2815 253 9 ) ) PUNCT cana-2815 253 10 ( ( PUNCT cana-2815 253 11 22 22 NUM cana-2815 253 12 ) ) PUNCT cana-2815 253 13 ( ( PUNCT cana-2815 253 14 23 23 X cana-2815 253 15 ) ) PUNCT cana-2815 253 16 communications communication NOUN cana-2815 253 17 on on ADP cana-2815 253 18 applied apply VERB cana-2815 253 19 nonlinear nonlinear ADJ cana-2815 253 20 analysis analysis NOUN cana-2815 253 21 issn issn NOUN cana-2815 253 22 : : PUNCT cana-2815 253 23 1074 1074 NUM cana-2815 253 24 - - PUNCT cana-2815 253 25 133x 133x NUM cana-2815 253 26 vol vol NOUN cana-2815 253 27 32 32 NUM cana-2815 253 28 no no NOUN cana-2815 253 29 . . PUNCT cana-2815 254 1 4s 4s NUM cana-2815 254 2 ( ( PUNCT cana-2815 254 3 2025 2025 NUM cana-2815 254 4 ) ) PUNCT cana-2815 254 5 318 318 NUM cana-2815 254 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 254 7 βˆ… βˆ… NOUN cana-2815 254 8 ( ( PUNCT cana-2815 254 9 π‘₯ π‘₯ INTJ cana-2815 254 10 , , PUNCT cana-2815 254 11 𝑑 𝑑 PROPN cana-2815 254 12 ) ) PUNCT cana-2815 254 13 = = PUNCT cana-2815 254 14 π‘₯+𝑏 π‘₯+𝑏 PROPN cana-2815 254 15 2𝑐 2𝑐 NUM cana-2815 254 16 + + CCONJ cana-2815 254 17 ( ( PUNCT cana-2815 254 18 π‘₯+𝑏 π‘₯+𝑏 NOUN cana-2815 254 19 ) ) PUNCT cana-2815 254 20 4𝑐3 4𝑐3 NUM cana-2815 255 1 𝑑𝛼 𝑑𝛼 PROPN cana-2815 255 2 ⌈(𝛼+1 ⌈(𝛼+1 PROPN cana-2815 255 3 ) ) PUNCT cana-2815 256 1 + + CCONJ cana-2815 256 2 3(π‘₯+𝑏 3(π‘₯+𝑏 NUM cana-2815 256 3 ) ) PUNCT cana-2815 256 4 8𝑐5 8𝑐5 NUM cana-2815 256 5 𝑑2𝛼 𝑑2𝛼 PUNCT cana-2815 256 6 ⌈(2𝛼+1 ⌈(2𝛼+1 NOUN cana-2815 256 7 ) ) PUNCT cana-2815 257 1 + + CCONJ cana-2815 257 2 4 4 NUM cana-2815 257 3 ( ( PUNCT cana-2815 257 4 π‘₯+𝑏 π‘₯+𝑏 NOUN cana-2815 257 5 ) ) PUNCT cana-2815 257 6 16 16 NUM cana-2815 257 7 𝑐5 𝑐5 ADJ cana-2815 257 8 𝑑3𝛼 𝑑3𝛼 NOUN cana-2815 257 9 ⌈(3𝛼+1 ⌈(3𝛼+1 NOUN cana-2815 257 10 ) ) PUNCT cana-2815 258 1 + + VERB cana-2815 258 2 β‹― β‹― NOUN cana-2815 258 3 …… …… NOUN cana-2815 258 4 . . PUNCT cana-2815 259 1 in in ADP cana-2815 259 2 particular particular ADJ cana-2815 259 3 when when SCONJ cana-2815 259 4 𝛼 𝛼 X cana-2815 259 5 = = SYM cana-2815 259 6 1 1 NUM cana-2815 259 7 , , PUNCT cana-2815 259 8 the the DET cana-2815 259 9 solution solution NOUN cana-2815 259 10 is be AUX cana-2815 259 11 of of ADP cana-2815 259 12 the the DET cana-2815 259 13 form form NOUN cana-2815 259 14 : : PUNCT cana-2815 259 15 βˆ… βˆ… NOUN cana-2815 259 16 ( ( PUNCT cana-2815 259 17 π‘₯ π‘₯ NOUN cana-2815 259 18 , , PUNCT cana-2815 259 19 𝑑 𝑑 NOUN cana-2815 259 20 ) ) PUNCT cana-2815 259 21 = = NOUN cana-2815 260 1 [ [ PUNCT cana-2815 260 2 π‘₯+𝑏 π‘₯+𝑏 X cana-2815 260 3 2𝑐 2𝑐 NUM cana-2815 260 4 + + CCONJ cana-2815 260 5 ( ( PUNCT cana-2815 260 6 π‘₯+𝑏 π‘₯+𝑏 NOUN cana-2815 260 7 ) ) PUNCT cana-2815 260 8 𝑑 𝑑 VERB cana-2815 260 9 4𝑐3 4𝑐3 PROPN cana-2815 260 10 + + CCONJ cana-2815 260 11 3(π‘₯+𝑏 3(π‘₯+𝑏 NUM cana-2815 260 12 ) ) PUNCT cana-2815 260 13 𝑑2 𝑑2 NOUN cana-2815 260 14 16 16 NUM cana-2815 260 15 𝑐5 𝑐5 NOUN cana-2815 260 16 + + X cana-2815 260 17 4 4 NUM cana-2815 260 18 ( ( PUNCT cana-2815 260 19 π‘₯+𝑏 π‘₯+𝑏 NOUN cana-2815 260 20 ) ) PUNCT cana-2815 260 21 𝑑3 𝑑3 PROPN cana-2815 260 22 64 64 NUM cana-2815 260 23 𝑐7 𝑐7 NOUN cana-2815 260 24 + + CCONJ cana-2815 260 25 β‹― β‹― PROPN cana-2815 260 26 … … PUNCT cana-2815 260 27 ] ] PUNCT cana-2815 260 28 the the DET cana-2815 260 29 exact exact ADJ cana-2815 260 30 solution solution NOUN cana-2815 260 31 for for ADP cana-2815 260 32 equation equation NOUN cana-2815 260 33 ( ( PUNCT cana-2815 260 34 19 19 NUM cana-2815 260 35 ) ) PUNCT cana-2815 260 36 is be AUX cana-2815 260 37 : : PUNCT cana-2815 260 38 βˆ… βˆ… NOUN cana-2815 260 39 ( ( PUNCT cana-2815 260 40 π‘₯ π‘₯ NOUN cana-2815 260 41 , , PUNCT cana-2815 260 42 𝑑 𝑑 NOUN cana-2815 260 43 ) ) PUNCT cana-2815 260 44 = = SYM cana-2815 260 45 π‘₯+𝑏 π‘₯+𝑏 NUM cana-2815 260 46 2βˆšπ‘2βˆ’ 2βˆšπ‘2βˆ’ NUM cana-2815 260 47 𝑑 𝑑 NOUN cana-2815 260 48 , , PUNCT cana-2815 260 49 𝑑 𝑑 X cana-2815 260 50 < < X cana-2815 260 51 𝑐2 𝑐2 ADJ cana-2815 260 52 example example NOUN cana-2815 260 53 3 3 X cana-2815 260 54 . . X cana-2815 260 55 illustrate illustrate VERB cana-2815 260 56 the the DET cana-2815 260 57 following follow VERB cana-2815 260 58 fractional fractional ADJ cana-2815 260 59 - - PUNCT cana-2815 260 60 order order NOUN cana-2815 260 61 one one NUM cana-2815 260 62 dimensional dimensional ADJ cana-2815 260 63 telegraph telegraph NOUN cana-2815 260 64 equation equation NOUN cana-2815 260 65 [ [ X cana-2815 260 66 23 23 NUM cana-2815 260 67 ] ] X cana-2815 260 68 𝐷𝑑 𝐷𝑑 PROPN cana-2815 260 69 2π›Όβˆ…(π‘₯ 2π›Όβˆ…(π‘₯ NUM cana-2815 260 70 , , PUNCT cana-2815 260 71 𝑑 𝑑 X cana-2815 260 72 ) ) PUNCT cana-2815 260 73 + + CCONJ cana-2815 260 74 2 2 NUM cana-2815 260 75 𝐷𝑑 𝐷𝑑 PROPN cana-2815 260 76 π›Όβˆ…(π‘₯ π›Όβˆ…(π‘₯ NOUN cana-2815 260 77 , , PUNCT cana-2815 260 78 𝑑 𝑑 NOUN cana-2815 260 79 ) ) PUNCT cana-2815 260 80 + + CCONJ cana-2815 260 81 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 260 82 , , PUNCT cana-2815 260 83 𝑑 𝑑 PROPN cana-2815 260 84 ) ) PUNCT cana-2815 260 85 = = SYM cana-2815 260 86 βˆ…π‘₯π‘₯(π‘₯ βˆ…π‘₯π‘₯(π‘₯ X cana-2815 260 87 , , PUNCT cana-2815 260 88 𝑑 𝑑 X cana-2815 260 89 ) ) PUNCT cana-2815 260 90 0 0 PUNCT cana-2815 261 1 < < X cana-2815 261 2 βˆβ‰€ βˆβ‰€ X cana-2815 261 3 1 1 NUM cana-2815 261 4 , , PUNCT cana-2815 261 5 π‘₯ π‘₯ NOUN cana-2815 261 6 = = SYM cana-2815 261 7 1 1 NUM cana-2815 261 8 with with ADP cana-2815 261 9 initial initial ADJ cana-2815 261 10 conditions condition NOUN cana-2815 261 11 : : PUNCT cana-2815 261 12 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 261 13 , , PUNCT cana-2815 261 14 0 0 NUM cana-2815 261 15 ) ) PUNCT cana-2815 261 16 = = SYM cana-2815 261 17 𝑒π‘₯ 𝑒π‘₯ NOUN cana-2815 261 18 , , PUNCT cana-2815 261 19 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 261 20 , , PUNCT cana-2815 261 21 0 0 NUM cana-2815 261 22 ) ) PUNCT cana-2815 261 23 = = NOUN cana-2815 262 1 βˆ’2𝑒π‘₯ βˆ’2𝑒π‘₯ NOUN cana-2815 262 2 using use VERB cana-2815 262 3 the the DET cana-2815 262 4 elzaki elzaki NOUN cana-2815 262 5 transformation transformation NOUN cana-2815 262 6 of of ADP cana-2815 262 7 equation equation NOUN cana-2815 262 8 ( ( PUNCT cana-2815 262 9 27 27 NUM cana-2815 262 10 ) ) PUNCT cana-2815 262 11 , , PUNCT cana-2815 262 12 𝐸 𝐸 PROPN cana-2815 262 13 [ [ PUNCT cana-2815 262 14 𝐷𝑑 𝐷𝑑 PROPN cana-2815 262 15 2π›Όβˆ… 2π›Όβˆ… PROPN cana-2815 262 16 + + CCONJ cana-2815 262 17 2 2 NUM cana-2815 262 18 𝐷𝑑 𝐷𝑑 PROPN cana-2815 262 19 π›Όβˆ… π›Όβˆ… PUNCT cana-2815 262 20 + + NOUN cana-2815 262 21 βˆ… βˆ… NOUN cana-2815 262 22 ] ] PUNCT cana-2815 262 23 = = SYM cana-2815 262 24 𝐸[βˆ…π‘₯π‘₯ 𝐸[βˆ…π‘₯π‘₯ NOUN cana-2815 262 25 ] ] PUNCT cana-2815 262 26 applying apply VERB cana-2815 262 27 elzaki elzaki NOUN cana-2815 262 28 transform transform NOUN cana-2815 262 29 on on ADP cana-2815 262 30 above above ADP cana-2815 262 31 equation equation NOUN cana-2815 262 32 we we PRON cana-2815 262 33 get get VERB cana-2815 262 34 , , PUNCT cana-2815 262 35 1 1 NUM cana-2815 262 36 𝑣𝛼 𝑣𝛼 ADP cana-2815 262 37 𝐸[βˆ…(π‘₯ 𝐸[βˆ…(π‘₯ NOUN cana-2815 262 38 , , PUNCT cana-2815 262 39 𝑑 𝑑 NOUN cana-2815 262 40 ) ) PUNCT cana-2815 262 41 ] ] PUNCT cana-2815 262 42 βˆ’ βˆ’ PROPN cana-2815 262 43 𝑣2βˆ’π›Ό 𝑣2βˆ’π›Ό PROPN cana-2815 262 44 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 262 45 , , PUNCT cana-2815 262 46 0 0 NUM cana-2815 262 47 ) ) PUNCT cana-2815 262 48 βˆ’ βˆ’ PROPN cana-2815 263 1 𝑣3βˆ’π›Ό 𝑣3βˆ’π›Ό PROPN cana-2815 263 2 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ PROPN cana-2815 263 3 , , PUNCT cana-2815 263 4 0 0 NUM cana-2815 263 5 ) ) PUNCT cana-2815 263 6 = = PUNCT cana-2815 264 1 βˆ’πΈ[(βˆ… βˆ’πΈ[(βˆ… PRON cana-2815 264 2 βˆ’ βˆ’ PROPN cana-2815 264 3 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ ADJ cana-2815 264 4 ) ) PUNCT cana-2815 264 5 βˆ’ βˆ’ PROPN cana-2815 265 1 2 2 NUM cana-2815 265 2 𝐷𝑑 𝐷𝑑 PROPN cana-2815 265 3 π›Όβˆ… π›Όβˆ… X cana-2815 265 4 ] ] PUNCT cana-2815 265 5 𝐸[βˆ…(π‘₯ 𝐸[βˆ…(π‘₯ NOUN cana-2815 265 6 , , PUNCT cana-2815 265 7 𝑑 𝑑 NOUN cana-2815 265 8 ) ) PUNCT cana-2815 265 9 ] ] PUNCT cana-2815 266 1 = = SYM cana-2815 266 2 𝑣2 𝑣2 NUM cana-2815 266 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 266 4 , , PUNCT cana-2815 266 5 0 0 NUM cana-2815 266 6 ) ) PUNCT cana-2815 267 1 + + CCONJ cana-2815 267 2 𝑣3 𝑣3 ADJ cana-2815 267 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 267 4 , , PUNCT cana-2815 267 5 0 0 NUM cana-2815 267 6 ) ) PUNCT cana-2815 267 7 βˆ’ βˆ’ PROPN cana-2815 267 8 𝑣 𝑣 DET cana-2815 267 9 𝛼 𝛼 NOUN cana-2815 267 10 𝐸[(βˆ… 𝐸[(βˆ… NOUN cana-2815 267 11 βˆ’ βˆ’ PROPN cana-2815 267 12 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ PROPN cana-2815 267 13 ) ) PUNCT cana-2815 267 14 βˆ’ βˆ’ PROPN cana-2815 267 15 2 2 NUM cana-2815 267 16 𝐷𝑑 𝐷𝑑 PROPN cana-2815 267 17 π›Όβˆ… π›Όβˆ… X cana-2815 267 18 ] ] PUNCT cana-2815 267 19 applying apply VERB cana-2815 267 20 inverse inverse NOUN cana-2815 267 21 elzaki elzaki NOUN cana-2815 267 22 transform transform VERB cana-2815 267 23 on on ADP cana-2815 267 24 above above ADP cana-2815 267 25 equation equation NOUN cana-2815 267 26 πΈβˆ’1[𝐸 πΈβˆ’1[𝐸 NOUN cana-2815 267 27 [ [ X cana-2815 267 28 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 267 29 , , PUNCT cana-2815 267 30 𝑑 𝑑 PROPN cana-2815 267 31 ) ) PUNCT cana-2815 267 32 ] ] PUNCT cana-2815 267 33 ] ] X cana-2815 268 1 = = PUNCT cana-2815 268 2 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 269 1 [ [ X cana-2815 269 2 𝑣2 𝑣2 NOUN cana-2815 269 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 269 4 , , PUNCT cana-2815 269 5 0 0 NUM cana-2815 269 6 ) ) PUNCT cana-2815 269 7 + + CCONJ cana-2815 269 8 𝑣3 𝑣3 ADJ cana-2815 269 9 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 269 10 , , PUNCT cana-2815 269 11 0 0 NUM cana-2815 269 12 ) ) PUNCT cana-2815 269 13 βˆ’ βˆ’ PROPN cana-2815 269 14 𝑣 𝑣 DET cana-2815 269 15 𝛼 𝛼 NOUN cana-2815 269 16 𝐸[(βˆ… 𝐸[(βˆ… NOUN cana-2815 269 17 βˆ’ βˆ’ PROPN cana-2815 269 18 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ PROPN cana-2815 269 19 ) ) PUNCT cana-2815 269 20 βˆ’ βˆ’ PROPN cana-2815 269 21 2 2 NUM cana-2815 269 22 𝐷𝑑 𝐷𝑑 PROPN cana-2815 269 23 π›Όβˆ… π›Όβˆ… X cana-2815 269 24 ] ] PUNCT cana-2815 269 25 ] ] X cana-2815 269 26 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 269 27 , , PUNCT cana-2815 269 28 𝑑 𝑑 PROPN cana-2815 269 29 ) ) PUNCT cana-2815 269 30 = = PUNCT cana-2815 269 31 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 270 1 [ [ X cana-2815 270 2 𝑣2 𝑣2 NOUN cana-2815 270 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 270 4 , , PUNCT cana-2815 270 5 0 0 NUM cana-2815 270 6 ) ) PUNCT cana-2815 270 7 + + CCONJ cana-2815 270 8 𝑣3 𝑣3 ADJ cana-2815 270 9 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 270 10 , , PUNCT cana-2815 270 11 0 0 NUM cana-2815 270 12 ) ) PUNCT cana-2815 270 13 βˆ’ βˆ’ NOUN cana-2815 271 1 𝑣 𝑣 ADP cana-2815 271 2 𝛼 𝛼 X cana-2815 271 3 𝐸[𝐿(βˆ… 𝐸[𝐿(βˆ… NOUN cana-2815 271 4 ) ) PUNCT cana-2815 271 5 βˆ’ βˆ’ PROPN cana-2815 271 6 2 2 NUM cana-2815 271 7 𝐷𝑑 𝐷𝑑 PROPN cana-2815 271 8 π›Όβˆ… π›Όβˆ… NOUN cana-2815 271 9 ] ] PUNCT cana-2815 271 10 ] ] PUNCT cana-2815 271 11 appling apple VERB cana-2815 271 12 the the DET cana-2815 271 13 madetm madetm NOUN cana-2815 271 14 process process NOUN cana-2815 271 15 on on ADP cana-2815 271 16 above above ADP cana-2815 271 17 equation equation NOUN cana-2815 271 18 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 271 19 , , PUNCT cana-2815 271 20 𝑑 𝑑 X cana-2815 271 21 ) ) PUNCT cana-2815 271 22 = = PUNCT cana-2815 271 23 βˆ…(0 βˆ…(0 PROPN cana-2815 271 24 ) ) PUNCT cana-2815 271 25 = = PUNCT cana-2815 271 26 𝑒 𝑒 X cana-2815 271 27 π‘₯(1βˆ’ π‘₯(1βˆ’ NOUN cana-2815 271 28 2𝑑 2𝑑 NUM cana-2815 271 29 ) ) PUNCT cana-2815 271 30 ( ( PUNCT cana-2815 271 31 29 29 NUM cana-2815 271 32 ) ) PUNCT cana-2815 271 33 using use VERB cana-2815 271 34 the the DET cana-2815 271 35 recursive recursive ADJ cana-2815 271 36 series series NOUN cana-2815 271 37 as as SCONJ cana-2815 271 38 shown show VERB cana-2815 271 39 in in ADP cana-2815 271 40 equation equation NOUN cana-2815 271 41 ( ( PUNCT cana-2815 271 42 10 10 NUM cana-2815 271 43 ) ) PUNCT cana-2815 271 44 , , PUNCT cana-2815 271 45 βˆ…π‘›+1(π‘₯ βˆ…π‘›+1(π‘₯ NUM cana-2815 271 46 , , PUNCT cana-2815 271 47 𝑑 𝑑 NOUN cana-2815 271 48 ) ) PUNCT cana-2815 271 49 = = PUNCT cana-2815 271 50 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 271 51 [ [ X cana-2815 271 52 𝐿 𝐿 PROPN cana-2815 271 53 ( ( PUNCT cana-2815 271 54 βˆ…π‘› βˆ…π‘› NOUN cana-2815 271 55 ) ) PUNCT cana-2815 271 56 βˆ’ βˆ’ PROPN cana-2815 271 57 2 2 NUM cana-2815 271 58 𝐷𝑑 𝐷𝑑 NOUN cana-2815 271 59 𝛼 𝛼 NOUN cana-2815 271 60 βˆ…π‘› βˆ…π‘› NOUN cana-2815 271 61 ] ] PUNCT cana-2815 271 62 ] ] PUNCT cana-2815 271 63 for for ADP cana-2815 271 64 𝑛 𝑛 PROPN cana-2815 271 65 = = SYM cana-2815 271 66 0 0 NUM cana-2815 271 67 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 271 68 , , PUNCT cana-2815 271 69 𝑑 𝑑 X cana-2815 271 70 ) ) PUNCT cana-2815 271 71 = = SYM cana-2815 271 72 𝐸 𝐸 PROPN cana-2815 271 73 βˆ’1[𝑣𝛼𝐸 βˆ’1[𝑣𝛼𝐸 PROPN cana-2815 271 74 [ [ X cana-2815 271 75 𝐿 𝐿 PROPN cana-2815 271 76 ( ( PUNCT cana-2815 271 77 βˆ…0 βˆ…0 NOUN cana-2815 271 78 ) ) PUNCT cana-2815 271 79 βˆ’ βˆ’ PROPN cana-2815 271 80 2 2 NUM cana-2815 272 1 𝐷𝑑 𝐷𝑑 NOUN cana-2815 272 2 𝛼 𝛼 NOUN cana-2815 272 3 βˆ…0 βˆ…0 NOUN cana-2815 272 4 ] ] PUNCT cana-2815 272 5 ] ] PUNCT cana-2815 272 6 here here ADV cana-2815 272 7 , , PUNCT cana-2815 272 8 𝐿 𝐿 PROPN cana-2815 273 1 [ [ X cana-2815 273 2 βˆ…0 βˆ…0 NOUN cana-2815 273 3 ] ] X cana-2815 273 4 = = SYM cana-2815 273 5 βˆ…0π‘₯π‘₯ βˆ…0π‘₯π‘₯ VERB cana-2815 273 6 βˆ’ βˆ’ NOUN cana-2815 273 7 βˆ…0 βˆ…0 NOUN cana-2815 273 8 = = SYM cana-2815 273 9 0 0 NUM cana-2815 273 10 therefore therefore ADV cana-2815 273 11 , , PUNCT cana-2815 273 12 above above ADP cana-2815 273 13 equation equation NOUN cana-2815 273 14 implies imply VERB cana-2815 273 15 , , PUNCT cana-2815 273 16 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 273 17 , , PUNCT cana-2815 273 18 𝑑 𝑑 X cana-2815 273 19 ) ) PUNCT cana-2815 273 20 = = PUNCT cana-2815 273 21 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 274 1 [ [ X cana-2815 274 2 βˆ’2 βˆ’2 X cana-2815 274 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 274 4 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 274 5 ] ] PUNCT cana-2815 274 6 ] ] X cana-2815 274 7 consider consider VERB cana-2815 274 8 , , PUNCT cana-2815 274 9 𝐸 𝐸 PROPN cana-2815 275 1 [ [ X cana-2815 275 2 βˆ’2 βˆ’2 X cana-2815 275 3 𝐷𝑑 𝐷𝑑 NOUN cana-2815 275 4 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 275 5 ] ] PUNCT cana-2815 275 6 = = SYM cana-2815 275 7 βˆ’2 βˆ’2 NOUN cana-2815 275 8 [ [ PUNCT cana-2815 275 9 1 1 NUM cana-2815 275 10 𝑣𝛼 𝑣𝛼 NOUN cana-2815 275 11 𝐸[βˆ…0 𝐸[βˆ…0 PROPN cana-2815 275 12 ] ] X cana-2815 276 1 βˆ’ βˆ’ PROPN cana-2815 276 2 𝑣 𝑣 PRON cana-2815 276 3 2βˆ’π›Ό 2βˆ’π›Ό NUM cana-2815 276 4 βˆ…0(0 βˆ…0(0 PROPN cana-2815 276 5 ) ) PUNCT cana-2815 276 6 ] ] PUNCT cana-2815 277 1 𝐸 𝐸 PROPN cana-2815 278 1 [ [ X cana-2815 278 2 βˆ’2 βˆ’2 X cana-2815 278 3 𝐷𝑑 𝐷𝑑 NOUN cana-2815 278 4 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 278 5 ] ] PUNCT cana-2815 278 6 = = SYM cana-2815 279 1 βˆ’2 βˆ’2 NOUN cana-2815 280 1 [ [ X cana-2815 280 2 [ [ PUNCT cana-2815 280 3 1 1 NUM cana-2815 280 4 𝑣𝛼 𝑣𝛼 NOUN cana-2815 280 5 ] ] PUNCT cana-2815 280 6 𝐸[𝑒π‘₯(1βˆ’ 𝐸[𝑒π‘₯(1βˆ’ ADJ cana-2815 280 7 2𝑑 2𝑑 NOUN cana-2815 280 8 ) ) PUNCT cana-2815 280 9 ] ] PUNCT cana-2815 281 1 βˆ’ βˆ’ PROPN cana-2815 281 2 𝑣2 𝑣2 NUM cana-2815 281 3 𝑒π‘₯ 𝑒π‘₯ NOUN cana-2815 281 4 ] ] X cana-2815 281 5 𝐸 𝐸 PROPN cana-2815 281 6 [ [ X cana-2815 281 7 βˆ’2 βˆ’2 X cana-2815 281 8 𝐷𝑑 𝐷𝑑 NOUN cana-2815 281 9 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 281 10 ] ] PUNCT cana-2815 281 11 = = SYM cana-2815 281 12 βˆ’2𝑒π‘₯ βˆ’2𝑒π‘₯ NOUN cana-2815 282 1 [ [ X cana-2815 282 2 [ [ PUNCT cana-2815 282 3 1 1 NUM cana-2815 282 4 𝑣𝛼 𝑣𝛼 NOUN cana-2815 282 5 ] ] PUNCT cana-2815 283 1 [ [ X cana-2815 283 2 𝐸(1 𝐸(1 NUM cana-2815 283 3 ) ) PUNCT cana-2815 283 4 βˆ’ βˆ’ ADP cana-2815 284 1 2𝐸(𝑑 2𝐸(𝑑 NUM cana-2815 284 2 ) ) PUNCT cana-2815 284 3 ] ] PUNCT cana-2815 285 1 βˆ’ βˆ’ PROPN cana-2815 285 2 𝑣2 𝑣2 X cana-2815 285 3 ] ] X cana-2815 285 4 = = PUNCT cana-2815 285 5 βˆ’2𝑒π‘₯ βˆ’2𝑒π‘₯ NOUN cana-2815 286 1 [ [ X cana-2815 286 2 [ [ PUNCT cana-2815 286 3 1 1 NUM cana-2815 286 4 𝑣𝛼 𝑣𝛼 NOUN cana-2815 286 5 ] ] PUNCT cana-2815 287 1 [ [ X cana-2815 287 2 𝑣2 𝑣2 NUM cana-2815 287 3 βˆ’ βˆ’ PROPN cana-2815 287 4 2𝑣3 2𝑣3 NUM cana-2815 287 5 ] ] PUNCT cana-2815 287 6 βˆ’ βˆ’ PROPN cana-2815 287 7 𝑣2 𝑣2 X cana-2815 287 8 ] ] PUNCT cana-2815 287 9 ( ( PUNCT cana-2815 287 10 27 27 NUM cana-2815 287 11 ) ) PUNCT cana-2815 287 12 ( ( PUNCT cana-2815 287 13 26 26 NUM cana-2815 287 14 ) ) PUNCT cana-2815 287 15 ( ( PUNCT cana-2815 287 16 28 28 NUM cana-2815 287 17 ) ) PUNCT cana-2815 287 18 ( ( PUNCT cana-2815 287 19 25 25 NUM cana-2815 287 20 ) ) PUNCT cana-2815 287 21 ( ( PUNCT cana-2815 287 22 30 30 NUM cana-2815 287 23 ) ) PUNCT cana-2815 287 24 ( ( PUNCT cana-2815 287 25 29 29 NUM cana-2815 287 26 ) ) PUNCT cana-2815 287 27 https://example.com https://example.com X cana-2815 287 28 communications communication NOUN cana-2815 287 29 on on ADP cana-2815 287 30 applied apply VERB cana-2815 287 31 nonlinear nonlinear ADJ cana-2815 287 32 analysis analysis NOUN cana-2815 287 33 issn issn NOUN cana-2815 287 34 : : PUNCT cana-2815 287 35 1074 1074 NUM cana-2815 287 36 - - PUNCT cana-2815 287 37 133x 133x NUM cana-2815 287 38 vol vol NOUN cana-2815 287 39 32 32 NUM cana-2815 287 40 no no NOUN cana-2815 287 41 . . PUNCT cana-2815 288 1 4s 4s NUM cana-2815 288 2 ( ( PUNCT cana-2815 288 3 2025 2025 NUM cana-2815 288 4 ) ) PUNCT cana-2815 288 5 319 319 NUM cana-2815 288 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 288 7 𝐸 𝐸 PROPN cana-2815 289 1 [ [ X cana-2815 289 2 βˆ’2 βˆ’2 X cana-2815 289 3 𝐷𝑑 𝐷𝑑 NOUN cana-2815 289 4 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 289 5 ] ] PUNCT cana-2815 289 6 = = SYM cana-2815 289 7 βˆ’2𝑒π‘₯ βˆ’2𝑒π‘₯ NOUN cana-2815 290 1 [ [ X cana-2815 290 2 [ [ PUNCT cana-2815 290 3 1 1 NUM cana-2815 290 4 𝑣𝛼 𝑣𝛼 NOUN cana-2815 290 5 ] ] PUNCT cana-2815 291 1 [ [ X cana-2815 291 2 βˆ’2𝑣3 βˆ’2𝑣3 X cana-2815 291 3 ] ] X cana-2815 291 4 ] ] X cana-2815 291 5 = = SYM cana-2815 291 6 4𝑒π‘₯ 4𝑒π‘₯ NOUN cana-2815 291 7 [ [ PUNCT cana-2815 291 8 𝑣3βˆ’π›Ό 𝑣3βˆ’π›Ό PROPN cana-2815 291 9 ] ] PUNCT cana-2815 291 10 therefore therefore ADV cana-2815 291 11 , , PUNCT cana-2815 291 12 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 291 13 , , PUNCT cana-2815 291 14 𝑑 𝑑 PROPN cana-2815 291 15 ) ) PUNCT cana-2815 291 16 = = SYM cana-2815 291 17 𝐸 𝐸 PROPN cana-2815 291 18 βˆ’1 βˆ’1 NOUN cana-2815 292 1 [ [ X cana-2815 292 2 𝑣𝛼 𝑣𝛼 NOUN cana-2815 292 3 [ [ X cana-2815 292 4 4𝑒π‘₯(𝑣3βˆ’π›Ό 4𝑒π‘₯(𝑣3βˆ’π›Ό NUM cana-2815 292 5 ) ) PUNCT cana-2815 292 6 ] ] PUNCT cana-2815 293 1 ] ] PUNCT cana-2815 293 2 = = SYM cana-2815 293 3 4𝑒π‘₯πΈβˆ’1 4𝑒π‘₯πΈβˆ’1 X cana-2815 293 4 [ [ PUNCT cana-2815 293 5 ( ( PUNCT cana-2815 293 6 𝑣3+𝛼 𝑣3+𝛼 PROPN cana-2815 293 7 ) ) PUNCT cana-2815 293 8 ] ] PUNCT cana-2815 293 9 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 293 10 , , PUNCT cana-2815 293 11 𝑑 𝑑 X cana-2815 293 12 ) ) PUNCT cana-2815 293 13 = = SYM cana-2815 293 14 4𝑒π‘₯ 4𝑒π‘₯ NOUN cana-2815 293 15 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 293 16 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 293 17 ) ) PUNCT cana-2815 293 18 for for ADP cana-2815 293 19 𝑛 𝑛 NOUN cana-2815 293 20 = = SYM cana-2815 293 21 1,2,3 1,2,3 NUM cana-2815 293 22 … … NUM cana-2815 293 23 . . PUNCT cana-2815 294 1 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 294 2 , , PUNCT cana-2815 294 3 𝑑 𝑑 X cana-2815 294 4 ) ) PUNCT cana-2815 294 5 = = PUNCT cana-2815 294 6 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 295 1 [ [ X cana-2815 295 2 βˆ’2 βˆ’2 X cana-2815 295 3 𝐷𝑑 𝐷𝑑 NOUN cana-2815 295 4 π›Όβˆ…1 π›Όβˆ…1 NOUN cana-2815 295 5 ] ] X cana-2815 295 6 ] ] X cana-2815 295 7 = = SYM cana-2815 295 8 βˆ’8𝑒π‘₯ βˆ’8𝑒π‘₯ X cana-2815 295 9 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 295 10 ⌈(2𝛼+2 ⌈(2𝛼+2 X cana-2815 295 11 ) ) PUNCT cana-2815 295 12 βˆ…3(π‘₯ βˆ…3(π‘₯ NOUN cana-2815 295 13 , , PUNCT cana-2815 295 14 𝑑 𝑑 X cana-2815 295 15 ) ) PUNCT cana-2815 295 16 = = PUNCT cana-2815 295 17 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 296 1 [ [ X cana-2815 296 2 βˆ’2 βˆ’2 X cana-2815 296 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 296 4 π›Όβˆ…2 π›Όβˆ…2 NOUN cana-2815 296 5 ] ] X cana-2815 296 6 ] ] X cana-2815 296 7 = = SYM cana-2815 296 8 16𝑒π‘₯ 16𝑒π‘₯ NOUN cana-2815 296 9 𝑑3𝛼+1 𝑑3𝛼+1 ADJ cana-2815 296 10 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 296 11 ) ) PUNCT cana-2815 296 12 βˆ…4(π‘₯ βˆ…4(π‘₯ NOUN cana-2815 296 13 , , PUNCT cana-2815 296 14 𝑑 𝑑 NOUN cana-2815 296 15 ) ) PUNCT cana-2815 296 16 = = PUNCT cana-2815 296 17 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 297 1 [ [ X cana-2815 297 2 βˆ’2 βˆ’2 X cana-2815 297 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 297 4 π›Όβˆ…3 π›Όβˆ…3 NOUN cana-2815 297 5 ] ] X cana-2815 297 6 ] ] PUNCT cana-2815 297 7 = = PUNCT cana-2815 297 8 βˆ’32𝑒π‘₯ βˆ’32𝑒π‘₯ PROPN cana-2815 297 9 𝑑4𝛼+1 𝑑4𝛼+1 ADV cana-2815 297 10 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 297 11 ) ) PUNCT cana-2815 297 12 . . PUNCT cana-2815 298 1 : : PUNCT cana-2815 298 2 : : PUNCT cana-2815 298 3 therefore therefore ADV cana-2815 298 4 , , PUNCT cana-2815 298 5 series series NOUN cana-2815 298 6 representation representation NOUN cana-2815 298 7 of of ADP cana-2815 298 8 the the DET cana-2815 298 9 solution solution NOUN cana-2815 298 10 βˆ… βˆ… NOUN cana-2815 298 11 ( ( PUNCT cana-2815 298 12 π‘₯ π‘₯ NOUN cana-2815 298 13 , , PUNCT cana-2815 298 14 𝑑 𝑑 PROPN cana-2815 298 15 ) ) PUNCT cana-2815 298 16 is be AUX cana-2815 298 17 as as SCONJ cana-2815 298 18 follows follow VERB cana-2815 298 19 : : PUNCT cana-2815 298 20 βˆ… βˆ… NOUN cana-2815 298 21 ( ( PUNCT cana-2815 298 22 π‘₯ π‘₯ NOUN cana-2815 298 23 , , PUNCT cana-2815 298 24 𝑑 𝑑 NOUN cana-2815 298 25 ) ) PUNCT cana-2815 298 26 = = SYM cana-2815 298 27 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 298 28 , , PUNCT cana-2815 298 29 𝑑)+βˆ…1(π‘₯ 𝑑)+βˆ…1(π‘₯ NOUN cana-2815 298 30 , , PUNCT cana-2815 298 31 𝑑)+βˆ…2(π‘₯ 𝑑)+βˆ…2(π‘₯ NUM cana-2815 298 32 , , PUNCT cana-2815 298 33 𝑑 𝑑 X cana-2815 298 34 ) ) PUNCT cana-2815 298 35 + + CCONJ cana-2815 298 36 βˆ…3(π‘₯ βˆ…3(π‘₯ NUM cana-2815 298 37 , , PUNCT cana-2815 298 38 𝑑 𝑑 X cana-2815 298 39 ) ) PUNCT cana-2815 298 40 + + CCONJ cana-2815 298 41 βˆ…4(π‘₯ βˆ…4(π‘₯ NOUN cana-2815 298 42 , , PUNCT cana-2815 298 43 𝑑 𝑑 X cana-2815 298 44 ) ) PUNCT cana-2815 299 1 + + X cana-2815 299 2 β‹― β‹― NOUN cana-2815 299 3 …… …… NOUN cana-2815 299 4 βˆ… βˆ… NOUN cana-2815 299 5 ( ( PUNCT cana-2815 299 6 π‘₯ π‘₯ NOUN cana-2815 299 7 , , PUNCT cana-2815 299 8 𝑑 𝑑 NOUN cana-2815 299 9 ) ) PUNCT cana-2815 299 10 = = SYM cana-2815 299 11 𝑒π‘₯(1βˆ’ 𝑒π‘₯(1βˆ’ NUM cana-2815 299 12 2𝑑 2𝑑 NUM cana-2815 299 13 ) ) PUNCT cana-2815 300 1 + + CCONJ cana-2815 300 2 4𝑒π‘₯ 4𝑒π‘₯ ADJ cana-2815 300 3 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 300 4 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 300 5 ) ) PUNCT cana-2815 300 6 βˆ’ βˆ’ PROPN cana-2815 300 7 8𝑒π‘₯ 8𝑒π‘₯ ADJ cana-2815 300 8 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 300 9 ⌈(2𝛼+2 ⌈(2𝛼+2 X cana-2815 300 10 ) ) PUNCT cana-2815 301 1 + + CCONJ cana-2815 301 2 16𝑒π‘₯ 16𝑒π‘₯ ADJ cana-2815 301 3 𝑑3𝛼+1 𝑑3𝛼+1 ADJ cana-2815 301 4 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 301 5 ) ) PUNCT cana-2815 301 6 βˆ’ βˆ’ PROPN cana-2815 301 7 32𝑒π‘₯ 32𝑒π‘₯ PROPN cana-2815 301 8 𝑑4𝛼+1 𝑑4𝛼+1 ADV cana-2815 301 9 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 301 10 ) ) PUNCT cana-2815 301 11 … … PUNCT cana-2815 301 12 … … PUNCT cana-2815 301 13 … … SYM cana-2815 301 14 . . PUNCT cana-2815 302 1 βˆ… βˆ… NOUN cana-2815 302 2 ( ( PUNCT cana-2815 302 3 π‘₯ π‘₯ NOUN cana-2815 302 4 , , PUNCT cana-2815 302 5 𝑑 𝑑 NOUN cana-2815 302 6 ) ) PUNCT cana-2815 302 7 = = SYM cana-2815 303 1 𝑒π‘₯ 𝑒π‘₯ NOUN cana-2815 304 1 [ [ X cana-2815 304 2 ( ( PUNCT cana-2815 304 3 1βˆ’ 1βˆ’ NUM cana-2815 304 4 2𝑑 2𝑑 NUM cana-2815 304 5 ) ) PUNCT cana-2815 305 1 + + CCONJ cana-2815 305 2 4 4 NUM cana-2815 305 3 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 305 4 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 305 5 ) ) PUNCT cana-2815 306 1 βˆ’ βˆ’ PROPN cana-2815 306 2 8 8 NUM cana-2815 306 3 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 306 4 ⌈(2𝛼+2 ⌈(2𝛼+2 PROPN cana-2815 306 5 ) ) PUNCT cana-2815 307 1 + + CCONJ cana-2815 307 2 16 16 NUM cana-2815 307 3 𝑑3𝛼+1 𝑑3𝛼+1 PUNCT cana-2815 307 4 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 307 5 ) ) PUNCT cana-2815 307 6 βˆ’ βˆ’ PROPN cana-2815 307 7 32 32 NUM cana-2815 307 8 𝑑4𝛼+1 𝑑4𝛼+1 ADV cana-2815 307 9 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 307 10 ) ) PUNCT cana-2815 307 11 … … PUNCT cana-2815 307 12 …… …… X cana-2815 307 13 . . PUNCT cana-2815 307 14 ] ] PUNCT cana-2815 308 1 in in ADP cana-2815 308 2 particular particular ADJ cana-2815 308 3 when when SCONJ cana-2815 308 4 𝛼 𝛼 X cana-2815 308 5 = = SYM cana-2815 308 6 1 1 NUM cana-2815 308 7 , , PUNCT cana-2815 308 8 the the DET cana-2815 308 9 solution solution NOUN cana-2815 308 10 is be AUX cana-2815 308 11 in in ADP cana-2815 308 12 the the DET cana-2815 308 13 form form NOUN cana-2815 308 14 : : PUNCT cana-2815 308 15 βˆ… βˆ… NOUN cana-2815 308 16 ( ( PUNCT cana-2815 308 17 π‘₯ π‘₯ NOUN cana-2815 308 18 , , PUNCT cana-2815 308 19 𝑑 𝑑 NOUN cana-2815 308 20 ) ) PUNCT cana-2815 308 21 = = SYM cana-2815 309 1 𝑒π‘₯ 𝑒π‘₯ NOUN cana-2815 310 1 [ [ X cana-2815 310 2 1βˆ’ 1βˆ’ NUM cana-2815 310 3 2𝑑 2𝑑 NUM cana-2815 310 4 1 1 NUM cana-2815 310 5 ! ! PUNCT cana-2815 311 1 + + CCONJ cana-2815 311 2 ( ( PUNCT cana-2815 311 3 2𝑑)2 2𝑑)2 NUM cana-2815 311 4 2 2 NUM cana-2815 311 5 ! ! PUNCT cana-2815 311 6 βˆ’ βˆ’ PROPN cana-2815 311 7 ( ( PUNCT cana-2815 311 8 2𝑑)3 2𝑑)3 NUM cana-2815 311 9 3 3 NUM cana-2815 311 10 ! ! PUNCT cana-2815 312 1 + + CCONJ cana-2815 312 2 ( ( PUNCT cana-2815 312 3 2𝑑)4 2𝑑)4 NUM cana-2815 312 4 4 4 NUM cana-2815 312 5 ! ! PUNCT cana-2815 312 6 βˆ’ βˆ’ PROPN cana-2815 312 7 ( ( PUNCT cana-2815 312 8 2𝑑)5 2𝑑)5 NUM cana-2815 312 9 5 5 NUM cana-2815 312 10 ! ! PUNCT cana-2815 312 11 … … PUNCT cana-2815 312 12 . . PUNCT cana-2815 312 13 . . PUNCT cana-2815 313 1 ] ] PUNCT cana-2815 314 1 the the DET cana-2815 314 2 exact exact ADJ cana-2815 314 3 solution solution NOUN cana-2815 314 4 for for ADP cana-2815 314 5 equation equation NOUN cana-2815 314 6 ( ( PUNCT cana-2815 314 7 27 27 NUM cana-2815 314 8 ) ) PUNCT cana-2815 314 9 is be AUX cana-2815 314 10 : : PUNCT cana-2815 314 11 βˆ… βˆ… NOUN cana-2815 314 12 ( ( PUNCT cana-2815 314 13 π‘₯ π‘₯ NOUN cana-2815 314 14 , , PUNCT cana-2815 314 15 𝑑 𝑑 NOUN cana-2815 314 16 ) ) PUNCT cana-2815 314 17 = = SYM cana-2815 315 1 𝑒π‘₯βˆ’2𝑑 𝑒π‘₯βˆ’2𝑑 NUM cana-2815 315 2 two two NUM cana-2815 315 3 - - PUNCT cana-2815 315 4 dimensional dimensional ADJ cana-2815 315 5 fractional fractional ADJ cana-2815 315 6 telegraph telegraph NOUN cana-2815 315 7 equation equation NOUN cana-2815 315 8 : : PUNCT cana-2815 315 9 example example NOUN cana-2815 315 10 4 4 NUM cana-2815 315 11 . . PUNCT cana-2815 315 12 considering consider VERB cana-2815 315 13 the the DET cana-2815 315 14 two two NUM cana-2815 315 15 - - PUNCT cana-2815 315 16 dimensional dimensional ADJ cana-2815 315 17 fractional fractional ADJ cana-2815 315 18 telegraph telegraph NOUN cana-2815 315 19 equation equation NOUN cana-2815 315 20 as as SCONJ cana-2815 315 21 follows follow VERB cana-2815 315 22 [ [ X cana-2815 315 23 23 23 NUM cana-2815 315 24 ] ] X cana-2815 315 25 : : PUNCT cana-2815 316 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 316 2 2π›Όβˆ… 2π›Όβˆ… NUM cana-2815 316 3 + + CCONJ cana-2815 316 4 3 3 NUM cana-2815 316 5 𝐷𝑑 𝐷𝑑 PROPN cana-2815 316 6 π›Όβˆ… π›Όβˆ… X cana-2815 316 7 + + NOUN cana-2815 316 8 2βˆ… 2βˆ… NUM cana-2815 316 9 = = SYM cana-2815 316 10 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ INTJ cana-2815 317 1 + + CCONJ cana-2815 317 2 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ X cana-2815 317 3 0 0 NUM cana-2815 317 4 < < X cana-2815 317 5 ∝ ∝ X cana-2815 317 6 ≀ ≀ NUM cana-2815 317 7 1 1 NUM cana-2815 317 8 initial initial ADJ cana-2815 317 9 conditions condition NOUN cana-2815 317 10 : : PUNCT cana-2815 317 11 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 317 12 , , PUNCT cana-2815 317 13 𝑦 𝑦 NOUN cana-2815 317 14 , , PUNCT cana-2815 317 15 0 0 NUM cana-2815 317 16 ) ) PUNCT cana-2815 317 17 = = SYM cana-2815 317 18 𝑒 𝑒 PROPN cana-2815 317 19 π‘₯+𝑦 π‘₯+𝑦 PROPN cana-2815 317 20 , , PUNCT cana-2815 317 21 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 317 22 , , PUNCT cana-2815 317 23 𝑦 𝑦 NOUN cana-2815 317 24 , , PUNCT cana-2815 317 25 0 0 NUM cana-2815 317 26 ) ) PUNCT cana-2815 318 1 = = SYM cana-2815 318 2 βˆ’3𝑒π‘₯+𝑦 βˆ’3𝑒π‘₯+𝑦 NOUN cana-2815 318 3 ( ( PUNCT cana-2815 318 4 35 35 NUM cana-2815 318 5 ) ) PUNCT cana-2815 318 6 apply apply VERB cana-2815 318 7 the the DET cana-2815 318 8 elzaki elzaki NOUN cana-2815 318 9 transformation transformation NOUN cana-2815 318 10 of of ADP cana-2815 318 11 equation equation NOUN cana-2815 318 12 ( ( PUNCT cana-2815 318 13 34 34 NUM cana-2815 318 14 ) ) PUNCT cana-2815 318 15 , , PUNCT cana-2815 318 16 𝐸 𝐸 PROPN cana-2815 318 17 [ [ PUNCT cana-2815 318 18 𝐷𝑑 𝐷𝑑 PROPN cana-2815 318 19 2π›Όβˆ… 2π›Όβˆ… PROPN cana-2815 318 20 + + CCONJ cana-2815 318 21 3 3 NUM cana-2815 318 22 𝐷𝑑 𝐷𝑑 PROPN cana-2815 318 23 π›Όβˆ… π›Όβˆ… PUNCT cana-2815 318 24 + + NOUN cana-2815 319 1 2βˆ… 2βˆ… NUM cana-2815 319 2 ] ] X cana-2815 319 3 = = PUNCT cana-2815 320 1 𝐸[βˆ… 𝐸[βˆ… NOUN cana-2815 320 2 π‘₯π‘₯ π‘₯π‘₯ ADP cana-2815 320 3 + + NUM cana-2815 320 4 βˆ… βˆ… NOUN cana-2815 320 5 𝑦𝑦 𝑦𝑦 PROPN cana-2815 320 6 ] ] X cana-2815 320 7 𝐸 𝐸 PROPN cana-2815 320 8 [ [ PUNCT cana-2815 320 9 𝐷𝑑 𝐷𝑑 PROPN cana-2815 320 10 2π›Όβˆ… 2π›Όβˆ… NUM cana-2815 320 11 ] ] X cana-2815 320 12 = = PUNCT cana-2815 321 1 βˆ’πΈ[βˆ…π‘₯π‘₯ βˆ’πΈ[βˆ…π‘₯π‘₯ NOUN cana-2815 321 2 + + PUNCT cana-2815 321 3 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ ADV cana-2815 321 4 βˆ’ βˆ’ PROPN cana-2815 321 5 3 3 NUM cana-2815 321 6 𝐷𝑑 𝐷𝑑 PROPN cana-2815 321 7 π›Όβˆ… π›Όβˆ… PUNCT cana-2815 321 8 βˆ’ βˆ’ NOUN cana-2815 322 1 2βˆ… 2βˆ… NUM cana-2815 322 2 ] ] X cana-2815 322 3 ( ( PUNCT cana-2815 322 4 31 31 NUM cana-2815 322 5 ) ) PUNCT cana-2815 322 6 ( ( PUNCT cana-2815 322 7 34 34 NUM cana-2815 322 8 ) ) PUNCT cana-2815 322 9 ( ( PUNCT cana-2815 322 10 32 32 NUM cana-2815 322 11 ) ) PUNCT cana-2815 322 12 ( ( PUNCT cana-2815 322 13 33 33 NUM cana-2815 322 14 ) ) PUNCT cana-2815 322 15 ( ( PUNCT cana-2815 322 16 35 35 NUM cana-2815 322 17 ) ) PUNCT cana-2815 322 18 communications communication NOUN cana-2815 322 19 on on ADP cana-2815 322 20 applied apply VERB cana-2815 322 21 nonlinear nonlinear ADJ cana-2815 322 22 analysis analysis NOUN cana-2815 322 23 issn issn NOUN cana-2815 322 24 : : PUNCT cana-2815 322 25 1074 1074 NUM cana-2815 322 26 - - PUNCT cana-2815 322 27 133x 133x NUM cana-2815 322 28 vol vol NOUN cana-2815 322 29 32 32 NUM cana-2815 322 30 no no NOUN cana-2815 322 31 . . PUNCT cana-2815 323 1 4s 4s NUM cana-2815 323 2 ( ( PUNCT cana-2815 323 3 2025 2025 NUM cana-2815 323 4 ) ) PUNCT cana-2815 323 5 320 320 NUM cana-2815 323 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 323 7 1 1 NUM cana-2815 323 8 𝑣𝛼 𝑣𝛼 ADP cana-2815 323 9 𝐸[βˆ…(π‘₯ 𝐸[βˆ…(π‘₯ PROPN cana-2815 323 10 , , PUNCT cana-2815 323 11 y y PROPN cana-2815 323 12 , , PUNCT cana-2815 323 13 𝑑 𝑑 PROPN cana-2815 323 14 ) ) PUNCT cana-2815 323 15 ] ] PUNCT cana-2815 324 1 βˆ’ βˆ’ PROPN cana-2815 324 2 𝑣2βˆ’π›Ό 𝑣2βˆ’π›Ό PROPN cana-2815 324 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 324 4 , , PUNCT cana-2815 324 5 𝑦 𝑦 NOUN cana-2815 324 6 , , PUNCT cana-2815 324 7 0 0 NUM cana-2815 324 8 ) ) PUNCT cana-2815 324 9 βˆ’ βˆ’ PROPN cana-2815 324 10 𝑣3βˆ’π›Ό 𝑣3βˆ’π›Ό PROPN cana-2815 324 11 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ PROPN cana-2815 324 12 , , PUNCT cana-2815 324 13 𝑦 𝑦 NOUN cana-2815 324 14 , , PUNCT cana-2815 324 15 0 0 NUM cana-2815 324 16 ) ) PUNCT cana-2815 324 17 = = VERB cana-2815 325 1 βˆ’πΈ[(βˆ…π‘₯π‘₯ βˆ’πΈ[(βˆ…π‘₯π‘₯ NOUN cana-2815 325 2 + + CCONJ cana-2815 325 3 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ ADV cana-2815 325 4 βˆ’ βˆ’ PROPN cana-2815 325 5 2βˆ… 2βˆ… NUM cana-2815 325 6 ) ) PUNCT cana-2815 325 7 βˆ’ βˆ’ PROPN cana-2815 325 8 3 3 NUM cana-2815 326 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 326 2 π›Όβˆ… π›Όβˆ… X cana-2815 326 3 ] ] PUNCT cana-2815 326 4 𝐸[βˆ…(π‘₯ 𝐸[βˆ…(π‘₯ PROPN cana-2815 326 5 , , PUNCT cana-2815 326 6 y y PROPN cana-2815 326 7 , , PUNCT cana-2815 326 8 𝑑 𝑑 PROPN cana-2815 326 9 ) ) PUNCT cana-2815 326 10 ] ] PUNCT cana-2815 327 1 = = SYM cana-2815 327 2 𝑣2 𝑣2 NUM cana-2815 327 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 327 4 , , PUNCT cana-2815 327 5 𝑦 𝑦 NOUN cana-2815 327 6 , , PUNCT cana-2815 327 7 0 0 NUM cana-2815 327 8 ) ) PUNCT cana-2815 327 9 + + CCONJ cana-2815 327 10 𝑣3 𝑣3 ADJ cana-2815 327 11 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 327 12 , , PUNCT cana-2815 327 13 𝑦 𝑦 NOUN cana-2815 327 14 , , PUNCT cana-2815 327 15 0 0 NUM cana-2815 327 16 ) ) PUNCT cana-2815 327 17 βˆ’ βˆ’ PROPN cana-2815 328 1 𝑣 𝑣 DET cana-2815 328 2 𝛼 𝛼 X cana-2815 328 3 𝐸[(βˆ…π‘₯π‘₯ 𝐸[(βˆ…π‘₯π‘₯ NOUN cana-2815 328 4 + + CCONJ cana-2815 328 5 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ VERB cana-2815 328 6 βˆ’ βˆ’ PROPN cana-2815 328 7 2βˆ… 2βˆ… NUM cana-2815 328 8 ) ) PUNCT cana-2815 328 9 βˆ’ βˆ’ PROPN cana-2815 328 10 3 3 NUM cana-2815 328 11 𝐷𝑑 𝐷𝑑 PROPN cana-2815 328 12 π›Όβˆ… π›Όβˆ… X cana-2815 328 13 ] ] PUNCT cana-2815 328 14 applying apply VERB cana-2815 328 15 inverse inverse NOUN cana-2815 328 16 elzaki elzaki NOUN cana-2815 328 17 transform transform NOUN cana-2815 328 18 . . PUNCT cana-2815 329 1 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 330 1 [ [ X cana-2815 330 2 𝐸 𝐸 NOUN cana-2815 330 3 [ [ X cana-2815 330 4 βˆ…(π‘₯ βˆ…(π‘₯ INTJ cana-2815 330 5 , , PUNCT cana-2815 330 6 y y PROPN cana-2815 330 7 , , PUNCT cana-2815 330 8 𝑑 𝑑 PROPN cana-2815 330 9 ) ) PUNCT cana-2815 330 10 ] ] PUNCT cana-2815 330 11 ] ] X cana-2815 330 12 = = PUNCT cana-2815 330 13 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 331 1 [ [ X cana-2815 331 2 𝑣2 𝑣2 NOUN cana-2815 331 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 331 4 , , PUNCT cana-2815 331 5 𝑦 𝑦 NOUN cana-2815 331 6 , , PUNCT cana-2815 331 7 0 0 NUM cana-2815 331 8 ) ) PUNCT cana-2815 331 9 + + CCONJ cana-2815 331 10 𝑣3 𝑣3 ADJ cana-2815 331 11 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 331 12 , , PUNCT cana-2815 331 13 𝑦 𝑦 NOUN cana-2815 331 14 , , PUNCT cana-2815 331 15 0 0 NUM cana-2815 331 16 ) ) PUNCT cana-2815 331 17 βˆ’ βˆ’ PROPN cana-2815 332 1 𝑣 𝑣 DET cana-2815 332 2 𝛼 𝛼 X cana-2815 332 3 𝐸[(βˆ…π‘₯π‘₯ 𝐸[(βˆ…π‘₯π‘₯ NOUN cana-2815 332 4 + + CCONJ cana-2815 332 5 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ VERB cana-2815 332 6 βˆ’ βˆ’ PROPN cana-2815 332 7 2βˆ… 2βˆ… NUM cana-2815 332 8 ) ) PUNCT cana-2815 332 9 βˆ’ βˆ’ PROPN cana-2815 332 10 3 3 NUM cana-2815 332 11 𝐷𝑑 𝐷𝑑 PROPN cana-2815 332 12 π›Όβˆ… π›Όβˆ… NOUN cana-2815 332 13 ] ] PUNCT cana-2815 332 14 ] ] X cana-2815 332 15 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 332 16 , , PUNCT cana-2815 332 17 y y PROPN cana-2815 332 18 , , PUNCT cana-2815 332 19 𝑑 𝑑 PROPN cana-2815 332 20 ) ) PUNCT cana-2815 332 21 = = PUNCT cana-2815 332 22 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 333 1 [ [ X cana-2815 333 2 𝑣2 𝑣2 NOUN cana-2815 333 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 333 4 , , PUNCT cana-2815 333 5 𝑦 𝑦 NOUN cana-2815 333 6 , , PUNCT cana-2815 333 7 0 0 NUM cana-2815 333 8 ) ) PUNCT cana-2815 333 9 + + CCONJ cana-2815 333 10 𝑣3 𝑣3 ADJ cana-2815 333 11 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 333 12 , , PUNCT cana-2815 333 13 𝑦 𝑦 NOUN cana-2815 333 14 , , PUNCT cana-2815 333 15 0 0 NUM cana-2815 333 16 ) ) PUNCT cana-2815 333 17 βˆ’ βˆ’ PROPN cana-2815 334 1 𝑣 𝑣 DET cana-2815 334 2 𝛼 𝛼 X cana-2815 334 3 𝐸[(βˆ…π‘₯π‘₯ 𝐸[(βˆ…π‘₯π‘₯ NOUN cana-2815 334 4 + + CCONJ cana-2815 334 5 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ VERB cana-2815 334 6 βˆ’ βˆ’ PROPN cana-2815 334 7 2βˆ… 2βˆ… NUM cana-2815 334 8 ) ) PUNCT cana-2815 334 9 βˆ’ βˆ’ PROPN cana-2815 334 10 3 3 NUM cana-2815 334 11 𝐷𝑑 𝐷𝑑 PROPN cana-2815 334 12 π›Όβˆ… π›Όβˆ… NOUN cana-2815 334 13 ] ] PUNCT cana-2815 334 14 ] ] X cana-2815 334 15 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 334 16 , , PUNCT cana-2815 334 17 y y PROPN cana-2815 334 18 , , PUNCT cana-2815 334 19 𝑑 𝑑 PROPN cana-2815 334 20 ) ) PUNCT cana-2815 334 21 = = SYM cana-2815 334 22 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 X cana-2815 334 23 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 334 24 , , PUNCT cana-2815 334 25 𝑦 𝑦 NOUN cana-2815 334 26 , , PUNCT cana-2815 334 27 0 0 NUM cana-2815 334 28 ) ) PUNCT cana-2815 334 29 + + CCONJ cana-2815 334 30 𝑣3 𝑣3 ADJ cana-2815 334 31 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 334 32 , , PUNCT cana-2815 334 33 𝑦 𝑦 NOUN cana-2815 334 34 , , PUNCT cana-2815 334 35 0 0 NUM cana-2815 334 36 ) ) PUNCT cana-2815 334 37 βˆ’ βˆ’ NOUN cana-2815 334 38 𝑣 𝑣 ADP cana-2815 334 39 𝛼 𝛼 X cana-2815 334 40 𝐸[𝐿(βˆ… 𝐸[𝐿(βˆ… NOUN cana-2815 334 41 ) ) PUNCT cana-2815 334 42 βˆ’ βˆ’ PROPN cana-2815 335 1 3 3 NUM cana-2815 335 2 𝐷𝑑 𝐷𝑑 PROPN cana-2815 335 3 π›Όβˆ… π›Όβˆ… NOUN cana-2815 335 4 ] ] PUNCT cana-2815 335 5 ] ] PUNCT cana-2815 335 6 appling apple VERB cana-2815 335 7 the the DET cana-2815 335 8 madetm madetm NOUN cana-2815 335 9 process process NOUN cana-2815 335 10 on on ADP cana-2815 335 11 equation equation NOUN cana-2815 335 12 ( ( PUNCT cana-2815 335 13 36 36 NUM cana-2815 335 14 ) ) PUNCT cana-2815 335 15 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 335 16 , , PUNCT cana-2815 335 17 y y PROPN cana-2815 335 18 , , PUNCT cana-2815 335 19 𝑑 𝑑 PROPN cana-2815 335 20 ) ) PUNCT cana-2815 335 21 = = SYM cana-2815 335 22 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 ADP cana-2815 335 23 βˆ…(π‘₯ βˆ…(π‘₯ NUM cana-2815 335 24 , , PUNCT cana-2815 335 25 𝑦 𝑦 NOUN cana-2815 335 26 , , PUNCT cana-2815 335 27 0 0 NUM cana-2815 335 28 ) ) PUNCT cana-2815 336 1 + + CCONJ cana-2815 336 2 𝑣3 𝑣3 ADJ cana-2815 336 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 336 4 , , PUNCT cana-2815 336 5 𝑦 𝑦 NOUN cana-2815 336 6 , , PUNCT cana-2815 336 7 0 0 NUM cana-2815 336 8 ) ) PUNCT cana-2815 336 9 ] ] PUNCT cana-2815 336 10 βˆ…0 βˆ…0 NOUN cana-2815 336 11 ( ( PUNCT cana-2815 336 12 π‘₯ π‘₯ PROPN cana-2815 336 13 , , PUNCT cana-2815 336 14 y y PROPN cana-2815 336 15 , , PUNCT cana-2815 336 16 𝑑 𝑑 PROPN cana-2815 336 17 ) ) PUNCT cana-2815 336 18 = = SYM cana-2815 336 19 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 PART cana-2815 336 20 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 NOUN cana-2815 336 21 + + SYM cana-2815 336 22 𝑣3 𝑣3 ADJ cana-2815 336 23 ( ( PUNCT cana-2815 336 24 βˆ’3𝑒π‘₯+𝑦 βˆ’3𝑒π‘₯+𝑦 NOUN cana-2815 336 25 ) ) PUNCT cana-2815 336 26 ] ] PUNCT cana-2815 336 27 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 336 28 , , PUNCT cana-2815 336 29 y y PROPN cana-2815 336 30 , , PUNCT cana-2815 336 31 𝑑 𝑑 PROPN cana-2815 336 32 ) ) PUNCT cana-2815 336 33 = = SYM cana-2815 336 34 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 ADP cana-2815 336 35 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 NUM cana-2815 336 36 βˆ’ βˆ’ PROPN cana-2815 336 37 𝑣3 𝑣3 PROPN cana-2815 336 38 ( ( PUNCT cana-2815 336 39 3𝑒π‘₯+𝑦 3𝑒π‘₯+𝑦 NOUN cana-2815 336 40 ) ) PUNCT cana-2815 336 41 ] ] PUNCT cana-2815 336 42 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 336 43 , , PUNCT cana-2815 336 44 y y PROPN cana-2815 336 45 , , PUNCT cana-2815 336 46 𝑑 𝑑 PROPN cana-2815 336 47 ) ) PUNCT cana-2815 336 48 = = PUNCT cana-2815 336 49 βˆ…(0 βˆ…(0 PROPN cana-2815 336 50 ) ) PUNCT cana-2815 336 51 = = PUNCT cana-2815 336 52 𝑒 𝑒 PROPN cana-2815 336 53 π‘₯+𝑦(1βˆ’ π‘₯+𝑦(1βˆ’ NOUN cana-2815 336 54 3𝑑 3𝑑 NUM cana-2815 336 55 ) ) PUNCT cana-2815 336 56 βˆ… βˆ… NOUN cana-2815 336 57 𝑛+1(π‘₯ 𝑛+1(π‘₯ PROPN cana-2815 336 58 , , PUNCT cana-2815 336 59 𝑦 𝑦 NOUN cana-2815 336 60 , , PUNCT cana-2815 336 61 𝑑 𝑑 NOUN cana-2815 336 62 ) ) PUNCT cana-2815 336 63 = = PUNCT cana-2815 336 64 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 336 65 [ [ X cana-2815 336 66 𝐿 𝐿 PROPN cana-2815 336 67 ( ( PUNCT cana-2815 336 68 βˆ…π‘› βˆ…π‘› NOUN cana-2815 336 69 ) ) PUNCT cana-2815 336 70 βˆ’ βˆ’ PROPN cana-2815 337 1 3 3 NUM cana-2815 338 1 𝐷𝑑 𝐷𝑑 NOUN cana-2815 338 2 𝛼 𝛼 NOUN cana-2815 338 3 βˆ…π‘› βˆ…π‘› NOUN cana-2815 338 4 ] ] PUNCT cana-2815 338 5 ] ] PUNCT cana-2815 338 6 for for ADP cana-2815 338 7 𝑛 𝑛 PROPN cana-2815 338 8 = = SYM cana-2815 338 9 0 0 NUM cana-2815 338 10 βˆ…1 βˆ…1 NOUN cana-2815 338 11 ( ( PUNCT cana-2815 338 12 π‘₯ π‘₯ PROPN cana-2815 338 13 , , PUNCT cana-2815 338 14 𝑦 𝑦 NOUN cana-2815 338 15 , , PUNCT cana-2815 338 16 𝑑 𝑑 NOUN cana-2815 338 17 ) ) PUNCT cana-2815 338 18 = = SYM cana-2815 338 19 𝐸 𝐸 PROPN cana-2815 338 20 βˆ’1 βˆ’1 NOUN cana-2815 339 1 [ [ X cana-2815 339 2 𝑣𝛼𝐸 𝑣𝛼𝐸 PROPN cana-2815 339 3 [ [ X cana-2815 339 4 𝐿 𝐿 PROPN cana-2815 339 5 ( ( PUNCT cana-2815 339 6 βˆ…0 βˆ…0 NOUN cana-2815 339 7 ) ) PUNCT cana-2815 339 8 βˆ’ βˆ’ NOUN cana-2815 339 9 3 3 NUM cana-2815 340 1 𝐷𝑑 𝐷𝑑 NOUN cana-2815 340 2 𝛼 𝛼 NOUN cana-2815 340 3 βˆ…0 βˆ…0 NOUN cana-2815 340 4 ] ] PUNCT cana-2815 340 5 ] ] PUNCT cana-2815 340 6 here here ADV cana-2815 340 7 , , PUNCT cana-2815 340 8 𝐿 𝐿 PROPN cana-2815 340 9 [ [ X cana-2815 340 10 βˆ…0 βˆ…0 NOUN cana-2815 340 11 ] ] X cana-2815 340 12 = = SYM cana-2815 340 13 ( ( PUNCT cana-2815 340 14 βˆ…0π‘₯π‘₯ βˆ…0π‘₯π‘₯ NOUN cana-2815 340 15 + + CCONJ cana-2815 340 16 βˆ…0𝑦𝑦 βˆ…0𝑦𝑦 ADP cana-2815 340 17 βˆ’ βˆ’ NOUN cana-2815 340 18 2βˆ…0 2βˆ…0 NUM cana-2815 340 19 ) ) PUNCT cana-2815 340 20 = = SYM cana-2815 340 21 0 0 PUNCT cana-2815 340 22 therefore therefore ADV cana-2815 340 23 , , PUNCT cana-2815 340 24 above above ADP cana-2815 340 25 equation equation NOUN cana-2815 340 26 implies imply VERB cana-2815 340 27 , , PUNCT cana-2815 340 28 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 340 29 , , PUNCT cana-2815 340 30 𝑦 𝑦 NOUN cana-2815 340 31 , , PUNCT cana-2815 340 32 𝑑 𝑑 NOUN cana-2815 340 33 ) ) PUNCT cana-2815 340 34 = = PUNCT cana-2815 341 1 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 342 1 [ [ X cana-2815 342 2 𝑣𝛼 𝑣𝛼 NOUN cana-2815 342 3 𝐸 𝐸 PROPN cana-2815 343 1 [ [ X cana-2815 343 2 βˆ’3 βˆ’3 NOUN cana-2815 343 3 𝐷𝑑 𝐷𝑑 NOUN cana-2815 343 4 𝛼 𝛼 NOUN cana-2815 343 5 βˆ…0 βˆ…0 NOUN cana-2815 343 6 ] ] PUNCT cana-2815 343 7 ] ] PUNCT cana-2815 343 8 βˆ…1(π‘₯ βˆ…1(π‘₯ PROPN cana-2815 343 9 , , PUNCT cana-2815 343 10 𝑦 𝑦 NOUN cana-2815 343 11 , , PUNCT cana-2815 343 12 𝑑 𝑑 NOUN cana-2815 343 13 ) ) PUNCT cana-2815 343 14 = = SYM cana-2815 344 1 βˆ’πΈ βˆ’πΈ NOUN cana-2815 344 2 βˆ’1[𝑣𝛼𝐸 βˆ’1[𝑣𝛼𝐸 PROPN cana-2815 345 1 [ [ X cana-2815 345 2 βˆ’3 βˆ’3 X cana-2815 345 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 345 4 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 345 5 ] ] PUNCT cana-2815 345 6 ] ] PUNCT cana-2815 345 7 = = SYM cana-2815 345 8 9𝑒π‘₯+𝑦 9𝑒π‘₯+𝑦 NOUN cana-2815 345 9 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 345 10 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 345 11 ) ) PUNCT cana-2815 345 12 for for ADP cana-2815 345 13 𝑛 𝑛 NOUN cana-2815 345 14 = = SYM cana-2815 345 15 1,2,3 1,2,3 NUM cana-2815 345 16 … … NUM cana-2815 345 17 . . PUNCT cana-2815 346 1 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 346 2 , , PUNCT cana-2815 346 3 𝑦 𝑦 NOUN cana-2815 346 4 , , PUNCT cana-2815 346 5 𝑑 𝑑 NOUN cana-2815 346 6 ) ) PUNCT cana-2815 346 7 = = SYM cana-2815 347 1 βˆ’πΈ βˆ’πΈ NOUN cana-2815 347 2 βˆ’1[𝑣𝛼𝐸 βˆ’1[𝑣𝛼𝐸 PROPN cana-2815 348 1 [ [ X cana-2815 348 2 βˆ’3 βˆ’3 PROPN cana-2815 348 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 348 4 π›Όβˆ…1 π›Όβˆ…1 NOUN cana-2815 348 5 ] ] X cana-2815 348 6 ] ] X cana-2815 348 7 = = SYM cana-2815 348 8 βˆ’27𝑒π‘₯+𝑦 βˆ’27𝑒π‘₯+𝑦 X cana-2815 348 9 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 348 10 ⌈(2𝛼+2 ⌈(2𝛼+2 X cana-2815 348 11 ) ) PUNCT cana-2815 348 12 βˆ…3(π‘₯ βˆ…3(π‘₯ NOUN cana-2815 348 13 , , PUNCT cana-2815 348 14 𝑦 𝑦 NOUN cana-2815 348 15 , , PUNCT cana-2815 348 16 𝑑 𝑑 NOUN cana-2815 348 17 ) ) PUNCT cana-2815 349 1 = = SYM cana-2815 349 2 βˆ’πΈ βˆ’πΈ NOUN cana-2815 349 3 βˆ’1[𝑣𝛼𝐸 βˆ’1[𝑣𝛼𝐸 PROPN cana-2815 350 1 [ [ X cana-2815 350 2 βˆ’3 βˆ’3 PROPN cana-2815 350 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 350 4 π›Όβˆ…2 π›Όβˆ…2 PROPN cana-2815 350 5 ] ] X cana-2815 350 6 ] ] X cana-2815 351 1 = = SYM cana-2815 351 2 81 81 NUM cana-2815 351 3 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 NOUN cana-2815 351 4 𝑑3𝛼+1 𝑑3𝛼+1 PUNCT cana-2815 351 5 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 351 6 ) ) PUNCT cana-2815 351 7 βˆ…4(π‘₯ βˆ…4(π‘₯ NOUN cana-2815 351 8 , , PUNCT cana-2815 351 9 𝑦 𝑦 NOUN cana-2815 351 10 , , PUNCT cana-2815 351 11 𝑑 𝑑 NOUN cana-2815 351 12 ) ) PUNCT cana-2815 351 13 = = SYM cana-2815 351 14 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 352 1 [ [ X cana-2815 352 2 βˆ’3 βˆ’3 X cana-2815 352 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 352 4 π›Όβˆ…3 π›Όβˆ…3 PROPN cana-2815 352 5 ] ] X cana-2815 352 6 ] ] X cana-2815 352 7 = = PUNCT cana-2815 352 8 βˆ’243 βˆ’243 PROPN cana-2815 352 9 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 NOUN cana-2815 352 10 𝑑4𝛼+1 𝑑4𝛼+1 ADV cana-2815 352 11 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 352 12 ) ) PUNCT cana-2815 352 13 . . PUNCT cana-2815 353 1 : : PUNCT cana-2815 353 2 : : PUNCT cana-2815 353 3 therefore therefore ADV cana-2815 353 4 , , PUNCT cana-2815 353 5 series series NOUN cana-2815 353 6 form form NOUN cana-2815 353 7 βˆ… βˆ… NOUN cana-2815 353 8 ( ( PUNCT cana-2815 353 9 π‘₯ π‘₯ INTJ cana-2815 353 10 , , PUNCT cana-2815 353 11 𝑦 𝑦 NOUN cana-2815 353 12 , , PUNCT cana-2815 353 13 𝑑 𝑑 PROPN cana-2815 353 14 ) ) PUNCT cana-2815 353 15 will will AUX cana-2815 353 16 be be AUX cana-2815 353 17 : : PUNCT cana-2815 353 18 βˆ… βˆ… NOUN cana-2815 353 19 ( ( PUNCT cana-2815 353 20 π‘₯ π‘₯ NOUN cana-2815 353 21 , , PUNCT cana-2815 353 22 𝑦 𝑦 PRON cana-2815 353 23 , , PUNCT cana-2815 353 24 𝑑 𝑑 PROPN cana-2815 353 25 ) ) PUNCT cana-2815 353 26 = = SYM cana-2815 353 27 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 353 28 , , PUNCT cana-2815 353 29 𝑦 𝑦 NOUN cana-2815 353 30 , , PUNCT cana-2815 353 31 𝑑)+βˆ…1(π‘₯ 𝑑)+βˆ…1(π‘₯ ADJ cana-2815 353 32 , , PUNCT cana-2815 353 33 𝑦 𝑦 NOUN cana-2815 353 34 , , PUNCT cana-2815 353 35 𝑑)+βˆ…2(π‘₯ 𝑑)+βˆ…2(π‘₯ X cana-2815 353 36 , , PUNCT cana-2815 353 37 𝑦 𝑦 NOUN cana-2815 353 38 , , PUNCT cana-2815 353 39 𝑑 𝑑 NOUN cana-2815 353 40 ) ) PUNCT cana-2815 353 41 + + CCONJ cana-2815 353 42 βˆ…3(π‘₯ βˆ…3(π‘₯ NUM cana-2815 353 43 , , PUNCT cana-2815 353 44 𝑦 𝑦 NOUN cana-2815 353 45 , , PUNCT cana-2815 353 46 𝑑 𝑑 NOUN cana-2815 353 47 ) ) PUNCT cana-2815 353 48 + + CCONJ cana-2815 353 49 βˆ…4(π‘₯ βˆ…4(π‘₯ NOUN cana-2815 353 50 , , PUNCT cana-2815 353 51 𝑦 𝑦 NOUN cana-2815 353 52 , , PUNCT cana-2815 353 53 𝑑 𝑑 NOUN cana-2815 353 54 ) ) PUNCT cana-2815 354 1 + + X cana-2815 354 2 β‹― β‹― NOUN cana-2815 354 3 …… …… NOUN cana-2815 354 4 ( ( PUNCT cana-2815 354 5 39 39 NUM cana-2815 354 6 ) ) PUNCT cana-2815 354 7 ( ( PUNCT cana-2815 354 8 36 36 NUM cana-2815 354 9 ) ) PUNCT cana-2815 354 10 ( ( PUNCT cana-2815 354 11 38 38 NUM cana-2815 354 12 ) ) PUNCT cana-2815 354 13 ( ( PUNCT cana-2815 354 14 37 37 NUM cana-2815 354 15 ) ) PUNCT cana-2815 354 16 communications communication NOUN cana-2815 354 17 on on ADP cana-2815 354 18 applied apply VERB cana-2815 354 19 nonlinear nonlinear ADJ cana-2815 354 20 analysis analysis NOUN cana-2815 354 21 issn issn NOUN cana-2815 354 22 : : PUNCT cana-2815 354 23 1074 1074 NUM cana-2815 354 24 - - PUNCT cana-2815 354 25 133x 133x NUM cana-2815 354 26 vol vol NOUN cana-2815 354 27 32 32 NUM cana-2815 354 28 no no NOUN cana-2815 354 29 . . PUNCT cana-2815 355 1 4s 4s NUM cana-2815 355 2 ( ( PUNCT cana-2815 355 3 2025 2025 NUM cana-2815 355 4 ) ) PUNCT cana-2815 355 5 321 321 NUM cana-2815 355 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 355 7 βˆ… βˆ… NOUN cana-2815 355 8 ( ( PUNCT cana-2815 355 9 π‘₯ π‘₯ NOUN cana-2815 355 10 , , PUNCT cana-2815 355 11 𝑦 𝑦 NOUN cana-2815 355 12 , , PUNCT cana-2815 355 13 𝑑 𝑑 NOUN cana-2815 355 14 ) ) PUNCT cana-2815 355 15 = = PUNCT cana-2815 355 16 𝑒π‘₯+𝑦(1βˆ’ 𝑒π‘₯+𝑦(1βˆ’ PROPN cana-2815 355 17 3𝑑 3𝑑 NUM cana-2815 355 18 ) ) PUNCT cana-2815 356 1 + + SYM cana-2815 356 2 9𝑒π‘₯+𝑦 9𝑒π‘₯+𝑦 NOUN cana-2815 356 3 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 356 4 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 356 5 ) ) PUNCT cana-2815 356 6 βˆ’ βˆ’ PROPN cana-2815 356 7 27𝑒π‘₯+𝑦 27𝑒π‘₯+𝑦 NUM cana-2815 356 8 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 356 9 ⌈(2𝛼+2 ⌈(2𝛼+2 X cana-2815 356 10 ) ) PUNCT cana-2815 357 1 + + CCONJ cana-2815 357 2 81 81 NUM cana-2815 357 3 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 NOUN cana-2815 357 4 𝑑3𝛼+1 𝑑3𝛼+1 PUNCT cana-2815 357 5 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 357 6 ) ) PUNCT cana-2815 358 1 βˆ’ βˆ’ PROPN cana-2815 358 2 243 243 NUM cana-2815 358 3 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 NOUN cana-2815 358 4 𝑑4𝛼+1 𝑑4𝛼+1 ADV cana-2815 358 5 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 358 6 ) ) PUNCT cana-2815 358 7 … … PUNCT cana-2815 358 8 … … PUNCT cana-2815 358 9 … … SYM cana-2815 358 10 . . PUNCT cana-2815 359 1 βˆ… βˆ… NOUN cana-2815 359 2 ( ( PUNCT cana-2815 359 3 π‘₯ π‘₯ NOUN cana-2815 359 4 , , PUNCT cana-2815 359 5 𝑦 𝑦 NOUN cana-2815 359 6 , , PUNCT cana-2815 359 7 𝑑 𝑑 NOUN cana-2815 359 8 ) ) PUNCT cana-2815 359 9 = = SYM cana-2815 359 10 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 PUNCT cana-2815 360 1 [ [ X cana-2815 360 2 ( ( PUNCT cana-2815 360 3 1βˆ’ 1βˆ’ NUM cana-2815 360 4 3𝑑 3𝑑 NUM cana-2815 360 5 ) ) PUNCT cana-2815 361 1 + + CCONJ cana-2815 361 2 9 9 NUM cana-2815 361 3 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 361 4 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 361 5 ) ) PUNCT cana-2815 361 6 βˆ’ βˆ’ PROPN cana-2815 361 7 27 27 NUM cana-2815 361 8 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 361 9 ⌈(2𝛼+2 ⌈(2𝛼+2 PROPN cana-2815 361 10 ) ) PUNCT cana-2815 362 1 + + CCONJ cana-2815 362 2 81 81 NUM cana-2815 362 3 𝑑3𝛼+1 𝑑3𝛼+1 ADJ cana-2815 362 4 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 362 5 ) ) PUNCT cana-2815 363 1 βˆ’ βˆ’ PROPN cana-2815 363 2 243 243 NUM cana-2815 363 3 𝑑4𝛼+1 𝑑4𝛼+1 ADV cana-2815 363 4 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 363 5 ) ) PUNCT cana-2815 363 6 … … PUNCT cana-2815 363 7 …… …… X cana-2815 363 8 . . PUNCT cana-2815 364 1 ] ] X cana-2815 364 2 when when SCONJ cana-2815 364 3 𝛼 𝛼 X cana-2815 364 4 = = SYM cana-2815 364 5 1 1 NUM cana-2815 364 6 , , PUNCT cana-2815 364 7 the the DET cana-2815 364 8 approximate approximate ADJ cana-2815 364 9 solution solution NOUN cana-2815 364 10 will will AUX cana-2815 364 11 be be AUX cana-2815 364 12 in in ADP cana-2815 364 13 the the DET cana-2815 364 14 form form NOUN cana-2815 364 15 : : PUNCT cana-2815 364 16 βˆ… βˆ… NOUN cana-2815 364 17 ( ( PUNCT cana-2815 364 18 π‘₯ π‘₯ NOUN cana-2815 364 19 , , PUNCT cana-2815 364 20 𝑦 𝑦 NOUN cana-2815 364 21 , , PUNCT cana-2815 364 22 𝑑 𝑑 NOUN cana-2815 364 23 ) ) PUNCT cana-2815 364 24 = = SYM cana-2815 364 25 𝑒π‘₯+𝑦 𝑒π‘₯+𝑦 PUNCT cana-2815 365 1 [ [ X cana-2815 365 2 1βˆ’ 1βˆ’ NUM cana-2815 365 3 3𝑑 3𝑑 NUM cana-2815 365 4 1 1 NUM cana-2815 365 5 ! ! PUNCT cana-2815 366 1 + + CCONJ cana-2815 366 2 ( ( PUNCT cana-2815 366 3 3𝑑 3𝑑 NUM cana-2815 366 4 ) ) PUNCT cana-2815 366 5 2 2 NUM cana-2815 366 6 2 2 NUM cana-2815 366 7 ! ! PUNCT cana-2815 367 1 βˆ’ βˆ’ PROPN cana-2815 368 1 ( ( PUNCT cana-2815 368 2 3𝑑 3𝑑 NUM cana-2815 368 3 ) ) PUNCT cana-2815 368 4 3 3 NUM cana-2815 368 5 3 3 NUM cana-2815 368 6 ! ! PUNCT cana-2815 369 1 + + CCONJ cana-2815 369 2 ( ( PUNCT cana-2815 369 3 3𝑑)4 3𝑑)4 NUM cana-2815 369 4 4 4 NUM cana-2815 369 5 ! ! PUNCT cana-2815 369 6 βˆ’ βˆ’ PROPN cana-2815 370 1 ( ( PUNCT cana-2815 370 2 3𝑑 3𝑑 NUM cana-2815 370 3 ) ) PUNCT cana-2815 370 4 5 5 NUM cana-2815 370 5 5 5 NUM cana-2815 370 6 ! ! PUNCT cana-2815 371 1 + + VERB cana-2815 371 2 β‹― β‹― PROPN cana-2815 371 3 . . PUNCT cana-2815 371 4 . . PUNCT cana-2815 372 1 ] ] PUNCT cana-2815 373 1 the the DET cana-2815 373 2 exact exact ADJ cana-2815 373 3 solution solution NOUN cana-2815 373 4 for for ADP cana-2815 373 5 equation equation NOUN cana-2815 373 6 ( ( PUNCT cana-2815 373 7 34 34 NUM cana-2815 373 8 ) ) PUNCT cana-2815 373 9 implies imply VERB cana-2815 373 10 βˆ… βˆ… NOUN cana-2815 373 11 ( ( PUNCT cana-2815 373 12 π‘₯ π‘₯ NOUN cana-2815 373 13 , , PUNCT cana-2815 373 14 𝑦 𝑦 NOUN cana-2815 373 15 , , PUNCT cana-2815 373 16 𝑑 𝑑 NOUN cana-2815 373 17 ) ) PUNCT cana-2815 373 18 = = SYM cana-2815 373 19 𝑒π‘₯+π‘¦βˆ’3𝑑 𝑒π‘₯+π‘¦βˆ’3𝑑 PROPN cana-2815 373 20 ( ( PUNCT cana-2815 373 21 41 41 NUM cana-2815 373 22 ) ) PUNCT cana-2815 373 23 three three NUM cana-2815 373 24 - - PUNCT cana-2815 373 25 dimensional dimensional ADJ cana-2815 373 26 fractional fractional ADJ cana-2815 373 27 telegraph telegraph NOUN cana-2815 373 28 equation equation NOUN cana-2815 373 29 : : PUNCT cana-2815 373 30 example example NOUN cana-2815 373 31 5 5 NUM cana-2815 373 32 . . PUNCT cana-2815 374 1 the the DET cana-2815 374 2 3d 3d PROPN cana-2815 374 3 telegraph telegraph NOUN cana-2815 374 4 equation equation NOUN cana-2815 374 5 of of ADP cana-2815 374 6 fractional fractional ADJ cana-2815 374 7 order order NOUN cana-2815 374 8 is be AUX cana-2815 374 9 to to PART cana-2815 374 10 be be AUX cana-2815 374 11 considered consider VERB cana-2815 374 12 [ [ X cana-2815 374 13 23 23 NUM cana-2815 374 14 ] ] X cana-2815 375 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 375 2 2π›Όβˆ… 2π›Όβˆ… NUM cana-2815 375 3 + + CCONJ cana-2815 375 4 2 2 NUM cana-2815 375 5 𝐷𝑑 𝐷𝑑 PROPN cana-2815 375 6 π›Όβˆ… π›Όβˆ… X cana-2815 375 7 + + NOUN cana-2815 375 8 3βˆ… 3βˆ… NUM cana-2815 375 9 = = SYM cana-2815 375 10 βˆ…π‘₯π‘₯ βˆ…π‘₯π‘₯ NOUN cana-2815 375 11 + + CCONJ cana-2815 375 12 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ NOUN cana-2815 375 13 + + X cana-2815 375 14 βˆ…π‘§π‘§ βˆ…π‘§π‘§ X cana-2815 375 15 0 0 PUNCT cana-2815 375 16 < < X cana-2815 375 17 ∝ ∝ PROPN cana-2815 375 18 ≀ ≀ ADV cana-2815 375 19 1 1 NUM cana-2815 375 20 with with ADP cana-2815 375 21 initial initial ADJ cana-2815 375 22 conditions condition NOUN cana-2815 375 23 : : PUNCT cana-2815 375 24 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-2815 375 25 , , PUNCT cana-2815 375 26 𝑦 𝑦 NOUN cana-2815 375 27 , , PUNCT cana-2815 375 28 𝑧 𝑧 PROPN cana-2815 375 29 , , PUNCT cana-2815 375 30 0 0 NUM cana-2815 375 31 ) ) PUNCT cana-2815 375 32 = = VERB cana-2815 376 1 sinh sinh NOUN cana-2815 376 2 π‘₯ π‘₯ DET cana-2815 376 3 sinh𝑦 sinh𝑦 ADJ cana-2815 376 4 sinh sinh VERB cana-2815 376 5 𝑧 𝑧 PROPN cana-2815 376 6 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ PROPN cana-2815 376 7 , , PUNCT cana-2815 376 8 𝑦 𝑦 NOUN cana-2815 376 9 , , PUNCT cana-2815 376 10 𝑧 𝑧 PROPN cana-2815 376 11 , , PUNCT cana-2815 376 12 0 0 NUM cana-2815 376 13 ) ) PUNCT cana-2815 376 14 = = SYM cana-2815 376 15 βˆ’ βˆ’ PROPN cana-2815 376 16 sinh sinh NOUN cana-2815 376 17 π‘₯ π‘₯ PROPN cana-2815 376 18 sinh sinh PROPN cana-2815 376 19 𝑦 𝑦 PRON cana-2815 376 20 sinh sinh NOUN cana-2815 376 21 𝑧 𝑧 PROPN cana-2815 376 22 apply apply VERB cana-2815 376 23 the the DET cana-2815 376 24 elzaki elzaki NOUN cana-2815 376 25 transformation transformation NOUN cana-2815 376 26 of of ADP cana-2815 376 27 equation equation NOUN cana-2815 376 28 ( ( PUNCT cana-2815 376 29 42 42 NUM cana-2815 376 30 ) ) PUNCT cana-2815 376 31 , , PUNCT cana-2815 376 32 𝐸 𝐸 PROPN cana-2815 376 33 [ [ PUNCT cana-2815 376 34 𝐷𝑑 𝐷𝑑 PROPN cana-2815 376 35 2π›Όβˆ… 2π›Όβˆ… PROPN cana-2815 376 36 + + CCONJ cana-2815 376 37 2 2 NUM cana-2815 376 38 𝐷𝑑 𝐷𝑑 PROPN cana-2815 376 39 π›Όβˆ… π›Όβˆ… PUNCT cana-2815 376 40 + + NOUN cana-2815 377 1 3βˆ… 3βˆ… NUM cana-2815 377 2 ] ] PUNCT cana-2815 377 3 = = PUNCT cana-2815 378 1 𝐸[βˆ… 𝐸[βˆ… NOUN cana-2815 378 2 π‘₯π‘₯ π‘₯π‘₯ ADP cana-2815 378 3 + + NUM cana-2815 378 4 βˆ… βˆ… NOUN cana-2815 378 5 𝑦𝑦 𝑦𝑦 ADP cana-2815 378 6 + + ADJ cana-2815 378 7 βˆ… βˆ… ADP cana-2815 378 8 𝑧𝑧 𝑧𝑧 NOUN cana-2815 378 9 ] ] X cana-2815 378 10 𝐸 𝐸 PROPN cana-2815 378 11 [ [ PUNCT cana-2815 378 12 𝐷𝑑 𝐷𝑑 PROPN cana-2815 378 13 2π›Όβˆ… 2π›Όβˆ… NUM cana-2815 378 14 ] ] X cana-2815 378 15 = = PUNCT cana-2815 379 1 βˆ’πΈ[βˆ…π‘₯π‘₯ βˆ’πΈ[βˆ…π‘₯π‘₯ NOUN cana-2815 379 2 + + CCONJ cana-2815 379 3 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ X cana-2815 379 4 + + X cana-2815 379 5 βˆ…π‘§π‘§ βˆ…π‘§π‘§ ADV cana-2815 379 6 βˆ’ βˆ’ PROPN cana-2815 379 7 2 2 NUM cana-2815 379 8 𝐷𝑑 𝐷𝑑 PROPN cana-2815 379 9 π›Όβˆ… π›Όβˆ… PUNCT cana-2815 379 10 βˆ’ βˆ’ NOUN cana-2815 380 1 3βˆ… 3βˆ… NUM cana-2815 380 2 ] ] SYM cana-2815 380 3 1 1 NUM cana-2815 380 4 𝑣𝛼 𝑣𝛼 NOUN cana-2815 380 5 𝐸[βˆ…(π‘₯ 𝐸[βˆ…(π‘₯ PROPN cana-2815 380 6 , , PUNCT cana-2815 380 7 y y PROPN cana-2815 380 8 , , PUNCT cana-2815 380 9 z z PROPN cana-2815 380 10 , , PUNCT cana-2815 380 11 𝑑 𝑑 PROPN cana-2815 380 12 ) ) PUNCT cana-2815 380 13 ] ] PUNCT cana-2815 381 1 βˆ’ βˆ’ PROPN cana-2815 381 2 𝑣2βˆ’π›Ό 𝑣2βˆ’π›Ό PROPN cana-2815 381 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 381 4 , , PUNCT cana-2815 381 5 𝑦 𝑦 NOUN cana-2815 381 6 , , PUNCT cana-2815 381 7 𝑧 𝑧 PROPN cana-2815 381 8 , , PUNCT cana-2815 381 9 0 0 NUM cana-2815 381 10 ) ) PUNCT cana-2815 381 11 βˆ’ βˆ’ PROPN cana-2815 381 12 𝑣3βˆ’π›Ό 𝑣3βˆ’π›Ό PROPN cana-2815 381 13 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ PROPN cana-2815 381 14 , , PUNCT cana-2815 381 15 𝑦 𝑦 NOUN cana-2815 381 16 , , PUNCT cana-2815 381 17 𝑧 𝑧 PROPN cana-2815 381 18 , , PUNCT cana-2815 381 19 0 0 NUM cana-2815 381 20 ) ) PUNCT cana-2815 381 21 = = VERB cana-2815 382 1 βˆ’πΈ[(βˆ…π‘₯π‘₯ βˆ’πΈ[(βˆ…π‘₯π‘₯ NOUN cana-2815 382 2 + + CCONJ cana-2815 382 3 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ NOUN cana-2815 382 4 + + X cana-2815 382 5 βˆ…π‘§π‘§ βˆ…π‘§π‘§ ADV cana-2815 382 6 βˆ’ βˆ’ PROPN cana-2815 382 7 3βˆ… 3βˆ… NUM cana-2815 382 8 ) ) PUNCT cana-2815 382 9 βˆ’ βˆ’ PROPN cana-2815 382 10 2 2 NUM cana-2815 383 1 𝐷𝑑 𝐷𝑑 PROPN cana-2815 383 2 π›Όβˆ… π›Όβˆ… X cana-2815 383 3 ] ] PUNCT cana-2815 383 4 𝐸[βˆ…(π‘₯ 𝐸[βˆ…(π‘₯ PROPN cana-2815 383 5 , , PUNCT cana-2815 383 6 y y PROPN cana-2815 383 7 , , PUNCT cana-2815 383 8 z z PROPN cana-2815 383 9 , , PUNCT cana-2815 383 10 𝑑 𝑑 NOUN cana-2815 383 11 ) ) PUNCT cana-2815 383 12 ] ] PUNCT cana-2815 384 1 = = SYM cana-2815 384 2 𝑣2 𝑣2 NUM cana-2815 384 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 384 4 , , PUNCT cana-2815 384 5 𝑦 𝑦 PRON cana-2815 384 6 , , PUNCT cana-2815 384 7 𝑧 𝑧 PROPN cana-2815 384 8 , , PUNCT cana-2815 384 9 0 0 NUM cana-2815 384 10 ) ) PUNCT cana-2815 385 1 + + CCONJ cana-2815 385 2 𝑣3 𝑣3 ADJ cana-2815 385 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 385 4 , , PUNCT cana-2815 385 5 𝑦 𝑦 NOUN cana-2815 385 6 , , PUNCT cana-2815 385 7 𝑧 𝑧 PROPN cana-2815 385 8 , , PUNCT cana-2815 385 9 0 0 NUM cana-2815 385 10 ) ) PUNCT cana-2815 385 11 βˆ’ βˆ’ PROPN cana-2815 386 1 𝑣 𝑣 DET cana-2815 386 2 𝛼 𝛼 X cana-2815 386 3 𝐸[(βˆ…π‘₯π‘₯ 𝐸[(βˆ…π‘₯π‘₯ NOUN cana-2815 386 4 + + CCONJ cana-2815 386 5 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ X cana-2815 386 6 + + X cana-2815 386 7 βˆ…π‘§π‘§ βˆ…π‘§π‘§ ADV cana-2815 386 8 βˆ’ βˆ’ PROPN cana-2815 386 9 3βˆ… 3βˆ… NUM cana-2815 386 10 ) ) PUNCT cana-2815 386 11 βˆ’ βˆ’ PROPN cana-2815 386 12 2 2 NUM cana-2815 386 13 𝐷𝑑 𝐷𝑑 PROPN cana-2815 386 14 π›Όβˆ… π›Όβˆ… X cana-2815 386 15 ] ] PUNCT cana-2815 386 16 applying apply VERB cana-2815 386 17 inverse inverse NOUN cana-2815 386 18 elzaki elzaki NOUN cana-2815 386 19 transform transform VERB cana-2815 386 20 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 387 1 [ [ X cana-2815 387 2 𝐸 𝐸 PROPN cana-2815 387 3 [ [ X cana-2815 387 4 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 387 5 , , PUNCT cana-2815 387 6 y y PROPN cana-2815 387 7 , , PUNCT cana-2815 387 8 z z PROPN cana-2815 387 9 , , PUNCT cana-2815 387 10 𝑑 𝑑 PROPN cana-2815 387 11 ) ) PUNCT cana-2815 387 12 ] ] PUNCT cana-2815 387 13 ] ] X cana-2815 387 14 = = SYM cana-2815 387 15 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 X cana-2815 387 16 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 387 17 , , PUNCT cana-2815 387 18 𝑦 𝑦 NOUN cana-2815 387 19 , , PUNCT cana-2815 387 20 𝑧 𝑧 PROPN cana-2815 387 21 , , PUNCT cana-2815 387 22 0 0 NUM cana-2815 387 23 ) ) PUNCT cana-2815 388 1 + + CCONJ cana-2815 388 2 𝑣3 𝑣3 ADJ cana-2815 388 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 388 4 , , PUNCT cana-2815 388 5 𝑦 𝑦 NOUN cana-2815 388 6 , , PUNCT cana-2815 388 7 𝑧 𝑧 PROPN cana-2815 388 8 , , PUNCT cana-2815 388 9 0 0 NUM cana-2815 388 10 ) ) PUNCT cana-2815 388 11 βˆ’ βˆ’ PROPN cana-2815 389 1 𝑣 𝑣 ADP cana-2815 389 2 𝛼 𝛼 NOUN cana-2815 389 3 𝐸(βˆ…π‘₯π‘₯ 𝐸(βˆ…π‘₯π‘₯ NOUN cana-2815 389 4 + + CCONJ cana-2815 389 5 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ X cana-2815 389 6 + + X cana-2815 389 7 βˆ…π‘§π‘§ βˆ…π‘§π‘§ ADV cana-2815 389 8 βˆ’ βˆ’ PROPN cana-2815 389 9 3βˆ… 3βˆ… NUM cana-2815 389 10 ) ) PUNCT cana-2815 389 11 βˆ’ βˆ’ PROPN cana-2815 389 12 2 2 NUM cana-2815 389 13 𝐷𝑑 𝐷𝑑 PROPN cana-2815 389 14 π›Όβˆ… π›Όβˆ… X cana-2815 389 15 ] ] X cana-2815 389 16 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 389 17 , , PUNCT cana-2815 389 18 y y PROPN cana-2815 389 19 , , PUNCT cana-2815 389 20 z z PROPN cana-2815 389 21 , , PUNCT cana-2815 389 22 𝑑 𝑑 NOUN cana-2815 389 23 ) ) PUNCT cana-2815 389 24 = = SYM cana-2815 389 25 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 X cana-2815 389 26 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 389 27 , , PUNCT cana-2815 389 28 𝑦 𝑦 NOUN cana-2815 389 29 , , PUNCT cana-2815 389 30 𝑧 𝑧 PROPN cana-2815 389 31 , , PUNCT cana-2815 389 32 0 0 NUM cana-2815 389 33 ) ) PUNCT cana-2815 390 1 + + CCONJ cana-2815 390 2 𝑣3 𝑣3 ADJ cana-2815 390 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 390 4 , , PUNCT cana-2815 390 5 𝑦 𝑦 NOUN cana-2815 390 6 , , PUNCT cana-2815 390 7 𝑧 𝑧 PROPN cana-2815 390 8 , , PUNCT cana-2815 390 9 0 0 NUM cana-2815 390 10 ) ) PUNCT cana-2815 390 11 βˆ’ βˆ’ PROPN cana-2815 391 1 𝑣 𝑣 ADP cana-2815 391 2 𝛼 𝛼 NOUN cana-2815 391 3 𝐸(βˆ…π‘₯π‘₯ 𝐸(βˆ…π‘₯π‘₯ NOUN cana-2815 391 4 + + CCONJ cana-2815 391 5 βˆ…π‘¦π‘¦ βˆ…π‘¦π‘¦ X cana-2815 391 6 + + X cana-2815 391 7 βˆ…π‘§π‘§ βˆ…π‘§π‘§ ADV cana-2815 391 8 βˆ’ βˆ’ PROPN cana-2815 391 9 3βˆ… 3βˆ… NUM cana-2815 391 10 ) ) PUNCT cana-2815 391 11 βˆ’ βˆ’ PROPN cana-2815 391 12 2 2 NUM cana-2815 391 13 𝐷𝑑 𝐷𝑑 PROPN cana-2815 391 14 π›Όβˆ… π›Όβˆ… X cana-2815 391 15 ] ] X cana-2815 391 16 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 391 17 , , PUNCT cana-2815 391 18 y y PROPN cana-2815 391 19 , , PUNCT cana-2815 391 20 z z PROPN cana-2815 391 21 , , PUNCT cana-2815 391 22 𝑑 𝑑 PROPN cana-2815 391 23 ) ) PUNCT cana-2815 391 24 = = PUNCT cana-2815 391 25 πΈβˆ’1 πΈβˆ’1 VERB cana-2815 392 1 [ [ X cana-2815 392 2 𝑣2 𝑣2 NOUN cana-2815 392 3 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 392 4 , , PUNCT cana-2815 392 5 𝑦 𝑦 NOUN cana-2815 392 6 , , PUNCT cana-2815 392 7 𝑧 𝑧 PROPN cana-2815 392 8 , , PUNCT cana-2815 392 9 0 0 NUM cana-2815 392 10 ) ) PUNCT cana-2815 392 11 + + CCONJ cana-2815 392 12 𝑣3 𝑣3 ADJ cana-2815 392 13 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 392 14 , , PUNCT cana-2815 392 15 𝑦 𝑦 NOUN cana-2815 392 16 , , PUNCT cana-2815 392 17 𝑧 𝑧 PROPN cana-2815 392 18 , , PUNCT cana-2815 392 19 0 0 NUM cana-2815 392 20 ) ) PUNCT cana-2815 392 21 βˆ’ βˆ’ NOUN cana-2815 393 1 𝑣 𝑣 ADP cana-2815 393 2 𝛼 𝛼 X cana-2815 393 3 𝐸[𝐿(βˆ… 𝐸[𝐿(βˆ… NOUN cana-2815 393 4 ) ) PUNCT cana-2815 393 5 βˆ’ βˆ’ PROPN cana-2815 393 6 2 2 NUM cana-2815 393 7 𝐷𝑑 𝐷𝑑 PROPN cana-2815 393 8 π›Όβˆ… π›Όβˆ… NOUN cana-2815 393 9 ] ] PUNCT cana-2815 393 10 ] ] PUNCT cana-2815 393 11 appling apple VERB cana-2815 393 12 the the DET cana-2815 393 13 madetm madetm NOUN cana-2815 393 14 process process NOUN cana-2815 393 15 on on ADP cana-2815 393 16 equation equation NOUN cana-2815 393 17 ( ( PUNCT cana-2815 393 18 44 44 NUM cana-2815 393 19 ) ) PUNCT cana-2815 393 20 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 393 21 , , PUNCT cana-2815 393 22 y y PROPN cana-2815 393 23 , , PUNCT cana-2815 393 24 z z PROPN cana-2815 393 25 , , PUNCT cana-2815 393 26 𝑑 𝑑 NOUN cana-2815 393 27 ) ) PUNCT cana-2815 393 28 = = SYM cana-2815 393 29 πΈβˆ’1[𝑣2 πΈβˆ’1[𝑣2 X cana-2815 393 30 βˆ…(π‘₯ βˆ…(π‘₯ PROPN cana-2815 393 31 , , PUNCT cana-2815 393 32 𝑦 𝑦 NOUN cana-2815 393 33 , , PUNCT cana-2815 393 34 𝑧 𝑧 PROPN cana-2815 393 35 , , PUNCT cana-2815 393 36 0 0 NUM cana-2815 393 37 ) ) PUNCT cana-2815 394 1 + + CCONJ cana-2815 394 2 𝑣3 𝑣3 ADJ cana-2815 394 3 βˆ…π‘‘(π‘₯ βˆ…π‘‘(π‘₯ NOUN cana-2815 394 4 , , PUNCT cana-2815 394 5 𝑦 𝑦 NOUN cana-2815 394 6 , , PUNCT cana-2815 394 7 𝑧 𝑧 PROPN cana-2815 394 8 , , PUNCT cana-2815 394 9 0 0 NUM cana-2815 394 10 ) ) PUNCT cana-2815 394 11 ] ] PUNCT cana-2815 394 12 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 394 13 , , PUNCT cana-2815 394 14 𝑦 𝑦 NOUN cana-2815 394 15 , , PUNCT cana-2815 394 16 𝑧 𝑧 NOUN cana-2815 394 17 , , PUNCT cana-2815 394 18 𝑑 𝑑 NOUN cana-2815 394 19 ) ) PUNCT cana-2815 394 20 = = SYM cana-2815 394 21 𝐸 𝐸 PROPN cana-2815 394 22 βˆ’1[𝑣2 βˆ’1[𝑣2 NOUN cana-2815 394 23 sinh sinh NOUN cana-2815 394 24 π‘₯ π‘₯ DET cana-2815 394 25 sinh𝑦 sinh𝑦 PROPN cana-2815 394 26 sinh sinh VERB cana-2815 394 27 𝑧 𝑧 PROPN cana-2815 395 1 + + X cana-2815 395 2 𝑣3 𝑣3 ADJ cana-2815 395 3 ( ( PUNCT cana-2815 395 4 βˆ’ βˆ’ PROPN cana-2815 395 5 sinh sinh NOUN cana-2815 395 6 π‘₯ π‘₯ DET cana-2815 395 7 sinh𝑦 sinh𝑦 NOUN cana-2815 395 8 sinh sinh VERB cana-2815 395 9 𝑧 𝑧 PROPN cana-2815 395 10 ) ) PUNCT cana-2815 395 11 ] ] PUNCT cana-2815 395 12 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 395 13 , , PUNCT cana-2815 395 14 𝑦 𝑦 NOUN cana-2815 395 15 , , PUNCT cana-2815 395 16 𝑧 𝑧 NOUN cana-2815 395 17 , , PUNCT cana-2815 395 18 𝑑 𝑑 NOUN cana-2815 395 19 ) ) PUNCT cana-2815 395 20 = = SYM cana-2815 395 21 𝐸 𝐸 PROPN cana-2815 395 22 βˆ’1[𝑣2 βˆ’1[𝑣2 NOUN cana-2815 395 23 sinh sinh NOUN cana-2815 395 24 π‘₯ π‘₯ DET cana-2815 395 25 sinh𝑦 sinh𝑦 PROPN cana-2815 395 26 sinh sinh VERB cana-2815 395 27 𝑧 𝑧 PRON cana-2815 395 28 βˆ’ βˆ’ PUNCT cana-2815 395 29 𝑣3 𝑣3 ADJ cana-2815 395 30 ( ( PUNCT cana-2815 395 31 sinh sinh NOUN cana-2815 395 32 π‘₯ π‘₯ DET cana-2815 395 33 sinh𝑦 sinh𝑦 ADJ cana-2815 395 34 sinh sinh VERB cana-2815 395 35 𝑧 𝑧 PROPN cana-2815 395 36 ) ) PUNCT cana-2815 395 37 ] ] PUNCT cana-2815 395 38 βˆ…0(π‘₯ βˆ…0(π‘₯ NUM cana-2815 395 39 , , PUNCT cana-2815 395 40 𝑦 𝑦 NOUN cana-2815 395 41 , , PUNCT cana-2815 395 42 𝑧 𝑧 NOUN cana-2815 395 43 , , PUNCT cana-2815 395 44 𝑑 𝑑 NOUN cana-2815 395 45 ) ) PUNCT cana-2815 395 46 = = PUNCT cana-2815 395 47 βˆ…(0 βˆ…(0 PROPN cana-2815 395 48 ) ) PUNCT cana-2815 395 49 = = VERB cana-2815 396 1 sinh sinh NOUN cana-2815 396 2 π‘₯ π‘₯ DET cana-2815 396 3 sinh𝑦 sinh𝑦 PROPN cana-2815 396 4 sinh sinh VERB cana-2815 396 5 𝑧 𝑧 PROPN cana-2815 396 6 ( ( PUNCT cana-2815 396 7 1βˆ’ 1βˆ’ NUM cana-2815 396 8 𝑑 𝑑 NOUN cana-2815 396 9 ) ) PUNCT cana-2815 396 10 ( ( PUNCT cana-2815 396 11 44 44 NUM cana-2815 396 12 ) ) PUNCT cana-2815 396 13 ( ( PUNCT cana-2815 396 14 40 40 NUM cana-2815 396 15 ) ) PUNCT cana-2815 396 16 ( ( PUNCT cana-2815 396 17 43 43 NUM cana-2815 396 18 ) ) PUNCT cana-2815 396 19 ( ( PUNCT cana-2815 396 20 42 42 NUM cana-2815 396 21 ) ) PUNCT cana-2815 396 22 ( ( PUNCT cana-2815 396 23 45 45 NUM cana-2815 396 24 ) ) PUNCT cana-2815 396 25 ( ( PUNCT cana-2815 396 26 41 41 NUM cana-2815 396 27 ) ) PUNCT cana-2815 396 28 communications communication NOUN cana-2815 396 29 on on ADP cana-2815 396 30 applied apply VERB cana-2815 396 31 nonlinear nonlinear ADJ cana-2815 396 32 analysis analysis NOUN cana-2815 396 33 issn issn NOUN cana-2815 396 34 : : PUNCT cana-2815 396 35 1074 1074 NUM cana-2815 396 36 - - PUNCT cana-2815 396 37 133x 133x NUM cana-2815 396 38 vol vol NOUN cana-2815 396 39 32 32 NUM cana-2815 396 40 no no NOUN cana-2815 396 41 . . PUNCT cana-2815 397 1 4s 4s NUM cana-2815 397 2 ( ( PUNCT cana-2815 397 3 2025 2025 NUM cana-2815 397 4 ) ) PUNCT cana-2815 397 5 322 322 NUM cana-2815 397 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 397 7 βˆ…π‘›+1(π‘₯ βˆ…π‘›+1(π‘₯ NUM cana-2815 397 8 , , PUNCT cana-2815 397 9 𝑦 𝑦 NOUN cana-2815 397 10 , , PUNCT cana-2815 397 11 𝑧 𝑧 NOUN cana-2815 397 12 , , PUNCT cana-2815 397 13 𝑑 𝑑 NOUN cana-2815 397 14 ) ) PUNCT cana-2815 397 15 = = PUNCT cana-2815 397 16 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 397 17 [ [ X cana-2815 397 18 𝐿 𝐿 PROPN cana-2815 397 19 ( ( PUNCT cana-2815 397 20 βˆ…π‘› βˆ…π‘› NOUN cana-2815 397 21 ) ) PUNCT cana-2815 397 22 βˆ’ βˆ’ PROPN cana-2815 398 1 2𝐷𝑑 2𝐷𝑑 NUM cana-2815 398 2 𝛼 𝛼 NOUN cana-2815 398 3 βˆ…π‘› βˆ…π‘› NOUN cana-2815 398 4 ] ] PUNCT cana-2815 398 5 ] ] PUNCT cana-2815 398 6 for for ADP cana-2815 398 7 𝑛 𝑛 PROPN cana-2815 398 8 = = SYM cana-2815 398 9 0 0 NUM cana-2815 398 10 βˆ…1(π‘₯ βˆ…1(π‘₯ PROPN cana-2815 398 11 , , PUNCT cana-2815 398 12 y y PROPN cana-2815 398 13 , , PUNCT cana-2815 398 14 z z PROPN cana-2815 398 15 , , PUNCT cana-2815 398 16 𝑑 𝑑 NOUN cana-2815 398 17 ) ) PUNCT cana-2815 398 18 = = PUNCT cana-2815 398 19 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 399 1 [ [ X cana-2815 399 2 𝐿 𝐿 PROPN cana-2815 399 3 ( ( PUNCT cana-2815 399 4 βˆ…0 βˆ…0 NOUN cana-2815 399 5 ) ) PUNCT cana-2815 399 6 βˆ’ βˆ’ PROPN cana-2815 399 7 2 2 NUM cana-2815 400 1 𝐷𝑑 𝐷𝑑 NOUN cana-2815 400 2 𝛼 𝛼 NOUN cana-2815 400 3 βˆ…0 βˆ…0 NOUN cana-2815 400 4 ] ] PUNCT cana-2815 400 5 ] ] PUNCT cana-2815 400 6 here here ADV cana-2815 400 7 , , PUNCT cana-2815 400 8 𝐿 𝐿 PROPN cana-2815 400 9 [ [ X cana-2815 400 10 βˆ…0 βˆ…0 NOUN cana-2815 400 11 ] ] X cana-2815 400 12 = = SYM cana-2815 400 13 ( ( PUNCT cana-2815 400 14 βˆ…0π‘₯π‘₯ βˆ…0π‘₯π‘₯ NOUN cana-2815 400 15 + + CCONJ cana-2815 400 16 βˆ…0𝑦𝑦 βˆ…0𝑦𝑦 X cana-2815 400 17 + + CCONJ cana-2815 400 18 βˆ…0𝑧𝑧 βˆ…0𝑧𝑧 NOUN cana-2815 400 19 βˆ’ βˆ’ PROPN cana-2815 400 20 3βˆ…0 3βˆ…0 NOUN cana-2815 400 21 ) ) PUNCT cana-2815 400 22 = = SYM cana-2815 400 23 0 0 PUNCT cana-2815 400 24 therefore therefore ADV cana-2815 400 25 , , PUNCT cana-2815 400 26 above above ADP cana-2815 400 27 equation equation NOUN cana-2815 400 28 implies imply VERB cana-2815 400 29 , , PUNCT cana-2815 400 30 βˆ…1(π‘₯ βˆ…1(π‘₯ NOUN cana-2815 400 31 , , PUNCT cana-2815 400 32 𝑦 𝑦 NOUN cana-2815 400 33 , , PUNCT cana-2815 400 34 𝑧 𝑧 NOUN cana-2815 400 35 , , PUNCT cana-2815 400 36 𝑑 𝑑 NOUN cana-2815 400 37 ) ) PUNCT cana-2815 400 38 = = PUNCT cana-2815 400 39 πΈβˆ’1[𝑣𝛼𝐸 πΈβˆ’1[𝑣𝛼𝐸 PROPN cana-2815 401 1 [ [ X cana-2815 401 2 βˆ’2 βˆ’2 X cana-2815 401 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 401 4 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 401 5 ] ] PUNCT cana-2815 401 6 ] ] PUNCT cana-2815 401 7 βˆ…1(π‘₯ βˆ…1(π‘₯ PROPN cana-2815 401 8 , , PUNCT cana-2815 401 9 𝑦 𝑦 NOUN cana-2815 401 10 , , PUNCT cana-2815 401 11 𝑧 𝑧 NOUN cana-2815 401 12 , , PUNCT cana-2815 401 13 𝑑 𝑑 NOUN cana-2815 401 14 ) ) PUNCT cana-2815 401 15 = = SYM cana-2815 401 16 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 402 1 [ [ X cana-2815 402 2 βˆ’2 βˆ’2 X cana-2815 402 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 402 4 π›Όβˆ…0 π›Όβˆ…0 NOUN cana-2815 402 5 ] ] PUNCT cana-2815 402 6 ] ] X cana-2815 402 7 = = SYM cana-2815 402 8 2 2 NUM cana-2815 402 9 sinh sinh NOUN cana-2815 402 10 π‘₯ π‘₯ DET cana-2815 402 11 sinh𝑦 sinh𝑦 NOUN cana-2815 402 12 sinh sinh VERB cana-2815 402 13 𝑧 𝑧 DET cana-2815 402 14 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 402 15 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 402 16 ) ) PUNCT cana-2815 402 17 for for ADP cana-2815 402 18 𝑛 𝑛 NOUN cana-2815 402 19 = = SYM cana-2815 402 20 1,2,3 1,2,3 NUM cana-2815 402 21 … … NUM cana-2815 402 22 . . PUNCT cana-2815 402 23 βˆ…2(π‘₯ βˆ…2(π‘₯ NOUN cana-2815 402 24 , , PUNCT cana-2815 402 25 𝑦 𝑦 PRON cana-2815 402 26 , , PUNCT cana-2815 402 27 𝑧 𝑧 NOUN cana-2815 402 28 , , PUNCT cana-2815 402 29 𝑑 𝑑 NOUN cana-2815 402 30 ) ) PUNCT cana-2815 402 31 = = SYM cana-2815 402 32 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 403 1 [ [ X cana-2815 403 2 βˆ’2 βˆ’2 X cana-2815 403 3 𝐷𝑑 𝐷𝑑 NOUN cana-2815 403 4 π›Όβˆ…1 π›Όβˆ…1 NOUN cana-2815 403 5 ] ] X cana-2815 403 6 ] ] X cana-2815 403 7 = = SYM cana-2815 403 8 βˆ’4 βˆ’4 PUNCT cana-2815 403 9 sinh sinh NOUN cana-2815 403 10 π‘₯ π‘₯ DET cana-2815 403 11 sinh𝑦 sinh𝑦 PROPN cana-2815 403 12 sinh sinh VERB cana-2815 403 13 𝑧 𝑧 DET cana-2815 403 14 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 403 15 ⌈(2𝛼+2 ⌈(2𝛼+2 X cana-2815 403 16 ) ) PUNCT cana-2815 403 17 βˆ…3(π‘₯ βˆ…3(π‘₯ NOUN cana-2815 403 18 , , PUNCT cana-2815 403 19 𝑦 𝑦 NOUN cana-2815 403 20 , , PUNCT cana-2815 403 21 𝑧 𝑧 NOUN cana-2815 403 22 , , PUNCT cana-2815 403 23 𝑑 𝑑 NOUN cana-2815 403 24 ) ) PUNCT cana-2815 403 25 = = SYM cana-2815 403 26 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 404 1 [ [ X cana-2815 404 2 βˆ’2 βˆ’2 X cana-2815 404 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 404 4 π›Όβˆ…2 π›Όβˆ…2 NOUN cana-2815 404 5 ] ] X cana-2815 404 6 ] ] X cana-2815 404 7 = = SYM cana-2815 404 8 8 8 NUM cana-2815 404 9 sinh sinh NOUN cana-2815 404 10 π‘₯ π‘₯ DET cana-2815 404 11 sinh𝑦 sinh𝑦 PROPN cana-2815 404 12 sinh sinh VERB cana-2815 404 13 𝑧 𝑧 PROPN cana-2815 404 14 𝑑3𝛼+1 𝑑3𝛼+1 ADJ cana-2815 404 15 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 404 16 ) ) PUNCT cana-2815 404 17 βˆ…4(π‘₯ βˆ…4(π‘₯ NOUN cana-2815 404 18 , , PUNCT cana-2815 404 19 𝑦 𝑦 NOUN cana-2815 404 20 , , PUNCT cana-2815 404 21 𝑧 𝑧 NOUN cana-2815 404 22 , , PUNCT cana-2815 404 23 𝑑 𝑑 NOUN cana-2815 404 24 ) ) PUNCT cana-2815 404 25 = = SYM cana-2815 404 26 βˆ’πΈβˆ’1[𝑣𝛼𝐸 βˆ’πΈβˆ’1[𝑣𝛼𝐸 X cana-2815 405 1 [ [ X cana-2815 405 2 βˆ’3 βˆ’3 X cana-2815 405 3 𝐷𝑑 𝐷𝑑 PROPN cana-2815 405 4 π›Όβˆ…3 π›Όβˆ…3 NOUN cana-2815 405 5 ] ] X cana-2815 405 6 ] ] X cana-2815 406 1 = = SYM cana-2815 406 2 βˆ’16 βˆ’16 CCONJ cana-2815 406 3 sinh sinh NOUN cana-2815 406 4 π‘₯ π‘₯ DET cana-2815 406 5 sinh𝑦 sinh𝑦 PROPN cana-2815 406 6 sinh sinh VERB cana-2815 406 7 𝑧 𝑧 PROPN cana-2815 406 8 𝑑4𝛼+1 𝑑4𝛼+1 ADV cana-2815 406 9 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 406 10 ) ) PUNCT cana-2815 406 11 . . PUNCT cana-2815 407 1 : : PUNCT cana-2815 407 2 : : PUNCT cana-2815 407 3 therefore therefore ADV cana-2815 407 4 , , PUNCT cana-2815 407 5 series series NOUN cana-2815 407 6 form form NOUN cana-2815 407 7 βˆ… βˆ… NOUN cana-2815 407 8 ( ( PUNCT cana-2815 407 9 π‘₯ π‘₯ NOUN cana-2815 407 10 , , PUNCT cana-2815 407 11 𝑦 𝑦 NOUN cana-2815 407 12 , , PUNCT cana-2815 407 13 𝑧 𝑧 PRON cana-2815 407 14 , , PUNCT cana-2815 407 15 𝑑 𝑑 X cana-2815 407 16 ) ) PUNCT cana-2815 407 17 will will AUX cana-2815 407 18 be be AUX cana-2815 407 19 : : PUNCT cana-2815 407 20 βˆ… βˆ… NOUN cana-2815 407 21 ( ( PUNCT cana-2815 407 22 π‘₯ π‘₯ NOUN cana-2815 407 23 , , PUNCT cana-2815 407 24 𝑦 𝑦 NOUN cana-2815 407 25 , , PUNCT cana-2815 407 26 𝑧 𝑧 NOUN cana-2815 407 27 , , PUNCT cana-2815 407 28 𝑑 𝑑 NOUN cana-2815 407 29 ) ) PUNCT cana-2815 407 30 = = SYM cana-2815 407 31 βˆ…0 βˆ…0 NOUN cana-2815 407 32 ( ( PUNCT cana-2815 407 33 π‘₯ π‘₯ PROPN cana-2815 407 34 , , PUNCT cana-2815 407 35 𝑦 𝑦 NOUN cana-2815 407 36 , , PUNCT cana-2815 407 37 𝑧 𝑧 PROPN cana-2815 407 38 , , PUNCT cana-2815 407 39 𝑑)+βˆ…1 𝑑)+βˆ…1 PROPN cana-2815 407 40 ( ( PUNCT cana-2815 407 41 π‘₯ π‘₯ PROPN cana-2815 407 42 , , PUNCT cana-2815 407 43 𝑦 𝑦 NOUN cana-2815 407 44 , , PUNCT cana-2815 407 45 𝑧 𝑧 PROPN cana-2815 407 46 , , PUNCT cana-2815 407 47 𝑑)+βˆ…2 𝑑)+βˆ…2 X cana-2815 407 48 ( ( PUNCT cana-2815 407 49 π‘₯ π‘₯ NOUN cana-2815 407 50 , , PUNCT cana-2815 407 51 𝑦 𝑦 NOUN cana-2815 407 52 , , PUNCT cana-2815 407 53 𝑧 𝑧 NOUN cana-2815 407 54 , , PUNCT cana-2815 407 55 𝑑 𝑑 NOUN cana-2815 407 56 ) ) PUNCT cana-2815 407 57 + + CCONJ cana-2815 407 58 βˆ…3(π‘₯ βˆ…3(π‘₯ NUM cana-2815 407 59 , , PUNCT cana-2815 407 60 𝑦 𝑦 NOUN cana-2815 407 61 , , PUNCT cana-2815 407 62 𝑧 𝑧 NOUN cana-2815 407 63 , , PUNCT cana-2815 407 64 𝑑 𝑑 NOUN cana-2815 407 65 ) ) PUNCT cana-2815 408 1 + + X cana-2815 408 2 β‹― β‹― NOUN cana-2815 408 3 …… …… NOUN cana-2815 408 4 βˆ… βˆ… NOUN cana-2815 408 5 ( ( PUNCT cana-2815 408 6 π‘₯ π‘₯ NOUN cana-2815 408 7 , , PUNCT cana-2815 408 8 𝑦 𝑦 NOUN cana-2815 408 9 , , PUNCT cana-2815 408 10 𝑧 𝑧 NOUN cana-2815 408 11 , , PUNCT cana-2815 408 12 𝑑 𝑑 NOUN cana-2815 408 13 ) ) PUNCT cana-2815 408 14 = = VERB cana-2815 408 15 sinh sinh NOUN cana-2815 408 16 π‘₯ π‘₯ DET cana-2815 408 17 sinh𝑦 sinh𝑦 PROPN cana-2815 408 18 sinh sinh VERB cana-2815 408 19 𝑧 𝑧 PROPN cana-2815 408 20 ( ( PUNCT cana-2815 408 21 1βˆ’ 1βˆ’ NUM cana-2815 408 22 𝑑 𝑑 NOUN cana-2815 408 23 ) ) PUNCT cana-2815 408 24 + + CCONJ cana-2815 408 25 2 2 NUM cana-2815 408 26 sinh sinh NOUN cana-2815 408 27 π‘₯ π‘₯ DET cana-2815 408 28 sinh𝑦 sinh𝑦 NOUN cana-2815 408 29 sinh sinh VERB cana-2815 408 30 𝑧 𝑧 PRON cana-2815 408 31 𝑑𝛼+1 𝑑𝛼+1 NUM cana-2815 408 32 ⌈(𝛼 ⌈(𝛼 PROPN cana-2815 409 1 + + CCONJ cana-2815 409 2 2 2 X cana-2815 409 3 ) ) PUNCT cana-2815 409 4 βˆ’ βˆ’ PROPN cana-2815 409 5 4 4 NUM cana-2815 409 6 sinh sinh NOUN cana-2815 409 7 π‘₯ π‘₯ DET cana-2815 409 8 sinh𝑦 sinh𝑦 PROPN cana-2815 409 9 sinh sinh VERB cana-2815 409 10 𝑧 𝑧 PRON cana-2815 409 11 𝑑2𝛼+1 𝑑2𝛼+1 PROPN cana-2815 409 12 ⌈(2𝛼 ⌈(2𝛼 NOUN cana-2815 409 13 + + CCONJ cana-2815 409 14 2 2 NUM cana-2815 409 15 ) ) PUNCT cana-2815 409 16 + + CCONJ cana-2815 409 17 8 8 NUM cana-2815 409 18 sinh sinh NOUN cana-2815 409 19 π‘₯ π‘₯ DET cana-2815 409 20 sinh𝑦 sinh𝑦 NOUN cana-2815 409 21 sinh sinh VERB cana-2815 409 22 𝑧 𝑧 PRON cana-2815 409 23 𝑑3𝛼+1 𝑑3𝛼+1 PUNCT cana-2815 409 24 ⌈(3𝛼 ⌈(3𝛼 PROPN cana-2815 409 25 + + CCONJ cana-2815 409 26 2 2 NUM cana-2815 409 27 ) ) PUNCT cana-2815 409 28 βˆ’ βˆ’ PROPN cana-2815 409 29 16 16 NUM cana-2815 409 30 sinh sinh NOUN cana-2815 409 31 π‘₯ π‘₯ DET cana-2815 409 32 sinh𝑦 sinh𝑦 PROPN cana-2815 409 33 sinh sinh VERB cana-2815 409 34 𝑧 𝑧 PROPN cana-2815 409 35 𝑑4𝛼+1 𝑑4𝛼+1 PROPN cana-2815 409 36 ⌈(4𝛼 ⌈(4𝛼 NUM cana-2815 410 1 + + CCONJ cana-2815 410 2 2 2 NUM cana-2815 410 3 ) ) PUNCT cana-2815 410 4 … … PUNCT cana-2815 410 5 … … PUNCT cana-2815 410 6 … … SYM cana-2815 410 7 . . PUNCT cana-2815 411 1 βˆ… βˆ… NOUN cana-2815 411 2 ( ( PUNCT cana-2815 411 3 π‘₯ π‘₯ NOUN cana-2815 411 4 , , PUNCT cana-2815 411 5 𝑦 𝑦 NOUN cana-2815 411 6 , , PUNCT cana-2815 411 7 𝑧 𝑧 NOUN cana-2815 411 8 , , PUNCT cana-2815 411 9 𝑑 𝑑 NOUN cana-2815 411 10 ) ) PUNCT cana-2815 411 11 = = VERB cana-2815 411 12 sinh sinh NOUN cana-2815 411 13 π‘₯ π‘₯ DET cana-2815 411 14 sinh𝑦 sinh𝑦 NOUN cana-2815 411 15 sinh sinh VERB cana-2815 411 16 𝑧 𝑧 PROPN cana-2815 412 1 [ [ X cana-2815 412 2 ( ( PUNCT cana-2815 412 3 1βˆ’ 1βˆ’ NUM cana-2815 412 4 𝑑 𝑑 NOUN cana-2815 412 5 ) ) PUNCT cana-2815 412 6 + + CCONJ cana-2815 412 7 2 2 NUM cana-2815 412 8 𝑑𝛼+1 𝑑𝛼+1 NOUN cana-2815 412 9 ⌈(𝛼+2 ⌈(𝛼+2 VERB cana-2815 412 10 ) ) PUNCT cana-2815 412 11 βˆ’ βˆ’ PROPN cana-2815 412 12 4 4 NUM cana-2815 412 13 𝑑2𝛼+1 𝑑2𝛼+1 NOUN cana-2815 412 14 ⌈(2𝛼+2 ⌈(2𝛼+2 PROPN cana-2815 412 15 ) ) PUNCT cana-2815 413 1 + + CCONJ cana-2815 413 2 8 8 NUM cana-2815 413 3 𝑑3𝛼+1 𝑑3𝛼+1 PUNCT cana-2815 413 4 ⌈(3𝛼+2 ⌈(3𝛼+2 ADJ cana-2815 413 5 ) ) PUNCT cana-2815 413 6 βˆ’βˆ’16 βˆ’βˆ’16 PROPN cana-2815 413 7 𝑑4𝛼+1 𝑑4𝛼+1 PUNCT cana-2815 413 8 ⌈(4𝛼+2 ⌈(4𝛼+2 PROPN cana-2815 413 9 ) ) PUNCT cana-2815 413 10 … … PUNCT cana-2815 413 11 …… …… X cana-2815 413 12 . . PUNCT cana-2815 414 1 ] ] X cana-2815 414 2 when when SCONJ cana-2815 414 3 𝛼 𝛼 X cana-2815 414 4 = = SYM cana-2815 414 5 1 1 NUM cana-2815 414 6 , , PUNCT cana-2815 414 7 the the DET cana-2815 414 8 following follow VERB cana-2815 414 9 approximate approximate ADJ cana-2815 414 10 solution solution NOUN cana-2815 414 11 will will AUX cana-2815 414 12 represent represent VERB cana-2815 414 13 as as ADP cana-2815 414 14 : : PUNCT cana-2815 414 15 βˆ… βˆ… NOUN cana-2815 414 16 ( ( PUNCT cana-2815 414 17 π‘₯ π‘₯ NOUN cana-2815 414 18 , , PUNCT cana-2815 414 19 𝑦 𝑦 NOUN cana-2815 414 20 , , PUNCT cana-2815 414 21 𝑧 𝑧 NOUN cana-2815 414 22 , , PUNCT cana-2815 414 23 𝑑 𝑑 NOUN cana-2815 414 24 ) ) PUNCT cana-2815 414 25 = = VERB cana-2815 414 26 sinh sinh NOUN cana-2815 414 27 π‘₯ π‘₯ DET cana-2815 414 28 sinh𝑦 sinh𝑦 NOUN cana-2815 414 29 sinh sinh VERB cana-2815 414 30 𝑧 𝑧 PROPN cana-2815 415 1 [ [ X cana-2815 415 2 1βˆ’ 1βˆ’ NUM cana-2815 415 3 𝑑 𝑑 PRON cana-2815 415 4 + + NOUN cana-2815 415 5 2t2 2t2 NUM cana-2815 415 6 2 2 NUM cana-2815 415 7 ! ! PUNCT cana-2815 415 8 βˆ’ βˆ’ NOUN cana-2815 415 9 4 4 NUM cana-2815 415 10 𝑑3 𝑑3 PROPN cana-2815 415 11 3 3 NUM cana-2815 415 12 ! ! PUNCT cana-2815 416 1 + + CCONJ cana-2815 416 2 8𝑑4 8𝑑4 NUM cana-2815 416 3 4 4 NUM cana-2815 416 4 ! ! PUNCT cana-2815 417 1 βˆ’ βˆ’ NOUN cana-2815 418 1 16t5 16t5 NUM cana-2815 418 2 5 5 NUM cana-2815 418 3 ! ! PUNCT cana-2815 418 4 … … PUNCT cana-2815 418 5 . . PUNCT cana-2815 418 6 . . PUNCT cana-2815 419 1 ] ] PUNCT cana-2815 419 2 ( ( PUNCT cana-2815 419 3 48 48 NUM cana-2815 419 4 ) ) PUNCT cana-2815 419 5 βˆ… βˆ… NOUN cana-2815 419 6 ( ( PUNCT cana-2815 419 7 π‘₯ π‘₯ NOUN cana-2815 419 8 , , PUNCT cana-2815 419 9 𝑦 𝑦 NOUN cana-2815 419 10 , , PUNCT cana-2815 419 11 𝑧 𝑧 NOUN cana-2815 419 12 , , PUNCT cana-2815 419 13 𝑑 𝑑 NOUN cana-2815 419 14 ) ) PUNCT cana-2815 419 15 = = VERB cana-2815 419 16 sinh sinh NOUN cana-2815 419 17 π‘₯ π‘₯ DET cana-2815 419 18 sinh𝑦 sinh𝑦 NOUN cana-2815 419 19 sinh sinh VERB cana-2815 419 20 𝑧 𝑧 PROPN cana-2815 419 21 2 2 NUM cana-2815 420 1 [ [ X cana-2815 420 2 2βˆ’ 2βˆ’ NUM cana-2815 420 3 2 2 NUM cana-2815 420 4 t t NOUN cana-2815 420 5 1 1 NUM cana-2815 420 6 ! ! PUNCT cana-2815 421 1 + + CCONJ cana-2815 421 2 4 4 NUM cana-2815 421 3 t t NOUN cana-2815 421 4 2 2 NUM cana-2815 421 5 2 2 NUM cana-2815 421 6 ! ! PUNCT cana-2815 422 1 βˆ’ βˆ’ NOUN cana-2815 422 2 8 8 NUM cana-2815 422 3 𝑑3 𝑑3 PROPN cana-2815 422 4 3 3 NUM cana-2815 422 5 ! ! PUNCT cana-2815 423 1 + + CCONJ cana-2815 423 2 16𝑑4 16𝑑4 NUM cana-2815 423 3 4 4 NUM cana-2815 423 4 ! ! PUNCT cana-2815 424 1 βˆ’ βˆ’ PROPN cana-2815 424 2 32 32 NUM cana-2815 424 3 𝑑5 𝑑5 VERB cana-2815 424 4 5 5 NUM cana-2815 424 5 ! ! PUNCT cana-2815 424 6 … … PUNCT cana-2815 424 7 . . PUNCT cana-2815 424 8 . . PUNCT cana-2815 425 1 ] ] PUNCT cana-2815 425 2 ( ( PUNCT cana-2815 425 3 47 47 NUM cana-2815 425 4 ) ) PUNCT cana-2815 425 5 ( ( PUNCT cana-2815 425 6 46 46 NUM cana-2815 425 7 ) ) PUNCT cana-2815 425 8 ( ( PUNCT cana-2815 425 9 48 48 NUM cana-2815 425 10 ) ) PUNCT cana-2815 425 11 communications communication NOUN cana-2815 425 12 on on ADP cana-2815 425 13 applied apply VERB cana-2815 425 14 nonlinear nonlinear ADJ cana-2815 425 15 analysis analysis NOUN cana-2815 425 16 issn issn NOUN cana-2815 425 17 : : PUNCT cana-2815 425 18 1074 1074 NUM cana-2815 425 19 - - PUNCT cana-2815 425 20 133x 133x NUM cana-2815 425 21 vol vol NOUN cana-2815 425 22 32 32 NUM cana-2815 425 23 no no NOUN cana-2815 425 24 . . PUNCT cana-2815 426 1 4s 4s NUM cana-2815 426 2 ( ( PUNCT cana-2815 426 3 2025 2025 NUM cana-2815 426 4 ) ) PUNCT cana-2815 426 5 323 323 NUM cana-2815 426 6 https://internationalpubls.com https://internationalpubls.com X cana-2815 426 7 βˆ… βˆ… NOUN cana-2815 426 8 ( ( PUNCT cana-2815 426 9 π‘₯ π‘₯ NOUN cana-2815 426 10 , , PUNCT cana-2815 426 11 𝑦 𝑦 NOUN cana-2815 426 12 , , PUNCT cana-2815 426 13 𝑧 𝑧 NOUN cana-2815 426 14 , , PUNCT cana-2815 426 15 𝑑 𝑑 NOUN cana-2815 426 16 ) ) PUNCT cana-2815 426 17 = = VERB cana-2815 426 18 sinh sinh NOUN cana-2815 426 19 π‘₯ π‘₯ DET cana-2815 426 20 sinh𝑦 sinh𝑦 NOUN cana-2815 426 21 sinh sinh VERB cana-2815 426 22 𝑧 𝑧 PROPN cana-2815 426 23 2 2 NUM cana-2815 427 1 [ [ SYM cana-2815 427 2 2 2 NUM cana-2815 427 3 βˆ’ βˆ’ NUM cana-2815 427 4 2 2 NUM cana-2815 427 5 t t NOUN cana-2815 427 6 1 1 NUM cana-2815 427 7 ! ! PUNCT cana-2815 428 1 + + CCONJ cana-2815 428 2 ( ( PUNCT cana-2815 428 3 2𝑑)2 2𝑑)2 NUM cana-2815 428 4 2 2 NUM cana-2815 428 5 ! ! PUNCT cana-2815 428 6 βˆ’ βˆ’ PROPN cana-2815 428 7 ( ( PUNCT cana-2815 428 8 2𝑑 2𝑑 NOUN cana-2815 428 9 ) ) PUNCT cana-2815 428 10 3 3 NUM cana-2815 428 11 3 3 NUM cana-2815 428 12 ! ! PUNCT cana-2815 429 1 + + CCONJ cana-2815 429 2 ( ( PUNCT cana-2815 429 3 2𝑑)4 2𝑑)4 NUM cana-2815 429 4 4 4 NUM cana-2815 429 5 ! ! PUNCT cana-2815 429 6 βˆ’ βˆ’ PROPN cana-2815 429 7 ( ( PUNCT cana-2815 429 8 2𝑑 2𝑑 NOUN cana-2815 429 9 ) ) PUNCT cana-2815 429 10 5 5 NUM cana-2815 429 11 5 5 NUM cana-2815 429 12 ! ! PUNCT cana-2815 429 13 … … PUNCT cana-2815 429 14 . . PUNCT cana-2815 429 15 . . PUNCT cana-2815 430 1 ] ] PUNCT cana-2815 430 2 βˆ… βˆ… NOUN cana-2815 430 3 ( ( PUNCT cana-2815 430 4 π‘₯ π‘₯ NOUN cana-2815 430 5 , , PUNCT cana-2815 430 6 𝑦 𝑦 NOUN cana-2815 430 7 , , PUNCT cana-2815 430 8 𝑧 𝑧 NOUN cana-2815 430 9 , , PUNCT cana-2815 430 10 𝑑 𝑑 NOUN cana-2815 430 11 ) ) PUNCT cana-2815 430 12 = = VERB cana-2815 430 13 sinhπ‘₯ sinhπ‘₯ PROPN cana-2815 430 14 sinh𝑦 sinh𝑦 PROPN cana-2815 430 15 sinh sinh VERB cana-2815 430 16 𝑧 𝑧 PROPN cana-2815 430 17 2 2 NUM cana-2815 430 18 ( ( PUNCT cana-2815 430 19 1 1 NUM cana-2815 430 20 + + CCONJ cana-2815 430 21 𝑒1βˆ’2𝑑 𝑒1βˆ’2𝑑 NOUN cana-2815 430 22 ) ) PUNCT cana-2815 430 23 the the DET cana-2815 430 24 precise precise ADJ cana-2815 430 25 answer answer NOUN cana-2815 430 26 to to ADP cana-2815 430 27 the the DET cana-2815 430 28 equation equation NOUN cana-2815 430 29 ( ( PUNCT cana-2815 430 30 42 42 NUM cana-2815 430 31 ) ) PUNCT cana-2815 430 32 is be AUX cana-2815 430 33 : : PUNCT cana-2815 430 34 βˆ… βˆ… NOUN cana-2815 430 35 ( ( PUNCT cana-2815 430 36 π‘₯ π‘₯ NOUN cana-2815 430 37 , , PUNCT cana-2815 430 38 𝑦 𝑦 NOUN cana-2815 430 39 , , PUNCT cana-2815 430 40 𝑑 𝑑 NOUN cana-2815 430 41 ) ) PUNCT cana-2815 430 42 = = VERB cana-2815 430 43 sinhπ‘₯ sinhπ‘₯ PROPN cana-2815 430 44 sinh𝑦 sinh𝑦 PROPN cana-2815 430 45 sinh sinh VERB cana-2815 430 46 𝑧 𝑧 PROPN cana-2815 430 47 2 2 NUM cana-2815 430 48 ( ( PUNCT cana-2815 430 49 1 1 NUM cana-2815 430 50 + + CCONJ cana-2815 430 51 𝑒1βˆ’2𝑑 𝑒1βˆ’2𝑑 NOUN cana-2815 430 52 ) ) PUNCT cana-2815 430 53 5 5 NUM cana-2815 430 54 . . PUNCT cana-2815 430 55 graphical graphical ADJ cana-2815 430 56 discussion discussion NOUN cana-2815 430 57 in in ADP cana-2815 430 58 this this DET cana-2815 430 59 discussion discussion NOUN cana-2815 430 60 , , PUNCT cana-2815 430 61 the the DET cana-2815 430 62 graphical graphical ADJ cana-2815 430 63 simulation simulation NOUN cana-2815 430 64 is be AUX cana-2815 430 65 shown show VERB cana-2815 430 66 to to PART cana-2815 430 67 validate validate VERB cana-2815 430 68 the the DET cana-2815 430 69 results result NOUN cana-2815 430 70 between between ADP cana-2815 430 71 the the DET cana-2815 430 72 approximate approximate ADJ cana-2815 430 73 solution solution NOUN cana-2815 430 74 calculated calculate VERB cana-2815 430 75 by by ADP cana-2815 430 76 adopted adopt VERB cana-2815 430 77 technique technique NOUN cana-2815 430 78 and and CCONJ cana-2815 430 79 exact exact ADJ cana-2815 430 80 solution solution NOUN cana-2815 430 81 exist exist NOUN cana-2815 430 82 are be AUX cana-2815 430 83 expressed express VERB cana-2815 430 84 for for ADP cana-2815 430 85 said say VERB cana-2815 430 86 applications application NOUN cana-2815 430 87 . . PUNCT cana-2815 431 1 example example NOUN cana-2815 431 2 1 1 NUM cana-2815 432 1 , , PUNCT cana-2815 432 2 the the DET cana-2815 432 3 approximation approximation NOUN cana-2815 432 4 solution solution NOUN cana-2815 432 5 and and CCONJ cana-2815 432 6 exact exact ADJ cana-2815 432 7 solution solution NOUN cana-2815 432 8 outcomes outcome NOUN cana-2815 432 9 are be AUX cana-2815 432 10 compared compare VERB cana-2815 432 11 at at ADP cana-2815 432 12 𝑑 𝑑 X cana-2815 432 13 = = SYM cana-2815 432 14 1 1 NUM cana-2815 432 15 , , PUNCT cana-2815 432 16 2 2 NUM cana-2815 432 17 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-2815 432 18 3 3 NUM cana-2815 432 19 at at ADP cana-2815 432 20 𝛼 𝛼 NOUN cana-2815 432 21 = = SYM cana-2815 432 22 2 2 NUM cana-2815 432 23 shown show VERB cana-2815 432 24 in in ADP cana-2815 432 25 figure figure NOUN cana-2815 432 26 1 1 NUM cana-2815 432 27 . . PUNCT cana-2815 432 28 figure figure NOUN cana-2815 432 29 2 2 NUM cana-2815 432 30 . . PUNCT cana-2815 432 31 . . PUNCT cana-2815 433 1 shows show VERB cana-2815 433 2 the the DET cana-2815 433 3 surface surface NOUN cana-2815 433 4 graph graph NOUN cana-2815 433 5 of of ADP cana-2815 433 6 the the DET cana-2815 433 7 approximate approximate ADJ cana-2815 433 8 and and CCONJ cana-2815 433 9 exact exact ADJ cana-2815 433 10 solutions solution NOUN cana-2815 433 11 for for ADP cana-2815 433 12 example example NOUN cana-2815 433 13 1 1 NUM cana-2815 433 14 at at ADP cana-2815 433 15 𝛼 𝛼 NOUN cana-2815 433 16 = = SYM cana-2815 433 17 2 2 NUM cana-2815 433 18 . . PUNCT cana-2815 434 1 the the DET cana-2815 434 2 error error NOUN cana-2815 434 3 surface surface NOUN cana-2815 434 4 graph graph NOUN cana-2815 434 5 for for ADP cana-2815 434 6 example example NOUN cana-2815 434 7 1 1 NUM cana-2815 434 8 is be AUX cana-2815 434 9 shown show VERB cana-2815 434 10 in in ADP cana-2815 434 11 figure figure NOUN cana-2815 434 12 2 2 NUM cana-2815 434 13 .. .. PUNCT cana-2815 434 14 additionally additionally ADV cana-2815 434 15 , , PUNCT cana-2815 434 16 figure figure NOUN cana-2815 434 17 2 2 NUM cana-2815 434 18 . . PUNCT cana-2815 435 1 a a PRON cana-2815 435 2 illustrates illustrate VERB cana-2815 435 3 a a DET cana-2815 435 4 line line NOUN cana-2815 435 5 graph graph NOUN cana-2815 435 6 for for ADP cana-2815 435 7 example example NOUN cana-2815 435 8 1 1 NUM cana-2815 435 9 , , PUNCT cana-2815 435 10 displaying display VERB cana-2815 435 11 the the DET cana-2815 435 12 approximate approximate ADJ cana-2815 435 13 solution solution NOUN cana-2815 435 14 , , PUNCT cana-2815 435 15 exact exact ADJ cana-2815 435 16 solution solution NOUN cana-2815 435 17 , , PUNCT cana-2815 435 18 and and CCONJ cana-2815 435 19 the the DET cana-2815 435 20 absolute absolute ADJ cana-2815 435 21 error error NOUN cana-2815 435 22 considering consider VERB cana-2815 435 23 𝑑 𝑑 PROPN cana-2815 435 24 = = NOUN cana-2815 435 25 1 1 X cana-2815 435 26 . . X cana-2815 435 27 figure figure NOUN cana-2815 435 28 3 3 NUM cana-2815 435 29 . . PUNCT cana-2815 435 30 displays display VERB cana-2815 435 31 the the DET cana-2815 435 32 surface surface NOUN cana-2815 435 33 graph graph NOUN cana-2815 435 34 for for ADP cana-2815 435 35 example example NOUN cana-2815 435 36 2 2 NUM cana-2815 435 37 , , PUNCT cana-2815 435 38 showcasing showcase VERB cana-2815 435 39 both both CCONJ cana-2815 435 40 the the DET cana-2815 435 41 approximate approximate ADJ cana-2815 435 42 and and CCONJ cana-2815 435 43 exact exact ADJ cana-2815 435 44 solutions solution NOUN cana-2815 435 45 at at ADP cana-2815 435 46 𝛼 𝛼 NOUN cana-2815 435 47 = = SYM cana-2815 435 48 1 1 NUM cana-2815 435 49 figure figure NOUN cana-2815 435 50 3 3 NUM cana-2815 435 51 . . PUNCT cana-2815 435 52 represents represent VERB cana-2815 435 53 the the DET cana-2815 435 54 corresponding correspond VERB cana-2815 435 55 error error NOUN cana-2815 435 56 surface surface NOUN cana-2815 435 57 graph graph NOUN cana-2815 435 58 for for ADP cana-2815 435 59 example example NOUN cana-2815 436 1 2 2 NUM cana-2815 436 2 . . PUNCT cana-2815 437 1 furthermore furthermore ADV cana-2815 437 2 , , PUNCT cana-2815 437 3 figure figure VERB cana-2815 437 4 3 3 NUM cana-2815 437 5 . . PUNCT cana-2815 437 6 features feature VERB cana-2815 437 7 a a DET cana-2815 437 8 line line NOUN cana-2815 437 9 graph graph NOUN cana-2815 437 10 for for ADP cana-2815 437 11 example example NOUN cana-2815 437 12 2 2 NUM cana-2815 437 13 , , PUNCT cana-2815 437 14 which which PRON cana-2815 437 15 highlights highlight VERB cana-2815 437 16 the the DET cana-2815 437 17 approximate approximate ADJ cana-2815 437 18 and and CCONJ cana-2815 437 19 exact exact ADJ cana-2815 437 20 solutions solution NOUN cana-2815 437 21 , , PUNCT cana-2815 437 22 as as ADV cana-2815 437 23 well well ADV cana-2815 437 24 as as ADP cana-2815 437 25 the the DET cana-2815 437 26 absolute absolute ADJ cana-2815 437 27 error error NOUN cana-2815 437 28 , , PUNCT cana-2815 437 29 evaluated evaluate VERB cana-2815 437 30 at at ADP cana-2815 437 31 𝑑 𝑑 X cana-2815 437 32 = = NOUN cana-2815 437 33 1 1 X cana-2815 437 34 . . PUNCT cana-2815 438 1 in in ADP cana-2815 438 2 figure figure NOUN cana-2815 438 3 4 4 NUM cana-2815 438 4 . . PUNCT cana-2815 439 1 a a DET cana-2815 439 2 : : PUNCT cana-2815 439 3 comparison comparison NOUN cana-2815 439 4 of of ADP cana-2815 439 5 approximate approximate ADJ cana-2815 439 6 solutions solution NOUN cana-2815 439 7 and and CCONJ cana-2815 439 8 exact exact ADJ cana-2815 439 9 solutions solution NOUN cana-2815 439 10 at at ADP cana-2815 439 11 𝛼 𝛼 NOUN cana-2815 439 12 = = SYM cana-2815 439 13 1 1 NUM cana-2815 439 14 for for ADP cana-2815 439 15 example example NOUN cana-2815 439 16 3 3 NUM cana-2815 439 17 figure figure NOUN cana-2815 439 18 2 2 NUM cana-2815 439 19 . . PUNCT cana-2815 440 1 a a DET cana-2815 440 2 : : PUNCT cana-2815 440 3 line line NOUN cana-2815 440 4 plot plot NOUN cana-2815 440 5 for for ADP cana-2815 440 6 exact exact ADJ cana-2815 440 7 , , PUNCT cana-2815 440 8 approximate approximate ADJ cana-2815 440 9 & & CCONJ cana-2815 440 10 absolute absolute ADJ cana-2815 440 11 error error NOUN cana-2815 440 12 example example NOUN cana-2815 440 13 1 1 NUM cana-2815 440 14 figure figure NOUN cana-2815 440 15 3 3 NUM cana-2815 440 16 . . PUNCT cana-2815 441 1 c c NOUN cana-2815 441 2 : : PUNCT cana-2815 441 3 line line NOUN cana-2815 441 4 plot plot NOUN cana-2815 441 5 for for ADP cana-2815 441 6 exact exact ADJ cana-2815 441 7 , , PUNCT cana-2815 441 8 approximate approximate ADJ cana-2815 441 9 & & CCONJ cana-2815 441 10 absolute absolute ADJ cana-2815 441 11 error error NOUN cana-2815 441 12 example example NOUN cana-2815 441 13 2 2 NUM cana-2815 441 14 ( ( PUNCT cana-2815 441 15 49 49 NUM cana-2815 441 16 ) ) PUNCT cana-2815 441 17 324 324 NUM cana-2815 441 18 c c NOUN cana-2815 441 19 325 325 NUM cana-2815 441 20 2 2 NUM cana-2815 441 21 326 326 NUM cana-2815 441 22 c c NOUN cana-2815 441 23 communications communication NOUN cana-2815 441 24 on on ADP cana-2815 441 25 applied apply VERB cana-2815 441 26 nonlinear nonlinear ADJ cana-2815 441 27 analysis analysis NOUN cana-2815 441 28 issn issn NOUN cana-2815 441 29 : : PUNCT cana-2815 441 30 1074 1074 NUM cana-2815 441 31 - - PUNCT cana-2815 441 32 133x 133x NUM cana-2815 441 33 vol vol NOUN cana-2815 441 34 32 32 NUM cana-2815 441 35 no no NOUN cana-2815 441 36 . . PUNCT cana-2815 442 1 4s 4s NUM cana-2815 442 2 ( ( PUNCT cana-2815 442 3 2025 2025 NUM cana-2815 442 4 ) ) PUNCT cana-2815 443 1 https://internationalpubls.com https://internationalpubls.com X cana-2815 443 2 figure figure NOUN cana-2815 443 3 5 5 NUM cana-2815 443 4 . . PUNCT cana-2815 444 1 a a DET cana-2815 444 2 : : PUNCT cana-2815 444 3 comparison comparison NOUN cana-2815 444 4 of of ADP cana-2815 444 5 approximate approximate ADJ cana-2815 444 6 solution solution NOUN cana-2815 444 7 and and CCONJ cana-2815 444 8 exact exact ADJ cana-2815 444 9 solution solution NOUN cana-2815 444 10 profiles profile NOUN cana-2815 444 11 at at ADP cana-2815 444 12 𝛼 𝛼 NOUN cana-2815 444 13 = = SYM cana-2815 444 14 1 1 NUM cana-2815 444 15 for for ADP cana-2815 444 16 example example NOUN cana-2815 444 17 4 4 NUM cana-2815 444 18 figure figure NOUN cana-2815 444 19 6 6 NUM cana-2815 444 20 . . PUNCT cana-2815 445 1 a a DET cana-2815 445 2 : : PUNCT cana-2815 445 3 comparing compare VERB cana-2815 445 4 of of ADP cana-2815 445 5 approximate approximate ADJ cana-2815 445 6 solution solution NOUN cana-2815 445 7 and and CCONJ cana-2815 445 8 exact exact ADJ cana-2815 445 9 solutions solution NOUN cana-2815 445 10 at at ADP cana-2815 445 11 𝛼 𝛼 NOUN cana-2815 445 12 = = SYM cana-2815 445 13 1 1 NUM cana-2815 445 14 for for ADP cana-2815 445 15 example example NOUN cana-2815 445 16 5 5 NUM cana-2815 445 17 figure figure NOUN cana-2815 445 18 5 5 NUM cana-2815 445 19 . . PUNCT cana-2815 446 1 c c NOUN cana-2815 446 2 : : PUNCT cana-2815 446 3 line line NOUN cana-2815 446 4 plot plot NOUN cana-2815 446 5 for for ADP cana-2815 446 6 exact exact ADJ cana-2815 446 7 , , PUNCT cana-2815 446 8 approximate approximate ADJ cana-2815 446 9 & & CCONJ cana-2815 446 10 absolute absolute ADJ cana-2815 446 11 error error NOUN cana-2815 446 12 example example NOUN cana-2815 446 13 4 4 NUM cana-2815 446 14 figure figure NOUN cana-2815 446 15 5 5 NUM cana-2815 446 16 . . PUNCT cana-2815 447 1 c c NOUN cana-2815 447 2 : : PUNCT cana-2815 447 3 line line NOUN cana-2815 447 4 plot plot NOUN cana-2815 447 5 for for ADP cana-2815 447 6 exact exact ADJ cana-2815 447 7 , , PUNCT cana-2815 447 8 approximate approximate ADJ cana-2815 447 9 & & CCONJ cana-2815 447 10 absolute absolute ADJ cana-2815 447 11 error error NOUN cana-2815 447 12 example example NOUN cana-2815 447 13 4 4 NUM cana-2815 447 14 figure figure NOUN cana-2815 447 15 6 6 NUM cana-2815 447 16 . . PUNCT cana-2815 448 1 b b X cana-2815 448 2 : : PUNCT cana-2815 448 3 error error NOUN cana-2815 448 4 plot plot NOUN cana-2815 448 5 between between ADP cana-2815 448 6 exactappro exactappro ADJ cana-2815 448 7 . . PUNCT cana-2815 449 1 solution solution NOUN cana-2815 449 2 example example NOUN cana-2815 449 3 5 5 NUM cana-2815 449 4 327 327 NUM cana-2815 449 5 4 4 NUM cana-2815 449 6 328 328 NUM cana-2815 449 7 329 329 NUM cana-2815 449 8 330 330 NUM cana-2815 449 9 331 331 NUM