id sid tid token lemma pos iajs-3251 1 1 ihjpas ihjpas PROPN iajs-3251 1 2 . . PUNCT iajs-3251 2 1 36(2)2023 36(2)2023 NUM iajs-3251 2 2 430 430 NUM iajs-3251 2 3 this this DET iajs-3251 2 4 work work NOUN iajs-3251 2 5 is be AUX iajs-3251 2 6 licensed license VERB iajs-3251 2 7 under under ADP iajs-3251 2 8 a a DET iajs-3251 2 9 creative creative ADJ iajs-3251 2 10 commons common NOUN iajs-3251 2 11 attribution attribution NOUN iajs-3251 2 12 4.0 4.0 NUM iajs-3251 2 13 international international ADJ iajs-3251 2 14 license license NOUN iajs-3251 2 15 abstract abstract NOUN iajs-3251 2 16 the the DET iajs-3251 2 17 integral integral ADJ iajs-3251 2 18 transformations transformation NOUN iajs-3251 2 19 is be AUX iajs-3251 2 20 a a DET iajs-3251 2 21 complicated complicated ADJ iajs-3251 2 22 function function NOUN iajs-3251 2 23 from from ADP iajs-3251 2 24 a a DET iajs-3251 2 25 function function NOUN iajs-3251 2 26 space space NOUN iajs-3251 2 27 into into ADP iajs-3251 2 28 a a DET iajs-3251 2 29 simple simple ADJ iajs-3251 2 30 function function NOUN iajs-3251 2 31 in in ADP iajs-3251 2 32 transformed transform VERB iajs-3251 2 33 space space NOUN iajs-3251 2 34 . . PUNCT iajs-3251 3 1 where where SCONJ iajs-3251 3 2 the the DET iajs-3251 3 3 function function NOUN iajs-3251 3 4 being be AUX iajs-3251 3 5 characterized characterize VERB iajs-3251 3 6 easily easily ADV iajs-3251 3 7 and and CCONJ iajs-3251 3 8 manipulated manipulate VERB iajs-3251 3 9 through through ADP iajs-3251 3 10 integration integration NOUN iajs-3251 3 11 in in ADP iajs-3251 3 12 transformed transform VERB iajs-3251 3 13 function function NOUN iajs-3251 3 14 space space NOUN iajs-3251 3 15 . . PUNCT iajs-3251 4 1 the the DET iajs-3251 4 2 two two NUM iajs-3251 4 3 parametric parametric ADJ iajs-3251 4 4 form form NOUN iajs-3251 4 5 of of ADP iajs-3251 4 6 see see VERB iajs-3251 4 7 transformation transformation NOUN iajs-3251 4 8 and and CCONJ iajs-3251 4 9 its its PRON iajs-3251 4 10 basic basic ADJ iajs-3251 4 11 characteristics characteristic NOUN iajs-3251 4 12 have have AUX iajs-3251 4 13 been be AUX iajs-3251 4 14 demonstrated demonstrate VERB iajs-3251 4 15 in in ADP iajs-3251 4 16 this this DET iajs-3251 4 17 study study NOUN iajs-3251 4 18 . . PUNCT iajs-3251 5 1 the the DET iajs-3251 5 2 transformed transform VERB iajs-3251 5 3 function function NOUN iajs-3251 5 4 of of ADP iajs-3251 5 5 a a DET iajs-3251 5 6 few few ADJ iajs-3251 5 7 fundamental fundamental ADJ iajs-3251 5 8 functions function NOUN iajs-3251 5 9 along along ADP iajs-3251 5 10 with with ADP iajs-3251 5 11 its its PRON iajs-3251 5 12 time time NOUN iajs-3251 5 13 derivative derivative ADJ iajs-3251 5 14 rule rule NOUN iajs-3251 5 15 is be AUX iajs-3251 5 16 shown show VERB iajs-3251 5 17 . . PUNCT iajs-3251 6 1 it it PRON iajs-3251 6 2 has have AUX iajs-3251 6 3 been be AUX iajs-3251 6 4 demonstrated demonstrate VERB iajs-3251 6 5 how how SCONJ iajs-3251 6 6 two two NUM iajs-3251 6 7 parametric parametric ADJ iajs-3251 6 8 see see NOUN iajs-3251 6 9 transformations transformation NOUN iajs-3251 6 10 can can AUX iajs-3251 6 11 be be AUX iajs-3251 6 12 used use VERB iajs-3251 6 13 to to PART iajs-3251 6 14 solve solve VERB iajs-3251 6 15 linear linear ADJ iajs-3251 6 16 differential differential NOUN iajs-3251 6 17 equations equation NOUN iajs-3251 6 18 . . PUNCT iajs-3251 7 1 this this DET iajs-3251 7 2 research research NOUN iajs-3251 7 3 provides provide VERB iajs-3251 7 4 a a DET iajs-3251 7 5 solution solution NOUN iajs-3251 7 6 to to ADP iajs-3251 7 7 population population NOUN iajs-3251 7 8 growth growth NOUN iajs-3251 7 9 rate rate NOUN iajs-3251 7 10 equation equation NOUN iajs-3251 7 11 . . PUNCT iajs-3251 8 1 one one PRON iajs-3251 8 2 can can AUX iajs-3251 8 3 contrast contrast VERB iajs-3251 8 4 these these DET iajs-3251 8 5 outcomes outcome NOUN iajs-3251 8 6 with with ADP iajs-3251 8 7 different different ADJ iajs-3251 8 8 laplace laplace NOUN iajs-3251 8 9 type type NOUN iajs-3251 8 10 transformations transformation NOUN iajs-3251 8 11 . . PUNCT iajs-3251 9 1 keywords keyword NOUN iajs-3251 9 2 : : PUNCT iajs-3251 9 3 two two NUM iajs-3251 9 4 parametric parametric ADJ iajs-3251 9 5 see see NOUN iajs-3251 9 6 transformation transformation NOUN iajs-3251 9 7 , , PUNCT iajs-3251 9 8 population population NOUN iajs-3251 9 9 growth growth NOUN iajs-3251 9 10 rate rate NOUN iajs-3251 9 11 equation equation NOUN iajs-3251 9 12 , , PUNCT iajs-3251 9 13 linear linear ADJ iajs-3251 9 14 differential differential NOUN iajs-3251 9 15 equations equation NOUN iajs-3251 9 16 , , PUNCT iajs-3251 9 17 time time NOUN iajs-3251 9 18 derivative derivative ADJ iajs-3251 9 19 rule rule NOUN iajs-3251 9 20 . . PUNCT iajs-3251 10 1 1.introduction 1.introduction NUM iajs-3251 10 2 as as SCONJ iajs-3251 10 3 the the DET iajs-3251 10 4 transformations transformation NOUN iajs-3251 10 5 have have VERB iajs-3251 10 6 numerous numerous ADJ iajs-3251 10 7 applications application NOUN iajs-3251 10 8 in in ADP iajs-3251 10 9 fields field NOUN iajs-3251 10 10 of of ADP iajs-3251 10 11 engineering engineering NOUN iajs-3251 10 12 and and CCONJ iajs-3251 10 13 other other ADJ iajs-3251 10 14 sciences science NOUN iajs-3251 10 15 . . PUNCT iajs-3251 11 1 numerous numerous ADJ iajs-3251 11 2 mathematicians mathematician NOUN iajs-3251 11 3 presented present VERB iajs-3251 11 4 the the DET iajs-3251 11 5 fourier fourier NOUN iajs-3251 11 6 and and CCONJ iajs-3251 11 7 laplace laplace NOUN iajs-3251 11 8 transformations transformation NOUN iajs-3251 11 9 theory theory NOUN iajs-3251 11 10 . . PUNCT iajs-3251 12 1 in in ADP iajs-3251 12 2 [ [ X iajs-3251 12 3 1 1 NUM iajs-3251 12 4 ] ] PUNCT iajs-3251 12 5 authors author NOUN iajs-3251 12 6 used use VERB iajs-3251 12 7 this this DET iajs-3251 12 8 technique technique NOUN iajs-3251 12 9 on on ADP iajs-3251 12 10 delay delay NOUN iajs-3251 12 11 differential differential ADJ iajs-3251 12 12 equations equation NOUN iajs-3251 12 13 . . PUNCT iajs-3251 13 1 the the DET iajs-3251 13 2 modified modify VERB iajs-3251 13 3 laplace laplace NOUN iajs-3251 13 4 series series PROPN iajs-3251 13 5 findings finding VERB iajs-3251 13 6 from from ADP iajs-3251 13 7 the the DET iajs-3251 13 8 adomian adomian NOUN iajs-3251 13 9 polynomial polynomial ADJ iajs-3251 13 10 decomposition decomposition NOUN iajs-3251 13 11 method method NOUN iajs-3251 13 12 were be AUX iajs-3251 13 13 used use VERB iajs-3251 13 14 to to PART iajs-3251 13 15 solve solve VERB iajs-3251 13 16 the the DET iajs-3251 13 17 differentialintegral differentialintegral ADJ iajs-3251 13 18 equations equation NOUN iajs-3251 13 19 can can AUX iajs-3251 13 20 be be AUX iajs-3251 13 21 seen see VERB iajs-3251 13 22 in in ADP iajs-3251 13 23 [ [ X iajs-3251 13 24 2 2 NUM iajs-3251 13 25 ] ] PUNCT iajs-3251 13 26 . . PUNCT iajs-3251 14 1 the the PRON iajs-3251 14 2 non non ADJ iajs-3251 14 3 - - ADJ iajs-3251 14 4 linearity linearity NOUN iajs-3251 14 5 was be AUX iajs-3251 14 6 circumvented circumvent VERB iajs-3251 14 7 by by ADP iajs-3251 14 8 using use VERB iajs-3251 14 9 standard standard ADJ iajs-3251 14 10 adomian adomian NOUN iajs-3251 14 11 polynomials polynomial NOUN iajs-3251 14 12 . . PUNCT iajs-3251 15 1 the the DET iajs-3251 15 2 laplace laplace NOUN iajs-3251 15 3 decomposition decomposition NOUN iajs-3251 15 4 approach approach NOUN iajs-3251 15 5 and and CCONJ iajs-3251 15 6 the the DET iajs-3251 15 7 pade pade NOUN iajs-3251 15 8 approximation approximation NOUN iajs-3251 15 9 were be AUX iajs-3251 15 10 put put VERB iajs-3251 15 11 to to PART iajs-3251 15 12 practise practise VERB iajs-3251 15 13 in in ADP iajs-3251 15 14 order order NOUN iajs-3251 15 15 to to PART iajs-3251 15 16 determine determine VERB iajs-3251 15 17 the the DET iajs-3251 15 18 numerically numerically ADV iajs-3251 15 19 approximated approximate VERB iajs-3251 15 20 results result NOUN iajs-3251 15 21 of of ADP iajs-3251 15 22 the the DET iajs-3251 15 23 system system NOUN iajs-3251 15 24 of of ADP iajs-3251 15 25 partial partial ADJ iajs-3251 15 26 on on ADP iajs-3251 15 27 linear linear PROPN iajs-3251 15 28 differential differential ADJ iajs-3251 15 29 equations equation NOUN iajs-3251 15 30 in in ADP iajs-3251 15 31 [ [ X iajs-3251 15 32 3 3 NUM iajs-3251 15 33 ] ] PUNCT iajs-3251 15 34 . . PUNCT iajs-3251 16 1 the the DET iajs-3251 16 2 laplace laplace NOUN iajs-3251 16 3 decomposition decomposition NOUN iajs-3251 16 4 approach approach NOUN iajs-3251 16 5 , , PUNCT iajs-3251 16 6 which which PRON iajs-3251 16 7 was be AUX iajs-3251 16 8 improved improve VERB iajs-3251 16 9 utilised utilise VERB iajs-3251 16 10 in in ADP iajs-3251 16 11 [ [ X iajs-3251 16 12 4 4 NUM iajs-3251 16 13 ] ] PUNCT iajs-3251 16 14 for for ADP iajs-3251 16 15 differential differential ADJ iajs-3251 16 16 equalities equality NOUN iajs-3251 16 17 of of ADP iajs-3251 16 18 the the DET iajs-3251 16 19 emden emden ADJ iajs-3251 16 20 - - PUNCT iajs-3251 16 21 lane lane NOUN iajs-3251 16 22 type type NOUN iajs-3251 16 23 . . PUNCT iajs-3251 17 1 the the DET iajs-3251 17 2 author author NOUN iajs-3251 17 3 used use VERB iajs-3251 17 4 an an DET iajs-3251 17 5 analytical analytical ADJ iajs-3251 17 6 doi.org/10.30526/36.2.3251 doi.org/10.30526/36.2.3251 NOUN iajs-3251 17 7 article article NOUN iajs-3251 17 8 history history NOUN iajs-3251 17 9 : : PUNCT iajs-3251 17 10 received receive VERB iajs-3251 17 11 30 30 NUM iajs-3251 17 12 january january PROPN iajs-3251 17 13 2023 2023 NUM iajs-3251 17 14 , , PUNCT iajs-3251 17 15 accepted accept VERB iajs-3251 17 16 26 26 NUM iajs-3251 17 17 february february NOUN iajs-3251 17 18 2023 2023 NUM iajs-3251 17 19 , , PUNCT iajs-3251 17 20 published publish VERB iajs-3251 17 21 in in ADP iajs-3251 17 22 april april PROPN iajs-3251 17 23 2023 2023 NUM iajs-3251 17 24 . . PUNCT iajs-3251 18 1 ibn ibn PROPN iajs-3251 18 2 al al PROPN iajs-3251 18 3 - - PUNCT iajs-3251 18 4 haitham haitham PROPN iajs-3251 18 5 journal journal PROPN iajs-3251 18 6 for for ADP iajs-3251 18 7 pure pure ADJ iajs-3251 18 8 and and CCONJ iajs-3251 18 9 applied applied ADJ iajs-3251 18 10 sciences sciences PROPN iajs-3251 18 11 journal journal PROPN iajs-3251 18 12 homepage homepage NOUN iajs-3251 18 13 : : PUNCT iajs-3251 18 14 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq NOUN iajs-3251 18 15 solution solution NOUN iajs-3251 18 16 of of ADP iajs-3251 18 17 population population NOUN iajs-3251 18 18 growth growth NOUN iajs-3251 18 19 rate rate NOUN iajs-3251 18 20 linear linear PROPN iajs-3251 18 21 differential differential NOUN iajs-3251 18 22 model model NOUN iajs-3251 18 23 via via ADP iajs-3251 18 24 two two NUM iajs-3251 18 25 parametric parametric ADJ iajs-3251 18 26 see see NOUN iajs-3251 18 27 transformation transformation NOUN iajs-3251 18 28 ali ali PROPN iajs-3251 18 29 moazzam moazzam PROPN iajs-3251 18 30 university university PROPN iajs-3251 18 31 of of ADP iajs-3251 18 32 agriculture agriculture PROPN iajs-3251 18 33 faisalabad faisalabad PROPN iajs-3251 18 34 , , PUNCT iajs-3251 18 35 pakistan pakistan PROPN iajs-3251 18 36 . . PUNCT iajs-3251 19 1 alimoazzam7309723@gmail.com alimoazzam7309723@gmail.com X iajs-3251 20 1 hira hira PROPN iajs-3251 20 2 aslam aslam PROPN iajs-3251 20 3 university university PROPN iajs-3251 20 4 of of ADP iajs-3251 20 5 agriculture agriculture PROPN iajs-3251 20 6 , , PUNCT iajs-3251 20 7 faisalabad faisalabad PROPN iajs-3251 20 8 , , PUNCT iajs-3251 20 9 pakistan pakistan PROPN iajs-3251 20 10 . . PUNCT iajs-3251 21 1 hiraaslam053@gmail.com hiraaslam053@gmail.com X iajs-3251 21 2 . . PUNCT iajs-3251 22 1 naila naila PROPN iajs-3251 22 2 tabassum tabassum PROPN iajs-3251 22 3 university university PROPN iajs-3251 22 4 of of ADP iajs-3251 22 5 agriculture agriculture PROPN iajs-3251 22 6 faisalabad faisalabad PROPN iajs-3251 22 7 , , PUNCT iajs-3251 22 8 pakistan pakistan PROPN iajs-3251 22 9 . . PUNCT iajs-3251 23 1 nailatabassumgp@gmail.com nailatabassumgp@gmail.com X iajs-3251 23 2 emad emad PROPN iajs-3251 23 3 a. a. PROPN iajs-3251 23 4 kuffi kuffi PROPN iajs-3251 23 5 college college PROPN iajs-3251 23 6 of of ADP iajs-3251 23 7 engineering engineering PROPN iajs-3251 23 8 , , PUNCT iajs-3251 23 9 al al PROPN iajs-3251 23 10 - - PUNCT iajs-3251 23 11 qadisiyah qadisiyah PROPN iajs-3251 23 12 university university NOUN iajs-3251 23 13 , , PUNCT iajs-3251 23 14 , , PUNCT iajs-3251 23 15 iraq iraq PROPN iajs-3251 23 16 . . PUNCT iajs-3251 24 1 emad.abbas@qu.edu.iq emad.abbas@qu.edu.iq PROPN iajs-3251 24 2 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3251 24 3 mailto:alimoazzam7309723@gmail.com mailto:alimoazzam7309723@gmail.com PROPN iajs-3251 24 4 mailto:hiraaslam053@gmail.com mailto:hiraaslam053@gmail.com PROPN iajs-3251 24 5 mailto:nailatabassumgp@gmail.com mailto:nailatabassumgp@gmail.com PROPN iajs-3251 25 1 mailto:emad.abbas@qu.edu.iq mailto:emad.abbas@qu.edu.iq PROPN iajs-3251 25 2 ihjpas ihjpa VERB iajs-3251 25 3 . . PUNCT iajs-3251 26 1 36(2)2023 36(2)2023 NUM iajs-3251 26 2 431 431 NUM iajs-3251 26 3 approach approach NOUN iajs-3251 26 4 to to PART iajs-3251 26 5 resolve resolve VERB iajs-3251 26 6 the the DET iajs-3251 26 7 first first ADJ iajs-3251 26 8 and and CCONJ iajs-3251 26 9 second second ADJ iajs-3251 26 10 types type NOUN iajs-3251 26 11 of of ADP iajs-3251 26 12 non non ADJ iajs-3251 26 13 - - ADJ iajs-3251 26 14 linear linear ADJ iajs-3251 26 15 volterra volterra NOUN iajs-3251 26 16 - - PUNCT iajs-3251 26 17 fredholm fredholm NOUN iajs-3251 26 18 integro integro PROPN iajs-3251 26 19 differential differential ADJ iajs-3251 26 20 equations equation NOUN iajs-3251 26 21 in in ADP iajs-3251 26 22 [ [ X iajs-3251 26 23 5 5 NUM iajs-3251 26 24 ] ] PUNCT iajs-3251 26 25 by by ADP iajs-3251 26 26 combining combine VERB iajs-3251 26 27 the the DET iajs-3251 26 28 modified modify VERB iajs-3251 26 29 laplace laplace NOUN iajs-3251 26 30 adomian adomian NOUN iajs-3251 26 31 substitution substitution NOUN iajs-3251 26 32 and and CCONJ iajs-3251 26 33 laplace laplace NOUN iajs-3251 26 34 adomian adomian NOUN iajs-3251 26 35 polynomial polynomial ADJ iajs-3251 26 36 decomposition decomposition NOUN iajs-3251 26 37 methods method NOUN iajs-3251 26 38 ( ( PUNCT iajs-3251 26 39 ladm ladm PROPN iajs-3251 26 40 ) ) PUNCT iajs-3251 26 41 . . PUNCT iajs-3251 27 1 laplace laplace PROPN iajs-3251 27 2 integral integral ADJ iajs-3251 27 3 techniques technique NOUN iajs-3251 27 4 are be AUX iajs-3251 27 5 frequently frequently ADV iajs-3251 27 6 utilised utilise VERB iajs-3251 27 7 in in ADP iajs-3251 27 8 the the DET iajs-3251 27 9 engineering engineering NOUN iajs-3251 27 10 and and CCONJ iajs-3251 27 11 scientific scientific ADJ iajs-3251 27 12 fields field NOUN iajs-3251 27 13 . . PUNCT iajs-3251 28 1 the the DET iajs-3251 28 2 use use NOUN iajs-3251 28 3 of of ADP iajs-3251 28 4 integral integral ADJ iajs-3251 28 5 transformations transformation NOUN iajs-3251 28 6 to to PART iajs-3251 28 7 correct correct VERB iajs-3251 28 8 erroneous erroneous ADJ iajs-3251 28 9 integrals integral NOUN iajs-3251 28 10 that that PRON iajs-3251 28 11 contained contain VERB iajs-3251 28 12 error error NOUN iajs-3251 28 13 functions function NOUN iajs-3251 28 14 has have AUX iajs-3251 28 15 been be AUX iajs-3251 28 16 covered cover VERB iajs-3251 28 17 by by ADP iajs-3251 28 18 a a DET iajs-3251 28 19 number number NOUN iajs-3251 28 20 of of ADP iajs-3251 28 21 authors author NOUN iajs-3251 28 22 [ [ X iajs-3251 28 23 6–9 6–9 NOUN iajs-3251 28 24 ] ] PUNCT iajs-3251 28 25 . . PUNCT iajs-3251 29 1 in in ADP iajs-3251 29 2 [ [ X iajs-3251 29 3 10 10 NUM iajs-3251 29 4 ] ] PUNCT iajs-3251 29 5 authors author NOUN iajs-3251 29 6 obtained obtain VERB iajs-3251 29 7 the the DET iajs-3251 29 8 solutions solution NOUN iajs-3251 29 9 to to ADP iajs-3251 29 10 abel abel PROPN iajs-3251 29 11 's 's PART iajs-3251 29 12 equation equation NOUN iajs-3251 29 13 using use VERB iajs-3251 29 14 the the DET iajs-3251 29 15 kamal kamal ADJ iajs-3251 29 16 integral integral ADJ iajs-3251 29 17 method method NOUN iajs-3251 29 18 . . PUNCT iajs-3251 30 1 solution solution NOUN iajs-3251 30 2 to to ADP iajs-3251 30 3 the the DET iajs-3251 30 4 error error NOUN iajs-3251 30 5 function function NOUN iajs-3251 30 6 based base VERB iajs-3251 30 7 on on ADP iajs-3251 30 8 the the DET iajs-3251 30 9 kamal kamal ADJ iajs-3251 30 10 integral integral ADJ iajs-3251 30 11 approach approach NOUN iajs-3251 30 12 has have AUX iajs-3251 30 13 shown show VERB iajs-3251 30 14 in in ADP iajs-3251 30 15 [ [ X iajs-3251 30 16 11 11 NUM iajs-3251 30 17 ] ] PUNCT iajs-3251 30 18 . . PUNCT iajs-3251 31 1 to to PART iajs-3251 31 2 solve solve VERB iajs-3251 31 3 linear linear ADJ iajs-3251 31 4 partial partial ADJ iajs-3251 31 5 differential differential NOUN iajs-3251 31 6 equations equation NOUN iajs-3251 31 7 with with ADP iajs-3251 31 8 more more ADJ iajs-3251 31 9 than than ADP iajs-3251 31 10 two two NUM iajs-3251 31 11 independent independent ADJ iajs-3251 31 12 variables variable NOUN iajs-3251 31 13 , , PUNCT iajs-3251 31 14 writers writer NOUN iajs-3251 31 15 in in ADP iajs-3251 31 16 [ [ X iajs-3251 31 17 12 12 NUM iajs-3251 31 18 ] ] PUNCT iajs-3251 31 19 described describe VERB iajs-3251 31 20 the the DET iajs-3251 31 21 substitution substitution NOUN iajs-3251 31 22 approach approach NOUN iajs-3251 31 23 of of ADP iajs-3251 31 24 the the DET iajs-3251 31 25 laplace laplace NOUN iajs-3251 31 26 transformation transformation NOUN iajs-3251 31 27 . . PUNCT iajs-3251 32 1 in in ADP iajs-3251 32 2 [ [ X iajs-3251 32 3 13 13 NUM iajs-3251 32 4 ] ] PUNCT iajs-3251 32 5 authors author NOUN iajs-3251 32 6 used use VERB iajs-3251 32 7 the the DET iajs-3251 32 8 matlab matlab PROPN iajs-3251 32 9 programme programme NOUN iajs-3251 32 10 to to PART iajs-3251 32 11 illustrate illustrate VERB iajs-3251 32 12 how how SCONJ iajs-3251 32 13 the the DET iajs-3251 32 14 effects effect NOUN iajs-3251 32 15 of of ADP iajs-3251 32 16 bio bio NOUN iajs-3251 32 17 - - ADJ iajs-3251 32 18 convection convection ADJ iajs-3251 32 19 and and CCONJ iajs-3251 32 20 activation activation NOUN iajs-3251 32 21 energy energy NOUN iajs-3251 32 22 on on ADP iajs-3251 32 23 the the DET iajs-3251 32 24 maxwell maxwell PROPN iajs-3251 32 25 equation equation NOUN iajs-3251 32 26 on on ADP iajs-3251 32 27 nano nano NOUN iajs-3251 32 28 - - PUNCT iajs-3251 32 29 fluid fluid NOUN iajs-3251 32 30 . . PUNCT iajs-3251 33 1 with with ADP iajs-3251 33 2 the the DET iajs-3251 33 3 use use NOUN iajs-3251 33 4 of of ADP iajs-3251 33 5 the the DET iajs-3251 33 6 laplace laplace NOUN iajs-3251 33 7 technique technique NOUN iajs-3251 33 8 of of ADP iajs-3251 33 9 transformation transformation NOUN iajs-3251 33 10 , , PUNCT iajs-3251 33 11 an an DET iajs-3251 33 12 analytical analytical ADJ iajs-3251 33 13 solution solution NOUN iajs-3251 33 14 to to ADP iajs-3251 33 15 the the DET iajs-3251 33 16 advection advection NOUN iajs-3251 33 17 diffusion diffusion NOUN iajs-3251 33 18 equation equation NOUN iajs-3251 33 19 in in ADP iajs-3251 33 20 a a DET iajs-3251 33 21 one one NUM iajs-3251 33 22 - - PUNCT iajs-3251 33 23 dimensional dimensional ADJ iajs-3251 33 24 , , PUNCT iajs-3251 33 25 semi semi ADJ iajs-3251 33 26 - - ADJ iajs-3251 33 27 infinite infinite ADJ iajs-3251 33 28 media medium NOUN iajs-3251 33 29 has have AUX iajs-3251 33 30 been be AUX iajs-3251 33 31 developed develop VERB iajs-3251 33 32 by by ADP iajs-3251 33 33 authors author NOUN iajs-3251 33 34 of of ADP iajs-3251 33 35 [ [ X iajs-3251 33 36 14 14 NUM iajs-3251 33 37 ] ] PUNCT iajs-3251 33 38 . . PUNCT iajs-3251 34 1 in in ADP iajs-3251 34 2 [ [ X iajs-3251 34 3 15 15 NUM iajs-3251 34 4 and and CCONJ iajs-3251 34 5 16 16 NUM iajs-3251 34 6 ] ] X iajs-3251 34 7 new new ADJ iajs-3251 34 8 transformations transformation NOUN iajs-3251 34 9 to to PART iajs-3251 34 10 solve solve VERB iajs-3251 34 11 new new ADJ iajs-3251 34 12 type type NOUN iajs-3251 34 13 differential differential ADJ iajs-3251 34 14 equations equation NOUN iajs-3251 34 15 with with ADP iajs-3251 34 16 trigonometric trigonometric ADJ iajs-3251 34 17 coefficients coefficient NOUN iajs-3251 34 18 was be AUX iajs-3251 34 19 introduced introduce VERB iajs-3251 34 20 in in ADP iajs-3251 34 21 2021 2021 NUM iajs-3251 34 22 bymoazzam bymoazzam VERB iajs-3251 34 23 et et PROPN iajs-3251 34 24 . . PUNCT iajs-3251 35 1 al al PROPN iajs-3251 35 2 . . PUNCT iajs-3251 36 1 the the DET iajs-3251 36 2 solution solution NOUN iajs-3251 36 3 to to ADP iajs-3251 36 4 the the DET iajs-3251 36 5 temperature temperature NOUN iajs-3251 36 6 problem problem NOUN iajs-3251 36 7 and and CCONJ iajs-3251 36 8 some some PRON iajs-3251 36 9 of of ADP iajs-3251 36 10 the the DET iajs-3251 36 11 applications application NOUN iajs-3251 36 12 of of ADP iajs-3251 36 13 the the DET iajs-3251 36 14 kamal kamal PROPN iajs-3251 36 15 transformation transformation NOUN iajs-3251 36 16 in in ADP iajs-3251 36 17 thermal thermal ADJ iajs-3251 36 18 engineering engineering NOUN iajs-3251 36 19 are be AUX iajs-3251 36 20 shown show VERB iajs-3251 36 21 in in ADP iajs-3251 36 22 [ [ X iajs-3251 36 23 17 17 NUM iajs-3251 36 24 ] ] PUNCT iajs-3251 36 25 . . PUNCT iajs-3251 37 1 the the DET iajs-3251 37 2 see see ADJ iajs-3251 37 3 transformation transformation NOUN iajs-3251 37 4 , , PUNCT iajs-3251 37 5 a a DET iajs-3251 37 6 new new ADJ iajs-3251 37 7 transformation transformation NOUN iajs-3251 37 8 that that PRON iajs-3251 37 9 is be AUX iajs-3251 37 10 very very ADV iajs-3251 37 11 useful useful ADJ iajs-3251 37 12 in in ADP iajs-3251 37 13 the the DET iajs-3251 37 14 solution solution NOUN iajs-3251 37 15 of of ADP iajs-3251 37 16 differential differential ADJ iajs-3251 37 17 equations equation NOUN iajs-3251 37 18 and and CCONJ iajs-3251 37 19 systems system NOUN iajs-3251 37 20 of of ADP iajs-3251 37 21 differential differential ADJ iajs-3251 37 22 equations equation NOUN iajs-3251 37 23 , , PUNCT iajs-3251 37 24 has have AUX iajs-3251 37 25 been be AUX iajs-3251 37 26 demonstrated demonstrate VERB iajs-3251 37 27 in in ADP iajs-3251 37 28 [ [ X iajs-3251 37 29 18 18 NUM iajs-3251 37 30 ] ] PUNCT iajs-3251 37 31 . . PUNCT iajs-3251 38 1 some some DET iajs-3251 38 2 others other NOUN iajs-3251 38 3 transformations transformation NOUN iajs-3251 38 4 having have VERB iajs-3251 38 5 different different ADJ iajs-3251 38 6 kernels kernel NOUN iajs-3251 38 7 then then ADV iajs-3251 38 8 laplace laplace NOUN iajs-3251 38 9 transformation transformation NOUN iajs-3251 38 10 are be AUX iajs-3251 38 11 used use VERB iajs-3251 38 12 to to PART iajs-3251 38 13 find find VERB iajs-3251 38 14 the the DET iajs-3251 38 15 solutions solution NOUN iajs-3251 38 16 of of ADP iajs-3251 38 17 linear linear PROPN iajs-3251 38 18 differential differential ADJ iajs-3251 38 19 equations equation NOUN iajs-3251 38 20 as as SCONJ iajs-3251 38 21 al al PROPN iajs-3251 38 22 - - PUNCT iajs-3251 38 23 zughair zughair PROPN iajs-3251 38 24 transformation transformation NOUN iajs-3251 38 25 is be AUX iajs-3251 38 26 used use VERB iajs-3251 38 27 to to PART iajs-3251 38 28 demonstrate demonstrate VERB iajs-3251 38 29 the the DET iajs-3251 38 30 solution solution NOUN iajs-3251 38 31 of of ADP iajs-3251 38 32 moment moment NOUN iajs-3251 38 33 pareto pareto ADJ iajs-3251 38 34 differential differential ADJ iajs-3251 38 35 equations equation NOUN iajs-3251 38 36 with with ADP iajs-3251 38 37 logarithmic logarithmic ADJ iajs-3251 38 38 coefficients coefficient NOUN iajs-3251 38 39 in in ADP iajs-3251 38 40 [ [ X iajs-3251 38 41 19 19 NUM iajs-3251 38 42 ] ] PUNCT iajs-3251 38 43 . . PUNCT iajs-3251 39 1 in in ADP iajs-3251 39 2 paper paper NOUN iajs-3251 39 3 [ [ X iajs-3251 39 4 20 20 NUM iajs-3251 39 5 ] ] PUNCT iajs-3251 39 6 , , PUNCT iajs-3251 39 7 a a DET iajs-3251 39 8 new new ADJ iajs-3251 39 9 logarithmic logarithmic ADJ iajs-3251 39 10 kernel kernel NOUN iajs-3251 39 11 transformation transformation NOUN iajs-3251 39 12 was be AUX iajs-3251 39 13 demonstrated demonstrate VERB iajs-3251 39 14 which which PRON iajs-3251 39 15 is be AUX iajs-3251 39 16 useful useful ADJ iajs-3251 39 17 for for ADP iajs-3251 39 18 solving solve VERB iajs-3251 39 19 both both PRON iajs-3251 39 20 ordinary ordinary ADJ iajs-3251 39 21 differential differential ADJ iajs-3251 39 22 equations equation NOUN iajs-3251 39 23 and and CCONJ iajs-3251 39 24 differential differential ADJ iajs-3251 39 25 equations equation NOUN iajs-3251 39 26 with with ADP iajs-3251 39 27 a a DET iajs-3251 39 28 logarithmic logarithmic ADJ iajs-3251 39 29 coefficient coefficient NOUN iajs-3251 39 30 . . PUNCT iajs-3251 40 1 2.definitions 2.definitions NUM iajs-3251 40 2 and and CCONJ iajs-3251 40 3 preliminaries preliminary NOUN iajs-3251 40 4 in in ADP iajs-3251 40 5 this this DET iajs-3251 40 6 paper paper NOUN iajs-3251 40 7 , , PUNCT iajs-3251 40 8 we we PRON iajs-3251 40 9 have have AUX iajs-3251 40 10 introduced introduce VERB iajs-3251 40 11 the the DET iajs-3251 40 12 two two NUM iajs-3251 40 13 parametric parametric ADJ iajs-3251 40 14 see see NOUN iajs-3251 40 15 transformation transformation NOUN iajs-3251 40 16 , , PUNCT iajs-3251 40 17 which which PRON iajs-3251 40 18 is be AUX iajs-3251 40 19 a a DET iajs-3251 40 20 generalised generalised ADJ iajs-3251 40 21 version version NOUN iajs-3251 40 22 of of ADP iajs-3251 40 23 the the DET iajs-3251 40 24 see see ADJ iajs-3251 40 25 transformation transformation NOUN iajs-3251 40 26 and and CCONJ iajs-3251 40 27 is be AUX iajs-3251 40 28 used use VERB iajs-3251 40 29 in in ADP iajs-3251 40 30 this this DET iajs-3251 40 31 study study NOUN iajs-3251 40 32 to to PART iajs-3251 40 33 solve solve VERB iajs-3251 40 34 linear linear ADJ iajs-3251 40 35 differential differential NOUN iajs-3251 40 36 equations equation NOUN iajs-3251 40 37 . . PUNCT iajs-3251 41 1 the the DET iajs-3251 41 2 definition definition NOUN iajs-3251 41 3 of of ADP iajs-3251 41 4 two two NUM iajs-3251 41 5 parametric parametric ADJ iajs-3251 41 6 see see NOUN iajs-3251 41 7 transformations transformation NOUN iajs-3251 41 8 is be AUX iajs-3251 41 9 given give VERB iajs-3251 41 10 below below ADV iajs-3251 41 11 . . PUNCT iajs-3251 42 1 definition definition NOUN iajs-3251 42 2 1 1 NUM iajs-3251 42 3 : : PUNCT iajs-3251 42 4 two two NUM iajs-3251 42 5 parametric parametric ADJ iajs-3251 42 6 see see VERB iajs-3251 42 7 integral integral ADJ iajs-3251 42 8 transform transform NOUN iajs-3251 42 9 of of ADP iajs-3251 42 10 the the DET iajs-3251 42 11 function function NOUN iajs-3251 42 12 𝑓(𝑑 𝑓(𝑑 PROPN iajs-3251 42 13 ) ) PUNCT iajs-3251 42 14 for for SCONJ iajs-3251 42 15 all all DET iajs-3251 42 16 𝑑 𝑑 PROPN iajs-3251 42 17 β‰₯ β‰₯ X iajs-3251 42 18 0 0 NUM iajs-3251 42 19 is be AUX iajs-3251 42 20 defined define VERB iajs-3251 42 21 as as ADP iajs-3251 42 22 : : PUNCT iajs-3251 42 23 𝑆𝐸𝐸𝛼,𝛽{𝑓(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓(𝑑 NUM iajs-3251 42 24 ) ) PUNCT iajs-3251 42 25 } } PUNCT iajs-3251 42 26 = = SYM iajs-3251 43 1 π‘’βˆ’π›½π’— π‘’βˆ’π›½π’— PROPN iajs-3251 43 2 ( ( PUNCT iajs-3251 43 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 43 4 ∫ ∫ PROPN iajs-3251 43 5 𝑓(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 𝑓(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 PROPN iajs-3251 43 6 ∞ ∞ PROPN iajs-3251 43 7 0 0 PUNCT iajs-3251 44 1 = = SYM iajs-3251 44 2 𝐹 𝐹 PROPN iajs-3251 44 3 𝛼,𝛽(𝑣 𝛼,𝛽(𝑣 NUM iajs-3251 44 4 ) ) PUNCT iajs-3251 44 5 ( ( PUNCT iajs-3251 44 6 1 1 X iajs-3251 44 7 ) ) PUNCT iajs-3251 44 8 where where SCONJ iajs-3251 44 9 𝛼 𝛼 X iajs-3251 44 10 , , PUNCT iajs-3251 44 11 𝛽 𝛽 NOUN iajs-3251 44 12 are be AUX iajs-3251 44 13 real real ADJ iajs-3251 44 14 numbers number NOUN iajs-3251 44 15 , , PUNCT iajs-3251 44 16 𝑛 𝑛 PRON iajs-3251 44 17 is be AUX iajs-3251 44 18 the the DET iajs-3251 44 19 integer integer NOUN iajs-3251 44 20 and and CCONJ iajs-3251 44 21 𝑣 𝑣 PROPN iajs-3251 44 22 is be AUX iajs-3251 44 23 transformed transform VERB iajs-3251 44 24 variable variable ADJ iajs-3251 44 25 . . PUNCT iajs-3251 45 1 by by ADP iajs-3251 45 2 choosing choose VERB iajs-3251 45 3 the the DET iajs-3251 45 4 values value NOUN iajs-3251 45 5 of of ADP iajs-3251 45 6 parameters parameter NOUN iajs-3251 45 7 𝛼 𝛼 VERB iajs-3251 45 8 , , PUNCT iajs-3251 45 9 𝛽 𝛽 DET iajs-3251 45 10 real real ADJ iajs-3251 45 11 numbers number NOUN iajs-3251 45 12 and and CCONJ iajs-3251 45 13 integer integer PROPN iajs-3251 45 14 𝑛 𝑛 PROPN iajs-3251 45 15 , , PUNCT iajs-3251 45 16 we we PRON iajs-3251 45 17 can can AUX iajs-3251 45 18 get get VERB iajs-3251 45 19 the the DET iajs-3251 45 20 different different ADJ iajs-3251 45 21 transformations transformation NOUN iajs-3251 45 22 as as ADP iajs-3251 45 23 laplace laplace NOUN iajs-3251 45 24 , , PUNCT iajs-3251 45 25 kamal kamal PROPN iajs-3251 45 26 , , PUNCT iajs-3251 45 27 mohand mohand NOUN iajs-3251 45 28 , , PUNCT iajs-3251 45 29 see see VERB iajs-3251 45 30 , , PUNCT iajs-3251 45 31 sumudu sumudu NOUN iajs-3251 45 32 , , PUNCT iajs-3251 45 33 aboodh aboodh ADJ iajs-3251 45 34 , , PUNCT iajs-3251 45 35 and and CCONJ iajs-3251 45 36 many many ADJ iajs-3251 45 37 other other ADJ iajs-3251 45 38 laplace laplace NOUN iajs-3251 45 39 type type NOUN iajs-3251 45 40 transformations transformation NOUN iajs-3251 45 41 . . PUNCT iajs-3251 46 1 ihjpas ihjpas PROPN iajs-3251 46 2 . . PUNCT iajs-3251 47 1 36(2)2023 36(2)2023 NUM iajs-3251 47 2 432 432 NUM iajs-3251 47 3 table table NOUN iajs-3251 47 4 1 1 NUM iajs-3251 47 5 : : PUNCT iajs-3251 47 6 two two NUM iajs-3251 47 7 parametric parametric ADJ iajs-3251 47 8 see see VERB iajs-3251 47 9 transformation transformation NOUN iajs-3251 47 10 of of ADP iajs-3251 47 11 some some DET iajs-3251 47 12 function function NOUN iajs-3251 47 13 serial serial ADJ iajs-3251 47 14 numbers number NOUN iajs-3251 47 15 function function VERB iajs-3251 47 16 𝒇(𝒕 𝒇(𝒕 NUM iajs-3251 47 17 ) ) PUNCT iajs-3251 47 18 π‘Ίπ‘¬π‘¬πœΆ,𝜷{𝒇(𝒕 π‘Ίπ‘¬π‘¬πœΆ,𝜷{𝒇(𝒕 NUM iajs-3251 47 19 ) ) PUNCT iajs-3251 47 20 } } PUNCT iajs-3251 48 1 = = SYM iajs-3251 48 2 π’†βˆ’πœ·π’— π’†βˆ’πœ·π’— PROPN iajs-3251 48 3 ( ( PUNCT iajs-3251 48 4 πœΆπ’—)𝒏 πœΆπ’—)𝒏 PROPN iajs-3251 48 5 ∫ ∫ PROPN iajs-3251 48 6 𝒇(𝒕)π’†βˆ’(πœΆπ’—)𝒕𝒅𝒕 𝒇(𝒕)π’†βˆ’(πœΆπ’—)𝒕𝒅𝒕 NUM iajs-3251 48 7 ∞ ∞ PROPN iajs-3251 48 8 𝟎 𝟎 NUM iajs-3251 48 9 1 1 NUM iajs-3251 48 10 . . PUNCT iajs-3251 49 1 π‘˜ π‘˜ DET iajs-3251 49 2 π‘’βˆ’π›½π‘£π‘˜ π‘’βˆ’π›½π‘£π‘˜ NOUN iajs-3251 49 3 ( ( PUNCT iajs-3251 49 4 𝛼𝑣)𝑛+1 𝛼𝑣)𝑛+1 NOUN iajs-3251 49 5 2 2 NUM iajs-3251 49 6 . . NOUN iajs-3251 49 7 1 1 NUM iajs-3251 49 8 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 49 9 ( ( PUNCT iajs-3251 49 10 𝛼𝑣)𝑛+1 𝛼𝑣)𝑛+1 NOUN iajs-3251 49 11 3 3 X iajs-3251 49 12 . . PUNCT iajs-3251 49 13 𝑑 𝑑 X iajs-3251 49 14 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 49 15 ( ( PUNCT iajs-3251 49 16 𝛼𝑣)𝑛+2 𝛼𝑣)𝑛+2 NOUN iajs-3251 49 17 4 4 X iajs-3251 49 18 . . PUNCT iajs-3251 49 19 𝑑2 𝑑2 PROPN iajs-3251 49 20 π‘’βˆ’π›½π‘£2 π‘’βˆ’π›½π‘£2 NOUN iajs-3251 49 21 ! ! PUNCT iajs-3251 50 1 ( ( PUNCT iajs-3251 50 2 𝛼𝑣)𝑛+3 𝛼𝑣)𝑛+3 NOUN iajs-3251 50 3 5 5 NUM iajs-3251 50 4 . . PUNCT iajs-3251 50 5 π‘‘π‘š π‘‘π‘š VERB iajs-3251 50 6 π‘’βˆ’π›½π‘£π‘š π‘’βˆ’π›½π‘£π‘š PROPN iajs-3251 50 7 ! ! PUNCT iajs-3251 51 1 ( ( PUNCT iajs-3251 51 2 𝛼𝑣)𝑛+π‘š+1 𝛼𝑣)𝑛+π‘š+1 NUM iajs-3251 51 3 6 6 NUM iajs-3251 51 4 . . PUNCT iajs-3251 51 5 𝑒𝑒𝑑 𝑒𝑒𝑑 VERB iajs-3251 51 6 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 51 7 ( ( PUNCT iajs-3251 51 8 𝛼𝑣)𝑛(𝛼𝑣 𝛼𝑣)𝑛(𝛼𝑣 NOUN iajs-3251 51 9 βˆ’ βˆ’ NOUN iajs-3251 51 10 𝑒 𝑒 NOUN iajs-3251 51 11 ) ) PUNCT iajs-3251 51 12 7 7 NUM iajs-3251 52 1 . . X iajs-3251 52 2 𝑠𝑖𝑛 𝑠𝑖𝑛 PROPN iajs-3251 52 3 ( ( PUNCT iajs-3251 52 4 𝑒𝑑 𝑒𝑑 NOUN iajs-3251 52 5 ) ) PUNCT iajs-3251 52 6 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 52 7 ( ( PUNCT iajs-3251 52 8 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 52 9 [ [ PUNCT iajs-3251 52 10 𝑒 𝑒 X iajs-3251 52 11 ( ( PUNCT iajs-3251 52 12 𝛼𝑣)2 𝛼𝑣)2 PROPN iajs-3251 52 13 + + CCONJ iajs-3251 52 14 𝑒2 𝑒2 PROPN iajs-3251 52 15 ] ] PUNCT iajs-3251 52 16 8 8 NUM iajs-3251 52 17 . . PUNCT iajs-3251 52 18 π‘π‘œπ‘ (𝑒𝑑 π‘π‘œπ‘ (𝑒𝑑 NOUN iajs-3251 52 19 ) ) PUNCT iajs-3251 53 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 53 2 ( ( PUNCT iajs-3251 53 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 53 4 [ [ PUNCT iajs-3251 53 5 𝛼𝑣 𝛼𝑣 PROPN iajs-3251 53 6 ( ( PUNCT iajs-3251 53 7 𝛼𝑣)2 𝛼𝑣)2 PROPN iajs-3251 53 8 + + CCONJ iajs-3251 53 9 𝑒2 𝑒2 PROPN iajs-3251 53 10 ] ] PUNCT iajs-3251 53 11 9 9 X iajs-3251 53 12 . . X iajs-3251 53 13 π‘π‘œπ‘ β„Ž π‘π‘œπ‘ β„Ž NOUN iajs-3251 53 14 ( ( PUNCT iajs-3251 53 15 𝑒𝑑 𝑒𝑑 NOUN iajs-3251 53 16 ) ) PUNCT iajs-3251 53 17 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 53 18 ( ( PUNCT iajs-3251 53 19 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 53 20 [ [ PUNCT iajs-3251 53 21 𝑒 𝑒 X iajs-3251 53 22 ( ( PUNCT iajs-3251 53 23 𝛼𝑣)2 𝛼𝑣)2 PROPN iajs-3251 53 24 + + CCONJ iajs-3251 53 25 𝑒2 𝑒2 PROPN iajs-3251 53 26 ] ] PUNCT iajs-3251 53 27 10 10 NUM iajs-3251 53 28 . . PUNCT iajs-3251 54 1 π‘ π‘–π‘›β„Ž π‘ π‘–π‘›β„Ž PROPN iajs-3251 54 2 ( ( PUNCT iajs-3251 54 3 𝑒𝑑 𝑒𝑑 PROPN iajs-3251 54 4 ) ) PUNCT iajs-3251 54 5 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 54 6 ( ( PUNCT iajs-3251 54 7 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 54 8 [ [ PUNCT iajs-3251 54 9 𝛼𝑣 𝛼𝑣 PROPN iajs-3251 54 10 ( ( PUNCT iajs-3251 54 11 𝛼𝑣)2 𝛼𝑣)2 PROPN iajs-3251 54 12 βˆ’ βˆ’ PROPN iajs-3251 54 13 𝑒2 𝑒2 PROPN iajs-3251 54 14 ] ] PUNCT iajs-3251 54 15 theorem theorem VERB iajs-3251 54 16 1 1 NUM iajs-3251 54 17 : : PUNCT iajs-3251 54 18 two two NUM iajs-3251 54 19 parametric parametric ADJ iajs-3251 54 20 see see NOUN iajs-3251 54 21 transform transform NOUN iajs-3251 54 22 for for ADP iajs-3251 54 23 first first ADJ iajs-3251 54 24 , , PUNCT iajs-3251 54 25 second second ADJ iajs-3251 54 26 up up ADP iajs-3251 54 27 - - PUNCT iajs-3251 54 28 to to NOUN iajs-3251 54 29 nth nth NOUN iajs-3251 54 30 order order NOUN iajs-3251 54 31 derivative derivative NOUN iajs-3251 54 32 can can AUX iajs-3251 54 33 be be AUX iajs-3251 54 34 described describe VERB iajs-3251 54 35 as as ADP iajs-3251 54 36 : : PUNCT iajs-3251 54 37 1 1 NUM iajs-3251 54 38 . . X iajs-3251 54 39 𝑆𝐸𝐸𝛼,𝛽{𝑓′(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓′(𝑑 NOUN iajs-3251 54 40 ) ) PUNCT iajs-3251 54 41 } } PUNCT iajs-3251 54 42 = = PUNCT iajs-3251 55 1 βˆ’ βˆ’ NOUN iajs-3251 55 2 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 55 3 ( ( PUNCT iajs-3251 55 4 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 55 5 𝑓(0 𝑓(0 PROPN iajs-3251 55 6 ) ) PUNCT iajs-3251 55 7 + + CCONJ iajs-3251 55 8 ( ( PUNCT iajs-3251 55 9 𝛼𝑣)𝐹𝛼,𝛽(𝑣 𝛼𝑣)𝐹𝛼,𝛽(𝑣 NOUN iajs-3251 55 10 ) ) PUNCT iajs-3251 55 11 , , PUNCT iajs-3251 55 12 2 2 NUM iajs-3251 55 13 . . X iajs-3251 55 14 𝑆𝐸𝐸𝛼,𝛽{𝑓′′(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓′′(𝑑 NUM iajs-3251 55 15 ) ) PUNCT iajs-3251 55 16 } } PUNCT iajs-3251 55 17 = = PUNCT iajs-3251 55 18 βˆ’ βˆ’ NOUN iajs-3251 56 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 56 2 ( ( PUNCT iajs-3251 56 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 56 4 𝑓′(0 𝑓′(0 PROPN iajs-3251 56 5 ) ) PUNCT iajs-3251 56 6 βˆ’ βˆ’ ADV iajs-3251 57 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 57 2 ( ( PUNCT iajs-3251 57 3 𝛼𝑣)π‘›βˆ’1 𝛼𝑣)π‘›βˆ’1 PROPN iajs-3251 57 4 𝑓(0 𝑓(0 PROPN iajs-3251 57 5 ) ) PUNCT iajs-3251 57 6 + + CCONJ iajs-3251 57 7 ( ( PUNCT iajs-3251 57 8 𝛼𝑣)2𝐹𝛼,𝛽(𝑣 𝛼𝑣)2𝐹𝛼,𝛽(𝑣 PROPN iajs-3251 57 9 ) ) PUNCT iajs-3251 57 10 , , PUNCT iajs-3251 57 11 3 3 X iajs-3251 57 12 . . X iajs-3251 57 13 𝑆𝐸𝐸𝛼,𝛽{𝑓′′′(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓′′′(𝑑 NOUN iajs-3251 57 14 ) ) PUNCT iajs-3251 57 15 } } PUNCT iajs-3251 57 16 = = PUNCT iajs-3251 57 17 βˆ’ βˆ’ NOUN iajs-3251 57 18 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 57 19 ( ( PUNCT iajs-3251 57 20 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 57 21 𝑓′′(0 𝑓′′(0 NOUN iajs-3251 57 22 ) ) PUNCT iajs-3251 57 23 βˆ’ βˆ’ ADP iajs-3251 58 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 58 2 ( ( PUNCT iajs-3251 58 3 𝛼𝑣)π‘›βˆ’1 𝛼𝑣)π‘›βˆ’1 PROPN iajs-3251 58 4 𝑓′(0 𝑓′(0 PROPN iajs-3251 58 5 ) ) PUNCT iajs-3251 58 6 βˆ’ βˆ’ ADP iajs-3251 58 7 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 58 8 ( ( PUNCT iajs-3251 58 9 𝛼𝑣)π‘›βˆ’2 𝛼𝑣)π‘›βˆ’2 PROPN iajs-3251 58 10 𝑓(0 𝑓(0 PROPN iajs-3251 58 11 ) ) PUNCT iajs-3251 59 1 + + CCONJ iajs-3251 59 2 ( ( PUNCT iajs-3251 59 3 𝛼𝑣)3𝐹𝛼,𝛽(𝑣 𝛼𝑣)3𝐹𝛼,𝛽(𝑣 NOUN iajs-3251 59 4 ) ) PUNCT iajs-3251 59 5 , , PUNCT iajs-3251 59 6 4 4 NUM iajs-3251 59 7 . . PUNCT iajs-3251 59 8 𝑆𝐸𝐸𝛼,𝛽{𝑓(π‘š)(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓(π‘š)(𝑑 ADV iajs-3251 59 9 ) ) PUNCT iajs-3251 59 10 } } PUNCT iajs-3251 59 11 = = PUNCT iajs-3251 60 1 βˆ’ βˆ’ NOUN iajs-3251 60 2 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 60 3 ( ( PUNCT iajs-3251 60 4 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 60 5 𝑓(π‘šβˆ’1)(0 𝑓(π‘šβˆ’1)(0 NOUN iajs-3251 60 6 ) ) PUNCT iajs-3251 60 7 βˆ’ βˆ’ NOUN iajs-3251 61 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 62 1 ( ( PUNCT iajs-3251 62 2 𝛼𝑣)π‘›βˆ’1 𝛼𝑣)π‘›βˆ’1 PROPN iajs-3251 62 3 𝑓(π‘šβˆ’2)(0 𝑓(π‘šβˆ’2)(0 X iajs-3251 62 4 ) ) PUNCT iajs-3251 62 5 βˆ’ βˆ’ PROPN iajs-3251 62 6 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 62 7 ( ( PUNCT iajs-3251 62 8 𝛼𝑣)π‘›βˆ’2 𝛼𝑣)π‘›βˆ’2 PROPN iajs-3251 62 9 𝑓(π‘šβˆ’3)(0 𝑓(π‘šβˆ’3)(0 ADV iajs-3251 62 10 ) ) PUNCT iajs-3251 63 1 βˆ’ βˆ’ PROPN iajs-3251 63 2 β‹― β‹― ADP iajs-3251 63 3 βˆ’ βˆ’ PROPN iajs-3251 64 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 64 2 ( ( PUNCT iajs-3251 64 3 𝛼𝑣)π‘›βˆ’π‘š+1 𝛼𝑣)π‘›βˆ’π‘š+1 ADJ iajs-3251 64 4 𝑓(0 𝑓(0 NOUN iajs-3251 64 5 ) ) PUNCT iajs-3251 65 1 + + CCONJ iajs-3251 65 2 ( ( PUNCT iajs-3251 65 3 𝛼𝑣)π‘šπ‘†πΈπΈπ›Ό,𝛽{𝑓(𝑑 𝛼𝑣)π‘šπ‘†πΈπΈπ›Ό,𝛽{𝑓(𝑑 NUM iajs-3251 65 4 ) ) PUNCT iajs-3251 65 5 } } PUNCT iajs-3251 65 6 proof proof NOUN iajs-3251 65 7 : : PUNCT iajs-3251 65 8 to to PART iajs-3251 65 9 prove prove VERB iajs-3251 65 10 1 1 NUM iajs-3251 65 11 , , PUNCT iajs-3251 65 12 let let VERB iajs-3251 65 13 double double ADJ iajs-3251 65 14 parametric parametric ADJ iajs-3251 65 15 see see VERB iajs-3251 65 16 transformation transformation NOUN iajs-3251 65 17 for for ADP iajs-3251 65 18 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 65 19 ) ) PUNCT iajs-3251 65 20 = = SYM iajs-3251 65 21 𝑓′(𝑑 𝑓′(𝑑 NOUN iajs-3251 65 22 ) ) PUNCT iajs-3251 65 23 , , PUNCT iajs-3251 65 24 𝑆𝐸𝐸𝛼,𝛽{𝑓′(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓′(𝑑 NOUN iajs-3251 65 25 ) ) PUNCT iajs-3251 65 26 } } PUNCT iajs-3251 65 27 = = SYM iajs-3251 65 28 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 65 29 ( ( PUNCT iajs-3251 65 30 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 65 31 ∫ ∫ PROPN iajs-3251 65 32 𝑓′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 𝑓′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 PROPN iajs-3251 65 33 ∞ ∞ PROPN iajs-3251 65 34 0 0 NUM iajs-3251 66 1 = = SYM iajs-3251 66 2 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 66 3 ( ( PUNCT iajs-3251 66 4 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 66 5 𝑓(𝑑)|0 𝑓(𝑑)|0 PROPN iajs-3251 66 6 ∞ ∞ PROPN iajs-3251 67 1 + + CCONJ iajs-3251 67 2 ( ( PUNCT iajs-3251 67 3 𝛼𝑣 𝛼𝑣 NOUN iajs-3251 67 4 ) ) PUNCT iajs-3251 67 5 ∫ ∫ PROPN iajs-3251 67 6 𝑓(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 𝑓(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 PROPN iajs-3251 67 7 ∞ ∞ PROPN iajs-3251 67 8 0 0 NUM iajs-3251 68 1 = = PUNCT iajs-3251 68 2 βˆ’ βˆ’ NOUN iajs-3251 69 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 69 2 ( ( PUNCT iajs-3251 69 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 69 4 𝑓(0 𝑓(0 PROPN iajs-3251 69 5 ) ) PUNCT iajs-3251 69 6 + + CCONJ iajs-3251 69 7 ( ( PUNCT iajs-3251 69 8 𝛼𝑣)𝐹𝛼,𝛽(𝑣 𝛼𝑣)𝐹𝛼,𝛽(𝑣 NOUN iajs-3251 69 9 ) ) PUNCT iajs-3251 69 10 ( ( PUNCT iajs-3251 69 11 2 2 X iajs-3251 69 12 ) ) PUNCT iajs-3251 69 13 let let AUX iajs-3251 69 14 double double ADJ iajs-3251 69 15 parametric parametric ADJ iajs-3251 69 16 see see VERB iajs-3251 69 17 transformation transformation NOUN iajs-3251 69 18 for for ADP iajs-3251 69 19 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 69 20 ) ) PUNCT iajs-3251 69 21 = = SYM iajs-3251 69 22 𝑓′′(𝑑 𝑓′′(𝑑 NOUN iajs-3251 69 23 ) ) PUNCT iajs-3251 69 24 . . PUNCT iajs-3251 70 1 𝑆𝐸𝐸𝛼,𝛽{𝑓′′(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓′′(𝑑 X iajs-3251 70 2 ) ) PUNCT iajs-3251 70 3 } } PUNCT iajs-3251 71 1 = = SYM iajs-3251 71 2 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 71 3 ( ( PUNCT iajs-3251 71 4 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 71 5 ∫ ∫ NOUN iajs-3251 71 6 𝑓′′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 𝑓′′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 PROPN iajs-3251 71 7 ∞ ∞ PROPN iajs-3251 71 8 0 0 NUM iajs-3251 72 1 = = SYM iajs-3251 72 2 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 72 3 ( ( PUNCT iajs-3251 72 4 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 72 5 𝑓′(𝑑)|0 𝑓′(𝑑)|0 PROPN iajs-3251 72 6 ∞ ∞ PROPN iajs-3251 72 7 + + CCONJ iajs-3251 72 8 ( ( PUNCT iajs-3251 72 9 𝛼𝑣 𝛼𝑣 NOUN iajs-3251 72 10 ) ) PUNCT iajs-3251 72 11 ∫ ∫ PROPN iajs-3251 72 12 𝑓′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 𝑓′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 PROPN iajs-3251 72 13 ∞ ∞ PROPN iajs-3251 72 14 0 0 NUM iajs-3251 72 15 = = PUNCT iajs-3251 73 1 βˆ’ βˆ’ NOUN iajs-3251 73 2 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 73 3 ( ( PUNCT iajs-3251 73 4 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 73 5 𝑓′(0 𝑓′(0 PROPN iajs-3251 73 6 ) ) PUNCT iajs-3251 73 7 + + CCONJ iajs-3251 73 8 ( ( PUNCT iajs-3251 73 9 𝛼𝑣)𝑆𝐸𝐸𝛼,𝛽{𝑓′(𝑑 𝛼𝑣)𝑆𝐸𝐸𝛼,𝛽{𝑓′(𝑑 NOUN iajs-3251 73 10 ) ) PUNCT iajs-3251 73 11 } } PUNCT iajs-3251 73 12 . . PUNCT iajs-3251 74 1 ihjpas ihjpas PROPN iajs-3251 74 2 . . PUNCT iajs-3251 75 1 36(2)2023 36(2)2023 NUM iajs-3251 75 2 433 433 NUM iajs-3251 75 3 by by ADP iajs-3251 75 4 the the DET iajs-3251 75 5 help help NOUN iajs-3251 75 6 of of ADP iajs-3251 75 7 ( ( PUNCT iajs-3251 75 8 2 2 X iajs-3251 75 9 ) ) PUNCT iajs-3251 75 10 we we PRON iajs-3251 75 11 can can AUX iajs-3251 75 12 get get VERB iajs-3251 75 13 : : PUNCT iajs-3251 75 14 = = SYM iajs-3251 75 15 βˆ’ βˆ’ NOUN iajs-3251 76 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 76 2 ( ( PUNCT iajs-3251 76 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 76 4 𝑓′(0 𝑓′(0 PROPN iajs-3251 76 5 ) ) PUNCT iajs-3251 76 6 βˆ’ βˆ’ ADV iajs-3251 77 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 77 2 ( ( PUNCT iajs-3251 77 3 𝛼𝑣)π‘›βˆ’1 𝛼𝑣)π‘›βˆ’1 PROPN iajs-3251 77 4 𝑓(0 𝑓(0 PROPN iajs-3251 77 5 ) ) PUNCT iajs-3251 77 6 + + CCONJ iajs-3251 77 7 ( ( PUNCT iajs-3251 77 8 𝛼𝑣)2𝐹𝛼,𝛽(𝑣 𝛼𝑣)2𝐹𝛼,𝛽(𝑣 PROPN iajs-3251 77 9 ) ) PUNCT iajs-3251 77 10 ( ( PUNCT iajs-3251 77 11 3 3 X iajs-3251 77 12 ) ) PUNCT iajs-3251 77 13 let let VERB iajs-3251 77 14 double double ADJ iajs-3251 77 15 parametric parametric ADJ iajs-3251 77 16 see see VERB iajs-3251 77 17 transformation transformation NOUN iajs-3251 77 18 for for ADP iajs-3251 77 19 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 77 20 ) ) PUNCT iajs-3251 77 21 = = PUNCT iajs-3251 77 22 𝑓′′′(𝑑 𝑓′′′(𝑑 PROPN iajs-3251 77 23 ) ) PUNCT iajs-3251 77 24 . . PUNCT iajs-3251 78 1 𝑆𝐸𝐸𝛼,𝛽{𝑓′′′(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓′′′(𝑑 NOUN iajs-3251 78 2 ) ) PUNCT iajs-3251 78 3 } } PUNCT iajs-3251 78 4 = = SYM iajs-3251 78 5 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 78 6 ( ( PUNCT iajs-3251 78 7 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 78 8 ∫ ∫ NOUN iajs-3251 78 9 𝑓′′′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 𝑓′′′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 PROPN iajs-3251 78 10 ∞ ∞ PROPN iajs-3251 78 11 0 0 NUM iajs-3251 79 1 = = SYM iajs-3251 79 2 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ NOUN iajs-3251 79 3 ( ( PUNCT iajs-3251 79 4 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 79 5 𝑓′′(𝑑)|0 𝑓′′(𝑑)|0 PROPN iajs-3251 79 6 ∞ ∞ PROPN iajs-3251 79 7 + + CCONJ iajs-3251 79 8 ( ( PUNCT iajs-3251 79 9 𝛼𝑣 𝛼𝑣 NOUN iajs-3251 79 10 ) ) PUNCT iajs-3251 79 11 ∫ ∫ PROPN iajs-3251 80 1 𝑓′′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 𝑓′′(𝑑)π‘’βˆ’(𝛼𝑣)𝑑𝑑𝑑 PROPN iajs-3251 80 2 ∞ ∞ PROPN iajs-3251 80 3 0 0 NUM iajs-3251 81 1 = = PUNCT iajs-3251 81 2 βˆ’ βˆ’ NOUN iajs-3251 82 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 82 2 ( ( PUNCT iajs-3251 82 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 82 4 𝑓′′(0 𝑓′′(0 NOUN iajs-3251 82 5 ) ) PUNCT iajs-3251 83 1 + + CCONJ iajs-3251 83 2 ( ( PUNCT iajs-3251 83 3 𝛼𝑣)𝑆𝐸𝐸𝛼,𝛽{𝑓′′(𝑑 𝛼𝑣)𝑆𝐸𝐸𝛼,𝛽{𝑓′′(𝑑 NUM iajs-3251 83 4 ) ) PUNCT iajs-3251 83 5 } } PUNCT iajs-3251 83 6 . . PUNCT iajs-3251 84 1 by by ADP iajs-3251 84 2 the the DET iajs-3251 84 3 help help NOUN iajs-3251 84 4 of of ADP iajs-3251 84 5 ( ( PUNCT iajs-3251 84 6 3 3 X iajs-3251 84 7 ) ) PUNCT iajs-3251 84 8 we we PRON iajs-3251 84 9 can can AUX iajs-3251 84 10 get get VERB iajs-3251 84 11 : : PUNCT iajs-3251 84 12 = = SYM iajs-3251 84 13 βˆ’ βˆ’ NOUN iajs-3251 85 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 85 2 ( ( PUNCT iajs-3251 85 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 85 4 𝑓′′(0 𝑓′′(0 NOUN iajs-3251 85 5 ) ) PUNCT iajs-3251 85 6 βˆ’ βˆ’ ADP iajs-3251 85 7 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 85 8 ( ( PUNCT iajs-3251 85 9 𝛼𝑣)π‘›βˆ’1 𝛼𝑣)π‘›βˆ’1 PROPN iajs-3251 85 10 𝑓′(0 𝑓′(0 PROPN iajs-3251 85 11 ) ) PUNCT iajs-3251 85 12 βˆ’ βˆ’ ADP iajs-3251 86 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 86 2 ( ( PUNCT iajs-3251 86 3 𝛼𝑣)π‘›βˆ’2 𝛼𝑣)π‘›βˆ’2 PROPN iajs-3251 86 4 𝑓(0 𝑓(0 PROPN iajs-3251 86 5 ) ) PUNCT iajs-3251 86 6 + + CCONJ iajs-3251 86 7 ( ( PUNCT iajs-3251 86 8 𝛼𝑣)3𝐹𝛼,𝛽(𝑣 𝛼𝑣)3𝐹𝛼,𝛽(𝑣 NOUN iajs-3251 86 9 ) ) PUNCT iajs-3251 86 10 ( ( PUNCT iajs-3251 86 11 4 4 X iajs-3251 86 12 ) ) PUNCT iajs-3251 86 13 similarly similarly ADV iajs-3251 86 14 we we PRON iajs-3251 86 15 can can AUX iajs-3251 86 16 find find VERB iajs-3251 86 17 the the DET iajs-3251 86 18 result result NOUN iajs-3251 86 19 form form VERB iajs-3251 86 20 the the DET iajs-3251 86 21 statement statement NOUN iajs-3251 86 22 4 4 NUM iajs-3251 86 23 mentioned mention VERB iajs-3251 86 24 in in ADP iajs-3251 86 25 theorem theorem ADJ iajs-3251 86 26 : : PUNCT iajs-3251 86 27 𝑆𝐸𝐸𝛼,𝛽{𝑓(π‘š)(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓(π‘š)(𝑑 ADJ iajs-3251 86 28 ) ) PUNCT iajs-3251 86 29 } } PUNCT iajs-3251 87 1 = = PUNCT iajs-3251 87 2 βˆ’ βˆ’ NOUN iajs-3251 88 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 88 2 ( ( PUNCT iajs-3251 88 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 88 4 𝑓(π‘šβˆ’1)(0 𝑓(π‘šβˆ’1)(0 NOUN iajs-3251 88 5 ) ) PUNCT iajs-3251 88 6 βˆ’ βˆ’ NOUN iajs-3251 89 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 89 2 ( ( PUNCT iajs-3251 89 3 𝛼𝑣)π‘›βˆ’1 𝛼𝑣)π‘›βˆ’1 PROPN iajs-3251 89 4 𝑓(π‘šβˆ’2)(0 𝑓(π‘šβˆ’2)(0 X iajs-3251 89 5 ) ) PUNCT iajs-3251 89 6 βˆ’ βˆ’ PROPN iajs-3251 90 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 90 2 ( ( PUNCT iajs-3251 90 3 𝛼𝑣)π‘›βˆ’2 𝛼𝑣)π‘›βˆ’2 PROPN iajs-3251 90 4 𝑓(π‘šβˆ’3)(0 𝑓(π‘šβˆ’3)(0 ADV iajs-3251 90 5 ) ) PUNCT iajs-3251 90 6 βˆ’ βˆ’ PROPN iajs-3251 91 1 β‹― β‹― ADP iajs-3251 91 2 βˆ’ βˆ’ PROPN iajs-3251 92 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 92 2 ( ( PUNCT iajs-3251 92 3 𝛼𝑣)π‘›βˆ’π‘š+1 𝛼𝑣)π‘›βˆ’π‘š+1 ADJ iajs-3251 92 4 𝑓(0 𝑓(0 NOUN iajs-3251 92 5 ) ) PUNCT iajs-3251 93 1 + + CCONJ iajs-3251 93 2 ( ( PUNCT iajs-3251 93 3 𝛼𝑣)π‘šπ‘†πΈπΈπ›Ό,𝛽{𝑓(𝑑 𝛼𝑣)π‘šπ‘†πΈπΈπ›Ό,𝛽{𝑓(𝑑 NUM iajs-3251 93 4 ) ) PUNCT iajs-3251 93 5 } } PUNCT iajs-3251 93 6 ( ( PUNCT iajs-3251 93 7 5 5 X iajs-3251 93 8 ) ) PUNCT iajs-3251 93 9 linear linear ADJ iajs-3251 93 10 property property NOUN iajs-3251 93 11 : : PUNCT iajs-3251 93 12 let let VERB iajs-3251 93 13 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 93 14 ) ) PUNCT iajs-3251 93 15 and and CCONJ iajs-3251 93 16 𝑔(𝑑 𝑔(𝑑 NUM iajs-3251 93 17 ) ) PUNCT iajs-3251 93 18 be be VERB iajs-3251 93 19 two two NUM iajs-3251 93 20 functions function NOUN iajs-3251 93 21 then then ADV iajs-3251 93 22 linearity linearity NOUN iajs-3251 93 23 property property NOUN iajs-3251 93 24 of of ADP iajs-3251 93 25 two two NUM iajs-3251 93 26 parametric parametric ADJ iajs-3251 93 27 extensions extension NOUN iajs-3251 93 28 of of ADP iajs-3251 93 29 see see VERB iajs-3251 93 30 transformation transformation NOUN iajs-3251 93 31 define define VERB iajs-3251 93 32 as as ADP iajs-3251 93 33 : : PUNCT iajs-3251 93 34 𝑆𝐸𝐸𝛼,𝛽{𝑐1𝑓(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑐1𝑓(𝑑 NUM iajs-3251 93 35 ) ) PUNCT iajs-3251 93 36 Β± Β± NUM iajs-3251 93 37 𝑐2𝑔(𝑑 𝑐2𝑔(𝑑 PROPN iajs-3251 93 38 ) ) PUNCT iajs-3251 93 39 } } PUNCT iajs-3251 93 40 = = SYM iajs-3251 93 41 𝑐1𝑆𝐸𝐸𝛼,𝛽𝑓(𝑑 𝑐1𝑆𝐸𝐸𝛼,𝛽𝑓(𝑑 NUM iajs-3251 93 42 ) ) PUNCT iajs-3251 93 43 Β± Β± NUM iajs-3251 93 44 𝑐2𝑆𝐸𝐸𝛼,𝛽𝑔(𝑑 𝑐2𝑆𝐸𝐸𝛼,𝛽𝑔(𝑑 PROPN iajs-3251 93 45 ) ) PUNCT iajs-3251 93 46 ( ( PUNCT iajs-3251 93 47 6 6 NUM iajs-3251 93 48 ) ) PUNCT iajs-3251 93 49 where where SCONJ iajs-3251 93 50 𝑐1 𝑐1 NOUN iajs-3251 93 51 and and CCONJ iajs-3251 93 52 𝑐2 𝑐2 NOUN iajs-3251 93 53 are be AUX iajs-3251 93 54 constants constant NOUN iajs-3251 93 55 . . PUNCT iajs-3251 94 1 change change NOUN iajs-3251 94 2 of of ADP iajs-3251 94 3 scale scale NOUN iajs-3251 94 4 property property NOUN iajs-3251 94 5 : : PUNCT iajs-3251 94 6 let let VERB iajs-3251 94 7 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 94 8 ) ) PUNCT iajs-3251 94 9 be be AUX iajs-3251 94 10 a a DET iajs-3251 94 11 function function NOUN iajs-3251 94 12 and and CCONJ iajs-3251 94 13 𝑆𝐸𝐸𝛼,𝛽 𝑆𝐸𝐸𝛼,𝛽 NOUN iajs-3251 94 14 of of ADP iajs-3251 94 15 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 94 16 ) ) PUNCT iajs-3251 94 17 is be AUX iajs-3251 94 18 𝐹𝛼,𝛽(𝑣 𝐹𝛼,𝛽(𝑣 PRON iajs-3251 94 19 ) ) PUNCT iajs-3251 94 20 then then ADV iajs-3251 94 21 change change NOUN iajs-3251 94 22 of of ADP iajs-3251 94 23 scale scale NOUN iajs-3251 94 24 property property NOUN iajs-3251 94 25 of of ADP iajs-3251 94 26 two two NUM iajs-3251 94 27 parametric parametric ADJ iajs-3251 94 28 see see NOUN iajs-3251 94 29 transformation transformation NOUN iajs-3251 94 30 can can AUX iajs-3251 94 31 be be AUX iajs-3251 94 32 described describe VERB iajs-3251 94 33 as as ADP iajs-3251 94 34 : : PUNCT iajs-3251 94 35 𝑆𝐸𝐸𝛼,𝛽{𝑓(𝑐𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓(𝑐𝑑 PROPN iajs-3251 94 36 ) ) PUNCT iajs-3251 94 37 } } PUNCT iajs-3251 94 38 = = SYM iajs-3251 94 39 ( ( PUNCT iajs-3251 94 40 1 1 NUM iajs-3251 94 41 π‘π‘›βˆ’1 π‘π‘›βˆ’1 NOUN iajs-3251 95 1 ) ) PUNCT iajs-3251 95 2 𝐹𝛼,𝛽 𝐹𝛼,𝛽 PROPN iajs-3251 95 3 ( ( PUNCT iajs-3251 95 4 𝑣 𝑣 PART iajs-3251 95 5 𝑐 𝑐 PROPN iajs-3251 95 6 ) ) PUNCT iajs-3251 95 7 ( ( PUNCT iajs-3251 95 8 7 7 X iajs-3251 95 9 ) ) PUNCT iajs-3251 95 10 here here ADV iajs-3251 95 11 𝑐 𝑐 PROPN iajs-3251 95 12 is be AUX iajs-3251 95 13 a a DET iajs-3251 95 14 constant constant ADJ iajs-3251 95 15 value value NOUN iajs-3251 95 16 . . PUNCT iajs-3251 96 1 shifting shift VERB iajs-3251 96 2 property property NOUN iajs-3251 96 3 : : PUNCT iajs-3251 96 4 let let AUX iajs-3251 96 5 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 96 6 ) ) PUNCT iajs-3251 96 7 be be AUX iajs-3251 96 8 a a DET iajs-3251 96 9 function function NOUN iajs-3251 96 10 and and CCONJ iajs-3251 96 11 𝑆𝐸𝐸𝛼,𝛽 𝑆𝐸𝐸𝛼,𝛽 NOUN iajs-3251 96 12 of of ADP iajs-3251 96 13 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 96 14 ) ) PUNCT iajs-3251 96 15 is be AUX iajs-3251 96 16 𝐹𝛼,𝛽(𝑣 𝐹𝛼,𝛽(𝑣 ADP iajs-3251 96 17 ) ) PUNCT iajs-3251 96 18 then then ADV iajs-3251 96 19 shifting shift VERB iajs-3251 96 20 property property NOUN iajs-3251 96 21 of of ADP iajs-3251 96 22 two two NUM iajs-3251 96 23 parametric parametric ADJ iajs-3251 96 24 see see NOUN iajs-3251 96 25 transformation transformation NOUN iajs-3251 96 26 can can AUX iajs-3251 96 27 be be AUX iajs-3251 96 28 described describe VERB iajs-3251 96 29 as as ADP iajs-3251 96 30 : : PUNCT iajs-3251 96 31 𝑆𝐸𝐸𝛼,𝛽{𝑒𝑐𝑑𝑓(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑒𝑐𝑑𝑓(𝑑 ADJ iajs-3251 96 32 ) ) PUNCT iajs-3251 96 33 } } PUNCT iajs-3251 96 34 = = PUNCT iajs-3251 97 1 𝐹𝛼,𝛽(𝑣 𝐹𝛼,𝛽(𝑣 ADP iajs-3251 97 2 βˆ’ βˆ’ PROPN iajs-3251 97 3 𝑐 𝑐 NOUN iajs-3251 97 4 ) ) PUNCT iajs-3251 97 5 ( ( PUNCT iajs-3251 97 6 8) 8) NUM iajs-3251 97 7 theorem theorem NOUN iajs-3251 97 8 2 2 NUM iajs-3251 97 9 . . PUNCT iajs-3251 97 10 let let VERB iajs-3251 97 11 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 97 12 ) ) PUNCT iajs-3251 97 13 and and CCONJ iajs-3251 97 14 𝑔(𝑑 𝑔(𝑑 NUM iajs-3251 97 15 ) ) PUNCT iajs-3251 97 16 be be VERB iajs-3251 97 17 two two NUM iajs-3251 97 18 functions function NOUN iajs-3251 97 19 and and CCONJ iajs-3251 97 20 𝑆𝐸𝐸𝛼,𝛽 𝑆𝐸𝐸𝛼,𝛽 NOUN iajs-3251 97 21 of of ADP iajs-3251 97 22 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3251 97 23 ) ) PUNCT iajs-3251 97 24 is be AUX iajs-3251 97 25 𝐹𝛼,𝛽(𝑣 𝐹𝛼,𝛽(𝑣 NOUN iajs-3251 97 26 ) ) PUNCT iajs-3251 97 27 , , PUNCT iajs-3251 97 28 𝑆𝐸𝐸𝛼,𝛽 𝑆𝐸𝐸𝛼,𝛽 NUM iajs-3251 97 29 of of ADP iajs-3251 97 30 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-3251 97 31 ) ) PUNCT iajs-3251 97 32 is be AUX iajs-3251 97 33 𝐻𝛼,𝛽(𝑣 𝐻𝛼,𝛽(𝑣 NOUN iajs-3251 97 34 ) ) PUNCT iajs-3251 97 35 then then ADV iajs-3251 97 36 the the DET iajs-3251 97 37 convolution convolution NOUN iajs-3251 97 38 of of ADP iajs-3251 97 39 two two NUM iajs-3251 97 40 parametric parametric ADJ iajs-3251 97 41 𝑆𝐸𝐸 𝑆𝐸𝐸 PROPN iajs-3251 97 42 transform transform NOUN iajs-3251 97 43 can can AUX iajs-3251 97 44 be be AUX iajs-3251 97 45 described describe VERB iajs-3251 97 46 as as ADP iajs-3251 97 47 : : PUNCT iajs-3251 97 48 𝑆𝐸𝐸𝛼,𝛽{𝑓(𝑑 𝑆𝐸𝐸𝛼,𝛽{𝑓(𝑑 NUM iajs-3251 97 49 ) ) PUNCT iajs-3251 97 50 βˆ— βˆ— NOUN iajs-3251 97 51 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-3251 97 52 ) ) PUNCT iajs-3251 97 53 } } PUNCT iajs-3251 97 54 = = SYM iajs-3251 97 55 βˆ’(𝛼𝑣)𝑛𝐹𝛼,𝛽(𝑣 βˆ’(𝛼𝑣)𝑛𝐹𝛼,𝛽(𝑣 NOUN iajs-3251 97 56 ) ) PUNCT iajs-3251 97 57 . . PUNCT iajs-3251 98 1 𝐻𝛼,𝛽(𝑣 𝐻𝛼,𝛽(𝑣 PROPN iajs-3251 98 2 ) ) PUNCT iajs-3251 98 3 ( ( PUNCT iajs-3251 98 4 9 9 X iajs-3251 98 5 ) ) SYM iajs-3251 98 6 3.application 3.application NUM iajs-3251 98 7 population population NOUN iajs-3251 98 8 growth growth NOUN iajs-3251 98 9 rate rate NOUN iajs-3251 98 10 problem problem NOUN iajs-3251 98 11 : : PUNCT iajs-3251 98 12 exponential exponential ADJ iajs-3251 98 13 growth growth NOUN iajs-3251 98 14 is be AUX iajs-3251 98 15 linked link VERB iajs-3251 98 16 to to ADP iajs-3251 98 17 the the DET iajs-3251 98 18 process process NOUN iajs-3251 98 19 of of ADP iajs-3251 98 20 population population NOUN iajs-3251 98 21 growth growth NOUN iajs-3251 98 22 rate rate NOUN iajs-3251 98 23 equation equation NOUN iajs-3251 98 24 decline decline NOUN iajs-3251 98 25 . . PUNCT iajs-3251 99 1 authors author NOUN iajs-3251 99 2 in in ADP iajs-3251 99 3 [ [ X iajs-3251 99 4 21 21 NUM iajs-3251 99 5 ] ] PUNCT iajs-3251 99 6 have have AUX iajs-3251 99 7 mathematically mathematically ADV iajs-3251 99 8 described describe VERB iajs-3251 99 9 the the DET iajs-3251 99 10 decay decay NOUN iajs-3251 99 11 problems problem NOUN iajs-3251 99 12 of of ADP iajs-3251 99 13 a a DET iajs-3251 99 14 substance substance NOUN iajs-3251 99 15 with with ADP iajs-3251 99 16 the the DET iajs-3251 99 17 help help NOUN iajs-3251 99 18 of of ADP iajs-3251 99 19 first first ADJ iajs-3251 99 20 order order NOUN iajs-3251 99 21 ordinary ordinary ADJ iajs-3251 99 22 linear linear ADJ iajs-3251 99 23 differential differential NOUN iajs-3251 99 24 equation equation NOUN iajs-3251 99 25 : : PUNCT iajs-3251 99 26 𝑑𝑃 𝑑𝑃 PROPN iajs-3251 99 27 𝑑𝑑 𝑑𝑑 ADP iajs-3251 99 28 = = SYM iajs-3251 99 29 βˆ’πœ‡π‘ƒ βˆ’πœ‡π‘ƒ PROPN iajs-3251 99 30 ( ( PUNCT iajs-3251 99 31 10 10 NUM iajs-3251 99 32 ) ) PUNCT iajs-3251 99 33 with with ADP iajs-3251 99 34 the the DET iajs-3251 99 35 initial initial ADJ iajs-3251 99 36 condition condition NOUN iajs-3251 99 37 : : PUNCT iajs-3251 99 38 𝑃(0 𝑃(0 NUM iajs-3251 99 39 ) ) PUNCT iajs-3251 99 40 = = NOUN iajs-3251 99 41 𝑃0 𝑃0 NOUN iajs-3251 99 42 ( ( PUNCT iajs-3251 99 43 11 11 NUM iajs-3251 99 44 ) ) PUNCT iajs-3251 99 45 ihjpas ihjpa NOUN iajs-3251 99 46 . . PUNCT iajs-3251 100 1 36(2)2023 36(2)2023 NUM iajs-3251 100 2 434 434 NUM iajs-3251 100 3 where where SCONJ iajs-3251 100 4 πœ‡ πœ‡ ADV iajs-3251 100 5 is be AUX iajs-3251 100 6 a a DET iajs-3251 100 7 real real ADJ iajs-3251 100 8 positive positive ADJ iajs-3251 100 9 number number NOUN iajs-3251 100 10 , , PUNCT iajs-3251 100 11 𝑃 𝑃 PROPN iajs-3251 100 12 is be AUX iajs-3251 100 13 the the DET iajs-3251 100 14 value value NOUN iajs-3251 100 15 of of ADP iajs-3251 100 16 population population NOUN iajs-3251 100 17 at at ADP iajs-3251 100 18 𝑑-time 𝑑-time PROPN iajs-3251 100 19 and and CCONJ iajs-3251 100 20 𝑃0 𝑃0 NOUN iajs-3251 100 21 is be AUX iajs-3251 100 22 the the DET iajs-3251 100 23 initial initial ADJ iajs-3251 100 24 value value NOUN iajs-3251 100 25 of of ADP iajs-3251 100 26 population population NOUN iajs-3251 100 27 at at ADP iajs-3251 100 28 initial initial ADJ iajs-3251 100 29 time time NOUN iajs-3251 100 30 𝑑0 𝑑0 PROPN iajs-3251 100 31 . . PUNCT iajs-3251 101 1 sol sol PROPN iajs-3251 101 2 : : PUNCT iajs-3251 101 3 by by ADP iajs-3251 101 4 applying apply VERB iajs-3251 101 5 the the DET iajs-3251 101 6 two two NUM iajs-3251 101 7 parametric parametric ADJ iajs-3251 101 8 see see NOUN iajs-3251 101 9 transform transform NOUN iajs-3251 101 10 on on ADP iajs-3251 101 11 ( ( PUNCT iajs-3251 101 12 10 10 NUM iajs-3251 101 13 ) ) PUNCT iajs-3251 101 14 and and CCONJ iajs-3251 101 15 by by ADP iajs-3251 101 16 using use VERB iajs-3251 101 17 the the DET iajs-3251 101 18 rule rule NOUN iajs-3251 101 19 mentioned mention VERB iajs-3251 101 20 in in ADP iajs-3251 101 21 ( ( PUNCT iajs-3251 101 22 2 2 X iajs-3251 101 23 ) ) PUNCT iajs-3251 101 24 we we PRON iajs-3251 101 25 will will AUX iajs-3251 101 26 get get VERB iajs-3251 101 27 : : PUNCT iajs-3251 101 28 βˆ’ βˆ’ PUNCT iajs-3251 102 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 102 2 ( ( PUNCT iajs-3251 102 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 102 4 𝑃(0 𝑃(0 NUM iajs-3251 102 5 ) ) PUNCT iajs-3251 102 6 + + CCONJ iajs-3251 102 7 ( ( PUNCT iajs-3251 102 8 𝛼𝑣)𝑃𝛼,𝛽(𝑣 𝛼𝑣)𝑃𝛼,𝛽(𝑣 NOUN iajs-3251 102 9 ) ) PUNCT iajs-3251 102 10 = = SYM iajs-3251 102 11 βˆ’πœ‡π‘ƒπ›Ό,𝛽(𝑣 βˆ’πœ‡π‘ƒπ›Ό,𝛽(𝑣 NOUN iajs-3251 102 12 ) ) PUNCT iajs-3251 102 13 . . PUNCT iajs-3251 103 1 by by ADP iajs-3251 103 2 using use VERB iajs-3251 103 3 initial initial ADJ iajs-3251 103 4 condition condition NOUN iajs-3251 103 5 in in ADP iajs-3251 103 6 ( ( PUNCT iajs-3251 103 7 11 11 NUM iajs-3251 103 8 ) ) PUNCT iajs-3251 103 9 we we PRON iajs-3251 103 10 get get VERB iajs-3251 103 11 : : PUNCT iajs-3251 103 12 βˆ’ βˆ’ PUNCT iajs-3251 103 13 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 103 14 ( ( PUNCT iajs-3251 103 15 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 103 16 𝑃0 𝑃0 NOUN iajs-3251 103 17 + + CCONJ iajs-3251 103 18 ( ( PUNCT iajs-3251 103 19 𝛼𝑣)𝑃𝛼,𝛽(𝑣 𝛼𝑣)𝑃𝛼,𝛽(𝑣 NOUN iajs-3251 103 20 ) ) PUNCT iajs-3251 103 21 = = SYM iajs-3251 103 22 βˆ’πœ‡π‘ƒπ›Ό,𝛽(𝑣 βˆ’πœ‡π‘ƒπ›Ό,𝛽(𝑣 NOUN iajs-3251 103 23 ) ) PUNCT iajs-3251 104 1 [ [ X iajs-3251 104 2 ( ( PUNCT iajs-3251 104 3 𝛼𝑣 𝛼𝑣 NOUN iajs-3251 104 4 ) ) PUNCT iajs-3251 104 5 + + SYM iajs-3251 104 6 πœ‡]𝑃𝛼,𝛽(𝑣 πœ‡]𝑃𝛼,𝛽(𝑣 NOUN iajs-3251 104 7 ) ) PUNCT iajs-3251 104 8 = = X iajs-3251 105 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 105 2 ( ( PUNCT iajs-3251 105 3 𝛼𝑣)𝑛 𝛼𝑣)𝑛 PROPN iajs-3251 105 4 𝑃0 𝑃0 NOUN iajs-3251 105 5 𝑃𝛼,𝛽(𝑣 𝑃𝛼,𝛽(𝑣 ADV iajs-3251 105 6 ) ) PUNCT iajs-3251 105 7 = = SYM iajs-3251 106 1 π‘’βˆ’π›½π‘£ π‘’βˆ’π›½π‘£ ADV iajs-3251 106 2 ( ( PUNCT iajs-3251 106 3 𝛼𝑣)𝑛[(𝛼𝑣)+πœ‡ 𝛼𝑣)𝑛[(𝛼𝑣)+πœ‡ PROPN iajs-3251 106 4 ] ] X iajs-3251 106 5 𝑃0 𝑃0 NOUN iajs-3251 106 6 ( ( PUNCT iajs-3251 106 7 12 12 NUM iajs-3251 106 8 ) ) PUNCT iajs-3251 106 9 by by ADP iajs-3251 106 10 using use VERB iajs-3251 106 11 the the DET iajs-3251 106 12 definition definition NOUN iajs-3251 106 13 in in ADP iajs-3251 106 14 ( ( PUNCT iajs-3251 106 15 1 1 NUM iajs-3251 106 16 ) ) PUNCT iajs-3251 106 17 and and CCONJ iajs-3251 106 18 table table NOUN iajs-3251 106 19 values value NOUN iajs-3251 106 20 we we PRON iajs-3251 106 21 get get VERB iajs-3251 106 22 : : PUNCT iajs-3251 106 23 𝑃(𝑑 𝑃(𝑑 NUM iajs-3251 106 24 ) ) PUNCT iajs-3251 106 25 = = SYM iajs-3251 106 26 𝑃0π‘’βˆ’πœ‡π‘‘ 𝑃0π‘’βˆ’πœ‡π‘‘ NOUN iajs-3251 106 27 ( ( PUNCT iajs-3251 106 28 13 13 NUM iajs-3251 106 29 ) ) PUNCT iajs-3251 106 30 equation equation NOUN iajs-3251 106 31 13 13 NUM iajs-3251 106 32 provides provide VERB iajs-3251 106 33 the the DET iajs-3251 106 34 solution solution NOUN iajs-3251 106 35 of of ADP iajs-3251 106 36 population population NOUN iajs-3251 106 37 growth growth NOUN iajs-3251 106 38 rate rate NOUN iajs-3251 106 39 problem problem NOUN iajs-3251 106 40 . . PUNCT iajs-3251 107 1 4.conclusion 4.conclusion NUM iajs-3251 107 2 see see VERB iajs-3251 107 3 transformation transformation NOUN iajs-3251 107 4 with with ADP iajs-3251 107 5 two two NUM iajs-3251 107 6 parameter parameter NOUN iajs-3251 107 7 has have AUX iajs-3251 107 8 been be AUX iajs-3251 107 9 defined define VERB iajs-3251 107 10 in in ADP iajs-3251 107 11 this this DET iajs-3251 107 12 research research NOUN iajs-3251 107 13 . . PUNCT iajs-3251 108 1 the the DET iajs-3251 108 2 solution solution NOUN iajs-3251 108 3 of of ADP iajs-3251 108 4 population population NOUN iajs-3251 108 5 growth growth NOUN iajs-3251 108 6 rate rate NOUN iajs-3251 108 7 equation equation NOUN iajs-3251 108 8 has have AUX iajs-3251 108 9 been be AUX iajs-3251 108 10 described describe VERB iajs-3251 108 11 to to PART iajs-3251 108 12 show show VERB iajs-3251 108 13 the the DET iajs-3251 108 14 efficiency efficiency NOUN iajs-3251 108 15 of of ADP iajs-3251 108 16 this this DET iajs-3251 108 17 proposed propose VERB iajs-3251 108 18 technique technique NOUN iajs-3251 108 19 which which PRON iajs-3251 108 20 can can AUX iajs-3251 108 21 be be AUX iajs-3251 108 22 compared compare VERB iajs-3251 108 23 with with ADP iajs-3251 108 24 other other ADJ iajs-3251 108 25 laplace laplace NOUN iajs-3251 108 26 type type NOUN iajs-3251 108 27 transformation transformation NOUN iajs-3251 108 28 . . PUNCT iajs-3251 109 1 it it PRON iajs-3251 109 2 can can AUX iajs-3251 109 3 be be AUX iajs-3251 109 4 seen see VERB iajs-3251 109 5 that that SCONJ iajs-3251 109 6 it it PRON iajs-3251 109 7 is be AUX iajs-3251 109 8 helpful helpful ADJ iajs-3251 109 9 in in ADP iajs-3251 109 10 solving solve VERB iajs-3251 109 11 linear linear ADJ iajs-3251 109 12 differential differential NOUN iajs-3251 109 13 models model NOUN iajs-3251 109 14 . . PUNCT iajs-3251 110 1 the the DET iajs-3251 110 2 proposed propose VERB iajs-3251 110 3 transformation transformation NOUN iajs-3251 110 4 can can AUX iajs-3251 110 5 turned turn VERB iajs-3251 110 6 into into ADP iajs-3251 110 7 ordinary ordinary ADJ iajs-3251 110 8 see see NOUN iajs-3251 110 9 transformation transformation NOUN iajs-3251 110 10 by by ADP iajs-3251 110 11 adjusting adjust VERB iajs-3251 110 12 the the DET iajs-3251 110 13 values value NOUN iajs-3251 110 14 of of ADP iajs-3251 110 15 parameters parameter NOUN iajs-3251 110 16 . . PUNCT iajs-3251 111 1 references reference NOUN iajs-3251 111 2 : : PUNCT iajs-3251 111 3 1.evans 1.evans NUM iajs-3251 111 4 d.j d.j PROPN iajs-3251 111 5 . . PROPN iajs-3251 111 6 ; ; PUNCT iajs-3251 111 7 raslan raslan PROPN iajs-3251 111 8 k.r k.r PROPN iajs-3251 111 9 . . PUNCT iajs-3251 112 1 the the DET iajs-3251 112 2 adomian adomian NOUN iajs-3251 112 3 decomposition decomposition NOUN iajs-3251 112 4 methhod methhod NOUN iajs-3251 112 5 for for ADP iajs-3251 112 6 solving solve VERB iajs-3251 112 7 delay delay NOUN iajs-3251 112 8 differential differential ADJ iajs-3251 112 9 equation equation NOUN iajs-3251 112 10 . . PUNCT iajs-3251 113 1 international international ADJ iajs-3251 113 2 journal journal PROPN iajs-3251 113 3 of of ADP iajs-3251 113 4 computer computer NOUN iajs-3251 113 5 mathematics mathematic NOUN iajs-3251 113 6 , , PUNCT iajs-3251 113 7 2005 2005 NUM iajs-3251 113 8 , , PUNCT iajs-3251 113 9 82(1 82(1 NOUN iajs-3251 113 10 ) ) PUNCT iajs-3251 113 11 , , PUNCT iajs-3251 113 12 49 49 NUM iajs-3251 113 13 - - SYM iajs-3251 113 14 54 54 NUM iajs-3251 113 15 . . PUNCT iajs-3251 114 1 2.shah 2.shah NUM iajs-3251 114 2 , , PUNCT iajs-3251 114 3 k. k. PROPN iajs-3251 114 4 ; ; PUNCT iajs-3251 114 5 khalil khalil PROPN iajs-3251 114 6 , , PUNCT iajs-3251 114 7 h. h. PROPN iajs-3251 114 8 ; ; PUNCT iajs-3251 114 9 khan khan PROPN iajs-3251 114 10 , , PUNCT iajs-3251 114 11 ra ra PROPN iajs-3251 114 12 . . PUNCT iajs-3251 115 1 a a DET iajs-3251 115 2 generalized generalized ADJ iajs-3251 115 3 scheme scheme NOUN iajs-3251 115 4 based base VERB iajs-3251 115 5 on on ADP iajs-3251 115 6 shifted shift VERB iajs-3251 115 7 jacobi jacobi PROPN iajs-3251 115 8 polynomials polynomial NOUN iajs-3251 115 9 for for ADP iajs-3251 115 10 numerical numerical ADJ iajs-3251 115 11 simulation simulation NOUN iajs-3251 115 12 of of ADP iajs-3251 115 13 coupled couple VERB iajs-3251 115 14 systems system NOUN iajs-3251 115 15 of of ADP iajs-3251 115 16 multi multi ADJ iajs-3251 115 17 - - ADJ iajs-3251 115 18 term term ADJ iajs-3251 115 19 fractional fractional ADJ iajs-3251 115 20 - - PUNCT iajs-3251 115 21 order order NOUN iajs-3251 115 22 partial partial ADJ iajs-3251 115 23 differential differential NOUN iajs-3251 115 24 equations equation NOUN iajs-3251 115 25 . . PUNCT iajs-3251 116 1 lms lm NOUN iajs-3251 116 2 journal journal PROPN iajs-3251 116 3 of of ADP iajs-3251 116 4 computation computation NOUN iajs-3251 116 5 and and CCONJ iajs-3251 116 6 mathematics mathematic NOUN iajs-3251 116 7 , , PUNCT iajs-3251 116 8 2017 2017 NUM iajs-3251 116 9 , , PUNCT iajs-3251 116 10 20(1 20(1 NUM iajs-3251 116 11 ) ) PUNCT iajs-3251 116 12 , , PUNCT iajs-3251 116 13 11 11 NUM iajs-3251 116 14 - - SYM iajs-3251 116 15 29 29 NUM iajs-3251 116 16 . . PUNCT iajs-3251 117 1 3.abassy 3.abassy NUM iajs-3251 117 2 , , PUNCT iajs-3251 117 3 t.a t.a PROPN iajs-3251 117 4 . . PROPN iajs-3251 117 5 ; ; PUNCT iajs-3251 117 6 el el PROPN iajs-3251 117 7 - - PUNCT iajs-3251 117 8 tawil tawil PROPN iajs-3251 117 9 , , PUNCT iajs-3251 117 10 m.a m.a PROPN iajs-3251 117 11 . . PROPN iajs-3251 117 12 ; ; PUNCT iajs-3251 117 13 el el PROPN iajs-3251 117 14 - - PROPN iajs-3251 117 15 zoheiry zoheiry PROPN iajs-3251 117 16 , , PUNCT iajs-3251 117 17 h. h. PROPN iajs-3251 117 18 exact exact ADJ iajs-3251 117 19 solutions solution NOUN iajs-3251 117 20 of of ADP iajs-3251 117 21 some some DET iajs-3251 117 22 nonlinear nonlinear ADJ iajs-3251 117 23 partial partial ADJ iajs-3251 117 24 differential differential NOUN iajs-3251 117 25 equations equation NOUN iajs-3251 117 26 using use VERB iajs-3251 117 27 the the DET iajs-3251 117 28 variational variational ADJ iajs-3251 117 29 iteration iteration NOUN iajs-3251 117 30 method method NOUN iajs-3251 117 31 linked link VERB iajs-3251 117 32 with with ADP iajs-3251 117 33 laplace laplace NOUN iajs-3251 117 34 transforms transform VERB iajs-3251 117 35 and and CCONJ iajs-3251 117 36 the the DET iajs-3251 117 37 padΓ© padΓ© NOUN iajs-3251 117 38 technique technique NOUN iajs-3251 117 39 . . PUNCT iajs-3251 118 1 computers computer NOUN iajs-3251 118 2 and and CCONJ iajs-3251 118 3 mathematics mathematic NOUN iajs-3251 118 4 with with ADP iajs-3251 118 5 applications application NOUN iajs-3251 118 6 . . PUNCT iajs-3251 119 1 2007 2007 NUM iajs-3251 119 2 , , PUNCT iajs-3251 119 3 54(7 54(7 NOUN iajs-3251 119 4 - - PUNCT iajs-3251 119 5 8) 8) NUM iajs-3251 119 6 , , PUNCT iajs-3251 119 7 940 940 NUM iajs-3251 119 8 - - SYM iajs-3251 119 9 54 54 NUM iajs-3251 119 10 . . PUNCT iajs-3251 120 1 4.yindoula 4.yindoula NUM iajs-3251 120 2 , , PUNCT iajs-3251 120 3 j.b j.b PROPN iajs-3251 120 4 . . PROPN iajs-3251 120 5 ; ; PUNCT iajs-3251 120 6 youssouf youssouf PROPN iajs-3251 120 7 , , PUNCT iajs-3251 120 8 p. p. PROPN iajs-3251 120 9 ; ; PUNCT iajs-3251 120 10 bissanga bissanga ADV iajs-3251 120 11 , , PUNCT iajs-3251 120 12 g. g. PROPN iajs-3251 120 13 ; ; PUNCT iajs-3251 120 14 bassono bassono PROPN iajs-3251 120 15 , , PUNCT iajs-3251 120 16 f. f. PROPN iajs-3251 120 17 some some DET iajs-3251 120 18 b. b. PROPN iajs-3251 120 19 application application NOUN iajs-3251 120 20 of of ADP iajs-3251 120 21 the the DET iajs-3251 120 22 adomian adomian NOUN iajs-3251 120 23 decomposition decomposition NOUN iajs-3251 120 24 method method NOUN iajs-3251 120 25 and and CCONJ iajs-3251 120 26 laplace laplace NOUN iajs-3251 120 27 transform transform NOUN iajs-3251 120 28 method method NOUN iajs-3251 120 29 to to ADP iajs-3251 120 30 solving solve VERB iajs-3251 120 31 the the DET iajs-3251 120 32 convection convection NOUN iajs-3251 120 33 diffusiondissipation diffusiondissipation NOUN iajs-3251 120 34 equation equation NOUN iajs-3251 120 35 . . PUNCT iajs-3251 121 1 international international ADJ iajs-3251 121 2 journal journal NOUN iajs-3251 121 3 of of ADP iajs-3251 121 4 applied apply VERB iajs-3251 121 5 mathematics mathematic NOUN iajs-3251 121 6 research research NOUN iajs-3251 121 7 . . PUNCT iajs-3251 122 1 2014 2014 NUM iajs-3251 122 2 , , PUNCT iajs-3251 122 3 3(1 3(1 NUM iajs-3251 122 4 ) ) PUNCT iajs-3251 122 5 , , PUNCT iajs-3251 122 6 30 30 NUM iajs-3251 122 7 - - SYM iajs-3251 122 8 35 35 NUM iajs-3251 122 9 . . PUNCT iajs-3251 123 1 5.hamoud 5.hamoud NUM iajs-3251 123 2 , , PUNCT iajs-3251 123 3 a.a a.a PROPN iajs-3251 123 4 . . PROPN iajs-3251 123 5 ; ; PUNCT iajs-3251 123 6 ghadle ghadle PROPN iajs-3251 123 7 , , PUNCT iajs-3251 123 8 k.p k.p PROPN iajs-3251 123 9 . . PROPN iajs-3251 124 1 the the DET iajs-3251 124 2 combined combine VERB iajs-3251 124 3 modified modify VERB iajs-3251 124 4 laplace laplace NOUN iajs-3251 124 5 with with ADP iajs-3251 124 6 adomian adomian NOUN iajs-3251 124 7 decomposition decomposition NOUN iajs-3251 124 8 method method NOUN iajs-3251 124 9 for for ADP iajs-3251 124 10 solving solve VERB iajs-3251 124 11 the the DET iajs-3251 124 12 nonlinear nonlinear PROPN iajs-3251 124 13 volterra volterra NOUN iajs-3251 124 14 - - PUNCT iajs-3251 124 15 fredholm fredholm NOUN iajs-3251 124 16 integro integro PROPN iajs-3251 124 17 differential differential ADJ iajs-3251 124 18 equations equation NOUN iajs-3251 124 19 . . PUNCT iajs-3251 125 1 journal journal NOUN iajs-3251 125 2 of of ADP iajs-3251 125 3 the the DET iajs-3251 125 4 korean korean ADJ iajs-3251 125 5 society society NOUN iajs-3251 125 6 for for ADP iajs-3251 125 7 industrial industrial ADJ iajs-3251 125 8 and and CCONJ iajs-3251 125 9 applied applied ADJ iajs-3251 125 10 mathematics mathematic NOUN iajs-3251 125 11 , , PUNCT iajs-3251 125 12 2017 2017 NUM iajs-3251 125 13 , , PUNCT iajs-3251 125 14 21(1 21(1 NUM iajs-3251 125 15 ) ) PUNCT iajs-3251 125 16 , , PUNCT iajs-3251 125 17 17 17 NUM iajs-3251 125 18 - - SYM iajs-3251 125 19 28 28 NUM iajs-3251 125 20 . . PUNCT iajs-3251 126 1 6.aggarwal 6.aggarwal NUM iajs-3251 126 2 , , PUNCT iajs-3251 126 3 s. s. PROPN iajs-3251 126 4 ; ; PUNCT iajs-3251 126 5 gupta gupta PROPN iajs-3251 126 6 , , PUNCT iajs-3251 126 7 a.r a.r PROPN iajs-3251 126 8 . . PROPN iajs-3251 126 9 ; ; PUNCT iajs-3251 126 10 kumar kumar PROPN iajs-3251 126 11 , , PUNCT iajs-3251 126 12 mohand mohand NOUN iajs-3251 126 13 , , PUNCT iajs-3251 126 14 d. d. PROPN iajs-3251 126 15 transform transform NOUN iajs-3251 126 16 of of ADP iajs-3251 126 17 error error NOUN iajs-3251 126 18 function function NOUN iajs-3251 126 19 . . PUNCT iajs-3251 127 1 international international ADJ iajs-3251 127 2 journal journal PROPN iajs-3251 127 3 of of ADP iajs-3251 127 4 research research NOUN iajs-3251 127 5 in in ADP iajs-3251 127 6 advent advent ADJ iajs-3251 127 7 technology technology NOUN iajs-3251 127 8 , , PUNCT iajs-3251 127 9 2019 2019 NUM iajs-3251 127 10 , , PUNCT iajs-3251 127 11 7(5 7(5 NUM iajs-3251 127 12 ) ) PUNCT iajs-3251 127 13 , , PUNCT iajs-3251 127 14 224 224 NUM iajs-3251 127 15 - - SYM iajs-3251 127 16 231 231 NUM iajs-3251 127 17 . . PUNCT iajs-3251 128 1 7.aggarwal 7.aggarwal NUM iajs-3251 128 2 , , PUNCT iajs-3251 128 3 s. s. PROPN iajs-3251 128 4 ; ; PUNCT iajs-3251 128 5 sharma sharma PROPN iajs-3251 128 6 , , PUNCT iajs-3251 128 7 s.d s.d PROPN iajs-3251 128 8 . . PROPN iajs-3251 128 9 sumudu sumudu PROPN iajs-3251 128 10 transform transform NOUN iajs-3251 128 11 of of ADP iajs-3251 128 12 error error NOUN iajs-3251 128 13 function function NOUN iajs-3251 128 14 . . PUNCT iajs-3251 129 1 journal journal NOUN iajs-3251 129 2 of of ADP iajs-3251 129 3 applied apply VERB iajs-3251 129 4 science science NOUN iajs-3251 129 5 and and CCONJ iajs-3251 129 6 computations computation NOUN iajs-3251 129 7 . . PUNCT iajs-3251 130 1 2019 2019 NUM iajs-3251 130 2 , , PUNCT iajs-3251 130 3 6(6 6(6 NOUN iajs-3251 130 4 ) ) PUNCT iajs-3251 130 5 , , PUNCT iajs-3251 130 6 1222 1222 NUM iajs-3251 130 7 - - SYM iajs-3251 130 8 1231 1231 NUM iajs-3251 130 9 . . PUNCT iajs-3251 131 1 ihjpas ihjpas PROPN iajs-3251 131 2 . . PUNCT iajs-3251 132 1 36(2)2023 36(2)2023 NUM iajs-3251 132 2 435 435 NUM iajs-3251 132 3 8.aggarwal 8.aggarwal NUM iajs-3251 132 4 , , PUNCT iajs-3251 132 5 s. s. PROPN iajs-3251 132 6 ; ; PUNCT iajs-3251 132 7 bhatnagar bhatnagar PROPN iajs-3251 132 8 , , PUNCT iajs-3251 132 9 k. k. PROPN iajs-3251 132 10 ; ; PUNCT iajs-3251 132 11 dua dua PROPN iajs-3251 132 12 , , PUNCT iajs-3251 132 13 a. a. NOUN iajs-3251 132 14 dualities duality NOUN iajs-3251 132 15 between between ADP iajs-3251 132 16 elzaki elzaki NOUN iajs-3251 132 17 transform transform NOUN iajs-3251 132 18 and and CCONJ iajs-3251 132 19 some some DET iajs-3251 132 20 useful useful ADJ iajs-3251 132 21 integral integral ADJ iajs-3251 132 22 transforms transform NOUN iajs-3251 132 23 . . PUNCT iajs-3251 133 1 international international ADJ iajs-3251 133 2 journal journal NOUN iajs-3251 133 3 of of ADP iajs-3251 133 4 innovative innovative ADJ iajs-3251 133 5 technology technology NOUN iajs-3251 133 6 and and CCONJ iajs-3251 133 7 exploring explore VERB iajs-3251 133 8 engineering engineering NOUN iajs-3251 133 9 , , PUNCT iajs-3251 133 10 2019 2019 NUM iajs-3251 133 11 , , PUNCT iajs-3251 133 12 8(12 8(12 NUM iajs-3251 133 13 ) ) PUNCT iajs-3251 133 14 , , PUNCT iajs-3251 133 15 4312 4312 NUM iajs-3251 133 16 - - SYM iajs-3251 133 17 4318 4318 NUM iajs-3251 133 18 . . PUNCT iajs-3251 134 1 9.aggarwal 9.aggarwal NUM iajs-3251 134 2 , , PUNCT iajs-3251 134 3 s. s. PROPN iajs-3251 134 4 ; ; PUNCT iajs-3251 134 5 gupta gupta PROPN iajs-3251 134 6 , , PUNCT iajs-3251 134 7 a.r a.r PROPN iajs-3251 134 8 . . PROPN iajs-3251 134 9 ; ; PUNCT iajs-3251 134 10 sharma sharma PROPN iajs-3251 134 11 , , PUNCT iajs-3251 134 12 s.d s.d PROPN iajs-3251 134 13 . . PROPN iajs-3251 134 14 ; ; PUNCT iajs-3251 134 15 chauhan chauhan PROPN iajs-3251 134 16 , , PUNCT iajs-3251 134 17 r. r. PROPN iajs-3251 134 18 ; ; PUNCT iajs-3251 134 19 sharma sharma PROPN iajs-3251 134 20 , , PUNCT iajs-3251 134 21 n. n. PROPN iajs-3251 134 22 mahgoub mahgoub PROPN iajs-3251 134 23 transform transform VERB iajs-3251 134 24 ( ( PUNCT iajs-3251 134 25 laplace laplace NOUN iajs-3251 134 26 - - PUNCT iajs-3251 134 27 carson carson PROPN iajs-3251 134 28 transform transform NOUN iajs-3251 134 29 ) ) PUNCT iajs-3251 134 30 of of ADP iajs-3251 134 31 error error NOUN iajs-3251 134 32 function function NOUN iajs-3251 134 33 . . PUNCT iajs-3251 135 1 international international ADJ iajs-3251 135 2 journal journal NOUN iajs-3251 135 3 of of ADP iajs-3251 135 4 latest late ADJ iajs-3251 135 5 technology technology NOUN iajs-3251 135 6 in in ADP iajs-3251 135 7 engineering engineering NOUN iajs-3251 135 8 , , PUNCT iajs-3251 135 9 management management NOUN iajs-3251 135 10 & & CCONJ iajs-3251 135 11 applied apply VERB iajs-3251 135 12 science science NOUN iajs-3251 135 13 , , PUNCT iajs-3251 135 14 2019 2019 NUM iajs-3251 135 15 , , PUNCT iajs-3251 135 16 8(4 8(4 NUM iajs-3251 135 17 ) ) PUNCT iajs-3251 135 18 , , PUNCT iajs-3251 135 19 92 92 NUM iajs-3251 135 20 - - SYM iajs-3251 135 21 98 98 NUM iajs-3251 135 22 . . PUNCT iajs-3251 136 1 10.aggarwal 10.aggarwal NUM iajs-3251 136 2 , , PUNCT iajs-3251 136 3 s. s. PROPN iajs-3251 136 4 ; ; PUNCT iajs-3251 136 5 sharma sharma PROPN iajs-3251 136 6 , , PUNCT iajs-3251 136 7 s.d s.d PROPN iajs-3251 136 8 . . PROPN iajs-3251 136 9 application application NOUN iajs-3251 136 10 of of ADP iajs-3251 136 11 kamal kamal PROPN iajs-3251 136 12 transform transform NOUN iajs-3251 136 13 for for ADP iajs-3251 136 14 solving solve VERB iajs-3251 136 15 abel abel NOUN iajs-3251 136 16 ’s ’s PART iajs-3251 136 17 integral integral ADJ iajs-3251 136 18 equation equation NOUN iajs-3251 136 19 . . PUNCT iajs-3251 137 1 global global ADJ iajs-3251 137 2 journal journal PROPN iajs-3251 137 3 of of ADP iajs-3251 137 4 engineering engineering NOUN iajs-3251 137 5 science science NOUN iajs-3251 137 6 and and CCONJ iajs-3251 137 7 researches research NOUN iajs-3251 137 8 , , PUNCT iajs-3251 137 9 2019 2019 NUM iajs-3251 137 10 , , PUNCT iajs-3251 137 11 6(3 6(3 NUM iajs-3251 137 12 ) ) PUNCT iajs-3251 137 13 , , PUNCT iajs-3251 137 14 82 82 NUM iajs-3251 137 15 - - SYM iajs-3251 137 16 90 90 NUM iajs-3251 137 17 . . PUNCT iajs-3251 138 1 11.aggarwal 11.aggarwal NUM iajs-3251 138 2 , , PUNCT iajs-3251 138 3 s. s. PROPN iajs-3251 138 4 ; ; PUNCT iajs-3251 138 5 singh singh PROPN iajs-3251 138 6 , , PUNCT iajs-3251 138 7 g.p g.p PROPN iajs-3251 138 8 . . PROPN iajs-3251 138 9 kamal kamal PROPN iajs-3251 138 10 transform transform NOUN iajs-3251 138 11 of of ADP iajs-3251 138 12 error error NOUN iajs-3251 138 13 function function NOUN iajs-3251 138 14 . . PUNCT iajs-3251 139 1 journal journal NOUN iajs-3251 139 2 of of ADP iajs-3251 139 3 applied apply VERB iajs-3251 139 4 science science NOUN iajs-3251 139 5 and and CCONJ iajs-3251 139 6 computations computation NOUN iajs-3251 139 7 , , PUNCT iajs-3251 139 8 2019 2019 NUM iajs-3251 139 9 , , PUNCT iajs-3251 139 10 6(5 6(5 NUM iajs-3251 139 11 ) ) PUNCT iajs-3251 139 12 , , PUNCT iajs-3251 139 13 2223 2223 NUM iajs-3251 139 14 - - SYM iajs-3251 139 15 2235 2235 NUM iajs-3251 139 16 . . PUNCT iajs-3251 140 1 12.fariha 12.fariha NUM iajs-3251 140 2 , , PUNCT iajs-3251 140 3 m. m. NOUN iajs-3251 140 4 ; ; PUNCT iajs-3251 140 5 muhammad muhammad PROPN iajs-3251 140 6 , , PUNCT iajs-3251 140 7 z.i z.i PROPN iajs-3251 140 8 . . PROPN iajs-3251 140 9 ; ; PUNCT iajs-3251 140 10 moazzam moazzam NOUN iajs-3251 140 11 , , PUNCT iajs-3251 140 12 a. a. NOUN iajs-3251 140 13 ; ; PUNCT iajs-3251 140 14 usman usman PROPN iajs-3251 140 15 , , PUNCT iajs-3251 140 16 a. a. PROPN iajs-3251 140 17 ; ; PUNCT iajs-3251 140 18 muhammad muhammad PROPN iajs-3251 140 19 , , PUNCT iajs-3251 140 20 u.n u.n PROPN iajs-3251 140 21 . . PROPN iajs-3251 140 22 subtituition subtituition NOUN iajs-3251 140 23 method method NOUN iajs-3251 140 24 using use VERB iajs-3251 140 25 the the DET iajs-3251 140 26 laplace laplace NOUN iajs-3251 140 27 transformation transformation NOUN iajs-3251 140 28 for for ADP iajs-3251 140 29 solving solve VERB iajs-3251 140 30 partial partial ADJ iajs-3251 140 31 differential differential ADJ iajs-3251 140 32 equations equation NOUN iajs-3251 140 33 involving involve VERB iajs-3251 140 34 more more ADJ iajs-3251 140 35 than than ADP iajs-3251 140 36 two two NUM iajs-3251 140 37 independent independent ADJ iajs-3251 140 38 variables variable NOUN iajs-3251 140 39 . . PUNCT iajs-3251 141 1 bulletin bulletin NOUN iajs-3251 141 2 of of ADP iajs-3251 141 3 mathematics mathematic NOUN iajs-3251 141 4 and and CCONJ iajs-3251 141 5 statistics statistic NOUN iajs-3251 141 6 research research NOUN iajs-3251 141 7 . . PUNCT iajs-3251 142 1 2021 2021 NUM iajs-3251 142 2 , , PUNCT iajs-3251 142 3 9(3 9(3 NUM iajs-3251 142 4 ) ) PUNCT iajs-3251 142 5 , , PUNCT iajs-3251 142 6 104 104 NUM iajs-3251 142 7 - - SYM iajs-3251 142 8 116 116 NUM iajs-3251 142 9 . . PUNCT iajs-3251 143 1 13muhammad 13muhammad NUM iajs-3251 143 2 , , PUNCT iajs-3251 143 3 i.k i.k PROPN iajs-3251 143 4 . . PROPN iajs-3251 143 5 ; ; PUNCT iajs-3251 143 6 khurrem khurrem PROPN iajs-3251 143 7 , , PUNCT iajs-3251 143 8 s. s. PROPN iajs-3251 143 9 ; ; PUNCT iajs-3251 143 10 muhammad muhammad PROPN iajs-3251 143 11 , , PUNCT iajs-3251 143 12 m.u.r m.u.r PROPN iajs-3251 143 13 . . PUNCT iajs-3251 143 14 ; ; PUNCT iajs-3251 144 1 maria maria PROPN iajs-3251 144 2 ; ; PUNCT iajs-3251 144 3 moazzam moazzam NOUN iajs-3251 144 4 , , PUNCT iajs-3251 144 5 a. a. NOUN iajs-3251 144 6 influence influence NOUN iajs-3251 144 7 of of ADP iajs-3251 144 8 bioconvection bioconvection NOUN iajs-3251 144 9 and and CCONJ iajs-3251 144 10 activation activation NOUN iajs-3251 144 11 energy energy NOUN iajs-3251 144 12 on on ADP iajs-3251 144 13 maxwell maxwell PROPN iajs-3251 144 14 flow flow NOUN iajs-3251 144 15 of of ADP iajs-3251 144 16 nano nano NOUN iajs-3251 144 17 - - PUNCT iajs-3251 144 18 fluid fluid NOUN iajs-3251 144 19 in in ADP iajs-3251 144 20 the the DET iajs-3251 144 21 existence existence NOUN iajs-3251 144 22 of of ADP iajs-3251 144 23 motile motile ADJ iajs-3251 144 24 microorganisms microorganism NOUN iajs-3251 144 25 over over ADP iajs-3251 144 26 a a DET iajs-3251 144 27 cylinder cylinder NOUN iajs-3251 144 28 . . PUNCT iajs-3251 145 1 international international ADJ iajs-3251 145 2 journal journal NOUN iajs-3251 145 3 of of ADP iajs-3251 145 4 multidisciplinary multidisciplinary ADJ iajs-3251 145 5 research research NOUN iajs-3251 145 6 and and CCONJ iajs-3251 145 7 growth growth NOUN iajs-3251 145 8 evaluation evaluation NOUN iajs-3251 145 9 . . PUNCT iajs-3251 146 1 2021 2021 NUM iajs-3251 146 2 , , PUNCT iajs-3251 146 3 2(5 2(5 NUM iajs-3251 146 4 ) ) PUNCT iajs-3251 146 5 , , PUNCT iajs-3251 146 6 135 135 NUM iajs-3251 146 7 - - SYM iajs-3251 146 8 144 144 NUM iajs-3251 146 9 . . PUNCT iajs-3251 147 1 14.ammara 14.ammara NUM iajs-3251 147 2 , , PUNCT iajs-3251 147 3 a. a. NOUN iajs-3251 147 4 ; ; PUNCT iajs-3251 147 5 madiha madiha PROPN iajs-3251 147 6 , , PUNCT iajs-3251 147 7 g. g. PROPN iajs-3251 147 8 ; ; PUNCT iajs-3251 147 9 amina amina PROPN iajs-3251 147 10 , , PUNCT iajs-3251 147 11 a. a. NOUN iajs-3251 147 12 ; ; PUNCT iajs-3251 147 13 moazzam moazzam NOUN iajs-3251 147 14 , , PUNCT iajs-3251 147 15 a. a. NOUN iajs-3251 147 16 laplace laplace NOUN iajs-3251 147 17 transforms transform VERB iajs-3251 147 18 techniques technique NOUN iajs-3251 147 19 on on ADP iajs-3251 147 20 equation equation NOUN iajs-3251 147 21 of of ADP iajs-3251 147 22 advection advection NOUN iajs-3251 147 23 - - PUNCT iajs-3251 147 24 diffussion diffussion NOUN iajs-3251 147 25 in in ADP iajs-3251 147 26 one one NUM iajs-3251 147 27 - - PUNCT iajs-3251 147 28 dimensional dimensional ADJ iajs-3251 147 29 with with ADP iajs-3251 147 30 semi semi ADJ iajs-3251 147 31 - - ADJ iajs-3251 147 32 infinite infinite ADJ iajs-3251 147 33 medium medium NOUN iajs-3251 147 34 to to PART iajs-3251 147 35 find find VERB iajs-3251 147 36 the the DET iajs-3251 147 37 analytical analytical ADJ iajs-3251 147 38 solution solution NOUN iajs-3251 147 39 . . PUNCT iajs-3251 148 1 bulletin bulletin NOUN iajs-3251 148 2 of of ADP iajs-3251 148 3 mathematics mathematic NOUN iajs-3251 148 4 and and CCONJ iajs-3251 148 5 statistics statistic NOUN iajs-3251 148 6 research research NOUN iajs-3251 148 7 . . PUNCT iajs-3251 149 1 2021 2021 NUM iajs-3251 149 2 , , PUNCT iajs-3251 149 3 9(3 9(3 NUM iajs-3251 149 4 ) ) PUNCT iajs-3251 149 5 , , PUNCT iajs-3251 149 6 86 86 NUM iajs-3251 149 7 - - SYM iajs-3251 149 8 96 96 NUM iajs-3251 149 9 . . PUNCT iajs-3251 150 1 15.muhammad 15.muhammad NUM iajs-3251 150 2 , , PUNCT iajs-3251 150 3 k. k. PROPN iajs-3251 150 4 ; ; PUNCT iajs-3251 150 5 khurrem khurrem PROPN iajs-3251 150 6 , , PUNCT iajs-3251 150 7 s. s. PROPN iajs-3251 150 8 ; ; PUNCT iajs-3251 150 9 moazzam moazzam NOUN iajs-3251 150 10 , , PUNCT iajs-3251 150 11 a. a. NOUN iajs-3251 150 12 ; ; PUNCT iajs-3251 150 13 ammara ammara ADV iajs-3251 150 14 , , PUNCT iajs-3251 150 15 a. a. NOUN iajs-3251 150 16 ; ; PUNCT iajs-3251 150 17 mariyam mariyam NOUN iajs-3251 150 18 , , PUNCT iajs-3251 150 19 b. b. PROPN iajs-3251 150 20 new new ADJ iajs-3251 150 21 transformation transformation NOUN iajs-3251 150 22 β€˜ ' PUNCT iajs-3251 150 23 amk amk ADJ iajs-3251 150 24 - - PUNCT iajs-3251 150 25 transformation transformation NOUN iajs-3251 150 26 ’ ' PUNCT iajs-3251 150 27 to to PART iajs-3251 150 28 solve solve VERB iajs-3251 150 29 ordinary ordinary ADJ iajs-3251 150 30 linear linear ADJ iajs-3251 150 31 differential differential NOUN iajs-3251 150 32 equation equation NOUN iajs-3251 150 33 of of ADP iajs-3251 150 34 moment moment NOUN iajs-3251 150 35 pareto pareto ADJ iajs-3251 150 36 distribution distribution NOUN iajs-3251 150 37 . . PUNCT iajs-3251 151 1 international international ADJ iajs-3251 151 2 journal journal NOUN iajs-3251 151 3 of of ADP iajs-3251 151 4 multidisciplinary multidisciplinary ADJ iajs-3251 151 5 research research NOUN iajs-3251 151 6 and and CCONJ iajs-3251 151 7 growth growth NOUN iajs-3251 151 8 evaluation evaluation NOUN iajs-3251 151 9 . . PUNCT iajs-3251 152 1 2021 2021 NUM iajs-3251 152 2 , , PUNCT iajs-3251 152 3 2(5 2(5 NUM iajs-3251 152 4 ) ) PUNCT iajs-3251 152 5 , , PUNCT iajs-3251 152 6 125 125 NUM iajs-3251 152 7 - - SYM iajs-3251 152 8 134 134 NUM iajs-3251 152 9 . . PUNCT iajs-3251 153 1 16.moazzam 16.moazzam NUM iajs-3251 153 2 , , PUNCT iajs-3251 153 3 a. a. NOUN iajs-3251 153 4 ; ; PUNCT iajs-3251 153 5 kashif kashif PROPN iajs-3251 153 6 , , PUNCT iajs-3251 153 7 m. m. NOUN iajs-3251 153 8 ; ; PUNCT iajs-3251 153 9 amjed amjed PROPN iajs-3251 153 10 , , PUNCT iajs-3251 153 11 u. u. PROPN iajs-3251 153 12 ; ; PUNCT iajs-3251 153 13 khawar khawar PROPN iajs-3251 153 14 , , PUNCT iajs-3251 153 15 m.i m.i PROPN iajs-3251 153 16 . . PROPN iajs-3251 153 17 development development NOUN iajs-3251 153 18 of of ADP iajs-3251 153 19 a a DET iajs-3251 153 20 new new ADJ iajs-3251 153 21 transformation transformation NOUN iajs-3251 153 22 to to PART iajs-3251 153 23 solve solve VERB iajs-3251 153 24 a a DET iajs-3251 153 25 new new ADJ iajs-3251 153 26 type type NOUN iajs-3251 153 27 of of ADP iajs-3251 153 28 ordinary ordinary ADJ iajs-3251 153 29 linear linear ADJ iajs-3251 153 30 differential differential NOUN iajs-3251 153 31 equation equation NOUN iajs-3251 153 32 . . PUNCT iajs-3251 154 1 bulletin bulletin NOUN iajs-3251 154 2 of of ADP iajs-3251 154 3 mathematics mathematic NOUN iajs-3251 154 4 and and CCONJ iajs-3251 154 5 statistics statistics PROPN iajs-3251 154 6 research research NOUN iajs-3251 154 7 ( ( PUNCT iajs-3251 154 8 bomsr bomsr PROPN iajs-3251 154 9 ) ) PUNCT iajs-3251 154 10 . . PUNCT iajs-3251 155 1 2021 2021 NUM iajs-3251 155 2 , , PUNCT iajs-3251 155 3 9(3 9(3 NUM iajs-3251 155 4 ) ) PUNCT iajs-3251 155 5 , , PUNCT iajs-3251 155 6 56 56 NUM iajs-3251 155 7 - - SYM iajs-3251 155 8 60 60 NUM iajs-3251 155 9 . . PUNCT iajs-3251 156 1 17.muhammad 17.muhammad NUM iajs-3251 156 2 , , PUNCT iajs-3251 156 3 w. w. PROPN iajs-3251 156 4 ; ; PUNCT iajs-3251 156 5 khurrem khurrem PROPN iajs-3251 156 6 , , PUNCT iajs-3251 156 7 s. s. PROPN iajs-3251 156 8 ; ; PUNCT iajs-3251 156 9 moazzam moazzam NOUN iajs-3251 156 10 , , PUNCT iajs-3251 156 11 a. a. NOUN iajs-3251 156 12 ; ; PUNCT iajs-3251 156 13 alizay alizay PROPN iajs-3251 156 14 , , PUNCT iajs-3251 156 15 b. b. PROPN iajs-3251 156 16 applications application NOUN iajs-3251 156 17 of of ADP iajs-3251 156 18 kamal kamal ADJ iajs-3251 156 19 transformation transformation NOUN iajs-3251 156 20 in in ADP iajs-3251 156 21 temperature temperature NOUN iajs-3251 156 22 problems problem NOUN iajs-3251 156 23 . . PUNCT iajs-3251 157 1 scholar scholar NOUN iajs-3251 157 2 journal journal PROPN iajs-3251 157 3 engineering engineering NOUN iajs-3251 157 4 and and CCONJ iajs-3251 157 5 technology technology NOUN iajs-3251 157 6 . . PUNCT iajs-3251 158 1 2022 2022 NUM iajs-3251 158 2 , , PUNCT iajs-3251 158 3 10(2 10(2 NUM iajs-3251 158 4 ) ) PUNCT iajs-3251 158 5 , , PUNCT iajs-3251 158 6 5 5 NUM iajs-3251 158 7 - - SYM iajs-3251 158 8 8 8 NUM iajs-3251 158 9 . . PUNCT iajs-3251 159 1 18.eman 18.eman NUM iajs-3251 159 2 , , PUNCT iajs-3251 159 3 a. a. NOUN iajs-3251 159 4 ; ; PUNCT iajs-3251 159 5 mansour mansour PROPN iajs-3251 159 6 ; ; PUNCT iajs-3251 159 7 emad emad PROPN iajs-3251 159 8 , , PUNCT iajs-3251 159 9 a. a. PROPN iajs-3251 159 10 k. k. PROPN iajs-3251 159 11 ; ; PUNCT iajs-3251 160 1 sadiq sadiq PROPN iajs-3251 160 2 , , PUNCT iajs-3251 160 3 a. a. NOUN iajs-3251 160 4 ; ; PUNCT iajs-3251 160 5 mehdi mehdi PROPN iajs-3251 160 6 . . PUNCT iajs-3251 161 1 on on ADP iajs-3251 161 2 the the DET iajs-3251 161 3 see see NOUN iajs-3251 161 4 transform transform NOUN iajs-3251 161 5 and and CCONJ iajs-3251 161 6 system system NOUN iajs-3251 161 7 of of ADP iajs-3251 161 8 ordinary ordinary ADJ iajs-3251 161 9 differential differential ADJ iajs-3251 161 10 equations equation NOUN iajs-3251 161 11 . . PUNCT iajs-3251 162 1 periodicals periodical NOUN iajs-3251 162 2 of of ADP iajs-3251 162 3 engineering engineering NOUN iajs-3251 162 4 and and CCONJ iajs-3251 162 5 natural natural ADJ iajs-3251 162 6 sciences science NOUN iajs-3251 162 7 , , PUNCT iajs-3251 162 8 2021 2021 NUM iajs-3251 162 9 , , PUNCT iajs-3251 162 10 9(3 9(3 NUM iajs-3251 162 11 ) ) PUNCT iajs-3251 162 12 , , PUNCT iajs-3251 162 13 277281 277281 NUM iajs-3251 162 14 . . PUNCT iajs-3251 163 1 19.moazzam 19.moazzam NUM iajs-3251 163 2 , , PUNCT iajs-3251 163 3 a. a. NOUN iajs-3251 163 4 ; ; PUNCT iajs-3251 163 5 muhammad muhammad PROPN iajs-3251 163 6 , , PUNCT iajs-3251 163 7 z.i z.i PROPN iajs-3251 163 8 . . PROPN iajs-3251 163 9 ; ; PUNCT iajs-3251 163 10 al al PROPN iajs-3251 163 11 - - PUNCT iajs-3251 163 12 zughair zughair NOUN iajs-3251 163 13 transformations transformation NOUN iajs-3251 163 14 on on ADP iajs-3251 163 15 linear linear ADJ iajs-3251 163 16 differential differential ADJ iajs-3251 163 17 equations equation NOUN iajs-3251 163 18 of of ADP iajs-3251 163 19 moment moment NOUN iajs-3251 163 20 pareto pareto ADJ iajs-3251 163 21 distribution distribution NOUN iajs-3251 163 22 . . PUNCT iajs-3251 164 1 proc proc NOUN iajs-3251 164 2 . . 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