Through the fractional Riemann Liouville integral x
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1 Volue 7 Issue 5 M 7 ISSN: 77 8X Ierol ourl o Advced Reserch Copuer Scece d Sowre geerg Reserch Pper Avlle ole : wwwjrcsseco se he Soluo o Frcol erel quos wh Trscedel Fucos Mukesh Grover r Aru Kur Toer Asss Proessor epre o Mhecs GZS Cpus College o geerg & Techolog Mhrj Rj Sgh Puj Techcl Uvers Bhd Puj Id Assoce Proessor epre o Mhecs SMRS College Phko Puj Id OI: 956/jrcsse/SV7I5/9 Asrc: I hs prese pper we hve oed he soluo o Frcol order derel equos wh he help o rcol dervve uder codo he ge soluos wh Rel dsc d repeed d cople correspodg roos lke s rgooerc hperolc The oucoes or o soluos re solvg he ers o Re-Louvlle ucol Mg-Leler uco Cpuo deo d ore dsc ucos lke rcol rgooerc ucos Kewords: erel rsor ehod Re-Louvlle Cpuo Moded Re-Louvlle deos Mg-Leler ucos rcol derel equos I INTROUCTION M eecve dere echques or solvg s uercl d lcl soluos o Fs hve ee preseed Re-Louvelle rcol dervves Cpuo ure dervve re helps or eloro o develop he soluo o Fs We hve used oos or ure pe rcol dervve operor here There re pes o rcol egrl d derel operors Ou o hese oe s he Re Louvlle deo whch dels scece d egrg wh oe lo o hs s dervve o cos uco s o ero Sce he deo o rcol dervve s odg o rele wh he clsscl dervve Re-Louvlle deo he rcol dervve o cos s o-ero whch crees dcul o rele ewee he sc clculus To overcoe hs dcul ure oded he deo o rcol dervve o Re-Louvelle pe d wh hs ew orulo we o he dervve o cos s ero erel equos o rcol order were he prcple rge o he sudes ecuse o her coo pperce vrous pplcos lud echcs sceces d egeerg These ds ssve qu o lerures dvced cocerg wh Fs oler dcs [-5-5 O he grouds h u rcol derel equos do ow o hve geue lc swers pproo d uercl echques cosequel he re used susll lel he Ado decoposo echque d vrol ew relese ehod were used or decso lrge rge o proles [78 II RIMANN- LIOUVILL AN RIVATIVS IN FRACTIONAL CALCULUS ch deo kes use o Re Louvlle rcol egro d dervves o whole order Le : R R deoe couous uco d le he pro h he ervl [ Through he rcol Re Louvlle egrl I d d d d he oded Re-Louvlle dervve s deed s where [ The derece ewee he wo deos s he order o evluo Re Louvlle rcol egro o order α s ouled s d 7 IARCSS All Rghs Reserved Pge 97
2 Grover e l Ierol ourl o Advced Reserch Copuer Scece d Sowre geerg 75 M- 7 pp IARCSS All Rghs Reserved Pge 97 d d * d d where d N For ow he Cpuo rcol -produc c e deoed usg * o cler derece wh he Re Louvlle rcol dervve III MITTAG-LFFLR FUNCTION Mg-Leler Fuco helps o d ou he soluo o dsc equos lke s rcol order derel egrl derece equos ec C C Soe rscedel uco re dscussed wh he help o Mg-Leler uco e ep rgooer hperolc ucos d ore e cos Fro he ure deo o rcol dervve we hve C wh he help o order o ure dervve wh he l po R sh s cos Sh Wh he help o ove epresso we c prove h ollowg resuls or o rs dervves o rgooerc d hperolc ucos re s ollows: sh sh IV FRACTIONAL ORR IFFRNTIAL QUATIONS WITH ARBITRARY ORR Α For dg he coplee soluo o No-Hoogeous Frcol order derel equos wh rrr order α For dg he geerl soluo wh s rel & dsc rel& repeed cople roos he su o s copleer soluo d prculr egrl wh order equo derel The rcol B A where A d B re coss equo derel The rcol c c c c c c C B A where AB d C re coss
3 Grover e l Ierol ourl o Advced Reserch Copuer Scece d Sowre geerg 75 M- 7 pp IARCSS All Rghs Reserved Pge 97 For he evluo o he prculr egrl or vlue o s u or less h u he ppl ol epso d he cse o rgooerc ucos lke s s replce Now prculr egrl or we ge Fll ge he coplee soluo wll e B A + Now d he prculr egrl o s s p s Slrl o d he prculr egrl o cos sh We hve dscussed he pleeo o he ove eoed echques wh rgooerc ucos d hperolc uco s ollows Frs we hve dscussed cse o hperolc uco o sh sh sh sh We kow h uler s heore sh sh Slrl we c dscuss cse o hperolc uco o h Thrdl we hve dscussed cse o rgooerc uco o e We kow h uler s heore
4 Grover e l Ierol ourl o Advced Reserch Copuer Scece d Sowre geerg 75 M- 7 pp IARCSS All Rghs Reserved Pge 97 V ILLUSTRATIV XAMPLS I hs seco he ehod preseed ove s ppled I gves coplee soluo o rcol derel equo wh rrr order Here we hve dscussed rel d dsc roos rel d repeed roos he or o copleer uco Le us suppose h vlue o les ewee [ wh equl-spced ervls o legh ple As he s eple we cosder he ollowg rcol derel equo wh oudr codos = = 5 sh [ 5 : [ Soluo Copleer Fuco s B A Geerl Soluo wll e sh 6 B A 5 ple we cosder he ollowg rcol derel equo wh oudr codos = = 5 s 9 [ 9 : [ Soluo Copleer Fuco s B S A 55 Geerl Soluo wll e s 8 B S A 56 ple We cosder he ollowg rcol derel equo wh oudr codos = = 57 [ [ [ [ : Soluo Copleer Fuco s B S A 58 Geerl Soluo wll e S B S A
5 Grover e l Ierol ourl o Advced Reserch Copuer Scece d Sowre geerg 75 M- 7 pp Wh The pproe soluos or = 9 re show Fgure he help o sowre MATHMATICA We hve see hese grphs prculr rge Fgure- 5 Seres Seres Fgure Seres Seres -5 Fgure Seres Seres - - VI CONCLUSIONS Ths prese lss ehs he perece o he purposed ehodolog o urvel sses o derel equos o rcol order The soluos re oed here ers o Mg-Leler uco d rgooerc hperolc ucos wh he help o sowre MATHMATICA Here we've go red v usge o ure rcol dervve h or geg he prculr egrls or orcg perors scled uco o vrle eses he sreg d h we ge cojugo wh clsscl echque o urvel clsscl o-hoogeeous derel equos We hve developed here durg hs pper o erchge he operor s were uled clsscl clculus provdes ese evlug eplc egrls The proposed ehod s prosg d pplcle o rod clss o ler d oler proles he heor o rcol clculus RFRNCS [ M Cpuo Ler odels o dsspo whose Q s los requec depede II Geophs Ro Asroo Soc IARCSS All Rghs Reserved Pge 975
6 Grover e l Ierol ourl o Advced Reserch Copuer Scece d Sowre geerg 75 M- 7 pp [ V rdr-gejj H r A erve ehod or solvg oler ucol equos Mh Al Appl [ V rdr-gejj H r Alss o sse o ouooous rcol derel equos volvg Cpuo dervves Mh Alss Appl [ H r V rdr-gejj Solvg sse o oler rcol derel equos usg Ado decoposo Copuer Appl Mh [5 S Mo Z Od Nuercl pproch o derel equos o rcol order Copu Appl Mh 6 do: 6/jc675 [6 S Mo Z Od Nuercl coprso o ehods or solvg ler derel equos o rcol order Chos Sol Frc [7 Z Od S Mo Approe soluos or oudr vlue proles o e-rcol wve equo Appl Mh Copu [8 N Shwgeh Alcl pproe soluos or oler rcol derel equos Appl Mh Copu [9 U Ghosh S Segup S Srkr d S s Alc soluo o ler rcol derel equo wh ure dervve er o Mg-Leler uco Aerc ourl o Mhecl Alss [ G ure Fourer s Trsoro o rcol order v Mg-Leler uco d oded Re- Louvlle dervves Appl Mh & orcs [ Grover Mukesh Toer Aru kur Nuercl Soluo o Ler Ordr erel quos o Hgher Order erel Trsoro Mehod IST - Ierol ourl o Scece Techolog & geerg Volue Issue 6 eceer 6 ISSN ole: 9-78X [ K ehel N Ford d A Freed A predcor-correcor pproch or he uercl soluo o rcol derel equos Noler cs 9 - [ Grover Mukesh Toer Aru kur A New Techque o Iculce he Prculr Soluo o Frcol Order α { or or } erel quos Wh Boudr Codos Ierol ourl o Cop & Mhecl Sceces Vol 6Issue-5 7 pp67-75 [ F B Add d Cresso Frcol erel equos d he Schrodger equo App Mh d Cop65-5 [5 VV Ah d R Mcvsh Frcol erel quos ve Lev Nose o Appl Mh d Soch Al 6:97-9 [6 K ehel d Nevlle Ford Alss o Frcol erel quos Mh Alss Appl Volue 65 Issue 5 ur Pges 9-8 [7 A A M Ar S Z Rd H Mohed Hooop Alss Mehod or Solvg Bologcl Populo Model Coucos Theorecl Phscs vol56 No5 7 IARCSS All Rghs Reserved Pge 976
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