An Analytical Study on Fractional Partial Differential Equations by Laplace Transform Operator Method

Transcription

1 Iraioal Joural o Applid Egirig Rsarch ISSN Volum 3 Numbr (8 pp Rsarch Idia Publicaios hp://wwwripublicaiocom A Aalical Sud o Fracioal Parial Dirial Euaios b aplac Trasorm Opraor Mhod SKElaga M Sad 3 ad MHigaz 3 Mahmaics Dparm Facul o ScicTai Uivrsi Tai Saudi Arabia Mahmaics Dparm Facul o Scic Moia Uivrsi Shbi Elkom Egp 3 Dparm o Egirig Mahmaics Facul o Elcroic Egirig Mouia Uivrsi Mou 395 Egp Absrac Alhough a vr xsiv liraur icludig paprs o aplac rasorm o a ucio o a sigl variabl bu a vr lil is a vailabl o h doubl aplac rasorm This papr dals wih h doubl aplac rasorms ad hir applicaio o obai a xac aalic soluio o ohomogous spacim-racioal lgraph uaio W usd Tichmarsh ad Rsidu horms o g w xac soluios Th mai ida o his papr is o dvlop h mhod o h doubl aplac rasorm mhod o solv iiial ad boudar valu problms i mahmaical phsics Kwords Rima-iouvill racioal igrals; doubl aplac rasorm ; Tichmarsh horm; Rsidu horm Mahmaics Subjc Classiicaio: 6A33; 33C; 34A5 INTRODUCTION I h pas wo dcads h widl ivsigad subjc o racioal calculus has rmark ablgaid imporac ad populari du o is dmosrad applicaios i umrous divrs ilds o scic ad girig Ths coribuios o h ilds o scic ad girig ar basd o h mahmaical aalsis I covrs h widl kow classical ilds such as Abl s igral uaio ad viscolasici Also icludig h aalsis o dback ampliirs capacior hor gralizd volag dividrs racioal-ordr Chua- Harl ssms lcrod-lcrol irac modls lcric coducac o biological ssms racioal-ordr modls o uros iig o xprimal daa ad h ilds o spcial ucios [-6] Svral mhods hav b usd o solv racioal dirial uaios racioal parial dirial uaios racioal igro-dirial uaios ad damic ssms coaiig racioal drivaivs such as Adomia s dcomposiio mhod [7 ] H s variaioal iraio mhod [ 6] homoop prurbaio mhod [7 9] homoop aalsis mhod [] spcral mhods [ 4] ad ohr mhods [5 7] This papr is orgaizd as ollows: W bgi b iroducig som cssar diiios ad mahmaical prlimiaris o h racioal calculus hor I scio 3 h doubl aplac rasorm ad h ivrs o doubl aplac rasorm is dmosrad I scio 4 h xisc codiios o h doubl aplac rasorm is proposd I scio 5 w iroducd h covoluio horm o doubl aplac rasorm I h las scio w appl h mhod o doubl aplac rasorm o solv h ohomogous spac- im-racioal lgraph uaio PREIMINARIES AND NOTATIONS I his scio w giv som basic diiios ad propris o racioal calculus hor which ar urhr usd i his papr Diiio A ral ucio h( h spac is said o b i C i hr xiss a ral umbr p p such ha h( h ( whr h ( C[ ad i is said o b i h spac ( C i ad ol i C h Diiio Th Rima-iouvill racioal igral opraor ( J o ordr o a ucio h C is did as J h( ( h( d ( ( J h( h( ( is h wll- kow Gamma ucio Som o h propris o h opraor ( J J h( J h( ( J J h( J J h( ( (3 ( J ar as ollows: J whr ad Diiio 3 Th racioal drivaiv ( i h Capuo's ss is did as D h( ( ( D o h ( h ( or ad h C ( d Th ollowig ar hr basic propris o Capuo's racioal Drivaiv [4]: ( h C Th D h is wll D h C did ad 545

2 Iraioal Joural o Applid Egirig Rsarch ISSN Volum 3 Numbr (8 pp Rsarch Idia Publicaios hp://wwwripublicaiocom ( ad h C Th ( J D h( h( h ( k k (! k k (3 Th racioal drivaiv o ( i h Capuo ss is did as D m m ( J D ( or m m m N ad C DEFINITION OF THE DOUBE APACE TRANSFORM I [8] h doubl aplac rasorm o a ucio ( x o wo variabls x ad did i h irs uadra o h x pla is did b h doubl igral i h orm ( p = [ ( x ] = providd h igral xiss whr x ; x p px p ; ( x p dos h aplac rasorm o p x ; x p x ad o di b px x xdx Rp dxd ad is usd hroughou his papr Similarl is usd o do h ivrs aplac rasormaio o p ad o di b px x p p dp c i c i ci Th ivrs doubl aplac rasorm ( p x is did b h complx doubl igral ormula ( p x i c i px c i dp i d i dci px ( p d whr ( p mus b aalic ucio or all p ad i h rgio did b h iualiis R p c ad R d whr c ad d ar ral cosas o b chos suiabl I is as o s ha is liar igral rasormaio so saisis h liar propr ( p a ( p b g( p a ( p b g( p Whr a ad b ar cosas This shows ha is also a liar rasormaio EXISTANCE CONDITIONS FOR THE DOUBE APACE TRANSFORM Th x is said o b o xpoial ordr b o x a ad i hr xiss a x X ad Y posiiv cosa K such ha or all ad w wri Or uivall lim x K ax by x K ax by x O as x lim x x Such a ucio x ordr as x ax b a b is simpl calld a xpoial x ad clarl i dos o grow K xp ax b as x asr ha Th ollowig horm givs h codiios which mak h x xiss doubl aplac rasorm o 546

3 Iraioal Joural o Applid Egirig Rsarch ISSN Volum 3 Numbr (8 pp Rsarch Idia Publicaios hp://wwwripublicaiocom Thorm 4 I h ucio x is a coiuous ucio i vr ii irvals X ad Y ad xpoial ordr xp ax b h h doubl aplac rasorm o x xiss or all p ad providd R p a ad R b Th ollowig horm compus h ivrs aplac rasorm or som complicad ucios Thorm 4 ( Tichmarsh Thorm F p b a aalic ucio havig o sigulariis i h cu pla l valus Assumig ha F p F p ad h limiig F lim i F F F xis or almos all (i F p o or p ad F p o p or p uiorml i a scor arg p (ii Thr xiss such ha or vr F r i F r i a r r ar dos' dpd o ad Whr ar r Th or a F s Im F d COVOUTION THEOREM OF THE DOUBE APACE TRANSFORM Diiio 5 [9] Th covoluio o g x is dod b g x ad did b x g x x g d d x ad W obsrv ha h covoluio is commuaiv ha is g x g x I x ( p ad gx g( p h g x x g x ( p g ( p Or uival ( p g ( p g x NONHOMOGENOUS SPACE-TIME FRACTIONA TEEGRAPH EQUATION I his scio w cosidr h ollowig ohomogous spac- im racioal lgraph uaio u u u u x Wih h iiial ad boudar codiios u u u x x u x x x Now applig wo dimsioal aplac rasorm wih rspc o x w g p U p p U p U x U p U p U p U p U p U p I which p U p r u x : x p U u : U x u x : U p u x : x p U p u x : x p Ar simpliig w hav: p p U p p p p p Takig ivrs aplac rasorm irs wih rspc o p ad h w hav p p u x p p p Thorm 5 (Covoluio Thorm B usig Tichmarsh Thorm w hav 547

4 Iraioal Joural o Applid Egirig Rsarch ISSN Volum 3 Numbr (8 pp Rsarch Idia Publicaios hp://wwwripublicaiocom p rx p p x r si r dr Similarl w hav: p p p p rx r si r rx dr r si r dr x x r r si r r dr r si r dr x r r si r dr x x x ( ( ( ( ( ad h x x x ( ( ( ( ( J x x Now w id h doubl ivrs aplac rasorm or h ucio wh I p p So h doubl igral bcoms ci ci px I dp d i ci ci p ad h p igral has a pol a p Hc assumig ha c has b a suiabl chos wih p c w ca valua h p igral b akig h rsidu ad obai ci p x x c i x x I d d i c i i ci Obsrv ha his cao irval bcaus o procd wriig b s as a aplac ivrsio x wh x Ahow w x x x x x x 4x ad subsiuig o obai z x x x 3x 4x c i x z x I z dz i c i x Tak c which is allowd bcaus hr ar o sigular pois i h igrad Th w obsrv ha h xpoial ucio i h igrad is v i z Hc h odd rm z i h igrad will o giv a coribuio ad x 3x 4x x i x z I dz x i x 3x 4x x i x i x x 3x 4 4x x 3 x Th h doubl covoluio I J x rads x I J x u x I x J d d x I J x d d So i h prs cas 3 4 x x u x d d 3 x CONCUSIONS W hav applid doubl aplac rasorm mhod o obai h xac soluios o ohomogous spac- im-racioal lgraph uaio Our xampl shows ha doubl aplac rasorm mhod is capabl o rducig h volum o compuaioal work as compard o ohr mhods REFERENCES [] I Podlub Fracioal dirial uaios Acadmic Prss Sa Digo

5 Iraioal Joural o Applid Egirig Rsarch ISSN Volum 3 Numbr (8 pp Rsarch Idia Publicaios hp://wwwripublicaiocom [] I Podlub Gomric ad phsical irpraio o racioal igraio ad racioal diriaio FracCalculus Appl Aal 5(: [3] J H Noliar oscillaio wih racioal drivaiv ad is applicaios Iraioal Corc o Vibraig Egirig Dalia Chia (998:88-9 [4] J H Som applicaios o oliar racioal dirial uaios ad hir approximaios Bull Sci Tchol 5(999:86-9 [5] J H Approxima aalical soluio or spag low wih racioal drivaivs i porous mdia Compu Mhods Appl Mch Eg 67(998:57-68 [6] I Grigorko E Grigorko Chaoic damics o h racioal lorz ssm Phs Rv 9(3:34 [7] S Momai N T Shawagh Dcomposiio mhod or solvig racioal Riccai dirial uaios ApplMah Compu 8(6:83-9 [8] S Momai M A Noor Numrical mhods or ourh-ordr racioal igrodirial uaios Appl Mah Compu 8(6: [9] V D Gjji H Jaari Solvig a muli-ordr racioal dirial uaio Appl Mah Compu 89(7: [] S S Ra K S Chaudhuri R K Bra Aalical approxima soluio o oliar damic ssm coaiig racioal drivaiv b modiid dcomposiio mhod Appl Mah Compu 8(6: [] QWag Numrical soluios or racioal KdV- Burgrs uaio b Adomia dcomposiio mhod Appl MahCompu 8(6:48-55 [] M Ic Th approxima ad xac soluios o h spac- ad im-racioal Burgrs uaios wih iiial codiios b variaioal iraio mhod JMah Aal Appl 345(8: [3] S Momai Z Odiba Aalical approach o liar racioal parial dirial uaios arisig i luid mchaics Phs A 355(6:7-79 [4] Z Odiba S Momai Applicaio o variaioal iraio mhod o oliar dirial uaios o racioal ordr I J Noliar Sci Numr Simul 7 (6:7-79 [5] S Momai Z Odiba Homoop prurbaio mhod or oliar parial dirial uaios o racioal ordr Phs A 365(7: [6] N H Swilam M M Khadr R F Al-Bar Numrical sudis or a muli-ordr racioal dirial uaio Phs A 37(7:6-33 [7] K A Gprl Th homoop prurbaio mhod applid o h oliar racioal Kolmogorov- Provskii-PISkuov uaios Applid Mahmaics rs 4(8(: [8] M A Hrzallah ad K A Gprl Approxima soluio o im spac racioal cubic oliar Schrodigr uaio Applid Mahmaical Modlig 36((: [9] M S Mohamd Aalical ram o Abl igral uaios b opimal homoop aalsis rasorm mhod Joural o Iormaio ad Compuig Scic; ( (5: 9-8 [] Mohamd S Mohamd Khald A Gprl Faisal Al- Malki Maha Al-humai Approxima soluios o h gralizd Abl's igral uaios usig h xsio Kha's homoop aalsis rasormaio mhod Joural o Applid Mahmaics ( Hidawi Publishig Corporaio (Volum 5 Aricl ID pags [] K A Gprl ad M S Mohamd Aalical approxima soluio or oliar spac-im racioal Kli Gordo uaio Chis phsics B (( 3:-6 [] K Hmida M S Mohamd Numrical simulaio o h gralizd Huxl uaio b homoop aalsis mhod Joural o applid ucioal aalsis 5(4(: [3] K A Gprl ad M S Mohamd A opimal homoop aalsis mhod oliar racioal dirial uaio Joural o Advacd Rsarch i Damical ad Corol Ssms 6((4 : [4] K A Gprl ad S Omra Exac soluios or oliar parial racioal dirial uaios Chis Phsics B ((: 4 [5] R Churchill Opraioal Mahmaics (3rd diio McGraw-HillNw York 97 [6] DG Du Trasorm Mhods or Solvig Parial Dirial Chapma ad Hall/CRC 4 [7] G Dosch Iroducio o h Thor ad Applicaios o h aplac Trasormaio Sprigr- Vrlag Brli 974 [8] SKElaga O h Ivalidi o Smigroup Propr or h Miag-lr Fucio wih Two Paramrs" Egpia Joural o Mahmaical Soci 4 ( (6: -3 [9] okah Dbah Th Doubl aplac Trasorms ad Thir Propris wih Applicaios o Fucioal Igral ad Parial Dirial Euaios I J Appl Compu (