Fractional Complex Transform for Solving the Fractional Differential Equations

Transcription

1 Global Joral of Pr ad Applid Mahmaics. SSN Volm Nmbr 8 pp. 7-7 Rsarch dia Pblicaios hp:// Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios A. M. S. Mahdy ad G. M. A. Marai Dparm of Mahmaics Facly of Scic Zagazig Uivrsiy Zagazig Egyp Dparm of Mahmaics ad Saisics Facly of Scicaif Uivrsiy Sadi Arabia. Dparm of Mahmaics Facly of Scic Bghazi Uivrsiy Bghazi Libya. Absrac his papr fracioal compl rasform FC wih hlp of Nw raiv Mhod NM is sd o obai mrical ad aalyical solios for h fracioal Fokkr-plack qaio Fracioal Noliar Gas Dyamics qaio ad h oliar im-fracioal Fishr s qaio ad fracioal lgraph qaio. Fracioal compl rasform FC is proposd o covr fracioal diffrial qaios o is diffrial parr ad h applid NM o h w obaid qaios. Svral ampls ar giv ad h rsls ar compard o ac solios. h rsls rval ha h mhod is vry ffciv ad simpl. Kywords: Fracioal compl rasform Nw iraiv mhod fracioal Fokkr-plack qaio fracioal oliar Gas Dyamics qaio ad h oliar im-fracioal Fishr s qaio ad fracioal lgraph qaio.. NRODUCON Fracioal modls hav b show by may sciiss o adqaly dscrib h opraio of variy of physical ad biological procsss ad sysms. Co sqly cosidrabl aio has b giv o h solio of fracioal ordiary diffrial qaios igral qaios ad fracioal parial diffrial qaios of physical irs. Sic mos fracioal diffrial qaios do o hav ac aalyic solios approimaio ad mrical chiqs hrfor ar sd sivly Nmrical ad aalyical mhods hav icldd fi diffrc mhod 7 7 Adomia dcomposiio mhod -.

2 8 A. M. S. Mahdy ad G. M. A. Marai rasform is a impora mhod o solv mahmaical problms. May sfl rasforms for solvig varios problms wr appard i op lirar sch as h ravllig wav rasform h Laplac rasform h Forir rasform 8 h Bückld rasformaio 8 h igral rasform 9 ad h local fracioal igral rasforms 9. Vry rcly h fracioal compl rasform - - was sggsd o covr fracioal ordr diffrialqaios wih modifid Rima-Liovill drivaivs io igr ordr diffrial qaios ad h rsla qaios ca b solvd by advacd calcls. h im-fracioal Fokkr-Plack qaio srvs as a mahmaical modl for a mbr of problms i physical ad biological scics. arissfrom a diffsio approimaio of som sochasic procsss rgarcd as Markovia ad coios. is a bralizd diffsio qaio govrig h volio of h probabiliy dsiy i im. For h wo-variabl cas o whilch aio is rsricd hr. is commoly kow ha h qaio of Gas dyamics is h mahmaical prssios of cosrvaio laws which is i girig pracics sch as cosrvaio of mass cosrvaio of momm cosrvaio of rgy c. h oliar qaios of idal gas dyamics ar applicabl for hr yps of oliar wavs lik shock fros rarfacios ad coac discoiiis. 98 Sgr ad Warmig addrssd ha h cosrvaio-law form of h iviscid gas dyamic qaio posssss a rmarkabl propry by vir of which h oliar fl vcors ar homogos fcios of dgr o which prmis h spliig of fl vcors io sbvcors by similariy rasformaios 5. h im-fracioal Fishr s qaio FFE which is a mahmaical modl for a wid rag of impora physical phoma is a parial diffrial qaio obaid from h classical Fishr qaio by rplacig h im drivaiv wih a fracioal drivaiv of ordr. h lgraph qaio dvlopd by Olivr Havisid i 88 is widly sd i Scic ad Egirig. s applicaios aris i sigal aalysis for rasmissio ad propagaio of lcrical sigals ad also modllig racio diffsio. rc yars gra irs has b dvlopd i fracioal diffrial qaio bcas of is frq apparac i flid mchaics mahmaical biology lcrochmisry ad physics. A spac-im fracioal lgraph qaio is obaid from h classical lgraph qaio by rplacig h im ad spac drivaiv rms by fracioal drivaivs ad compl rasform mhod -. h papr is orgaizd as follows: scio w provid h Basic Compl rasform. Scio Basic da of Nw raiv Mhod NM. Scios Applicaios.Scios 5 Coclsio.

3 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios 9. FRACONAL COMPLE RANSFORM FC Cosidr h followig gral fracioal dffrial qaio f.... y z y z whr y z / dos h modifid Rima-Liovill drivaiv. < < < <. rodcig h followig rasforms q. p. k Y. l Z.5 whr p ; q; k ad l ar cosas. Usig h abov rasforms w ca covr fracioal drivaivs io classical drivaivs: q. p.7

4 A. M. S. Mahdy ad G. M. A. Marai Y k y.8 Z l z.8 hrfor w ca asily covr h fracioal diffrial qaios io parial diffrial qaios so ha vryo familiar wih advacd calcls ca dal wih fracioal calcls wiho ay diffcly. For ampl cosidr a fracioal diffrial qaio. 5 z y.9 By sig h abov rasformaios w g:. 5 Z k Y k p q. which ca b solvd by Nw raiv mhod.. NEW ERAVE MEHODNM o dscrib h ida of h NM cosidr h followig gral fcioal qaio : N f. whr N is a oliar opraor from a Baach spac B B ad f is a kow fcio. W ar lookig for a solio of. havig h sris form i i. h oliar opraor N ca b dcomposd as follows j j j j j i i N N N N. From Eqs.. ad. Eq.. is qival o i j j j j i i N N N f.

5 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios W dfi h rcrrc rlaio: f.5a N.5b.... N... N c h:... N..... i f Ni.. i i f N is a coracio i.. N N y k y < k < N... N....7 k... k... ad h sris i absolly ad iformly covrgs o a solio of. which i is iq i viw of h Baach fid poi horm. h k-rm approima solio of. ad. is giv by i. Rliabl Algorihm Afr h abov prsaio of h NM w irodc a rliabl algorihm for solvig oliar parial diffrial qaios sig h NM. Cosidr h followig oliar parial diffrial qaio of arbirary ordr: D A B N.8a k i wih h iiial codiios h m... m.8b whr A is a oliar fcio of ad parial drivaivs of wih rspc o ad ad B is h sorc fcio. viw of h igral opraors h iiial

6 A. M. S. Mahdy ad G. M. A. Marai val problm.8a ad.8b is qival o h followig igral qaio! N f A B m h m m m.9 Whr! B m h f m m m. ad A N. whr is a igral opraor of fold. W g h solio of.9 by mployig h algorihm.5.. APPLCAONS his scio W apply h w iraiv mhod approach o sdy fiv ampls. Eampl W cosidr h Fracioal Fokkr-plack qaio:. sbjc o h iiial codiio. o apply FC o Eq.. w s h abov rasformaios: q So w hav h followig parial diffrial qaio: q. For simpliciy w s q so w g.

7 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios Now w solv Eq.. by mas of NM. o apply NM o. w cosrc h corrcio fcioal as follows:!! By h fracioal compl rasform W hav

8 A. M. S. Mahdy ad G. M. A. Marai

9 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios 5. Eampl 5 W cosidr h followig oliar im- fracioal gas dyamics qaio of h form: D > <.5 sbjc o h iiial codiio. o apply FC o Eq..5 w s h abov rasformaios: q so w hav h followig parial diffrial qaio: q.7

10 A. M. S. Mahdy ad G. M. A. Marai For simpliciy w s q so w g.8 Now w solv Eq..8 by mas of NM. o apply NM o.8 w cosrc h corrcio fcioal as follows:.9....!!

11 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios 7 By h fracioal compl raform W hav

12 8 A. M. S. Mahdy ad G. M. A. Marai " U S E D E F " ; v a lid f il " F " ; w id h. 8 i ; h ig h. 8 i ; d p h i ; o r ig i a l w id h. 7 i ; o r ig i a l h i " G R A P H C " ; m a i a i a s p c r a io R U E ; d is p la y " U S E D E F " ; v a lid il " F " ; w id h. 8 i ; h ig h. f

13 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios 9. Eampl W cosidr h followigh im-fracioal Fishr s qaio FFE of h form: < > D. sbjc o h iiial codiio. o apply FC o Eq.. w s h abov rasformaios: q so w hav h followig parial diffrial qaio: q. For simpliciy w s q so w g. Now w solv Eq.. by mas of NM. o apply NM o. w cosrc h corrcio fcioal as follows:

14 A. M. S. Mahdy ad G. M. A. Marai By h fracioal compl raform w hav ; ;. 8 ;. 8 ; " " ; " " ; w i o r ig i a l i d p h i h ig h i w id h F il v a lid U S E D E F d is p la y r a io R U E s p c f

15 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios

16 A. M. S. Mahdy ad G. M. A. Marai. Eampl W cosidr h im-fracioal lgraph qaio: D. sbjc o h iiial codiio.5 o apply FC o Eq.. w s h abov rasformaios: p so w hav h followig parial diffrial qaio: p. For simpliciy w s p so w g.7 Now w solv Eq..7 by mas of NM. o apply NM o.7 w cosrc h corrcio fcioal as follows:

17 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios By h fracioal compl raform w hav

18 A. M. S. Mahdy ad G. M. A. Marai

19 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios 5 5. CONCLUSON his papr w hav sccssflly dvlopd FC wih hlp of NM o obai approima solio of h fracioal diffrial qaios. h fracioal compl rasform ca asily covr a fracioal diffrial qaios o is diffrial parr so ha is Nw raiv algorihm ca b simply cosrcd. h fracioal compl rasform is rmly simpl b ffciv for solvig fracioal diffrial qaios. h mhod is accssibl o all wih basic kowldg of Advacd Calcls ad wih lil fracioal calcls. may b cocldd ha FC NM is vry powrfl ad ffici i fidig aalyical as wll as mrical solios for wid classs of fracioal diffrial qaios. REFERENCES A. Bkir 9 Nw ac ravllig wav solios of som compl oliar qaios Commicaios i Noliar Scic ad Nmrical Simlaio A. Hmda Formlaio ad solio of h-ordr drivaiv fzzy igro-diffrial qaio sig w iraiv mhod wih a rliabl algorihm. Joral of Applid Mahmaics Aricl D 57 7 pags. A. Jrri 999 rodcio o gral Eqaios wih Applicaios. Scod Ed. Wily. rscic. A. R. Mohamd ad S. A. Mohamd Nw raiv Mhod for Solvig h Forbrg-Whiham Eqaio ad Compariso wih Homoopy Prrbaio rasform MhodBriish Joral of Mahmaics & Compr Scic 9: A. S. Abdl-Rady S. Z. Rida A. A. M. Arafa H. R. Abdl-Rahim Variaioal raio Smd rasform Mhod for Solvig Fracioal Noliar Gas Dyamics Eqaio raioal Joral of Rsarch Sdis i Scic Egirig ad chology A. S. Mohamd A. M. S. Mahdy ad A. A. H. Mawa 5 Aproima aalyical solio o a im-fracioal Fokkr-Plack qaio BOHALA Joral C. adjra M. M. Mrschar 7 A scod-ordr accra mrical mhod for h wo-dimsioal fracioal diffsio qaio J. Comp. Phys E. Sjdi Djrovi Fracioal Forir rasform as a sigal procssig ool: A ovrviw of rc dvlopms Sigal Procssig F. Maiardi 997 Fracioal calcls: Som basic problms i coim ad saisical mchaics Sprigr-Vrlag Nw York pp Podlby 999 Fracioal Diffrial Eqaios Acadmic Prss Nw York.

20 A. M. S. Mahdy ad G. M. A. Marai. Podlby Gomric ad physical irpraio of fracioal igraio ad fracioal diffriaio Frac. Calc. Appl. Aal J. H ad Z. Li Covrig Fracioal diffria qaios io parial diffrial qaios hrmal Scic DO REFERENCE:.98/SC58H J. H A Shor Rmark o Fracioal Variaioal raio Mhod Physics Lrs A DO:./j.physla.7. J. H. H 998 Noliar oscillaio wih fracioal drivaiv ad is applicaios. : raioal Cofrc o Vibraig Egirig 98 Dalia Chia J. H. H 999 Som applicaios of oliar fracioal diffrial qaios ad hir approimaios Bll. Sci. chol M. Cassol S. Worma ad U. Rizza 9 Aalyic modlig of wodimsioal rasi amosphric polla disprsio by dobl G ad Laplac rasform chiqs Eviromal Modllig & Sofwar M. M. Mrschar ad C. adjra Fii diffrc approimaios for wo sidd spac-fracioal parial diffrial qaios Appl. Nmr. Mah P. Gordoa A. Pickrig ad Z. Zh Bckld rasformaios for a mari scod Pailv qaio Physics Lrs A R. Coa ad M. Mikhailov 99 gral rasform mhod Applid Mahmaical Modllig R. Ylia Molliq ad B. Baiha Approima aalyic solios of fracioal Zakharov Kzsov qaios by fracioal compl rasform raioal Joral of Egirig ad chology -. R. Ci ad Y. H Fracioal powr sris mhod for solvig fracioal diffrmial qaio Joral of Advacs i Mahmaics. S. Bhalkar ad V. Dafardar-Gjji 8 Nw iraiv mhod: applicaio o parial diffrial qaios. Applid Mahmaics ad Compaio S. Bhalkar ad V. Dafardar-Gjji Solvig volio qaios sig a w iraiv mhod Nmrical Mhods for Parial Diffrial Eqaios 9-9. S. Momai ad K. Al-Khald 5 Nmrical solios for sysms of fracioal diffrial qaios by h dcomposiio mhod Appl. Mah. Comp S. Momai 5 A plici ad mrical solios of h fracioal KdV qaio Mah. Comp. Siml S. Momai ad Z. Odiba 7 Homoopy prrbaio mhod for oliar parial diffrial qaios of fracioal ordr Physics Lrs A

21 Fracioal Compl rasform for Solvig h Fracioal Diffrial Eqaios V. Dafardar-Gjji ad H. Jafari A iraiv mhod for solvig oliar fcioal qaios Joral of Mahmaical Aalysis ad Applicaios V. Dafardar-Gjji ad S. Bhalkar Solvig fracioal bodary val problms wih Dirichl bodary codiios sig a w iraiv mhod Comprs & Mahmaics wih Applicaios Yag Local Fracioal gral rasforms Progrss i Noliar Scic -5. Y. Chrral 989 Covrgc of Adomia s mhod. Kybrs8-8. Z. Li ad J.H Fracioal Compl rasform for Fracioal Diffrial Eqaios Mahmaical ad Compaioal Applicaios Z. Li A Edd Fracioal Compl rasform Joral of Noliar Scic ad Nmrical Simlaio 5-7. A. Saravaa ad N. Magsh A ffici compaioal chiq for solvig h Fokkr Plack qaio wih spac ad im fracioal drivaivs Joral of Kig Sad Uivrsiy Scic 8 -. Y. C. Esiz ad A. Kr Nw Fracioal Compl rasform for Coformabl Fracioal Parial Diffrial Eqaios JAMS O. Ozka 7 Approima Aalyical Solios of Sysms of Fracioal Parial DiffrialEqaios Karalmas F v Müh. Drg O. Gr ad A. Bkir 8 Solvig oliar spac-im fracioal diffrial qaios via asaz mhod Compaioal Mhods for Diffrial Eqaios -.

22 8 A. M. S. Mahdy ad G. M. A. Marai