New Method to Solve Partial Fractional Differential Equations
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- Miles Fox
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1 Global Joal of Pe ad Applied Mahemaics ISSN Volme 3 Nmbe 9 7 pp eseach Idia Pblicaios hp://ipblicaiocom Ne Mehod o Solve Paial Facioal iffeeial Eqaios M iahi E Edfa 3 ad K El ashid 4 5 epame of Mahemaics ad Saisics Facl of Sciece Taif ivesi 974 Kigdom of Sadi Aabia Caage ivesi Tisia 3 epame of Mahemaics ad Saisics Facl of Sciece Assi ivesi Egp 4 Mahemaics epame College of as ad Sciece Taif ivesi aah Sadi Aabia 5 Mahemaics epame Facl of Sciece Bei-Sef ivesi Egp * Coespodig aho Absac I his pape e esablish a modified edced diffeeial asfom mehod hich ae sccessfll applied o obai he aalical solios of he imespace facioal Navie-Soes eqaios The facioal deivaive is ae i Capo sese The obaied esls sho ha he poposed echiqes ae simple efficie ad eas o impleme fo facioal diffeeial eqaios We mae he Figes o compae beee he appoimae solios We compae beee he appoimae solios ad he eac solios fo he paial facioal diffeeial eqaios he α β INTOCTION Noliea diffeeial eqaios descibe ma phsical pheomea ad aalic solio o hese eqaios is impoa becase he sall coai global ifomaio o he solio sces of hese oliea eqaios ] Ma aalic appoaches fo solvig oliea diffeeial eqaios have bee poposed ad he mos osadig oe is he homoop aalsis mehod HAM I ece
2 4736 M iahi E Edfa ad K El ashid eas ma ahos have paid aeio o sdig he solios of oliea paial diffeeial eqaios b vaios mehods 3-4] I his pape e coside he sead flo of a viscos flid i a be he veloci field is a fcio of ol oe space coodiae ad he ime is a depede vaiable This id of ime-space facioal Navie-Soes eqaio has bee sdied b Momai ad Odiba 5] Kma e al 6 7] ad Kha 8] b sig he Adomia decomposiio mehod AM he homoop pebaio asfommehod HPTM hemodified Laplace decomposiio mehod MLM he vaiaioal ieaio mehod VIM ad he homoop pebaio mehod HPM especivel I 6 afada-gejji ad Jafai 9] ee fis o popose he Gejji-Jafai ieaio mehod fo solvig a liea ad oliea facioal diffeeial eqaio The Gejji- Jafai ieaio mehod is eas o impleme ad obais a highl accae esl The edced diffeeial asfom mehod TM as fis poposed b Kesi ad Oac ] The TM as also applied b ma eseaches o hadle oliea eqaios aisig i sciece ad egieeig I ece eas Kma e al 8] sed vaios mehods o sd he solios of liea ad oliea facioal diffeeial eqaio combied ih a Laplace asfom The mai aim of his aicle is o pese appoimae aalical solios of imespace facioal Navie Soes eqaio b sig he edced diffeeial asfom mehod MTM The Navie Soes eqaio NSE ih ime-space facioal deivaives is ie i opeao fom as: p Whee is paamee descibig he ode of he ime facioal deivaives I he case of he facioal eqaio edces o he sadad Navie Soes eqaio BASIC EFINITION OF FACTIONAL CALCLS I his secio e give some basic defiiios ad popeies of facioal calcls heo hich shall be sed i his pape: efiiio The facioal deivaive of f i he Capo s sese is defied as 9]
3 Ne Mehod o Solve Paial Facioal iffeeial Eqaios 4737 f f d fo efiiio Fo he Capo facioal deivaive of ode o he hole space deoed b c c f is defied b 3] f d Pope Some sefl fomla ad impoa popeies fo he modified iema- Lioville deivaive as follos 3-34]: f g f g g f f g f g g g 3 edced diffeeial asfom mehod TM f g f g g ] g I his secio e iodce he basic defiiios of he edced diffeeial asfomaios Coside a fcio of hee vaiables ad assme ha i ca be epeseed as a podc -] F G Based o he popeies of oe dimesioal diffeeial asfom he fcio ca be epeseed as i i j i i F i i G j W i i i i j i i j hee W i i F i i G j is called he specm of Le deoes he edced diffeeial asfom opeao ad he ivese edced diffeeial asfom opeao The basic defiiio ad opeaio of he TM mehod is descibed belo efiiio 3 If is aalic ad coiosl diffeeiable ih espec o space vaiables fcio ad ime vaiable i he domai of iees he he specm j
4 4738 M iahi E Edfa ad K El ashid ] ] W 7 is he edced asfomed fcio of The diffeeial ivese edced asfom of W is defied as ] W W 8 Combiig Eqs 7 ad 8 e ge ] 9 Whe Eq 9 edces o ] Fom he Eq 8 i ca be see ha he cocep of he edced diffeeial asfom is deived fom he poe seies epasio of he fcio efiiio 3 If ] ] V v ad he covolio deoes he edced diffeeial asfom vesio of he mliplicaio he he fdameal opeaios of he edced diffeeial asfom ae sho i he Table Table - Fdameal opeaios of he edced diffeeial asfom mehod Oigial fcio edced diffeeial asfom fcio ] v V V ] v V ] N N N N
5 Ne Mehod o Solve Paial Facioal iffeeial Eqaios 4739 ] s m s m!! s s m m ] e! 4 solvig The ime ad space facioal of paial diffeeial eqaios b edced diffeeial asfom mehod TM Eample Coside he folloig ime-space facioal Navie-Soes eqaio: p sbjec o he iiial codiio Applig he TM o Eqs e obai he folloig ecece elaios ] ] P 3 sig he TM o he iiial codiios e ge 4 Applig he iiial codiios 4 io Eqs3 a β e have p 5 So he geeal solios of Eqs ae : a 6
![474 M iahi E Edfa ad K El ashid So he solio of eqaio a β is give as: P 4 7 The esl is he same as AM HPTM HPM VIM ad HAM b 5] Kma e al 6 7]](/docs-images/77/75220684/images/6-1.jpg "8] ad 35] Fige : The behavio of he solios fo diffee vale of α β a p = ad = Fige : The behavio of he solios fo a: α = β ad b: α = 5 β a p =")
6 474 M iahi E Edfa ad K El ashid So he solio of eqaio a β is give as: P 4 7 The esl is he same as AM HPTM HPM VIM ad HAM b 5] Kma e al 6 7] 8] ad 35] Fige : The behavio of he solios fo diffee vale of α β a p = ad = Fige : The behavio of he solios fo a: α = β ad b: α = 5 β a p =
7 Ne Mehod o Solve Paial Facioal iffeeial Eqaios 474 Eample Coside he folloig ime-space facioal Navie-Soes eqaio: 8 sbjec o he iiial codiio 9 Applig he TM o Eqs 8 e obai he folloig ecece elaios ] ] sig he TM o he iiial codiios 9 e ge Applig he iiial codiios io Eqs e have 4 The geeal solios of Eqs 8 ae : So e have: 4 3 ] 3 3 a 3 4 So he solio of eqaio 8 a β is give as: 3-3 5
![474 M iahi E Edfa ad K El ashid Whe eqaio 5 is he same as he eac solio of he Navie-Soes eqaio 5] 3-3 The esl is he same as AM HPTM HPM VIM ad HAM b Momai ad](/docs-images/77/75220684/images/8-0.jpg "Odiba 5] Kma e al 6 7] Kha 8] ad 35] 6 Fige 3: The behavio of he solios fo a: α = β ad b: α = 5 β Fige 4: The behavio of he solios fo diffee vales of β α = 5")
8 474 M iahi E Edfa ad K El ashid Whe eqaio 5 is he same as he eac solio of he Navie-Soes eqaio 5] 3-3 The esl is he same as AM HPTM HPM VIM ad HAM b Momai ad Odiba 5] Kma e al 6 7] Kha 8] ad 35] 6 Fige 3: The behavio of he solios fo a: α = β ad b: α = 5 β Fige 4: The behavio of he solios fo diffee vales of β α = 5 a =
9 Ne Mehod o Solve Paial Facioal iffeeial Eqaios 4743 Fige 5: The behavio of he solios fo diffee vales of β α a = 6 CONCLSION I his pape FTM has bee implemeed fo he Capo ime-space facioal ode Navie-Soes eqaio The poposed appoimaed solios of Navie-Soes eqaio ih a appopiae iiial codiio ae obaied i ems of a poe seies iho sig a id of disceizaio pebaio o esicive codiios ec To eamples ae illsaed o sd he effeciveess ad accaeess of FTM I is fod ha FTM solios ae i ecelle ageeme ih hose obaied sig AM HPTM HPM VIM HAM TM ad FHATM Hoeve compaios sho ha he FTM is ve eas o impleme ad eeds small size of compaio coa o AM TM ad FHATM This shos ha FTM is ve effecive ad efficie poefl mahemaical ool hich is easil applicable i fidig o he appoimae aalic solios of a ide age of eal old poblems aisig i egieeig ad allied scieces Mahemaica has bee sed fo compaios i his pape
10 4744 M iahi E Edfa ad K El ashid EFEENCE ] J Biaza H Ghazvii Covegece of he homoop pebaio mehod fo paial diffeeial eqaios Noliea Aalsis: eal Wold Applicaios 9: ] A M Wazaz ad A Gogis Eac solios fo hea-lie ad ave-lie eqaios ih vaiable coefficies Applied Mahemaics ad Compaio 49 4:5--9 3] J H He Homoop pebaio echiqe Compe Mehods i Applied Mechaics ad Egieeig 78999: ] J H He Acoplig mehod of a homoop echiqe ad a pebaio echiqe fo o-liea poblems Ieaioal Joal of No-Liea Mechaics 35: ] J H He Applicaio of homoop pebaio mehod o oliea ave eqaio Chaos Solios ad Facals 65: ] Gaji A Sadighi Applicaio of He's homoop pebaio mehod o oliea copled ssems of eacio diffsio eqaios Ieaioal Joal of Noliea Scieces ad Nmeical Simlaio 76: ] J H He Homoop pebaio mehod fo solvig boda vale poblems Phsics Lees A Vol 356: ] A M Wazaz A compaiso beee he vaiaioal ieaio mehod ad Adomia decomposiio mehod Joal of Compaioal ad Applied Mahemaics 7 7: ] A M Wazaz A e algoihm fo calclaig Adomia polomials fo oliea opeaos Applied Mahemaics ad Compaio : ] A M Wazaz The vaiaioal ieaio mehod: a poefl scheme fo hadlig liea ad oliea diffsio eqaios Compes & Mahemaics ih Applicaios 5477: ] Y Kha ad F Asi Applicaio of he Laplace decomposiio mehod o oliea homogeeos ad o-homogeeos advecio eqaios Zeischif fe Nafoschg A 65: -5 ] M Madai M Fahizadeh Y Kha ad A Yildiim O he coplig of he homoop pebaio mehod ad Laplace asfomaio Mahemaical ad Compe Modellig 539:
11 Ne Mehod o Solve Paial Facioal iffeeial Eqaios ] Y Kha ad Q W Homoop pebaio asfom mehod fo oliea eqaios sig He's polomials Compes ad Mahemaics ih Applicaios 6: ] F Abidi K Omai The homoop aalsis mehod fo solvig he Fobeg- Whiham eqaio ad compaiso ih Adomia's decomposiio mehod Compes ad Mahemaics ih Applicaios 59: ] S Momai Z Odiba Aalical solio of a ime-facioal Navie-Soes eqaio b Adomia decomposiio mehod Appl Mah Comp 77 6: ] Kma J Sigh ad S Kma A facioal model of Navie-Soes eqaio aisig i sead flo of a viscos flid J Assoc Aab iv Basic Appl Sci ] Kma S Kma S Abbasbad ad MM ashidi Aalical solio of facioal Navie-Soes eqaio b sig modified Laplace decomposiio mehod Ai Shams Eg J 54: ] NA Kha Aalical sd of Navie-Soes eqaio ih facioal odes sig He s homoop pebaio ad vaiaioal ieaio mehods I J Noliea Sci Nme Siml ] M S Mohamed K A Gepeel edced diffeeial asfom mehod fo oliea iegal membe of Kadomsev- Peviashvili Hieach diffeeial eqaios Joal of he Egpia Mahemaical Socie ] Y Kesi G Oac The edced diffeeial asfom mehod fo paial diffeeial eqaios I J Noliea Sci Nme Siml 6 9: ] Y Kesi G Oac The edced diffeeial asfom mehod fo solvig liea ad oliea ave eqaios Ia J Sci Techol 34 : 3- ] S Kma A Yildiim Y Kha ad L Wei A facioal model of he diffsio eqaio ad is aalical solio sig Laplace asfom Sci Ia 94 : 7-3 3] M Kha MA Godal ad S Kma A e aalical pocede fo oliea iegal eqaio Mah Comp Model 55: ] AM Wazaz The combied Laplace asfom-adomia decomposiio mehod fo hadlig oliea Volea iego-diffeeial eqaios Appl Mah Comp 64 : 34-39
12 4746 M iahi E Edfa ad K El ashid 5] Y Kha N Faaz S Kma A Yildiim A coplig mehod of homoop pebaio ad Laplace asfom fo facioal models Sci Bll Polieh iv Bcha Se A Appl Mah Phs ] MA Godal M Kha Homoop pebaio mehod fo oliea epoeial boda Lae eqaio sig Laplace asfomaio He s polomials ad Pade echolog He s polomials ad Pade echolog I J Noliea Sci Nme Siml : ] M Madai M Fahizadeh Y Kha ad A Yildiim O he coplig of he homoop pebaio mehod ad Laplace asfomaio Mah Comp Model 53 : ] S Kma A e aalical modellig fo facioal elegaph eqaio via Laplace asfom Appl Mah Model 384: ] K A Gepeel M S Mohamed aalical appoimae solio fo oliea ime-space facioal Klei Godo eqaio MSC 4F35 6A33 34A8 3] M AE Hezallah K A Gepeel appoimae solio o he ime space facioal cbic oliea Schodige eqaio Applied Mahemaical Modelig 3] G Jmaie Ne sochasic facioal models fo Malhsia goh he Poissoiabih pocess ad opimal maageme of poplaios MahComp Modellig 446 :3-54 3] G Jmaie Laplace's asfom of facioal ode via he Miag-Le e fcioad modified iemalioville deivaive Appl Mah Le 9: ] Almeida AB Maliosa ad FM Toes A facioal calcls of vaiaios fo mliple iegals ih applicaio o vibaig sig J Mah Phs5 : ] G C W ad EWM Lee Facioal vaiaioal ieaio mehod ad is applicaio Phs Le A 374 : ] A A agab K M Hemida Mohamed S Mohamed ad M A Abd El Salam Solio of ime-facioal Navie-Soes eqaio b sig homoop aalsis mehod Ge Mah Noes 3:3-
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