Analytical Solution of Time-Fractional Advection Dispersion Equation

Transcription

1 Aville h://vu.edu/ Al. Al. Mh. ISSN: Vol. 4 Iue (June 9) (Previoul Vol. 4 No. ) Alicion nd Alied Mheic: An Inernionl Journl (AAM) Anlicl Soluion of Tie-Frcionl Advecion Dierion Equion Triq O. Sli Deren of Mheic Al-Azhr Univeri-Gz P. O. Box 77 Gz Pleine rli@hoo.co Ahd El-Khlou Al-Qud Oen Univeri Norh Gz Brnch Bei Lhi Pleine Received: Ocoer 3 8; Acceed: Ferur 7 9 Arc In hi er we ge exc oluion of he ie-frcionl dvecion-dierion equion wih recion er where he Cuo frcionl derivive i conidered of order. The oluion i chieved uing funcion rnfor Fourier nd Llce rnfor o ge he forul of he fundenl oluion which re exreed exlicil in er of Fox H- funcion king ue of he relionhi eween Fourier nd Mellin rnfor. A ecil ce he exc oluion of ie-frcionl diffuion nd wve equion re lo oined nd he oluion of he ineger order equion re enioned. Keword: Frcionl Derivive; Llce Trnfor; Fourier Trnfor; Mellin Trnfor; Fox H-funcion AMS () No.: 6A33 49K 44A. Inroducion Tie frcionl ril differenil equion oined relcing he fir order ie derivive frcionl derivive (of order in Cuo ene) hve een reed in differen conex nuer of reercher. Minrdi nd Pgnini (3) udied he ie frcionl 76

2 AAM: Inern. J. Vol. 4 Iue (June 9) [Previoul Vol. 4 No. ] 77 diffuion equion nd he fundenl oluion (Green funcion) uing Fourier-Llce rnfor. Liu Anh Turner nd Zhng (3) conidered he ie frcionl dvecion dierion equion relcing he fir order derivive in ie frcionl derivive of order ( ) nd he hve ued vrile rnforion Mellin nd Llce rnfor o chieve colee oluion. Sxen Mhi nd Huold ( c) nd Huold Mhi nd Sxen (7) ued inegrl rnfor ehod o oin exc oluion for he frcionl kineic diffuion nd recion diffuion equion.oher reerch ricle hndling ie frcionl ril differenil equion uing inegrl rnfor cn e found in he lierure nuer of uhor ee e.g. he review in Minrdi (996) Minrdi Luchko nd Pgnini () Minrdi Pgnini nd Sxen (5) Minrdi nd Pgnini (7) Minrdi Pgnini nd Gorenflo (7) Moni nd Odi (7) nd Wng Xu nd Li (7) nd he reference herein. In hi er we ud he ie frcionl dvecion dierion equion wih recion C x Cx Cx D x x C x (.) which decrie he rnien rnor of olue hrough hoogeneou oil where 3 C: i he olue concenrion ( ML ) : i he ie (T ) x : i he oil deh ( L ) ( ) : i he ore wer veloci ( LT ) ( D ) : i he dierion coefficien ( L T ) ( ) : i he fir order recion re coefficien ( T ) nd ( ) i he order of he ie frcionl derivive which i inended in he Cuo ene. For deiled dicuion on hi frcionl derivive we refer o Podlun (999). When i no ineger ( ) he Cuo frcionl derivive i wrien C x Cx if C x if (.) nd if i n ineger ( ) he Cuo frcionl derivive i idenicl o he correonding ril derivive of ineger order. Now uing he relion (ee Gorenflo nd Minrdi (997))

3 78 Sli nd Khlou J Cx ( ) Cx ( ) if Cx Cx ( ) Cx ( ) C ( x ) if (.3) where I f J f x i he Rienn-Liouville frcionl inegrl oeror defined f d (.4) hen we cn eliine he ie frcionl derivive in (.) nd oin he inegro-differenil equion C x C x x x (.5) Cx Cx D C x d if nd C x C x C x Cx C x D C x d x x (.6) if. In order o correcl forule nd olve he Cuch role for (.) we hve o elec exlici iniil condiion concerningc x if nd Cx C x if. If x nd x denoe ufficienl well-ehved rel funcion defined on he Cuch role coni in finding he oluion of (.) ueced o he iniil condiion C x x x if (.7) C x x C x x x if. (.7) Now we give oe ic definiion of he Llce rnfor nd he Fourier rnfor nd oe required forul. The Llce rnfor of funcion f () on i defined Ar (999) ~ (.8) f f e d f Re( )

4 AAM: Inern. J. Vol. 4 Iue (June 9) [Previoul Vol. 4 No. ] 79 nd he Llce rnfor for he Cuo derivive f ( ) i inended o e f f if f ; f f f if (.9) nd he invere Llce rnfor i wrien f () f e f d Re. (.) i i i The Fourier rnfor of funcion f (x) on i defined Minrdi nd Pgnini (3) f ( ); fˆ( ) i e f ( ) d (.) nd i invere i wrien f fˆ e d i. (.) In licion of Fourier rnfor o hicl role i i ueful o hve he forul: n n f x; i f (.3) where x n n d f f x nd i. n dx. The Green Funcion To reduce (.5) nd (.6) o ore filir for we ue he following funcion rnfor. Le C x. (.) D D x u ex Then (.5) nd (.6) ield he inegro-differenil equion

5 8 Sli nd Khlou u u u d (.) ( ) for nd u for nd u u d (.3) ( ) where D ex D ex. Now ling he Llce rnfor wih reec o nd he Fourier rnfor wih reec o hen equion (.) nd (.3) ield nd ˆ~ u ˆ for (.4) u ˆ ˆ for ˆ~. (.5) B fundenl oluion (or Green funcion) of he ove Cuch role we en he oluion correonding o iniil condiion G for (.6) where ˆ G G G G ( for (.6) i he Dirc-del generlized funcion whoe Fourier rnfor i known o e one ). Thu he Fourier-Llce rnfor of hee Green funcion re ˆ~ G. (.7)

6 AAM: Inern. J. Vol. 4 Iue (June 9) [Previoul Vol. 4 No. ] 8 Noe h king ue of he forul [ee Ar (999)] ; e (.8) nd eing we ge. (.9) ~ G e I i oviou fro (.9) h G ~ ~ G /. So G () G d. (.) Thi en h i i enough o oin G ince he oher Green funcion G cn e oined uing (.) if G i known. Unforunel invering he Llce rnfor fro equion (.9) i roleic. However we cn inver he Llce rnfor in equion (.7) fir rewriing he for ˆ~ G in ˆ~ G. (.) Now exnding he econd frcion nd ilifing we ge ˆ~ G. (.) Then king ue of he invering Llce rnfor forul (ee Podlun (999)) ; E (.3)!

7 8 Sli nd Khlou where!! d E d E (.4) i he derivive of he Mig-Leffler funcion E we oin E G! ˆ. (.5) Lnglnd (6) h hown h he Fourier invere of he derivive of he Mig-Leffler funcion in equion (.5) cn e oined fir rewriing he derivive in er of H- funcion H E. (.6) In order o evlue he invere Fourier rnfor of he H-funcion we need he following relionhi eween Fourier rnfor nd Mellin rnfor (ee Lnglnd (6)) x f M f M ); ( co ; ˆ (.7) where he Mellin rnfor of f(x) i wrien ; dx x f x x f M. (.8) To find he Mellin rnfor of equion (.6) we noe h he Mellin rnfor of Fox H- funcion i given Kil nd Sigo (4) (ee lo Srivv Gu nd Gol (98) nd Mhi nd Sxen (978)). q n n q q n q x H M ; (.9) when he following condiion re ified

8 AAM: Inern. J. Vol. 4 Iue (June 9) [Previoul Vol. 4 No. ] 83 q n q A n A rg nd in Re Re( ) in Re. n Anoher ueful ideni i [ee Oerheinger (974)] M / x f x ; M f x ; (.) for. We now inver equion (.5) uing equion (.9) nd (.) long wih equion (.7) o ge M G ; 4!. (.) Coring wih equion (9) we fined invering he Mellin rnfor h / / / / / / G H 33. (.) 4! / / Now ling he relion [ee Srivv Gu nd Gol (98)] n x x H q zx x dx q H z n q o equion (.) nd eliining we ge q (.3)

9 84 Sli nd Khlou / 33 // / /. 4! / / G H (.4) 3. The Colee Soluion nd Secil Ce The Green funcion llow u o rereen he oluion of he Cuch role hrough he relevn inegrl forul d G u (3.) d G G u. (3.) So uiuing equion (.) nd (.4) in (3.) nd (3.) reecivel we ge / / 33 4! / / / / / / u H d (3.3) for nd / 33 / / 33 // / / / / 4! // / / / / H u H d (3.4)

10 AAM: Inern. J. Vol. 4 Iue (June 9) [Previoul Vol. 4 No. ] 85 for. Now ling he uiuion in (.) llow u o ge exlici for for C ( x ) C x x D / e 4! // / / / D H 33 x De d // (3.5) for nd C x e x D / 4! // / / / H 33 x D / / D e d (3.6) // / / / H 33 x D / / for nd. 4 D Now if we conider he liiing ce (i.e. nd ) hen we oin he oluion of he ie-frcionl diffuion equion C x Cx D x. (3.7) Suec o iniil condiioncx x direcl fro equion (3.5)

11 86 Sli nd Khlou / // / / Cx H x Dd. 4 // / / 33 (3.8) Mking ue of he known roer of he G funcion hen equion (3.8) cn e wrien ( z) ( z) nd ilif inz C x / H / / x D d. (3.9) Siilrl he oluion of he ie-frcionl wve equion C x Cx D x (3.) wih iniil condiion Cx x nd x x C cn e chieved direcl fro equion (3.6) nd fer ilifing i cn e wrien C x / H / / x D d / H / / x D d. (3.) The reul of equion (3.9) nd equion (3.) re in full greeen wih he reul of equion (3.) nd equion (4.9) recenl oined Minrdi nd Pgnini (3) nd Minrdi Pgnini nd Sxen (5) reecivel. The ineger order dvecion-dierion equion i oined eing in equion (.) nd i oluion ield direcl fro equion (3.5) C x D / e x H / / / / 33 4! / x D e D d (3.)

12 AAM: Inern. J. Vol. 4 Iue (June 9) [Previoul Vol. 4 No. ] 87 nd he oluion of he ineger order diffuion nd wve equion cn e oined eing in equion (3.9) nd in equion (3.). Acknowledgen The uhor re highl greful o oh he referee for heir conrucive coen nd vlule uggeion. REFERENCES Ar N. (999). Pril Differenil Equion nd Boundr Vlue Prole. Prenic Hll New Jere. Gorenflo R. nd Minrdi F. (997). Frcionl clculu: Inegrl nd differenil equion of frcionl order in: Crineri A Minrdi F. (Ed) Frcl nd frcionl Clculu in Coninuu Mechnic. Sringer Verlg Wien nd New York 3-76.Rerin ville fro h:// Kil A.A. nd Sigo M. (4). H-rnfor: Theor nd Alicion. Chn & Hll/CRC London New York. Lglnd T. A. M. (6). Soluion of odified frcionl diffuion equion Phic A Liu F. Anh V. V. Turner I. nd Zhung P. (3). Tie frcionl dvecion dierion equion. J. Al. Mh. Cou Minrdi F. (996). The Fundenl oluion for he frcionl diffuion-wve equion. Al. Mh. Le Minrdi F. Luchko Y. nd Pgnini G. (). The fundenl oluion of he ce ie frcionl diffuion equion. Frc. Clc. Al.Anl Minrdi F. Pgnini G. nd Sxen R.K. (5). Fox H-funcion in frcionl diffuion. J. Co. Al. Mh. 78 No Minrdi F. nd Pgnini G. (3). The Wrigh funcion oluion of he ie-frcionl diffuion equion. Al. Mh. And Co Minrdi F. nd Pgnini G. (7). The role of he Fox-Wrigh funcion in frcionl udiffuion of diriued order. J. Co. Al. Mh Minrdi F. Pgnini G. nd Gorenflo R. (7). Soe ec of frcionl diffuion equion of ingle nd diriued order. Al. Mh. Cou Mhi A.M. nd Sxen R.K. (978). The H-funcion wih Alicion in Siic nd Oher Diceline. Wile Eern New Delhi. Moni S. nd Odi Z. (7). Frcionl Green Funcion for liner ie-frcionl inhoogeneou ril differenil equion in fluid echnic. J. Al. Mh. Couing 4 No Oerheinger F. (974). Tle of Mellin Trnfor. Sringer Berlin. Podlun I. (999). Frcionl differenil equion. Acdeic Pre New York 999.

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