The Variance of Fractional Partial Differential. Equation of Distributed Order

Transcription

1 In. J. Conem. Mah. Sciences Vol. 8 3 no HIKARI Ld.m-hikari.com h://dx.doi.org/.988/ijcms The Variance of Fracional Parial Differenial Equaion of Disribued Order Ahmad El-Kahlou Al-Quds Oen Universiy Norh Gaza ranch ei Lahia Palesine ahmadk_7@homail.com Coyrigh 3 Ahmad El-Kahlou. This is an oen access aricle disribued under he Creaive Commons Aribuion License hich ermis unresriced use disribuion and reroducion in any medium rovided he original ork is roerly cied. Absrac The aim of his aer is o exress variance of fracional arial differenial equaion of disribued order here he Cauo fracional derivaive is considered of order < α. This exression is achieved by using a funcion ransform Fourier and Lalace ransforms. In aricular he asymoic behavior of he fundamenal soluion and is variance are sudied. As secial case he variance of he fundamenal soluion of ime fracional diffusion equaion is obained. Keyords: Fracional derivaives Lalace ransform Fourier ransform variance disribued order Cauo fracional derivaive. Inroducion The arial equaion of Gaussian diffusion is generalized by using he ime fracional derivaive of disribued order here he order of his ye is considered in he Cauo sense[4789]. In his aer e ill ake fracional advecion disersion equaion and fracional diffusion equaion as o forms of fracional arial differenial equaion. Also e derive some resuls concerning variance and asymoic behaviors of fracional advecion disersion equaion of disribued order.

2 994 Ahmad El-Kahlou Then e ill sudy ho Mainardi and Pagnini [3] and Mainardi Pagnini and Gorenflo [] derived he variance and asymoic behaviors of he soluion of fracional diffusion equaion of disribued order as secial case of fracional advecion disersion equaion. The lan of he aer is as follos: In secion e rie don he general form of he ime fracional advecion disersion equaion ih disribued order ih Cauo derivaive and Fourier-Lalace reresenaion of he corresonding fundamenal soluion. For his urose e need o inroduce a osiive funcion ha acs as a discree or coninuous disribuion of orders. We sudy he fracional advecion disersion equaion of disribued order as a discree disribuion so e ake o disinc orders : < <. Secion.3 is devoed o he ime evoluion of variance hich is obained from he Fourier-Lalace reresenaion of he corresonding fundamenal soluion by invering only he Lalace ransform. We ell visible from heir asymoic exressions for small and large imes. Secion 3 e ake as secial case he variance of he fundamenal soluion of he ime fracional diffusion equaion of disribued order.. The Fracional Advecion Disersion Equaion of Disribued Order: The fracional advecion-disersion equaion of single order is rien as C x C x C x D b λc x. here x > > < and is he fracional derivaive of Cauo ye; D > b and λ. The iniial condiions is assumed o be C x C x. Eq.. can be generalized by using he noion of fracional derivaive of disribued order in ime. For his urose e need o consider a funcion ha acs as a eigh funcion for he order of differeniaion ] such ha and d.3 The ime fracional advecion-disersion equaion of disribued order can be rien as C x C x C x d D b λc x.4 Subjeced o iniial condiions..

3 The variance of fracional PDE of disribued order 995 Noe ha in case b λ and D hen Eq..4 reduces o he diffusion equaion of fracional disribued order.4 ih C x u x. No e assume he funcion ransform x b C x u ζ ex μζ ζ μ hen Eq..4 reduces o D D ζ u ζ u d u ζ.5 ζ here μ λ. Assuming ha C x δ xex b / D Dirac-dela funcion hen u ζ δ ζ is he iniial condiion of Eq..5.. The Cauo Form in Fourier-Lalace domain: here δ x is he The fundamenal soluion of he ime-fracional advecion disersion equaion.5 can be obained by alying in sequence he Fourier and Lalace ransforms o hem. The researcher rie for generic funcion fx hese ransforms []: I l { f x; κ} fˆ κ : f x dx ; κ R { f x; } f : e f d ; D e iκx No alying he Lalace ransform ih resec o hen Eq..5 imlies [ u ] ζ u u ζ u ζ d ζ ζ.6 And alying he Fourier ransform ih resec o ζ yields [ u k ]ˆ d u k [ d ]ˆ u k k ˆ.7 hich reduces o ˆ / u k k.8 here d and uˆ k ˆ δ k.. The variance of he fundamenal soluion: In his secion e orh o ouline he exression of he second momen variance of he fundamenal soluion of he fracional advecion disersion equaion of disribued order by a single Lalace inversion as i is shon here afer. Denoing for he form

4 996 Ahmad El-Kahlou dx x u x : e easily recognize ha u ˆ κ κ As a consequence he researcher need o inver only he Lalace ransforms aking ino accoun he behavior of he Fourier ransform for κ near zero e ge from equaion.8 κ κ u.9.3 Fracional advecion disersion equaion of double order: The execed sub-linear groh ih ime is shon in he folloing secial case of reaed in [3]. No le us assume he double order case of he second momen by assuming ha δ δ. here > > < <. Then is rien as. Subsiuing ino Eq..9 e ge.3 Invering via he inverse Lalace ransform by making use of he formula } ;! { λ η λ η λ η λ c E c n l n n n.4 And alying he Tauberian heory e obain E E.5 here E is he ell-knon Miag-Leffler funcion [564] and E is is firs derivaive ih resec o hich formula is

5 The variance of fracional PDE of disribued order 997 j j E.6 j j! Γ j hich for erminaes for all erms exce hen j and in his case i is reduced o Γ. Also E is reduced o Γ hen. 3. The ime fracional diffusion equaion of disribued order: The fracional diffusion Equaion u x u x x R < 3. is a secial case of Fracional advecion disersion equaion by aking in equaion.5. We no consider he so-call ime-fracional diffusion equaion of disribued order see [-]. since u x d ih x R x u d d b is a eigh funcion. Then x u u x d No e ill ake he Lalace ransform of Eq.3.4 as [ u ] x u x u x d bu since uxδx hen Eq..5 leads o u x u x δ x here d 3.7 The Fourier inegral ransform of Eq.3.6 ih resec o x can be rien as u ˆ κ ˆ δ κ κ uˆ κ 3.8

6 998 Ahmad El-Kahlou bu since ˆ δ κ. Then Eq.3.8 yields ˆ / u κ Re > κ R κ 3.9 efore rying o ge he soluion in he sace-ime domain i is orh o ouline he exression of is second momen as i can be derived from Eq. 3.9 so using he formula ˆ u κ < Γ 3. Then κ uˆ κ κ 3. and uˆ κ / κ 4κ κ / κ 4 3. uˆ κ κ 3.3 so u κ 3.4 Then from 3.4he researcher is alloed o drive he asymoic behaviors of for and from he asymoic behaviors of for and resecively. The execed sub-linear groh ih ime is shon in he folloing secial case of b reaed in [3]. This case is slo diffusion oer-la groh here δ δ 3.5 here < < > > In fac d [ δ δ ] here d d 3.6

7 The variance of fracional PDE of disribued order 999 δ δ 3.7 Subsiue 3.7 in Eq.3.4 e ge 3.8 So he inversion Lalace ransform of asymoic formula 3.8 is Γ Γ 3.9 This resuls is aken from [3] and i s a secial case of resuls of fracional advecion disersion equaion hich e achieved 4. Conclusion: I is concluded o exress variance of fracional arial differenial equaion of disribued order and asymoic behaviors of. References [] Gorenflu R. Louhcko J. and Luchko Yu. Comuaion of he Miag-Leffler funcion Eαz and is derivaives Fracional Calculus Al. Anal [] Podlubny I. Fracional Differenial Equaions Acadmic Press San Diego 999. [3] Chechkin V. A. Gorenflo R. Sokolov M. I. and Gonchar Yu. V. Disribued order ime fracional diffusion equaion Frac. Calc. Al. Anal [4] Chechkin V. A. Klafer J. and Sokolov M. I. Fracional Fokker-Planck equaion for ulraslo kineics Eurohys. Le

8 Ahmad El-Kahlou [5] Gorenflo R. Luchko Yu. and Mainardi F. Wrigh funcions as scale-invarian soluions of he diffusion-ave equaion J. Comu. Al. Mah [6] Gorenflo R. and Mainardi F. Simly and mulily scaled diffusion limis for coninuous ime random alks in: S. enkadda X. Leoncini G. Zaslavsky eds. Proceedings of he Inernaional Worksho on Chaoic Transor and Comlexiy in Fluids and Plasmas Carry Le Roue France -5 June 4 IOP Insiue of Physics Journal of Physics:Conference Series [7] Kilbas A. A. Srivasava M. H. and Trujillo J. J. Theory and Alicaions of Fracional Differenial Equaions Norh-Holland Mahemaics Sudies 4 Elsevier Science Publishers Amserdam Heidelberg and Ne York 6. [8] Saichev A. and Zaslavsky G. Fracional kineic equaions: soluions and alicaions Choas [9] Samko G. S. Kilbas A. A. and Marichev I. O. Fracional inegrals and derivaives: Theory and Alicaions Gordon and reach Ne York 993 Translaion from Russian ediion Nauka I Tekhnika Minsk 987. [] Sokolov M. I. Chechkin V. A. and Klafer J. Disribued order fracional kineics Aca Phys. Polon [] Mainardi.F Mura A. and Pagnini G. Time-fracional diffusion of disribued order. J Vibraions & Conrol [] Mainardi.FPagnini G. and Gorenflo R. Some asecs of fracional diffusion equaions of single and disribued order. Al. Mah. Comu [3] Mainardi.F and Pagnini G. The role of he Fox-Wrigh funcions in fracional sub-diffusion of disribued order. J. Comu Al. Mah [4] Mainardi F. The fundamenal soluions of he fracional diffusion-ave equaions. Al. Mah. Le. Vol. 9 No Received: Seember 9 3