ON THE FRACTIONAL-ORDER DIFFUSION-WAVE PROCESS

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1 ON THE FRCTIONL-ORDER DIFFUSION-WVE PROCESS MOHMED.E. HERZLLH, HMED M.. EL-SYED, DUMITRU BLENU 3 Faculy f Sciece, Zagazig Uiversiy, Zagazig, Egyp m_herzallah75@hmail.cm Faculy f Sciece, lexaria Uiversiy, Egyp amasaye@hmail.cm 3 Deparme f Mahemaics a Cmpuer Sciece Çakaya Uiversiy, 653 kara, Turkey a Isiue f Space Scieces, P.O.BOX, MG-3, RO-775, Magurele-Buchares, Rmaia umiru@cakaya.eu.r Receive pril 4, 9 Oe f he mai applicais f he fracial calculus, iegrai a iffereiai f arbirary rers is he mellig f he iermeiae physical prcesses. Here we frmulae a mre geeral mel which represes he iffusi wave prcess i all is cases, a give sme examples iscussig hese iffere cases. Key wrs: Evlui equai, fracial rer erivaive, Diffusi-Wave equai.. INTRODUCTION Fracial Calculus is a fiel f applie mahemaics ha eals wih erivaives a iegrals f arbirary rers. Firs here were alms pracical applicais f his fiel, a i was csiere by may as a absrac area caiig ly mahemaical maipulais f lile r use. Nearly 3 years ag, he paraigm bega shif frm pure mahemaical frmulais applicais i varius fiels. Durig he las ecae Fracial Calculus has bee applie alms every fiel f sciece, egieerig, a mahemaics. Several fiels f applicai f fracial iffereiai a fracial iegrai are alreay well esablishe, sme hers have jus sare. May applicais f fracial calculus ca be fu i urbulece a flui yamics, schasic yamical sysem, plasma physics a crlle hermuclear fusi, liear crl hery, image prcessig, liear bilgical sysems, asrphysics [ 5]. I rece years, here has bee a grea eal f ieres i fracial iffereial equais. Hisrical summaries f he evelpmes f fracial calculus ca be fu i [ 4]. Rm. Jur. Phys., Vl. 55, Ns. 3 4, P , Buchares,

2 O he fracial-rer iffusi-wave prcess 75 Oe f he mai applicais f he fracial calculus is mellig f he iermeiae physical prcess. very impra mel is he fracial iffusi a wave equais. May auhrs rie mel iffusi a wave equais frm he classical iffusi r wave equai by replacig he firs r sec-rer ime erivaive by a fracial erivaive f rer wih < <, (see [5, 6,, 6, 7, 8, 9, ]. Maiari (see [6] efie he fracial iffusi equai by u u = D, < <, D> ( x a he fracial wave equai by u u = D, < <, D> ( x where is he Riemma-Liuville fracial erivaive. He iscusse he w basic prblems fr bh iffusi equai a wave equai The Cauchy prblem is a iiial value prblem whe he aa are assige a = + he space axis < x <. u(x, = g(x, < x < a u( ±, =, > The Sigallig prblem, csiere i he mai x,, is a iiial buary value prblem whe he aa are assige bh a = + he semi-ifiie ime axis x > (iiial aa a x = + he semi-ifiie ime axis > (buary aa. u(x, =, x >, a u(, = h(, u(, =, > If < < he fracial wave prblem we ee a a iiial cii u (x, =. El-Saye (see [5] efie he absrac fracial rer prblem Du ( = u (, (, T] wih u( = u, (, (3 a he absrac fracial rer wave prblem D u( = u(, (, T] wih u( = u, u( =, (, (4 wih usig he Capu erivaive a prve he exisece a uiqueess f he slui uer sme ciis. Grefl a Maiari (see [7] efie he Feller space-fracial iffusi equai by u = x Dθ u, x R,, θ mi {, }, (, (5 where x D θ is he Riesz Feller space fracial erivaive. Maiari (see [8] replace he Riemma Liuville fracial erivaive i his wrk [6] by he Capu erivaive. Maiari, Luchk, Pagii (see [] gave he efiii f he space-ime fracial iffusi equai

3 76 Mhama.E. Herzallah, hme M.. El-Saye, Dumiru Baleau 3 ( ( D* u x, = xdθ u x,, x R, (6 where,, θ are real parameers resrice as fllws <, θ mi,, <. { } x D θ is he Riesz-Feller space fracial erivaive f rer a skewess θ, a D * is he Capu ime fracial erivaive f rer. El-Saye a M. ly (see [6] frmulae a mre accurae mel f he absrac iffusi wave prblem as D u( = h( s u( s s, >, (, ], u( = u (7 gave is slui a prve ha i is a geeral mel f iffusi-wave prblem. W. Che a S. Hlm (see [9] efie he fracial iffusi wave equai as u λ = k( / u, < λ, (8 where is he Laplacia perar, a k ees a physical csa. λ a ca be arbirary real umber. Ne ha i each e f he previus fracial D-W prcess here are w equais e each fr he fracial iffusi prblem a he fracial wave prblem. The mai purpse f his paper is give a mre geeral absrac mel f hmgeeus D-W equai which represes he D-W prcess i all cases. The paper is rgaize as fllws: I Seci, we give he pricipal efiiis a herems use i his paper. I Seci 3, we suy ur absrac fracial rer D-W mel u ( = I u(, u( = u, (,] (9 where is a clse liear perar wih ese mai D( = X X, X is a Baach space, wih givig sme examples illusrae ur mel. Our cclusi is give i Seci 4. Le f L( J, R. PRELIMINRIES a le be a psiive real umber. Defiii (fracial Riemma-Liuville iegral.. The fracial iegral f rer f he fuci f( is efie by (see [ 4] ( ( Γ( s I f = f s s= f ( ( ( φ

4 4 O he fracial-rer iffusi-wave prcess 77 where φ ( = Γ ( fr > a φ ( = fr, fuci as. Fr he fracial rer erivaive Defiii (Riemma-Liuville erivaive. a φ ( δ( (he ela The (Riemma-Liuville fracial erivaive f he fuci f( f rer, is efie by (see [ 4] ( Defiii (Capu erivaive.3 ( s Γ( f ( = D I f ( = f ( s s. The (Capu fracial erivaive f rer (, f he fuci g( is efie by (see [, 3] D g I Dg, D. ( = ( The fracial erivaive f rer (, f g( is efie by D g( = I D g(, D =. Csier w he fracial rer evlui prblem D u( = u( + f (, (,, u( = u. ( Usig he resuls f [] we have (see [] Defiii.3. fuci u C( J; X u C( J X a ( hls J; ; is calle a srg slui f ( J if Therem.4. Le, (,, u D(, f W ( J, X geerar f a bue C -semigrup { (, }. slui u C( J, X f ( give by ( ( ( ( ( φ k where S C ( J; B( X ( φ ( ( a is he ifiiesimal T The here is a uique srg ( ( ( ( δ( ( ( u = es u es u + es f + f f ( is he reslve perar wih he reslve equai ( ( es x= ex+ es x, x D. ( This slui saisfies he ciuai prpery ( = ( lim u u.

5 78 Mhama.E. Herzallah, hme M.. El-Saye, Dumiru Baleau 5 where u ( is he slui f he evlui equai u ( = u( + f(, u( = u. (3 Csier w he fracial rer evlui prblem D u = u + f,,, u = u, u = (4 we have (see []. ( ( ( ( ( ( Defiii.5. fuci u C( J; X u C( J; X C ( J; x C (, T] ; X ( a (4 hls J., Therem.6. Le (,, u D(, f W ( J, X is calle a srg slui f (4 J if a is he ifiiesimal geerar f a bue C -semigrup { T(, }. If arg ( λ + he here is a uique srg slui ( ( ( ] (4 give by π θ < ( u C J, X C J; X C, T ; X f u ( es ( u es ( u es ( φ ( ( f( δ ( f ( f( where S ( ( C J; B X C (, T] ; B( X = + + (5 reslve equai ( is he reslve perar wih he ( = + φ ( (, ( es z ez es z z D his slui saisfies he ciuai prperies ( = ( ( = ( lim u u, lim u u + where u ( is he slui f he evlui equai (3, a u ( is he slui f he Cauchy prblem (wave equai u ( = u ( + f (, u ( = u, u ( =. (6 3. BSTRCT DIFFUSION-WVE PROBLEM Csier he Cauchy prblem (9 where is a clse liear perar wih D = X X ese mai (. Defiii 3.. fuci u C( J; X u C( J; X C ( J; X a (9 hls J. is calle a srg slui f (9 J if

6 6 O he fracial-rer iffusi-wave prcess 79 Therem 3.. Le ( u D( bue C semigrup { (; }. u C( J; X C ( J; X,,, a is he ifiiesimal geerar f a f (9 give by where S C ( J; B( X Prf. T The here is a uique srg slui ( ( ( u = e S u e S u (7 is he reslve perar wih he reslve equai (i Fr (, ( φ ( ( (, ( S x= x+ e S x x D (8 we fi ha u u = D( φ ( u ( = φ ( u + φ ( Operaig bh sies by he cvlui f φ we ge ( ( ( Du = u, u = u which by Therem.4 wih f( = has he slui give by (7 wih he S C J; B X give by (8. This slui cverges, as reslve perar ( ( he slui f he hmgeeus iffusi prblem u u(, u ( u. = = (9 (ii Fr = (, we have u = DI u = I u = u Differeiaig bh sies we ge ( ( ( ( ( φ = ( φ ( u + φ Du(. u Operaig by he cvlui f ( hus we ge a we e ha u φ bh sies we ge ( ( D u( = ( ( u + Du( φ φ φ φ Du ( = u ( = ( φ ( ( = u = =

7 8 Mhama.E. Herzallah, hme M.. El-Saye, Dumiru Baleau 7 Thus we ge ( ( ( ( Du = u, u = u, u = which by Therem.6 wih f( = has he slui give by (7 wih he reslve perar S C ( J; B( X give by (8, which cverges, as +, he slui f he hmgeeus iffusi prblem (9 a cverges, as, he slui f he hmgeeus wave prblem u = u (, u ( = u, u( =. ( Fially, we prve ha he fuci u ( give by (7 is he slui f ur prblem (9. Usig (8 we ge: ( u ( = es ( u es ( u = e u + e S u e u + e S u = ( φ ( e S ( u + u + φ ( e S ( u φ ( = ( φ ( ( es ( u ( + es u + u I = u (. we have u = e S u e S u = u ( ( ( ( ( ( ( φ φ φ ( ( ( ( = which cmplees he prf. Nw we prve ha ur mel represes he fracial rer iffusi wave prcess i all cases. Therem 3.3. If, fr (,, he slui f ur mel (9 a he fracial rer iffusi prblem (3 exis he hey are equivale. Prf. We prve i Therem 3. ha if u ( is he slui f (9 he i is he slui f (3. Cversely, le u ( be he slui f (3 he ge D u = Du = u ( ( ( ( φ Operaig by he cvlui f u ( he bh sies we ge Differeiae bh sies we ge ( = φ ( ( u u u

8 8 O he fracial-rer iffusi-wave prcess 8 u ( ( u ( = φ = I u which cmplees he prf. Therem 3.4. If, fr ( ( (,,, he slui f ur mel (9 a he fracial rer wave prblem (4 exis he hey are equivale. Prf. We prve i Therem 3. ha if u ( is he slui f (9 he i is he slui f (4. Cversely, le u ( be he slui f (4 he we ge φ ( D u ( = u ( Operaig by he cvlui f φ ( bh sies we ge φ ( D u( = φ ( u( φ( Du ( Du( = = φ ( u ( Differeiae bh sies we ge u ( = I u (, u( = u, which cmplees he prf. We fiish his seci wih givig sme examples f ur mel. Example 3.5. Le he perar be efie by { } ( ( ( ( ( ( D = u x, C,, lim u x, =, u x, = u x,. x ± x pplyig Therem 3. he (9 has a uique srg slui. Takig Furier rasfrm fr he variable x, wih he parameer ν, a Laplace rasfrm fr he variable, wih he parameer λ, give U λ, ( ν λ = U ( ν λ + ν where U(ν, is he Furier rasfrm f u(x,, a U ( ν, λ rasfrm f U(ν,. Nw we iscuss he iffere cases f as fllws: ( is he Laplace Case. Fr (, (,, a akig he Laplace iverse rasfrm a he Furier iverse rasfrm ( we ge he slui i he frm u( x = G( x u ( x where,, iνx (, =, ( G x e E ν ν π

9 8 Mhama.E. Herzallah, hme M.. El-Saye, Dumiru Baleau 9 where E, ( ν is he Miag-Leffler fuci (see [3]. Case. Fr =, we ge he slui i he frm u( x, = G( x, u ( x where x ν iνx G 4 ( x, = e e ν e. π = π Which is he kw slui he iffusi prblem u u = wih u ( x, = u ( x, lim u ( x, =. x x ± Case 3. Fr =, Laplace iverse rasfrmai rasfrm ( ( ν, = ( ν ( ν = cs( ν ( ν U E U U akig he Furier iverse rasfrm, we ge iν x u( x, = e cs( ν U ( ν ν π, iν( x+ iν( x = ( ( e U e U π + π = ( u( x+ + u( x ν ν ν ν which is he kw D lember s slui he wave prblem u u = wih u ( x, = u ( x, u ( x, =, lim u ( x, =. x x ± Example 3.6. Le he perar be efie by {,,,,, lim, }, (, (, ( ( ( ( ( ( D = u x C u = h u x = u x = u x x x which gives he Sigallig prblem wih u(x, = u. Takig Laplace rasfrm wih respec wih parameer λ, we ge he riary iffereial equai u ( x, u ( x λ λ, λ = λ u, u (, λ = h ( λ, u (, λ =. x Puig = µ, we ge he geeral slui f his O.D.E. i he frm µ λ x u( x, λ = h( λ u e + u. λ λ

10 O he fracial-rer iffusi-wave prcess 83 Takig he iverse Laplace rasfrm, we ge x x u( x, ( h( u M, u fr µ µ µ = + < µ < + where M(z, µ is give by (see [7] ( z ( µ ( µ = = ( z ( M ( z, µ = = Γ( si (, z C,.! π! µ πµ < µ < Γ + Example 3.7. Le he perar be efie by { } ( ( ( ( ( ( ( D = u x, C, L, u, = u L, =, u x, = u x,. x Usig separai f variable meh, we ge ur slui i he frm m π mπx u( x, = cme, si, < m= L L where L mπ x cm = u( x si x. L L 4. CONCLUSION I his paper we give a fracial rer iffusi-wave mel which is mre accurae prvig he exisece, uiqueess a ciuai f he slui a ge he slui sme special cases which give he riary slui f he riary iffusi prblem D lember s slui he riary wave prblem. REFERENCES... Kilbas, H.M. Srivasava, a J.J. Trujill, Thery a pplicais f Fracial Differeial Equais, Nrh-Hlla Mahemaical Suies, Vl. 4, Elsevier, mseram, 6.. K.S. Miller a B. Rss, Iruci he Fracial Calculus a Fracial Differeial Equais, Jh Wiley&Ss. Ic., New Yrk I. Pluby, Fracial Differeial Equai, ca. Press, Sa Dieg-New Yrk-L, S.G. Samk,.. Kilbas, a O.I. Marichev, Fracial Iegrals a Derivaives: Thery a pplicais, Gr a Breach, Laghre, M.. El-Saye, Fracial-rer iffusi-wave equai, I. J. Ther. Phys., 35(, ( M.. El-Saye, a M..E. ly, O he ciuai f fracial rer evluiary iegral equais a sme applicais, 9(, (, R.L. Magi, Fracial Calculus i Biegieerig, Begell Huse Publisher, Ic. Cecicu, 6.

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