Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.The method of integral transforms based on using a fractional generalization of the Fourier transform and the classical Laplace transform is
applied for solving Cauchy-type problem for the time-space fractional diffusion equation expressed in terms of the Caputo time-fractional derivative and a generalized Riemann-Liouville space-fractional derivative

Boyadjiev, Lyubomir

Nikolova, Yanka

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

INTEGRAL TRANSFORMS METHOD TO SOLVEA TIME-SPACE FRACTIONAL DIFFUSION EQUATIONYanka Nikolova 1, Lyubomir Boyadjiev 2AbstractThe method of integral transforms based on using a fractional gener-alization of the Fourier transform and the classical Laplace transform isapplied for solving Cauchy-type problem for the time-space fractional dif-fusion equation expressed in terms of the Caputo time-fractional derivativeand a generalized Riemann-Liouville space-fractional derivative.Mathematical Subject Classification 2010 : 35R11, 42A38, 26A33, 33E12Key Words and Phrases: Caputo fractional derivative, fractional diffu-sion equation, Laplace transform, fractional Fourier transform1. IntroductionIt was Namias [15] who introduced officially in 1980 the FractionalFourier transform (FRFT) as a way to solve certain classes of ordinaryand partial differential equations appearing in quantum mechanics. Mostprobably Namias was unaware of Wiener’s paper [25] published in 1929,where the FRFT in the form of fractional powers of Fourier operator hasbeen already introduced. Namias’ results were later improved by McBrideand Kerr [11], who also developed an operational calculus for the FRFT.The FRFT became much popularity after 1995 because of its numerousapplications in quantum mechanics, chemistry, optics, dynamical systems,stochastic processes and signal processing.c© 2010, FCAA. All rights reserved.58 Y. Nikolova, L. BoyadjievBecause of the importance of the FRFT, a number of its generalizationshave been introduced [1]. In [26], Zayed extended the FRFT to largerclasses of functions and generalized functions, whereas in [3] and [20] discreteversions of the FRFT were introduced. In [27] a systematic and unifiedapproach to fractional integral transforms was presented and a new class offractional integral transforms that includes the FRFT, the fractional Hankeltransforms and the fractional integration and differentiation operators asspecial cases was studied.A detailed survey on the mathematical background, properties and ap-plications related to the FRFT is provided in [19]. The authors of the bookcited present practically independent different definitions of the FRFT andshow that every particular modeling process or mathematical problem re-quires the most suitable definition of the FRFT among the existing to betaken.The modeling of diffusion in a specific type of porous medium is one ofthe most significant applications of fractional derivatives [8], [21]. Two typesof partial differential equations of fractional order deserve special attention.The first type is a generalization of the fractional partial differential equa-tion suggested by Oldham and Spanier as a replacement of Fick’s law [18].The fractional-order diffusion equation suggested by Metzler, Glo¨ckle andNonnenmacher [12] is an example of the second type of fractional diffusionequation. Another example of the second type is the fractional diffusionequation deduced by Nigmatullin [16], [17] also known as the fractionaldiffusion-wave equation.A space-time fractional diffusion equation, obtained from the standarddiffusion equation by replacing the second order space-derivative by a frac-tional Riesz derivative and the first order time-derivative by a Caputo frac-tional derivative, has been treated by Saichev and Zaslavsky [22], Uchajkinand Zolotarev [24], Gorenflo, Iskenderov and Luchko [6], Scalas, Gorenfloand Mainardi [23], Metzler and Klafter [13]. The results obtained in [6] arecomplemented in [9], where the space-time fractional diffusion equation ex-pressed by the Riesz-Feller space-fractional derivative and the Caputo time-fractional derivative is considered. The fundamental solution (the Greenfunction) of the corresponding Cauchy problem is found in the cited paperby means of Fourier-Laplace transform. Based on Mellin-Barnes integralrepresentation, the fundamental solutions of the Cauchy problem are alsoexpressed in terms of proper Fox H-functions, [10].In this paper we develop some properties of the FRFT introduced in [7].INTEGRAL TRANSFORMS METHOD TO SOLVE . . . 59We apply it and the Laplace transformL[f(t); s] =∫ ∞0e−stf(t)dt (1)for solving the space-time fractional diffusion equation as well as a general-ization of it.2. PreliminariesFor a function u of the class S of a rapidly decreasing test functions onthe real axis R, the Fourier transform is defined asuˆ(ω) = F [u(x);ω] =∫ +∞−∞eiωxu(x)dx, ω ∈ R, (2)whereas the inverse Fourier transform has the formu(x) = F−1[uˆ(ω);x] =12pi∫ +∞−∞e−iωxuˆ(ω)dω, x ∈ R. (3)Denote by V (R) the set of functions v(x) ∈ S satisfying the conditions:dnvdxn∣∣∣∣x=0= 0, n = 0, 1, 2, ...The Fourier pre-image of the space V (R), i.e.Φ(R) = {ϕ ∈ S; ϕˆ ∈ V (R)}is called the Lizorkin space. As it is stated in [7], the Lizorkin space isinvariant with respect to the fractional integration and differentiation oper-ators.In this paper we adopt the following FRFT as introduced in [7].Definition 2.1. For a function u ∈ Φ(R), the FRFT uˆα of the orderα (0 < α ≤ 1) is defined asuˆα(ω) = Fα[u(x);ω] =∫ +∞−∞eα(ω, x)u(x)dx, ω ∈ R, (4)whereeα(ω, x) :={e−i|ω|1α x, ω ≤ 0ei|ω|1α x, ω > 0. (5)60 Y. Nikolova, L. BoyadjievEvidently if α = 1, the kernel (5) reduces to the kernel of (2). Therelation between the FRFT (4) and the classical Fourier transform (2) isgiven by the equalityuˆα(ω) = Fα[u(x);ω] = F1[u(x); w] = uˆ(w), (6)wherew :={−|ω| 1α , ω ≤ 0|ω| 1α , ω > 0 . (7)Thus, ifFα[u(x);ω] = F1[u(x); w] = ϕ(w),thenu(x) := F−1α [uˆα(ω);x] = F−11 [ϕ(w);x]. (8)In this paper we study the fractional diffusion equation in terms of theCaputo fractional derivative [4]:Dα∗ f(t) =1Γ(n− α)∫ t0f (n)(τ)(t− τ)α+1−ndτ, n− 1 < α < ndnf(t)dtn, α = n, (9)where n is positive integer. The method we follow makes the rule of theLaplace transform of Caputo derivative very important, see e.g. [21]:L[Dα∗ f(x); s] = sαL[f(t); s]−n−1∑k=0f (k)(0)sα−1−k, n− 1 < α ≤ n. (10)For generalization of the time-space diffusion equation we use the fractionalderivative operator of the formDαβu(x) := (1− β)Dα+u(x)− βDα−u(x), 0 < α ≤ 1, β ∈ R, (11)where Dα+ and Dα− are the Riemann-Liouville fractional derivatives on thereal axis given asDα+u(x) =1Γ(1− α)ddx∫ x−∞(x− τ)α−1u(τ)dτandDα−u(x) = −1Γ(1− α)ddx∫ ∞x(τ − x)α−1u(τ)dτ.INTEGRAL TRANSFORMS METHOD TO SOLVE . . . 61A key role in our considerations is given to a relation established in [7]according to which for 0 < α ≤ 1, any value of β and a function u(x) ∈ Φ(R),Fα[Dαβu(x);ω] = (−icαω)Fα[u(x);ω], ω ∈ R, (12)where cα = sin(αpi/2) + isignω(1− 2β) cos(αpi/2).The one-parameter generalization of the exponential function was intro-duced by Mittag-Leﬄer [14] asEα(z) =∞∑k=0zkΓ(αk + 1).Its further generalization was done by Agarwal [2] who defined the two-parameter function of the Mittag-Leﬄer type in the formEα,β(z) =∞∑k=0zkΓ(αk + β), α > 0, β > 0. (13)Let us remind also that the effect of the application of the Laplace transform(1) on the functions (13) is described by the formulas [21, 1.2.2, (1.80)]:L[tαk+β−1E(k)α,β(±atα); s] =k!sα−β(sα ∓ a)k+1 , Re s > |a|1α . (14)3. Some properties of the FRFTDue to the relations (7) and (8), it is not difficult to prove the validityof the following statement.Theorem 3.1. If 0 < α ≤ 1, u(x) ∈ Φ(R) and Fα[u(x);ω] = uˆα(ω),then:(a) Fα[u(x− a);ω] = eα(ω, a)uˆα(ω) (Shifting)(b) Fα[u(ax);ω] = 1|a| uˆα(|ω|a ), a 6= 0 (Scaling)(c) Fα[u(−x);ω] = uˆα(ω) (Conjugate)(d) Fα[uˆα(x);ω] = u(−ω) (Duality)(e) If g(x) ∈ Φ(R) and Fα[g(x);ω] = gˆα(ω), then+∞∫−∞uˆα(ω)g(ω)eα(ω, t)dω =+∞∫−∞u(x)gˆα(x+ t)dx (Composition).62 Y. Nikolova, L. BoyadjievTheorem 3.2. If 0 ≤ α < 1 and u(n)(x) ∈ Φ(R), thenFα[u(n)(x);ω] = (−isignω|ω| 1α )n uˆα(ω), ω ∈ R.P r o o f. Let us take n = 1 and ω ≤ 0. Then according to (4) and (5),the integration by parts leads toFα[u′(x);ω] =∫ +∞−∞e−i|ω|1α xu′(x)dx = u(x)e−i|ω|1α x∣∣∣∣+∞−∞+i|ω| 1α∫ +∞−∞e−i|ω|1α xu(x)dx = (−isignω|ω| 1α )uˆα(ω). (15)The case ω > 0 is considered similarly. By induction, formula (15) yieldsthe desired result.Theorem 3.3. (Convolution theorem) If a < α ≤ 1 and u(x), v(x) ∈Φ(R), thenFα[(u ∗ v)(x);ω] = uˆα(ω)vˆα(ω),where(u ∗ v)(x) =∫ +∞−∞u(x− ξ)v(ξ)dξ,and Fα[u(x);ω] = uˆα(ω), Fα[v(x);ω] = vˆα(ω).P r o o f. Consider first the case ω ≤ 0. According to (4) and (5) wehaveFα[(u ∗ v)(x);ω] =∫ +∞−∞e−i|ω|1α x[∫ +∞−∞u(x− ξ)v(ξ)dξ]dx=∫ +∞−∞e−i|ω|1α ξv(ξ)[∫ +∞−∞e−i|ω|1α (x−ξ)u(x− ξ)dx]dξ=∫ +∞−∞e−i|ω|1α ξv(ξ)dξ∫ +∞−∞e−i|ω|1α ηu(η)dη = uˆα(ω)vˆα(ω).Likewise, if ω > 0,Fα[(u ∗ v)(x);ω] =∫ +∞−∞ei|ω|1α x[∫ +∞−∞u(x− ξ)v(ξ)dξ]dx=∫ +∞−∞ei|ω|1α ξv(ξ)[∫ +∞∞ei|ω|1α (x−ξ)u(x− ξ)dx]dξ=∫ +∞−∞ei|ω|1α ξv(ξ)dξ∫ +∞−∞ei|ω|1α ηu(η)dη = uˆα(ω)vˆα(ω).This accomplishes the proof.INTEGRAL TRANSFORMS METHOD TO SOLVE . . . 634. Fractional diffusion equationIn this section we apply the FRFT (4) for solving the Cauchy-typeproblem for the fractional diffusion equationDα∗ u(x, t) = µ∂2u∂x2, x ∈ R, t > 0 (16)subject to the initial conditionu(x, t)|t=0 = f(x), (17)where Dα∗ is the Caputo time-fractional derivative (9) of order α, f(x) ∈Φ(R) and µ is a diffusivity constant.Theorem 4.1. If 0 < α ≤ 1, the Cauchy-type problem (16)–(17) issolvable and its solution u(x, t) is given by the integralu(x, t) =∫ +∞−∞G(x− ξ, t)f(ξ)dξ, (18)whereG(x, t) =12pi∫ +∞−∞e−iwxEα(−µ|w| 2α tα)dw.P r o o f. Denote Fα[u(x, t);ω] = uˆα(ω, t). According to Theorem 3.2,the application of the FRFT (4) to the equation (16) and the initial condi-tion (17) transforms the Cauchy-type problem (16)–(17) to the equationDα∗ uˆα(ω, t) = −µ|ω|2α uˆα(ω, t) (19)subject to the initial conditionuˆα(ω, t)|t=0 = fˆα(ω). (20)The formulas (10) and (20) make clear that applying the Laplace transform(1) to (19), we obtainL[uˆα(ω, t); s] =sα−1sα + µ|ω| 2αfˆα(ω). (21)Taking into account that by the formula (14),sα−1sα + µ|ω| 2α= L[Eα(−µ|ω| 2α tα); s],64 Y. Nikolova, L. Boyadjievwe apply the convolution theorem for the Laplace transform and thus obtainuˆα(ω, t) = Eα(−µ|ω| 2α tα)fˆα(ω).Because of (6) and (7), the latest equality gives:uˆ(w, t) = Eα(−µ|w| 2α tα)fˆ(w).Now the validity of the statement follows from the above equation andTheorem 3.3 taken for the particular case α = 1.By means of the formula [5, p.611, (5)],F−11[√piae−ω2/4a;x]= e−ax2,it might be seen that the solution in Theorem 4.1 occurs as a generalizationof the fundamental solution of the classical diffusion problem.Corollary 4.1. If α = 1, the solution of the Cauchy-type problem(16)–(17) is given by the integralu(x, t) =1√4piµt∫ +∞−∞e−(x−ξ)2/4µtf(ξ)dξ.5. Generalized fractional diffusion equationConsider now the Cauchy-type problem that refers to the generalizedfractional diffusion equationDγ∗u(x, t) = µDα+1β u(x, t) (22)subject to the initial condition (17), where Dγ∗ is the Caputo time-fractionalderivative (9) of order γ, whereasDα+1β is the space-fractional derivative (11)we can refer to as generalized Riemann-Liouville space-fractional derivative.Evidently the equation (22) reduces to (16) if α = 1.Theorem 5.1. If f(x) ∈ Φ(R), 0 < γ ≤ 1, 0 < α ≤ 1 and for every valueof β ∈ R, the Cauchy-type problem (22)–(17) is solvable and its solutionu(x, t) is given by the integral (18), whereG(x, t) =12pi∫ +∞−∞e−iwxEγ(−iµcα+1wtγ)dw.INTEGRAL TRANSFORMS METHOD TO SOLVE . . . 65P r o o f. According to (12) it is clear that the application of the FRFTFα+1 to the equation (22) and the initial condition (17) results to the equa-tionDγ∗ uˆα+1(ω, t) = (−iµcα+1ω)uˆα+1(ω, t) (23)subject to the conditionuˆα+1(ω, t)|t=0 = fˆα+1(ω). (24)The Laplace transform (1) applied then to (23) and (24) impliesL[uˆα+1(ω, t); s] =sγ−1sγ + iµcα+1ωfˆα+1(ω). (25)The formula (14) enables us to conclude from (25) thatuˆ(w, t) = Eγ(−iµcα+1wtγ)fˆ(w).By Theorem 3.3 (for α = 1) we obtain from the latest equation that thesolution desired is indeed given by (18), whereG(x, t) =12pi∫ ∞−∞e−iwxEγ(−iµcα+1wtγ)dw.It is worth to notice that the solution of the Cauchy-type problem (22)–(17) reduces for µ = 1 and β = 12 to the solution of the space-time diffusionequation studied in [7].Acknowledgements. This survey paper is partially supported byProject D ID 02 / 25 / 2009 ”Integral Transform Methods, Special Func-tions and Applications”, by National Science Fund - Ministry of Education,Youth and Science, Bulgaria.References[1] A. Abe and J. Sheridan, Optical operations on wave functions as theAbelian subgroups of the special affine Fourier transformation. Opt.Lett. 19 (1994), 1801-1803.[2] R.P. Agarwal, A propos d’une note de M. Pierre Humbert. 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Pagnini, The fundamental solution of thespace-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4,No 2 (2001), 153-192.[10] F. Mainardi, G. Pagnini, R.K. Saxena, Fox H-functions in fractionaldiffusion. J. of Comput. and Appl. Mathematics 178 (2005), 321-331.[11] A. McBride and F. Kerr, On Namias’s fractional Fourier transforms.IMA J. Appl. Math. 39 (1987), 159-175.[12] R. Metzler, W.G. Glo¨ckle, T.F. Nonnenmacher, Fractional model equa-tion for anomalous diffusion. Physics A211 (1994), 13-24.[13] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion:A fractional dynamic approach. Phys. Reports 339 (2000), 1-77.[14] G.M. Mittag-Leﬄer, Sur la nouvelle fonction Eα(x). C.R. Acad. Sci.Paris 137 (1903), 554-558.[15] V. Namias, The fractional order Fourier and its application to quantummechanics. J. Inst. Math. Appl. 25 (1980), 180-191.[16] R.R. Nigmatullin, To the theoretical explanation of the “universal re-sponse”. Phys. Sta. Sol. 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Zaslavsky, Fractional kinetic equations: Solutions andapplications. Chaos 7 (1997), 753-764.[23] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance. Physica A 284 (2000), 37-384.[24] V.V. Uchajkin and V.M. Zolotarev, Chance and Stability. Stable Dis-tributions and Their Applications. VSP, Utrecht (1999).[25] N. Wiener, Hermitian polynomials and Fourier analysis. J. Math. Phys.MIT 8 (1929), 70-73.[26] A.I. Zayed, Fractional Fourier transform of generalized functions. In-tegr. Transforms Special Func. 7 (1998), 299-312.[27] A.I. Zayed, A class of fractional integral transforms: A generalizationof the fractional Fourier transform. IEEE Trans. Signal Processing 50,No 3 (2002), 619-627.1,2 Faculty of Applied Mathematics and InformaticsTechnical University of SofiaSofia 1156, BULGARIA Received: December 21, 2009e-mails: 1 jvr@abv.bg , 2 boyadjievl@yahoo.com

Integral Transforms Method to Solve a Time-Space Fractional Diffusion Equation

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Integral Transforms Method to Solve a Time-Space Fractional Diffusion Equation

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