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The fundamental solution of the space-time fractional advection-dispersion equation. (English) Zbl 1086.35003
Summary: A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order $\alpha \in \left(0,1\right]$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $\beta \in \left(0,2\right]$. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.
##### MSC:
 35A08 Fundamental solutions of PDE 35K57 Reaction-diffusion equations 26A33 Fractional derivatives and integrals (real functions) 49K20 Optimal control problems with PDE (optimality conditions) 44A10 Laplace transform
##### References:
 [1] V.V. Anh and N.N. Leonenka,Spectral analysis of fractional kinetic equations with random data, J. Stat. Physics,104, N5/6 (2001), 1349–1387. · Zbl 1034.82044 · doi:10.1023/A:1010474332598 [2] V.V. Anh and N.N. Leonenko,Renormalization and homogenization of fractional diffusion equations with random data, Probab. Theory Rel. Fields,124 (2002), 381–408. · Zbl 1031.60043 · doi:10.1007/s004400200217 [3] V.V. Anh and N.N. Leonenko,Harmmonic analysis of fractional diffusion-wave equations, Applied Math. Comput.,48(3) (2003), 239–252. [4] M. BAsu and D.P. Acharya,On quadratic fractional generalized solid bi-criterion, J. Appl. Math. and Computing(old:KJCAM)2(2002), 131–144. · Zbl 1007.90038 · doi:10.1007/BF02936212 [5] D.A. Benson,The fractional advection dispersion equation: Development and application, Ph.D. thesis, Univ. of Nev. Reno, 1998. [6] D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert,Application of a fractional advection-dispersion equation, Water Resources Research,36(6) (2000), 1403–1412. · doi:10.1029/2000WR900031 [7] M. Caputo,Linear model of dissipation whose Q is almost frequency indepent-H, Geophys. J. R. Astr. Soc.,13 (1967), 529–539. [8] M.M. Djrbashian,Integral transforms and representations of functions in the complex plane, Nauka, 1966 (in russian). [9] A.M.A. El-Sayed and M.A.E. Aly,Continuation theorem of fractionalorder evolutionary integral equations, J. Appl. Math. and Computing (old:KJCAM)2(2002), 525–534. [10] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi,Higer transcendental functions,3, New York, McGraw-Hill, (1953-1954). [11] Y. Fujita,Integro differential equation which interpolates the heat equation and the wave equation, Osaka, J. Math.27 (2004), 309–321. [12] A.A. Kilbas, T. Pierantozzi, J. Trujillo,On the solution of fractional evolution equations, J. Phys. A: Math. Gen.37 (2004), 3271–3283. · Zbl 1059.35030 · doi:10.1088/0305-4470/37/9/015 [13] F. Liu, I. Turner and V. Anh,An unstructured mesh finite volume method for modelling saltwater intrusion into coatal aquifer, J. Appl. Math. and Computing (old:KJCAM)9 (2002), 391–407. [14] F. Liu, L.W. Turner, V. Anh and N. Su,A two-dimensional finite volume method for transient simulation of time-, scale-and density-dependent transport in heterogeneous aquifer systems, J. Appl. Math. and Computing11 (2003a), 215–241. · Zbl 1145.76407 · doi:10.1007/BF02935733 [15] F. Liu, I.W. Turner, V. Anh and P. Zhuang,Time fractional advection-dispersion equation, J. Appl. Math. and Computing13(2003b), 233–245. · Zbl 1068.26006 · doi:10.1007/BF02936089 [16] F. Liu, V.V. Anh and I. Turner:Numerical solution of the space fractional Fokker-Plank Equation, J. Comp. Appl. Math.166 2004, 209–319. · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028 [17] W. Feller,On a generalization of Marcel Riesz’s potentials and the semigroups generated by them, Meddekanden lunds Universitets Matematiska Seminarium (Comm. Sém. M.athém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, (1952) 73–81. [18] R. Gorenflo and F. Mainardi,Approximation of Lévy-Feller diffusion by random walk, ZAA,18 (1999), 231–246. [19] R. Gorenflo, Yu. Luchko and F. Mainardi,Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math.118 (2000), 175–191. · Zbl 0973.35012 · doi:10.1016/S0377-0427(00)00288-0 [20] R. Gorenflo and F. Mainardi,Fractional calculus: integral and differential equations of fractional order, in A. Carpinteri and Mainardi (Editors) Fractals and Fractional Calculus in Continuum Mechanics, Wien and New York, Springer Verlag, (1997), 223–273. [21] F. Huang and F. Liu,The time fractional diffusion equation and advection-dispersion equation, Submitted to the Australian and New Zealand Industrial and Applied Mathematic Journal (ANZIAM), 2004, in press. [22] F. Mainardi,Fraction calculus: some basic problems in continuum, and statistical mechanics (A. Carpinteri, F. Mainardi, Eds.),Fractal and Fractional Colin Continuum Mechanics, Springer, Wien (1997), 291–348. [23] F. Mainardi, Y. Luchko, G. Pagnini,The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis,4 (2001), 153–1925. [24] M.M. Meerschaert, D.A. Benson and B. Bäumer,Multidimensional advection and fractional dispersion, Phys. Rev. E.59(5), (1999), 5026–5028. · doi:10.1103/PhysRevE.59.5026 [25] M.M. Meerschaert and C. Tadjeran,Finite difference approximations for fractional advection-dispersion equations. [26] I. Podlubny,Fractional differential equations, Academic press, San Diego, 1999. [27] R. Schunner, D.A. Benson, M.M. Meerschaert, S.W. Wheatcraft,Eulerian derivation of the factional adverction-dispersion equation, Journal of Contaninant Hydrology48 (2001), 69–88. · doi:10.1016/S0169-7722(00)00170-4 [28] A. Saichev and G. Zaslavsky,Fractional kinetic, equations: solutions and applications, Chaos7 (1997), 753–764. · Zbl 0933.37029 · doi:10.1063/1.166272