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Zbl 1086.35003
Huang, F.; Liu, F.
The fundamental solution of the space-time fractional advection-dispersion equation. (English)
[J] J. Appl. Math. Comput. 18, No. 1-2, 339-350 (2005). ISSN 1598-5865; ISSN 1865-2085/e

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 $\alpha\in(0,1]$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $\beta\in(0,2]$. 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 2000:: *35A08 Fundamental solutions of PDE
35K57 Reaction-diffusion equations
26A33 Fractional derivatives and integrals (real functions)
49K20 Optimal control problems with PDE (nec./ suff.)
44A10 Laplace transform

Keywords: Mittag-Leffler functions; Caputo derivative; Riesz-Feller derivative; Fourier-Laplace transforms

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Zentralblatt MATH Berlin [Germany]