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Zbl 1068.26006
Liu, F.; Anh, V.V.; Turner, I.; Zhuang, P.
Time fractional advection-dispersion equation. (English)
[J] J. Appl. Math. Comput. 13, No. 1-2, 233-245 (2003). ISSN 1598-5865; ISSN 1865-2085/e

Applying first a simplifying substitution of the dependent variable (which in the end must be inverted) and then using the transforms of Laplace and Mellin the authors construct in terms of H-functions a representation of the fundamental solution to the linear time-fractional diffusion-convection equation (called by them ``advection dispersion-equation") with constant coefficients. In this way they generalize the solution known for the special case of pure diffusion (dispersion) without convection (advection). By ``time-fractional" they mean, as has become common language, replacement of the first time derivative by a fractional time derivative of order $\alpha \in (0, 1]$. Actually, they use the fractional derivative suitably regularized at zero, usually now called the ``Caputo fractional derivative" which is appropriate in handling Cauchy problems.
[Rudolf Gorenflo (Berlin)]

MSC 2000:: *26A33 Fractional derivatives and integrals (real functions)
33D15 Basic hypergeometric functions of one variable
44A10 Laplace transform
44A15 Special transforms
45K05 Integro-partial differential equations
35K57 Reaction-diffusion equations

Keywords: time-fractional advection-dispersion; Mellin transform; Laplace transform; H-functions

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