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Time fractional advection-dispersion equation. (English) Zbl 1068.26006

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.

MSC:

26A33 Fractional derivatives and integrals
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
44A10 Laplace transform
44A15 Special integral transforms (Legendre, Hilbert, etc.)
45K05 Integro-partial differential equations
35K57 Reaction-diffusion equations
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