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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.

26A33Fractional derivatives and integrals (real functions)
33D15Basic hypergeometric functions of one variable, ${}_r\phi_s$
44A10Laplace transform
44A15Special transforms (Legendre, Hilbert, etc.)
45K05Integro-partial differential equations
35K57Reaction-diffusion equations
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