Time fractional advection-dispersion equation (Q1429351)

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.