Progress in Fractional Differentiation & Applications

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Motivated by recent applications of fractional calculus, in this paper, we derive analytical solutions of fractional advection - dispersion equation with retardation by replacing the integer order partial derivatives with fractional Riesz - Feller derivative for space variable and Caputo fractional derivative for time variable. The Laplace and Fourier transforms are applied to obtain the solution in terms of the Mittag - Leffler function. Some interesting special cases of the time - space fractional advection - dispersion equation with retardation are also considered. The composition formulas for Green function has been evaluated which enables us to express the solution of the space time fractional advection dispersion equation in terms of the solution of space fractional advection dispersion equation and time fractional advection dispersion equation. Furthermore, from this representation we derive explicit formulae, which enable us to plot the probability densities in space for the different values of the relevant parameters.

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