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The fundamental solution of the space-time fractional advection-dispersion equation. (English) Zbl 1086.35003
Summary: A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order $$\alpha\in(0,1]$$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $$\beta\in(0,2]$$. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.

##### MSC:
 35A08 Fundamental solutions to PDEs 35K57 Reaction-diffusion equations 26A33 Fractional derivatives and integrals 49K20 Optimality conditions for problems involving partial differential equations 44A10 Laplace transform
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