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Analytical study on the fractional anomalous diffusion in a half-plane. (English) Zbl 1205.82108
Summary: In this study, anomalous diffusion in a half-plane with a constant source and a perfect sink at each half of the boundary is considered. The discontinuity of the boundary condition is erased by decomposing the solution into two parts – a symmetric part and an antisymmetric part. The symmetric part which has been studied extensively can be solved by an integral transform method, Green’s function method or others. To obtain the solution of the antisymmetric part, a separable similarity solution is assumed and the Erdélyi-Kober-type fractional derivative is used. By doing so, the partial differential equation reduces to an ordinary one. Using the Mellin transform method, the solution of the antisymmetric part in terms of a Fox-$H$ function is obtained. Some figures are given to show the characters of the diffusion process and the influences of different orders of fractional derivatives.
##### MSC:
 82C24 Interface problems (dynamic and non-equilibrium); diffusion-limited aggregation 44A15 Special transforms (Legendre, Hilbert, etc.) 26A33 Fractional derivatives and integrals (real functions)