Anisotropic non-linear time-fractional diffusion equation with a source term: Classification via Lie point symmetries, analytic solutions and numerical simulation

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The non-linear anomalous diffusion in anisotropic media is using in various fields, e.g. in molecular dynamics, hydrology, financial systems, porous media analysis, etc. The Lie group method is developed to study the time-fractional diffusion equation with a source term in anisotropic media. As an application, the complete Lie group classification is performed up to the equivalence transformations for all special cases of the coefficients. Some similarity reductions are obtained for implicit cases. The analytical invariant subspace method is used in order to find some exact solutions. The work is concluded by the fractional Chebyshev pseudo-spectral (FCPS) method for constructing a numerical simulation for some of the reduced equations in the symmetry analysis section.