One- and two-dimensional analytical solutions of the fractional Richards equation for testing numerical solvers

This study provides analytical solutions of the fractional Richards equation for modeling water subdiffusion in porous media. Our aim is to provide some nontrivial analytical solutions which can serve as benchmark references to test numerical methods for solving the fractional Richards equation. The derivation of the analytical solutions of the fractional Richards equation is based on the use of the Gardner model for capillary pressure to allow the linearization of the problem. The solutions are derived for the Caputo and Atangana-Baleanu-Caputo fractional derivatives using the method of separation of variables and Fourier series expansion. In our case, we used a nonlinear form of the equation through another bounded auxiliary variable. Comparisons are conducted between the analytical solutions and the corresponding numerical ones which are obtained using a numerical scheme based on the localized radial basis function (LRBF) method across various porous media having different physical characteristics.