Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line

Nonlocality in time is an important property of systems in which their present state depends on the history of the whole evolution. Combined with the nonlinearity of the process it poses serious difficulties in both analytical and numerical treatment. We investigate a time-fractional porous medium equation that has proved to be important in many applications, notably in hydrology and material sciences. We show that the solution of both free boundary Dirichlet, Neumann, and Robin problems on the half-line satisfies a Volterra integral equation with non-Lipschitz nonlinearity. Based on this result we prove existence, uniqueness, and construct a family of numerical methods that solve these equations outperforming the usual na\"ive finite difference approach. Moreover, we prove the convergence of these methods and illustrate the theory with several numerical examples.