Solving time-fractional differential equations via rational approximation

Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Their nonlocal structure requires taking into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature for the history integral is to approximate the fractional kernel with a sum of exponentials, which is equivalent to considering the FDE solution as a sum of solutions to a system of ordinary differential equations. One possibility to construct this system is to approximate the Laplace spectrum of the fractional kernel with a rational function. In this paper we use the adaptive Antoulas-Anderson algorithm for the rational approximation of the kernel spectrum, which yields only a small number of real-valued poles. We propose a numerical scheme based on this idea and study its stability and convergence properties. In addition, we apply the algorithm to a time-fractional Cahn-Hilliard problem.