Efficient Implementation of Rational Approximations to Fractional Differential Operators

This paper deals with some numerical issues about the rational approximation to fractional differential operators provided by the Pad ́e approximants. In particular, the attention is focused on the fractional Laplacian and on the Caputo derivative which, in this context, occur into the definition of anomalous diffusion problems and of time fractional differential equations (FDEs), respectively. The paper provides the algorithms for an efficient implementation of the IMEX schemes for semi-discrete anomalous diffusion problems and of the short-memory-FBDF methods for Caputo’s FDEs.