A Numerical Scheme for Time-Space Fractional diffusion Models

Time-space fractional models are proved to be highly efficient in characterizing anomalous diffusion in many intricate systems. This study introduces a numerical scheme employing a matrix transform technique to discretize the fractional spectral Laplacian and subsequently applying generalized exponential time differencing to the semi-discrete system. The resulting second-order scheme is implemented efficiently using rational approximations for the two-parameter Mittag-Leffler function and the partial fraction decomposition of these approximations. The advantage of this approach is its applicability to different types of homogeneous boundary conditions including Robin boundary conditions. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed numerical scheme.