An efficient hybrid Chebyshev spectral method for multidimensional time-fractional partial integro-differential equations

The aim of this paper is to implement a high-order Chebyshev spectral collocation method for solving partial integro-differential equations (PIDEs) in multiple dimensions. Time marching methods and transform-based methods (non-time marching) can be used to discretize temporal terms. The combination of a small time step size in time marching methods and a high-order scheme in space with many degrees of freedom requires a considerable amount of computational time. To address this limitation, we propose an algorithm based on a combination of the Laplace transform and Chebyshev spectral collocation for solving second-order time-fractional PIDEs with weakly singular kernels. Unlike time-marching approaches, the transform-based method has a much lower computational complexity and can take advantage of parallel computing. Numerical experiments in one, two, and three dimensions validate the accuracy, efficiency, and applicability of the proposed approach.