Computational study based on the Laplace transform and local discontinuous Galerkin methods for solving fourth-order time-fractional partial integro-differential equations with weakly singular kernels

The aim of this paper is to implement the high-order local discontinuous Galerkin method (LDGM) for solving partial integro-differential equations (PIDEs) in two dimensions. Time marching method and the transform-based method known as the non-time marching method 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. In order to address this limitation, we propose an algorithm based on a combination of the Laplace transform and LDGM for solving fourth-order time-fractional (TF) 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. A numerical experiment validates the accuracy and applicability of the proposed temporal discretization approach and also shows that k-degree LDG solutions have a \(k + 1\) $k+1$ convergence rate.