Local discontinuous Galerkin method for the Riesz space distributed-order Sobolev equation

Abstract

We present and analyze a local discontinuous Galerkin finite element method to solve the Riesz space distributed-order Sobolev equation. To approximate the distributed-order Riesz space derivative, the Gauss quadrature as the high computational accuracy method is proposed. A multi-term fractional equation is then constructed from the considered equation by approximating the Riesz space derivative. Moreover, stability analysis is provided for a semi-discrete scheme. We provide numerical results to justify the theoretical analysis by solving the ordinary differential equation. We obtain this after implementing the local discontinuous Galerkin scheme on the Riesz space distributed-order Sobolev equation with the Crank–Nicolson scheme as a time marching method and the Laplace transform technique.