A linear B-spline interpolation/Galerkin finite element method for the two-dimensional Riesz space distributed-order diffusion-wave equation with error analysis

Abstract

This paper focuses on the distributed-order time-fractional diffusion-wave equations with the Riesz space fractional derivatives. A combined method based on the midpoint quadrature rule, linear B-spline interpolation, and the Galerkin finite element method is proposed to obtain the approximate solution. Two steps are used to calculate the approximate solution to this type of equation. The first step approximates the temporal direction by combining a midpoint quadrature rule and linear B-spline interpolation. In the second step, a Galerkin finite element method in the space direction is applied to compute a full-discrete method. Furthermore, the error estimate has been displayed to demonstrate unconditional stability and convergence. Finally, two numerical examples are reported to show the simplicity and efficiency of the proposed method.