Finite Volume Method for the Solution of Space‐Time Tempered Fractional Diffusion‐Wave Equation

ABSTRACT In the current paper, we investigate the numerical solution of space‐time fractional tempered diffusion‐wave equation. We use the Crank‐Nicolson scheme to discretize this equation in temporal direction and a finite difference method of order to approximate the tempered fractional integral operator. In spatial direction, we first propose a finite volume method using piecewise‐linear basis functions to obtain fully discrete scheme. Then, by employing some concepts based on the eigenvalues of matrices, we prove the convergence and unconditional stability of the proposed method. To increase the accuracy, we propose a quadratic finite volume method using piecewise quadratic basis functions. To show the accuracy, superiority, and efficiency of the proposed method, we present two test problems and compare the results with some methods developed in literature.