Generalized finite difference/spectral Galerkin approximations for the time-fractional telegraph equation

Abstract

Abstract We discuss the numerical solution of the time-fractional telegraph equation. The main purpose of this work is to construct and analyze stable and high-order scheme for solving the time-fractional telegraph equation efficiently. The proposed method is based on a generalized finite difference scheme in time and Legendre spectral Galerkin method in space. Stability and convergence of the method are established rigorously. We prove that the temporal discretization scheme is unconditionally stable and the numerical solution converges to the exact one with order O ( τ 2 − α + N 1 − ω ) $\mathcal {O}(\tau^{2-\alpha}+N^{1-\omega})$ , where τ , N $\tau, N $ , and ω are the time step size, polynomial degree, and regularity of the exact solution, respectively. Numerical experiments are carried out to verify the theoretical claims.

Keywords

• time-fractional telegraph equation
• generalized finite difference scheme
• Legendre spectral Galerkin method