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 mathcal {O}(tau^{2-alpha}+N^{1-omega}), where 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.
Highlights
In recent decades, fractional partial differential equations have attracted increasing interest mainly due to their potential applications in various realms of science and engineering [ – ]
Wang et al [ ] discussed and analyzed an H -Galerkin mixed finite element method to look for the numerical solution of time-fractional telegraph equation
We study the method resulting from a generalized finite difference method for the temporal discretization and a Legendre spectral Galerkin method for the spatial discretization of the following time-fractional telegraph problem:
Summary
Fractional partial differential equations have attracted increasing interest mainly due to their potential applications in various realms of science and engineering [ – ]. Liu et al [ ] considered the analytical solution of the time-fractional telegraph equation by the method of separating variables. Wang et al [ ] discussed and analyzed an H -Galerkin mixed finite element method to look for the numerical solution of time-fractional telegraph equation. Ford et al [ ] considered a finite difference method for the two-parameter fractional telegraph equation and obtained a stability condition of the numerical method. Many authors have studied the numerical solution of the fractional telegraph equation, they did not strictly prove the unconditional stability and convergence in time direction. The temporal discretization scheme of the time-fractional telegraph equation and its stability and convergence are discussed. In Section , we derive a full discretization scheme of the time-fractional telegraph equation and obtain error estimates.
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