Fractional difference/finite element approximations for the time–space fractional telegraph equation

In this paper, we study the numerical solution of the time–space fractional order (fractional for simplicity) telegraph equation, which can be used in signal analysis for transmission and propagation of electrical signals, also the modeling of the reaction diffusion and the random walk of suspension flows and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the spatial Riemann–Liouville fractional derivative with order 1+β∈(1,2)1+β∈(1,2) and the finite difference schemes for the temporal Caputo derivatives with orders α∈(1/2,1)α∈(1/2,1) and 2α2α are analyzed respectively. Results on the existence and uniqueness of the solution, the numerical stability, and the error estimates are displayed in details. Numerical examples are included to confirm the theoretical analysis.