An Effective Algorithm for Delay Fractional Convection-Diffusion Wave Equation Based on Reversible Exponential Recovery Method

In this paper, we investigate a linearized finite difference scheme for the variable coefficient semi-linear fractional convection-diffusion wave equation with delay. Based on reversible recovery technique, the original problems are transformed into an equivalent variable coefficient semi-linear fractional delay reaction-diffusion equation. Then, the temporal Caputo derivative is discreted by using
$L_{1}$
approximation and the second-order spatial derivative is approximated by the centered finite difference scheme. The numerical solution can be obtained by an inverse exponential recovery method. By introducing a new weighted norm and applying discrete Gronwall inequality, the solvability, unconditionally stability, and convergence in the sense of
$L_{2}$
- and
$L_{\infty }$
- norms are proved rigorously. Finally, we present a numerical example to verify the effectiveness of our algorithm.