Spectral method for time-space fractional partial differential equation

The diffusion equation is one of the fundamental equations in physics. This paper investigates the numerical solution of spectral method for a time-space fractional diffusion equation. In the article, the temporal semi-discrete scheme is constructed using the L1 interpolation approximation scheme of the Caputo fractional order. The existence of uniqueness and stability of the solution in this semi-discrete scheme is demonstrated, and the error analysis of the semi-discrete scheme is rigorously discussed. On the basis of this semi-discrete scheme, the fully discrete scheme is obtained by discretizing it in the spatial direction using the Legendre spectral method. It is further shown that the solution of this fully discrete scheme exists, is unique, and unconditionally stable. Conclusions on the error between the numerical and exact solutions is given and rigorously discussed in the article.