Galerkin spectral method for a multi-term time-fractional diffusion equation and an application to inverse source problem

In this paper, we employ the Galerkin spectral method to handle a multi-term time-fractional diffusion equation, and investigate the numerical stability and convergence of the proposed method. In addition, we find an interesting application of the Galerkin spectral method to solving an inverse source problem efficiently from the noisy final data in a general bounded domain, and the uniqueness and the ill-posedness for the inverse problem are proved based on expression of the solution. Furthermore, we compare the calculation results of spectral method and finite difference method without any regularization method, and get a norm estimate of the coefficient matrix of a spectral method discretized. And for that we conclude that the spectral method itself can act as a regularization method for some inverse problem (such as inverse source problem). Finally, several numerical examples are used to illustrate the effectiveness and accuracy of the method.