The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains

This study uses Laguerre-Hermite spectral Galerkin and spectral collocation methods for solving time-fractional sub-diffusion equations on unbounded domains. In the time domain, the generalized associated Laguerre functions of the first kind are employed as basis functions. In the Galerkin method, the fractional derivative of the basis functions can be obtained using the Laplace transform and its inverse. In the collocation method, the solution is expanded in terms of suitable global basis functions, and collocation conditions are imposed on the Gauss points. The corresponding errors are estimated in the time-space domain, and numerical examples verify the results and provide additional insight into the convergence behavior of the proposed method.