Exponential/fractional Jacobi spectral Galerkin and collocation schemes for the multi-dimensional time-fractional diffusion equations on the semi-infinite domain

The usual spectral Galerkin and collocation methods using classical orthogonal polynomials as trial functions enjoy high-order accuracy for partial differential equations with smooth solutions. However, their accuracy and efficiency can deteriorate when the solutions exhibit weakly singular behaviors. In this paper, we consider efficient spectral schemes for classes of time-fractional diffusion equations in one- and two-dimensional semi-infinite domains with the time-fractional derivative described in Caputo sense. Since the solutions of these equations usually exhibit singularities at the beginning of time, then they can not be well approximated using the usual spectral schemes. We construct fractional spectral Galerkin and collocation schemes using the fractional Jacobi and the exponential Jacobi functions as basis functions. The use of fractional Jacobi functions allows us to deal with the usual singularity of solutions at $t = 0$, while the use of exponential Jacobi functions allows us to deal with the problem on the semi-infinite domain. The schemes are successfully extended to the two-dimensional case. Various one- and two-dimensional numerical examples are performed to show the validity of the discretization of the non-local terms.