Solving time-fractional diffusion equations with Robin boundary conditions via fractional Hamiltonian boundary value methods

In this paper, we propose a novel numerical scheme for solving time-fractional reaction-diffusion problems with Robin boundary conditions, where the time derivative is in the Caputo sense of order $\alpha\in(0,1)$. The existence and uniqueness of the solution is proved. Our proposed method is based on the spectral collocation method in space and Fractional Hamiltonian boundary value methods in time. For the considered spectral collocation method, the basis functions used are not the standard polynomial basis functions, but rather adapt to Robin boundary conditions, and the exponential convergence property is provided. The proposed procedure achieves spectral accuracy in space and is also capable of getting spectral accuracy in time. Some numerical examples are provided to support the theoretical results.