Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation

In this paper, we investigate initial boundary value problems of the space-time fractional diffusion equation and its numerical solutions. Two definitions, i.e., Riemann-Liouville definition and Caputo one, of the fractional derivative are considered in parallel. In both cases, we establish the well-posedness of the weak solution. Moveover, based on the proposed weak formulation, we construct an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: First, a theoretical framework for the variational solutions of the space-time fractional diffusion equation is developed. We find suitable functional spaces and norms in which the space-time fractional diffusion problem can be formulated into an elliptic weak problem, and the existence and uniqueness of the weak solution are then proved by using existing theory for elliptic problems. Secondly, we show that in the case of Riemann-Liouville definition, the well-posedness of the space-time fractional diffusion equation does not require any initial conditions. This contrasts with the case of Caputo definition, in which the initial condition has to be integrated into the weak formulation in order to establish the well-posedness. Finally, thanks to the weak formulation, we are able to construct an efficient numerical method for solving the space-time fractional diffusion problem.