Numerical Approximation of a Fractional-In-Space Diffusion Equation, I

Abstract

This paper provides a new method and corresponding numerical schemes to approximate a fractional-in-space diffusion equation on a bounded domain under boundary conditions of the Dirichlet, Neumann or Robin type.The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power.Numerical results are provided to gauge the performance of the proposed method relative to exact analytical solutions determined using a spectral representation of the fractional derivative.Initial results for a variety of one-dimensional test problems appear promising.Furthermore, the proposed strategy can be generalised to higher dimensions.