Analytical and numerical solutions of a one-dimensional fractional-in-space diffusion equation in a composite medium

In this paper a one-dimensional space fractional diffusion equation in a composite medium consisting of two layers in contact is studied both analytically and numerically. Since domain decomposition is the only approach available to solve this problem, we at first investigate analytical and numerical strategies for a composite medium with the same fractal dimension in each layer to ascertain which domain decomposition approach is the most accurate and consistent with a global solution methodology, which is available in this case. We utilise a matrix representation of the fractional-in-space operator to generate a system of linear ODEs with the matrix raised to the same fractional exponent. We show that the global and domain decomposition numerical strategies for this problem produce simulation results that are in good agreement with their analytic counterparts and conclude that the domain decomposition that imposes the Neumann condition at the interface produces the most consistent results. Finally, we carry this finding to study the composite problem with different fractal dimensions, where we again favourably compare analytic and numerical solutions. The resulting method can be naturally extended to space fractional diffusion in a composite medium consisting of more than two layers.