Abstract
The distributed-order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power-law scaling over the whole time-domain. An important application of distributed-order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate. To broaden the range of applicability of distributed-order fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we develop spectral tau schemes to discretize the fractional diffusion equation with distributed-order fractional derivative in time and Dirichlet boundary conditions. The model solution is expanded in multi-dimensions in terms of Legendre polynomials and the discrete equations are obtained with the tau method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth. Our results confirm that nonlocal numerical methods are best suited to discretize distributed-order fractional differential equations as they naturally take the global behavior of the solution into account.