Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure

A one-dimensional space-time fractional diffusion equation describes anomalous diffusion on fractals in one dimension. In this paper, this equation is discretized by finite difference schemes based on the Gr¨unwald-Letnikov approximation for Riemann-Liouville and Caputo’s fractional derivatives. It turns out that the discretized equations can be put into a compact form, i.e., a linear system with a block lower-triangular coefficient matrix. To solve the linear system, we formulate a matrix iterative algorithm based on gradient-descent technique. In particular, we work out for the space fractional diffusion equation. Theoretically, the proposed solver is always applicable with satisfactory convergence rate and error estimates. Simulations are presented numerically and graphically to illustrate the accuracy, the efficiency, and the performance of the algorithm, compared to other iterative procedures for linear systems.