A fast algorithm for two-dimensional distributed-order time-space fractional diffusion equations.

• Two-dimensional distributed-order time-space fractional diffusion problem is considered and its finite difference discretization is studied. • The stability and convergence of the scheme are investigated. • The spatial second-order convergence and the temporal optimal convergence are obtained. • A fast and memory saving algorithm for solving DO time-space fractional diffusion equation is developed through Gauss quadrature formula, ESA method and PCG method. • Numerical experiments show strong effectiveness and efficiency of the method. In this paper, a fast algorithm is proposed for solving distributed-order time-space fractional diffusion equations. Integral terms in time and space directions are discretized by the Gauss-Legendre quadrature formula. The Caputo fractional derivatives are approximated by the exponential-sum-approximation method, and the finite difference method is applied for spatial approximation. The coefficient matrix of the discretized linear system is symmetric positive definite and possesses block-Toeplitz-Toeplitz-block structure. The preconditioned conjugate gradient method with a block-circulant-circulant-block preconditioner is employed to solve the linear system. Theoretically, the stability and convergence of the proposed scheme are discussed. Numerical experiments are carried out to demonstrate the effectiveness of the scheme. [ABSTRACT FROM AUTHOR]