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Recently, several problems in mathematics, physics, and engineering have been modeled via distributed-order fractional diffusion equations. In this paper, a new class of time distributed-order and space fractional diffusion equations with variable coefficients on bounded domains and Dirichlet boundary conditions is considered. By performing numerical integration we transform the time distributed-order fractional diffusion equations into multiterm time-space fractional diffusion equations. An implicit difference scheme for the multiterm time-space fractional diffusion equations is proposed along with a discussion about the unconditional stability and convergence. Then, the fast Krylov subspace methods with suitable circulant preconditioners are developed to solve the resultant linear system in light of their Toeplitz-like structures. The aforementioned methods are proved to acquire the capability to reduce the memory storage of the proposed implicit difference scheme from \(\mathcal{O}(M^{2})\) to \(\mathcal {O}(M)\) and the computational cost from \(\mathcal{O}(M^{3})\) to \(\mathcal{O}(M\log M)\) during iteration procedures, where M is the number of grid nodes. Finally, numerical experiments are employed to support the theoretical findings and show the efficiency of the proposed methods.

A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients