Abstract
The fractional PDEs based upon the distributed-order fractional derivative have several applications in physics. The two-dimensional time-space distributed-order weakly singular integro-partial differential model is investigated by a combination of finite difference and Galerkin spectral methods. A second-order finite difference formula is employed to approximate the temporal variable. In this stage, the stability and convergence of the semi-discrete scheme are proved. Then, the Galerkin spectral method based on the modified Jacobi polynomials is applied to discrete the space variable. Also, in this step, the error estimate of the full-discrete scheme is studied. Finally, two test problems have been presented to confirm the theoretical results.