Crankxe2x80x93NicolsonGalerkin spectral method for solving twodimensional timespace distributedorder weakly singular integropartial differential equation Abstract   The fractional PDEs based upon the distributedorder fractional derivative have several applications in physics. The twodimensional timespace distributedorder weakly singular integropartial differential model is investigated by a combination of finite difference and Galekrin spectral methods. A secondorder finite difference formula is employed to approximate the temporal variable. In this stage, the stability and convergence of the semidiscrete 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 fulldiscrete scheme is studied. Finally, two test problems have been presented to confirm the theoretical results.

An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarellixe2x80x93Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the (d1) dimensional problem to a sequence of ddimension Poissontype problems by using the matrix diagonalizational method; (ii) resolving the singularity by applying the enriched spectral method in the extended dimension. We carry out rigorous analysis for the proposed numerical method, and provide abundant numerical examples to verify the theoretical results and illustrate effectiveness of the proposed method.