The fractional diffusion equation involving the fractional Laplacian is used to govern fractional random walk dynamics on network, which allowing long-range displacements. This paper develops a high accuracy numerical method for the computation of nonlinear fractional diffusion equation. The main idea is to approximate the spatial domain with a spectral Galerkin method based on Fourier-like basis functions, and then to discretize time by the general linear methods which contains Runge-Kutta methods, multistep methods, and many new classes of methods. For $(k,l)$-algebraically stable general linear methods with general stage order $p$, the nonlinear term satisfies the locally Lipschitz condition, and the proposed method is proved to be well-posed, stable, and convergent with order $p$ in time. Moreover, an optimal spatial error estimate is established, whose convergence rate is independent of the fractional parameter $\alpha$. Finally, several numerical experiments are presented to verify and support the theoretical results.

This paper presents a high-order numerical method for nonlinear Volterra integral equations of the third kind (VIE3) with weakly singular kernels. To overcome the accuracy loss caused by the unbounded derivatives of the solution near the origin, we propose a fractional Adams-Simpson-type method on a graded mesh. The stability and convergence of the proposed scheme, along with detailed error estimates, are rigorously established. It is shown that the method achieves an optimal convergence order of $4-\alpha -\lambda \beta$, provided the mesh grading exponent $\lambda$ is chosen appropriately. Numerical experiments are presented to validate the theoretical convergence results and to demonstrate the effectiveness of the method in resolving initial singularities.