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Fourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive noises in one-dimensional case are introduced and analyzed. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The space-time noise is approximated by the piecewise constant functions in the time direction and by some appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. For the linear problem, we obtain the precise error estimates in the L2 norm and find the relations between the error bounds and the fractional powers. For the nonlinear problem, we introduce the numerical algorithms and MATLAB codes based on the FFT transforms. Our numerical algorithms can be adapted easily to solve other stochastic space fractional partial differential equations with multiplicative noises. Numerical examples for the semilinear stochastic space fractional partial differential equations are given. [ABSTRACT FROM AUTHOR]

We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin, Lazarov, and Zhou [IMA J. Numer. Anal., 36 (2016), pp. 197-221) established an O(k) convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where k denotes the time step size. We show that the modified L1 scheme has a convergence rate of O(k²-α), 0 < α < 1 for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discontinuous Galerkin time stepping method for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinuous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q − 1 , q ≥ 1 , which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given. [ABSTRACT FROM AUTHOR]