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An analysis of the modified L1 scheme for timefractional partial differential equations with nonsmooth dataYan, Yubin; Khan, Monzorul; Ford, Neville J.; University of Chester (Society for Industrial and Applied Mathematics, 20180111)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 \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197221) 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 convergence rate $O(k^{2\alpha}), 0< \alpha <1$ 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.

Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equationsLiu, Yanmei; Yan, Yubin; Khan, Monzorul; LuLiang University; University of Chester (Elsevier, 20170123)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 $q1, q \geq 1$, 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.

Fourier spectral methods for some linear stochastic spacefractional partial differential equationsLiu, Yanmei; Khan, Monzorul; Yan, Yubin; LuLiang University; University of Chester (MDPI, 20160701)Fourier spectral methods for solving some linear stochastic spacefractional partial differential equations perturbed by spacetime white noises in onedimensional case are introduced and analyzed. The spacefractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the spacetime white noise by using piecewise constant functions and obtain the approximated stochastic spacefractional partial differential equations. The approximated stochastic spacefractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in $L^{2}$ norm are obtained. Numerical examples are given.

Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noisesLiu, Fang; Yan, Yubin; Khan, Monzorul; Lvliang University, University of Chester (EudoxusPress, 20180228)Fourier spectral methods for solving stochastic space fractional partial differential equations driven by special additive noises in onedimensional 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 spacetime 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 $L_{2}$ 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.