Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

• Published in: Mathematics (Basel) Vol. 4; no. 3; p. 45
• Main Authors: Liu, Yanmei, Khan, Monzorul, Yan, Yubin
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2016
• Subjects: Approximation; Derivatives; Differential equations; error estimates; Fourier analysis; Fourier spectral method; Mathematical models; Mathematics; Partial differential equations; space-fractional partial differential equations; Spectral methods; stochastic partial differential equations; Stochasticity; White noise
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math4030045

Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the L 2 -norm are obtained, and numerical examples are given.