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

Yanmei Liu,Yubin Yan,Monzorul Khan

doi:10.3390/math4030045

Abstract

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.

Highlights

IntroductionWe will consider a Fourier spectral method for solving the following linear stochastic space fractional partial differential equation:
In this paper, we will consider a Fourier spectral method for solving the following linear stochastic space fractional partial differential equation: ∂u(t, x ) ∂2 W (t, x )+ (−∆)α u(t, x ) = ∂t∂t∂x u(t, 0) = u(t, 1) = 0, 0 < t < T u(0, x ) = u0 ( x ),0 < t < T, 0 < x < 1 (1) (2) 0
We will present the computational issues for solving the following stochastic space-fractional parabolic partial differential equations by using the spectral method developed in the previous section, with 1/2 < α ≤ 1

Summary

Introduction

We will consider a Fourier spectral method for solving the following linear stochastic space fractional partial differential equation:. The numerical methods for solving space-fractional stochastic partial differential equations are quite restricted even for the case 1/2 < α ≤ 1. We will use Fourier spectral methods to solve the approximated stochastic space-fractional partial differential equations. Note that ∂t∂x is a function in L2 ((0, T ) × (0, 1)), and we can solve (8)–(10) by using any appropriate numerical method for deterministic space-fractional partial differential equations.

Approximate White Noise and Regularity

Fourier Spectral Method

Numerical Simulations

Conclusions