A Finite Difference Method for Stochastic Nonlinear Second-Order Boundary-Value Problems Driven by Additive Noises
Year: 2020
Author: Mahboub Baccouch
International Journal of Numerical Analysis and Modeling, Vol. 17 (2020), Iss. 3 : pp. 368–389
Abstract
In this paper, we present a finite difference method for stochastic nonlinear second-order boundary-value problems (BVPs) driven by additive noises. We first approximate the white noise process with its piecewise constant approximation to obtain an approximate stochastic BVP. The solution to the new BVP is shown to converge to the solution of the original BVP at $\mathcal{O}$($h$) in the mean-square sense. The approximate BVP is shown to have certain regularity properties which are not true in general for the solution to the original stochastic BVP. The standard finite difference method for deterministic BVPs is then applied to approximate the solution of the new stochastic BVP. Convergence analysis is presented for the numerical solution based on the standard finite difference method. We prove that the finite difference solution converges to the solution to the original stochastic BVP at $\mathcal{O}$($h$) in the mean-square sense. Finally, we perform several numerical examples to validate the theoretical results.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2020-IJNAM-16864
International Journal of Numerical Analysis and Modeling, Vol. 17 (2020), Iss. 3 : pp. 368–389
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Stochastic nonlinear boundary-value problems finite difference method additive white noise mean-square convergence order of convergence.