TY - JOUR
T1 - A Finite Difference Method for Stochastic Nonlinear Second-Order Boundary-Value Problems Driven by Additive Noises
AU - Baccouch , Mahboub
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 368
EP - 389
PY - 2020
DA - 2020/05
SN - 17
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/16864.html
KW - Stochastic nonlinear boundary-value problems, finite difference method, additive white noise, mean-square convergence, order of convergence.
AB - <p style="text-align: justify;">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.</p>