TY - JOUR

T1 - A finite difference method for stochastic nonlinear second-order boundary-value problems driven by additive noises

AU - Baccouch, Mahboub

N1 - Funding Information:
This research was partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha.
Publisher Copyright:
© 2020 Institute for Scientific Computing and Information.

PY - 2020

Y1 - 2020

N2 - In this paper, we present a finite difference method for stochastic nonlinear secondorder 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 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 O(h) in the mean-square sense. Finally, we perform several numerical examples to validate the theoretical results.

AB - In this paper, we present a finite difference method for stochastic nonlinear secondorder 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 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 O(h) in the mean-square sense. Finally, we perform several numerical examples to validate the theoretical results.

KW - Additive white noise

KW - Finite difference method

KW - Mean-square convergence

KW - Order of convergence

KW - Stochastic nonlinear boundary-value problems

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M3 - Article

AN - SCOPUS:85085303882

VL - 17

SP - 368

EP - 389

JO - International Journal of Numerical Analysis and Modeling

JF - International Journal of Numerical Analysis and Modeling

SN - 1705-5105

IS - 3

ER -