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 -