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Volume 40, Issue 2
Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

Haiyan Yuan

J. Comp. Math., 40 (2022), pp. 177-204.

Published online: 2022-01

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  • Abstract

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and  can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

  • Keywords

Semi-linear stochastic delay integro-differential equation, Exponential Euler method, Mean-square exponential stability, Trapezoidal rule.

  • AMS Subject Headings

60H35, 65C20, 65C30, 65L20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yhy82_47@163.com (Haiyan Yuan)

  • BibTex
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  • TXT
@Article{JCM-40-177, author = {Haiyan and Yuan and yhy82_47@163.com and 6833 and Department of Mathematics, Heilongjiang Institute of Technology, Harbin, China and Haiyan Yuan}, title = {Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {2}, pages = {177--204}, abstract = {

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and  can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2010-m2019-0200}, url = {http://global-sci.org/intro/article_detail/jcm/20183.html} }
TY - JOUR T1 - Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations AU - Yuan , Haiyan JO - Journal of Computational Mathematics VL - 2 SP - 177 EP - 204 PY - 2022 DA - 2022/01 SN - 40 DO - http://doi.org/10.4208/jcm.2010-m2019-0200 UR - https://global-sci.org/intro/article_detail/jcm/20183.html KW - Semi-linear stochastic delay integro-differential equation, Exponential Euler method, Mean-square exponential stability, Trapezoidal rule. AB -

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and  can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

Haiyan Yuan. (2022). Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations. Journal of Computational Mathematics. 40 (2). 177-204. doi:10.4208/jcm.2010-m2019-0200
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