{"title":"A Note on Single-Step Difference Scheme for the Solution of Stochastic Differential Equation","authors":"A. Ashyralyev, U. Okur, C. Ashyralyyev","doi":"10.1134/s1995080224601164","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This is a discuss on the application of operator approach to stochastic partial differential equations with dependent coefficients. Single step difference schemes generated by exact difference scheme for an abstract Cauchy problem for the solution of stochastic differential equation in a Hilbert space with the time-dependent positive operator are presented. The main theorems of the convergence of these difference schemes for the approximate solutions of the time-dependent abstract Cauchy problem for the parabolic equations are established. In applications, the convergence estimates for the solution of difference schemes for stochastic parabolic differential equations are obtained. Numerical results for the <span>\\({1}/{2}\\)</span> order of accuracy difference schemes of the approximate solution of mixed problems for stochastic parabolic equations with Dirichlet and Neumann conditions are provided. Numerical results are given.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lobachevskii Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1995080224601164","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This is a discuss on the application of operator approach to stochastic partial differential equations with dependent coefficients. Single step difference schemes generated by exact difference scheme for an abstract Cauchy problem for the solution of stochastic differential equation in a Hilbert space with the time-dependent positive operator are presented. The main theorems of the convergence of these difference schemes for the approximate solutions of the time-dependent abstract Cauchy problem for the parabolic equations are established. In applications, the convergence estimates for the solution of difference schemes for stochastic parabolic differential equations are obtained. Numerical results for the \({1}/{2}\) order of accuracy difference schemes of the approximate solution of mixed problems for stochastic parabolic equations with Dirichlet and Neumann conditions are provided. Numerical results are given.
期刊介绍：
Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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