{"title":"On the Non-local Problem for a Boussinesq Type Equations","authors":"Kh. T. Dekhkonov, Yu. E. Fayziev, R. R. Ashurov","doi":"10.1134/s1995080224600808","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The problem of finding a solution, satisfying the non-local condition <span>\\(u(\\xi_{0})=\\alpha u(+0)+\\varphi\\)</span> in time for the Boussinesq type equation of the form <span>\\(u_{tt}+Au_{tt}+Au=f\\)</span> is studied in the article. Here <span>\\(\\alpha\\)</span> and <span>\\(\\xi_{0}\\)</span>, <span>\\(\\xi_{0}\\in(0,T],\\)</span> are the given numbers, <span>\\(A:H\\rightarrow H\\)</span> is the self-adjoint, unbounded, positive operator defined in the Hilbert separable space <span>\\(H\\)</span>. By using the Fourier method, it was shown that the solution to the problem exists and is unique. The effect of parameter <span>\\(\\alpha\\)</span> on the existence and uniqueness of the solution is studied in the article. The inverse problem of determining the right-hand side of the equation is also considered.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-07-19","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/s1995080224600808","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
The problem of finding a solution, satisfying the non-local condition \(u(\xi_{0})=\alpha u(+0)+\varphi\) in time for the Boussinesq type equation of the form \(u_{tt}+Au_{tt}+Au=f\) is studied in the article. Here \(\alpha\) and \(\xi_{0}\), \(\xi_{0}\in(0,T],\) are the given numbers, \(A:H\rightarrow H\) is the self-adjoint, unbounded, positive operator defined in the Hilbert separable space \(H\). By using the Fourier method, it was shown that the solution to the problem exists and is unique. The effect of parameter \(\alpha\) on the existence and uniqueness of the solution is studied in the article. The inverse problem of determining the right-hand side of the equation is also considered.
Abstract The problem of finding a solution, satisfying the non-local condition \(u(\xi_{0})=\alpha u(+0)+\varphi\) in time for the Boussinesq type equation of the form \(u_{tt}+Au_{tt}+Au=f\) is studied in the article.这里\(\alpha\)和\(\xi_{0}\), \(\xi_{0}\in(0,T],\)是给定的数，\(A:H\rightarrow H\) 是定义在希尔伯特可分离空间 \(H\)中的自交、无界、正算子。通过使用傅立叶方法，证明了问题的解是存在的，并且是唯一的。文章研究了参数 \（α）对解的存在性和唯一性的影响。还考虑了确定方程右边的逆问题。
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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