{"title":"Inverse Problem for a Non-Homogeneous Integro-Differential Equation of the Hyperbolic Type","authors":"J. Sh. Safarov","doi":"10.1134/s1063454124010114","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The inverse problem of determining the solution and one-dimensional kernel of the integral term in an inhomogeneous integro-differential equation of hyperbolic type from the conditions that make up the direct problem and some additional condition is considered. First, the direct problem is investigated, while the kernel of the integral term is assumed to be known. By integrating over the characteristics, the given integro-differential equation is reduced to a Volterra integral equation of the second kind and is solved by the method of successive approximations. Further, using additional information about the solution of the direct problem, we obtain an integral equation with respect to the kernel of the integral <i>h</i>(<i>t</i>) of the integral term. Using additional information about the solution of the direct problem, we obtain an integral equation of the second kind with respect to the kernel of the integral h(t) of the integral term. Thus, the problem is reduced to solving a system of integral equations of the Volterra type of the second kind. The resulting system is written as an operator equation. To prove the global, unique solvability of this problem, the method of contraction mappings in the space of continuous functions with weighted norms is used. In addition the theorem of the conditional stability of the solution of the inverse problem is proved, while the method of estimating integrals and Gronwall’s inequality is used.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik St Petersburg University-Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1063454124010114","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The inverse problem of determining the solution and one-dimensional kernel of the integral term in an inhomogeneous integro-differential equation of hyperbolic type from the conditions that make up the direct problem and some additional condition is considered. First, the direct problem is investigated, while the kernel of the integral term is assumed to be known. By integrating over the characteristics, the given integro-differential equation is reduced to a Volterra integral equation of the second kind and is solved by the method of successive approximations. Further, using additional information about the solution of the direct problem, we obtain an integral equation with respect to the kernel of the integral h(t) of the integral term. Using additional information about the solution of the direct problem, we obtain an integral equation of the second kind with respect to the kernel of the integral h(t) of the integral term. Thus, the problem is reduced to solving a system of integral equations of the Volterra type of the second kind. The resulting system is written as an operator equation. To prove the global, unique solvability of this problem, the method of contraction mappings in the space of continuous functions with weighted norms is used. In addition the theorem of the conditional stability of the solution of the inverse problem is proved, while the method of estimating integrals and Gronwall’s inequality is used.
期刊介绍：
Vestnik St. Petersburg University, Mathematics is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.
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