{"title":"On the Solvability of Fredholm Boundary Integral Equations of the First Kind for the Three-Dimensional Transmission Problem on the Spectrum","authors":"A. A. Kashirin, S. I. Smagin","doi":"10.1134/s0012266124020058","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper considers two weakly singular Fredholm boundary integral equations of the first\nkind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The\nproperties of these equations are studied on the spectra, where they are ill posed. For the first\nequation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the\ntransmission problem. The second equation in this case always has infinitely many solutions, with\nonly one of them giving a solution of the transmission problem. The interpolation method for\nfinding approximate solutions of the integral equations and the transmission problem in question\nis discussed.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124020058","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The paper considers two weakly singular Fredholm boundary integral equations of the first
kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The
properties of these equations are studied on the spectra, where they are ill posed. For the first
equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the
transmission problem. The second equation in this case always has infinitely many solutions, with
only one of them giving a solution of the transmission problem. The interpolation method for
finding approximate solutions of the integral equations and the transmission problem in question
is discussed.
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