{"title":"Solution of a Boundary Value Problem for an Elliptic Equation with a Small Noninteger Order Degeneracy","authors":"D. P. Emel’yanov","doi":"10.1134/s0012266124030066","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the Dirichlet boundary value problem for an elliptic type equation with\nirregular noninteger-order degeneration in a rectangle. The coefficients of the differential operator\nare supposed to be analytic. We construct a formal solution by using the method of spectral\nseparation of singularities in the form of a series; the character of the nonanalytic dependence of\nthe solution on the variable <span>\\(y\\)</span> in a neighborhood of\n<span>\\(y=0 \\)</span> is written out explicitly. We prove the convergence\nof the series to the classical solution using the Green’s function method.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"144 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124030066","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the Dirichlet boundary value problem for an elliptic type equation with
irregular noninteger-order degeneration in a rectangle. The coefficients of the differential operator
are supposed to be analytic. We construct a formal solution by using the method of spectral
separation of singularities in the form of a series; the character of the nonanalytic dependence of
the solution on the variable \(y\) in a neighborhood of
\(y=0 \) is written out explicitly. We prove the convergence
of the series to the classical solution using the Green’s function method.
期刊介绍：
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.
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