{"title":"Gellerstedt Problem with a Nonlocal Oddness Boundary Condition for the Lavrent’ev–Bitsadze Equation","authors":"T. E. Moiseev","doi":"10.1134/s00122661230100051","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness\nboundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions\nare obtained in closed form. It is proved that the system of eigenfunctions is complete in the\nelliptic part of the domain and incomplete in the entire domain. The unique solvability of the\nproblem is also proved; the solution is written in the form of a series if the spectral parameter is\nnot equal to an eigenvalue. For the spectral parameter coinciding with an eigenvalue, solvability\nconditions are obtained under which the family of solutions is found in the form of a series. A\ncondition for the solvability of the problem depending on the eigenvalues is obtained. The\nconstructed analytical solutions can be used efficiently in numerical modeling of transonic gas\ndynamics problems.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-23","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/s00122661230100051","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
We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness
boundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions
are obtained in closed form. It is proved that the system of eigenfunctions is complete in the
elliptic part of the domain and incomplete in the entire domain. The unique solvability of the
problem is also proved; the solution is written in the form of a series if the spectral parameter is
not equal to an eigenvalue. For the spectral parameter coinciding with an eigenvalue, solvability
conditions are obtained under which the family of solutions is found in the form of a series. A
condition for the solvability of the problem depending on the eigenvalues is obtained. The
constructed analytical solutions can be used efficiently in numerical modeling of transonic gas
dynamics problems.
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