ON A SPECTRAL PROBLEM FOR THE LAPLACE OPERATOR WITH MORE GENERAL BOUNDARY CONDITIONS
ЖАЛПЫЛАНҒАН ШЕКТІК ШАРТТАРЫ БАР ЛАПЛАС ОПЕРАТОРЫ ҮШІН ҚОЙЫЛҒАН СПЕКТРАЛДЫҚ ЕСЕП ТУРАЛЫ
ОБ ОДНОЙ СПЕКТРАЛЬНОЙ ЗАДАЧЕ ДЛЯ ОПЕРАТОРА ЛАПЛАСА С БОЛЕЕ ОБЩИМИ ГРАНИЧНЫМИ УСЛОВИЯМИ
Dukenbayeva A. Sadybekov M.
2024Kazakh-British Technical University
Herald of the Kazakh British Technical UNiversity
2024#21Issue 4146 - 152 pp.
In this paper, we consider a spectral problem for the Laplace operator with more general boundary conditions in a unit disk B1 . In the special cases, the boundary conditions inlude periodic and Samarskii-Ionkin type boundary conditions. The main important property of our problem is its non-self-adjointness, which causes number of difficulties in their analytical and numerical solutions. For example, the Fourier method of separation of variables cannot be applied directly to our problem. Therefore, the possibility of separation of variables is justified in this paper. Namely, we present a method that reduces solution of the problem to a sequential solution of two classical local boundary value problems. By using this method, we construct all eigenfunctions and eigenvalues of the problem in explicit forms. Moreover, completeness of the system of the eigenfunctions is proved in L2(B1 ). Notably, our result generalises the special case of the result on the two-dimensional periodic boundary value problem for the Laplace operator obtained in [1–2].
eigenfunctions , eigenvalues , Laplace operator , Samarskii-Ionkin type problem
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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling
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