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This article is cited in 4 scientific papers (total in 4 papers)
Differential Equations and Mathematical Physics
On the correctness of boundary value problems for the mixed type equation of the second kind
K. B. Sabitova, I. P. Egorovab a Sterlitamak Branch of Bashkir State University, Sterlitamak, 453103, Russian Federation
b Samara State Technical University, Samara, 443100, Russian Federation
(published under the terms of the Creative Commons Attribution 4.0 International License)
Abstract:
In this paper, the intervals of change in the exponent of the degree of degeneration of a mixed-type equation with characteristic degeneration are established.
The first boundary problem and the modified boundary problem (analogue of the Keldysh problem) with the conditions of periodicity are correctly set. In the case of the first problem, a criterion for the uniqueness of its solution is manifested. It is shown that the solution of the analogue of the Keldysh problem is unique up to a term of a linear function. Solutions are constructed as the sum of series of eigenfunctions of the corresponding one-dimensional spectral problem. In justifying the convergence of a series representing the solution of the first boundary-value problem, the problem of small denominators of a more complex structure arises in the class of regular solutions of this equation than in previously known works.
The estimate on separation from zero is established with the corresponding asymptotic. Based on this estimate, sufficient conditions are found for the boundary functions to substantiate the uniform convergence of the series and their derivatives up to the second order inclusive.
Keywords:
equations of mixed type, characteristic degeneration, boundary-value problems, periodicity conditions, spectral method, uniqueness, small denominators, existence.
Received: July 1, 2019 Revised: August 21, 2019 Accepted: September 16, 2019 First online: October 11, 2019
Citation:
K. B. Sabitov, I. P. Egorova, “On the correctness of boundary value problems for the mixed type equation of the second kind”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:3 (2019), 430–451
Linking options:
https://www.mathnet.ru/eng/vsgtu1718 https://www.mathnet.ru/eng/vsgtu/v223/i3/p430
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