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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 2, Pages 79–112 (Mi izv8596)  

This article is cited in 3 scientific papers (total in 3 papers)

The first boundary-value problem for an equation of mixed type with a singular coefficient

K. B. Sabitovab, R. M. Safinac

a Institute of Applied Research, Sterlitamak
b Sterlitamak Branch of Bashkir State University
c Volga Region State Academy of Physical Culture, Sport and Tourism

Abstract: We study the first boundary-value problem in a rectangle for an equation of mixed type with a singular coefficient. We establish a criterion for the uniqueness of solutions and construct the solution as the sum of a series in the system of eigenfunctions of a one-dimensional eigenvalue problem. Justifying the uniform convergence of the series encounters a problem of small denominators. To deal with this we obtain bounds for the separation of the small denominators from zero along with the corresponding asymptotic results. These bounds enable us to justify the convergence of the series in the class of regular solutions of the equation.

Keywords: equation of mixed type, singular coefficient, Dirichlet problem, Keldysh problem, survey, uniqueness, orthogonal series, small denominators, bounds, existence, stability.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-31-50008 мол\_нр
17-41-020516 р\_а
This paper was written with the support of the Russian Foundation for Basic Research (grant no.~16-31-50008~mol\_nr), Russian Foundation for Basic Research and Republic Bashkortostan (grant no.~17-41-020516~r\_a).


DOI: https://doi.org/10.4213/im8596

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English version:
Izvestiya: Mathematics, 2018, 82:2, 318–350

Bibliographic databases:

UDC: 517.95
MSC: 35M12
Received: 02.09.2016

Citation: K. B. Sabitov, R. M. Safina, “The first boundary-value problem for an equation of mixed type with a singular coefficient”, Izv. RAN. Ser. Mat., 82:2 (2018), 79–112; Izv. Math., 82:2 (2018), 318–350

Citation in format AMSBIB
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\paper The first boundary-value problem for an equation of mixed type with a~singular coefficient
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\vol 82
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\pages 318--350
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Zaitseva V N., “First Initial-Boundary Value Problem For B-Hyperbolic Equation”, Lobachevskii J. Math., 40:2 (2019), 240–247  crossref  isi
    2. A. A. Abashkin, I. P. Egorova, “Zadacha Dirikhle v parallelepipede dlya ellipticheskogo uravneniya s tremya singulyarnymi koeffitsientami”, Izv. vuzov. Matem., 2019, no. 10, 3–14  mathnet  crossref
    3. K. B. Sabitov, N. V. Zaitseva, “Vtoraya nachalno-granichnaya zadacha dlya $B$-giperbolicheskogo uravneniya”, Izv. vuzov. Matem., 2019, no. 10, 75–86  mathnet  crossref
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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