This article is cited in 1 paper

Differential Equations and Mathematical Physics

The Dirichlet problem for a mixed-type equation with strong characteristic degeneracy and a singular coefficient

R. M. Safina

Volga Region State Academy of Physical Culture, Sport and Tourism, Kazan, 420010, Russian Federation

Abstract: In this paper we consider the first boundary value problem in a rectangular area for a mixed-type equation of the second kind with a singular coefficient. The criterion of the uniqueness of the problem solution is determined. The uniqueness of the problem solution is proved on the basis of completeness of the system of eigenfunctions of the corresponding one-dimensional spectral problem. The solution of the problem is built explicitly as a sum of Fourier–Bessel. There is the problem of the small denominators that appears when justifying the uniform convergence of the constructed series. In this regard, an evaluation of separateness from zero with a corresponding small denominator asymptotic behavior is found. This estimate has allowed to prove the convergence of the series and its derivatives up to the second order, and the existence theorem for the class of regular solutions of this equation.

Keywords: mixed-type equation, Dirichlet problem, singular coefficient, spectral method, uniqueness, Fourier–Bessel series, small denominators, existence.

UDC: 517.956.6

Received: June 3, 2016
Revised: January 17, 2017
Accepted: March 13, 2017
First online: April 27, 2017

DOI: 10.14498/vsgtu1495

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