This article is cited in 7 papers

Differential Equations and Mathematical Physics

A boundary value problem for a third order hyperbolic equation with degeneration of order inside the domain

R. Kh. Makaova

Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Centre of RAS, Nal’chik, 360000, Russian Federation

Abstract: In this paper we study the boundary value problem for a degenerating third order equation of hyperbolic type in a mixed domain. The equation under consideration in the positive part of the domain coincides with the Hallaire equation, which is a pseudoparabolic type equation. Moreover, in the negative part of the domain it coincides with a degenerating hyperbolic equation of the first kind, the particular case of the Bitsadze–Lykov equation. The existence and uniqueness theorem for the solution is proved. The uniqueness of the solution to the problem is proved with the Tricomi method. Using the functional relationships of the positive and negative parts of the domain on the degeneration line, we arrive at the convolution type Volterra integral equation of the 2nd kind with respect to the desired solution by a derivative trace. With the Laplace transform method, we obtain the solution of the integral equation in its explicit form. At last, the solution to the problem under study is written out explicitly as the solution of the second boundary-value problem in the positive part of the domain for the Hallaire equation and as the solution to the Cauchy problem in the negative part of the domain for a degenerate hyperbolic equation of the first kind.

Keywords: boundary-value problem, third order hyperbolic equation, Hallaire equation.

UDC: 517.956.326

MSC: 35L25, 35L80

Received: October 27, 2017
Revised: December 11, 2017
Accepted: December 18, 2017
First online: December 28, 2017

DOI: 10.14498/vsgtu1574

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