This article is cited in 7 papers

Differential Equations and Mathematical Physics

Goursat problem for loaded degenerate second order hyperbolic equation with Gellerstedt operator in principal part

A. H. Attaev

Institution of Applied Mathematics and Automation, Nal'chik, 360000, Russian Federation

Abstract: In the paper we study a loaded degenerate hyperbolic equation of the second order with variable coefficients. The principal part of the equation is the Gellerstedt operator. The loaded term is given in the form of the trace of desired solution on the degenerate line. The latter is located in the inner part of the domain. We investigate a boundary value problem. The boundary conditions are given on a characteristics line of the equation under study. For the model equation (when all subordinated coefficients are zero) we construct an explicit representation for solution of the Goursat problem. In the general case, we reduce the problem to an integral Volterra equation of the second kind. We apply the method realized by Sven Gellerstedt solving the second Darboux problem. In both cases, model and general, we use widely properties of the Green–Hadamard function.

Keywords: Goursat problem, loaded equation, hyperbolic equation, degenerate equation, Gellerstedt operator, the Green–Hadamard's function method.

UDC: 517.956.3

MSC: 35L80

Original article submitted 13/X/2015
revision submitted – 23/X/2015

DOI: 10.14498/vsgtu1452

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