This article is cited in 24 papers

The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain

K. B. Sabitov, A. Kh. Suleimanova

Laboratory of Differential Equations, Academy of Sciences of Bashkortostan Republic, Sterlitamak Branch, Sterlitamak, Russia

Abstract: We study the first boundary problem for the following mixed-type equation of the second kind:
$$ u_{xx}+yu_{yy}+au_y-b^2u=0 $$
in the domain $\{(x,y)\mid0<x<1,\ -\alpha<y<\beta\}$, where $a,b,\alpha$, and $\beta$ are given real numbers, and $0<a<1$, $b\geq0$, $\alpha>0$, $\beta>0$. Based on the completeness of the system of eigenfunctions of one-dimensional spectral problem we establish a uniqueness criterion. We construct a solution to the problem as the sum of the series in eigenfunctions.

Keywords: Dirichlet problem, mixed-type equation, spectral method, uniqueness, existence.

UDC: 517.95

Received: 05.07.2007

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2009, 53:11, 37–45

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