Fifth Mississippi State Conference on Differential Equations and Computational Simulations,
Electron. J. Diff. Eqns., Conf. 10, 2003, pp. 153-161.

A selfadjoint hyperbolic boundary-value problem

Nezam Iraniparast

Abstract:
We consider the eigenvalue wave equation
$$u_{tt} - u_{ss} = \lambda pu,$$
subject to $ u(s,0) = 0$, where $u\in\mathbb{R}$, is a function of $(s, t) \in \mathbb{R}^2$, with $t\ge 0$. In the characteristic triangle $T =\{(s,t):0\leq t\leq 1, t\leq s\leq 2-t\}$ we impose a boundary condition along characteristics so that
$$ \alpha u(t,t)-\beta \frac{\partial u}{\partial n_1}(t,t) = \alpha u(1+t,1-t) +\beta\frac{\partial u}{\partial n_2}(1+t,1-t),\quad 0\leq t\leq1. $$
The parameters $\alpha$ and $\beta$ are arbitrary except for the condition that they are not both zero. The two vectors $n_1$ and $n_2$ are the exterior unit normals to the characteristic boundaries and $\frac{\partial u}{\partial n_1}$, $\frac{\partial u}{\partial n_2}$ are the normal derivatives in those directions. When $p\equiv 1$ we will show that the above characteristic boundary value problem has real, discrete eigenvalues and corresponding eigenfunctions that are complete and orthogonal in $L_2(T)$. We will also investigate the case where $p\geq 0$ is an arbitrary continuous function in $T$.

Published February 28, 2003.
Subject classifications: 35L05, 35L20, 35P99.
Key words: Characteristics, eigenvalues, eigenfunctions, Green's function, Fredholm alternative.

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Nezam Iraniparast
Department of Mathematics
Western Kentucky University
E-mail: nezam.iraniparast@wku.edu

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