On the numerical solution of the first initial-boundary value problem for telegraph equation in the case of open boundary (Q2742786)

This paper deals with the initial-boundary value problem for the equation \({1\over a^2}{\partial^2 u\over\partial t^2}+ b{\partial u\over\partial t}=\Delta u\), \((x,t)\in Q_{\infty}\), with initial conditions \(u(x,0)={\partial u(x,0)\over\partial t}=0\), \(x\in D\), Dirichlet boundary condition \(u=F\), \((x,t)\in\Sigma_{\infty}\), and consistency condition \(F(x,0)={\partial F(x,0)\over\partial t}=0\), \(x\in\Gamma\). Here \(\Gamma\subset R^2\) is an open curve; \(D=R^2\setminus\Gamma\); \(\Sigma_{\infty}=\Gamma\times(0,\infty)\); \(Q_{\infty}=D\times(0,\infty)\). By the Laguerre transform on time this non-stationary problem is reduced to a system of boundary value problems for the Helmholtz equations. Using the potential representation of a solution an equivalent system of first kind integral equations is obtained and the existence of solution of this system in modified Hölder spaces is proved. Applying trigonometric interpolation the numerical solution of the integral equation is obtained by the quadrature method.