A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations. (English) Zbl 1065.35021

Ashyralyev, Allaberen; Aggez, Necmettin

Numer. Funct. Anal. Optimization 25, No. 5-6, 439-462 (2004).

The authors consider the nonlocal boundary-value problem for hyperbolic equations \[ \frac{d^2 u(t)}{d t^2}+Au(t) =f(t)\quad (0\leq t\leq l), \qquad u(0) = \alpha u (1) +\varphi,\qquad u'(0)=\beta' u' (1)+\psi \] in a Hilbert space \(H\) with self-adjoint positive definite operator \(A\). The stability estimates are obtained. The first and second order difference schemes generated by the integer power of \(A\) for approximately solving this nonlocal boundary-value are presented. The stability estimates for the difference schemes are obtained. The theoretical statements for the solution of these difference schemes are illustrated by numerical example.

Reviewer: Qin Mengzhao (Beijing)

MSC:

35A35	Theoretical approximation in context of PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35L20	Initial-boundary value problems for second-order hyperbolic equations
34G10	Linear differential equations in abstract spaces
35L90	Abstract hyperbolic equations

Keywords:

stability; numerical example

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