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

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
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