A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations

The nonlocal boundary-value problem for hyperbolic equations d(2)u(t)/dt(2) + Au(t) = f(t) (0 less than or equal to t less than or equal to 1), u(0) = alphau(1) + phi, u'(0) = betau'(1) + psi in a Hilbert space H with the self-adjoint positive definite operator A is considered. Applying the operator approach, we establish the stability estimates for solution of this nonlocal boundary-value problem. In applications, the stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained. The first and second order of accuracy difference schemes generated by the integer power of A for approximately solving this abstract nonlocal boundary-value problem are presented. The stability estimates for the solution of these difference schemes are obtained. The theoretical statements for the solution of this difference schemes are supported by the results of numerical experiments.