Second order of accuracy stable difference schemes for hyperbolic problems subject to nonlocal conditions with self-adjoint operator

Abstract

In the present paper, two new second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value problem

{ d(2)u(t)/dt(2) + Au(t) = f(t) (0 <= t <= 1),

u(0) = Sigma(n)(j=1) alpha(j)u(lambda(j)) + phi, u(t)(0) = Sigma(n)(j=1) beta(j)u(t)(lambda(j)) + psi,

0 < lambda(1) < lambda(2) < ... < lambda(n) <= 1

for differential equations in a Hilbert space H with the self-adjoint positive definite operator A. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of difference schemes for the nonlocal boundary value hyperbolic problems are obtained and the numerical results are presented to support our theoretical statements.