On the Asymptotic Formula for the Solution of Nonlocal Boundary Value Perturbation Problems for Hyperbolic Equations

Abstract

In the present paper we consider the nonlocal boundary value perturbation problem {epsilon(2) partial derivative(2)u(t,x)/partial derivative t(2 )- (a(x)u(x)(t,x))(x )+ delta u(t,x) = f(t,x), 0 < t < T,x is an element of (0,l), u(0,x) = alpha u(T,x) + phi(x),x is an element of [0,l], u'(0,x) = beta u'(T,x) + psi(x), x is an element of [0,l], u(t,0) = u(t,l),u(x)(t,0) = u(x)(t,l), 0 <= t <= T for hyperbolic equation with an arbitrary epsilon is an element of (0,infinity) parameter multiplying the derivative term. An asymptotic formula for the solution of this problem with a small epsilon parameter is presented.