Asymptotic expansion of the solution for nonlinear wave equation with the mixed homogeneous conditions

Abstract

An asymptotic expansion of solution in the small parameter ε of the following initial and boundary value problem: utt - uxx = Fε(x,t,u,ux,ut), x ∈ ω = (0,1), 0 < t < T,; ux(0,t) - h0u(0,t) = ux(1,t) + h1u(1,t) = 0; and u(x,0) = u0(x), ut(x,0) = u1(x) was studied, where h0, h1 are the given nonnegative constants with h0 + h1 > 0; u0, u1 are the given functions with u0 ∈ H2(ω), u1 ∈ H1 (ω) and Fε is a given function.