High-order iterative methods for a nonlinear kirchhoff wave equation

Abstract

In this paper we consider the following nonlinear wave equation (1) {utt-α/αx(μ(x,t,∥ux(t)∥2)ux) = f(x,t,u), 0 < x < 1,0 < t < T, u(0,t) = u(1,t) = 0, u(x,0) = ũ0(x),ut(x,0) = ũ1(x), where n, μ, f, ũ0,ũ1 are given functions satisfying conditions specified later. In Eq. (l)1, the nonlinear term μ,(x,t, ∥ux∥2) depends on the integral ∥ux(t) ∥2 = ∫10ux(x,i)|2 dx. In this paper we associate with equation (l)1 a recurrent sequence {um} defined by (2) α2um/αt2 - α/ αx (μ(x,t,∥umx(t)∥2umx) = N-1Σi=01/zi αi∫/ αui (x,t, um-1) um - um-1)i,0 < x < 1, 0 < t < T, with um satisfying (1)2,3. The first term uo is chosen as uo ≡ 0. If ∫ ∈ CN([0,1] × ℝ+ × ℝ), we prove that the sequence {um} converges at a rate of order N to a unique weak solution of problem (1).