A High Order Iterative Scheme for a Nonlinear Kirchhoff Wave Equation in the Unit Membrane

Le Thı Phuong Ngoc ,Nguyễn Thành Long

doi:10.1155/2011/679528

Abstract

A high‐order iterative scheme is established in order to get a convergent sequence at a rate of order N (N ≥ 1) to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2.

Highlights

In this paper we consider the initial and boundary value problem utt − B ur2 0 urr 1 r ur f r, t, u, 0 < r < 1, 0 < t < T, √ lim r→0 rur r, t< ∞, 1.1 ur 1, t hu 1, t 0, u r, 0 u0 r, ut r, 0 u1 r, where B, f, u0, u1 are given functions satisfying conditions specified later, ur 1 0 r |ur r, t|2 dr, and h > 0 is a given constant.International Journal of Differential EquationsEquation 1.1 1 is the bidimensional nonlinear wave equation describing nonlinear vibrations of the unit membrane Ω1 { x, y : x2 y2 < 1}
The boundary condition |limr → 0 rur r, t | < ∞ is satisfied automatically if u is a classical solution of the problem 1.1, for example, with u ∈ C1 0, 1 × 0, T ∩ C2 0, 1 ×
This condition is used in connection with Sobolev spaces with weight r see 1

Summary

International Journal of Differential Equations

Equation 1.1 1 is the bidimensional nonlinear wave equation describing nonlinear vibrations of the unit membrane Ω1 { x, y : x2 y2 < 1}. There exists an orthonormal Hilbert basis {wj } of the space V0 consisting of eigenfunctions wj corresponding to eigenvalues λj such that i 0 < λ1 ≤ λ2 ≤ · · · ≤ λj ↑ ∞ as j → ∞, ii a wj , v λj wj , v for all v ∈ V1 and j ∈ N Note that it follows from (ii) that {wj / λj } is automatically an orthonormal set in V1 with respect to a ·, · as inner product. With B and f satisfying assumptions H2 and H3 , respectively, for each M > 0 given, we introduce the following constants: KM B sup B η.

It follows that

We deduce that

Smk t

Fmk s

Em t