Communicated by W. Eckhaus
Solutions of weakly non-linear wave quations can be approximated using Galerkin's procedure combined with the averaging method. In this paper existence and uniqueness of solutions are proved in suitably chosen function spaces. Erroretimates lead us to results on asymptotic validity of the approximations. Some applications are indicated.
where u: [O, 13 x Iw-rR. Conditions onf, In this equation E is a small parameter, O < c 4 1 , the term t$(x,u,ut, u,) can be regarded as a perturbation term.
Examples of these equations were studied by Luke" and Whitham16 forf= V'(u), a potential problem; a Van der Pol-like perturbation f= u, -3.: describes galloping oscillations of overhead transition lines, as studied by Chikwendu and Kevorkian* as a special case off= H(ux, u,) and by van Horssen.6 The unperturbed equation (c=O) is the wave equation. Solutions of (1.1) can be expanded in Fourier series. This method is convenient for various reasons: the functions sin (jnx) are the eigenfunctions of the operator d2/dx2; Fourier expansion turns (1.1) into an infinite-dimensional system of ordinary differential equations. This infinite-dimensional system can be truncated into a finite one by taking only a finite part of the Fourier expansions of u and5 This truncation method is known as Galerkin's method.
We would like to know in what sense this method gives us an approximation of the solutions of (1.1). This question will be answered by giving error estimates concerning convergence of Fourier expansions and truncation effects.