Validity of approximations for time periodic solutions of a forced non–linear hyperbolic differential equation

Abstract

For the following class of boundary value problems: where p > 0 and E are real parameters and I&( < E O , E O small enough, it can beshown that when f fulfills certain conditions, a unique 2~-time-periodic solution u(t, z ; c ) = u(t + 2*, 2 ; E ) exists. It is shown that this 2~p e r i o d i c solution u ( t , 2 ; E ) is analytic with respect to E for (61 < EO. This result will be used to justify a formal perturbation method for the construction of approximations to the time periodic solution. Integral expressions for the terms in the series representation d u i ( t , 2 ) for the time-periodic solution are derived with this perturbation method. As the time-periodic solution is now known to be analytic the series representation converges with anon zero radius of convergence. This series can be truncated to obtain approximations to the time periodic solution of O(cN+'). In a number of practical applications the integral expressions for the terms in the series representation for the time periodic solution can be evaluated by means of Fourier series. As the computation of the coefficients in these Fourier series is in general rather complicated these Fourier series are also truncated. In this way formal approximations can be obtained, formal in the sense that every u i ( t , z ) , i = 1,2, ..., N of the truncated series zz0 &ui(t, z ) is now represented by truncated Fourier-series. A Maple Computer Algebra program is given in the appendix to compute these formal approximations for the time-periodic solution of O(E**+'), where E' = E / E O when the first term uo has already been calculated or approximated. This program may be used to find an estimate for the radius of convergence for the series expansion in 6. As an illustration explicit examples are presented.

The asymptotic theory of global solutions for semilinear wave equations in three space dimensions

“…In [1][2][3][4], the asymptotic theory for validation of the formal approximations of the solutions to the initial-boundary value problems associated with the second-order semilinear wave equation in one space dimension with the time order function T = O(le] -1) has been presented. But for x E R 1, there still exist some unsolved open problems, as given in [1][2][3], on the asymptotic theory of initial value problems associated with second-order nonlinear wave equations.…”