A unified Krylov-Bogoliubov-Mitropolskii method for solving hyperbolic-type nonlinear partial differential systems

Abstract

A general asymptotic solution is presented for investigating the transient response of non-linear systems modeled by hyperbolic-type partial differential equations with small nonlinearities. The method covers all the cases when eigen-values of the corresponding unperturbed systems are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the eigen-values in the general result that the solution corresponding to each of the three cases can be obtained. The method is an extension of the unified Krylov-Bogoliubov-Mitropolskii method, which was initially developed for un-darnped, under-clamped and over-clamped cases of the second order ordinary differential equation. The methods also cover a special condition of the over-damped case in which the general solution is useless.