Asymptotic Solutions of Inhomogeneous Initial Boundary Value Problems for Weakly Nonlinear Partial Differential Equations

Abstract

The two-time perturbation procedure is applied to a class of inhomogeneous initial boundary value problems for weakly nonlinear partial differential equations of wave type. An operator calculus approach is adopted to treat the problem as an initial value problem in Hilbert space. The first order equation is applied to two problems in the forced vibration of a nonlinear string. They are governed by the generalized Duffing and van der Pol equations respectively. In the case of the Duffing equation, the results show that there exists a stable, periodic vibration when small damping is present and how the system approaches this final state in the course of time. In the absence of damping, the system does not in general possess a periodic motion. For the van der Pol equation, our results generalize that of Kelley and Kogelman [1]. Here we find that there exist self excited motions in the form of many periodic motions superimposed upon an externally excited periodic motion. The dependence of the final motions on the initial conditions and the forcing function is shown explicitly. A sufficient condition is given for the existence of certain doubly periodic motions as a stable limiting solution. In the Appendix the validity of the method is proved.