Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

It is shown that the Krylov–Bogoliubov methods that give the approximate oscillatory solution of the equation in terms of Jacobi elliptic functions are applicable not only when c 1>0 and c 3>0, but also when c 1>0 and c 3 0. In particular, the most precise of these methods, the Christopher-Brocklehurst method, is discussed in detail. Its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters.

Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions

We construct analytical approximations for the transition from strongly nonlinear, early-time oscillations to weakly nonlinear, late-time motions of single degree of freedom, damped, nonlinear oscillators. Two methods are developed. The first relies on (a) derivation of an analytic solution for the initial value problem of an exactly integrable damped system, (b) development of separate early- and late-time approximations to the damped motion using the integrable solution, and (c) patching of the two approximations in the time domain by imposing continuity conditions on the composite solution at the point of matching. The second approach relies on a multiple-scales application of the method of nonsmooth transformations first developed by Pilipchuck, but complemented with a corrected frequency-amplitude relation. This improved relation is obtained by developing two separate frequency-amplitude asymptotic expansions in the frequency-amplitude plane, that are valid for large and small amplitudes, respectively, and then matching them using two-point diagonal Pade approximants. Comparisons between analytical approximations and numerical results validate the two approaches developed

Transitions from strongly to weakly nonlinear motions of damped nonlinear oscillators

Comments on the method of harmonic balance in which Jacobi elliptic functions are used

Amplitude decay of damped non-linear oscillators studied with Jacobian elliptic functions

An approximate solution, in terms of Jacobian elliptic functions, of the equation $ with $ was developed in Christopher (1973). A generalization of this method, applicable to the class of equations $ with $, is developed. When applied to the case where $ the method is not subject to the restriction c 1 − b 2/4 > 0. A numerical comparison for this latter case shows that the now technique is still accurate over a wide range of values of b and c 3. Some falling off of accuracy is experienced when b/c 3 > 0·5.

A. Brocklehurst

Papers

A generalized form of an approximate solution to a strongly non-linear, second-order, differential equation