Periodic problems of unforced/periodically-forced nonlinear oscillators: Theoretical formulation and boundary shape function method

For the second-order conservative nonlinear oscillator and the Liénard equation we derive integral-type theoretical formulas to compute the periods through a generalized conservation law equipped with a weight function. Minimizing the error to satisfy periodicity conditions the optimal value of parameter is determined; hence, very accurate value of the period can be obtained for examples in testing. Three methods to solve periods and periodic solutions are developed for the Liénard equations without/with periodically-forcing terms. The periodicity conditions and nonlinear differential equation for the oscillatory motion constitute a special kind boundary value problem (BVP), rather than the initial value problem (IVP). This study is concerned with two possible cases: (a) given initial values, and (b) unknown initial values. For case (a) the boundary values at two end points of a time interval are known but with an unknown period, while for case (b) the resulting BVP is more difficult with both unknown period and boundary values on an unknown time interval. By means of boundary shape function method (BSFM) we can transform the BVP to an IVP for an easy computation of periodic problem, where the initial values for the new variable are given and derived explicitly; however, the period and terminal values of the new variable are unknown to be determined iteratively. BSFM's periodic solutions automatically satisfy the periodicity conditions. Numerical examples disclose the merit of BSFM, which is convergent fast to offer very accurate period and periodic solution.