Generalized decomposition methods for singular oscillators

Generalized decomposition methods based on a Volterra integral equation, the introduction of an ordering parameter and a power series expansion of the solution in terms of the ordering parameter are developed and used to determine the solution and the frequency of oscillation of a singular, nonlinear oscillator with an odd nonlinearity. It is shown that these techniques provide solutions which are free from secularities if the unknown frequency of oscillation is also expanded in power series of the ordering parameter, require that the nonlinearities be analytic functions of their arguments, and, at leading-order, provide the same frequency of oscillation as two-level iterative techniques, the homotopy perturbation method if the constants that appear in the governing equation are expanded in power series of the ordering parameter, and modified artificial parameter – Linstedt–Poincaré procedures.