Abstract :
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.
