A HYBRID PERTURBATION GALERKIN TECHNIQUE THAT COMBINES MULTIPLE EXPANSIONS

A two-step hybrid perturbation-Galerkin method for the solution of a variety of differential equations-type problems is found to give better results when multiple perturbation expansions are employed. The method assumes that there is a parameter in the problem formulation and that a perturbation method can be used to construct one or more expansions in this parameter. An approximate solution is constructed in the form of a sum of perturbation coefficient functions multiplied by computed amplitudes. In step one, regular and/or singular perturbation methods are used to determine the perturbation coefficient functions. The results of step one are in the form of one or more expansions, each expressed as a sum of perturbation coefficient functions multiplied by a priori known gauge functions. In step two, the classical Bubnov-Galerkin method uses the perturbation coefficient functions computed in step one to determine a set of amplitudes that replace and improve upon the guage functions. The hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Galerkin methods as applied seperately, while combining some of their better features. In this study the proposed method is applied, with two perturbation expansions in each case, to a variety of model ordinary differential equations problems including: a family of linear two-point boundary-value problems, a nonlinear two-point boundary-value problem, a quantum mechanical eigenvalue problem, and a nonlinear free oscillation problem. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.