Hybrid Pade´-Galerkin Technique for Differential Equations

A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade´ expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter ε associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade´ approximation in the form of a rational function in the parameter ε. In the third step, the various powers of ε which appear in the Pade´ approximation are replaced by new (unknown) parameters {δ_j}. These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade´ approximations fail to do so. The method is discussed and topics for future investigations are indicated.