Application of Topological Technology to Construction of a Perturbation System for a Strongly Nonlinear Equation

The homotopy perturbation method proposed by the present author is further improved in this paper, which is proved to be effective and convenient to solving nonlinear equations. Introduction There exist many difficulties encountered in the application of the traditional perturbation techniques to solving strongly nonlinear equations. Of these, one of the most frustrating is the fact that even a simple practical system may not posses the so-called “small parameter”, which is the theoretical base of the perturbation methods. The determination of a small parameter in an equation seems to be a special art requiring special techniques. An appropriate choice of a small parameter leads to an ideal result, however, an unsuitable choice of a small parameter results in badly effects, sometimes seriously. Furthermore, even if there exists a suitable small parameter, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameter. To overcome the shortcomings of the perturbation techniques, we will apply homotopy in topological technique to construct a perturbation system. The basic 2000 Mathematics Subject Classification. 55U40, 34E10.