Homotopy Continuation Method for Solving a Class of Slightly Perturbed Equations

Abstract

In the present paper, an iterative method of second order convergence for solving a class of slightly perturbed equations is established using homotopy continuation technique. The accuracy of the method is very high and verified through two applications, in the first application we fix all the input parameters of our algorithm and allow the homotopy function to vary. For this application we considered as a typical example, Kepler equation of elliptical motion. In the second applications, the function is fixed while the other input parameters of the algorithm are varied, and we used for these applications seven highly transcendental slightly perturbed equations. The most important characteristic of the method is that it does not need any priori knowledge of the initial guess, a property which avoids the critical situations between divergent to very slow convergent solutions, that may exist in the application of other numerical methods which depending on initial guess. Moreover, the method allows freedom in choosing the function in order to secure any required accuracy.