Algorithms of Nonlinear Equations by Adomian Polynomials

Abstract

This work is carried out in order to improve and optimize the iterative processes of approximation of solutions of non-linear equations. Newton's method is an iterative algorithm that allows solving these types of equations. The research carried out consisted of finding new schemes and iterative methods equivalent to Newton's method. This thesis raises the natural relationship that exists between Adomian Decomposition Methods and Variational Iterative Techniques, establishing the mathematical links developed in both spheres of knowledge. For the demonstrations of the new schemes and iterative methods, it was based on the Adomian Polynomial scheme and then combined with variational iterative techniques, obtaining in these ways new iterative formulas for calculating roots nonlinear equations. In all cases, a familiar auxiliary function of exponential functions was used, since they have the particularity of being    functions. The main objective is to demonstrate these iterative formulas and to show that the mathematical theory developed in this scientific field are theoretically and analytically based by logical methods and procedures, which allow the development of new iterative schemes and techniques. The algorithms are generated by the procedures of the Adomian Polynomials and the Variational Iterative Technique. This work presents fifteen new algorithms that allow finding solutions to nonlinear equations in a smaller number of iterations than Newton's method and therefore are more efficient than said method. All these algorithms were programmed in the Python programming language and the object-oriented programming paradigm (OOP) was used. All these new algorithms present convergence in said solution. The ideas of this work can be extended to generate new algorithms with the Abbasbandy and Cisneros Methods in the search for more efficient algorithms.