Algorithms of Nonlinear Equations by Adomian Polynomials

Abstract

This work was carried out in order to improve and optimize the iterative processes of approximation of solutions to nonlinear equations. Newton's method is an iterative algorithm that allows solving these types of equations. The research developed consisted in finding new schemes and iterative methods equivalent or superior in the number of iterations to Newton's method. This scientific article raises the natural relationship that exists between the Adomian Decomposition Methods and the 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 the iterative variational techniques, obtaining in these ways new iterative formulas for calculating the roots of nonlinear equations. In all cases an auxiliary function of the exponential function family was used, since they have the particularity of being functions C^∞. The main objective is to demonstrate these iterative formulas and to show that the mathematical theory developed in this scientific field is theoretically and analytically based on logical methods and procedures, which allow the development of new schemes, methods and iterative techniques. The algorithms are generated by means of the procedures of the Adomian Polynomials and the Variational Iterative Technique. This work presents three new algorithms that allow finding solutions to nonlinear equations in fewer iterations than Newton's method and therefore are more efficient than Newton's method. All these algorithms were programmed in the Python programming language and the object-oriented programming (OOP) paradigm was used. All these new algorithms show convergence in such a solution. The ideas of this work can be extended to generate new algorithms with the Abbasbandy and Cisneros Method in the search for more efficient algorithms.