The Novel Integral Homotopy Expansive Method

Uriel Filobello-Nino,Darwin Mayorga-Cruz,Hector Vazquez-Leal,Jaime Ramirez-Angulo,Rogelio Alejandro Callejas-Molina,Jesus Huerta-Chua

doi:10.3390/math9111204

Abstract

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

Highlights

Modeling natural processes in the mathematical realm is certainly a complex task.Most of these processes are nonlinear; there is a need to use complicated mathematical calculations to acquire an approximate value, which is not always the desired result
Unlike linear ordinary differential equations (ODES), in the case of nonlinear differential equations, an exact solution to a given nonlinear problem can rarely be obtained [2], this work will show the potentiality of the proposed method finding both exact and approximate solutions
One advantage of the method over other iterative methods is that, due to the way Integral Homotopy Expansive Method (IHEM) presents the different iterations, it requires only integrations that are elementary most of the time, such as those that occurred with the discussed case studies, while methods as homotopy perturbation method (HPM) and PM have to solve a set of coupled differential equations for the different iterations

Summary

Introduction

Modeling natural processes in the mathematical realm is certainly a complex task Most of these processes are nonlinear; there is a need to use complicated mathematical calculations to acquire an approximate value, which is not always the desired result. Natural processes give rise to scientific problems and to the proposal of new methods in order to reach exact and approximate solutions to the differential equations that govern said nonlinear problems; the search for such solutions is not an easy task and justifies all the research efforts carried out on this topic. Unlike linear ODES, in the case of nonlinear differential equations, an exact solution to a given nonlinear problem can rarely be obtained [2], this work will show the potentiality of the proposed method finding both exact and approximate solutions. The diversity of nonlinear problems has led to the proposal of several methods as alternatives to classical methods, with the aim to produce various types of nonlinear differential equations

Methods

Discussion

Conclusion