Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

Daniel Olvera,Alex Elías-Zúñiga

doi:10.1155/2014/486509

Abstract

We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameterhby following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

Highlights

With the recent progress in nonlinear problems research, there has been an increasing interest in analytical techniques to solve the corresponding nonlinear equations
At the end of the paper, we will derive the approximate solutions of some nonlinear differential equations and compare their numerical predictions with respect to their corresponding numerical integration solutions obtained by using the fourth order Runge-Kutta method
Before we examine the application of the enhanced multistage homotopy perturbation method (EMHPM) to derive approximate solutions of nonlinear differential equation, we will address some convergence issues related to our proposed approach

Summary

Introduction

With the recent progress in nonlinear problems research, there has been an increasing interest in analytical techniques to solve the corresponding nonlinear equations. Abbasbandy in [11] concluded that one advantage of the HPM when compared to the ADM is related to its capability of achieving the approximate solution of the quadratic Riccati differential equation by considering all the Taylor expansion series terms. These results were confirmed by Pamuk in [12]. At the end of the paper, we will derive the approximate solutions of some nonlinear differential equations and compare their numerical predictions with respect to their corresponding numerical integration solutions obtained by using the fourth order Runge-Kutta method

Description of the Homotopy Perturbation Method

Description of the Enhanced Multistage Homotopy Perturbation Method

Approximate Solutions of Some Nonlinear Ordinary Differential Equations

Conflict of Interests

Conclusions