Solutions of the Duffing and Painlevé-Gambier Equations by Generalized Sundman Transformation

Damien Kolawolé Kêgnidé Adjaï,Lucas Hervé Koudahoun,Jean Akande,Marc Delphin Monsia,Yélomè Judicaë Fernando Kpomahou

doi:10.3844/jmssp.2018.241.252

Damien Kolawolé Kêgnidé Adjaï, Lucas Hervé Koudahoun + Show 3 more

Open Access

https://doi.org/10.3844/jmssp.2018.241.252

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Abstract

A new approach using the generalized Sundman transformation to solve explicitly and exactly in a straightforward manner the cubic elliptic Duffing equation is proposed in this study. The method has the advantage to closely relate this equation to the linear harmonic oscillator equation and to be applied to solve other nonlinear differential equations. As a result, explicit and exact general periodic solutions to some Painleve-Gambier type equations have been established and in particular, it is shown that a reduced Painleve-Gambier XII equation can exhibit trigonometric solutions, but with a shift factor.

Highlights

A large part of physics, engineering and applied science problems is solved by using linear models where the principle of superposition may be applied
The present theory will have the advantage of generalizing the approach developed by Akande et al (2017a) and other methods that exist in literature in the perspective of detecting large class of nonlinear differential equations of the Liénard type for which explicit and exact general periodic solutions can be computed by a well-documented linearization method, that is the generalized Sundman transformation
The explicit and exact general solutions to the cubic Duffing equation as well as for some Painlevé- Gambier type equations are in a straightforward manner determined

Summary

Introduction

A large part of physics, engineering and applied science problems is solved by using linear models where the principle of superposition may be applied. The present theory will have the advantage of generalizing the approach developed by Akande et al (2017a) and other methods that exist in literature in the perspective of detecting large class of nonlinear differential equations of the Liénard type for which explicit and exact general periodic solutions can be computed by a well-documented linearization method, that is the generalized Sundman transformation. This theory will have the advantage to closely relate for.

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Discussion

Conclusion