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ISSN: 1549-3644 (Print)
ISSN: 1558-6359 (Online)
Research Article Open Access

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

Damien Kolawolé Kêgnidé Adjaï1, Lucas Hervé Koudahoun1, Jean Akande1, Yélomè Judicaël Fernando Kpomahou2 and Marc Delphin Monsia1
  • 1 University of Abomey-Calavi, Benin
  • 2 University of Abomey, Benin

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 Painlevé-Gambier type equations have been established and in particular, it is shown that a reduced Painlevé-Gambier XII equation can exhibit trigonometric solutions, but with a shift factor.

Journal of Mathematics and Statistics
Volume 14 No. 1, 2018, 241-252

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

Submitted On: 20 March 2018 Published On: 8 November 2018

How to Cite: Kêgnidé Adjaï, D. K., Koudahoun, L. H., Akande, J., Kpomahou, Y. J. F. & Monsia, M. D. (2018). Solutions of the Duffing and Painlevé-Gambier Equations by Generalized Sundman Transformation. Journal of Mathematics and Statistics, 14(1), 241-252. https://doi.org/10.3844/jmssp.2018.241.252

Open AccessCopyright: © 2018 Damien Kolawolé Kêgnidé Adjaï, Lucas Hervé Koudahoun, Jean Akande, Yélomè Judicaël Fernando Kpomahou and Marc Delphin Monsia. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Keywords

  • Cubic Duffing Equation
  • Painlevé-Gambier Equations
  • Jacobian Elliptic Functions
  • Exact Periodic Solution
  • Generalized Sundman Transformation