Abstract
We investigate the connection between the linear harmonic oscillator equation and some classes of second-order nonlinear ordinary differential equations of Liénard and generalized Liénard type, which physically describe important oscillator systems. By means of a method inspired by quantum mechanics, and which consists of the deformation of the phase space coordinates of the harmonic oscillator, we generalize the equation of motion of the classical linear harmonic oscillator to several classes of strongly nonlinear differential equations. The first integrals, and a number of exact solutions of the corresponding equations, are explicitly obtained. The devised method can be further generalized to derive explicit general solutions of nonlinear second-order differential equations unrelated to the harmonic oscillator. Applications of the obtained results for the study of the traveling wave solutions of the reaction–convection–diffusion equations, and of the large amplitude-free vibrations of a uniform cantilever beam, are also presented.
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Acknowledgments
The authors would like to thank to the three anonymous referees for comments and suggestions that helped us to significantly improve our manuscript. The authors are very grateful to Dr. Christian Böhmer for his careful reading of the manuscript.
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Harko, T., Liang, SD. Exact solutions of the Liénard- and generalized Liénard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator. J Eng Math 98, 93–111 (2016). https://doi.org/10.1007/s10665-015-9812-z
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DOI: https://doi.org/10.1007/s10665-015-9812-z