|Applications of the Aboodh Transform and the Homotopy Perturbation Method to the Nonlinear Oscillators|
|P.K. Bera1 , S.K. Das2 , P. Bera3|
1 Dept. of Physics, Dumkal College, Murshidabad, India.
2 Dept. of Mechanical Engineering, IIT Ropar, Rupnagar, India.
3 School of Electronics Engineering, VIT University, Vellore, India.
|Correspondence should be addressed to: firstname.lastname@example.org.|
Section:Research Paper, Product Type: Journal Paper
Volume-6 , Issue-1 , Page no. 1-10, Jan-2018
Online published on Jan 31, 2018
Copyright © P.K. Bera, S.K. Das, P. Bera . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
|View this paper at Google Scholar | DPI Digital Library|
|XML View||PDF Download|
IEEE Style Citation: P.K. Bera, S.K. Das, P. Bera, “Applications of the Aboodh Transform and the Homotopy Perturbation Method to the Nonlinear Oscillators”, International Journal of Computer Sciences and Engineering, Vol.6, Issue.1, pp.1-10, 2018.
MLA Style Citation: P.K. Bera, S.K. Das, P. Bera "Applications of the Aboodh Transform and the Homotopy Perturbation Method to the Nonlinear Oscillators." International Journal of Computer Sciences and Engineering 6.1 (2018): 1-10.
APA Style Citation: P.K. Bera, S.K. Das, P. Bera, (2018). Applications of the Aboodh Transform and the Homotopy Perturbation Method to the Nonlinear Oscillators. International Journal of Computer Sciences and Engineering, 6(1), 1-10.
|86||155 downloads||18 downloads|
|In this paper, the differential equation of motion of the classical Helmholtz-Duffing oscillator, Van der Pol, Duffing oscillator and Duffing-Van der Pol oscillator equations have been solved analytically with the help of a new integral transform named Aboodh transform and homotopy perturbation method. By recasting the governing equations as nonlinear eigenvalue problems, we have obtained the excellent approximate analytical solution of the displacement and the relation between amplitude and angular frequency. We have also compared our results with exact numerical results graphically for few cases. Here, we have also demonstrated the sophistication and simplicity of this technique.|
|Key-Words / Index Term :|
|Aboodh Transform, Homotopy Perturbation Method, Helmholtz-Duffing Oscillator, Van der Pol, Duffing Oscillator, Duffing-Van der Pol Oscillator, Approximate Analytical Solution|
 A.H. Nayfeh, D.T. Mook, “Nonlinear Oscillations”, John Willey and Sons., New York, 1979.
 N.N. Bogoliubov, Y.A. Mitropolsky, “Asymptotic Methods in the Theory of Nonlinear Oscillations” Hindustan Publishing Company, Delhi, Chap. I, 1961.
 V.P. Agrwal, H. Denman, “Weighted linearization technique for period approximation in large amplitude Nonlinear Oscillations”, J. Sound Vib., Vol.57, pp.463-473, 1985.
 S.H. Chen, Y.K. Cheung, S.L. Lau, “On perturbation procedure for limit cycle analysis”, Int. J. Nonlinera Mech., Vol.26, pp.125-133, 1991.
 Y.K. Cheung, S.H. Chen, S.L. Lau, “A modified Lindstedt-Pioncare method for certain strong nonlinear oscillations”, Int. J. Non-Linear Mech., Vol.26, pp.367-378, 1991.
 G. Adomain, “A review of the decomposition method in applied mathematics”, J. Math. Anal. and Appl., Vol.135, pp.501-544, 1998.
 G.L. Lau, “New research direction in singular perturbation theory, artificial parameter approach and inverse-perturbation technique”, In the proceedings of the 1997 National Conference on 7th Modern Mathematics and Mechanics, pp.47-53.
 A.F. Nikiforov, V.B. Uvarov, “Special functions of mathematical physics”, Birkhauser, Basel, 1988.
 P.K. Bera, T. Sil, “Exact solutions of Feinberg-Horodecki equation for Time dependent anharmonic oscillator”, Pramana-J. Phys., Vol.80, pp.31-39, 2013.
 J.H. He, “Homotopy perturbation technique”, Comp. Methods in Appl. Mech. and Engg., Vol.178, pp.257-262, 1999.
 J.H. He, “A coupling method of a homotopy technique and a perturbation technique for nonlinear problems”, Int. J. Non-Linear Mech., Vol.3, pp.37-43, 2000.
 J. Biazar, M. Eslami, “A new homotopy perturbation method for solving systems of partial differential equations”, Comp. and Math. with Appli., Vol.62, pp.225–234, 2011.
 A. Yildirim, “Homotopy perturbation method to obtain exact special solutions with solitary pattern for Boussinesq-like B(m,n) equations with fully nonlinear dispersion”, J. Math. Phys., Vol.50, pp.5–10, 2009.
 M. Gover, A.K. Tomer, “Comparison of Optimal Homotopy Asymptotic Method with Homotopy Perturbation Method of Twelfth Order Boundary Value Problems”, International Journal of Computer Sciences and Engineering, Vol.3, pp.2739-2747, 2011.
 P.K. Bera, T. Sil, “Homotopy perturbation method in quantum mechanical problems”, Applied Math. and Comp., Vol.219, pp. 3272–3278, 2012.
 K.S. Aboodh, “The New integral Transform Aboodh Transform”, Global Journal of Pure and Applied Mathematics, Vol.9, pp.35-43, 2013.
 K. Abdelilah, S. Hassan, M. Mohand, M. Abdelrahim, A.S.S. Muneer, “An application of the new integral transform in Cryptography”, Pure and Applied Mathematics Journal, Vol.5, pp.151-154, 2016.
 P.K. Bera, S.K. Das, P. Bera, “A Study of Nonlinear Vibration of Euler-Bernoulli Beams Using Coupling Between The Aboodh Transform And The Homotopy Perturbation Method”, International Journal of Computer Sciences and Engineering, Vol.5, pp.84-93, 2017.
 B. Bulbul, M. Sezer, “Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method”, Article ID691614.
 M. Najafi, M. Moghimi, H. Massah, H. Khoramishad, M. Daemi, “On the Application of Adomian Decomposition Method and Oscillation Equations”, In the 9th International Conference on Applied Mathematics, Istanbul, Turkey, 2006.
 Gh. Asadi Cordshooli , A.R. Vahidi, “Silution of Duffing –Van der Pol Equation using Decomposition Method”, Adv. Studies Theor. Phys., Vol.5, pp.121-129, 2011.
 R.E. Mickens, “Iteration method solutions for conservative and limit-cycle x^(1/3) force oscillators”, Journal of Sound and Vibration, Vol.292, pp.964-968, 2006.
 I.S. Gradshteyn, I.S. Ryzhik, “Table of Integrals, Series and Products”, Academic Press, New York, 1980.
 H.Li Zhang, “Periodic solutions for some strongly nonlinear oscillations by He’s energy balance method”, Computer & Mathematics with Applications, Vol.58, pp.2480-2485, 2009.
 C.W. Lim, S.K. Lai, “Accrate higher order analytical approximate solutions to nonconservative nonlinear oscillators and applications to van der Pol damped oscillators”, International Journal of Mechanical Sciences, Vol.48, pp.483-892, 2006.
 J.J. Stoker, “Nonlinear Vibration in Mechanical and Electrical Systems”, Wiley, New York, 1992.