A New Modification of The HPM for The Duffing Equation With High Nonlinearity

Authors

  • Ahmed Khdir Faculty of Technology of Mathematical Sciences and Statistics Alneelain University, Algamhoria Street, P.O. Box 12702, Khartoum - Sudan

DOI:

https://doi.org/10.24297/jap.v2i2.2099

Keywords:

Homotopy perturbation method, Chebyshev spectral method, Duffing equation

Abstract

In this work we introduce a new modification of the homotopy perturbation method for solving nonlinear ordinary differential equations. The technique is based on the blending of the Chebyshev pseudo-spectral methods and the homotopy perturbation method (HPM). The method is tested by solving the strongly nonlinear Duffing equation for undamped oscillators. Comparison is made between the proposed technique, the standard HPM, an earlier modification of the HPM and the numerical solutions to demonstrate the high accuracy, applicability and validity of the present approach.

Downloads

Download data is not yet available.

References

G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Comp.

and Math. Appl., Vol. 21, 1991, pp. 101 - 27.

A. Beléndez, C. Pascual, M. Ortuño, T. Beléndez, S. Gallego, Application of a modified He's homotopy

perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities,

Nonlinear Analysis: Real World Applications, Vol. 10, 2009, pp. 601 - 610.

A. Beléndez, C. Pascual, S. Gallego, M. Ortuño, C. Neipp, Application of a modified He's homotopy

perturbation method to obtain higher-order approximations of an

/3 x force nonlinear oscillator, Phys.

Lett. A, Vol. 371, 2007, pp. 421-426.

A. Beléndez, C. Pascual, T. Beléndez, A. Hernández, Solution for an anti-symmetric quadratic nonlinear

oscillator by a modified He's homotopy perturbation method, Nonlinear Analysis: Real World

Applications, Vol. 10, 2009, pp. 416-427.

A. Beléndez, T. Beléndez, A. Marquez, C. Neipp, Application of He's homotopy perturbation method to

conservative truly nonlinear oscillators, Chaos, Solitons & Fractals, Vol. 37, 2008, pp. 770 - 780.

W. S. Don, A. Solomonoff, Accuracy and speed in Computing the Chebyshev Collocation Derivative, SIAM J.

Sci. Comput, Vol. 16, 1995, pp. 1253-1268.

D. D. Ganji, A.R. Sahouli, M. Famouri, A new modification of He's homotopy perturbation method for rapid

convergence of nonlinear undamped oscillators, J Appl Math Comput, Vol. 30, 2009, pp. 181-192.

S. Ghosh, A. Roy, D. Roy, An adaptation of Adomian decomposition for numeric–analytic integration of

strongly nonlinear and chaotic oscillators, Comput. Methods Appl. Mech. Eng., Vol. 196, 2007, pp. 1133

- 1153.

J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., Vol. 178, 1999, pp. 257-

J. H. He, A coupling method for homotopy technique and perturbation technique for non-linear problems,

International Journal of Non-Linear Mechanics, Vol. 35, 2000, pp. 37 - 43.

J. H. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., Vol.

, 2003, pp. 73-79.

J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A, Vol. 350,

, pp. 87 - 88.

J. H. He, X.-H. Wu, Variational iteration method: New development and applications, Comput.Math. Appl.,

Vol. 54, 2007, pp. 881 - 894.

S. J. Liao, Beyond perturbation: Introduction to homotopy analysis method, Chapman & Hall/CRC Press,

R. E. Mickens, Analysis of non-linear oscillators having non-polynomial elastic terms, J. Sound Vibration,

Vol. 255, 2002, pp. 789 - 792.

R. E. Mickens, Oscillations in an

/3 x potential, J. Sound Vibration, Vol. 246, 2001, pp. 375 - 378.

Z. M. Odibat, A new modification of the homotopy perturbation method for linear and nonlinear operators,

Appl. Math. Comput., Vol. 189, 2007, pp. 746 - 753.

Z. M. Odibat, S. Momani, S, Modified homotopy perturbation method: Application to quadratic Riccati

differential equation of fractional order, Chaos, Solitons & Fractals, Vol. 36, 2008, pp. 167-174.

J. I. Ramos, An artificial parameter-decomposition method for nonlinear oscillators: Applications to

oscillators with odd nonlinearities, J. Sound Vib., Vol. 307, 2007, pp. 312–329.

J. I. Ramos, On the variational iteration method and other iterative techniques for nonlinear differential

equations, Appl. Math. Comput., Vol. 199, 2008, pp. 39 - 69.

L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, PA, 2000.

L. Xu, Determination of limit cycle by He's parameter-expanding method for strongly nonlinear oscillators, J.

Sound Vib., Vol. 302, 2007, pp. 178 - 184.

S. J. Liao, An approximate solution technique not depending on small parameters: a special example, Int.

J. Non-linear Mechanics, 30(3),1995, pp. 371-380.

S. J. Liao, Boundary element method for general nonliear differential operator, Engineering Analysis with

boundary element, 20(2), 1997, pp. 91-99.

Downloads

Published

2013-10-06

How to Cite

Khdir, A. (2013). A New Modification of The HPM for The Duffing Equation With High Nonlinearity. JOURNAL OF ADVANCES IN PHYSICS, 2(2), 124–133. https://doi.org/10.24297/jap.v2i2.2099

Issue

Section

Articles