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homotopy approaches are able to estimate the frequency
and period of the oscillations in a better way. Moreover,
unlike the solution of He, no singularity appears in the
displacement solution. Thus it can be concluded that the
homotopy method presented here can safely handle the
situations of high nonlinearity occurring in the Duffing
problems.
5. Concluding Remarks
In this paper the nonlinear problem of Duffing equation
has been considered with the homotopy analysis tech-
nique. As compared to the perturbation methods, the
homotopy treatment employed here does not require any
small parameter, yielding explicit formulas for the
quantities of physical desire, such as the displacement,
frequency and the period of the oscillations.
Four homotopy approaches have been pursued in this
analysis. First, the homotopy method of He in [15] has
been reinvestigated further calculating the second-order
contribution that was missed in [15]. Taking the
advantage of free selection of the linear operator and the
initial approximation to the solution, two more approa-
ches have been proposed here. The homotopy methods
with these choices are proven to generate analytic
approximations that are more accurate than the result
presented in [15]. Moreover, the approximate solutions
obtained by these techniques are valid not only for small
parameters but also very large parameters that are in-
volved in the degree of the nonlinearity. The purely
explicit analytical formulas obtained for the frequency
and period agree excellently with the exact values even
for infinitely large parameters, the discrepancy of the
approximate period with respect to the exact one is as
low as 0.26%. In addition to this, the displacement
function is uniformly valid and does not exhibit any
singularities as compared to the solution in [15].
The homotopy technique proposed here can be safely
adopted for sets of fully coupled, highly nonlinear
equations governing other physical problems in science
and engineering. Analytical solutions obtained here also
provide a good scientific base for the validation of the
numerically computed values using different schemes in
the literature.
6. References
[1] A. H. Nayfeh and D. T. Mook, “Nonlinear Oscillations,”
John Willey and Sons, New York, 1979.
[2] A. H. Nayfeh, “Problems in Perturbations,” John Willey
and Sons, New York, 1985.
[3] Y. K. Cheung, S. H. Chen and S. L. Lau, “A Modified
Lindstedt-Poincaré Method for Certain Strongly Non-Linear
Oscillators,” International Journal of Non-Linear Mechanics,
Vol. 26, No. 3-4, 1991, pp. 367-378.
doi:10.1016/0020-7462(91)90066-3
[4] J. I. Ramos, “On Lindstedt-Poincaré Techniques for the
Quintic Duffing Equation,” Applied Mathematics and
Computation, Vol. 193, No. 2, 2007, pp. 303-310.
doi:10.1016/j.amc.2007.03.050
[5] S. Momani and S. Abuasad, “Application of He’s Varia-
tional Iteration Method to Helmholtz Equation,” Chaos,
Solitons & Fractals, Vol. 27, No. 25, 2006, pp. 1119-1123.
doi:10.1016/j.chaos.2005.04.113
[6] W. S. Loud, “On Periodic Solutions of Duffing’s Equa-
tion with Damping,” Journal of Mathematics and Physics,
Vol. 34, No. 2, 1955, pp. 173-178.
[7] G. R. Liu and T. Y. Wu, “Numerical Solution for Differ-
ential Equations of Duffing-Type Non-Linearity Using
the Generalized Differential Quadrature Rule,” Journal of
Sound and Vibration, Vol. 237, No. 5, 2000, pp. 805-817.
doi:10.1006/jsvi.2000.3050
[8] D. M. Wu and Z. C. Wang, “A Mathematica Program for
the Approximate Analytical Solution to a Nonlinear Un-
damped Duffing Equation by a New Approximate Ap-
proach,” Computer Physics Communications, Vol. 174,
No. 6, 2006, pp. 447-463. doi:10.1016/j.cpc.2005.09.006
[9] E. Yusufoğlu, “Numerical Solution of Duffing Equation
by the Laplace Decomposition Algorithm,” Applied
Mathematics and Computation, Vol. 177, No. 2, 2006, pp.
572-580. doi:10.1016/j.amc.2005.07.072
[10] V. Marinca and N. Herişanu, “Periodic Solutions of
Duffing Equation with Strong Non-Linearity,” Chaos,
Solitons & Fractals, Vol. 37, No. 1, 2008, pp. 144-149.
doi:10.1016/j.chaos.2006.08.033
[11] S. J. Liao, “The Proposed Homotopy Analysis Technique
for the Solution of Nonlinear Problems,” Ph.D. Thesis,
Shanghai Jiao Tong University, Shanghai, 1992.
[12] S. J. Liao, “Beyond Perturbation: Introduction to Homo-
topy Analysis Method,” Chapman & Hall/CRC, Boca
Raton-London-New York-Washington, DC, 2003.
doi:10.1201/9780203491164
[13] S. J. Liao, “An Explicit, Totally Analytic Approximation
of Blasius’ Viscous Flow Problems,” International Jour-
nal of Non-Linear Mechanics, Vol. 34, No. 4, 1999, pp.
759-778. doi:10.1016/S0020-7462(98)00056-0
[14] S. J. Liao and Y. Tan, “A General Approach to Obtain
Series Solutions of Nonlinear Differential Equations,”
Studies in Applied Mathematics, Vol. 119, No. 4, 2007,
pp. 297-254. doi:10.1111/j.1467-9590.2007.00387.x
[15] J. H. He, “An Approximate Solution Technique Depend-
ing on an Artificial Parameter: A Special Example,”
Communications in Nonlinear Science and Numerical
Simulation, Vol. 3, No. 2, 1998, pp. 92-96.
doi:10.1016/S1007-5704(98)90070-3
[16] M. Sajid and T. Hayat, “Comparison of Ham and Hpm
Methods in Nonlinear Heat Conduction and Convection
Equations,” Nonlinear Analysis: Real World Applications,
Vol. 9, No. 5, 2008, pp. 2296-2301.
doi:10.1016/j.nonrwa.2007.08.007