The Selection of Proper Auxiliary Parameters in the Homotopy Analysis Method. A Case Study: Uniform Solutions of Undamped Duffing Equation

Author(s)
Mustafa Turkyilmazoglu

ABSTRACT

The present paper is concerned with two novel approximate analytic solutions of the undamped Duffing equation. Instead of the traditional perturbation or asymptotic methods, a homotopy technique is employed, which does not require a small perturbation parameter or a large parameter for an asymptotic expansion. It is shown that proper choices of an auxiliary linear operator and also an initial approximation during the implementation of the homotopy analysis method can yield uniformly valid and accurate solutions. The obtained explicit analytical expressions for the solution predict the displacement, frequency and period of the oscillations much more accurate than the previously known asymptotic or perturbation formulas.

The present paper is concerned with two novel approximate analytic solutions of the undamped Duffing equation. Instead of the traditional perturbation or asymptotic methods, a homotopy technique is employed, which does not require a small perturbation parameter or a large parameter for an asymptotic expansion. It is shown that proper choices of an auxiliary linear operator and also an initial approximation during the implementation of the homotopy analysis method can yield uniformly valid and accurate solutions. The obtained explicit analytical expressions for the solution predict the displacement, frequency and period of the oscillations much more accurate than the previously known asymptotic or perturbation formulas.

KEYWORDS

Homotopy Analysis Method, Auxiliary Linear Operator, Initial Approximation, Duffing Oscillator, Uniform Solution

Homotopy Analysis Method, Auxiliary Linear Operator, Initial Approximation, Duffing Oscillator, Uniform Solution

Cite this paper

nullM. Turkyilmazoglu, "The Selection of Proper Auxiliary Parameters in the Homotopy Analysis Method. A Case Study: Uniform Solutions of Undamped Duffing Equation,"*Applied Mathematics*, Vol. 2 No. 6, 2011, pp. 783-790. doi: 10.4236/am.2011.26105.

nullM. Turkyilmazoglu, "The Selection of Proper Auxiliary Parameters in the Homotopy Analysis Method. A Case Study: Uniform Solutions of Undamped Duffing Equation,"

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 Variational 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 Equation 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 Differential 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 Undamped Duffing Equation by a New Approximate Approach,” Computer Physics Communications, Vol. 174, No. 6, 2006, pp. 447-463. doi:10.1016/j.cpc.2005.09.006

[9] E. Yusufo?lu, “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. Heri?anu, “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 Homotopy 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 Journal 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 Depending 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

[17] S. Abbasbandy, “The Application of the Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer,” Physics Letter A, Vol. 360, No. 1, 2006, pp. 109-113. doi:10.1016/j.physleta.2006.07.065

[18] R. A. Van Gorder and K. Vajravelu, “Analytic and Numerical Solutions to the Lane-Emden Equation,” Physics letter A, Vol. 372, No. 39, 2008, pp. 6060-6065. doi:10.1016/j.physleta.2008.08.002

[19] S. Liang and D. J. Jeffrey, “Comparison of Homotopy Analysis Method and Homotopy Perturbation Method through an Evolution Equation,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 12, 2009, pp. 4057-4064. doi:10.1016/j.cnsns.2009.02.016

[20] J. H. He, “Variational Iteration Method—Some Recent Results and New Interpretations,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 3-17. doi:10.1016/j.cam.2006.07.009

[21] S. J. Liao, “A Uniformly Valid Analytic Solution of Two-Dimensional Viscous Flow over a Semi-Infinite Flat Plate,” Journal of Fluid Mechanics, Vol. 385, 1999, pp. 101-128.

[22] S. J. Liao and I. Pop, “Explicit Analytic Solution for Similarity Boundary Layer Equations,” International Journal of Heat and Mass Transfer, Vol. 47, No. 1, 2004, pp. 75-85. doi:10.1016/S0017-9310(03)00405-8

[23] J. H. He, “Some Asymptotic Methods for Strongly Nonlinear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199. doi:10.1142/S0217979206033796

[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 Variational 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 Equation 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 Differential 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 Undamped Duffing Equation by a New Approximate Approach,” Computer Physics Communications, Vol. 174, No. 6, 2006, pp. 447-463. doi:10.1016/j.cpc.2005.09.006

[9] E. Yusufo?lu, “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. Heri?anu, “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 Homotopy 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 Journal 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 Depending 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

[17] S. Abbasbandy, “The Application of the Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer,” Physics Letter A, Vol. 360, No. 1, 2006, pp. 109-113. doi:10.1016/j.physleta.2006.07.065

[18] R. A. Van Gorder and K. Vajravelu, “Analytic and Numerical Solutions to the Lane-Emden Equation,” Physics letter A, Vol. 372, No. 39, 2008, pp. 6060-6065. doi:10.1016/j.physleta.2008.08.002

[19] S. Liang and D. J. Jeffrey, “Comparison of Homotopy Analysis Method and Homotopy Perturbation Method through an Evolution Equation,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 12, 2009, pp. 4057-4064. doi:10.1016/j.cnsns.2009.02.016

[20] J. H. He, “Variational Iteration Method—Some Recent Results and New Interpretations,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 3-17. doi:10.1016/j.cam.2006.07.009

[21] S. J. Liao, “A Uniformly Valid Analytic Solution of Two-Dimensional Viscous Flow over a Semi-Infinite Flat Plate,” Journal of Fluid Mechanics, Vol. 385, 1999, pp. 101-128.

[22] S. J. Liao and I. Pop, “Explicit Analytic Solution for Similarity Boundary Layer Equations,” International Journal of Heat and Mass Transfer, Vol. 47, No. 1, 2004, pp. 75-85. doi:10.1016/S0017-9310(03)00405-8

[23] J. H. He, “Some Asymptotic Methods for Strongly Nonlinear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199. doi:10.1142/S0217979206033796