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Asymptotic Methods for Solitary Solutions and Compactons

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  • Ji-Huan He

Abstract

This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential‐difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter‐expansion method, the Yang‐Laplace transform, the Yang‐Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.

Suggested Citation

  • Ji-Huan He, 2012. "Asymptotic Methods for Solitary Solutions and Compactons," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:916793
    DOI: 10.1155/2012/916793
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    References listed on IDEAS

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    1. Hamid Khan & S. Islam & Javed Ali & Inayat Ali Shah, 2012. "Comparison of Different Analytic Solutions to Axisymmetric Squeezing Fluid Flow between Two Infinite Parallel Plates with Slip Boundary Conditions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. He, Ji-Huan, 2007. "Variational approach for nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1430-1439.
    3. Ji-Huan He, 2012. "Homotopy Perturbation Method with an Auxiliary Term," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-7, February.
    4. Hamid Khan & S. Islam & Javed Ali & Inayat Ali Shah, 2012. "Comparison of Different Analytic Solutions to Axisymmetric Squeezing Fluid Flow between Two Infinite Parallel Plates with Slip Boundary Conditions," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-18, February.
    5. He, Ji-Huan & Wu, Xu-Hong, 2006. "Construction of solitary solution and compacton-like solution by variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 108-113.
    6. He, Ji-Huan, 2005. "Application of homotopy perturbation method to nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 695-700.
    7. Ji-Huan He, 2012. "Homotopy Perturbation Method with an Auxiliary Term," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    8. Weiguo Zhang & Xiang Li, 2011. "Approximate Damped Oscillatory Solutions for Generalized KdV‐Burgers Equation and Their Error Estimates," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
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    15. Junqiang Song & Fukang Yin & Xiaoqun Cao & Fengshun Lu, 2013. "Fractional Variational Iteration Method versus Adomian’s Decomposition Method in Some Fractional Partial Differential Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    16. Jagdev Singh & Devendra Kumar & A. Kılıçman, 2013. "Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    17. Li Yao & Yun-Jie Yang & Xing-Wei Zhou, 2013. "A Note on the Semi‐Inverse Method and a Variational Principle for the Generalized KdV‐mKdV Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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    19. Kai Liu & Ren-Jie Hu & Carlo Cattani & Gong-Nan Xie & Xiao-Jun Yang & Yang Zhao, 2014. "Local Fractional Z‐Transforms with Applications to Signals on Cantor Sets," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    20. Wei Wei & H. M. Srivastava & Yunyi Zhang & Lei Wang & Peiyi Shen & Jing Zhang, 2014. "A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson’s Inequality," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
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