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Rational Biparameter Homotopy Perturbation Method and Laplace‐Padé Coupled Version

Author

Listed:
  • Hector Vazquez-Leal
  • Arturo Sarmiento-Reyes
  • Yasir Khan
  • Uriel Filobello-Nino
  • Alejandro Diaz-Sanchez

Abstract

The fact that most of the physical phenomena are modelled by nonlinear differential equations underlines the importance of having reliable methods for solving them. This work presents the rational biparameter homotopy perturbation method (RBHPM) as a novel tool with the potential to find approximate solutions for nonlinear differential equations. The method generates the solutions in the form of a quotient of two power series of different homotopy parameters. Besides, in order to improve accuracy, we propose the Laplace‐Padé rational biparameter homotopy perturbation method (LPRBHPM), when the solution is expressed as the quotient of two truncated power series. The usage of the method is illustrated with two case studies. On one side, a Ricatti nonlinear differential equation is solved and a comparison with the homotopy perturbation method (HPM) is presented. On the other side, a nonforced Van der Pol Oscillator is analysed and we compare results obtained with RBHPM, LPRBHPM, and HPM in order to conclude that the LPRBHPM and RBHPM methods generate the most accurate approximated solutions.

Suggested Citation

  • Hector Vazquez-Leal & Arturo Sarmiento-Reyes & Yasir Khan & Uriel Filobello-Nino & Alejandro Diaz-Sanchez, 2012. "Rational Biparameter Homotopy Perturbation Method and Laplace‐Padé Coupled Version," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:923975
    DOI: 10.1155/2012/923975
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    References listed on IDEAS

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    1. Biazar, J. & Ghazvini, H. & Eslami, M., 2009. "He’s homotopy perturbation method for systems of integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1253-1258.
    2. Yasir Khan & Hector Vázquez-Leal & Luis Hernandez-Martínez, 2012. "Removal of Noise Oscillation Term Appearing in the Nonlinear Equation Solution," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Kubler, Felix & Schmedders, Karl, 2000. "Computing Equilibria in Stochastic Finance Economies," Computational Economics, Springer;Society for Computational Economics, vol. 15(1-2), pages 145-172, April.
    4. Biazar, J. & Eslami, M. & Aminikhah, H., 2009. "Application of homotopy perturbation method for systems of Volterra integral equations of the first kind," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3020-3026.
    5. He, Ji-Huan, 2005. "Application of homotopy perturbation method to nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 695-700.
    6. Öziş, Turgut & Yıldırım, Ahmet, 2007. "A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 34(3), pages 989-991.
    7. Yasir Khan & Hector Vázquez-Leal & Luis Hernandez-Martínez, 2012. "Removal of Noise Oscillation Term Appearing in the Nonlinear Equation Solution," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-9, August.
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    Cited by:

    1. Hector Vazquez-Leal & Yasir Khan & Uriel Filobello-Nino & Arturo Sarmiento-Reyes & Alejandro Diaz-Sanchez & Luis-F. Cisneros-Sinencio, 2013. "Fixed‐Term Homotopy," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    2. Abdon Atangana & Suares Clovis Oukouomi Noutchie, 2014. "Novel Approach for Dealing with Partial Differential Equations with Mixed Derivatives," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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