##
**Solving the fractional Rosenau-Hyman equation via variational iteration method and homotopy perturbation method.**
*(English)*
Zbl 1267.35244

Summary: In this study, fractional Rosenau-Hynam equations is considered. We implement relatively new analytical techniques, the variational iteration method and the homotopy perturbation method, for solving this equation. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity.

PDF
BibTeX
XML
Cite

\textit{R. Yulita Molliq} and \textit{M. S. M. Noorani}, Int. J. Differ. Equ. 2012, Article ID 472030, 14 p. (2012; Zbl 1267.35244)

Full Text:
DOI

### References:

[1] | J. H. He, “Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics,” International Journal of Turbo and Jet Engines, vol. 14, no. 1, pp. 23-28, 1997. |

[2] | J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86-90, 1999. |

[3] | J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57-68, 1998. · Zbl 0942.76077 |

[4] | I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 |

[5] | K. Al-Khaled and S. Momani, “An approximate solution for a fractional diffusion-wave equation using the decomposition method,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 473-483, 2005. · Zbl 1071.65135 |

[6] | F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153-192, 2001. · Zbl 1054.35156 |

[7] | A. Hanyga, “Multidimensional solutions of time-fractional diffusion-wave equations,” Proceedings of the Royal Society of London, Series A, vol. 458, no. 2020, pp. 933-957, 2002. · Zbl 1153.35347 |

[8] | F. Huang and F. Liu, “The time fractional diffusion equation and the advection-dispersion equation,” The Australian & New Zealand Industrial and Applied Mathematics Journal, vol. 46, no. 3, pp. 317-330, 2005. · Zbl 1072.35218 |

[9] | F. Huang and F. Liu, “The fundamental solution of the space-time fractional advection-dispersion equation,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 339-350, 2005. · Zbl 1086.35003 |

[10] | S. Momani, “An explicit and numerical solutions of the fractional KdV equation,” Mathematics and Computers in Simulation, vol. 70, no. 2, pp. 110-118, 2005. · Zbl 1119.65394 |

[11] | L. Debnath and D. D. Bhatta, “Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics,” Fractional Calculus & Applied Analysis, vol. 7, no. 1, pp. 21-36, 2004. · Zbl 1076.35096 |

[12] | J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017 |

[13] | J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618 |

[14] | J. H. He, “Periodic solutions and bifurcations of delay-differential equations,” Physics Letters A, vol. 347, no. 4-6, pp. 228-230, 2005. · Zbl 1195.34116 |

[15] | J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695-700, 2005. · Zbl 1072.35502 |

[16] | J. H. He, “Limit cycle and bifurcation of nonlinear problems,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 827-833, 2005. · Zbl 1093.34520 |

[17] | J. H. He, Non-perturbative methods for strongly nonlinear problems [Dissertation], de-Verlag im Internet GmbH, Berlin, Germany, 2006. |

[18] | J. H. He, “A new approach to nonlinear partial differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, pp. 230-235, 1997. · Zbl 0923.35046 |

[19] | J. H. He, “Variational iteration method for delay differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, pp. 235-236, 1997. · Zbl 0924.34063 |

[20] | M. Tatari and M. Dehghan, “On the convergence of He’s variational iteration method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 121-128, 2007. · Zbl 1120.65112 |

[21] | Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations of fractional order,” Applied Mathematical Modelling, vol. 32, no. 1, pp. 28-39, 2008. · Zbl 1133.65116 |

[22] | R. Yulita Molliq, M. S. M. Noorani, and I. Hashim, “Variational iteration method for fractional heat- and wave-like equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1854-1869, 2009. · Zbl 1172.35302 |

[23] | R. Yulita Molliq, M. S. M. Noorani, I. Hashim, and R. R. Ahmad, “Approximate solutions of fractional Zakharov-Kuznetsov equations by VIM,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 103-108, 2009. · Zbl 1173.65066 |

[24] | C. Chun, “Variational iteration method for a reliable treatment of heat equations with ill-defined initial data,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 435-440, 2008. · Zbl 06942368 |

[25] | M. G. Porshokouhi and B. Ghanbari, “Application of He’s variational iteration method for solution of the family of Kuramoto-Sivashinsky equations,” Journal of King Saud University - Science, vol. 23, no. 4, pp. 407-411, 2011. |

[26] | S. T. Mohyud-Din and A. Yildirim, “An algorithm for solving the fractional vibration equation,” Computational Mathematics and Modeling, vol. 23, no. 2, pp. 228-237, 2012. · Zbl 1283.74020 |

[27] | A. M. A. El-Sayed, A. Elsaid, I. L. El-Kalla, and D. Hammad, “A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8329-8340, 2012. · Zbl 1245.65141 |

[28] | P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,” Physical Review Letters, vol. 70, no. 5, pp. 564-567, 1993. · Zbl 0952.35502 |

[29] | R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus, A. Carpinteri and F. Mainardi, Eds., Springer, New York, NY, USA, 1997. · Zbl 0934.35008 |

[30] | M. Caputo, “Linear models of dissipation whose Q is almost frequency independent II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, no. 5, pp. 529-539, 1967. |

[31] | M. Inokuti, H. Sekine, and T. Mura, “General use of the Lagrange multiplier in nonlinear mathematical physics,” in Variational Method in the Mechanics of Solids, S. Nemat Nasser, Ed., pp. 156-162, Pergamon Press, New York, NY, USA, 1978. |

[32] | B. A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York, NY, USA, 1972. · Zbl 0319.49020 |

[33] | P. A. Clarkson, E. L. Mansfield, and T. J. Priestley, “Symmetries of a class of nonlinear third-order partial differential equations,” Mathematical and Computer Modelling, vol. 25, no. 8-9, pp. 195-212, 1997. · Zbl 0879.35005 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.