##
**A coupled method of Laplace transform and Legendre wavelets for Lane-Emden-type differential equations.**
*(English)*
Zbl 1264.65227

Summary: A coupled method of Laplace transform and Legendre wavelets is presented to obtain exact solutions of Lane-Emden-type equations. By employing properties of Laplace transform, a new operator is first introduced and then its Legendre wavelets operational matrix is derived to convert the Lane-Emden equations into a system of algebraic equations. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The results show that the proposed method is very effective and easy to implement.

### MSC:

65T60 | Numerical methods for wavelets |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

PDF
BibTeX
XML
Cite

\textit{F. Yin} et al., J. Appl. Math. 2012, Article ID 163821, 16 p. (2012; Zbl 1264.65227)

Full Text:
DOI

### References:

[1] | S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover, New York, NY, USA, 1967. · Zbl 0123.23604 |

[2] | H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, NY, USA, 1962. · Zbl 0217.54002 |

[3] | O. U. Richardson, The Emission of Electricity From Hot Bodies, Zongmans Green and Company, London, UK, 1921. · JFM 48.0118.05 |

[4] | N. T. Shawagfeh, “Nonperturbative approximate solution for Lane-Emden equation,” Journal of Mathematical Physics, vol. 34, no. 9, pp. 4364-4369, 1993. · Zbl 0780.34007 |

[5] | A.-M. Wazwaz, “A new algorithm for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 287-310, 2001. · Zbl 1023.65067 |

[6] | A.-M. Wazwaz, “A new method for solving singular initial value problems in the second-order ordinary differential equations,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 45-57, 2002. · Zbl 1030.34004 |

[7] | F. Olga and \vS. Zden\vek, “A domian decomposition method for certain singular initial value problems,” Journal of Applied Mathematics, vol. 3, no. 2, pp. 91-98, 2010. |

[8] | J. I. Ramos, “Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 38, no. 2, pp. 400-408, 2008. · Zbl 1146.34300 |

[9] | A. Yildirim and T. Özi\cs, “Solutions of singular IVP’s of Lane-Emden type by homotopy pertutbation method,” Physics Letters A, vol. 369, pp. 70-76, 2007. · Zbl 1209.65120 |

[10] | M. S. H. Chowdhury and I. Hashim, “Solutions of a class of singular second-order IVPs by homotopy-perturbation method,” Physics Letters A, vol. 365, no. 5-6, pp. 439-447, 2007. · Zbl 1203.65124 |

[11] | J.-H. He, “Variational approach to the Lane-Emden equation,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 539-541, 2003. · Zbl 1022.65076 |

[12] | M. Dehghan and F. Shakeri, “Approximate solution of a differential equation arising in astrophysics using the variational iteration method,” New Astronomy, vol. 13, no. 1, pp. 53-59, 2008. |

[13] | A. Yildirım and T. Özi\cs, “Solutions of singular IVPs of Lane-Emden type by the variational iteration method,” Nonlinear Analysis, vol. 70, no. 6, pp. 2480-2484, 2009. · Zbl 1162.34005 |

[14] | S. Liao, “A new analytic algorithm of Lane-Emden type equations,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 1-16, 2003. · Zbl 1022.65078 |

[15] | A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Homotopy analysis method for singular IVPs of Emden-Fowler type,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1121-1131, 2009. · Zbl 1221.65197 |

[16] | C. Mohan and A. R. Al-Bayaty, “Power-series solutions of the Lane-Emden equation,” Astrophysics and Space Science, vol. 73, no. 1, pp. 227-239, 1980. · Zbl 0457.76039 |

[17] | V. S. Ertürk, “Differential transformation method for solving differential equations of Lane-Emden type,” Mathematical & Computational Applications, vol. 12, no. 3, pp. 135-139, 2007. · Zbl 1175.34007 |

[18] | S. Mukherjee, B. Roy, and P. K. Chaterjee, “Solution of Lane-Emden equation by differential transform method,” International Journal of Nonlinear Science, vol. 12, no. 4, pp. 478-484, 2011. · Zbl 1394.34026 |

[19] | K. Parand and M. Razzaghi, “Rational legendre approximation for solving some physical problems on semi-infinite intervals,” Physica Scripta, vol. 69, no. 5, pp. 353-357, 2004. · Zbl 1063.65146 |

[20] | K. Parand and A. Pirkhedri, “Sinc-Collocation method for solving astrophysics equations,” New Astronomy, vol. 15, no. 6, pp. 533-537, 2010. |

[21] | K. Parand, A. R. Rezaei, and A. Taghavi, “Lagrangian method for solving Lane-Emden type equation arising in astrophysics on semi-infinite domains,” Acta Astronautica, vol. 67, no. 7-8, pp. 673-680, 2010. |

[22] | S. S. Motsa and P. Sibanda, “A new algorithm for solving singular IVPs of Lane-Emden type,” in Proceedings of the 4th International Conference on Applied Mathematics, Simulation, Modelling (ASM ’10), pp. 176-180, NAUN International Conferences, Corfu Island, Greece, July 2010. · Zbl 1343.65087 |

[23] | S. Karimi Vanani and A. Aminataei, “On the numerical solution of differential equations of Lane-Emden type,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2815-2820, 2010. · Zbl 1193.65151 |

[24] | C. Yang and J. Hou, “A numerical method for Lane-Emden equations using hybrid functions and the collocation method,” Journal of Applied Mathematics, vol. 2012, Article ID 316534, 9 pages, 2012. · Zbl 1235.65107 |

[25] | S. A. Yousefi, “Legendre wavelets method for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1417-1422, 2006. · Zbl 1105.65080 |

[26] | R. K. Pandey, N. Kumar, A. Bhardwaj, and G. Dutta, “Solution of Lane-Emden type equations using legendre operational matrix of differentiation,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7629-7637, 2012. · Zbl 1246.65115 |

[27] | J. P. Boyd, “Chebyshev spectral methods and the Lane-Emden problem,” Numerical Mathematics, vol. 4, no. 2, pp. 142-157, 2011. · Zbl 1249.65158 |

[28] | S. A. Khuri, “A Laplace decomposition algorithm applied to a class of nonlinear differential equations,” Journal of Applied Mathematics, vol. 1, no. 4, pp. 141-155, 2001. · Zbl 0996.65068 |

[29] | Y. Khan, “An effective modification of the laplace decomposition method for nonlinear equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11-12, pp. 1373-1376, 2009. |

[30] | M. Khan and M. A. Gondal, “A new analytical solution of foam drainage equation by laplace decomposition method,” Journal of Advanced Research in Scientific Computing, vol. 2, no. 2, pp. 53-64, 2010. |

[31] | S. Abbasbandy, “Application of He’s homotopy perturbation method for Laplace transform,” Chaos, Solitons and Fractals, vol. 30, no. 5, pp. 1206-1212, 2006. · Zbl 1142.65417 |

[32] | S. A. Khuri and A. Sayfy, “A laplace variational iteration strategy for the solution of differential equations,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2298-2305, 2012. · Zbl 1252.65128 |

[33] | M. Madani and M. Fathizadeh, “Homotopy perturbation algorithm using laplace transformation,” Nonlinear Science Letters A, vol. 1, pp. 263-267, 2010. |

[34] | M. Khan, M. A. Gondal, and S. Karimi Vanani, “On the coupling of homotopy perturbation and laplace transformation for system of partial differential equations,” Applied Mathematical Sciences, vol. 6, no. 9-12, pp. 467-478, 2012. · Zbl 1247.65136 |

[35] | M. Madani, M. Fathizadeh, Y. Khan, and A. Yildirim, “On the coupling of the homotopy perturbation method and laplace transformation,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1937-1945, 2011. · Zbl 1219.65121 |

[36] | Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He’s polynomials,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 1963-1967, 2011. · Zbl 1219.65119 |

[37] | V. G. Gupta and S. Gupta, “Homotopy perturbation transform method for solving initial boundary value problems of variable coefficients,” International Journal of Nonlinear Science, vol. 12, no. 3, pp. 270-277, 2011. · Zbl 1257.35011 |

[38] | J. Singh, D. Kumar, and S. Rathore, “Application of homotopy perturbation transform method for solving linear and nonlinear Klein-Gordon equations,” Journal of Information and Computing Science, vol. 7, pp. 131-139, 2012. |

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.