Scientific Research

An Academic Publisher

Sinc-Collocation Method for Solving Linear and Nonlinear System of Second-Order Boundary Value Problems

**Author(s)**Leave a comment

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the sinc-collocation method for solving linear and nonlinear system of second order differential equation. The method is then tested on linear and nonlinear examples and a comparison with B-spline method is made. It is shown that the sinc-collocation method yields better results.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

M. El-Gamel, "Sinc-Collocation Method for Solving Linear and Nonlinear System of Second-Order Boundary Value Problems,"

*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1627-1633. doi: 10.4236/am.2012.311225.

[1] | K.W. Tomantschger, “Series Solutions of Coupled Differential Equations with One Regular Singular Point,” Journal of Computational and Applied Mathematics, Vol. 140, No. 1-2, 2002, pp. 773-783. doi:10.1016/S0377-0427(01)00598-2 |

[2] | C. Wafo Soh and F. M. Mahomed, “Linearization Criteria for a System of Second-Order Ordinary Differential Equations,” International Journal of Non-Linear Mechanics, Vol. 36, No. 4, 2001, pp. 671-677. doi:10.1016/S0020-7462(00)00032-9 |

[3] | N. Caglar and H. Caglar, “B-Spline Method for Solving Linear System of Second-Order Boundary Value Problems,” Computers & Mathematics with Applications, Vol. 57, No. 5, 2009, pp. 757-762. doi:10.1016/j.camwa.2008.09.033 |

[4] | S. H. Chen, J. Hu, L. Chen and C. P. Wang, “Existence Results for n-Point Boundary Value Problem of Second Order Ordinary Differential Equations,” Journal of Computational and Applied Mathematics, Vol. 180, No. 2, 2005, pp. 425-432. doi:10.1016/j.cam.2004.11.010 |

[5] | X. Y. Cheng and C. K. Zhong, “Existence of Positive Solutions for a Second-Order Ordinary Differential System,” Journal of Mathematical Analysis and Applications, Vol. 312, No. 1, 2005, pp. 14-23. doi:10.1016/j.jmaa.2005.03.016 |

[6] | A. Lomtatidze and L. Malaguti, “On a Two-Point Boundary Value Problem for the Second-Order Ordinary Differential Equations with Singularities,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 52, No. 6, 2003, pp. 1553-1567. doi:10.1016/S0362-546X(01)00148-1 |

[7] | H. Thompson and C. Tisdell, “Boundary Value Problems for Systems of Difference Equations Associated with Systems of Second-Order Ordinary Differential Equations,” Applied Mathematics Letters, Vol. 15, No. 6, 2002, pp. 761-766. doi:10.1016/S0893-9659(02)00039-3 |

[8] | H. Thompson and C. Tisdell, “The Nonexistence of Spurious Solutions to Discrete, Two-Point Boundary Value Problems,” Applied Mathematics Letters, Vol. 16, No. 1, 2003, pp. 79-84. doi:10.1016/S0893-9659(02)00147-7 |

[9] | F. Z. Geng and M. G. Cui, “Solving a Nonlinear System of Second-Order Boundary Value Problems,” Journal of Mathematical Analysis and Applications, Vol. 327, No. 2, 2007, pp. 1167-1181. doi:10.1016/j.jmaa.2006.05.011 |

[10] | J. F. Lu, “Variational Iteration Method for Solving a Nonlinear System of Second-Order Boundary Value Problems,” Computers & Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 1133-1138. doi:10.1016/j.camwa.2006.12.060 |

[11] | A. Bataineh, M. S. M. Noorani and I. Hashim, “Modified Homotopy Analysis Method for Solving Systems of Second-Order BVPs,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 2, 2009, pp. 430-442. doi:10.1016/j.cnsns.2007.09.012 |

[12] | M. Dehghan and A. Saadatmandi, “The Numerical Solution of a Nonlinear System of Second-Order Boundary Value Problems Using the Sinc-Collocation Method,” Mathematical and Computer Modelling, Vol. 46, No. 11-12, 2007, pp. 1434-1441. doi:10.1016/j.mcm.2007.02.002 |

[13] | B. Bialecki, “Sinc-Collocation Methods for Two-Point Boundary Value Problems,” IMA Journal of Numerical Analysis, Vol. 11, No. 3, 1991, pp. 357-375. doi:10.1093/imanum/11.3.357 |

[14] | M. El-Gamel and A. I. Zayed, “Sinc-Galerkin Method for Solving Nonlinear Boundary-Value Problems,” Computers & Mathematics with Applications, Vol. 48, No. 9, 2004, pp. 1285-1298. doi:10.1016/j.camwa.2004.10.021 |

[15] | M. El-Gamel, J. Cannon and A. Zayed, “Sinc-Galerkin Method for Solving Linear Sixth Order Boundary-Value Problems,” Mathematics of Computation, Vol. 73, No. 247, 2004, pp. 1325-1343. |

[16] | M. El-Gamel, S. H. Behiry and H. Hashish, “Numerical Method for the Solution of Special Nonlinear FourthOrder Boundary Value Problems,” Applied Mathematics and Computation, Vol. 145, No. 2-3, 2003, pp. 717-734. doi:10.1016/S0096-3003(03)00269-8 |

[17] | M. El-Gamel and J. Cannon, “On the Solution of Second Order Singularly-Perturbed Boundary Value Problem by the Sinc-Galerkin Method,” Zeitschrift für Angewandte Mathematik und Physik (ZAMP), Vol. 56, No. 1, 2005, pp. 45-58. doi:10.1007/s00033-004-3002-6 |

[18] | A. Mohsen and M. El-Gamel, “A Sinc-Collocation Method for the Linear Fredholm Integro-Differential Equations,” Zeitschrift für Angewandte Mathematik und Physik (ZAMP), Vol. 58, No. 3, 2007, pp. 380-390. doi:10.1007/s00033-006-5124-5 |

[19] | A. Mohsen and M. El-Gamel, “On the Galerkin and Collocation Methods for Two-Point Boundary Value Problems Using Sinc Bases,” Computers & Mathematics with Applications, Vol. 56, No. 4, 2008, pp. 930-941. doi:10.1016/j.camwa.2008.01.023 |

[20] | A. Mohsen and M. El-Gamel, “On the Numerical Solution of Linear and Nonlinear Volterra Integral and IntegroDifferential Equations,” Applied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3330-3337. doi:10.1016/j.amc.2010.08.065 |

[21] | R. Smith, G. Bogar, K. Bowers and J. Lund, “The SincGalerkin Method for Fourth-Order Differential Equations,” SIAM Journal on Numerical Analysis, Vol. 28, No. 3, 1991, pp. 760-788. doi:10.1137/0728041 |

[22] | G. Y. Yin, “Sinc-Collocation Method with Orthogonalization for Singular Poisson-Like Problem,” Mathematics of Computation, Vol. 62, No. 205, 1994, pp. 21-40. doi:10.1090/S0025-5718-1994-1203738-7 |

[23] | J. Lund and K. L. Bowers, “Sinc Methods for Quadrature and Differential Equations,” Society for Industry and Applied Mathematics (SIAM), Philadelphia, 1992. doi:10.1137/1.9781611971637 |

[24] | F. Stenger, “Numerical Methods Based on Sinc and Analytic Functions,” Springer, New York, 1993. doi:10.1007/978-1-4612-2706-9 |

Copyright © 2018 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.