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**A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems.**
*(English)*
Zbl 1166.65358

Summary: This paper presents a new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between our method and other methods for solving an open fourth-order boundary value problem presented by M. R. Scott and H. A. Watts [SUPPORT – A computer code for two-point boundary-value problems via orthonormalization, SAND75-0198, Sandia Laboratories, Albuquerque, NM (1975)]. The method is also applied to a nonlinear fourth-order boundary value problem. The numerical results demonstrate that the new method is quite accurate and efficient for fourth-order boundary value problems.

### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

Full Text:
DOI

### References:

[1] | S.M. Momani, Some problems in non-Newtonian fluid mechanics, Ph.D. Thesis, Walse University, United Kingdom, 1991.; S.M. Momani, Some problems in non-Newtonian fluid mechanics, Ph.D. Thesis, Walse University, United Kingdom, 1991. |

[2] | Ma, T. F.; Silva, J., Iteration solution for a beam equation with nonlinear boundary conditions of third order, Applied Mathematics and Computation, 159, 1, 11-18 (2004) · Zbl 1095.74018 |

[3] | Chawla, M. M.; Katti, C. P., Finite difference methods for two-point boundary-value problems involving higher order differential equation, BIT, 19, 27-33 (1979) · Zbl 0401.65053 |

[4] | M.R. Scott, H.A.Watts, SUPORT-A computer code for two-point boundary-value problems via orthonormalization, SAND75-0198, Sandia Laboratories, Albuquerque, NM, 1975.; M.R. Scott, H.A.Watts, SUPORT-A computer code for two-point boundary-value problems via orthonormalization, SAND75-0198, Sandia Laboratories, Albuquerque, NM, 1975. |

[5] | Noor, M. A.; Mohyud-Din, S. T., Variational iteration technique for solving higher order boundary value problems, Applied Mathematics and Computation, 189, 2, 1929-1942 (2007) · Zbl 1122.65374 |

[6] | Momani, S.; Noor, M. A., Numerical comparison of methods for solving a special fourth-order boundary value problem, Applied Mathematics and Computation, 191, 218-224 (2007) · Zbl 1193.65135 |

[7] | Doedel, E., Finite difference method for nonlinear two-point boundary-value problems, SIAM Journal on Numerical Analysis, 16, 173-185 (1979) · Zbl 0438.65068 |

[8] | Wazwaz, A. M., The numerical solution of special fourth-order boundary value problems by the modified decomposition method, International Journal of Computer Mathematics, 79, 3, 345-356 (2002) · Zbl 0995.65082 |

[9] | Erturk, V. S.; Momani, S., Comparing numerical method for solving fourth-order boundary value problems, Applied Mathematics and Computation, 188, 2, 1963-1968 (2007) · Zbl 1119.65066 |

[10] | Mohyud-Din, S. T.; Noor, M. A., Homotopy perturbation method for solving fourth-order boundary value problems, Mathematical problems in Engineering, 2007, 1-15 (2007) · Zbl 1144.65311 |

[11] | Daniel, Alpay, Reproducing Kernel Spaces and Applications (2003), Springer · Zbl 1021.00005 |

[12] | Berlinet, Alain; Thomas-Agnan, Christine, Reproducing Kernel Hilbert Space in Probability and Statistics (2004), Kluwer Academic Publishers · Zbl 1145.62002 |

[13] | Geng, Fazhan; Cui, Minggen, Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Applied Mathematics and Computation, 192, 389-398 (2007) · Zbl 1193.34017 |

[14] | Geng, Fazhan; Cui, Minggen, Solving a nonlinear system of second order boundary value problems, Journal of Mathematical Analysis and Applications, 327, 1167-1181 (2007) · Zbl 1113.34009 |

[15] | Geng, Fazhan; Cui, Minggen, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the Korean Mathematical Society, 45, 3 (2008) · Zbl 1154.34012 |

[16] | Cui, Minggen; Geng, Fazhan, Solving singular two-point boundary value problem in reproducing kernel space, Journal of Computational and Applied Mathematics, 205, 6-15 (2007) · Zbl 1149.65057 |

[17] | Cui, Minggen; Geng, Fazhan, A computational method for solving one-dimensional variable-coefficient Burgers equation, Applied Mathematics and Computation, 188, 1389-1401 (2007) · Zbl 1118.35348 |

[18] | Cui, Minggen; Lin, Yingzhen, A new method of solving the coefficient inverse problem of differential equation, Science in China Series A, 50, 4, 561-572 (2007) · Zbl 1125.35418 |

[19] | Cui, Minggen; Chen, Zhong, The exact solution of nonlinear age-structured population model, Nonlinear Analysis: Real World Applications, 8, 1096-1112 (2007) · Zbl 1124.35030 |

[20] | Li, Chunli; Cui, Minggen, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied Mathematics and Computation, 143, 2-3, 393-399 (2003) · Zbl 1034.47030 |

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