##
**A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space.**
*(English)*
Zbl 1206.65187

Summary: A new numerical algorithm is provided to solve nonlinear multi-point boundary value problems in a very favorable reproducing kernel space, which satisfies all complex boundary conditions. Its reproducing kernel function is discussed in detail. The theorem proves that the approximate solution and its first- and second-order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear multi-point boundary value problems.

### MSC:

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

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |

### Keywords:

nonlinear problems; multi-point boundary value conditions; reproducing kernel space; convergence; algorithm; numerical experiments
PDFBibTeX
XMLCite

\textit{Y. Lin} and \textit{M. Cui}, Math. Methods Appl. Sci. 34, No. 1, 44--47 (2011; Zbl 1206.65187)

Full Text:
DOI

### References:

[1] | Geng, Solving nonlinear multi-point boundary value problems by combining homotopy perturbation and iteration methods, International Jounal of Nonlinear Sciences and Numerical Simulation 10 (5) pp 597– (2009) |

[2] | Denche, A three-point boundary value problem with an integral condition for parabolic equations with the bessel operator, Applied Mathematics Letters 13 pp 85– (2000) · Zbl 0956.35072 |

[3] | Momani SM Some problems in non-Newtonian fluid mechanics 1991 1 15 |

[4] | Dehghan, Numerical techniques for a parabolic equation subject to an overspecified boundary condition, Applied Mathematics and Computation 132 pp 299– (2002) · Zbl 1024.65088 |

[5] | Geng, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the Korean Mathematical Society 45 pp 631– (2008) · Zbl 1154.34012 |

[6] | Geng, Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Applied Mathematics and Computation 192 pp 389– (2007) · Zbl 1193.34017 |

[7] | Chawla, Finite difference methods for two-point boundary-value problems involving higher order differential equation, BIT 19 pp 27– (1979) · Zbl 0401.65053 |

[8] | Ma, Iteration solution for a beam equation with nonlinear boundary conditions of third order, Applied Mathematics and Computation 159 (1) pp 11– (2004) |

[9] | Cui, Nonlinear numerical analysis in reproducing kernel hilbert space pp 1– (2009) |

[10] | Zhou, Numerical algorithm for parabolic problems with non-classical conditions, Journal of Computational and Applied Mathematics 230 pp 770– (2009) · Zbl 1190.65136 |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.