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**Application of reproducing kernel method to third order three-point boundary value problems.**
*(English)*
Zbl 1204.65090

Summary: We investigate the analytical approximate solutions of third order three-point boundary value problems using the reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions.

This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.

This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.

### MSC:

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

34B05 | Linear boundary value problems for ordinary differential equations |

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

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

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

### Keywords:

three-point boundary value problem; reproducing kernel method; series solution; numerical examples
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\textit{B. Wu} and \textit{X. Li}, Appl. Math. Comput. 217, No. 7, 3425--3428 (2010; Zbl 1204.65090)

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### References:

[1] | Agarwal, R. P.; Thompson, H. B.; Tisdell, C. C., Three-point boundary value problems for second-order discrete equations, Computer and Mathematics with Applications, 45, 1429-1435 (2003) · Zbl 1055.39024 |

[2] | Agarwal, R. P.; Kiguradze, I., On multi-point boundary value problems for linear ordinary differential equations with singularities, Journal of Mathematical Analysis and Applications, 297, 131-151 (2004) · Zbl 1058.34012 |

[3] | Thompson, H. B.; Tisdell, C., Three-point boundary value problems for second-order, ordinary, differential equation, Mathematical and Computer Modelling, 34, 311-318 (2001) · Zbl 0998.34011 |

[4] | Lepin, A. Ya.; Ponomarev, V. D., On a positive solution of a three-point boundary value problem, Differential Equations, 42, 2, 291-293 (2006) · Zbl 1294.34021 |

[5] | Sun, Y. P., Positive solutions for third-order three-point nonhomogeneous boundary value problems, Applied Mathematics Letters, 22, 1, 45-51 (2009) · Zbl 1163.34313 |

[6] | Sun, Y. P., Existence of triple positive solutions for a third-order three-point boundary value problem, Journal of Computational and Applied Mathematics, 221, 1, 194-201 (2008) · Zbl 1157.34311 |

[7] | Moorti, V. R.G.; Garner, J. B., Existence and uniqueness theorems for three-point boundary value problems for third order differential equations, Journal of Mathematical Analysis and Applications, 70, 2, 370-385 (1979) · Zbl 0418.34025 |

[8] | Anderson, D. R.; Smyrlis, G., Solvability for a third-order three-point BVP on time scales, Mathematical and Computer Modelling (2009) · Zbl 1171.34306 |

[9] | 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 |

[10] | 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 |

[11] | 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, 77-87 (2008) · Zbl 1154.34012 |

[12] | 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 |

[13] | 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 |

[14] | 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 |

[15] | 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 |

[16] | 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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