Application of reproducing kernel method to third order three-point boundary value problems.

*(English)*Zbl 1204.65090Summary: 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:

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