﻿ 一类Fredholm积分微分方程边值问题的数值方法 Numerical Algorithm for a Class of Fredholm Integro-Differential Boundary Value Problems

Vol.06 No.04(2017), Article ID:21533,7 pages
10.12677/AAM.2017.64075

Numerical Algorithm for a Class of Fredholm Integro-Differential Boundary Value Problems

Yongfang Zhou1, Lihua Mu2, Jinghe Li1, Lijun Ma1

1School of Science, Hebei University of Technology, Tianjin

2School of Science, Heilongjiang University of Science and Technology, Harbin Heilongjiang

Received: Jul. 8th, 2017; accepted: Jul. 24th, 2017; published: Jul. 27th, 2017

ABSTRACT

This paper discusses the numerical method for a class of Fredholm integro-differential boundary value problems. By constructing the reproducing kernel space which satisfies the boundary conditions, the simple reproducing kernel numerical approximate method is established. The paper describes both the exact solution obtained in the form of series and the approximate solution obtained by truncating the series representation of the exact solution. Error estimation of the method was discussed. The results of numerical simulation demonstrate the validity of the method in the paper.

Keywords:Boundary Value Problems, Integro-Differential Equation, Reproducing Kernel Space

1河北工业大学理学院，天津

2黑龙江科技大学理学院，黑龙江 哈尔滨

1. 引言

Fredholm积分微分方程边值问题广泛地出现在力学、物理学、化学、天文学、生物学、经济学以及静电学 [1] 等科学和工程问题 [2] 之中。这类问题解的存在性与唯一性研究可以参见文献 [3] 。许多学者致力于方程(1)的数值方法研究，这些方法包括Adomian分解法、变分迭代法、同伦分析法、小波法、凸方法、泰勒级数展开法 [4] [5] [6] 等。这些方法各有优缺点，不断地寻找更加有效，简单的数值方法是学者们一直关心的热点问题。

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2. 再生核空间

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3. 精确解和近似解的构造

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4. 误差估计

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Table 1. Numerical results of example 1

5. 数值算例

6. 结论

Numerical Algorithm for a Class of Fredholm Integro-Differential Boundary Value Problems[J]. 应用数学进展, 2017, 06(04): 644-650. http://dx.doi.org/10.12677/AAM.2017.64075

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