王奇生，賴嘉導(dǎo)

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二維Volterra-Fredholm型積分方程問題Taylor配置解法及誤差分析

王奇生，賴嘉導(dǎo)

（五邑大學(xué) 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院，廣東 江門 529020）

利用Taylor配置方法，研究二維Volterra-Fredholm型積分方程問題的數(shù)值解. 即對研究的積分方程問題進(jìn)行Taylor配置離散，將積分方程問題轉(zhuǎn)化為代數(shù)方程進(jìn)行求解，建立了Taylor逼近解的求解格式，給出了配置解與精確解的誤差估計(jì)結(jié)果以及闡述理論分析的3個(gè)數(shù)值例子.

二維Volterra-Fredholm型積分方程；Taylor配置解；誤差分析

積分方程在自然科學(xué)領(lǐng)域應(yīng)用十分廣泛，Volterra和Fredholm型積分方程是其中兩種最常見的類型，許多實(shí)際問題都可以轉(zhuǎn)化為這兩類方程來求解. 有關(guān)積分方程解的存在性理論已經(jīng)很完善，但在實(shí)際應(yīng)用中，除了一些極為特殊的情形，積分方程解的解析形式難以求出，因此必須求助于近似方法. 而另一方面，Taylor配置方法是求解積分方程問題數(shù)值解的有力工具. 在過去的20年，Taylor配置方法被廣泛應(yīng)用于求解積分方程近似解的研究中：利用Taylor配置方法M.sezer[1]解決了Volterra型積分方程的解拆，S.Yalinbas等[2-3]將其應(yīng)用于求解高階線性、非線性一維Volterra-Fredholm型積分-微分方程問題，P.Darania等[4]研究了二維Volterra型積分-微分方程Taylor配置方法求解，陳少軍等[5-6]研究了二維Volterra積分方程問題的Chebyshev和Legendre譜配置解法及收斂性，M.T.Rashed等[7-9]研究了積分方程、微分方程、積分-微分方程問題的Lagrange配置方法，陳艷萍等[10-11]研究了帶奇異核Volterra型積分問題的Chebyshev和Legendre譜配置解法及收斂性. 本文利用Taylor配置方法求解二維Volterra-Fredholm積分方程問題，給出了Taylor配置解的求解格式和誤差分析的結(jié)果，同時(shí)給出了闡述理論分析結(jié)果的數(shù)值例子.

考慮二維Volterra-Fredholm型積分方程問題：

1 方法介紹

因?yàn)?/p>

再把式（6）、（7）代入式（3），得：

可簡寫成

2 誤差分析

對二維積分方程問題（1），定義誤差函數(shù)：

用式（1）減去式（4），且由式（13）可得：

3 數(shù)值例子

例1 求解二維線性Volterra-Fredholm型積分方程

表1 例2的誤差函數(shù)變化趨勢

圖1 例2數(shù)值解與誤差函數(shù)的圖像

表2 例3的誤差函數(shù)變化趨勢

圖2 例3數(shù)值解與誤差函數(shù)的圖像

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[責(zé)任編輯：熊玉濤]

Taylor Collocation Solution and Error Analysis for 2-Dimensional Volterra-Fredholm Integral Equations

WANGQi-sheng, LAIJia-dao

(School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China)

An approximate method for solving 2-dimensional Volterra-Fredholm integral equations is presented by Taylor collocation method. That is, the Volterra-Fredholm integral equations are discretized by Taylor collocation method，which are transformed for the algebraic property systems，the format of Taylor collocation method are obtained. The results of error analysis are given between the collocation solution and the exact solution. Moreover，the effectiveness of this method are illustrated by means of 3 numerical examples.

2-dimensional Volterra-Fredholm integral equations; Taylor collocation solution; error analysis

1006-7302（2013）01-0001-05

O189.1

A

2012-09-19

廣東省計(jì)算科學(xué)重點(diǎn)實(shí)驗(yàn)室開放基金資助項(xiàng)目（201206007）

王奇生（1961—），男，湖南衡陽人，教授，博士，碩士生導(dǎo)師，研究方向?yàn)槲⒎e分方程數(shù)值解法.