分?jǐn)?shù)階微分方程邊值問題的Picards迭代方法

孫宇鋒　曾廣釗

摘 要 從分?jǐn)?shù)階微分方程邊值問題的近似解出發(fā)，應(yīng)用Picards迭代方法證明了其存在唯一解；研究了非線性函數(shù)f（t；x（t），x′（t））由一個函數(shù)序列{fm（t；x（t），x′（t））}近似代替時，邊值問題解的Picards迭代序列滿足的形式及其存在唯一解的充要條件；討論了這類邊值問題不考慮近似解以及非線性函數(shù)Lipschitz類的因素時，其解的一般性存在條件；最后通過兩個數(shù)值算例驗(yàn)證了這類邊值問題解的存在性以及解與其迭代序列的誤差估計(jì).

關(guān)鍵詞 分?jǐn)?shù)階微分方程；迭代方法；近似解；誤差估計(jì)

中圖分類號 O175.8，O241.81文獻(xiàn)標(biāo)識碼 A文章編號 10002537（2016）02008208

Picards Iterative Method for the Boundary Value Problem of

a Class of the Fractional Order Differential Equation

SUN Yufeng*， ZENG Guangzhao

（College of Mathematics and Statistics， Shaoguan University， Shaoguan 512005， China）

Abstract In this article the existence and uniqueness of the solution for the boundary value problem of a class of fractional differential equations is proved by the Picards iterative method starting form the approximate solution of boundary value problems of these equations. We also proved the existence and uniqueners of the solution and provided the sufficient conditions for the boundary value problem by the Picards iterative methods when the nonlinear function f（t；x（t），x′（t）） is approximated instead of by a sequence of functions {fm（t；x（t），x′（t））}. The general condition for the existence of its solution is discussed without considering factors like the approximate solution of such boundary value problems and nonlinear function Lipschitzclass. Finally， the existence of the solution of such boundary value problems and the estimation of error between the accurate solution and the solution of iterative sequence are verified by two numerical examples.

Key words fractional differential equations； iterative method； approximate solution； estimation of error

本文在文獻(xiàn)[1～7]的基礎(chǔ)上，討論基于Caputos分?jǐn)?shù)導(dǎo)數(shù)的一類分?jǐn)?shù)階微分方程的邊值問題， 并通過其近似解的Picards迭代序列，得到相應(yīng)的解的存在性和唯一性定理.

考慮如下分?jǐn)?shù)階微分方程的邊值問題

致謝 感謝安徽大學(xué)鄭祖庥教授、中科院俞元洪研究員的教誨和指導(dǎo)！

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（編輯 HWJ）