分数阶微分方程边值问题的Picard’s迭代方法

Picard's Iterative Method for the Boundary Value Problem of a Class of the Fractional Order Differential Equation

从分数阶微分方程边值问题的近似解出发,应用Picard’s迭代方法证明了其存在唯一解;研究了非线性函数f(t;x(t),x'(t))由一个函数序列{fm(t;x(t),x'(t))}近似代替时,边值问题解的Picard’s迭代序列满足的形式及其存在唯一解的充要条件;讨论了这类边值问题不考虑近似解以及非线性函数Lipschitz类的因素时,其解的一般性存在条件;最后通过两个数值算例验证了这类边值问题解的存在性以及解与其迭代序列的误差估计.

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 Picard＇s 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 Picard＇s 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 Lipschitz-class. 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.