一类分数阶微分方程初值问题解的存在唯一性

The Existence of Uniqueness in the Solution of a Class of Initial Value Problem for Fractional Differential Equations

摘要: 分数阶微积分在数学和工程方面已经成为人们特别熟知的概念，其是整数阶微积分的推广。分数阶微积分有好多种形式，譬如，Riemann-Liouville、Caputo分数阶微积分，带有一个函数的分数阶微积分是Riemann-Liouville分数阶微积分的推广形式。在本文中，基于带有一个函数的分数阶微积分的基本性质和Picard迭代方法，我们将讨论一类以带有一个函数的分数阶导数表示的微分方程初值问题解的存在唯一性。同时通过本文的研究，我们不仅将Picard迭代法应用于一类以带有一个函数的分数阶导数表示的微分方程初值问题解的存在唯一性的论证中，还提供了求解此类分数阶微分方程初值问题近似解的一种思路。

Abstract: Fractional calculus has become a particularly well-known concept in mathematics and engineering, and is a generalization of integer-order calculus. There are many forms of fractional calculus, for example, Riemann-Liouville, Caputo fractional calculus, and fractional calculus with a function is a generalized form of Riemann-Liouville fractional calculus. In this article, based on the fundamental properties of fractional calculus with a function and the Picard iterative method, we will discuss the uniqueness of a class of solutions to the initial value problem of differential equations represented by fractional derivatives with a function. At the same time, through the research of this paper, we not only apply the Picard iterative method to the demonstration of the uniqueness of the solution of the initial value problem of differential equations represented by a fractional derivative with a function, but also provide an idea for solving the approximate solution of the initial value problem of such fractional order differential equations.