高阶分数阶微分方程系统的解的注记

A Note on Solutions to Systems of Fractional Differential Equations with Higher Order

分数阶导数在描述不同物质的记忆与遗传性质方面提供了有力的工具.在科学和工程的不同领域,都用分数阶微分方程组来描述动力系统.本文主要探讨分数阶微分方程系统初值问题局部解的存在性与唯一性.对于线性系统,运用Schur分解定理,给出其局部解的存在性与唯一性,并通过举例说明该方法是有效的.对于非线性系统,利用Schauder不动点定理,给出了解的存在性;运用Banach不动点定理,给出了解的唯一性.

Fractional-order derivatives provide a powerful instrument for describing the mem-ory and hereditary of different substances. A growing number of works from various fields throughout science and engineering deal with dynamical systems expressed by fractional-order equations. In this paper, the existence and uniqueness of the solution to the initial problem for systems of fractional differential equations are derived. Based on the Schur theorem, the existence and uniqueness of the local solution to the linear system are given. Furthermore, the effectiveness of the proposed method is verified by a simulated example. For a nonlinear system, the existence and uniqueness of the solution are obtained by the Schauder fixed point theorem and the Banach fixed point theorem, respectively.