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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (1): 61-71.doi: 10.3969/j.issn.1005-3085.2015.01.007

• • 上一篇    下一篇

高阶分数阶微分方程系统的解的注记

郑艳萍1,   李胜利2   

  1. 1- 太原师范学院数学系，太原 030012
    2- 太原理工大学数学学院，太原 030024
  • 收稿日期:2013-10-14 接受日期:2014-07-18 出版日期:2015-02-15 发布日期:2015-04-15
  • 基金资助:
    国家自然科学基金 (11361147)；山西省归国留学人员基金 (2013-102).

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

ZHENG Yan-ping1,  LI Sheng-li2   

  1. 1- Department of Mathematics, Taiyuan Normal University, Taiyuan 030012
    2- College of Mathematics, Taiyuan University of Technology, Taiyuan 030024
  • Received:2013-10-14 Accepted:2014-07-18 Online:2015-02-15 Published:2015-04-15
  • Supported by:
    The National Natural Science Foundation of China (11361147); the Scientific Research Foundation for the Returned Overseas Chinese Scholars of Shanxi Province (2013-102).

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摘要： 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．

关键词: Riemann-Liouville型导数, 局部解, 三角形

Abstract:

Fractional-order derivatives provide a powerful instrument for describing the memory 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.

Key words: Riemann-Liouville differentiation, local solution, triangle

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