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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 ›› 2025, Vol. 42 ›› Issue (5): 905-917.doi: 10.3969/j.issn.1005-3085.2025.05.008cstr: 32411.14.cjem.CN61-1269/O1.2025.05.008

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一类Atangana-Baleanu-Caputo型分数阶微分耦合系统的解与数值模拟

蔺学凡1,2,3,   胡卫敏1,4,   苏有慧2,   贠永震2   

  1. 1. 伊犁师范大学数学与统计学院，伊宁 835000

    2. 徐州工程学院数学与统计学院，徐州 221018

    3. 盐城市时杨中学，盐城 224035

    4. 伊犁师范大学应用数学研究所，伊宁 835000

  • 收稿日期:2022-12-26 接受日期:2023-03-29 出版日期:2025-10-15 发布日期:2025-12-15
  • 通讯作者: 胡卫敏 E-mail: hwm680702@163.com
  • 基金资助:
    新疆维吾尔自治区自然科学基金 (2023D01C51)；伊犁师范大学高级别培育项目 (YSPY2022014)；伊犁师范大学科研创新团队培育计划 (CXZK2021016).

Solutions and Numerical Simulation of a Fractional Differential Coupling System with Atangana-Baleanu-Caputo Derivative

LIN Xuefan1,2,3,   HU Weimin1,4,   SU Youhui2,   YUN Yongzhen2   

  1. 1. School of Mathematics and Statistics, Yili Normal University, Yining 835000
    2. School of Mathematics and Statistics, Xuzhou Institute of Technology, Xuzhou 221018
    3. Shiyang Middle School, Yancheng 224035
    4. Institute of Applied Mathematics, Yili Normal University, Yining 835000
  • Received:2022-12-26 Accepted:2023-03-29 Online:2025-10-15 Published:2025-12-15
  • Contact: W. Hu. E-mail address: hwm680702@163.com
  • Supported by:
    The Natural Science Foundation of Xinjiang Uygur Autonomous Region (2023D01C51); the High-level Cultivation Project of Yili Normal University (YSPY2022014); the Research and Innovation Team of Yili Normal University (CXZK2021016).

摘要：

主要研究了一类带有Atangana-Baleanu-Caputo型分数阶导数的三点微分耦合系统解的存在唯一性问题。首先，利用上下解技术和单调迭代方法得到了所研究边值系统最大最小解存在唯一性的充分条件。其次，利用Green函数及其性质，证明了系统最大最小解的存 在唯一性并给出误差估计。最后，为了说明所得理论结果的有效性和实用性，给出了一个具体的应用实例，验证了该系统在实际问题中的适用性。此外，还对该系统进行了数值模拟，进一步验证了理论分析的正确性。

关键词: 分数阶微分方程, 数值模拟, Atangana-Baleanu-Caputo导数, 存在唯一性, 迭代法

Abstract:

In this paper, we are concerned with the existence and uniqueness of solutions for a class of three-point boundary value problems involving coupled differential systems with Atangana-Baleanu-Caputo fractional derivatives. The research employs upper-lower solution techniques combined with monotone iteration methods to establish sufficient conditions for the existence and uniqueness of maximal and minimal solutions. Furthermore, by constructing Green's functions and analyzing their properties, we rigorously prove the existence and uniqueness of extremal solutions while providing precise error estimates. To demonstrate the practical significance of our theoretical findings, we present a concrete application case that validates the system's effectiveness in real-world scenarios. Numerical simulations are also conducted to provide additional verification of the theoretical analysis, confirming the reliability of our proposed approach.

Key words: fractional differential equations, numerical simulation, Atangana-Baleanu-Caputo derivative, existence and uniqueness, iterative method

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