带有变系数的分数阶线性微分方程的显式解  

Explicit Solutions for Fractional Linear Differential Equations with Variable Coefficients

作　　者：马奎奎 高磊 MA Kui-kui;GAO Lei(College of Information Science and Engineering/Shandong Agricultural University,Tai'an 271018,China)

机构地区：[1]山东农业大学信息科学与工程学院,山东泰安271018

出　　处：《山东农业大学学报（自然科学版）》2024年第6期874-880,共7页Journal of Shandong Agricultural University：Natural Science Edition

基　　金：山东省自然科学基金(ZR2020QF066)。

摘　　要：分数阶微积分理论是传统整数阶微积分理论的推广和延伸。相比较于传统整数阶微积分,分数阶微积分具有遗传和记忆功能,可以更加准确地模拟现实生活中的复杂现象。许多农业机械控制的研究指出,分数阶微积分可以大大提升控制系统设计过程中的灵活度,使系统具有更好的控制性能。可见,分数阶微积分理论在农业机械控制和农业信息化等方面起到了不可或缺的作用。分数阶线性微分方程作为基础和常见的分数阶系统,其显式解虽然得到了一些研究,但仍然不够成熟,致使后续应用工作受阻。本文讨论了带有变系数的分数阶线性微分方程的初值问题,通过逐步逼近方法和广义Mittag-Leffler函数,得到了在齐次和非齐次两种情况下的显式解,并给出了通俗易记的表达式。齐次情况下的显式解与现有研究结果保持一致。非齐次情况下的显式解修正并改进了B.Sambandham等人在文献[1]中的论述。另外,当阶数ν→1时,整数阶的结果可作为特殊情况推导得出。本文期待能为交叉学科的发展提供一定的理论参考。Fractional calculus theory is a generalization and extension of traditional integer-order calculus.Compared to traditional integer order calculus,fractional calculus possesses hereditary and memory functions,enabling more accurate simulation of complex phenomena in real life.Many studies on agricultural machinery control indicate that fractional calculus can significantly enhance the flexibility in the design process of control systems,resulting in better control performance.Thus,fractional calculus theory plays an indispensable role in agricultural machinery control and agricultural informatization.Fractional linear differential equations,as a fundamental and common fractional system,have been studied for their explicit solutions,but the research is still not mature enough,hindering subsequent application work.This paper discusses the initial value problem of fractional linear differential equations with variable coefficients,and by using stepwise approximation methods and generalized Mittag-Leffler functions,explicit solutions for both homogeneous and nonhomogeneous cases are obtained,with user-friendly expressions provided.The explicit solution for the homogeneous case is consistent with existing research results.The explicit solution for the non-homogeneous case corrects and revise the statement of B.Sambandham et al.in [1].Furthermore,the integer results can be derived as a special case as the order ν → 1.This paper aims to provide a theoretical reference for the development of interdisciplinary fields.

关 键 词：分数阶微分方程 显式解 Mittag-Leffler函数 算子级数的收敛 

分 类 号：O175.1[理学—数学]