山东农业大学学报（自然科学版）2024，Vol.55Issue(6)：874-880,7.DOI:10.3969/j.issn.1000-2324.2024.06.007
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
摘要
Abstract
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 non-homogeneous 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Key words
Fractional differential equations/explicit solutions/Mittag-Leffler functions/convergence for operator series分类
数理科学引用本文复制引用
马奎奎,高磊..带有变系数的分数阶线性微分方程的显式解[J].山东农业大学学报（自然科学版）,2024,55(6):874-880,7.基金项目
山东省自然科学基金(ZR2020QF066) （ZR2020QF066）
