Explicit Solutions for Fractional Linear Differential Equations with Variable Coefficients
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.
Fractional differential equationsexplicit solutionsMittag-Leffler functionsconvergence for operator series
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