Abstract
In this paper, we find formulas of general solution for a kind of impulsive differential equations with Hadamard fractional derivative of order $q \in(1, 2)$ by analysis of the limit case (as the impulse tends to zero) and provide an example to illustrate the importance of our results.
Highlights
Fractional differential equations are an excellent tool in the modeling of many phenomena in various fields of science and engineering [ – ], and the subject of fractional differential equations is gaining much attention
Impulsive effects exist widely in many processes in which their states can be described by impulsive differential equations, and the subject of impulsive Caputo fractional differential equations is widely studied; impulsive fractional partial differential equations are considered
Example Let us consider the general solution of the impulsive fractional system
Summary
Fractional differential equations are an excellent tool in the modeling of many phenomena in various fields of science and engineering [ – ], and the subject of fractional differential equations is gaining much attention (see [ – ] and the references therein).Recently, Hadamard fractional derivative was studied in [ – ], and Klimek [ ] studied the existence and uniqueness of the solution of a sequential fractional differential equation with Hadamard derivative by using the contraction principle and a new equivalent norm and metric. Motivated by the above-mentioned works, we consider the following impulsive system with Hadamard fractional derivative: Zhang et al Advances in Difference Equations (2016) 2016:14 where a > , H Dqa+ denotes left-sided Hadamard fractional derivative of order q, f : J × R → R is an appropriate continuous function, a = t < t < · · · < tm < tm+ = T In Section , we give formulas of a general solution for a kind of impulsive differential equations with Hadamard fractional derivative of order q ∈ ( , ).
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