The Dirac delta function and its integer-order derivative are widely used to solve integer-order differential/integral equation and integer-order system in related fields. On the other hand, the fractional-order system gets more and more attention. This paper investigates the fractional derivative of the Dirac delta function and its Laplace transform to explore the solution for fractional-order system. The paper presents the Riemann-Liouville and the Caputo fractional derivative of the Dirac delta function, and their analytic expression. The Laplace transform of the fractional derivative of the Dirac delta function is given later. The proposed fractional derivative of the Dirac delta function and its Laplace transform are effectively used to solve fractional-order integral equation and fractional-order system, the correctness of each solution is also verified.

中文翻译：

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Dirac Delta函数的分数阶导数及其应用

Dirac delta函数及其整数阶导数在相关领域中广泛用于求解整数阶微分/积分方程和整数阶系统。另一方面，分数阶系统越来越受到关注。本文研究了Dirac delta函数的分数导数及其Laplace变换，以探索分数阶系统的解。本文介绍了狄拉克三角洲函数的黎曼-利维尔和卡普托分数阶导数，以及它们的解析表达。稍后给出狄拉克δ函数的分数导数的拉普拉斯变换。所提出的Dirac delta函数的分数导数及其Laplace变换可有效地求解分数阶积分方程和分数阶系统，