In this paper, we modify the ( k , s ) fractional integral operator involving k -Mittag-Leffler function and discuss its properties. We originate a new fractional operator named ( k , s )-Prabhakar derivative and obtained some classical fractional operators as a special case of the newly proposed derivative. Some properties of the introduced operator are also part of the present work. The generalized Laplace transform is employed to study the characteristics of fractional operators. We modeled the free-electron laser (FEL) equation by involving the proposed derivative and can find the solution by using the said Laplace transform.

中文翻译：

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关于某些分数微积分算子及其在数学物理学中的应用

在本文中，我们修改了涉及k -Mittag-Leffler函数的（k，s）分数积分算子，并讨论了其性质。我们起源于一个名为（k，s）-Prabhakar导数的新分数阶算子，并获得了一些经典的分数阶算子作为新提出的导数的特例。引入的运算符的某些属性也是当前工作的一部分。广义拉普拉斯变换用于研究分数算子的特征。我们通过涉及拟议的导数对自由电子激光（FEL）方程进行建模，并可以通过使用所述拉普拉斯变换找到解决方案。