In this paper, the Laplace operator is used with Caputo-Type Marichev–Saigo–Maeda (MSM) fractional differentiation of the extended Mittag-Leffler function in terms of the Laplace function. Further in this paper, some corollaries and consequences are shown which are the special cases of our main findings. We apply the Laplace operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. We also apply the Laplace operator on the right-sided MSM fractional differential operator with the Mittag-Leffler function and the left-sided MSM fractional differential operator with the Mittag-Leffler function.

中文翻译：

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具有扩展Mittag-Leffler函数的Caputo型Marichev–Saigo–Maeda分数阶微分算子的Laplace算子

在本文中，拉普拉斯算子与扩展Mittag-Leffler函数的Caputo型Marichev-Saigo-Maeda（MSM）分数微分一起使用。在本文的进一步部分，将显示一些推论和结果，这些是我们主要发现的特例。我们将Laplace运算符应用于右侧MSM分数微分运算符和左侧MSM分数微分运算符。我们还将Laplace运算符应用于具有Mittag-Leffler函数的右侧MSM分数微分算符和具有Mittag-Leffler函数的左侧MSM分数微分算符。