In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations.

中文翻译：

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关于广义拉普拉斯变换

在本文中，我们介绍了广义Laplace变换，以便与非常通用的分数导数一起工作，并获得了此新变换的性质。我们还包括相应的卷积和逆公式。特别地，对此广义拉普拉斯变换的卷积定义改善了先前的结果。此外，我们处理了广义谐振子方程，表明该变换及其性质允许求解分数阶微分方程。