In this work, we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced in both Riemann–Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.

中文翻译：

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广义实分数阶算子与柯西问题的适定性

在这项工作中，我们集中于在分数阶异常扩散建模中起重要作用的大量分数积分和微分算子。我们引入了一个新的广义实数分数积分。在黎曼-利维尔和卡普托意义上也引入了分数阶导数的一般化。此外，我们分析了这些运算符的基本属性。最后，我们考虑了一类广义的实质分数阶微分方程，并讨论了解对初始数据的存在性，唯一性和连续依赖性。