In this paper, we explore the existence and uniqueness theorem for a problem of the fractional Cauchy form, with dependence on the generalized proportional Caputo derivative. Furthermore, a new numerical technique is presented based on a decomposition formula for the generalized proportional Caputo derivative. Convergence analysis of the proposed technique is proved. Finally, numerical results are obtained to confirm the validity of the proposed method.

中文翻译：

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广义比例分数柯西问题的分析和数值解

在本文中，我们探索了分数柯西形式问题的存在性和唯一性定理，并依赖于广义比例Caputo导数。此外，提出了一种基于分解公式的广义比例Caputo导数的新数值技术。证明了所提技术的收敛性。最后，获得了数值结果，验证了所提方法的有效性。