AI 中文总结
本文给出模糊 Caputo-Katugampola 导数新定义及相关概念，用于建立分数阶模糊变分问题最优性条件，借助模糊数水平隶属函数表示提升计算效率，还研究了最优性条件并举例说明，非模糊情况能用特殊函数表示精确解。
AI 中文摘要
在本文中，我们给出了模糊函数类的模糊 Caputo-Katugampola 导数的新定义及其相关概念，并将其应用于建立分数阶模糊变分问题的充分必要最优性条件。通过采用模糊数的水平隶属函数表示，颗粒 Caputo-Katugampola 导数在某些情况下比以前的方法具有更高的计算效率。还研究了颗粒模糊 Caputo 分数导数下模糊分数变分问题的最优性条件，并用详细例子进行说明。即使在非模糊情况下，我们结果的另一个新颖之处是应用特殊函数来表示模糊分数变分问题的精确解。
英文摘要
In this article, we present the new definition of the fuzzy Caputo-Katugampola derivative and its related concepts for the class of fuzzy functions and apply them to establish both necessary and sufficient optimality conditions for fractional fuzzy variational problems. By adopting the horizontal membership functional representation of fuzzy numbers, the granular Caputo-Katugampola derivatives offer higher computational efficiency than the previous approaches in certain scenarios. The optimality conditions for the fuzzy fractional variational problems under granular fuzzy Caputo fractional derivatives are also examined and illustrated with detailed examples. Another novel aspect of our results, even in the non-fuzzy case, is the application of special functions to represent the exact solutions of the fuzzy fractional variational problems.
