This work aims to solve a fuzzy initial value problem for fractional difference equations and to study a class of discrete fractional cobweb models with fuzzy data under the Caputo granular difference operator. Based on relative-distance-measure fuzzy interval arithmetic, we first present several new concepts for fuzzy functions in the field of fuzzy discrete fractional calculus, such as the forward granular difference operator, Riemann-Liouville fractional granular sum, Riemann-Liouville and Caputo granular differences. The composition rules and Leibniz laws used to solve a fuzzy initial value problem for fractional difference equations are also presented. As applications, we obtain the solutions of fuzzy discrete Caputo fractional cobweb models, provide conditions for the convergence of the solution to the equilibrium value, and discuss different cases of how the trajectory of the granular solution converges to the equilibrium value. The developed results are also illustrated through several numerical examples.

中文翻译：

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模糊离散分数阶颗粒演算及其在分数阶蜘蛛网模型中的应用


这项工作旨在解决分数差分方程的模糊初始值问题，并研究一类在 Caputo 粒度差分算子下具有模糊数据的离散分数级蜘蛛网模型。基于相对距离测度模糊区间算法，我们首先提出了模糊离散分数阶演算领域中模糊函数的几个新概念，如前向粒度差集算子、Riemann-Liouville 分数粒子和、Riemann-Liouville 和 Caputo 粒度差集。还介绍了用于解决分数差分方程的模糊初始值问题的组成规则和莱布尼茨定律。作为应用，我们获得了模糊离散 Caputo 分数蜘蛛网模型的解，为解收敛到平衡值提供了条件，并讨论了颗粒解的轨迹如何收敛到平衡值的不同情况。开发的结果也通过几个数值示例进行了说明。