An effective strategy for addressing the limitations inherent in the fuzzy number space involves the construction of a quotient space, denoted as \({\mathbb {R}}_{{\mathcal {F}}(A)}\) , specifically for linearly correlated fuzzy numbers, where A may be either symmetric or non-symmetric. While previous studies have primarily concentrated on the calculus of linearly correlated functions within \({\mathbb {R}}_{{\mathcal {F}}(A)}\) for non-symmetric fuzzy numbers, the symmetric case remains underexplored, particularly with respect to the development of analytical operations and the formulation of dynamic systems for functions within this space. This paper aims to bridge that gap by systematically establishing and proving analytical operations for linearly correlated functions in \({\mathbb {R}}_{{\mathcal {F}}(A)}\) for any fuzzy number A. We introduce relevant concepts and properties associated with Riemann-Liouville fractional integrals and Caputo fractional derivatives in the context of linearly correlated fuzzy functions. The two highlighted contributions of this study are the demonstration of existence and uniqueness of solutions to the Cauchy problem, along with the investigation of Ulam stability, encompassing both Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy fractional evolution equations taking values in \({\mathbb {R}}_{{\mathcal {F}}(A)}\) . Furthermore, the paper includes specific illustrative examples that elucidate the significance and applicability of these results.
