Certain phenomena with uncertain properties that take the shape of intricate mathematical modeling are known to have fuzzy system integro-differential equations (FSIDEs). The methods used to roughly solve FSIDEs seek to provide open-form solutions that are regarded as solutions for polynomial series. However, for many FSIDEs, the polynomial series solutions are not easily derived, especially in nonlinear forms. Meanwhile, some existing approximate techniques cannot guarantee convergence of the series solution. Nevertheless, to solve second-kind fuzzy Fredholm integro-differential equations (FFSIDEs), there exist perturbation techniques based on the standard Homotopy Analysis Method (HAM) that have the ability to control and rate solution convergence. Therefore, this study focused on modifying new approximate techniques, fuzzy Fredholm HAM (HAMFF), for solving second-kind FFSIDEs subject to initial and boundary value problems. In the theoretical framework modification, the establishment of the series solution convergence was done based on combining some fuzzy sets theory concepts and convergence-control parameters from standard HAM. The HAMFF was not only able to solve linear systems but also difficult nonlinear systems with proper accuracy. The demonstration of the modified technique's performance was shown in comparison to other methods, where HAMFF was individually superior in terms of accuracy for solving linear and nonlinear test problems of FFSIDEs.

In the present paper, we employ the generalized Mittag-Leffler function to investigate several fuzzy differential subordination results associated with suitable families of admissible functions in the open unit disk. By utilizing a refined analytic framework, we derive new inclusion relationships and establish sufficient conditions for fuzzy subordinating functions defined via Mittag-Leffler-type operators. Furthermore, the obtained results unify and extend a number of earlier findings in the theory of fuzzy analytic functions. These developments provide a deeper insight into the interaction between generalized special functions and the structure of fuzzy differential subordinations, offering potential applications to broader subclasses of analytic and bi-univalent functions, as well as to various operator-defined families in geometric function theory.