A Fuzzy Fractional Initial Value Problem with Applications Under New Conformable Fractional Granular Differentiability

This paper introduces a novel definition of the fractional order derivative for fuzzy setvalued functions (FSVF), utilizing the concept of granular difference, which we refer to as the new conformable fractional granular derivative (NCFGD).It also presents the corresponding integral, termed the new conformable fractional granular integral (NCFGI).In terms of results, the basic properties of both the NCFGD and NCFGI are rigorously established and proven, providing a strong mathematical foundation for these new operators.Several illustrative examples demonstrate the effectiveness and applicability of the proposed definitions.Additionally, the paper explores methods for solving the new conformable fractional granular initial value problem (NCFG IVP), which is integral to understanding the behavior of fuzzy set-valued functions under fractional calculus.We apply these methods to solve new conformable fractional granular differential equations (NCFG DEqs) related to growth and decay models, offering a more flexible approach to modeling dynamic systems with fuzzy uncertainty.This approach offers a more flexible and adaptive framework for modeling dynamic systems characterized by fuzzy uncertainty, as it allows for fractional order differentiation.The incorporation of these concepts into the solution of fuzzy fractional differential equations ensures more precise and adaptable models.Ultimately, these findings contribute significantly to advancing both fuzzy calculus and fractional calculus, enhancing the understanding and modeling of complex systems with uncertainty.