Preliminary Observations on the Novel Approaches in Fuzzy Conformable Differential Calculus

Abstract

This paper presents a novel extension of fuzzy calculus by integrating it with conformable calculus to introduce the fuzzy conformable derivative, a mathematical tool designed to handle the complexities of systems characterized by both uncertainty and fractional-order dynamics. The study begins by defining the fuzzy conformable derivative of order Ψ, which combines the ability of fuzzy calculus to model vagueness with the flexibility of conformable derivatives to capture non-integer order behaviors. This concept is further extended to higher orders, specifically the second order (2Ψ) and arbitrary order pΨ, enabling the modeling of more complex, multi-order dynamics in fuzzy-valued functions. Key contributions include the formal definitions of these derivatives, the establishment of their properties, and the development of operational rules that extend classical calculus operations to fuzzy systems. Additionally, the paper demonstrates how these derivatives can be applied to solve fuzzy differential equations and approximate functions using Taylor series expansions. While the fuzzy conformable derivative offers a powerful framework for modeling complex systems, the study also identifies certain limitations, such as potential unboundedness of solutions and the non-adherence to classical mathematical laws in higher-order cases. Overall, this work provides a comprehensive approach to differentiating fuzzy-valued functions in systems where uncertainty and fractional-order dynamics play a critical role. The proposed methods open up new avenues for research and application in fields such as control systems, economics, and engineering, where traditional calculus methods may not suffice. Future research directions include refining these methods, exploring computational techniques, and applying the framework to a broader range of real-world problems.