The Hadamard \(\psi \)-Caputo tempered fractional derivative in various types of fuzzy fractional differential equations

Different types of fractional calculus have been defined, grouped into categories based on their properties. Two types particularly studied are Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two types by expressing one in terms of the other, using the theory of fuzzy fractional calculus with respect to functions. By naturally extending this connection, a generalization is developed that unifies several existing fractional operators, including Riemann–Liouville, Caputo, \(\psi \) $\psi$ -Caputo, classical Hadamard, Hadamard-type, tempered, and all these operators considered with respect to functions via the Hadamard \(\psi \) $\psi$ -Caputo tempered fractional derivative (CTFD) using the Mittag-Leffler function. In addition, we explore the Cauchy problem for fuzzy fractional differential equations (FDEs) with this kind of derivative and demonstrate various results of existence and uniqueness using fixed-point theorems of Krasnoselskii, Schaüder’s, and Banach. Applications further serve as proof of the utility of our findings, which converge, expand, and generalize several previously reported results in research.