An Innovative Application of the Fuzzy Generalized ZZ Transform to First‐ and Second‐Order Linear Fuzzy Differential Systems

Source abstract

Classical differential equations are not always sufficient to model real‐world phenomena that involve uncertainty. In this paper, we propose an innovative solution to a system of first‐ and second‐order linear fuzzy differential equations based on the fuzzy Generalized ZZ transform, using the concept of generalized Hukuhara (gH) differentiability. The main contribution of this paper is the systematic and rigorous treatment of all four combinations of gH differentiability for coupled two‐dimensional fuzzy systems, which have not been previously addressed with this transform. For each combination of differentiability, we obtain the corresponding transformed algebraic systems and derive closed‐form solutions using the inverse transform. Moreover, we present an elegant reduction method that transforms second‐order fuzzy equations into equivalent first‐order systems, without deriving higher‐order transform formulas separately. The approach is validated by explicit back‐substitution of the resulting solutions, detailed numerical examples, and full verification of the properties of fuzzy numbers. The results show that the fuzzy Generalized ZZ transform offers a general and scalable analytical tool that includes the Laplace transform and Sumudu transform as special cases and could be applied to science, engineering, and biological modeling in general.