Arbitrary Order Differential Equations with Fuzzy Parameters

Abstract

In the last decades, some generalization of theory of ordinary differential equations has been considered to the arbitrary order differential equations by many researchers, the so-called theory of arbitrary order differential equations (often called as fractional order differential equations [FDEs]). Because of the ability for modeling real phenomena, arbitrary order differential equations have been applied in various fields such as control systems, biosciences, bioengineering, and references therein. In this chapter, the authors propose arbitrary order differential equations with respect to another function using fuzzy parameters (initial values and the unknown solutions). The generalized fuzzy Laplace transform is applied to obtain the Laplace transform of arbitrary order integral and derivative of fuzzy-valued functions to solve linear FDEs. To obtain the large class of solutions for FDEs, the concept of generalized Hukuhara differentiability is applied. © 2025 Elsevier B.V., All rights reserved.