The Generic and Generalized Neutrosophic Differential Equations for Modeling the Exponential Growth and Decay Process

Abstract

This work investigates the concept of Neutrosophic Differential Equations (NDEs) as an extension of classical and fuzzy differential equation models for handling uncertainty in dynamic systems. Classical differential equations assume precise knowledge of parameters and initial conditions, while fuzzy differential equations account for vagueness through degrees of membership.  However, both approaches fail to explicitly model indeterminacy, which arises from incomplete, inconsistent, or conflicting information. Neutrosophic theory introduces three independent components—Truth T, Indeterminacy I, and Falsity F—which provide a more flexible framework for representing uncertainty. In this study, neutrosophic-valued functions are defined, and the concept of the neutrosophic derivative is developed. First-order neutrosophic differential equations are formulated and transformed into systems of classical differential equations using decomposition techniques. Analytical and numerical solution methods, including the Euler method and Runge–Kutta schemes, are applied to solve neutrosophic differential equations. The models are further illustrated through applications such as population growth and decay processes, where the evolution of truth, indeterminacy, and falsity components are analyzed. A comparative analysis between classical, fuzzy, and neutrosophic models demonstrates that neutrosophic differential equations provide a more comprehensive representation of uncertainty by explicitly capturing indeterminacy. The results show that neutrosophic models reveal hidden uncertainty that is not observable in traditional approaches. This study contributes to the growing field of neutrosophic mathematics and provides a foundation for further research in dynamic systems modeling, control theory, and applied sciences.