Z-Differential Equations

This paper is devoted to make a framework for studying a class of uncertain differential equations called Z-differential equations. In order to achieve the purpose, we first introduce four basic operations on Z+-numbers based on semigranular function. Then, the limit and continuity concepts of a Z-number-valued function are given, under a definition of a metric on the space of Z+-numbers. Moreover, the concepts of Z-differentiability, Z-integral, and Z-Laplace transform of a Z-number-valued function are introduced. In addition, by giving some theories proved in this paper, a basis for calculus-Z-calculus-is established. We further give theories based on which existence and uniqueness of Z-differential equations are investigated. A conceptual unity between Z-differential equations and Z+-numbers is also shown. The conceptual unity demonstrates that a Z-differential equation may be expressed as a bimodal differential equation combining a fuzzy differential equation (FDE) and a random differential equation. Moreover, the concept of a bimodal cut called $ (s, mu)-cut is introduced and its relation to other new concepts such as acceptable time and acceptable information area is explained. Using an example, the application of Z-differential equations in medicine is clarified. It is demonstrated that Z-differential equations outperform FDEs in making a decision under uncertainty.