Numerical Solution of Complex Fuzzy Differential Equations by Euler and Taylor Methods

Abstract

In 1965, Zadeh introduced the concept of fuzzy set, which is a class of objects with a continuum of grades of membership, such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. In 1987 O. Kaleva defined the concept of fuzzy differential equations and present some basic notions of differential equations such as differentiability, integrability, existence and uniqueness theorem for a solution to a fuzzy differential equation. He also, in 1990 studied the Cauchy problem for fuzzy differential equations and showed that it has a solution if and only if there is a subset and its locally compact. Later, M. Ma, M. Friedman, and A. Kandel 1999 introduced numerical solutions of fuzzy differential equations. In this paper we incorporate the above ideas to introduce numerical solution of complex fuzzy differential equations by Euler and Taylor methods by extending the codomain of membership function of fuzzy topological space from [0, 1] to the unit disk in the complex plane. This extension allows us getting more range and flexibility to represent objects with uncertainty and periodicity semantics without losing the full meaning of information. Also, we considered the definitions of complex fuzzy sets, cartesian and polar representation of complex membership, and Cauchy problem for CFDEs. We then found the exact solutions and approximations for Taylor and Euler methods for CFDEs by levels and  where , and provide examples of the results we obtained.