Advanced Numerical and Semi-Analytical Methods for Differential Equations

20Differential Equations with Fuzzy Uncertainty

20.1 Introduction

In this chapter, a system of fuzzy linear differential equations is studied. Recently, a new technique using the triangular fuzzy numbers (TFNs) [1,2] is illustrated to model the fuzzy linear differential equations. The solution of linear differential equations with fuzzy initial conditions may be studied as a set of intervals by varying α‐cut. The term fuzzy differential equations were first introduced by Chang and Zadeh [3,4]. Later, Bede introduced a strongly generalized differentiability of fuzzy functions in Refs. [5,6]. Recently, various researchers viz. Allahviranloo et al. [7], Chakraverty et al. [8], Tapaswini and Chakraverty [9] have studied fuzzy differential equations. As such, a geometric approach to solve fuzzy linear systems of differential equations have been studied by Gasilov et al. [ 1 , 2 ,10,11]. The difference between this method and the methods offered to handle the system of fuzzy linear differential equation is that at any time the solution consists a fuzzy region in the coordinate space. In this regard, the following section presents a procedure to solve fuzzy linear system of differential equations.

20.2 Solving Fuzzy Linear System of Differential Equations

20.2.1 α‐Cut of TFN

To understand the preliminary concepts of fuzzy set theory, one can refer Refs. [12,13]. There exist various types of fuzzy numbers and among them the TFN is found to be mostly used by different authors.

A TFN ...