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 is a convex normalized fuzzy set of the real line ℜ [ 3 , 4 ] such that,

1. ∃ exactly one x₀ ∈ R with where x₀ is referred to as the mean value of and is the membership function of the fuzzy set such that .
2. is piecewise continuous.

The membership function of is defined (Figure 20.1) as follows:

where c ≠ a and c ≠ b.

Using the α‐cut approach, the TFN can further be represented by an ordered pair of functions, , where α ∈ [0, 1].

20.2.2 Fuzzy Linear System of Differential Equations (FLSDEs)

Let a_ij are crisp numbers, h_i(t) are given crisp functions, for 0 ≤ α ≤ 1, and 1 ≤ i, j ≤ n, (are fuzzy numbers). Fuzzy Linear System of Differential Equations (FLSDEs) [ 1 , 2 ] may be written as

(20.1)

subject to the fuzzy initial conditions

(20.2)

One can formulate Eqs. (20.1) and (20.2) in matrix notation as follows:

(20.3)

where A = (a_ij) is n × n crisp matrix, H(t) = (h₁(t), h₂(t), …, h_n(t))^T is the crisp vector function, and is the vector of fuzzy numbers.

20.2.3 Solution Procedure for FLSDE

Let us express the initial condition vector as , where (b_cr)_i denotes the crisp part of and denotes the uncertain part. Solution of such system may be considered of the form (crisp solution + uncertainty). Here, x_cr(t) is a solution of the nonhomogeneous crisp problem as given in Refs. [ 1 , 2 ]

(20.4)

whereas is the solution of the homogeneous system with fuzzy initial conditions,

(20.5)

It is possible to compute x_cr(t) by means of known analytical or numerical methods. Our aim is to solve Eq. (20.3) with fuzzy initial conditions. In this regard, fuzzy uncertainty in terms of TFN is taken into consideration. Let be the TFN, then is written as , where q_i designates the crisp part and designates the TFN.

Below, we present an example problem for clear understanding of the described method.

Exercise

Solve the following system of differential equations:

1. 1
2. 2
3. 3

References

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