On new solutions of fuzzy differential equations

doi:10.1016/j.chaos.2006.10.043

Chaos, Solitons & Fractals

Volume 38, Issue 1, October 2008, Pages 112-119

https://doi.org/10.1016/j.chaos.2006.10.043 Get rights and content

Abstract

We study fuzzy differential equations (FDE) using the concept of generalized H-differentiability. This concept is based in the enlargement of the class of differentiable fuzzy mappings and, for this, we consider the lateral Hukuhara derivatives. We will see that both derivatives are different and they lead us to different solutions from a FDE. Also, some illustrative examples are given and some comparisons with other methods for solving FDE are made.

Introduction

Fuzzy set theory is a powerful tool for modelling uncertainty and for processing vague or subjective information in mathematical models. Their main directions of development have been diverse and its applications to the very varied real problems, for instance, in the golden mean [8], particle systems [11], quantum optics and gravity [12], synnchronize hyperchaotic sytems [29], chaotic system [13], [26], [27], medicine [1], [3], engineering problems [17]. Particularly, fuzzy differential equation is a topic very important as much of the theoretical point of view (see [9], [14], [19], [20], [22], [24], [28]) as well as of their applications, for example, in population models [15], [16], civil engineering [23] and in modeling hydraulic [6].

Initially, the derivative for fuzzy valued mappings was developed by Puri and Ralescu [24], that generalized and extended the concept of Hukuhara differentiability (H-derivative) for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative, Kaleva [19] started to develop a theory for FDE. In the last few years, many works have been done by several authors in theoretical and applied fields (see [6], [7], [9], [19], [22], [25], [28]). In some cases this approach suffers certain disadvantages since the diameter diam(x(t)) of the solution is unbounded as time t increases [9], [14]. This problem demonstrates that this interpretation is not a good generalization of the associated crisp case and we assume that this problem is due to the fuzzification of the derivative utilized in the formulation of the FDE. In this direction, Bede and Gal in [4], [5] introduce a more general definition of derivative for fuzzy mappings enlarging the class of differentiable fuzzy mappings.

Following this idea, in this paper we define the fuzzy lateral H-derivative for a fuzzy mapping $F : (a, b) \to F^{n}$ as the existence of one of the following limits: $(1) \lim_{h \to 0^{+}} \frac{F (t_{0} + h) - F (t_{0})}{h} = \lim_{h \to 0^{+}} \frac{F (t_{0}) - F (t_{0} - h)}{h} = F^{'} (t_{0})$ or $(2) \lim_{h \to 0^{-}} \frac{F (t_{0} + h) - F (t_{0})}{h} = \lim_{h \to 0^{-}} \frac{F (t_{0}) - F (t_{0} - h)}{h} = F^{'} (t_{0}) .$ Note that the first equality (1) is the classic definition of the fuzzy H-derivative (or differentiability in the sense of Hukuhara). Adding the form (2), we enlarge the class of differentiable fuzzy mappings. Also, note that (1) and (2) are a generalization of the classical definitions of lateral derivative.

This paper has been organized as follows: Section 2 contains the basic material to be used in the rest of the article, in Section 3 we discuss some properties of the forms (1) and (2) of lateral H-derivatives and the their relationships. In particular we shall show that there are fuzzy mappings which are differentiable in the first form (1) and are not differentiable in the second form (2), and vice versa. In Section 4, we shall see that this definition of fuzzy lateral H-derivative leads us to interpret a FDE in two different forms, generating new solutions for FDE. Now, we does not have unique solution, which it seems natural in the fuzzy context because the fuzzy lateral H-derivatives breaks the restriction associated to the H-difference of fuzzy sets and solves the deficiency produced when we only consider the first form. Finally, in Section 5, we study the connection between the solutions of a FDE found using properties 1 and 2 and the solutions found via differential inclusions.

Section snippets

Basic concepts

We denote by $K^{n}$ the family of all nonempty compact subsets of $R^{n}$ , the n-dimensional Euclidean space. If $A, B \in K^{n}$ and $λ \in R$ , then the operations of addition and scalar multiplication are defined as $A + B = {a + b / a \in A, b \in B} λ A = {λ a / a \in A} .$ If $A \in K^{n}$ we define the “ϵ-neighbourhood of A″ as the set $N (A, ϵ) = {x \in R^{n} / d (x, A) < ϵ},$ where d(x, A) = inf_a∈A∥x − a∥ and ∥·∥ is the usual Euclidean norm on $R^{n}$ . The Hausdorff separation ρ(A, B) of $A, B \in K (X)$ is defined by $ρ (A, B) = \inf {ϵ > 0 / A \subseteq N (B, ϵ)},$ and the Hausdorff metric on $K^{n}$ is defined by $h (A, B) =$

The fuzzy derivative

It is well-known that the H-derivative (differentiability in the sense of Hukuhara) for fuzzy mappings was initially introduced by Puri and Ralescu [24] and it is based in the H-difference of sets, as follows.

Definition 1

Let be $u, v \in F^{n}$ . If there exists $w \in F^{n}$ such that u = v + w, then w is called the H-difference of u and v and it is denoted by u − v.

Definition 2

[24]

Let be T = (a, b) and consider a fuzzy mapping $F : (a, b) \to F^{n}$ . We say that F is differentiable at t₀ ∈ T if there exists an element $F^{'} (t_{0}) \in F^{n}$ such that the limits $\lim_{h \to 0^{+}} F (t_{0} + h) -$

Solving fuzzy differential equations

In this section, we study the fuzzy initial value problem $x^{'} (t) = F (t, x (t)), x (0) = x_{0},$ where $F : [0, a] \times F \to F$ is a continuous fuzzy mapping and x₀ is a fuzzy number. Following Kaleva [20], we observe that the relations (3), (4) in Theorem 5 give us an useful procedure to solve the fuzzy differential equation (5). In fact, denote $[x (t)]^{α} = [u_{α} (t), v_{α} (t)], [x_{0}]^{α} = [u_{α}^{0}, v_{α}^{0}]$ and $[F (t, x (t))]^{α} = [f_{α} (t, u_{α} (t), v_{α} (t)), g_{α} (t, u_{α} (t), v_{α} (t))] .$ Then, we have the following alternatives for solving the problem (5):

Case I

If we consider x^′(t)

Constructing solutions via differential inclusions

We consider the fuzzy differential equation $x^{'} (t) = F (t, x (t)), x (0) = x_{0},$ where $F : [0, a] \times F \to F$ is obtained by Zadeh’s extension principle from a continuous function $f : [0, a] \times R \to R$ . Now F(t, x) can be computed levelwise, i.e., for each α ∈ [0, 1] $[F (t, x)]^{α} = f (t, [x]^{α})$ for all t ∈ [0, a] and $x \in F$ . Following the idea of Hullermeier [18] (see also [2], [9], [10]) we interpret the fuzzy initial value problem (8) as a family of differential inclusions: $y_{α}^{'} (t) = f (t, y_{α} (t)), y_{α} (0) \in [x_{0}]^{α}, α \in [0, 1] .$ Under suitable assumptions, the

Conclusions

In this work we introduce the concept of lateral fuzzy H-derivatives for fuzzy mappings (see Section 3), enlarging the class of differentiable fuzzy mappings. Subsequently, by using the laterel H-derivatives, we obtain two different interpretations of a FDE generating new solutions for a FDE (see Section 4).

Different studies and many examples have been developed by several authors by using the fuzzy derivative x′ in the first form (which is coincident with the classic H-derivative). For

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^☆: This work was partially supported by Fondecyt-Chile through Projects 1061244 and 1040303 and Dipog-UTA by Projects 4731-04 and 4731-05.

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On new solutions of fuzzy differential equations☆

Abstract

Introduction

Section snippets

Basic concepts

The fuzzy derivative

[24]

Solving fuzzy differential equations

Constructing solutions via differential inclusions

Conclusions

Fuzzy Set Syst

Chaos, Solitons & Fractals

Fuzzy Set Syst

Fuzzy Set Syst

Inform Sci

Chaos, Solitons& Fractals

Fuzzy Set Syst

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Math Comput Model

Fuzzy Set Syst

Fuzzy Set Syst

Nonlinear Anal

Chaos, Solitons & Fractals