Elsevier

Fuzzy Sets and Systems

Volume 358, 1 March 2019, Pages 84-96
Fuzzy Sets and Systems

On interactive fuzzy boundary value problems

https://doi.org/10.1016/j.fss.2018.07.009Get rights and content

Abstract

In this paper we use the concept of interactivity between fuzzy numbers for the solution to a linear fuzzy boundary value problem (FBVP). We show that a solution of a FBVP, with non-interactive fuzzy numbers as boundary values, can be obtained by the Zadeh's Extension Principle. In addition, we show it is possible to obtain a fuzzy solution by means of the extension principle based on joint possibility distributions for the case where the boundary values are given by interactive fuzzy numbers. Examples of FBVPs with both cases, interactive and non-interactive, are presented. Also from arithmetic operations for linearly correlated fuzzy numbers, we compare our solutions to the one proposed by Gasilov et al. We conclude that the fuzzy solution in the interactive case (when the boundary values are linearly correlated fuzzy numbers) is contained in the fuzzy solution for the non-interactive case. Finally, we present the fuzzy solution for a nonlinear FBVP with Gaussian fuzzy numbers as boundary values.

Introduction

One of the most important classes of differential equation problems is the boundary value problems (BVPs) which arise in many areas of knowledge such as in biology, chemistry, medicine, engineering, etc. Since some parameters of these problems can be uncertain, many researchers have studied BVPs where one or more parameters and/or state variables are given by fuzzy numbers. These problems are called of fuzzy boundary values problems (FBVPs). In this paper, we focus on FBVPs where the boundary values are given by possibly interactive fuzzy numbers.

Fuzzy boundary value problems have been studied since the early 2000s. O'Regan et al. investigated FBVPs based on Hukuhara derivatives by solving fuzzy integral equations given in terms of Aumann integrals [1]. However, O'Regan et al.'s approach can not be applied to a large class of FBVPs [2]. Khastan and Nieto associated FBVPs based on generalized Hukuhara derivatives with four classical BVPs [3]. Their approach may not produce fuzzy solution on the entire domain, that is, the solution obtained at a given instant t may not be a fuzzy number. Moreover, Gasilov et al. argued that the solutions obtained using Khastan and Nieto's method are difficult to interpret because the solutions of the four different problems may not reflect the nature of the studied phenomenon [4].

Other approaches to FBVPs are based on the theory of differential inclusion [5] and modification/adaptation of classical numerical methods for the fuzzy case, such as the undetermined coefficients method [6], the finite difference method [7], and the finite element method [8]. Gasilov et al. investigated FBVPs where the corresponding solutions are given by fuzzy sets in the class of real functions [4].

We show that the approach presented by Gasilov et al. in [4] corresponds to the application of Zadeh's extension principle for the solution of an associated classical BVP. Next, we deal with FBVPs where the boundary values are interactive fuzzy numbers. Recall that the relation of interactivity between two fuzzy numbers arises in the presence a joint possibility distribution J for them. In this case, the solution is obtained in terms of the sup-J extension principle of the solution of an associated classical BVP.

This paper is organized as follows. In Section 2, we review some basic concepts of fuzzy set theory and deterministic methods for solving classical BVPs. In Section 3, we determine solutions to fuzzy BVPs with interactive and non-interactive boundary values using extension principle. Finally, in Section 4, we employ our proposal to solve linear and nonlinear fuzzy BVPs with interactive and non-interactive boundary values in order to illustrate the effect of considering the existence of interactivity of the boundary values.

Section snippets

Solution for a deterministic boundary value problem

We begin by considering a second order linear non-homogeneous differential equation with boundary values x(0)=x0 and x(T)=xT, where x0,xTR, given by{x(t)+k(t)x(t)+p(t)x(t)=f(t),x(0)=x0,x(T)=xT.

Under certain conditions on the functions k(t), p(t) and f(t), the general solution of (1), using the superposition principle [9], is:x(t)=xP(t)+c1x1(t)+c2x2(t), where xP is a particular solution of (1) and x1, x2 are linearly independent solutions of the associated homogeneous problem (i.e., when f(t)=

Fuzzy boundary value problems (FBVPs)

In this section we study the fuzzy boundary value problem obtained by replacing the boundary values x0 and xT with fuzzy numbers A and B in (1). More precisely, let us consider the following FBVP:{x(t)+k(t)x(t)+p(t)x(t)=f(t),x(0)=A,x(T)=B.

Examples

Here we discuss examples of linear and non-linear FBVPs whose boundary values are possibly interactive. First, we study the case where the boundary values are non-interactive fuzzy numbers. In this case, we present the solution proposed by Gasilov et al. [4] that corresponds to the solution obtained by means of the Zadeh's extension principle according to Theorem 3.1. Second, we deal with the case where the boundary values are interactive fuzzy numbers. As expected from Theorem 3.2, the

Final remarks

The main contribution of this article is the study of second order (linear and non-linear) boundary value problems with boundary values given by interactive fuzzy numbers. Our analysis is based mainly on the idea of the generalization of the classical Zadeh's extension principle, namely sup-J extension principle. We obtain fuzzy solutions by means of the sup-J extension principle of deterministic solutions of the associated BVPs. In the case where the boundary values of the underlying FBVP are

Acknowledgements

This research was partially supported by CNPq under grant no. 306546/2017-5, and FAPESP under grant no. 2016/26040-7.

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    The fuzzy solutions of (51) obtained via the sup-J extension are depicted in Fig. 2. The solutions obtained from the gH-derivative are always contained in the solution produced via Zadeh's extension principle [27]. In fact, this result holds true because the gH-derivative is a particular case of an interactive derivative [29,32].

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