On interactive fuzzy boundary value problems
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 and , where , given by
Under certain conditions on the functions , and , the general solution of (1), using the superposition principle [9], is: where is a particular solution of (1) and , are linearly independent solutions of the associated homogeneous problem (i.e., when
Fuzzy boundary value problems (FBVPs)
In this section we study the fuzzy boundary value problem obtained by replacing the boundary values and with fuzzy numbers A and B in (1). More precisely, let us consider the following FBVP:
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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