Numerical solution of fuzzy initial value problem (FIVP) using optimization

In this paper, we introduce a new approach to obtain a novel numerical solution of fuzzy initial value problem (FIVP). This technique is based on optimization problem. In fact, the optimal solution of optimization problem is approximated solution of fuzzy initial value problem. Theoretical consideration is discussed and some examples are presented to show the ability of the method for fuzzy initial value problem.


Introduction
*In modeling of real physical phenomena, differential equations play an important role in many areas of science and engineering. In many cases, information about the physical phenomena involved is always pervaded with uncertainty. Fuzzy differential equations (FDEs) are a natural way to model dynamical systems under possibilistic uncertainty.
Also, in modeling real-word phenomena, fuzzy initial value problems (FIVP) appear naturally.
Fuzzy linear systems have recently been studied by a good number of researchers, but only a few of them are mentioned here. Friedman et al. (1998), Allahviranloo (2004, 2005a, 2005b and Annelies Vroman et al. (2008).
The concept of fuzzy derivative was first introduced by Chang and Zadeh in (1965) it was followed up by Dubios and prade (1982), who defined and used the extension principle. The fuzzy differential equation and the fuzzy initial value problem were regularly treated by Kaleva in (1987Kaleva in ( , 1990 and by Seikkala in (1987).
There are several approaches to the study of fuzzy differential equations. The approach based on H-derivative (Puri and Ralescu, 1983) has the disadvantage that any solution of a FDE has increasing length of its support (Diamond, 2000). This shortcoming was resolved by interpreting a FDE as a family of differential inclusions (Hullermeier, 1999). The main shortcoming of using differential inclusions is that we do not have a derivative of a fuzzy -number-valued function, and so, the numerical solutions of a FDE are difficult to be obtained. Strongly generalized differentiability of fuzzy -number-valued functions is introduced and studied in (Bede and Gal, 2005). The numerical method for solving fuzzy differential equations is introduced in (Ma et al., 1999;Abbasbandy and Allahviranloo, 2002;Abbasbandy et al., 2004;Allahviranloo et al., 2007;Allahviranloo et al., 2006) by standard Euler method.
In this paper, we have used a different method to obtain the numerical solution of fuzzy initial valued problem (FIVP). In this method, we reduce (FIVP) to an optimization problem. In fact, the optimal solution of this optimization problem is the numerical solution of FIVP. Also, this method is simple and it does not need any differentiability of the fuzzy functions.

Fuzzy sets
According to zadeh (1965), a fuzzy set is a generalization of a classical set that allows membership Definition 2.1.1: Let be a universal set. A fuzzy set in is defined by a membership function that maps every element in to the unit interval [0,1]. A fuzzy set in may also be presented as a set of ordered pairs of a generic element and its 1) It exist exactly one ∈ ℝ with ( ) = 1.
2) ( ) is piecewise continuous. Nowadays, definition 2.6 is very often modified. Definition 2.1.7: A triangular fuzzy number has the following form: and are called left and right spreads of the fuzzy number . Fig. 1 shows a triangular fuzzy number.

The extension principle
One of the most basic concepts of fuzzy set theory that can be used to generalize crisp mathematical concepts to fuzzy sets is the extension principle. Definition 2.2.1: Let = × × … × and , , … , be fuzzy sets in , , … , , respectively. is a mapping from to a universe , = ( , , … , ). Then the extension principle allows us to define a fuzzy set in by: and is the inverse of . For = 1 , the extension principle, of course, reduces to

Fuzzy Initial value problem (FIVP)
In this section, at first, we consider the initial value problem (IVP). To interpret the connection between (3.1) and (3.2) we refer to Mizukoshi et al. (2007) and Hullermeier (1999).
As we explained in "Introduction" section, some researchers have worked on numerical solution of fuzzy initial value problem FIVP. In this study, we have used a different approach for obtaining numerical solution of this FIVP.

New approach
We consider the following fuzzy initial value problem: As we explained in section 6, the optimal solution of this linear programming problem is the approximated solution of fuzzy initial value problem (FIVP). The accurate of solution is getting to improve as is increasing.

Numerical experiments
In this section, we present some experiments to show performance and accuracy of presented method. Example 5.1: Consider the following fuzzy initial value problem: We have solved this problem using new approach for n=10. The numerical solution is high accurate and the accuracy of the numerical solution is increasing as n is increasing (Table 2). We have shown the exact and approximated solutions at = 1 (Figs. 2-4).

Conclusion
In this paper, we have presented a new approach is based on an optimization problem. In fact, we transform fuzzy initial value problem to a linear programming problem. The optimal solution of this linear programming problem is approximated solution of fuzzy initial value problem. This new approach is high accurate and very simple. Since, it does not need any differentiability of the fuzzy functions, so we can use it to solve fuzzy nonsmooth systems.    Numerical solution of fuzzy differential equations by predictor-corrector method. Information Sciences, 177 (7): 1633-1647. Bede B and Gal SG (2005). Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems, 151(3): 581-599.