1. Introduction

Every physical problem is inherently biased by uncertainty. There is often a need to model, solve, and interpret the problems one encounters in the world of uncertainty. To overcome this uncertainty and vague, we may use the interval and fuzzy set theory. The topic of fuzzy differential equations (FDEs) forms a suitable setting for mathematical modelling of this physical problems. The concept of fuzzy derivative was first introduced by Chang and Zadeh (1972). Numerical solution for linear fuzzy differential equation was studied by many researcher ([1], [2], [3], [4], [5], [6], [7], [8], [9]). The solution of n-th order of fuzzy differential equation also derive by [10], [11], [12], [13] and [14]. The most frequently method to get the numerical solution is Runge Kutta method.

This paper studied a third order Runge Kutta method based on combination of arithmatics, harmonics and geometrics mean to solve n-th order of fuzzy initial value problem. In the Section 2, we begin with some preliminary results and concepts about fuzzy number and system of fuzzy initial value problem. In Section 3, we discuss the main idea to solve the problem. We also analyse the stability, convergence and the error, then we employ the method on test example. Finally, in Section 4 we give the conclusion of this study.

2. Preliminaries

2.1. A Fuzzy Number

An interval is denoted by on the set of real numbers R given by

In this paper, we have only considered closed intervals, although there exist various types of intervals such as open and half-open intervals. A fuzzy number is convex, normalized fuzzy set of the real line such that

where, is called the membership function of the fuzzy set, and it is piecewise continuous. A triangular fuzzy number is defined by three numbers , where the graph of , the member of function of the fuzzy number , is a triangle with the base on the interval and the vertex at . We specify as and

Let be a set of all the upper semicontinuous normal convex fuzzy numbers with bounded r-level sets. It means that if , then the -level set

is a closed bounded interval which is denoted by

Let be a real interval. The mapping is called fuzzy process and its -level set is denoted by

The derivative of the fuzzy process x is defined by

provided that this equation determines the fuzzy number.

Let be the set of all nonempty compact subset of and be the subset of consisting of nonempty convex compact sets. Recall that

is a distance of the point from and that the Hausdorff separation of is defined as

2.2. A n^th Fuzzy Initial Value Problem

Consider the fuzzy initial value problem

(1)

where is continuous mapping from into and are fuzzy numbers in . The -order fuzzy differential equation by changing variables

converts to the following fuzzy system

(2)

where are continuous mapping from into and are fuzzy numbers in with -level intervals for and

Now, we have to show that the solution of (2) is on a interval , if

For fixed value , we have a system of initial value problem in and we have intervals with a fuzzy number . Let

and

with respect to the indicators system (2) can be written as with assumption

(3)

With assumption and where

and where

Function is a fuzzy solution of (3) on an interval for all , if

(4)

Or

(5)

Theorem 2.1. If for are continuous function of and satisfies the Lipschitz condition in in the region with constant then the initial value problem (2) has unique solution in each case.

Proof. See [15]

By Theorem 3.1 the initial value problem (2) has a unique solution .

2.3. Runge Kutta Method

The basis of all Runge Kutta method of order m is to express the difference between the value of and as

where are constants and

The Runge Kutta method of order 3 based on combination of arithmetic, harmonic and geometric means is [16]

(6)

with

3. The Third Order of Runge Kutta Method Based on Combination of Arithmatics, Harmonics and Geometrics Mean

Define

With

By the third order Runge Kutta based on combination of means, we obtain

From Eq. (6), define

and

The discrete equally spaced grid points is a partition for interval . If the exact and the approximate solution in the i-th cut at are denoted by and respectively, then the numerical solution by third order Runge Kutta method based on combination of arithmetic, harmonics and geometrics means is

The approximate solution for -cut of Eq.(2) is

(7)

where

and

with

3.1. Stability, Convergence and Error Analysis

To analyse the stability, convergence and the error of the method, consider the next definition and theorem.

Definition 3.1. [15] A one-step method for approximating the solution of differential equation

with is a ordered as and is a method that can be written in the form

(8)

where the increment function is determined by .

Theorem 3.2. If satisfies a Lipschitz condition in then the method given by (8) is stable.

Theorem 3.3. In relation (2), if satisfies a Lipschitz condition in y then the method given by (7) is stable.

Theorem 3.4. If

where

Is a numerical method for approximation of differential equation (2), and are continuous in for and all , and if they satisfy a Lipschitz condition in the region , ,

the necessary and sufficient conditions for convergence is

Proof. See [15].

Then the method proposed by (6) is convergent to the solution of the system (2).

3.2. Numerical Examples

The next example show the performance the new method.

Figure 1

Example [15]. Consider the vibrating mass ( slug) in Fig.1. The spring constant is , there is no damping force and the forcing function is for . The differential equation of motion is

Let

The exact solution is

for

By using the new method, the numerical solution is in Table 1 and Table 2.

Table 1. The Solution of Example 1 for

Table 2. The Solution of Example 1 for

4. Conclusions

In this paper we presented a numerical approach to solve system of fuzzy differential equations with initial value. The scheme is based on the third order Runge Kutta method for solving n-th order of fuzzy initial value probrems. The stability, convergence and error analysis have been studied. Numerical simulation performs that the new method is an accurate method for n-th order of fuzzy initial value problems.

ACKNOWLEDGEMENTS

This works has fully supported by the Ministry of Research Technology and Higher Education of Indonesia.