Numerical Solution of N-order Fuzzy Differential Equations by Runge-kutta Method

In this paper we study a numerical method for n-th order fuzzy differential equations based on Seikkala derivative with initial value conditions. The Runge-Kutta method is used for the numerical solution of this problem and the convergence and stability of the method is proved. By this method, we can obtain strong fuzzy solution. This method is illustrated by solving some examples. The topic of fuzzy differential equations (FDEs) have been rapidly growing in recent years. The concept of the fuzzy derivative was first introduced by Chang and Zadeh [1], it was followed up by Dubois and Prade [2] by using the extension principle in their approach. Other methods have been discussed by Puri and Ralescu [3] and Goetschel and Voxman [4]. Kandel and Byatt [5] applied the concept of fuzzy differential equation (FDE) to the analysis of fuzzy dynamical problems. The FDE and the initial value problem (Cauchy problem) were rigorously treated by Kaleva [6,7], Seikkala [8], He and Yi [9], Kloeden [10] and by other researchers (see [11,15]). The numerical methods for solving fuzzy differential equations are introduced in [16-18]. Buckley and Feuring [20] introduced two analytical methods for solving n-th-order linear differential equations with fuzzy initial value conditions. Their first method of solution was to fuzzify the crisp solution and then check to see if it satisfies the differential equation with fuzzy initial conditions; and the second method was the reverse of the first method, they first solved the fuzzy initial value problem and the checked to see if it defined a fuzzy function. In this paper, a numerical method to solve n-th-order linear differential equations with fuzzy initial conditions is presented. The structure of the paper is organized as follows: In Section 2, we give some basic results on fuzzy numbers and define a fuzzy derivative and a fuzzy integral. Then the fuzzy initial values is treated in Section 3 using the extension principle of Zadeh and the concept of fuzzy derivative. It is shown that the fuzzy initial value problem has a unique fuzzy solution when f satisfies Lipschitz condition which guarantees a unique solution to the deterministic initial value problem. In Section 4, the Runge-Kutta method of order 4 for solving n-th order fuzzy differential

The topic of fuzzy differential equations (FDEs) have been rapidly growing in recent years.The concept of the fuzzy derivative was first introduced by Chang and Zadeh [1], it was followed up by Dubois and Prade [2] by using the extension principle in their approach.Other methods have been discussed by Puri and Ralescu [3] and Goetschel and Voxman [4].Kandel and Byatt [5] applied the concept of fuzzy differential equation (FDE) to the analysis of fuzzy dynamical problems.The FDE and the initial value problem (Cauchy problem) were rigorously treated by Kaleva [6,7], Seikkala [8], He and Yi [9], Kloeden [10] and by other researchers (see [11,15]).The numerical methods for solving fuzzy differential equations are introduced in [16][17][18].Buckley and Feuring [20] introduced two analytical methods for solving n-th-order linear differential equations with fuzzy initial value conditions.Their first method of solution was to fuzzify the crisp solution and then check to see if it satisfies the differential equation with fuzzy initial conditions; and the second method was the reverse of the first method, they first solved the fuzzy initial value problem and the checked to see if it defined a fuzzy function.In this paper, a numerical method to solve n-th-order linear differential equations with fuzzy initial conditions is presented.The structure of the paper is organized as follows: In Section 2, we give some basic results on fuzzy numbers and define a fuzzy derivative and a fuzzy integral.Then the fuzzy initial values is treated in Section 3 using the extension principle of Zadeh and the concept of fuzzy derivative.It is shown that the fuzzy initial value problem has a unique fuzzy solution when f satisfies Lipschitz condition which guarantees a unique solution to the deterministic initial value problem.In Section 4, the Runge-Kutta method of order 4 for solving n-th order fuzzy differential equations is introduced.In Section 5 convergence and stability are illustrated.In Section 6 the proposed method is illustrated by solving several examples, and the conclusion is drawn in Section 7.

PRELIMINARIES
An arbitrary fuzzy number is represented by an ordered pair of functions )) ( ), ( ( , which satisfy the following requirements [2]: is a bounded left continuous non-decreasing function over [0,1] , Let E be the set of all upper semi-continuous normal convex fuzzy numbers with bounded  -level intervals.

Lemma
be a given family of non-empty intervals.
, are  -level sets of a fuzzy number E v  , then the conditions (i) and (ii) hold true.[8] Let I be a real interval.A mapping E I v  : is called a fuzzy process and we [20] suppose u and v are fuzzy sets in E. Then their Hausdorff }, 0 { : (u,v) is maximal distance between  level sets of u and v.

Now we consider the initial value problem
where  is a continuous mapping from , converts to the following fuzzy system , with respect to the above mentioned indicators, system (3.2) can be writhen as with assumption Now we show that under the assumptions for functions i f , for i=1,…,n how we can obtain a unique fuzzy solution for system (3.2).

Theorem
for i=1,…,n are continuous function of t and satisfies the Lipschitz condition in converge uniformly on closed subintervals of  R to the solution of (3.16).In other word we have the following successive approximations where the integrals are the fuzzy integrals, define a sequence of fuzzy numbers For the proof of the property (ii) of Lemma (2.1), let and so y is a fuzzy solution of (3.1).The uniqueness follows from the uniqueness of the solution of (3.16).

THE RUNGE-KUTTA METHOD OF ORDER 4
With before assumptions, the initial values problem (3.2) has a unique solution, such as is a partition for interval [0,T].If the exact and approximate solution in the i-th  cut at then the numerical method for solution approximation in the i-th coordinate  cut, with the Runge-Kutta method is The Runge-Kutta method for solutions approximation  -cut of differential equation

Definition [24]
A one-step method for approximating the solution of a differential equation satisfies a Lipschitz condition in y, then the method given by (5.28) is stable.

. Theorem
In relation (3.5), if ) , ( y F t satisfies a Lipschitz condition in y, then the method given by (4.24) is stable.

Theorem
is a numerical method for approximation of differential equation (3.13), and 1 ψ and 2 ψ are continuous in t, y , h and all y , and if they satisfy a Lipschitz condition in the region }, ,..., satisfying the conditions of theory (3.1), then the following equation     .We will show that the numerical solutions given by (5.29) convergent to the ) (t y .By the mean value theorem, , with assumption and t n ) ,..., . From equation (5.29) obtain the following relations ), ), similarly, we can obtain the following relation so the relations (5.36) and (5.37) can be written as follow Then By virtue of lemma (5.7) , , so the numerical solution (5.29) converge to the solutions (5.31).Conversely, suppose that the numerical method (5.29) convergent to the solution of the system (5.31).With absurd hypothesis we suppose that (5.30)

Corollary
The Runge-Kutta proposed method by (4.24) and is convergent to the solution of the system (3.13)respectively. .

Example. Consider the following fuzzy differential equation with fuzzy initial value
The Runge-Kutta solution is as follows and Figures 1 and Table 1 show the obtained results: . ) 24 61 3

CONCLUTION
In this paper an numerical method for solving n-th order fuzzy linear differential equations with fuzzy initial conditions is presented.In this method a n-th order fuzzy linear differential equation is converted to a fuzzy system which will be solved with the Runge-Kutta method of order 4.