Vol.1,Issue.III/Sept11pp.1-4ISSN No-2031-5063

Research Paper

1Golden Research Thoughts

] Introduction :- The concept of Fuzzy differential equation was first introduced by Chang and Zadeh [1].Later Dubois and Prade [2] defined and used the extension principle. Other methods have been discussed by Puri and Ralescu [3].The existence of solutions of Fuzzy differential equation has been studied by several authors [4,5]. Abbasbandy and Allahviranloo [6] developed numerical algorithms for solving Fuzzy differential equation based on Seikala's derivative of the Fuzzy process introduced in [7]. In the present paper ,the Runge-Kutta method of order three to solve Fuzzy differential equation have been established. The structure of the paper is as follows. In the first three sections, Some concepts and introductory materials to deal with the Fuzzy initial value problem have been recalled. In section 4 and 5 Runge-Kutta method of order three and its iterative solution for solving Fuzzy differential equations are presented. The illustration of numerical example is finally given in section 6. 2] Basic concepts and definition:- {A triangular Fuzzy number µ is defined by three real numbers a<b<c. where the base of the triangle is the interval [a , c] and its vertex is at x=b }The most popular kind of Fuzzy numbers is triangular Fuzzy numbers .These numbers are defined as Δ=(a, b, c) and their membership functions are defined as follows

Remark:- A Crisp number a may be regarded as the triangular Fuzzy number (a, a, a). For a given possibility

A new approach to solve Fuzzy differential equation by using third order Runge-Kutta method.

AbstractA solution of first order Fuzzy differential equation is obtained by Runge-Kutta method of third order (iterative method).The new approach presented in this paper gives complete error analysis by very simple and innovative

techniqueKeywords:- Ordinary differential equation, Fuzzy differential equation, Runge- Kutta method.

N. M. DeshmukhDepartment of Engineering Mathematics,

Sanjivani rural education society's College of Engineering, Kopargaon

where b ? a, b ? c .

the left and right boundaries of cuts of a triangular Fuzzy number Δ = (a, b, c) are

given by respectively.

We will have :-1]

2]

3]

4]

Let denote by the class of fuzzy set subsets of the real axis (i.e. satisfying the

following properties.

a) is normal i.e. with

b) is convex fuzzy set

c) is upper semi continuous on

d) is compact ,where denotes the closure of A.

Then is called the space of Fuzzy number (see e.g.5)

Obviously Here is understood as

.

We define the r -level set

--------- (2)

is compact.

Then it is well known that for each is bounded closed interval

We denote by It is clear that the following statements are true

i] is a bounded left continuous non-decreasing function over [0,1]

ii] is a bounded right continuous non-increasing function over [0,1]

iii]

for more details see [6],[7].

Let

By Hausdroff distance between Fuzzy numbers , where

], ]

The following properties are well-known (see e.g.[11],[16])

and is complete metric space.

Lemma 2.1

Let the sequence of numbers satisfy ,for

the given positive constants A &B. Then .

Proof:- See[8]

Lemma 2.2

Let the sequence of numbers satisfy

, for

the given positive constants A and B.Then denoting where

Vol.1,Issue. 11.III/Sept

2Golden Research Thoughts

Lemma 2.3

Let F(t,u,v) and G(t,u,v) belong to C’( ) and the partial derivatives of F and G be bounded

over .Then for arbitrarily fixed where

L is a bound of partial derivatives of F & G and

.

Theorem 2.4

Let F(t,u,v) and G(t,u,v) belong to C’( ) and the partial derivatives of F and G be bounded

over .Then for arbitrar y fixed to numerical solutions of

converge to the exact solution uniformly in t.

Theorem 2.5

Let F(t,u,v) and G(t,u,v) belong to C’( ) and the partial derivatives of F and G be bounded

over and 2Lh<1.Then for arbitrary fixed the iterative numerical solution of

converge to the exact solution

When

3] Fuzzy Initial Value Problem

Here we introduced Fuzzy Initial Value Problem in following form

-------- ( 3)

Where y is Fuzzy function of t, f (t, y) is a Fuzzy function of crisp variable t and Fuzzy

variable y, y’ is the Fuzzy derivative of y and is a triangular shaped Fuzzy number ,

therefore we have a Fuzzy Cauchy problem.

We denote the Fuzzy function y by y .It mean the r-level set of y(t) for

,

By using the extension principle of Zadeh, we have the membership function

------- (4)

So is a Fuzzy number. From this it follows that

-------- (5)

Where ]}

]} -------- (6)

We define

. -------- (7)

Definition 3.1:- A function is called a Fuzzy function. If for arbitrary fixed

exist is said to be

continuous.

Throughout this work we also consider Fuzzy functions which are continuous in metric

space D .Then the continuity of f(t, y(t); r) guarantees the existence of Definitionate of

f(t, y(t); r) for

Therefore the function G & F can be definite too.

4 Runge- Kutta method of order three

Consider the initial value problem

------- 4.1

Assuming the following Runge - Kutta method with three slopes

------- 4.2

Where

The parameters and are chosen to make .

For determination of these values using Taylor’s series expansion.

Taylor’s series expansion about gives

------ [4.3]

If we set

}

Now,

=

=

Substituting in (4.2) we get ,

]+…. ------(4.4)

By matching coefficients with those of the Taylor series, follow system of equation obtain .

see (4.3 and 4.4)

------- (i)

------- ( ii)

------- (iii)

------- (iv)

------- (v)

------- (vi)

The third order Runge-Kutta method corresponds to the solution obtained by setting

Therefore from (vi)

We put in (iv)

Therefore

Therefore

Substitute in (ii)

Putting , and in (iv)

Therefore

Putting in equation (1), we get

Therefore Third order Runge-Kutta method is obtained as follows

Where ,

A new approach to solve Fuzzy differential equation by using third

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Golden Research Thoughts 3

5 Third order Runge-Kutta method for solving Fuzzy differential equation: -

Let be exact solution and be the approximated solution of the Fuzzy

initial value problem (3)

Throughout this argument the value of r is fixed . Then the exact and approximate solution of

are respectively denoted by

The grid points at which the solution is calculated

Then we obtain

Where

and,

Where,

We have,

Where,

and

Where

Clearly converges to respectively whenever

6 Numerical Examples :-

Example : Consider the Fuzzy differential equation

The exact solution is given by

At t = 1 we get

Using third order Runge-Kutta method approximation and denote

and by

Where I = 0, 1,------,N-1 and h=1/N

Now using these equation as an initial guess for the following iterative solutions respectively

Where,

and

Where,

And j = 1,2,3. Thus we have

( ) and .

Therefore,

(1 ) And are obtained.

The comparison of exact and approximate solution obtained by third order Runge -Kutta method

is also represented in graphical form as shown in Fig.1 and Fig.2.

Table 1: Exact solution

r Exact solution

0 2.310539554, 2.990110011

0.2 2.392088009, 2.935744375

0.4 2.473636464, 2.881378738

0.6 2.555184919, 2.827013102

0.8 2.636733374, 2.772647465

1 2.718281828, 2.718281828

5 Third order Runge-Kutta method for solving Fuzzy differential equation: -

Let be exact solution and be the approximated solution of the Fuzzy

initial value problem (3)

Throughout this argument the value of r is fixed . Then the exact and approximate solution of

are respectively denoted by

The grid points at which the solution is calculated

Then we obtain

Where

and,

Where,

We have,

Where,

and

Where

Clearly converges to respectively whenever

6 Numerical Examples :-

Example : Consider the Fuzzy differential equation

The exact solution is given by

At t = 1 we get

Using third order Runge-Kutta method approximation and denote

and by

Where I = 0, 1,------,N-1 and h=1/N

A new approach to solve Fuzzy differential equation by using third

Vol.1,Issue..III/Sept11

Golden Research Thoughts 3

Conclusion: The third order Runge-Kutta method is used to find a numerical solution of fuzzy differential equation. By minimizing the step size h, the solution by exact method and Runge-Kutta method almost coincides. Acknowledgment: Author is thankful to Management, Principal of SRES's College of Engineering, Kopargaon and Prof. B. R. Shinde and Prof. A. J. Deshpande for constant encouragement and help required from time to time. References[1] S. L. Chang and T. A. Zadeh , On Fuzzy mapping and control , IEEE Trans. Systems man cyber net, 2(1972)30-34.[2] D. Dubois and H.Prade ,”Towards Fuzzy differential calculus”, Differentiation Fuzzy sets and systems, Part 3,8(1982)225-233.[3] M.L.Puri and D.A.Ralescu, Differentials of Fuzzy functions, J.Math.Anal.Appl.91(1983)321-325.[4] K.Balachandran and P.Prakash , Existence of solutions of Fuzzy delay differential equation with nonlocal condition., J. of the Korea Soc. for Indust and Appl.Maths, 6(2002)88- 89.[5] K.Balachandran and Kanagarajan ,Existence of solution of Fuzzy delay integro differential equations with nonlocal condition , J. of Korea Soc. for Indust. and Appl Maths, 9(2005).[6] S.Abbasandy and T.Allahviranloo, Numerical solution of Fuzzy differential equations by Taylor method , J. of Computational Methods in Appl. Maths, 2(2002)113- 124.[7] S.Seikkala , On the Fuzzy initial value problem ,Fuzzy sets and systems ,24(1987)319- 330.[8] J.J.Buckley and E.Eslami , Introduction to Fuzzy Logic and Fuzzy sets Physica-Verlag, Heidelberg ,Germany 2001.

Now using these equation as an initial guess for the following iterative solutions respectively

Where,

and

Where,

And j = 1,2,3. Thus we have

( ) and .

Therefore,

(1 ) And are obtained.

The comparison of exact and approximate solution obtained by third order Runge -Kutta method

is also represented in graphical form as shown in Fig.1 and Fig.2.

Table 1: Exact solution

r Exact solution

0 2.310539554, 2.990110011

0.2 2.392088009, 2.935744375

0.4 2.473636464, 2.881378738

0.6 2.555184919, 2.827013102

0.8 2.636733374, 2.772647465

1 2.718281828, 2.718281828

Table 2: Approximated solution:

r \ h 0.1 0.01

0 2.0808807 , 2.692904523 2.31034788, 2.987949837

0.2 2.154323618, 2.643942623 2.391889656, 2.935525421

0.4 2.227766469, 2.594980723 2.473431349, 2.881139814

0.6 2.30120932, 2.5460188222 2.554973042, 2.826778685

0.8 2.374652171, 2.497056922 2.636046069, 2.772417553

1 2.692909543, 2.692909543 2.718056428, 2.718056428

Table 3 : Error for different values of r and h

r / h 0.1 0.01

0 0.526864342 0.002351848

0.2 0.529566143 0.000417307

0.4 0.532268010 0.000444039

0.6 0.534969879 0.000446294

0.8 0.537671746 0.000917217

1 0.050744570 0.000450800

A new approach to solve Fuzzy differential equation by using third