Solving Fuzzy Differential Equationsin RungeKutta Method of Order Three

Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 198-201 ISSN: 2278-2397

Solving Fuzzy Differential Equationsin RungeKutta Method of Order Three Sharmila, Josephine Mary, Maria Nancy Flora , Mahesh PG and Research Department of Mathematics St. Joseph’s College of Arts and Science (autonomous), Cuddalore , India.

Abstract- In this paper, we study numerical method for Fuzzy differential equations by Runge-Kutta method of order three.

The elementary properties of this method are given. We use the extended Runge-Kutta method of order three in order to enhance the order of accuracy of the solution. Thus we can obtain the strong Fuzzy solution.

Keywords- Fuzzy differential equations, Runge-Kutta method of order three, Trapezoidal Fuzzy number .

I.

PRELIMINARY

A trapezoidal fuzzy number u is defined by four real numbers   k   m  n where the base of the trapezoidal is the interval [k, n] and itsvertices x  , x  m . at Trapezoidal fuzzy number will be written as u  ( k , , m , n ) . The membership function for the trapezoidal fuzzy number u  ( k , , m , n ) follows : x  k  k x  n m  n

is defined as

, k  x 

u

is

convex

fuzzy

set(i.

e.

u ( tx  (1  t ) y )  min{ u ( x ), u ( y )},  t  [ 0 ,1 ], x , y  R )

(iii)  u  R F , u i s u p p e r s e m i c o n t i n u o u s o n R ; (iv) x 

R ; u ( x )  0  is c o m p a c t, w h e re A d e n o te s th e c lo s u re o f A .

Then RF is

called the space of fuzzy numbers.

R  {  { x } ; x is usual

real

number }

We define the r-level set, x  R ; [ u ]r   x \ u ( x )  r  ,

0  r  1;

clearly [ u ] 0   x \ u ( x )  0  is c o m p a c t, Theorem 2.1 Let F ( t , u , v ) and G ( t , u , v ) belong to C 1 ( R F ) and the partial derivatives of F and G be bounded over R F . Then for arbitrarily fixed y ( tn 1 ; r )

r , 0  r  1, the numerical solutions of

and y ( t n  1 ; r ) converge to the exact solutions

Y ( t ; r ) and Y ( t ; r ) uniformly in t.

Theorem 2.2 Let F ( t , u , v ) and G ( t , u , v ) belong to C 1 ( R F ) and the partial derivatives of F and G be bounded over R F and 2 L h  1 . Then for arbitrarily fixed 0  r  1, the iterative

 x  m

1 ,

u(x) 

(ii)  u  R F ,

Obviously R  R F . H e r e R  R F is u n d e r s to o d a s

INTRODUCTION

In this paper, we have introduced and studied a new technique forgetting the solution of fuzzy initial value problem. The organized paper is asfollows: In the first three sections, we recall some concepts in fuzzy initial value problem. In sections four and five,we present Runge-Kutta method of order three and its iterative solution forsolving Fuzzy differential equations. The proposed algorithm is illustrated by anexample in the last section.

II.

(i)  u  R F , u is n o r m a l, i.e .  x 0  R w ith u ( x 0 )  1

numerical solutions of y

, m  x  n

( j)

(1)

( j)

( t n ; r ) and y

to the numerical solutions

y ( tn ; r )

( t n ; r ) converge

and

in

y ( tn ; r )

t0  tn  t N , w h e n j   .

The results may be: (1) u  0 if

k  0;

(2) u  0 if

 0;

III.

Consider a first-order fuzzy initial value differential equation is given by

(3) u  0 if m  0; and

(4) u  0 if

FUZZY INITIAL VALUE PROBLEM

n  0;

y (t )  f (t, y (t )), '

t   t0 , T

 (3)

Let us denote RFbythe class of all fuzzy subsets of R (i.e. u :R  [0,1]) satisfying the following properties: 198

y ( t0 )  y 0

Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 198-201 ISSN: 2278-2397  hf ( t n  c 3 h , y ( t n )  a 31 K 1  a 32 K 2 )

We denote the fuzzy function y by y   y , y  . It means that

K

the r-level set of y(t) for t   t 0 , T

and the parameters W 1 , W 2 , W 3 , c 2 , c 3 , a 2 1 , a 3 1 & a 3 2



 y (t ) r



 is

 [ y ( t ; r ), y ( t ; r )],

3

chosen to make y n  1 closer to y ( t n  1 ) .

are

There are eight

parameters to be determined. Now, Taylor’s series expansion

[ y ( t n )] r  [ y ( t 0 ; r ), y ( t 0 ; r )], r  ( 0 ,1 ]

about t n gives

we write Because of y'  f ( t , y ) we have

y ( t n 1 )  y ( t n ) 

f ( t , ( y ( t ); r )  F [ t , y ( t ; r ), y ( t ; r )]

h

y (t n ) 

f ( t , ( y ( t ); r )  G [ t , y ( t ; r ), y ( t ; r )]

1!

h

y (t n )  '

1!

h

2

h

y (t n )  ''

2!

3

3!

y ( t n )  ... = '''

(4) 2

f ( t n , y ( t n )) 

By using the extension principle, we have the membership function f ( t , y ( t ))( s ) = sup { y ( t )(  ) \ s  f ( t ,  )}, s  R

h

2!

[ f t  ff

y

] t n  ....

(6)

so fuzzy number f ( t , y ( t ) ) . From this it follows that  a 21 f 2

[ f ( t , y ( t ))] r  [ f ( t , y ( t ); r ), f ( t , y ( t ); r )], r  ( 0 ,1]

where

2

f yy ] tn  .....}

(7)

h

f ( t , y ( t ) ; r )  min

 f ( t , u ) | u  [ y ( t )] r 

(8)

f ( t , y ( t ) ; r )  max

 f ( t , u ) | u  [ y ( t )] r  Definition - A function

R  R F is said to be fuzzy

continuous function, if for an arbitrary fixed t 0  R and   0 ,   0 such that

+… } Substituting the values of

| t  t 0 |   D [f(t), f(t0)] < 

Throughout this paper it is considered that fuzzy functions are continuous in metric D. Then the continuity of f(t,y(t);r)guarantees the existence of the definition f(t, y(t); r) for t  [t0,T] and r  [0,1]. Therefore, the functions G and F can be defined too.

IV.

RUNGE-KUTTA METHOD OF ORDER THREE

(13)

Consider the initial value problem y '( t )  f ( t , y ( t ) ) ,

+…

t  [ t0 , T ]

Comparing the coefficients of h,h2& h3, we obtain (10)

y ( t0 )  y 0

,

, ,

Assuming the following Runge-Kutta method with three slopes y ( t n 1 )  y ( t n )  W 1 K 1  W 2 K 2  W 3 K 3

,

(11)

,

where K 1  h f ( tn , y ( tn ))

K 2  h f ( tn  c 2 h , y ( tn )  a 21 K 1 )

(14) Then we immediately obtain from the fourth and fifth equations , that . Similarly the values of the remaining parameters are obtained.

199

Integrated Intelligent Research (IIR)

When c 2  c 3 , we get of

the

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 198-201 ISSN: 2278-2397

and

. We get the values

other

parameters .

Also we have y ( t n 1 ; r ) 

as

y (tn ; r ) 

1

[2 K

8

 3K

1

 3K

2

3

]

where K 1  h F [tn ,

Runge-Kutta method is obtained as

2h

K 2  h F [tn , 

y ( tn ; r ), y ( tn ; r )]

2

, y (tn ; r ) 

3

K 1 , y (tn ; r ) 

3

2 3

K1

(15) K

Where

3

2h

 h F [tn , 

,

3

y (tn ; r ) 

2 3

K 2 , y (tn ; r ) 

2

(18)

]

K

2

 3K

3

3

]

and y ( t n 1 ; r ) 

y (t n ; r ) 

K 1  h G [tn ,

K

K

V.

2

 h G [tn , 

3

 h G [tn , 

[2 K

8

1

 3K

2

],

where

y (tn ; r ), y ( tn ; r )]

2h 3

2h 3

Clearly,

RUNGE-KUTTA METHOD OF ORDER THREE FOR SOLVING FUZZY DIFFERENTIAL EQUATIONS

1

2

, y (tn ; r ) 

,

y (tn ; r ) 

K 1 , y (tn ; r ) 

3

2

K

3

and

2

, y (tn ; r ) 

converge

2 3

2 3

K1

K

] ]

2

to

and

respectivelywhen ever VI.

Let Y  [ Y , Y ] be the exact solution and y  [ y , y ] be the

NUMERICAL RESULTS

In this section, the exact solution and approximated solution are obtained byEuler’s method and Runge-Kutta method of order three.

approximated solution of the fuzzy initial value problem .

Let [ Y ( t )] r  [ Y ( t ; r ), Y ( t ; r )], [ y ( t )] r  [ y ( t ; r ), y ( t ; r )] Throughout

this argument, the value of r is fixed. Then the exact and

Example Consider the initial value problem

approximated solution at t n are respectively denoted by

y′(t)= f(t),

[Y ( t n )] r  [Y ( t n ; r ), Y ( t n ; r )] ,

t∈[0,1]

y(0)= (0.75+0.25r, 1.125 -0.125r)

[ y ( t n )] r  [ y ( t n ; r ), y ( t n ; r )]( 0  n  N ) .

The grid points at which the solution is calculated are h 

T  t0

Y(1;r)=[(0.75+0.125r)e,(1.125-0.125r)e], 0≤r≤1 Using iterative solution of Runge-Kutta method of order three, we have

, t i  t 0  ih , 0  i  N .

N

Then we obtain, Y ( t n 1 ; r )  Y ( t n ; r ) 

where K

K

K

1

1

[2 K

8

 3K

1

2

 3K

, ]

3

 hF [ t n , Y ( t n ; r ), Y ( t n ; r )]

2

 h F [tn 

3

 h F [tn 

2h 3 2h 3

2

, Y (tn ; r ) 

1

K 1 , Y (tn ; r ) 

3 2

, Y (tn ; r ) 

Y ( t n 1 ; r )  Y ( t n ; r ) 

3

K 2 ,Y (tn ; r ) 

[2 K

8

1

 3K

2

2

2

And by

K1]

3

K

3

 3K

3

2

]

],

2

 h G [tn , 

2h 3

, Y ( tn ; r ) 

2 3

K 1 , Y ( tn ; r ) 

where

Where i=0,1,2,...N-1 and h= . Now, using these equations as an initial guess for following iterative solutions respectively,

2 3

K1

] (17)

K

3

 h G [tn , 

2h 3

, Y ( tn ; r ) 

2 3

(16)

and

K 1  h G [tn , Y ( tn ; r ), Y ( tn ; r )] K

The exact solution at t=1 is given by

K 2 ,Y (tn ; r ) 

2 3

K

] 2

200

Integrated Intelligent Research (IIR)

International Journal of Computing Algorithm Volume: 03 Issue: 03 December 2014 Pages: 198-201 ISSN: 2278-2397 problem has been formulated. Numerical example shows that the exact and approximate solutions converge when h ď&#x201A;Ž 0 .

REFERENCES [1]. S.L. chang and L.A. Zadeh, on Fuzzy Mapping and control, IEEE Trans.

Systems Man Cybernet.,2 (1972) 30-34 [2]. M.L. Puri and D.A. Ralescu, Differentials of Fuzzy Functions, J.Math.

Anal.,91(1983) 321-325 [3]. J.J. Buckley and E. Eslami and T. Feuring, Fuzzy Mathematics in Economics and Engineering, Physica - Verlag, Heidelberg, Germany.

2002 [4]. R. Goetschel and W. Voxman, Elementary Calculus, Fuzzy sets and systems. 18 (1986) 31-43 [5]. J.J. Buckely and E. Eslami, Introduction to Fuzzy Logic and Fuzzy Sets, Physica - Verlag, Heidelberg, Germany. 2001 [6]. O. Kaleva, The Cauchy Problem for Fuzzy Differential Equations, Fuzzy sets and systems, 35 (1990) 389-396.

[7]. Duraisamy .C and Usha .B, Numerical Solution of Fuzzy Differential Equations by Runge-kutta Method, National conference on scientific computing and Applied Mathematics.

[8]. Duraisamy .C and Usha .B, Numerical Solution of Fuzzy Differential Equations by Taylor method, International journal of Mathematical Archive Vol.2, 2011.

And

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

Therefore,

and

and

are

obtained.

By minimizing the step size h, the solution by exact method and Runge- Kutta method almost coincides.

Table 1: Exact solution r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2.038711371 2.106668417 2.174625463 2.242582508 2.310539554 2.378496600 2.446453646 2.514410691 2.582367737 2.650324783 2.718281828

3.058067057 3.024088534 2.990110011 2.956131488 2.922152966 2.888174443 2.85419592 2.820217397 2.786238874 2.752260351 2.718281828

Table 2: Approximated solution r 0 0.1 0.2 0.3 0.

0.5 0.6 0.7 0.8 0.9 1

VII.

2.038633 2.106587 2.174542 2.242496 2.310451 2.378405 2.446360 2.514314 2.582260 2.650223 2.718177

3.057949 3.023972 2.989995 2.956018 2.922041 2.888063 2.854086 2.820109 2.786132 2.752154 2.718177

CONCLUSION

In this paper, numerical method for solving Fuzzy differential equations is considered. A scheme based on thirdorder Runge â&#x20AC;&#x201C; Kuttamethod to approximate the solution of fuzzy initial value 201