IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 3 (May. - Jun. 2013), PP 47-63 www.iosrjournals.org www.iosrjournals.org 47 | Page Numerical Solution of N th - Order Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Basesd On Centroidal Mean R. Gethsi Sharmila and E. C. Henry Amirtharaj Department of Mathematics, Bishop Heber College(Autonomous), Tiruchirappalli -17, India. Abstract: In this paper, a numerical method for N th - order fuzzy initial value problems (FIVP) based on Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean (RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits well to find the numerical solution of N th – order FIVPs. Keywords - Fuzzy numbers, N th - order Fuzzy Initial Value Problems, Runge-Kutta method, Centroidal Mean, Lipschitz condition. I. Introduction The research work on Fuzzy Differential Equations (FDEs) has been rapidly developing in recent years. The concept of the fuzzy derivative was first introduced by Chang and Zadeh[9],it was followed up by Dubois and Prade [10] by using the extension principle in their approach. Other methods have been discussed by Puri and Ralescu [23] and Goetschel and Voxman [16]. Kandel and Byatt [21] 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 [19, 20], Seikkala [24], He and Yi [17], and by other researchers (see [6, 8]). The numerical methods for solving fuzzy differential equations are introduced by Abbasbandy et.al. and Allahviranloo et.al. in [1, 2, 5]. Buckley and Feuring [7] 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. Allahviranloo et.al [3, 4] proposed the methods for solving N th – order fuzzy differential equations. Jayakumar et.al [18] used the Runge - Kutta Nystrom method for solving N th – order fuzzy differential equations. Gethsi Sharmila and Henry Amirtharaj [14, 15] introduced the explicit third order Runge-Kutta method based on Centroidal Mean (CeM) to solve IVPs and developed a numerical algorithm for finding the solution of Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Based on Contraharmonic Mean. In this paper, a new numerical method to solve N th - order linear fuzzy initial value problem is presented using the fourth order Runge – Kutta method based on Centroidal mean. The structure of the paper is organized as follows: In Section 2, some basic results on fuzzy numbers and fuzzy derivative are given. Then the fuzzy initial value problem 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 fourth order Runge-Kutta method based on centroidal mean for solving N th - order fuzzy initial value problems is introduced. In Section 5 convergence and stability are illustrated. In Section 6 the proposed method is illustrated by solving two examples, and the conclusion is drawn in Section 7. II. Preliminaries An arbitrary fuzzy number is represented by an ordered pair of functions , ur ur for all 0,1 r , which satisfy the following requirements [10]: (i) ur is a bounded left continuous non-decreasing function over [0,1] , (ii) ur is a bounded left continuous non-increasing function over [0,1] , (iii) , 0 1. ur ur r Let E be the set of all upper semi-continuous normal convex fuzzy numbers with bounded level intervals.
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R. Gethsi Sharmila and E. C. Henry Amirtharaj Department of Mathematics, Bishop Heber College(Autonomous), Tiruchirappalli -17, India.
Abstract: In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
Keywords - Fuzzy numbers, Nth - order Fuzzy Initial Value Problems, Runge-Kutta method, Centroidal Mean,
Lipschitz condition.
I. Introduction The research work on Fuzzy Differential Equations (FDEs) has been rapidly developing in recent
years. The concept of the fuzzy derivative was first introduced by Chang and Zadeh[9],it was followed up by Dubois and Prade [10] by using the extension principle in their approach. Other methods have been discussed by
Puri and Ralescu [23] and Goetschel and Voxman [16]. Kandel and Byatt [21] 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 [19, 20], Seikkala [24], He and Yi [17], and by
other researchers (see [6, 8]). The numerical methods for solving fuzzy differential equations are introduced by
Abbasbandy et.al. and Allahviranloo et.al. in [1, 2, 5]. Buckley and Feuring [7] introduced two analytical
methods for solving Nth - 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. Allahviranloo et.al [3, 4]
proposed the methods for solving Nth – order fuzzy differential equations. Jayakumar et.al [18] used the Runge
- Kutta Nystrom method for solving Nth – order fuzzy differential equations. Gethsi Sharmila and Henry Amirtharaj [14, 15] introduced the explicit third order Runge-Kutta method based on Centroidal Mean (CeM)
to solve IVPs and developed a numerical algorithm for finding the solution of Fuzzy Initial Value Problems by
Fourth Order Runge-Kutta Method Based on Contraharmonic Mean.
In this paper, a new numerical method to solve Nth - order linear fuzzy initial value problem is presented
using the fourth order Runge – Kutta method based on Centroidal mean. The structure of the paper is organized
as follows: In Section 2, some basic results on fuzzy numbers and fuzzy derivative are given. Then the fuzzy
initial value problem 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 fourth
order Runge-Kutta method based on centroidal mean for solving Nth - order fuzzy initial value problems is
introduced. In Section 5 convergence and stability are illustrated. In Section 6 the proposed method is illustrated by solving two examples, and the conclusion is drawn in Section 7.
II. Preliminaries An arbitrary fuzzy number is represented by an ordered pair of functions
,u r u r for all 0,1r , which satisfy the following requirements [10]:
(i) u r is a bounded left continuous non-decreasing function over [0,1] ,
(ii) u r is a bounded left continuous non-increasing function over [0,1] ,
(iii) , 0 1.u r u r r
Let E be the set of all upper semi-continuous normal convex fuzzy numbers with
bounded level intervals.
Numerical Solution Of Nth - Order Fuzzy Initial Value Problems By Fourth Order Runge-Kutta
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Lemma 2.1
Let ( ), ( ) , (0,1]v v be a given family of non-empty intervals. If
( ) [ ( ), ( )] [ ( ), ( )] for 0 < , and
( ) [lim ( ), lim ( )] [ ( ), ( )], k kk k
i v v v v
ii v v v v
whenever ( )k is a non-decreasing sequence converging to (0,1] , then the
family[ ( ), ( )], (0,1]v v , represent the level set of fuzzy number v in E. Conversely if
[ ( ), ( )], (0,1]v v , are level set of fuzzy number v E then the conditions (i) and (ii) hold true.
Definition 2.1
Let I be a real interval. A mapping :v I E is called a fuzzy process and we denoted the level set by
[ ] [ ( , ), ( , )].v t v t v t The Seikkala derivative 'v t of v is defined by
[ ' ] [ '( , ), '( , )],v t v t v t provided that is a equation defines a fuzzy number '( )v t E
Definition 2.2
Suppose u and v are fuzzy sets in E. Then their Hausdorff
: 0 ,D E E R [0, 1]
( , ) sup max ( ) ( ) , ( ) ( ) ,D u v u v u v
i.e ( , )D u v is maximal distance between α level sets of u and v .
III. Fuzzy Initial Value Problem Now we consider the initial value problem
1 ( 1)
1 ,( ) ( , , ,..., ), (0) ,..., (0)n n n
nx t t x x x x a x a (3.1)
where is a continuous mapping from nR R into R and (0 )ia i n are fuzzy numbers in E . The
mentioned Nth - order fuzzy differential equation by changing variables ( 1)
1 2( ) ( ), y ( ) ( ),..., y ( ) ( ),n
ny t x t t x t t x t
converts to the following fuzzy system
1 1 1
1
[0] [0]
1 1 1
( ) ( , ,..., y ),
( ) ( , ,..., y ),
(0) ,..., y (0) ,
n
n n n
n n n
y t f t y
y t f t y
y y a y a
(3.2)
where (1 i n)if are continuous mapping from nR R into R and
[0]
iy are fuzzy numbers in E with α -
level intervals. [0][0][0][ ] [ ( ), ( )] for i = 1,..., n and 0 1i ii
y y y
We call 1( ,..., y )T
ny y is a fuzzy solution of (3.2) on an interval I, if
1
1
( , ) = min ( , ,..., u ); u [ ( , ), ( , ) ] ( , ( , )),
( , ) = max ( , ,..., u ); u [ ( , ), ( , ) ] ( , ( , )) ,
i n j ji j i
i n ji j ij
y t f t u y t y t f t y t
y t f t u y t y t f t y t
(3.3)
and [0][0]
(0, ) ( ), (0, ) ( )i ii iy y y y
(3.4)
Numerical Solution Of Nth - Order Fuzzy Initial Value Problems By Fourth Order Runge-Kutta
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Thus for fixed we have a system of initial value problem in 2nR . If we can solve it (uniquely), we have only
to verify that the intervals, [ ( , ), ( , )]jj
y t y t define a fuzzy number ( ) .iy t E Now let
[0] [0] [0]
1( ) ( ( ),..., ( ))T
ny y y and
[0] [0] [0]
1( ) ( ( ),..., ( ))T
ny y y with respect to the above
mentioned indicators, system (3.2) can be written as with assumption
[0]
( ) ( , ( )),
(0) .n
y t F t y t
y y E
(3.5)
With assumption
( , ) [ ( , ), ( , )] y ( , ) [ ( , ), ( , )] wherey t y t y t and t y t y t
(3.6)
( , ) [ ( , ),..., ( , )] ,Ty t y t y t (3.7)
( , ) [ ( , ),..., ( , )] ,Ty t y t y t (3.8)
( , ) [ ( , ),..., ( , )] ,Ty t y t y t (3.9)
and with assumption ( , ( , )) [ ( , ( , ))), ( , ( , ))]F t y t F t y t F t y t , where
1( , ( , )) [ ( , ( , ))),..., ( , ( , ))]T
nF t y t f t y t f t y t , (3.10)
1( , ( , )) [ ( , ( , ))),..., ( , ( , ))]T
nF t y t f t y t f t y t , (3.11)
y (t) is a fuzzy solution of (3.5) on an interval I for all α (0,1], if
[0][0]
( , )) ( , ( , ));
( , )) ( , ( , ))
(0, ) ( ), (0, ) ( )
y t F t y t
y t F t y t
y y y y
(3.12)
or
[0]
( , )) ( , ( , )),
(0, ) ( ).
y t F t y t
y y
(3.13)
Now we show that under the assumptions for functions , i=1,..., nif for how we can
obtain a unique fuzzy solution for system (3.2).
Theorem 3.1
If 1( , ,..., u )i nf t u for 1,..., ni are continuous function of t and satisfies the
Lipschitz condition in 1 ( ,..., u )T
nu u in the region
, [0,1], i=1,..., niD t u t I u for with constant iL then the initial
value problem (3.2) has a unique fuzzy solution in each case.
Proof. Denote 11( , ) ( ,..., , ,..., )T T
nnG F F f f f f where
1( , ) min{ ( , ,..., ); u [ , ], 1,..., n}i n j ji jf t u f t u u y y for j ,
(3.14)
1( , ) max{ ( , ,..., ); u [ , ], 1,..., n}i i n j jjf t u f t u u y y for j ,
(3.15)
( , ) [ ( , ),..., ( , )] ,Ty t y t y t
Numerical Solution Of Nth - Order Fuzzy Initial Value Problems By Fourth Order Runge-Kutta
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Denote 2
1( , ) ( ,..., y , ,..., ) RT T n
n nny y y y y y . It can be shown that Lipschitz condition
of functions if imply
( , ) ( , )F t z F t z L z z
This guarantees the existence and uniqueness solution of
[0][0][0] 2 2
( )) ( , ( )),
(0) ( , ) t n
y t F t y t
y y y y R
(3.16)
Also for any continuous function [1] 2: ny R R the successive approximations
[ 1] [0] [ ]
0
( ) ( , ( )) , t 0, m =1,2,...
t
m my t y F s y s ds (3.17)
converge uniformly on closed subintervals of R to the solution of (3.16). In other
word we have the following successive approximations
[ 1] [0] [ ]( ) ( , ( )) , for i=1,..., n,
tm m
i i io
y t y f s y s ds
(3.18)
[ 1] [0][ ]( ) ( , ( )) , for i=1,..., n.
tm
m
i i i
o
y t y f s y s ds
(3.19)
By choosing[0][0][0] ( ( ), ( ))y y y in (3.16) we get a unique solution
( ) ( ( , ), ( , ))y t y t y t to (3.3) and (3.4) for each (0,1].
Next we will show that the ( , ) ( ( , ), ( , )),y t y t y t defines a fuzzy number innE for
each 0 ,t T i.e. that 1( ,..., y )T
ny y is a fuzzy solution to (3.14) and (3.15). Thus we
will show that the intervals [ ( , ), ( , )],ii
y t y t for 1,..., ni satisfy the conditions of Lemma (2.1). The
successive approximations [ ] [0] ,m ny y E
[ 1] [0] [ ]( ) ( , ( )) , t 0, m=1,2,...,
t
m m
o
y t y F s y s ds (3.20)
where the integrals are the fuzzy integrals, define a sequence of fuzzy