A Comparative Study of Numerical Solutions for the
First order Fuzzy Differential Equations using
Runge-Kutta methods with different kinds of Means
J.Christy kingston
#1D.Paul Dhayabaran
#2#1 Assistant Professor, PG and Research Department of Mathematics, Bishop Heber College (Autonomous), Tiruchirappalli -620 017
#2 Associate Professor & Principal, PG and Research Department of Mathematics, Bishop Heber College (Autonomous), Tiruchirappalli -620 017
Ph:99948-93364, Email: [email protected]
Abstract- The numerical solutions for the first order fuzzy initial value problems with different orders of Runge-Kutta methods based on various types of means using triangular and trapezoidal fuzzy number has been proposed by the authors using RK4EHeM, RK3MoCoHaM, RK3HaM, RK3CoHaM, RK2CoHaM, Ralston‟s Method, RK5HeM , RK4EHaM , RK4 Gill Method. The comparative study of all the proposed methods with h=0.1, 0.01, 0.001 has been carried out and it helps to list out the absolute error values of all the methods. From this comparison we concluded that the RK4EHaM and RK4 Gill method yield better solutions for the fuzzy Initial value problem. Among these two proposed methods, the RK4 Gill method is superior to RK4EHaM. Key words- Fuzzy initial value problems, RK4EHeM, RK3MoCoHaM, RK3HaM, RK3CoHaM, RK2CoHaM, Ralston‟s Method, RK5HeM , RK4EHaM , RK4 Gill Method, Triangular Fuzzy Number , Trapezoidal Fuzzy Number
1. INTRODUCTION
Fuzzy differential equations are a natural way to
model dynamical systems under uncertainty. First order linear fuzzy differential equations are one of the simplest fuzzy differential equations,which appear in many applications.The concept of fuzzy derivative was first introduced by S.L.Chang and L.A.Zadeh in[6]. The fuzzy differential equation and initial value problems were extensively studied by O.Kaleva[11,12] and by S.Seikkala[20].Recently many research papers are focused on numerical solution of fuzzy initial value problems (FIVPS).Numerical Solution of fuzzy differential equations has been introduced by M.Ma, M.Friedman, A.Kandel [14] through Euler method and by S.Abbasbandy and T.Allahviranloo [1] by Taylor method.Runge – Kutta methods have also been studied by authors [2,17].
This paper is organized as follows:In section 2 some basic results of fuzzy numbers and definitions of fuzzy derivative are given. In section 3 the fuzzy initial value problem is being discussed. Section 4 deals the numerical solutions for the first order fuzzy differential equations using Runge-Kutta methods with different kinds of means such as RK4EHeM, RK3MoCoHaM , RK3HaM, Ralston‟s Method,RK2CoHaM, RK3CoHaM, RK4 Gill Method , RK4EHaM, RK5HeM which was already proposed by the authors has been compared with one another to show their efficiency by using the
triangular and the trapezoidal fuzzy numbers with h = 0.1, 0.01, 0.001.
2. PRELIMINARIES 2.1. Fuzzy number
An arbitrary fuzzy number is represented by an ordered pair of functions
( ( ), ( ))
u r u r
for allr
0,1
which satisfy the following conditions.i)
u r
( )
is a bounded left continuous non-decreasing function over
0,1
with respect to any r.ii)
u r
( )
is a bounded right continuous non-decreasing function over
0,1
with respect to any r.iii)
( ( )
u r
u r
( ))
for allr
0,1
then ther-level set is
{
\ ( )
; 0
}
1
r
u
x u x
r
r
Clearly,
0
{
\ ( )
0
}
u
x u x
iscompact,which is a closed bounded interval and we denote by
u
r
( ( ), ( )).
u r u r
2.2.Triangular fuzzy number
A triangular fuzzy number „u‟ is a fuzzy set in E that is characterized by an ordered triple
3,
,
l c r
u
0
u u
l;
r
and
u
l
u
c .The membershipfunction of the triangular fuzzy number „u‟ is given by
;
( )
1
;
;
l l c c l c r c r r cx u
u
x
u
u
u
u x
x
u
u
x
u
x
u
u
u
we have :
(1)
u
0
if u
l
0
(2)
u
0
if u
l
0
(3)
u
0
if u
c
0
and(4)
u
0
if u
c
0
. 2.3. Trapezoidal fuzzy number:A trapezoidal fuzzy number u is defined by four real numbers
k
l m
n
, where the base of the trapezoidal is the interval [k, n] and its vertices at x =l, x = m. Trapezoidal fuzzy number will be written as u = ( k, l, m , n). The membership function for the trapezoidal fuzzy number u = ( k , l, m , n) is defined as the following :
1
x k
k
x
l
l
k
u x
l
x
m
x n
m
x
n
m n
we have :(1)
u
0
if k
0;
(2)u
0
if l
0;
(3)u
0
if m
0;
and (4)u
0
if n
0;
Let us denote RF by the class of all fuzzy
subsets of R (i.e. u : R → [0,1]) satisfying the following properties:
(i) u
RF, u is normal, i.e. x0 R with u(xo) = 1;(ii) u
RF, 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
RF, u is upper semi continuouson R;
(iv) { x
R; u( x ) > 0 } is compact, whereA denotes the closure of A. 2.4.
- Level setLet I be the real interval. A mapping
:
y I
E
is called a fuzzy process andits
- level Set is denoted by
y t
( )
[ (
y
t
;
)
,
y t
(
;
) ]
],t
I
,0
1
2.5. Seikkala derivativeThe Seikkala derivative
y t
'( )
of a fuzzyprocess is defined by
y t
'(
)
[ '(
y
t
;
)
,
'
y t
(
;
)
]
t
I
, 0 1 provided that this equation defines a fuzzy number, as in [24]2.6. Lemma
If the sequence of non-negative number
m0n n
W satisfy |Wn 1| A Wn| |B ,
0
n
N
1
for the given positiveconstants A and B, then
0
1
|
|
|
|
1
n nA
W
A
B
A
n
W
,0
n
N
The proof of Lemma (2.5) follows Lemma (2) of Ming Ma et al [109].
2.7. Lemma
If the sequence of non-negative numbers
m0n n
W
,
0 N n n
V
satisfy1
,
|
W
n| |
W
n|
A ma
x
{|
W
n| |
V
n|}
B
,
|
V
n1| |
V
n|
A ma
x
{|
W
n| |
,
V
n|
}
B
for the given positive constants A and B, then
n n n
U
W
V
,0
n
N
we have,0
1
1
n n nA
U
A U
B
A
0
n
N
where
A 1 2A
andB
2
B
. The proof of Lemma (2.6) follows by MingMa et al [109]. 2.8. Lemma
Let
F t u v
( , , )
andG t u v
( , , )
belong to'
(
F)
C R
and the partial derivatives of Fand G be bounded over
R
F. Then for arbitrarily fixed r,0
r
1
,0 1
2 1
( (
n)
,
(
n)
)
(1 2
)
D y t
y t
h L
C
where L is a bound of partial derivatives of
F and G, and
G t
N, ( ; ) , (
y t
Nr
y t
N 1; ) ,
0,1
C
Max
r
r
2.9. Theorem
Let
F t u v
( , , )
andG t u v
(
, ,
)
belong to'(
F)
C R
and the partial derivatives of Fnumerical solutions of y t(n1; )r and 1
(n ; )
y t r converge to the exact solutions
1
(
n; )
Y t
r
andY t
(
n1; )
r
uniformly in t.The proof of Theorem (1.7.16) follows Theorem (1) of Ming Ma et al [109].
2.10.Theorem
Let
F t u v
( , , )
and G t u v( , , )belong to1
(
)
F
C R
and the partial derivatives of Fand G be bounded over RFand
2
Lh
1
.Then for arbitrarily fixed
0
r
1
,the iterative numerical solutions of ( )( ; )
j n
y t r
and y( )j ( ; )t r n converge to the numerical
solutions
y t r
( ; )
n andy t r
( ; )
n in0 n N
t
t
t
, whenj
.3. FUZZY INITIAL VALUE PROBLEM Consider a first-order fuzzy initial
value differential equation is given by
00 0
( )
( , ( )),
,
( )
y t
f t y t
t
t T
y t
y
where y is a fuzzy
function of
t
,f t y
( , )
is a fuzzy function of the crisp variable' '
t
and the fuzzy variable y,y
'
is the fuzzy derivative of yand
0 0
( )
y t
y
is a trapezoidal or a trapezoidal shaped fuzzy number.We denote the fuzzy function y by
y
[
y
,
y
]
.It means that the r-level set ofy t
( )
for
0,
t
t T
is[
y t
( )
]
r
[ ( ;
y
t r
)
, ( ; ) ,
y t
r
]
y t
( )
0
r
0
; ,
0[ (
y t
r y t
) (
; )]
r
,r
(
0,
1
]
, we writef t y
( , )
[
f
( ,
t y
)
, (
f t
,
y
)
]
and
f t y
( , )
F t y y
[
,
,
]
,
( , )
f t y
G
[
t y y
, , ,
]
because ofy
f t y
( , )
we have
, ( );
F t y
[
,
(
;
)
, (
;
)
]
f t y t r
t
r y t r
, ( );
G t y
[
,
(
;
)
, (
;
)
]
f t y t r
t
r y t r
by using the extension principle, we have the membership function
f t y t
( , ( )( )
)
s
{ ( )( )
\
(
,
)}
,
sup y t
s
f t
s
R
so the fuzzy numberf t y t
, ( )
follows that
f t y t
(
,
( )
)
r
( , ( ); ), ( ,
[
f t y t r
f t y t r
( ); ) ,
]
r
(
0,
1
]
where
f t y t r
( , ( ); )
( , ) |
( ) }
{
rmin f t u
u y t
f t y t r
, ( );
=
( , ) |
( ) }
{
r
max f t u u y t
3.1. Definition
A function
f R
:
R
F is said to be fuzzy continuous function, if for an arbitrary fixedt
0
R
and
0
,
0
such that–
ot
t
D f t
( ), ( )
f t
0
exists.The fuzzy function considered are continuous in metric D and the continuity of
f t y t r
(
, (
);
)
guarantees the existence of the definition of, (
(
);
)
f t y t r
fort
t T
0,
andr
0,1
[10]. Therefore, the functions G and F can be definite too.4. A COMPARATIVE STUDY OF NUMERICAL SOLUTIONS FOR THE FIRST ORDER FUZZY DIFFERENTIAL EQUATIONS USING RUNGE-KUTTA METHODS WITH DIFFERENT KINDS OF MEANS
4.4.1 Comparison Of Approximate Solutions Of All The Proposed Methods Of Different Orders Of Runge-Kutta Method Based On Variety Of Means Using Triangular Fuzzy Number When H=0.1
The comparative study helps to list out the absolute error values for the first order fuzzy initial value problems with different orders of Runge-Kutta method based on various types of means.
Table: 4.4.1shows the absolute error values for thefirst order fuzzy initial value problems with different orders of Runge-Kutta method based on various types of means using triangular fuzzy number
Consider the fuzzy initial value problem
( )
( ),
0
(0)
(0.75 0.25 ,1.125 0.125 )
y t
y t
t
y
r
r
r
RK4EHeM (h=0.1) RK3MoCoHaM (h=0.1) RK3HaM(h=0.1)
;
(
)
y t r
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
0.0 1.693922e-001 3.803880e-002 2.899301e-002 4.348952e-002 1.254071e-002 1.881106e-002
0.1 1.616662e-001 4.286250e-002 2.995944e-002 4.300630e-002 1.295873e-002 1.860205e-002
0.2 1.537639e-001 4.766061e-002 3.092588e-002 4.252308e-002 1.337676e-002 1.839304e-002
0.3 1.456936e-001 5.243267e-002 3.189231e-002 4.203987e-002 1.379478e-002 1.818403e-002
0.4 1.374626e-001 5.717824e-002 3.285875e-002 4.155665e-002 1.421280e-002 1.797502e-002
0.5 1.290783e-001 6.189686e-002 3.382518e-002 4.107343e-002 1.463083e-002 1.776600e-002
0.6 1.205470e-001 6.658804e-002 3.479161e-002 4.059022e-002 1.504885e-002 1.755699e-002
0.7 1.118748e-001 7.125129e-002 3.575805e-002 4.010700e-002 1.546687e-002 1.734798e-002
0.8 1.030675e-001 7.588611e-002 3.672448e-002 3.962378e-002 1.588490e-002 1.713897e-002
0.9 9.413038e-002 8.049198e-002 3.769091e-002 3.914057e-002 1.630292e-002 1.692996e-002 1.0 8.506835e-002 8.506835e-002 3.865735e-002 3.865735e-002 1.672094e-002 1.672094e-002
r
Ralston’s Method(h=0.1) RK2CoHaM(h=0.1) RK3CoHaM(h=0.1)
;
(
)
y t r
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
0.0 3.150736e-003 4.726105e-003 1.239558e-003 1.859337e-003 1.072212e-004 1.608318e-004 0.1 3.255761e-003 4.673592e-003 1.280876e-003 1.838677e-003 1.107953e-004 1.590448e-004 0.2 3.360785e-003 4.621080e-003 1.322195e-003 1.818018e-003 1.143693e-004 1.572578e-004 0.3 3.465810e-003 4.568568e-003 1.363513e-003 1.797359e-003 1.179433e-004 1.554708e-004 0.4 3.570835e-003 4.516055e-003 1.404832e-003 1.776699e-003 1.215174e-004 1.536837e-004 0.5 3.675859e-003 4.463543e-003 1.446151e-003 1.756040e-003 1.250914e-004 1.518967e-004 0.6 3.780884e-003 4.411031e-003 1.487469e-003 1.735381e-003 1.286655e-004 1.501097e-004 0.7 3.885908e-003 4.358519e-003 1.528788e-003 1.714721e-003 1.322395e-004 1.483227e-004 0.8 3.990933e-003 4.306006e-003 1.570106e-003 1.694062e-003 1.358135e-004 1.465357e-004 0.9 4.095957e-003 4.253494e-003 1.611425e-003 1.673403e-003 1.393876e-004 1.447486e-004 1.0 4.200982e-003 4.200982e-003 1.652744e-003 1.652744e-003 1.429616e-004 1.429616e-004
r
RK5HeM (h=0.1) RK4EHaM (h=0.1) RK4 Gill Method (h=0.1)
;
(
)
y t r
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
0.0 8.857221e-005 1.328583e-004 4.314332e-006 6.471498e-006 1.563243e-006 2.344864e-006
0.1 9.152462e-005 1.313821e-004 4.458143e-006 6.399592e-006 1.615351e-006 2.318810e-006
0.2 9.447702e-005 1.299059e-004 4.601954e-006 6.327687e-006 1.667459e-006 2.292756e-006
0.3 9.742943e-005 1.284297e-004 4.745765e-006 6.255781e-006 1.719567e-006 2.266702e-006
0.4 1.003818e-004 1.269535e-004 4.889576e-006 6.183876e-006 1.771675e-006 2.240648e-006
0.5 1.033342e-004 1.254773e-004 5.033387e-006 6.111970e-006 1.823783e-006 2.214594e-006
0.6 1.062867e-004 1.240011e-004 5.177198e-006 6.040065e-006 1.875891e-006 2.188540e-006
0.7 1.092391e-004 1.225249e-004 5.321009e-006 5.968159e-006 1.928000e-006 2.162486e-006
Fig: 4.4.1 Absolute error values of all the proposed methods when t=1 and h=0.1 for the problem 4.4.1
4.4.2 Comparison Of Approximate Solutions Of All The Proposed Methods Of Different Orders Of Runge-Kutta Method Based On Variety Of Means Using Trapezoidal Fuzzy Number When H=0.1
Thefirst order fuzzy initial value problems with different orders of Runge-Kutta method based on various types of means has been compared and this study helps to tabulate the absolute error values.
Table: 4.4.2 shows the absolute error values for thefirst order fuzzy initial value problems with different orders of Runge-Kutta method based on various types of means using trapezoidal fuzzy number Consider the Fuzzy Initial value problem
( )
( ),
0
(0)
(0.8 0.125 ,1.1 0.1 )
y t
y t
t
y
r
r
Table: 4.4.2 Absolute error values of all the proposed methods when t=1 and h=0.1 for the problem 4.4.2 0.9 1.151439e-004 1.195725e-004 5.608632e-006 5.824348e-006 2.032216e-006 2.110378e-006
1.0 1.180963e-004 1.180963e-004 5.752443e-006 5.752443e-006 2.084324e-006 2.084324e-006
r
RK4EHeM (h=0.1) RK3MoCoHaM (h=0.1) RK3HaM (h=0.1)
;
(
)
y t r
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
0.0 1.537639e-001 4.766061e-002 3.092588e-002 4.252308e-002 1.337676e-002 1.839304e-002 0.1 1.497493e-001 5.148036e-002 3.140910e-002 4.213651e-002 1.358577e-002 1.822583e-002 0.2 1.456936e-001 5.528322e-002 3.189231e-002 4.174994e-002 1.379478e-002 1.805862e-002 0.3 1.415977e-001 5.906895e-002 3.237553e-002 4.136336e-002 1.400379e-002 1.789141e-002 0.4 1.374626e-001 6.283730e-002 3.285875e-002 4.097679e-002 1.421280e-002 1.772420e-002 0.5 1.332892e-001 6.658804e-002 3.334196e-002 4.059022e-002 1.442181e-002 1.755699e-002 0.6 1.290783e-001 7.032090e-002 3.382518e-002 4.020364e-002 1.463083e-002 1.738978e-002 0.7 1.248306e-001 7.403563e-002 3.430840e-002 3.981707e-002 1.483984e-002 1.722257e-002 0.8 1.205470e-001 7.773196e-002 3.479161e-002 3.943050e-002 1.504885e-002 1.705536e-002 0.9 1.162281e-001 8.140963e-002 3.527483e-002 3.904392e-002 1.525786e-002 1.688815e-002 1.0 1.118748e-001 8.506835e-002 3.575805e-002 3.865735e-002 1.546687e-002 1.672094e-002
r
Ralston’s Method (h=0.1) RK2CoHaM (h=0.1) RK3CoHaM (h=0.1)
;
(
)
y t r
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
Fig: 4.4.2 Absolute error values of all the proposed methods when t=1 and h=0.1 for the problem 4.4.2
5. CONCLUSION
In this chapter the numerical solutions of first order fuzzy differential equations have been obtained by Ralston‟s method and RK4 Gill method. The accuracy of the proposed methods has been demonstrated by suitable example. From the numerical example 5.4.1 and 5.4.2 it has been observed that by lowering the step size „h‟, the approximate solutions at different points of the solution curve tend to be nearer with the exact solutions. To execute the validity of the proposed methods, the numerical solutions for the first order fuzzy differential equations with triangular and trapezoidal fuzzy numbers have been determined to ascertain the accuracy of the methods discussed.
The two proposed methods namely, Ralston‟s method and fourth order Runge-Kutta Gill method have been studied for a common numerical illustration in section 5.4.3 to 5.4.6 using triangular and trapezoidal fuzzy numbers respectively. From the table 5.4.3 it is evident that when the step size h=0.1, the absolute error of RK4 Gill method is to the precision of , but
in the case of Ralston‟s method when the step size h=0.1 the absolute error is to the precision of
3
10
. Hence for the step size h=0.1 there is no much accuracy gained between the methods. It has been viewed from the table values 5.4.3 to 5.4.8, that RK4 Gill method gives better results than Ralston‟s method when the step size are of h=0.1, h=0.01 and h=0.001.By Comparing the results from the tables 5.4.3 to 5.4.8 and figures 5.4.3 and 5.4.4,when h=0.001 the absolute errors have been drastically reduced to
10
7for Ralston‟s method also for RK4 Gill method it is10
14.Hence it is concluded that as step size decreased to h=0.01 and h=0.001 it can be observed that the absolute error also decreases appreciably.Finally from the table 5.5.1 to 5.5.6 the absolute errors of all the proposed methods have been compared, from this comparison we concluded that the RK4EHaM and RK4 Gill method yield better solutions for the linear fuzzy Initial value problem. Among these two proposed methods, the RK4 Gill Method is superior to RK4EHaM and hence this
6
10
0.7 3.728371e-003 4.327011e-003 1.466810e-003 1.702326e-003 1.268784e-004 1.472505e-004 0.8 3.780884e-003 4.285001e-003 1.487469e-003 1.685798e-003 1.286655e-004 1.458209e-004 0.9 3.833396e-003 4.242992e-003 1.508129e-003 1.669271e-003 1.304525e-004 1.443912e-004 1.0 3.885908e-003 4.200982e-003 1.528788e-003 1.652744e-003 1.322395e-004 1.429616e-004
r
RK5HeM (h=0.1) RK4EHaM (h=0.1) RK4 Gill Method (h=0.1)
;
(
)
y t r
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
y t r
(
;
)
proposed method suits very well to solve the first order fuzzy differential equations.
REFERENCES
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