Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28) Spring 2011, pp 1-8
ISSN: 2008-594x
Improving the Solution of Fuzzy Differential Inclusions
T. Allahviranloo1*, Y. Nejatbakhsh2, M. Shafiee1, S.Hajihosseinlou2
1
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
2
Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran.
Received: 10 April 2010
Accepted: 22 June 2010
Abstract
In this paper a numerical method for solving 'fuzzy differential inclusions' is considered.
which is discussed in detail, more ever the extrapolation method for increasing the accuracy of
the approximate solution is applied. The method is illustrated by solving some linear and
nonlinear fuzzy initial value problems.
Keywords: Fuzzy Differential Inclusions, Runge-Kutta Method, Extrapolation Method.
1 Introduction
The topics of numerical methods for solving fuzzy differential equations have been
rapidly growing in recent years. The concept of fuzzy derivative was first introduced by
S. L. Chang, L.A. Zadeh in [4]. It was followed up by D. Dubois, H. Prade in [5], who
defined and used the extension principle. Other methods have been discussed by M. L
Puri, D. A. Ralescu in [6] and R. Goetschel, W. Voxman in [7]. The fuzzy differential
equation and the initial value problem were regularly treated by O. Kaleva in [8] and
[9], by S. Seikkala in [10],.... Recently Hüllermeier [12] suggested a different
formulation of FIVP based on a family of differential inclusions at each  -level,
0    1,
x (t )  [f (t , x (t ))] , x (0)  [x 0 ] ,
where now [f (, )] :[0,T ]  R n  cn , and  cn is the space of nonempty convex
compact subsets of R n . The numerical methods for solving fuzzy differential equations
are introduced in [1, 2, 3]. The paper is organized as follows:
In section 2 some basic definitions and results on fuzzy numbers along with a definition
of a fuzzy derivative, discussed by D. Vorobiev and S. Seikkala in [14] and Phil
Diamond in [16], are given. In section 3, we define the problem which is a fuzzy initial
value problem. The numerical method for fuzzy differential inclusions is discussed in
section 4. And the extrapolation method is discussed in section 5. A proposed algorithm
is illustrated by solving some examples in section 6 and conclusions are in section 7.
*Corresponding author
E-mail address: tofigh@allahviranloo.com
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T. Allahviranloo, et al, Improving the Solution of Fuzzy Differential Inclusion
2 Preliminaries
Denote by  n , the set of all nonempty compact subset of R n and by  cn the subset
of  n consisting of nonempty convex compact sets. Recall that
 (x , A ) = min x  a
is the distance of a point x  R
 ( A, B ) of A, B   n is defined as
n
from A  
aA
n
and that the Hausdorff separation
 ( A, B) = max  (a, B).
a A
Note that the notation is consistent, since  (a, B) =  ({a}, B) . Now,  is not a metric.
In fact,  ( A, B ) = 0 if and only if A  B. The Hausdorff metric d H on  n is defined
by
d H ( A, B) = max{ ( A, B),  ( B, A)}
n
and ( , d H ) is a complete metric space. An open  -neighborhood of A   n is the set
N ( A,  ) = {x  R n :  ( x, A) <  } = A  B n ,
where B n is the open unit ball in R n . A mapping F : R n   n is upper semicontinuous
(usc) at x0 if for all  > 0 there exits  =  ( , x0 ) such that
F ( x)  N ( F ( x0 ),  )) = F ( x0  B n )
for all x  N ( x0 ,  ). Let D n denote the set of upper semicontinuous normal fuzzy sets
on R n with the property of compact support. That is, u  D n , then u : R n  [0,1] is usc,
supp(u)= {x  R n : u( x) > 0} is compact and there exists at least one   supp (u ) for
which u ( ) = 1 . The  -level set of u, 0 <   1 is
[u ] = {x  R n : u (x )  }.
Clearly, for    , [u ]  [u ] . The level sets are nonempty from normality and
compact by usc and compact support. The metric d H is defined on D n as
d  (u ,v ) = sup{d H ([u ] ,[v ] ) : 0    1}, u ,v  D n
and (D n , d  ) is a complete metric space. Denote by E n the subset of fuzzy convex
elements of D n . The metric space ( E n , d  ) is also complete, [13]. Let I = [0, T ] be a
finite interval, y0  R n , and G be a map from I  R n into the set of all subsets of R n ,
one must find an absolutely continuous function x () on I such that
 x(t )  G(t , x(t ));

n
 x(0) = y0  Y0  R .
for almost all t  I ,
(1)
Recall that a continuous function x : I  Y  Rn is said to be absolutely continuous if
there exists a locally integrable function  such that
s
  ( )d = x(s)  x(t )
t
2
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28) Spring 2011, pp 1-8
for all t , s  I . The differential inclusion (1) is said to have a solution x(t ) on I . If x ()
is absolutely continuous, x(0) = y0 and x () satisfies the inclusion a.e. on I . Let
( y , ), be the reachable set that is,
0
( y , ) := {x : I  R
0
n
| x is solution of (2.1)}  C( I ),
and A( y0 , ) = {x( ) : x(.)  ( y0 , )} be the attainable set that is, the set of all points
x () that are ends of trajectories of (1). Obviously, A( y0 , ),0 <   T is a compact
subset of R n . As a rule, the set ( y0 , ), consist of more than one elements, that is we
have a bundle of trajectories. We use a finite difference scheme together with suitable
selection procedures resulting in a sequence of grid reachable sets
( y0 , t0 ), ( y0 , t1 ),, ( y0 , T ) , on a uniform grid 0 = t0  t1    t N = T with step
size h =
T  t0
= ti  ti 1 , i = 1, , N .
N
Definition 2.1 The fuzzy number X  E n is called pyramidal fuzzy number if its
 -level sets are n -dimensional rectangles for 0    1.
3 A fuzzy initial value problem
Let U  E n then U = (u1 ,, un )T where ui  E = E 1 , i = 1,, n. The parametric
form of ui is u i ( ) = (u i ( ), u i ( )),   [0,1] where u i ( ) = mi  i ( 1) is the
increasing function and u i ( ) = mi   i (1   ) is the decreasing function where mi is
core and i ,  i are left and right spreads respectively in (m i , i ,  i ) . Also ui (1) = ui (1) .
The parametric form of inclusion triangular's have the following increasing and
decreasing functions respectively,
u i ( ,  ) = m i  i (1   )(  1),
u i ( ,  ) = m i   i (1   )(1   ),  ,   [0,1].
(2)
The Convex Hull (C.H.) of the corners can be gotten with any variation of   [0,1] for
any   [0,1] . Then the components of one fuzzy triangular number U  E n are as
follows
U
= (u1, ,, un,  )
= (C.H .( A1 , B1 , C1 ), , C.H .( An , Bn , Cn ))T , 0    1.
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T. Allahviranloo, et al, Improving the Solution of Fuzzy Differential Inclusion
where
u i ,  = C .H .(Ai , B i ,C i ) =
{(u i ( ,  ),u i ( ,  )) | 0    1}.
0  1
Now let f : I  E n  E n and consider the fuzzy initial value problem (FIVP)
 x(t ) = f (t , x(t )), t  I = [0, T ],

n
 x(0) = Y0  E ,
(3)
interpreted as a family of differential inclusions. Set
{(f (t , x ,  ,  ), f (t , x ,  ,  )) | 0    1}:= F (t , x ,  )
0  1
and
{(Y 0 ( ,  )),Y 0 ( ,  ) | 0    1}:= Y 0 (  )
0  1
and identify the FIVP with the family of differential inclusions
x  (t )  F (t , x  (t );  ), t  I = [0,T ],

x  (0) =Y 0 Y 0 (  ), 0    1
where
F :   [0,1]   cn

and
(4)
is an open subset of
(0, Y0 (  )),  [0,1].
Denote the set of all solutions of (3) on I by
 (Y ( ), T )
0
I  En
containing
and the attainable set by
A  (Y0 (  ), T ) = {x(T ) : x()   (Y0 (  ), T )},
where f (random)  F (t , x;  ) and Y0 (random)  Y0 (  ) that
t  I = [0,T ],
x  (t ) = f (t , x  (t )),

n
x  (0) =Y 0  E , almost every where
(5)
that can be solved as  -levels. Let
Z T (R n ) = {x (.) C ([0,T ]; R n ) : x (.)  L ([0,T ]; R n )}.
These sets are not in general convex; they are acyclic which are stronger than simply
connected, [15].
Theorem 3.1 [16], Let Y0  E n and let  be an open set in R R n containing
{0}  supp (Y0 ) . Suppose that
f :   En
4
is usc and
F (t , x;  )   cn
for all
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28) Spring 2011, pp 1-8
(t , x,  )  R n1  [0,1]. Let the boundedness assumption hold for all y0  supp (Y0 ) and
the inclusion
x(t )  F (t , x;0), x(0)  supp (Y0 ).
Then the attainable sets A (Y0 (  ), T ),   [0,1], of the family of inclusions (3) are the
level sets of a fuzzy set A(Y0 , T )  D n . The solution sets
level sets of a fuzzy set
(Y , T ) included in Z
0
T
 (Y ( ), T )
0
of (3) are the
( R n ).
4 Numerical method
Let x (ti )  yi (  ) for all   [0,1], then the Numerical method for approximating
the reachable set of problem (5) is proposed as follows:
y 0,  Y 0 (  )  R n ,
y i 1,  =
(s   h (t i , s  ;  )), i = 0,
, N  1,  [0,1],
(6)
s  Y i (  )
where
{(Y i ( ,  )),Y i ( ,  ) | 0    1}:= Y i (  )
0  1
and
 (t i , s  ;  )) 
{( (t , s  ,  ,  ),  (t , s  ,  ,  )) | 0    1} = (t , s  ,  )
0  1
Lemma 4.1 Let a sequence of numbers {Wn }nN= 0 satisfy
| Wn1 | A | Wn |  B, 0  n  N  1,
for some given positive constants A and B . Then
| Wn | An | W0 |  B
An  1
, 0  n  N.
A 1
Proof. See [11].
Theorem 4.1 Let   C p 1 () in (6) be a compact convex valued mapping such that
satisfies Lipschitz condition in x with Lipschitz constant L > 0 and x  be a solution of
(3) then lim h0 y N (  ) = x (T ), for any   [0,1].
Proof. Let
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T. Allahviranloo, et al, Improving the Solution of Fuzzy Differential Inclusion

x (ti 1 ) =
x  X i (  )
( x  h (ti , x ;  )), a.e.
and
yi 1 (  ) =
  ( y  h (t , y ;  )  O(h
p 1
i
)),
y  Yi ( )
it is enough to prove
lim h 0 y i 1 (  )  x  (t i 1 ) = 0,
i = 0,
, N  1,   [0,1].
Since
yi 1 (  ) = yi (  )  h (ti , yi (  );  )  O(h 2 )
and
x (ti 1 )  x (ti )  h (ti , x (ti );  )
then
y i 1 (  )  x  (t i 1 )  y i (  )  x  (t i ) (1  Lh )  O (h p 1 ).
By using the lemma 4.1 for all ti in particular at T ,
1
y N (  )  x  (T )  O (h p )[e LT  1],
L
hence proof is completed.
5 Extrapolation method
According to Hullermeier’s interpretation, we have

x  (t ) = f t , x  t    F (t , x  (t );  ), t  I = [0,T ],

0   1

x  (0) = Y 0 Y 0 ( ),
(7)
Let H > 0 be a basic step size and choose a monotonically increasing sequence of
positive integers
n1 > n2 > n3 >    > nk
H
, i = 1,2,..., k .
ni
Choose a numerical method of order p and compute the solution of the initial value
problem by carrying out steps with step size hi to obtain:
and define the corresponding step sizes h1 > h2 > h3 >    > hk by hi =

  
Ti ,1 := y j ,h   = y j 1,h    hi t j 1 , y j 1,h  ; hi  O hip = 1, 2,3,..., k ,
i
i
i
6
j = 1, 2,3,..., k
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.8, No.1(28) Spring 2011, pp 1-8
Extrapolation is performed using the Aitken-Neville algorithm by building up a table of
values:
Ti , j   = Ti , j 1   
Ti , j 1    Ti 1, j 1  
i = 1,2,3,..., n,
p
 ni 

 1
n

i

j

1


j = 2,3,4,..., i
A dependency graph of the values illustrates the relationship as follows:
h1 : T11  
h2 : T21   T22  
h3 : T31   T32   T33  
h4 : T41   T42   T43   T44  
...:
...
...
...
...
...
6 Examples
Example 6.1 Consider the FIVP in E 1 , y = t  y, y(0) = Y0 where Y0 is a symmetric
triangular fuzzy number with support [-1,1].
Since  y  = { y  } is a singleton set in  c1 , then this is interpreted as a family of
differential inclusions
y (t ) = t  y (t ), y (0)  Y = (1   )[ 1,1],   [0,1],
which has the solution set
 (Y , t )
on
[0, t ]
comprising the functions
y (t ) = y (0)et  t  1, y (0) Y .
Now we obtain the reachable set and its approximation at t = 1 in table (6.1).
Table 6.1. The distance between reachable set and its approximations

Method
Rk 2
D T11 , Exact
Distance
0.7332


Rk 2
D T22 , Exact

0.3629
Example 6.2 Consider the FIVP in E 1 , y = exp(t )  2 y, y(0) = Y0 , where Y0 is a
symmetric triangular fuzzy number with support [-1,1].
This is interpreted as a family of differential inclusions
y (t ) = exp(t )  2 y (t ), y (0)  Y = (1   )[ 1,1],   [0,1],
which has the solution set
 (Y , t ) on [0, t ] comprising the functions
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T. Allahviranloo, et al, Improving the Solution of Fuzzy Differential Inclusion
1
1

y  (t ) =  y  (0)  e  2t  et
3
3

Now we obtain the reachable set and its approximation at t = 1 in table (6.2).
Table 6.2. The distance between reachable set and its approximations

Method
D T11Rk 2 , Exact
Distance
1.2943


D T22Rk 2 , Exact

0.7397
5 Conclusion
In this paper first of all, we approximated fuzzy differential inclusions by Runge-kutta
method of order 2, and then it was improved by extrapolation method. As a result,
approximated reachable set by extrapolation method is better than Runge-kutta method
of order 2.
References
[1] Allahviranloo, T., Ahmady, N., Ahmady, E., Numerical solution of fuzzy differential equations by
predictor–corrector method, Information Sciences 48 177 (2007) 1633-1647.
[2] Allahviranloo, T., Ahmady, E., Ahmady, N., Nth-order fuzzy linear differential equations,
Information Sciences 178 (2008) 1309-1324.
[3] Abbasbandy, S., Allahviranloo, T., L o pez-Pouso, O., Nieto, Numerical methods forfuzzy differential
inclusions, Computers and Mathematics with Applications 48 (2004) 1633-1641.
[4] Chang, S. L., Zadeh, L. A., On fuzzy mapping and control, IEEE Trans, Systems Man Cybernet. 2
(1972) 30-34.
[5] Dubois, D., Prade, H., Towards fuzzy differential calculus: Part 3, differentiation, Fuzzy Sets and
Systems 8 (1982) 225-233.
[6] Puri, M. L., Ralescu, D. A., Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 321-325.
[7] Goetschel, R., Voxman, W., Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986) 31-43.
[8] Kaleva, O., Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317.
[9] Kaleva, O., The Cauchy problem for Fuzzy differential equations, Fuzzy Sets and Systems 35 (1990)
389-396.
[10] Seikkala, S., On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987) 319-330.
[11] Friedman, M. Ma, Kandel, M. A., Numerical Solutions of fuzzy differential equations, Fuzzy Sets
and Systems 105 (1999) 133-138.
[12] Hüllermeier, E., An approach to modelling and simulation of uncertain dynamical systems, Int. J.
Uncertainty, Fuzziness & Knowledge-Bases Systems 35 (1990) 389-396.
[13] Diamond, P., Kloeden, P. Metric spaces of fuzzy sets, Singapore: World Scientific, 1994.
[14] Vorobiev, D., Seikkala, S., Towards the theory of fuzzy differential equations, FSS, 125 (2002) 231237.
[15] Blasi, F. S., Myjak, J., On the solution sets for differential inclusions , Bull Acad. Polon. Sci. vol.
33,pp. 17-23,1985.
[16] Diamond, P., Time dependent differential inclusions, cocycle attractors and fuzzy differential
equations, IEEE Trans. On Fuzzy Systems, 7 (1999) 734-740.
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