Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013
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First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment Sankar Prasad Mondal*, Tapan Kumar Roy Department of Mathematics, Bengal Engineering and Science University,Shibpur,Howrah711103, West Bengal, India *Corresponding authormail: sankar.res07@gmail.com Abstract In this paper, the solution procedure of a first order linear non homogeneous ordinary differential equation in fuzzy environment is described. It is discussed for three different cases. They are i) Ordinary Differential Equation with initial value as a fuzzy number, ii) Ordinary Differential Equation with coefficient as a fuzzy number and iii) Ordinary Differential Equation with initial value and coefficient are fuzzy numbers. Here fuzzy numbers are taken as Generalized Triangular Fuzzy Numbers (GTFNs). An elementary application of population dynamics model is illustrated with numerical example. Keywords: Fuzzy Ordinary Differential Equation (FODE), Generalized Triangular fuzzy number (GTFN), strong solution. 1. Introduction: The idea of fuzzy number and fuzzy arithmetic were first introduced by Zadeh [11] and Dubois and Parade [5]. The term “Fuzzy Differential Equation (FDE)” was conceptualized in 1978 by Kandel and Byatt [1] and right after two years, a larger version was published [2]. Kaleva [16] and Seikkala [17] are the first persons who formulated FDE. Kaleva showed the Cauchy problem of fuzzy sets in which the Peano theorem is valid. The Generalization of the Hukuhara derivative which is based on fuzzy derivative was defined by Seikkala, and brought that the fuzzy initial value problem (FIVP) , , 0 which has a unique fuzzy solution when f satisfies the generalized Lipschitz condition which confirms a unique solution of the deterministic initial value problem. Fuzzy differential equation and initial value problem were extensively treated by other researchers (see [4,18,19,13,8,9,10]). Recently FDE has also used in many models such as HIV model [7], decay model [6], predatorprey model [15], population models [12] ,civil engineering [14], modeling hydraulic [3] etc. In this paper we have considered 1st order linear non homogeneous fuzzy ordinary differential equation and have described its solution procedure in section3. In section4 we have applied it in a biomathematical model.
2. Preliminary concept:
Definition 2.1: Fuzzy Set: Let X be a universal set. The fuzzy set ⊆ is defined by the set of tuples as , : : → 0,1 . The membership function of a fuzzy set is a function with mapping : → 0,1 . So every element x in X has membership degree in 0,1 which is a real number. As closer the value of is to 1, so much x belongs to . implies relevance of in is greater than the relevance of in . If = 1, then we say exactly belongs to , if = 0 we say does not belong to , and if = a where 0 < a < 1. We say the membership value of in is a. When is always equal to 1 or 0 we get a crisp (classical) subset of X. Here the term “crisp” means not fuzzy. A crisp set is a classical set. A crisp number is a real number.
85
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013
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Definition 2.2: !Level or !cut of a fuzzy set: Let X be an universal set. Let , ⊆ be a fuzzy set. " cut of the fuzzy set is a crisp set. It is denoted by # . It is defined as # $ ∶ & " ∀ ∈ ) Note: # is a crisp set with its characteristic function * + (x) defined as * + (x) = 1 & " ∀ ∈ = 0 otherwise.
Definition 2.3: Convex fuzzy set: A fuzzy set , ⊆ is called convex fuzzy set if all sets i.e. for every element ∈ # and ∈ # and for every " ∈ 0,1 , ,  1 . , ∈ # Otherwise the fuzzy set is called non convex fuzzy set.
#
are convex ∀ , ∈ 0,1 .
Definition 2.4: Fuzzy Number: ∈ / 0 is called a fuzzy number where R denotes the set of whole real numbers if i. is normal i.e. ∈ 0 exists such that 1. ii. ∀! ∈ 0,1 # is a closed interval.
1 then If is a fuzzy number then is a convex fuzzy set and if non increasing for & . The membership function of a fuzzy number 2 , 2 , 23 , 24 is defined by 1 , ∈ 2 , 23 6 ϕ 5 8 , 2 1 1 2 0 , 23 1 1 24 Where L(x) denotes an increasing function and 0 9 8 010
9 1.
is non decreasing for
1
and
1 1 and R(x) denotes a decreasing function and
Definition 2.5: Generalized Fuzzy number (GFN): Generalized Fuzzy number as 2 , 2 , 23 , 24 ;: where 0 9 : 1 1, and 2 , 2 , 23 , 24 (2 9 2 9 23 9 24 are real numbers. The generalized fuzzy number is a fuzzy subset of real line R, whose membership function satisfies the following conditions: 1) 2) 3) 4) 5) 6)
: R → [0, 1] 0 for 1 2 is strictly increasing function for 2 1 1 2 : for 2 1 1 23 is strictly decreasing function for 23 1 1 24 0 for 24 1
Fig2.1: Membership function of a GFN Definition 2.6: Generalized triangular fuzzy number (GTFN) : A Generalized Fuzzy number is called a Generalized Triangular Fuzzy Number if it is defined by 2 , 2 , 23 ;: its membership function is given by 86
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013 0 , 1 2 > ?@AB < : AC@AB , 2 1 1 2 : , 2 = : AD@? , 2 1 1 2 3 < AD@AC ; 0 , & 23 ?@AB E2 FEGH F: , :, : AC @AB
or,
AD @?
AD @AC
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I , 0I
Definition 2.7: Fuzzy ordinary differential equation (FODE): Consider a simple 1st Order Linear nonhomogeneous Ordinary Differential Equation (ODE) as follows: J? JK
L 
M
with initial condition
The above ODE is called FODE if any one of the following three cases holds:
(i) Only M is a generalized fuzzy number (TypeI). (ii) Only k is a generalized fuzzy number (TypeII). (iii) Both k and M are generalized fuzzy numbers (TypeIII). Definition 2.8: Strong and Weak solution of FODE:
Consider the 1st order linear non homogeneous fuzzy ordinary differential equation Here k or (and) be generalized fuzzy number(s).
J? JK
L 
with
Let the solution of the above FODE be N and its "cut be ," ," , ," . If ," 1 , " ∀ " ∈ 0, : where 0 9 : 1 1 then N is called strong solution otherwise N weak solution and in that case the "cut of the solution is given by ," min$ ," , , " ) , max$ ," , ," ) . 3. Solution Procedure of 1st Order Linear Non Homogeneous FODE
.
is called
The solution procedure of 1st order linear non homogeneous FODE of TypeI, TypeII and TypeIII are described. Here fuzzy numbers are taken as GTFNs. 3.1. Solution Procedure of 1st Order Linear Non Homogeneous FODE of TypeI Consider the initial value problem
with Fuzzy Initial Condition (FIC) N Let N
Let
and MW
J? JK
V MW
be a solution of FODE (3.1.1) .
," #
where ab\
," ,
," ,
M . M and db\
,"
,"
be the "cut of N YM 
M3 . M
Here we solve the given problem for L
#Z[\ ]
M , M , M3 ; :
, M3 .
#^[\ ]
_ ∀ " ` 0, : , 0 9 : 1 1
0 and L 9 0 respecively. 87
………….(3.1.1)
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013 Case 3.1.1. When e
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0
The FODE (3.1.1) becomes a system of linear ODE L
J?f K,∝ JK
," 
h
for G
1,2
,"
with initial condition
M 
The solution of (3.1.2) is ,"
.
and Now
n
,"
n#
?\ j
,:
and
k
?\
.
j
?\ j
,"
.
 M Z[ \
?\ j
]
mj
k
?\ j
#Z[\ ]
l mj
#Z[\ ]
 k j\  M3 . ?
…………..(3.1.2)
K@K\
#^[\ ]
K@K\
 M l mj
n
M3 .
#^[\ ]
….……….(3.1.3)
l mj
0 ,
,"
and
K@K\
n#
.
………….(3.1.4)
,"
.
,: .
K@K\
^[\ ]
m j K@K\ 9 0
So the solution of FODE (3.1.1) is a generalized fuzzy number N . The "cut of the solution is ,"
.
Y
?\ j
?\
Case 3.1.2. when e 9 0 Let L
j
 FM 
#Z[\ ]
I,
?\ j
 M3 .
#^[\ ]
_ mj
K@K\
.
.E where m is a positive real number.
Then the FODE (3.1.1) becomes a system of ODE as follows pqB r,∝ s@t?C K,# u?\ pr pqB r,∝ s@t?C K,# u?\ pr
v
………… ,"
with initial condition
M 
The solution of (3.1.5) is ,"
?\
 k.
?\
 M  M3 
]
?\
 k.
?\
 M  M3 
]
t
and
,"
Here n
n# n
n#
t
,"
,"
]
]
t
t
ab\ . db\ m @t
ab\ . db\ m @t
#Z[\ ]
and
,"
M3 .
#^[\ ]
.
#
ab\ . db\ l m @t
K@K\
 F . 1I ab\  db\ m t
K@ K\
#
ab\ . db\ l m @t
K@K\
. F . 1I ab\  db\ m t
K@K\
K@K\
K@K\

.
]
]
ab\  db\ m t
ab\  db\ m t
#
] #
]
K@K\
88
K@K\
,
.
…(3.1.5)
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013 and
,:
?\ t
 F.
?\ t
 M I m @t
Here three cases arise. Case1: When wxy ∴ n#
,"
n
,:
and
]
zxy i.e., x {y
ab\  db\ m t
,:
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,:
K@K\
x , x} , x~ ; • is a symmetric GTFN 0 , n# n
K@K\
,"
.
]
ab\  db\ m t
K@K\
90
So the solution of the FODE (3.1.1) is a strong solution. {y Case2: When wxy 9 zxy i.e., x
Here but
n
,"
n
n#
9 0 and
,"
n#
0 implies
,:
x , x} , x~ ; • is a non symmetric GTFN

t
,:
log „
^[\ @Z[\ Z[\ u^[\
….
So the solution of the FODE (3.1.1) is a strong solution if Case3: When wxy
Here but
n
,"
n
n#
,"
n#
zxy i.e., x {y
9 0 and
9 0 implies
,:

t
log „
^[\ @Z[\ Z[\ u^[\
x , x} , x~ ; • is a non symmetric GTFN 
t
,:
log „
^[\ @Z[\ Z[\ u^[\
….
So the solution of the FODE (3.1.1) is a strong solution if

t
log „
… .
Z[\ @^[\ Z[\ u^[\
… .
3.2. Solution Procedure of 1st Order Linear Non Homogeneous FODE of TypeII J? JK
Consider the initial value problem M . Here L
with IC Let N Let L
#
L 
………….(3.2.1)
† , † , †3 ; , .
be the solution of FODE (3.2.1)
,"
where aj
," ,
L " ,L "
† . † and dj
,"
Y† 
be the "cut of the solution and the "cut of L be
#Z‡ ˆ
, †3 .
†3 . † .
Here we solve the given problem for L ‰ Case 3.2.1: when e
0
#^‡ ˆ
_ ∀ " ` 0, , , 0 9 , 1 1
0 and L 9 0 respecively.
The FODE (3.2.1) becomes a system of linear ODE 89
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013 Lh "
J?f K,∝ JK
with IC
h
M.
for G
,∝ 
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1,2
…………..(3.2.2)
The solution of (3.2.1) ,"
.
,"
.
?\
 •M 
?\
 •M 
+‹ ŠB u ‡ Œ
?\
+‹ ŠB u ‡ Œ
vm
+‹ ŠB u ‡ K@K\ Œ
and +Ž ŠD @ ‡ Œ
?\
+Ž ŠD @ ‡ Œ
vm
ŠD @
#Z•
, †3 .
+Ž‡ K@K\ Œ
.
‰90 Case 3.2.2: when • Let L .E W , where E W † , † , †3 ; , is a positive GTFN.
So E W
#
where at
E " ,E "
† . † and dt
Y† 
†3 . †
ˆ
#^• ˆ
_ ∀ " ` 0, , , 0 9 , 1 1
Then the FODE (3.2.1) becomes a system of ODE as follows pqB r,∝ s@tC # ?C K,# u?\ pr pqC r,∝ s@tB # ?B K,# u?\ pr
with IC
M
v
……………(3.2.3)
Thus the solution is ,"
"d > †3 . t 1 , “. M ‘1 . ’ "at 2= † , ;
"d > †3 . t 1 , “. M ‘1  ’ "at 2= †  , ;
,"
› #Z• 1 1 • ˜ m – ŠB u ˆ . "a †  t – †  "at † . "dt š , 3 ” , , —™
#^ ŠD @ • K@K\ ˆ
› #Z #^ 1 1 @–FŠB u • I ŠD @ • • ˜ ˆ ˆ m "a †  ,t – †  "at † . "dt š 3 ” , , —™
"at > "d †3 . t 1 †  , , “. . ’ M ‘1 . ’ "at 2 †3 . "dt = † , , ;
K@K\

"a †  ,t
› #Z• #^• 1 1 • ˜ m – ŠB u ˆ FŠD @ ˆ I K@K\ . "a †  t – †  "at † . "dt š , 3 ” , , —™ 90
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013  œ
+‹ ŠB u •
Œ +Ž ŠD @ • Œ
5M •1  œ
+Ž• Œ +‹ ŠB u • Œ
ŠD @
ž.
•
+‹ ŠB u • Œ

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+‹ +Ž – ŠB u • ŠD @ • Œ
Œ
žŸ m
+‹ +Ž @–FŠB u •I ŠD @ • K@K\ Œ Œ

?\
+Ž• Œ
ŠD @
3.3. Solution Procedure of 1st Order Linear Non Homogeneous FODE of TypeIII
With fuzzy IC N Let N Let
MW
‰ V
J? JK
Consider the initial value problem
M , M , M3 ; : , where L
† , † , †3 ; ,
be the solution of FODE (3.3.1) .
,"
Also L
where aj
," ,
Y† 
#
#Z‡ ˆ
,"
, †3 .
† . † and dj
and MW
#
," ,
Let
min ,, :
ˆ
be the "cut of the solution.
_ ∀ " ` 0, , , 0 9 , 1 1
†3 . †
,"
M . M and db\
where ab\
#^‡
YM 
M3 . M
Here we solve the given problem for L ‰ Case I: when e
………….(3.3.1)
#Z[\ ]
, M3 .
#^[\ ]
_ ∀ " ` 0, : , 0 9 : 1 1
0 and L 9 0 respecively.
0
The FODE (3.1.1) becomes a system of linear ODE Lh
J?f K,∝ JK
h
," 
for G
,"
with initial condition
1,2
M 
…………..(3.3.2)
#Z[\ ]
,"
and
M3 .
Therefore the solution is of (3.3.1)
And
,"
.
,"
.
?\
+‹ ŠB u ‡ ¡
 •FM 
?\
+Ž ŠD @ ‡ ¡
‰90 Case II: when e
Let L
.E W where E W
Then E W
#
Y† 
#Z[\ ¢
 •FM3 .
#Z• ˆ
I
#^[\ ¢
I
?\
+‹ ŠB u ‡ ¡
?\
v m
+Ž ŠD @ ‡ ¡
+‹ FŠB u ‡ I K@K\
vm
¡
+Ž‡ I K@K\ ¡
FŠD @
† , † , †3 ; , is a positive GTFN.
, †3 .
#^• ˆ
_ ∀ " ` 0, , , 0 9 , 1 1 91
#^[\ ]
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013
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min ,, :
Let
Then the FODE (3.3.1) becomes a system of ODE as follows pqB r,∝ s@tC £ ?C K,# u?\ pr pqC r,∝ s@tB £ ?B K,# u?\ pr
,"
with IC
v
M 
………………(3.3.3) #Z[\ ]
,"
and
M3 .
#^[\ ]
.
Therefore the solution of (3.3.1) is ,"
"d F†3 . t I > "a "db\ 1 <•M  b\ . ’ 1 1 ˜ M3 . • ˜ "at . ¤ ¥. F† I "at 2= "a "dt t F† I –F† I F† . I < 3 ” — 0 ;” —
"d > F†3 . t I "ab\ 1 ‘M ’ "a 2= F†  t I ;
and
M3 .
,"
"at > "d †3 . t "ab\ 1 †  ’ ‘M ’ "a 2 †3 . "dt = †  t ; ?\
+Ž• ¡
ŠD @
š < ™
m
#Z #^ –FŠBu • IFŠD @ •I K@K\ ¢ ¢
› #Z "dM0 1 1 @– ŠBu ¢• ˜ “.• m "a : "a "d š †  t – †  t †3 . t ” — ™
"at > "d †3 . t "ab\ 1 † . ’ ‘M . ’ "a 2 †3 . "dt = †  t ;

› <
M3 . M3 .
"db\ "db\
1 1 ˜ . "at "a "d t † – † †3 . t ” —
“.•
1 1 ˜ "at "a "d t † – † †3 . t ” —
“.•
› š ™ › š ™
#^ ŠD@ ¢• K@K\

† 
"at
m
#Z #^ – ŠB u • ŠD@ • K@K\ ¢ ¢
m
#Z #^ @– ŠBu ¢• ŠD @ ¢• K@K\
.
4. Application: Population Dynamics Model Bacteria are being cultured for the production of medication. Without harvesting the bacteria, the rate of change of the population is proportional to its current population, with a proportionality constant L per hour. Also, the bacteria are being harvested at a rate of § per hour. If there are initially ¨ bacteria in the culture, solve the initial value
problem:
J© JK
L ¨ . §, ¨ 0 (i) ¨ (ii) ¨ ¯ (iii) ¨
¨ when
7800,8000,8150; 0.8 and L 0.2, § 1000, 8000 and L 0.17,0.2,0.24; 0.6 , § 1000, 7900,8000,8200; 0.7 and L 0.16,0.2,0.23; 0.8 , § 92
1000.
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013
Solution: (i) Here ¨
7800  250", 8150 . 187.5" .
#
Therefore solution of the model ¨
,"
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5000  2800  250" m
. K
and ¨
,"
5000  3150 . 187.5" m
. K
.
Table1: Value of ± ², ! and ±} ², ! for different ! and t=3 " 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
¨ ," 10101.9326 10147.4856 10193.0386 10238.5916 10284.1445 10329.6975 10375.2505 10420.8034 10466.3564
¨ ," 10739.6742 10705.5095 10671.3448 10637.1800 10603.0153 10568.8506 10534.6859 10500.5211 10466.3564
From above table1 we see that for this particular value of t=3, ¨ , " is an increasing function, ¨ , " is a decreasing function and ¨ , 0.8 ¨ , 0.8 10466.3564. So Solution of above model for particular value of t is a strong solution. (ii) Here L
#
0.17  0.050", 0.26 . 0.066" .
Therefore solution of the model is ¨
,"
and ¨
,"
. ³u . ´ #
 k8000 .
. 4@ . µµ#
. ³u . ´ #
 k8000 .
lm
. 4@ . µµ#
. ³u . ´ # K
lm
. 4@ . µµ# K
.
Table2: Value of ± ², ! and ±} ², ! for different ! and t=3 " 0 0.1 0.2 0.3 0.4 0.5 0.6
¨ ," 9408.8519 9578.1917 9750.2390 9925.0361 10102.6256 10283.0510 10466.3564
93
¨ ," 12041.9940 11765.8600 11495.6109 11231.1346 10972.3224 10719.0687 10471.2714
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From above table1 we see that for this particular value of t=3, ¨ , " is an increasing function, ¨ , " is a decreasing function and ¨ , 0.8 9 ¨ , 0.8 . So Solution of above model for particular value of t is a strong solution.
Here
(iii) L
#
¯ 0.16  0.057", 0.23 . 0.042" and ¨
#
Therefore solution of the model is ¨
,"
and ¨
,"
. µu . ´³#
 k 7900  142.8" .
. 3@ . 4 #
7900  142.8", 8200 . 285.7"
. µu . ´³#
 k 8200 . 285.7" .
l m
. 3@ . 4 #
. µu . ´³# K
lm
. 3@ . 4 # K
Table3: Value of ± ², ! and ±} ², ! for different ! and t=3 " 0 0.1 0.2 0.3
¨ ," 8916.5228 9124.4346 9336.5274 9552.8820
¨ ," 12027.9651 11797.2103 11570.1611 11346.7614
0.4 0.5 0.6
9773.5808 9998.7076 10228.3479
11126.9561 10910.6908 10697.9120
0.7
10462.5889
10488.5669
From above table1 we see that for this particular value of t=3, ¨ , " is an increasing function, ¨ , " is a decreasing function and ¨ , 0.8 9 ¨ , 0.8 . So Solution of above model for particular value of t is a strong solution. 5. Conclusion: In this paper we have solved a first order linear non homogeneous ordinary differential equation in fuzzy environment. Here fuzzy numbers are taken as GTFNs. We have also discussed three possible cases. For further work the same problem can be solved by Generalized LR type Fuzzy Number. This process can be applied for any economical or biomathematical model and problems in engineering and physical sciences. References: [1] A. Kandel and W. J. Byatt, “Fuzzy differential equations,” in Proceedings of the International Conference on Cybernetics and Society, pp. 1213–1216, Tokyo, Japan, 1978. [2] A. Kandel and W. J. Byatt, “Fuzzy processes,” Fuzzy Sets and Systems, vol. 4, no. 2, pp. 117–152, 1980. [3] A. Bencsik, B. Bede, J. Tar, J. Fodor, Fuzzy differential equations in modeling hydraulic differential servo cylinders, in: Third Romanian_Hungarian Joint Symposium on Applied Computational Intelligence, SACI, Timisoara, Romania, 2006. 94
Mathematical Theory and Modeling ISSN 22245804 (Paper) ISSN 22250522 (Online) Vol.3, No.1, 2013
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[4] B. Bede, I.J. Rudas, A.L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences 177 (2007) 1648–1662. [5] Dubois, D and H.Parade 1978, Operation on Fuzzy Number. International Journal of Fuzzy system, 9:613626 [6] G.L. Diniz, J.F.R. Fernandes, J.F.C.A. Meyer, L.C. Barros, A fuzzy Cauchy problem modeling the decay of the biochemical oxygen demand in water,2001 IEEE. [7] Hassan Zarei, Ali Vahidian Kamyad, and Ali Akbar Heydari, Fuzzy Modeling and Control of HIV Infection, Computational and Mathematical Methods in Medicine Volume 2012, Article ID 893474, 17 pages. [8] J. J. Buckley and T. Feuring, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 110, no. 1, pp.43–54, 2000. [9] James J. Buckley, Thomas Feuring, Fuzzy initial value problem for Nthorder linear differential equations, Fuzzy Sets and Systems 121 (2001) 247–255. [10] J.J. Buckley, T. Feuring, Y. Hayashi, Linear System of first order ordinary differential equations: fuzzy initial condition, soft computing6 (2002)415421. [11] L. A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338353. [12] L.C. Barros, R.C. Bassanezi, P.A. Tonelli, Fuzzy modelling in population dynamics, Ecol. Model. 128 (2000) 2733. [13] L.J. Jowers, J.J. Buckley, K.D. Reilly, Simulating continuous fuzzy systems, Information Sciences 177 (2007) 436–448. [14] M. Oberguggenberger, S. Pittschmann, Differential equations with fuzzy parameters, Math. Modelling Syst. 5 (1999) 181202. [15] Muhammad Zaini Ahmad, Bernard De Baets, A PredatorPrey Model with Fuzzy Initial Populations, IFSAEUSFLAT 2009. [16] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301–317. [17] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987) 319–330. [18] W. Congxin, S. Shiji, Existence theorem to the Cauchy problem of fuzzy differential equations under compactnesstype conditions, Information Sciences 108 (1998) 123–134. [19] Z. Ding, M. Ma, A. Kandel, Existence of the solutions of fuzzy differential equations with parameters, Information Sciences 99 (1997) 205–217.
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