Numerical solution of fuzzy differential equations by Milne’s predictor-corrector method and the dependency problem

Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 2, August 2014
65
Numerical solution of fuzzy differential equations
by Milne’s predictor-corrector method and the
dependency problem
Kanagarajan K, Indrakumar S, Muthukumar S
Department of Mathematics,Sri Ramakrishna mission Vidyalaya College of Arts
& Science Coimbatore – 641020
ABSTRACT
The study of this paper suggests on dependency problem in fuzzy computational method by using the
numerical solution of Fuzzy differential equations(FDEs) in Milne’s predictor-corrector method. This
method is adopted to solve the dependency problem in fuzzy computation. We solve some fuzzy initial value
problems to illustrate the theory.
KEYWORDS
Fuzzy initial value problem, Dependency problem in fuzzy computation, Milnes predictor-corrector
method.
1. INTRODUCTION
Fuzzy Differential Equations (FDEs) are used in modeling problems in science and engineering.
Most of the problems in science and engineering require the solutions of FDEs which are satisfied
by fuzzy initial conditions, therefore a Fuzzy Initial Value Problem(FIVP) occurs and should be
solved. Fuzzy set was first introduced by Zadeh[22]. Since then, the theory has been developed
and it is now emerged as an independent branch of Applied Mathematics. The elementary fuzzy
calculus based on the extension principle was studied by Dubois and Prade [14]. Seikkala[21] and
Kaleva[16] have discussed FIVP. Buckley and Feuring[13] compared the solutions of FIVPs
which where obtained using different derivatives. The numerical solutions of FIVP by Euler's
method was studied by Ma et al.[18]. Abbasbandy and Allviranloo [1, 2] proposed the Taylor
method and the fourth order Runge-Kutta method for solving FIVPs. Palligkinis et al.[20] applied
the Runge-Kutta method for more general problems and proved the convergence for n-stage
Runge-Kutta method. Allahviranloo et. al.[8] and Barnabas Bed [10] to solve the numerical
solution of FDEs by predictor-corrector method. The dependency problem in fuzzy computation
was discussed by Ahmad and Hasan[4] and they used Euler's method based on Zadeh's extension
principle for finding the numerical solution of FIVPs. Omar and Hasan[7] adopted the same
computation method to derive the fourth order Runge-Kutta method for FIVP. Latterly Ahmad
and Hasan[4] investigate the dependency problem in fuzzy computation based on Zadeh
extension principle. In this paper we study the dependency problem in fuzzy computations by
using Milne's predictor-corrector method.

66
2. PRELIMINARY CONCEPTS
In this section, we give some basic definitions.
Definition 2.1 Subset Ã of a universal set Y is said to be a fuzzy set if a membership function
µÃ(y) takes each object in Y onto the interval [0,1]. The function µÃ(y) is the possibility degrees
to which each object is compatible with the properties that characterized the group.
A fuzzy set ‫ܣ‬ሚ ⊆ ܻ can also be presented as a set of ordered pairs
‫ܣ‬ሚ = ൛൫‫,ݕ‬ ߤ஺෨ሺ‫ݕ‬ሻ൯ ∶ ‫ݕ‬ ߳ ܻൟ, (1)
The support, the core and the height of A are respectively
‫݌݌ݑݏ‬൫‫ܣ‬ሚ ൯ = ሼ‫ݕ‬ ߳ ܻ: ‫ݕ‬ > ߤ஺෨ሺ‫ݕ‬ሻሽ, (2)
ܿ‫݁ݎ݋‬൫‫ܣ‬ሚ ൯ = ሼ‫ݕ‬ ߳ ܻ: ߤ஺෨ሺ‫ݕ‬ሻ = 1ሽ, (3)
ℎ݃‫ݐ‬൫‫ܣ‬ሚ൯ = sup௬ ఢ ௒ ߤ஺෨ሺ‫ݕ‬ሻ. (4)
Definition 2.2 A fuzzy number is a convex fuzzy subset A of R, for which the following
conditions are satisfied:
(i) ‫ܣ‬ሚ is normalized. i.e. ℎ݃‫ݐ‬൫‫ܣ‬ሚ൯ = 1;
(ii) ߤ஺෨ሺ‫ݕ‬ሻ are upper semicontinuous;
(iii) ሼ‫ݕ‬ ߳ ܴ: ߤ஺෨ሺ‫ݕ‬ሻ = ߙሽ are compact sets for 0 < ߙ ≤ 1, and
(iv) ሼ‫ݕ‬ ߳ ܴ: ߤ஺෨ሺ‫ݕ‬ሻ = ߙሽ are also compact sets for 0 < ߙ ≤ 1.
Definition 2.3 If ‫ܨ‬ሺܴሻ is the set of all fuzzy numbers, and ‫ܣ‬ሚ ߳ ‫ܨ‬ሺܴሻ, we can characterize ‫ܣ‬ሚ by
its α-levels by the following closed-bounded intervals:
[‫ܣ‬ሚ] ఈ
= ሼ‫ݕ‬ ߳ ܴ: ߤ஺෨ሺ‫ݕ‬ሻ ≥ ߙሽ = [ܽଵ
ఈ
, ܽଶ
ఈ], 0 < ߙ ≤ 1 (5)
[‫ܣ‬ሚ] ఈ
= ሼ‫ݕ‬ ߳ ܴ: ߤ஺෨ሺ‫ݕ‬ሻ ≥ ߙሽ = [ܽଵ
ఈ
, ܽଶ
ఈ
], 0 < ߙ ≤ 1 (6)
Operations on fuzzy numbers can be described as follows: If ‫ܣ‬ሚ , ‫ܤ‬෨ ߳ ‫ܨ‬ሺܴሻ, then for 0 < ߙ1
1. ൣ‫ܣ‬ሚ + ‫ܤ‬෨൧
ఈ
= [ܽଵ
ఈ
+ ܾଵ
ఈ
, ܽଶ
ఈ
+ ܾଶ
ఈ];
2. [‫ܣ‬ሚ − ‫ܤ‬෨] ఈ
= [ܽଵ
ఈ
− ܾଵ
ఈ
, ܽଶ
ఈ
− ܾଶ
ఈ];
3. ൣ‫ܣ‬ሚ ∙ ‫ܤ‬෨൧
ఈ
= [݉݅݊ሼܽଵ
ఈ
∙ ܾଵ
ఈ
, ܽଵ
ఈ
∙ ܾଶ
ఈ
, ܽଶ
ఈ
∙ ܾଵ
ఈ
, ܽଶ
ఈ
∙ ܾଶ
ఈሽ, ݉ܽ‫ݔ‬ሼܽଵ
ఈ
∙ ܾଵ
ఈ
, ܽଵ
ఈ
∙ ܾଶ
ఈ
, ܽଶ
ఈ
∙ ܾଵ
ఈ
, ܽଶ
ఈ
∙
ܾଶ
ఈሽ];
4. ቂ
஺
஻
ቃ
ఈ
= ቂ݉݅݊ ቄ
௔భ
ഀ
௕భ
ഀ ,
௔భ
ഀ
௕మ
ഀ ,
௔మ
ഀ
௕భ
ഀ ,
௔మ
ഀ
௕మ
ഀቅ , ݉ܽ‫ݔ‬ ቄ
௔భ
ഀ
௕భ
ഀ ,
௔భ
ഀ
௕మ
ഀ ,
௔మ
ഀ
௕భ
ഀ ,
௔మ
ഀ
௕మ
ഀቅቃ here 0 ∉ [‫ܤ‬෨] ఈ
;
5. [‫ܣݏ‬ሚ] ఈ
= ‫ܣ[ݏ‬ሚ] ఈ
where s is scalar and
6. ൣ ܽଵ
ఈ೔
, ܽଶ
ఈ೔
൧ = ൣ ܽଵ
ఈೕ
, ܽଶ
ఈೕ
൧ for 0 < ߙ௜ ≤ ߙ௝.
Definition 2.4 A fuzzy process is a mapping ‫ݕ‬෤: ‫ܫ‬ → ‫ܨ‬ሺܴሻ, where I is a real interval [17]. This
process can be denoted as:
[‫ݕ‬෤ሺ‫ݐ‬ሻ]ఈ = [‫ݕ‬ଵ
ఈሺ‫ݐ‬ሻ, ‫ݕ‬ଶ
ఈ
ሺ‫ݐ‬ሻ], ‫ݐ‬ ∈ ‫ܫ‬ ܽ݊݀ 0 < ߙ ≤ 1. (7)

67
The fuzzy derivative of a fuzzy process x(t) is defined by
[‫ݕ‬෤ሺ‫ݐ‬ሻ]ఈ
= ൣ‫ݕ‬ଵ
′ఈሺ‫ݐ‬ሻ, ‫ݕ‬ଶ
′ఈ
ሺ‫ݐ‬ሻ൧, ‫ݐ‬ ∈ ‫ܫ‬ ܽ݊݀ 0 < ߙ ≤ 1. (8)
Definition 2.5 Triangular fuzzy number are those fuzzy sets in F(R) in which are characterized
by an ordered triple ( ) 3
,, Ryyy rcl
∈ with rcl
yyy ≤≤ such that [ ] [ ]rl
yyU ,
0
= and
[ ] [ ]c
yU =
1
then
[ ] ( )( ) ( )( )[ ],1,1 crclcc
yyyyyyU −−+−−−= ααα
(9)
for any R∈α
3. FUZZY INITIAL VALUE PROBLEM
The FIVP can be considered as follows
( ) ,
~
)0(,)(,
)(
0Yytytf
dt
tdy
== (10)
Where ݂: ܴା × ܴ → ܴ is a continuous mapping and ܻ෨଴ ∈ ‫ܨ‬ሺܴሻ with ߙ-level interval
[‫ݕ‬෤଴]ఈ
= [‫ݕ‬଴ଵ
ఈ
, ‫ݕ‬଴ଶ
ఈ ] 0 < ߙ ≤ 1. (11)
When ‫ݕ‬ = ‫ݕ‬ሺ‫ݐ‬ሻ is a fuzzy number, the extension principle of Zadeh leads to the
following definition:
݂ሺ‫,ݐ‬ ‫ݕ‬ሻሺ‫ݏ‬ሻ = ‫݌ݑݏ‬ሼ‫ݕ‬ሺ߬ሻ ∶ ‫ݏ‬ = ݂ሺ‫,ݐ‬ ߬ሻሽ, ‫ݏ‬ ∈ ܴ (12)
It follows that
[݂ሺ‫,ݐ‬ ‫ݕ‬ሺ‫ݐ‬ሻሻ]ఈ
= [݂ଵ
ఈሺ‫,ݐ‬ ‫ݕ‬ሺ‫ݐ‬ሻሻ, ݂ଶ
ఈሺ‫,ݐ‬ ‫ݕ‬ሺ‫ݐ‬ሻሻ], 0 < ߙ ≤ 1, (13)
Where
݂ଵ
ఈሺ‫,ݐ‬ ‫ݕ‬ሺ‫ݐ‬ሻሻ = ݉݅݊ሼ݂ሺ‫,ݐ‬ ‫ݓ‬ሻ ∶ ‫ݓ‬ ߳[‫ݕ‬ଵ
ఈ
ሺ‫ݐ‬ሻ]ሽ, 0 < ߙ ≤ 1 (14)
݂ଶ
ఈሺ‫,ݐ‬ ‫ݕ‬ሺ‫ݐ‬ሻሻ = ݉ܽ‫ݔ‬ሼ݂ሺ‫,ݐ‬ ‫ݓ‬ሻ ∶ ‫ݓ‬ ߳[‫ݕ‬ଵ
ఈ
ሺ‫ݐ‬ሻ]ሽ, 0 < ߙ ≤ 1. (15)
Theorem 3.1 Let f satisfy
|݂ሺ‫,ݐ‬ ‫ݕ‬ሻ − ݂ሺ‫,ݐ‬ ‫ݕ‬∗
ሻ| ≤ ݃ሺ‫,ݐ‬ |‫ݕ‬ − ‫ݕ‬∗|ሻ, ‫ݐ‬ ≥ 0 ‫,ݕ‬ ‫ݕ‬∗
∈ ܴ (16)
Where ݃: ܴା × ܴା → ܴା is a continuous mapping such that ‫ݎ‬ → ݃ሺ‫,ݐ‬ ‫ݎ‬ሻ is non decreasing, the
IVP
‫ݖ‬′ሺ‫ݐ‬ሻ = ݃൫‫,ݐ‬ ‫ݖ‬ሺ‫ݐ‬ሻ൯, ‫ݖ‬ሺ0ሻ = ‫ݖ‬଴, (17)

68
Has a solution on ܴା for ‫ݖ‬଴ > 0 and that ‫ݖ‬ሺ‫ݐ‬ሻ ≡ 0 is the only solution of equation (17) for ‫ݖ‬଴ =
0. Then the FIVP (10) has a unique fuzzy solution.
Proof .See [17]
In the fuzzy computation, the dependency problem arises when we apply the straightforward
fuzzy interval arithmetic and Zadeh's extension principle by computing the interval separately.
For the dependency problem we refer [7].
4. THE MILNE’S PREDICTOR-CORRECTOR METHOD IN DEPENDENCY
PROBLEM
We consider the IVP in equation (10) but with crisp initial condition ‫ݕ‬ሺ‫ݐ‬଴ሻ = ‫ݕ‬଴ ߳ ܴ and
‫ݐ‬ ߳ [‫ݐ‬଴, ܶ]. The formula for Milne’s predictor-corrector method is follows:
( ) ( ) ( )[ ]
( )( ) ( )( ) ( )( )[ ]
,)(,)(,)(,)(
,,,4,
3
,,2,,2
3
4
112233
1,11111,1
22113,1
rrrrrrrr
rPrrrrrrrCr
rrrrrrrPr
ytyytyytyyty
tytftytftytf
h
yy
ytfytfytf
h
yy
====
+++=
+−+=
−−−−−−
+++−−−+
−−−−−+
(18)
Where .,,1,0,1
0
Nrtt
N
tT
h rr K=−=
−
= + We consider the right-hand side of equation (18),
we modify the Milne’s predictor-corrector method by using dependency problem in fuzzy
computation as one function
( ) ( )( ) ( )( ) ( )( )[ ],,,4,
3
)(,, 1,11111 +++−−− +++= rPrrrrrrrrr tytftytftytf
h
tyyhtV (19)
By the equivalent formula
( )
( )( ) ( )( )
( ) ( ) ( )[ ]
.
,2,,2
3
4
,
,4,
3 221131
11
1,1











+−++
+
+=
−−−−−+
−−
−+
rrrrrrrr
rrrr
rCr
ytfytfytf
h
ytf
tytftytf
h
tyy (20)
Now, let ܻ෨ ∈ ‫ܨ‬ሺܴሻ, the formula
ܸ൫‫ݐ‬௥, ℎ, ܻ෨௥൯ሺ‫ݒ‬௥ሻ = ቊ
‫݌ݑݏ‬௬ೝ∈௏షభሺ௧ೝ,௛,௩ೝሻ ܻ෨௥ሺ‫ݕ‬௥ሻ , ݂݅ ‫ݒ‬௥ ∈ ‫݁݃݊ܽݎ‬ሺܸሻ;
0, ݂݅ ‫ݒ‬௥ ∉ ‫݁݃݊ܽݎ‬ሺܸሻ,
(21)
Can extend equation (20) in the fuzzy setting.
Let [ܻ෨௥]ఈ
= ൣ‫ݕ‬௥,ଵ
ఈ
, ‫ݕ‬௥,ଶ
ఈ
൧ represent the ߙ-level of the fuzzy number defined in equation (21). We
rewrite equation (21) using the ߙ-level as follows:

69
ܸ൫‫ݐ‬௥, ℎ, [ܻ෨௥]ఈ
൯ = ൣ݉݅݊൛‫ݐ‬௥, ℎ, ‫ݕ‬ห‫ݕ‬ ∈ ‫ݕ‬௥,ଵ
ఈ
, ‫ݕ‬௥,ଶ
ఈ
ൟ, ݉ܽ‫ݔ‬൛‫ݐ‬௥, ℎ, ‫ݕ‬ห‫ݕ‬ ∈ ‫ݕ‬௥,ଵ
ఈ
, ‫ݕ‬௥,ଶ
ఈ
ൟ൧ (22)
By applying equation (22) in (18) we get
[ܻ෨௥ାଵ]ఈ
= ൣ‫ݕ‬௥ାଵ,ଵ
ఈ
, ‫ݕ‬௥ାଵ,ଶ
ఈ
൧, (23)
Where
‫ݕ‬௥ାଵ,ଵ
ఈ
= ݉݅݊൛ܸሺ‫ݐ‬௥, ℎ, ‫ݕ‬ሻห‫ݕ‬ ∈ [‫ݕ‬௥,ଵ
ఈ
, ‫ݕ‬௥,ଶ
ఈ
]ൟ, (24)
‫ݕ‬௥ାଵ,ଶ
ఈ
= ݉ܽ‫ݔ‬൛ܸሺ‫ݐ‬௥, ℎ, ‫ݕ‬ሻห‫ݕ‬ ∈ [‫ݕ‬௥,ଵ
ఈ
, ‫ݕ‬௥,ଶ
ఈ
]ൟ. (25)
Therefore
( )( ) ( )( ) ( )( )[ ] [ ] ,,,,4,
3
)(min 2,1,1,111111,1






∈+++= +++−−−+
ααα
rrrPrrrrrrrr yyytytftytftytf
h
tyy (26)
( )( ) ( )( ) ( )( )[ ] [ ] ,,,,4,
3
)(max 2,1,1,111112,1






∈+++= +++−−−+
ααα
rrrPrrrrrrrr yyytytftytftytf
h
tyy (27)
By using the computational method proposed in [5], we compute the minimum and
maximum in equations (26), (27) as follows
‫ݔ‬௥ାଵ,ଵ
ఈ೔
= ݉݅݊ ൥ ݉݅݊
௫∈ቂ௫ೝ,భ
ഀ೔ ,௫ೝ,భ
ഀ೔శభቃ
ܸሺ‫,ݐ‬ ℎ, ‫ݔ‬ሻ, ⋯ , ݉݅݊
௫∈ቂ௫ೝ,భ
ഀ೙,௫ೝ,మ
ഀ೙శభቃ
ܸሺ‫,ݐ‬ ℎ, ‫ݔ‬ሻ, ⋯ , ݉݅݊
௫∈ቂ௫ೝ,మ
ഀ೔శభ,௫ೝ,మ
ഀ೔ ቃ
ܸሺ‫,ݐ‬ ℎ, ‫ݔ‬ሻ൩
(28)
‫ݔ‬௥ାଵ,ଶ
ఈ೔
= ݉ܽ‫ݔ‬ ൥ ݉ܽ‫ݔ‬
௫∈ቂ௫ೝ,భ
ഀ೔ ,௫ೝ,భ
ഀ೔శభቃ
ܸሺ‫,ݐ‬ ℎ, ‫ݔ‬ሻ, ⋯ , ݉ܽ‫ݔ‬
௫∈ቂ௫ೝ,భ
ഀ೙,௫ೝ,మ
ഀ೙శభቃ
ܸሺ‫,ݐ‬ ℎ, ‫ݔ‬ሻ, ⋯ , ݉ܽ‫ݔ‬
௫∈ቂ௫ೝ,మ
ഀ೔శభ,௫ೝ,మ
ഀ೔ ቃ
ܸሺ‫,ݐ‬ ℎ, ‫ݔ‬ሻ൩ ሺ29ሻ
5. NUMERICAL EXAMPLES
In this section, we present some numerical examples including linear and nonlinear FIVPs.
Example 5.1 Consider the following FIVP.
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬ ‫ݔ‬′ሺ‫ݐ‬ሻ = ‫ݔ‬ሺ1 − 2‫ݐ‬ሻ, ‫ݐ‬ ∈ [0,2];
ܺ෨଴ሺ‫ݓ‬ሻ = ቐ
0, ݂݅ ‫ݓ‬ < −0.5;
1 − 4‫ݓ‬ଶ
, ݂݅ − 0.5 ≤ ‫ݓ‬ ≤ 0.5;
0, ݂݅ ‫ݓ‬ > 0.5;
(30)

70
The exact solution of equation (30) is given by
[ܺሺ‫ݐ‬ሻ]ఈ
= ൤൬−
ඥሺଵିఈሻ
ଶ
൰݁௧ି௧మ
, ൬
ඥሺଵିఈሻ
ଶ
൰ ݁௧ି௧మ
൨. (31)
The absolute results of the numerical fuzzy Milne's predictor-corrector method approximated
solutions at .220 =t See Table 1 and Figure 1 and 2.
TABLE 1
The error of the obtained results with the exact solution at t=2.
Figure 1: The approximation of fuzzy solution by Milne's predictor-corrector (h=0.1)

71
Figure 2: Comparison between the exact, Milne's predictor-corrector, predictor-corrector
In this example, the comparison of the absolute local error between Milne's predictor-corrector
method with the fuzzy exact solution is given in Table 1 for various values of α -level
1,9.0,,1.0,0 K=α and fixed value of ( ).220 =th The results shows that Milne's predictor-
corrector method is more accurate than predictor-corrector method shows the graphical
comparison of a fuzzy solution between exact, Milne's predictor-corrector, predictor-corrector at
fixed ( ).110 =th The behaviour solutions of the end points of the fuzzy intervals of a fuzzy exact
solution, Milne's predictor-corrector and predictor-corrector fuzzy approximated solutions are
plotted and compared in Figure 2 at .01 =α Figure 2 clearly show that Milne's predictor-
corrector provides a more accurate results than predictor-corrector method.
Example 5.2 Consider the following FIVP
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬‫ݔ‬′ሺ‫ݐ‬ሻ = ‫ݔ‬ሺ‫ݐ‬ଶ
− 4‫ݐ‬ + 3ሻ, ‫ݐ‬ ∈ [0,2];
ܺ෨଴ሺ‫ݓ‬ሻ = ቐ
0, ݂݅ ‫ݓ‬ < −0.5;
1 − 4‫ݓ‬ଶ
, ݂݅ − 0.5 ≤ ‫ݓ‬ ≤ 0.5;
0, ݂݅ ‫ݓ‬ > 0.5;
(32)
The exact solution of equation (32) is given by
[ܺሺ‫ݐ‬ሻ]ఈ
= ൤൬−
ඥሺଵିఈሻ
ଶ
൰݁
೟య
య
ିଶ௧మାଷ௧
, ൬
ඥሺଵିఈሻ
ଶ
൰݁
೟య
య
ିଶ௧మାଷ௧
൨. (33)
The absolute results of the numerical fuzzy Milne's predictor-corrector method approximated
solutions at .220 =t See Table 2 and Figure 3 and 4.

72
TABLE 2
The error of the obtained results with the exact solution at t=2.
Figure 3: The approximation of fuzzy solution by Milne's predictor-corrector (h=0.1)

73
Figure 4: Comparison between the exact, Milne's predictor-corrector, predictor-corrector
In this example, we compare the solution obtained by Milne's predictor-corrector method with the
exact solution and predictor-corrector. We have given the numerical values in Table 2 fixed value
of .220 =t and for different values of .α
6. CONCLUSION
In this paper we used the Milne's predictor-corrector method for solving FIVP by considering the
dependency problem in fuzzy computation. We compared the solutions obtained in two numerical
examples.
ACKNOWLEDGEMENTS
This work has been supported by “Tamilnadu State Council of Science and Technology”,
Tamilnadu, India.
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