10th & 11th January 2019
42
A Third Order Runge-Kutta Method Based on Linear
Combination of Arithmetic Mean, Harmonic Mean and
Geometric Mean for Hybrid Fuzzy Differential Equation
GethsiSharmila . R
1, Evangelin Diana Rajakumari . P
2,
Assistant Professor1,2, PG and Research Department of Mathematics1,2,Bishop Heber College, Tiruchirappalli-17.
[email protected] 1,[email protected]
Abstract-In this paper, the numerical solution of hybrid fuzzy differential equation by using the third order Runge-Kutta method based on linear combination of Arithmetic mean, Harmonic mean and Geometric mean is proposed. The exact, approximate and absolute error are calculated for the hybrid fuzzy differential equations and compared with the third order Runge-Kutta method based on Arithmetic mean, Harmonic mean and Geometric mean. A graphical representation of the solution is also given. The proposed method provides an efficient and powerful mathematical tool for solving hybrid fuzzy differential equations.
Index Terms-Hybrid fuzzy differential equations, Runge-Kutta method, Arithmetic mean, Harmonic mean, Geometric mean, Absolute error.
1. INTRODUCTION
Control systems that are capable of controlling complex systems which have discrete time dynamics and continuous time dynamics can be modeled by hybrid systems. The differential systems containing fuzzy valued functions and interaction with a discrete time controller are named as hybrid fuzzy differential systems.
Pederson and Sambandham [10, 11] used the Euler and Runge-Kutta methods for solving hybrid fuzzy differential equations. The stability properties and comparision theorems are studied in [10, 6, 7]. Prakash and Kalaiselvi [12] have studied the numerical solution of hybrid fuzzy differential equations by predictor-corrector three step method. Kanagarajan, Indrakumar and Muthukumar in [5] have studied the numerical solution of hybrid fuzzy differential equations by Runge-Kutta method of order five and the dependency problem in fuzzy
computation. The solution of hybrid fuzzy
differential equations using Adams-Bashforth,
Adams-Moulton and Predictor-Corrector methods are discussed in [3]. Narayanamoorthy et al in [9, 8] used Iterative procedure and Taylors series method to solve hybrid fuzzy differential equations. A numerical solution for hybrid fuzzy differential equations by Adams fifth order predictor-corrector method was studied by Jayakumar and Kanagarajan in [4]. GethsiSharmila have solved fourth order Runge-Kutta method based on geometric mean for hybrid fuzzy initial value problem and centroidal mean for fuzzy and hybrid fuzzy initial value problem in [1, 2].
In this work, a third order Runge-Kutta Method based on linear combination of Arithmetic
Mean, Harmonic Mean and Geometric Mean is proposed to solve Hybrid fuzzy differential equations.
2. PRELIMINARIES
Definition:2.1( Fuzzy Set )
If X is a collection of objects denoted generally by x, then a fuzzy set
A
~
in X is a set of ordered pairsA
~
(
x
,
A~(
x
))
/
x
X
, where)
(
~x
A
is called the membership function or gradeof membership of x in
A
~
.Definition: 2.2 ( Membership Function )
The membership function values need not always be described by discrete values. Sometimes, these turn out to be as described by a continuous function.
The notation commonly used to denote the
membership functions are:
(i) The membership function of a fuzzy set
A
~
isdenoted by ~
(
x
)
A
, i.e.,]
1
,
0
[
:
)
(
~
x
X
A
.(ii) The membership function of a fuzzy set
A
~
has the following form]
1
,
0
[
:
~
X
10th & 11th January 2019
43 Types of Fuzzy Numbers :
The types of fuzzy numbers are based on the shapes of the fuzzy numbers. Commonly we use the
triangular fuzzy number the most. Also we have ordinary fuzzy number (interval), trapezoidal fuzzy number, parallelogram fuzzy number and bell shaped fuzzy number.
Definition: 2.3 ( Triangular fuzzy number )
It is a fuzzy number represented with three points as follows:
A
~
(
a
1,
a
2,
a
3)
, this representation isinterpreted as membership functions and holds the following condition(i)
a
1 toa
2 is increasing function (ii)a
2 toa
3 is decreasing function (iii)a
1
a
2
a
3
3 3 2
2 3
3
2 1
1 2
1
1
~
, 0
, , , 0
) (
a x
a x a a
a x a
a x a a
a a x
a x
x
A
Now, if you get crisp interval by
- cut operation, intervalA
~
shall be obtainedas follow
[
0
,
1
]
from
2 3
) ( 3 3
1 2
1 ) ( 1
,
a
a
a
a
a
a
a
a
The Third order runge-kutta method based on a linear Combination of Arithmetic Mean (AM), Harmonic Mean (HAM) and Geometric Mean (GM) for IVP was introduced by Khattri as follows
45
)
,
(
32
)
,
(
)
,
(
14
)
,
(
1 22 1 2
1
2 1
k
k
GM
k
k
HM
k
k
AM
k
k
RM
For the initial value problem of the form
)
,
(
t
y
f
y
, the Third order Runge-Kuttamethods using variety of means can be written in the
form
2
1 1
2
i nn
Means
h
y
y
(2.3)where,
)
,
)
(
(
)
,
(
)
,
(
2 3 1 2 3
2 3
1 1 1
2 1
hk
a
hk
a
y
h
a
a
t
f
k
hk
a
y
h
a
t
f
k
y
t
f
k
n n
n n
n n
where the parameters for RK3MC:
9
10
,
9
4
,
3
2
3 2
1
a
a
a
3. THE HYBRID FUZZY DIFFERENTIAL EQUATIONS
The hybrid fuzzy differential equation
0 0
1
)
(
...
,
2
,
1
,
0
]
,
[
,
)
)
(
),
(
,
(
)
(
y
t
y
k
t
t
t
y
t
y
t
f
t
y
k k k k
(3.1)
Where
t
k
0
is strictly increasing and unbounded,k
y
denotesy
(
t
k)
,
f
:
[
t
0,
)
R
R
R
is continuous, and each
k:
R
R
is acontinuous function. A solution to (3.1) will be a function
y
:
[
t
0,
)
R
satisfying (3.1) . For k = 0, 1, 2, 3, … , letf
k:
[
t
k,
t
k1)
R
R
where,
)
)
(
,
)
(
,
(
)
)
(
,
(
t
y
kt
f
t
y
kt
ky
kf
.The hybrid fuzzy differential equation in (3.1) can be written in expanded form as
. .
.
. .
.
, ) ( , ) ) ( , ( ) ) ( ,) ( , ( ) (
. .
.
. .
.
, ) ( , ) ) ( , ( ) ) ( , ) ( , ( ) (
, ) ( , ) ) ( , ( ) ) ( ,) ( , ( ) (
) (
1 2 1 1 1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 0 0 0 0
k k k k k k k k k k
k t f t y t y f t y t y t y t t t
y
t t t y t y t y t f y t y t f t y
t t t y t y t y t f y t y t f t y
t y
(3.2)
and a solution of (3.1) can be expressed as
. .
. .
, )
(
. .
. .
, )
(
, )
(
) (
1 2 1
1 1
1 0
0 0
k k
k
k t y t t t
y
t t t y
t y
t t t y
t y
t y
(3.3)
10th & 11th January 2019
44
and piecewise differentiable over
[
t
0,
)
anddifferentiable on each interval
(
t
k,
t
k1)
for anyfixed
y
k
R
and k = 0, 1, 2, … .4. A THIRD ORDER RUNGE-KUTTA METHOD BASED ON LINEAR COMBINATION OF ARITHMETIC MEAN, HARMONIC MEAN AND GEOMETRIC MEAN FOR HYBRID FUZZY DIFFERENTIAL EQUATION
Consider the IVP (3.1) with crisp initial condition
y
(
t
0)
y
0
R
andt
[
t
0,
T
]
.Let the exact solution
Y
(
t
)
r
Y
(
t
;
r
)
,
Y
(
t
;
r
)
is approximatedby some
y
(
t
)
r
y
(
t
;
r
)
,
y
(
t
;
r
)
from the equation (2.3)
We define
3 1 , , , 1 , 3 1 , , , 1 ,)
(
;
(
)
(
)
(
)
(
;
(
)
(
)
(
i n k n k i i n k n k i n k n k i i n k n kr
y
t
k
w
r
y
r
y
r
y
t
k
w
r
y
r
y
where
w
1,
w
2,
w
3 are constants,10th & 11th January 2019 45
))
(
,
(
))
(
,
(
))
(
,
(
))
(
,
(
32
))
(
,
(
))
(
,
(
))
(
,
(
))
(
,
(
2
))
(
,
(
))
(
,
(
))
(
,
(
))
(
,
(
2
))
(
,
(
))
(
,
(
2
))
(
,
(
7
)
(
),
(
;
, , 3 , , 2 , , 2 , , 1 , , 3 , , 2 , , 3 , , 2 , , 2 , , 1 , , 2 , , 1 , , 3 , , 2 , , 1 , , ,r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
t
k
r
y
r
y
t
T
n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k kThe exact solutions at
t
k,n1 is given by
;
(
)
,
(
)
2
)
(
)
(
)
(
,
)
(
;
2
)
(
)
(
, , , , 1 , , , , , 1 ,r
Y
r
Y
t
T
h
r
Y
r
Y
r
Y
r
Y
t
S
h
r
Y
r
Y
n k n k n k k k n k n k n k n k n k k k n k n k
The approximate solutions at
t
k,n1 is given by
;
(
)
,
(
)
2
)
(
)
(
)
(
,
)
(
;
2
)
(
)
(
, , , , 1 , , , , , 1 ,r
y
r
y
t
T
h
r
y
r
y
r
y
r
y
t
S
h
r
y
r
y
n k n k n k k k n k n k n k n k n k k k n k n k
5. NUMERICAL EXAMPLES
Consider the following HFDEs
1
0
,
]
)
125
.
0
125
.
1
(
,
)
25
.
0
75
.
0
(
[
)
;
(
...
,
2
,
1
,
0
,
]
,
[
,
)
)
(
(
)
(
)
(
)
(
1r
e
r
e
r
r
t
y
k
k
t
t
t
t
t
y
t
m
t
y
t
y
t t k k k k k
(5.1) where
5
.
0
)
1
mod
(
,
)
)
1
mod
(
1
(
2
,
5
.
0
)
1
mod
(
,
)
)
1
mod
(
(
2
)
(
t
if
t
t
if
t
t
m
}
...
,
2
,
1
{
,
,
0
,
0
)
(
k
if
k
if
k
The exact solution for
t
[
0
,
2
]
is given by . ] 2 , 5 . 1 [ , ] ) 4 3 ( 2 2 [ ) ; 1 ( , ] 5 . 1 , 1 [ , ] 2 3 [ ) ; 1 ( , ] 1 , 0 [ , ] ) 125 . 0 125 . 1 ( , ) 25 . 0 75 . 0 ( [ ) ; ( 5 . 1 1 t e e t r Y t t e r Y t e r e r r t Y t t t t
The approximate solution, exact solution and absolute error using the third order Runge-Kutta method based on a linear combination of Arithmetic Mean, Harmonic Mean and Geometric Mean for the r-level set with h = 0.5 and t = 2 of the example (5.1) is Given below.
r t
Absolute Error RK3 AM Absolute Error RK3 HaM Absolute Error RK3 GM Absolute Error RK3 AM HAM GM
0 2
5.4 0E -02 8.1 0E -02 1.6 4E -01 2.4 6E -01 1.0 5E -01 1.5 8E -01 8.6 8E -02 1.3 0E -01 0 . 1 2 5.5 8E -02 8.0 1E -02 1.7 0E -01 2.4 4E -01 1.0 9E -01 1.5 6E -01 8.9 7E -02 1.2 9E -01 0 . 2 2 5.7 6E -02 7.9 2E -02 1.7 5E -01 2.4 1E -01 1.1 2E -01 1.5 4E -01 9.2 6E -02 1.2 7E -01 0 . 3 2 5.9 4E -02 7.8 3E -02 1.8 1E -01 2.3 8E -01 1.1 6E -01 1.5 2E -01 9.5 5E -02 1.2 6E -01 0 . 4 2 6.1 2E -02 7.7 4E -02 1.8 6E -01 2.3 5E -01 1.1 9E -01 1.5 1E -01 9.8 4E -02 1.2 4E -01 0 . 5 2 6.3 0E -02 7.6 5E -02 1.9 2E -01 2.3 3E -01 1.2 3E -01 1.4 9E -01 1.0 1E -01 1.2 3E -01 0 . 6 2 6.4 8E -02 7.5 6E -02 1.9 7E -01 2.3 0E -01 1.2 6E -01 1.4 7E -01 1.0 4E -01 1.2 2E -01 0 . 7 2 6.6 6E -02 7.4 7E -02 2.0 3E -01 2.2 7E -01 1.3 0E -01 1.4 5E -01 1.0 7E -01 1.2 0E -01 0 . 8 2 6.8 4E -02 7.3 8E -02 2.0 8E -01 2.2 4E -01 1.3 3E -01 1.4 4E -01 1.1 0E -01 1.1 9E -01 0 . 9 2 7.0 2E -02 7.2 9E -02 2.1 3E -01 2.2 2E -01 1.3 7E -01 1.4 2E -01 1.1 3E -01 1.1 7E -01
1 2
10th & 11th January 2019
46 The Graphical representation of the exact and
approximate solution of the third order Runge-Kutta method based on the linear combination of Arithmetic Mean, Harmonic Mean and Geometric Mean for example 5.1. where t = 2, h = 0.5 is given below
6. CONCLUSION
In this paper, a new numerical method for solving first order hybrid fuzzy differential equation is proposed. Here the first order hybrid fuzzy diffe rential equation is solved using the third order Runge-Kutta method based on the linear combination of Arithmetic Mean, Harmonic Mean and Geometric Mean. The Comparison of the solution
obtained by the proposed method with the existing methods. From the numerical example, we can conclude that the proposed method is suitable for solving hybrid fuzzy initial value problem.
REFERENCES
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Fourth Order Runge-Kutta Method based on Geometric Mean for Hybrid Fuzzy Initial Value Problems, in IOSR Journal of Mathematics, Vol. 13, (Mar.-Apr. 2017), 43-51.
[2] Henry Amirtharaj E. C. and GethsiSharmila R.,
Fourth Order Runge-Kutta Method based on Centroidal Mean for Fuzzy Initial Value Problems and Hybrid Fuzzy Initial Value Problems,
Emerging Trends in Mathematics and Mathematics Education, (2017), 63-80.
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Numerical Solution for Hybrid Fuzzy System by Adams fourth order predictor-corrector method,
International Journal of Mathematical
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Solution of Hybrid Fuzzy Differential Equations by Adams Fifth Order Predictor- Corrector Method, International Journal of Mathematics Trends and Technology Vol. X Issue x-(May 2014), ISSN: 2231-5373.
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Numerical Solution of hybrid fuzzy differential equations by runge-kutta method of order five
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International Journal of Pure and Applied Mathematics, Vol. 113, No. 12 (2017), 133-141.
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