JST Vol. 21 (1) Jan. 2013 - Pertanika Journal - Universiti Putra ...

Solving Delay Differential Equations by Using Implicit 2-Point Block Backward Differentiation Formula
The numerical results were then compared to the classical 1-point backward differentiation
formula (BDF). The corrector formula (Lambert, 1993) is given as:
2 9 18 6
yn+ 1= yn-2- yn- 1+ yn+ hf (5)
n+
1
11 11 11 11
In general, most numerical methods for solving differential equations produce only one
new approximation value. However, the implicit block method can produce two new values
simultaneously at each step. Thus, it is logically safe to presume that this particular method
can reduce the timing of the calculation and its computational cost as well.
IMPLEMENTATION OF THE METHOD
Newton iteration is implemented in the method to evaluate the values of y n+
1
and y n+
2 for solving equation [1]. The values for both y n+
1 and y n+
2 at ( i + 1)
( i+
1)
( i+
1)
th iterative are given as y n+
1 and y n+
2 . According to Ibrahim et al. (2007), the
Newton iteration takes the form of
and
Hence,
æ ö
ç f
ç
÷
ç
÷
è ø÷
( i)
F1 y
( i+ 1) ( i)
n+
1)
n+ 1 = n+ 1- ' ( i)
F1 yn+
1
y y
( )
( )
( i)
F2y ( i+ 1) ( i)
n+
2)
n+ 2 = n+ 2- ' ( i)
F2 yn+
2
y y
( )
( )
n+
1 ( i+ 1 ) ( i) () i () i () i
1-2h ( yn+ 1 - yn+ 1) =-yn+ 1- yn+ 2+ 2hfn+
1+ V 1
yn+
1
3
(8)
æ 6 ö
ç f ÷
18 6
ç ÷
ç
÷
è ÷ ø
n+
2 ( i+ 1 ) ( i) () i () i () i
1- h ( yn+ 2 - yn+ 2) = yn+ 1- yn+ 2+ hfn+
2+ V 2
11 yn+
2
11 11
(9)
where f
y is a Jacobian and 1 V , V 2 are the back values.
Let
Pertanika J. Sci. & Technol. 21 (1): 283 - 298 (2013)
2
e = y -y
(10)
( i+ 1) ( i+ 1) ( i)
n+ 1 n+ 1 n+
1
e = y -y
(11)
( i+ 1) ( i+ 1) ( i)
n+ 2 n+ 2 n+
2
(6)
(7)
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