Scholars Research Library
Archives of Applied Science Research, 2010, 2 (6):362-372 (http://scholarsresearchlibrary.com/archive.html)
ISSN 0975-508X CODEN (USA) AASRC9
On generalized 2-step continuous linear multistep method of hybrid type for the integration of second order ordinary differential
equations
1
J. O. Ehigie,
2S. A. Okunuga,
3A. B. Sofoluwe,
4M. A. Akanbi
1,2
Department of Mathematics, University of Lagos, Akoka, Lagos
3
Department of Computer Science, University of Lagos, Akoka, Lagos
4
Department of Mathematics, Lagos State University, Ojo. Lagos
______________________________________________________________________________
ABSTRACT:
This paper proposes a generalized 2-step continuous multistep method of hybrid type for the direct integration of second-order ordinary differential equations in a multistep collocation technique, which yields block methods. The scheme obtained is used as a single continuous form which serves as a family of formula involving ( x , s ) such that on substitution of an off-step point s a bi-hybrid continuous scheme is obtained. The discrete equivalent is also obtained thereafter from the continuous family of formula as a block method. It was discovered that Numerical schemes of StÖrmer-Cowell type were recovered via this technique. The scheme obtained is implemented to generate the numerical solution to second order ordinary differential equations.
The results obtained are compared with the Renowned Numerov method, known to be of optimal Keywords: Collocation, Block method, hybrid, StÖrmer-Cowell, Numerov, Optimal Order ______________________________________________________________________________
INTRODUCTION
Many field of applications, notably in Science and Engineering yields initial value problems of Second order ordinary differential equations are of the form,
) , , ( )
( x f x y y
y ′′ = ′ , y ( a ) = y
0, y ′ (a ) = β (1.1) Many of such problems may not be easily solved analytically, hence numerical schemes are developed to solve (1.1). These equations are usually reduced to 2 systems of first order ordinary differential equations and numerical methods of First Order differential equations are used to solve them. Linear Multistep Methods are powerful numerical methods for solving Differential equations.
Some researchers have attempted the solution of (1.1) using linear multistep methods without
reduction to systems of first order ordinary differential equations, they include Brown (1977),
Lambert (1991), Kayode (2005), Adee et al. (2005), Onumanyi et al. (2008), Yahaya et al (2009) to mention a few. Lambert (1973) also discussed about an optimal two step method called the Numerov’s Method.
In our previous paper, we developed generalized schemes of continuous linear multistep methods (CLMS) of hybrid type for solving first order ordinary differential equations, Ehigie et al.
(2010). Here, we extend our results to generalizing the solution to Second Order ordinary differential equation via multistep collocation technique. This method helps to provide a continuous numerical scheme which accommodates all hybrid points, so that on substitution of an off-step point s a bi-hybrid scheme is obtained for the direct integration of (1.1) without reduction to systems of differential equations.
In section 2, we present the theoretical procedure used as a generalization for all cases in this class. In section 3, the generalized method is derived as a special case of the theoretical procedure for m = 2 . Examples of schemes generated from the generalized methods were derived in section 4. The Schemes generated were implemented on some problems in section 5.
Finally, conclusion and future research areas were also stated.
1. Theoretical Procedure:
2.
Consider the system,
) , , ( )
( x f x y y
y ′′ = ′ , y ( x
0) = y
0, y ′ ( x
0) = y
0′ (2.1) The Numerical Solution to (2.1) can be obtained using a m − step explicit Linear Multistep Method (LMM) of the form,
∑
∑
−= +
= +
=
10 2 0
m
j
j k j m
j
j k
j
y h β f
α
(2.2)
for a second-order differential equation, Lambert (1973), but since we are interested in obtaining an hybrid scheme, 2 hybrid points, ( 1 − s ) and ( 1 + s ) were introduced as a generalization of all hybrid methods of this class so that (2.2) becomes,
∑
∑
−= + + + + − − +
= +
+ +
=
10
1 1 2 1 1 2 2
0
m
j
s k s s
k s j
k j j
k m
j
j
y h β f h β f h β f
α (2.3)
To obtain the CLMS for (2.1), it is necessary to use the power series,
∑
=
−
=
mj
j k
j
h
x a x
x Y
2
0
)
( (2.4)
where m is the m − step method of this class.
Let m and n be numbers of interpolations and collocations respectively, then we shall have a
system of ( 2 m + 1 ) equations of the form represented in the matrix below,
=
⋅
⋅
⋅
+
− + +
− +
+
− +
+
−
−
−
−
+
− +
− +
−
−
−
−
−
+ + +
+ + + +
+ +
+
+ + +
+ +
+ +
+ +
+
+ +
+
+ +
+
+ +
+
1 2
1 2
1 2
1 2
1 1
2 3 2 1 0
2 2 1
1 2
2 0
0
1 12 2 2
1 2 1 1 2 1 2 13
2 12
2 11 2
1 22 2
2 1
2 2
23 22
21
1 12 2
1 1
1 1
13 12
11
1 2 2
1 3
2 1
1 22 2
2 1
2 2
23 22
21
1 12 2
1 1
1 1
13 12
11
) (
) ( )
( ) ( )
( ) ( )
( ) ( )
( 0
) ( 0
0 0
s k
s k
m k
k m k
k k
m
m m m
m m
m m
m
m m m
m m
m m m m
m m
m m m
m m m m m m
m m
m m m
m m
m m m m
m m
m m mm
mm mm
m m
m
m m
m m
m m
m m
f h
f h
f h
f h Y Y
Y
a a a a a
s g
s f s
g s f s g
s f s g
s f s
g s f
d d
d d
d d
d
d d
d d
d d
d
d d
d d
d d
d
d d
d d
d d
d
d d
d d
d d
d
d d
d d
d d
d
M M
L L
L
M M L M
M M
L M
M
L L
L L
M M
M M
M M
L L
L L
(2.5) Where d are as derived in the interpolations and collocations respectively as derived in
ijOkunuga et al (2009) and f
p(s ) and g
p(s ) are polynomials of degree p . i.e,
) (s f
p= ∑
= p
i i i
s d
0
and, g
p(s ) = ∑
= p
i i i
s d
0
Expressing (2.5) further as,
DA = C
⇒
A = D
−1C
where A ⇒ a
j= ϕ ( s , h , Y
k+j, f
k+j, f
k−s+1, f
k+s+1) , For j = 0 , 1 , 2 , L , 2 m substituting: a ,
jj = 0 , 1 , 2 , L , 2 m in (2.4), we shall obtain,
= ) , ( x s Y
∑
∑
+ − + + + + − −+−1
( , ) +
2 1( , ) +
2 1( , )
1+
2 1( , )
1m
s k s
s k s
j k j
j k m
j
x s Y h β x s f h β x s f h β x s f
α
such that for any appropriate value of s , a new CLMS of hybrid type is generated and given as,
= ) (x
Y ∑
+∑
− + + + + − − +−1
( ) +
2 1( ) +
2 1( )
1+
2 1( )
1m
s k s s
k s
j k j j
k m
j
x Y h β x f h β x f h β x f
α
where,
m i
i
k ji
j
h
x x ∑ x
=
−
=
20
)
( α
α and ∑
=
−
=
mi
i k ji
j
h
x x x
2
0
)
( β
β
This can be used to generate block methods on evaluation of Y ( x ) at x = x
k+m, x
k−s+1and x
k+s+1respectively.
3. Derivation of the Generalized method for m = 2
For such a 2 − step hybrid method, i.e. setting m = 2 . Let the basis polynomial to the problem (2.1) be given as,
∑
=
−
=
mj
j k
j
h
x a x
x Y
2
0
) (
(3.1) Interpolating x
j: j = 0 , 1 and also collocating Y ′′ (x ) at x
j: j = 0 , 1 , ( 1 − s ) and ( 1 + s )
We shall obtain the system of equation express by the matrix,
=
+
−
−
+ + +
+
− +
+ +
+
1 1
1 1
4 3 2 1 0
2 2
) 12 24 12 ( ) 6 6 ( 2 0 0
12 6
2 0 0
) 12 24 12 ( ) 6 6 ( 2 0 0
1 1
1 1 1
0 0
0 0 1
s k
k s k
k k
f f f
Y Y
a a a a a
s s s
s s s
(3.2) Solving this system of equations with maple software. The hybrid term s was delayed so as to obtain a generalized schemes rather than an individual continuous formula which is limited to a particular s values as in Yahaya et al. (2009). We obtain,
Y
ka
0=
(
2 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2)
1 2
24 3 6 3 4 4 24 12
24
1 Y s h f h f h f h sf h f s Y s h f s
a = − s
k+
k+s+−
k++
k−s+−
k+s++
k−s+−
k++
k+(
1 1 1 1 1 1 2)
2 2
2
2 2
4 f f f sf f s f s
s
a = h
k+s+−
k++
k−s+−
k+s++
k−s++
k+( f f f sf f s )
s
a h
2 k s 1 k 1 k s 1 k s 1 k s 12
3
2 4 2
12
+ +−
++
− +−
+ ++
− +−
=
(
1 1 1)
2
4
2
24
+ +−
++
− += h f
k sf
kf
k sa
Substituting a ,
jj = 0 , 1 , 2 , 3 and 4 into (3.1), we obtain, )
, ( x s
Y =
4 1
2 3
2
2
2 2
2 1
24 1 6
1 12
1
4 1 4
1 8
1 6
1
1
+ + +
− +
−
−
+
−
−
+
−
−
+
−
+
−
−
k sk k
k k
k k k
k
f
h x x h s
x x s s
h x x s s
h x x s s
h h Y
x Y x
h x x
4 1
2 3
2
2
2 2
2
12 1 3
1
2 1 2 1 2
1 4
1
+
−
−
− +
−
− +
−
−
+
kk k
k k
f h
x x h s
x x s
h x x h s
x
x
h s
4 1
2 3
2
2
2 2
2
24 1 6
1 12
1
4 1 4
1 8
1 6
1
+
−
− +
−
+
−
−
+
+
−
+
−
+
k sk k
k k
f h
x x s h
x x s s
h x x s s h
x x s
h s .(3.3)
and its first derivative is, )
, ( x s
Y ′ =
3 1
2
2
2 2
2 1
6 1
2 1 4
1 2
1 2
1 8
1 6
1 1
1
+ + +
− +
−
−
+
−
−
+
−
+
+
−
k sk
k k
k
k
f
h x x s
h x x s s
h x x s s
s s h h Y
h Y
3 1
2 2
2
2 2
3 1 1
1 1 2 1 4
1
+
−
−
− +
−
− +
−
+
kk k
k
f h
x x s h
x x s
h x x s
h s (3.4)
3 1
2 2
2
2 2
6 1 2
1 4
1
2 1 2
1 8
1 6
1
+
−
− +
−
+
−
−
+
+
+
−
+
k sk k
k
f h
x x s h
x x s s
h x x s s s
h s
) , ( x s
Y is evaluated at x = x
k−s+1, x
k+s+1and x
k+mrespectively to obtain the generalized discrete hybrid schemes from (3.3) as,
k k
s
k
s Y sY
Y
− +1− [ 1 − ]
+1− =
− − +
+
− + +
+
− −
2 + +1 2 +1 2 − +12
8 24
1 12
1 12
5 12
1 2 24
24 1 12
1
s k k
s
k
s f
f s s s f s
s h s
k k
s
k
s Y sY
Y
+ +1− [ 1 + ]
+1+ =
+ −
+
− +
+
− + +
+ + + + +12 1
2 1
2 2
24 24
1 12
1 12
5 12
1 2 8
24 1 12
1
s k k
s
k
s f
f s s s f s
s h s
k k
k
Y Y
Y
+2− 2
+1+ =
+
−
+
+ − ++
+ 1 2 1 2 1
2 2
12 1 6
1 1 12
1
s k k
s
k
f
f s f s
h s
(3.5)
(3.3) which involves function of ( ) x, s is the generalized continuous formulation of a class of
−
2 step continuous hybrid scheme which yield the generalized discrete block methods (3.5) on
substitution of an appropriate value of s ∈ Q , s ∈ ( ) 0 , 1 . Similarly, the derivative (3.4) is
evaluated at x = x
k−s+1x
k+s+1, and x
k+mrespectively to obtain a generalized block scheme,
k k
s
k
Y
Y h
Y h 1 1
1
1
− +
′
− + +=
+ −
+
− −
+
− +
2 ++1 2 +1 2 − +112 5 24
1 12
1 12
1 3 2 2 1 24
1 12
1
12
k s ks f
k ss f s
s f s
s s h s
k k
s
k
Y
Y h
Y h 1 1
1
1
− +
′
+ + +=
+ −
+
− +
+
− + +
2 ++1 2 +1 2 − +112 24
1 12
1 3
2 12
1 2 1 24
1 12 5 12
1
s k k
s
k
s f
s f s
s f s
s s h s
k k
k
Y
Y h
Y h 1 1
1
2
− +
′
+ +=
−
+
− +
2+
+ +1 2 +1 2 − +16 1 24
5 12
5 2 3 6
1 24
5
s k k
s
k
f
s s s f
s f h s
(3.6)
which is used to obtain derivative values ( y′ ) present in the f
n+j= f ( x
n+j, y
n+j, y
n′
+j) in the main problem (2.1).
4. Examples of schemes derived from the generalized continuous scheme We shall now substitute into the generalized CLMS (3.3) at
4
= 1
s , we obtain the CLMS,
k k k
k
Y
h x Y x
h x x x
Y
−
− +
−
=
+1
)
(
1+
− +
−
−
− +
−
− +
−
−
− +
−
−
− +
− +
−
−
− +
−
−
+ + +
4 3 4 3
2
1 4 3
2
4 5 4 3
2
2
3 3 2
3 5 8
3 4 3
16 2
15 2
7
3 2 3
3 7 3
4
k k k
k k
k k k
k k
k k k
k k
h f x x h
x x h
x x h
x x
h f x x h
x x h
x x h
x x
h f x x h
x x h
x x h
x x
h
(4.1a) Substituting
4
= 1
s in the generalized discrete block scheme or evaluating (4.1a) at
+2
= x
kx , x
k+s+1and x
k−s+1yields,
k
k
Y
kY
Y 4
1 4
3
1 4
3
−
+−
+
=
− + −
+ +
+ 4
5 1
4 3 2
11 30
128 h 31 f
kf
kf
kk k k
Y Y
Y 4
1 4
5
1 4
5
−
++
+
=
− +
+ +
+ 4
5 1
4 3 2
35 70
384 h 95 f
kf
kf
kk k
k
Y Y
Y
+2− 2
+1+
=
− +
+ +
+ 4
5 1
4 3 2
4 5
3 4 f
kf
kf
kh
(4.1b)
Substituting again into the generalized CLMS (3.3) at 2
= 1
s , we obtain the CLMS,
k k k
k
Y
h x Y x
h x x x
Y
−
− +
−
=
+1
)
(
1+
− +
−
−
− +
−
− +
−
−
− +
−
−
− +
− +
−
−
− +
−
−
+ + +
2 1 4 3
2
1 4 3
2
2 3 4 3
2
2
6 1 6
5 2
3 6
5
3 1 3
4 2
3 2
1
6 1 2
1 2
1 6
1
k k k
k k
k k k
k k
k k k
k k
h f x x h
x x h
x x h
x x
h f x x h
x x h
x x h
x x
h f x x h
x x h
x x h
x x
h
(
4.2a)Substituting 2
= 1
s in the generalized discrete block scheme or evaluating (4.2a) at
+2
= x
kx , x
k+s+1and x
k−s+1yields,
k
k
Y
kY
Y 2
1 2
1
1 2
1
−
+−
+
=
− + −
+ +
+ 2
3 1 2
1 2
2
96 13 f
kf
kf
kh
k k k
Y Y
Y 2
1 2
3
1 2
3
−
++
+
=
+ +
+ +
+ 2
3 1 2
1 2
6
32 5 f
kf
kf
kh
k k
k
Y Y
Y
+2− 2
+1+ =
+ +
+ +
+ 2
3 1 2 1 2
3 f
kf
kf
kh
(4.2b) which are the schemes derived in Yahaya et al (2009).
Also substituting (3.3) at 4
= 3
s , we obtain the CLMS,
k k k
k
Y
h x Y x
h x x x
Y
−
− +
−
=
+1
)
(
1+
− +
−
−
− +
−
− +
−
−
− +
−
−
−
− +
− +
−
−
−
+ + +
4 1 4 3
2
1 4 3
2
4 7 4 3
2
2
27 2 27
11 9
7 9
4
27 4 27
16 18
7 18
1
27 2 27
5 9
1
k k k
k k
k k k
k k
k k k
k
h f x x h
x x h
x x h
x x
h f x x h
x x h
x x h
x x
h f x x h
x x h
x x
h
(4.3a) Substituting
4
= 3
s in the generalized discrete block scheme or evaluating (4.3a) at
+2
= x
kx , x
k+s+1and x
k−s+1yields,
=
−
−
++ k k
k
Y Y
Y 4
3 4
1
1 4
1
− − +
+ +
+ 4
7 1
4 1 2
5 34
1152 h 79 f
kf
kf
kk
k
Y
kY
Y 4
3 4
7
1 4
7
−
++
+
=
+ +
+ +
+ 4
7 1
4 1 2
49 574
1152 h 133 f
kf
kf
kk k
k
Y Y
Y
+2− 2
+1+
=
+ +
+ +
+ 4
7 1
4 1 2