# Block Hybrid Collocation Method with Application to Fourth Order Differential Equations

Authors: Lee Ken Yap; Fudziah Ismail
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.


---

## Body

## 1. Introduction

Fourth order ordinary differential equations (ODEs) arise in several fields such as fluid dynamics (see [1]), beam theory (see [2, 3]), electric circuits (see [4]), ship dynamics (see [5–7]), and neural networks (see [8]). Therefore, many theoretical and numerical studies dealing with such equations have appeared in the literature.Here, we consider general fourth order ordinary differential equations:(1)
y
(
4
)
=
f
t
,
y
,
y
′
,
y
′′
,
y
′′′
with the initial conditions
(2)
y
a
=
y
0
,
y
′
a
=
y
0
′
,
y
′′
a
=
y
0
′′
,
y
′′′
a
=
y
0
′′′
,
t
∈
a
,
b
.
Conventionally, fourth order problems (1) are reduced to system of first order ODEs and solved with the methods available in the literature. Many investigators [2, 9, 10] remarked the drawback of this approach as it requires heavier computational work and longer execution time. Thus, the direct approach on higher order ODEs has attracted considerable attention.Recent developments have led to the implementation of collocation method for the direct solution of fourth order ODEs (1). Awoyemi [9] proposed a multiderivative collocation method to obtain the approximation of y at t
n
+
4. Moreover, Kayode [11, 12] developed collocation methods for the approximation of y at t
n
+
5 with the predictor of orders five and six, respectively. These schemes [9, 11, 12] are implemented in predictor-corrector mode with the employment of Taylor series expansions for the computation of starting values. Jator [2] remarked that the implementation of these schemes is more costly since the subroutines for incorporating the starting values lead to lengthy computational time. Thus, some attempts have been made on the self-starting collocation method which eliminates the requirement of either predictors or starting values from other methods. Jator [2] derived a collocation multistep method and used it to generate a new self-starting finite difference method. On the other hand, Olabode and Alabi [13] developed a self-starting direct block method for the approximation of y at t
n
+
j, j
=
1,2
,
3,4.Here, we are going to derive a block hybrid collocation method for the direct solution of general fourth order ODEs (1). The method is extended from the line proposed by Jator [14] and Yap et al. [15]. We apply the interpolation and collocation technique on basic polynomials to derive the main and additional methods which are combined and used as block hybrid collocation method. This method generates the approximation of y at four main points and three off-step points concurrently.


## 2. Derivation of Block Hybrid Collocation Methods

The hybrid collocation method that generates the approximations to the general fourth order ODEs (1) is defined as follows:
(3)
∑
j
=
0
k
‍
α
j
y
n
+
j
+
∑
j
=
1
3
‍
α
ν
j
y
n
+
ν
j
=
h
4
∑
j
=
0
k
‍
β
j
f
n
+
j
+
∑
j
=
1
3
‍
β
ν
j
f
n
+
ν
j
.
We approximate the solution by considering the interpolating function
(4)
Y
t
=
∑
j
=
0
r
+
s
-
1
‍
ϕ
j
t
j
,
where t
∈
[
a
,
b
], ϕ
j are unknown coefficients to be determined, r is the number of interpolations for 4
≤
r
≤
k, and s is the number of distinct collocation points with s
>
0. The continuous approximation is constructed by imposing the conditions as follows:
(5)
Y
(
t
n
+
j
)
=
y
n
+
j
,
j
=
0,1
,
2
,
…
,
r
-
1
,
(6)
Y
4
t
n
+
μ
=
f
n
+
μ
,
μ
=
j
,
ν
1
,
ν
2
,
ν
3
,
j
=
0,1
,
2
,
…
,
k
,
where ν
1, ν
2, and ν
3 are not integers. By considering r
=
4 and s
=
8, we interpolate (5) at the points t
n, t
n
+
1, t
n
+
2, and t
n
+
3 and collocate (6) at the points t
n, t
n
+
1
/
2, t
n
+
1, t
n
+
3
/
2, t
n
+
2, t
n
+
5
/
2, t
n
+
3, and t
n
+
4. This leads to a system of twelve equations which is solved by Mathematica. The values of ϕ
j are substituted into (4) to develop the multistep method:
(7)
Y
t
=
∑
j
=
0
k
‍
α
j
y
n
+
j
+
h
4
∑
j
=
0
k
‍
β
j
f
n
+
j
+
∑
j
=
1
3
‍
β
ν
j
f
n
+
ν
j
,
where α
j, β
j, and β
ν
j are constant coefficients. Hence, the block hybrid collocation method can be derived as follows.Main Method. Consider the following
(8)
y
n
+
4
=
4
y
n
+
3
-
6
y
n
+
2
+
4
y
n
+
1
-
y
n
+
h
4
15120
×
19
f
n
+
1804
f
n
+
1
+
2560
f
n
+
3
/
2
+
6354
f
n
+
2
d
.
d
+
2560
f
n
+
5
/
2
+
1804
f
n
+
3
+
19
f
n
+
4
.Additional Method. Consider the following
(9)
y
n
+
1
/
2
=
1
16
y
n
+
3
-
5
16
y
n
+
2
+
15
16
y
n
+
1
+
5
16
y
n
+
h
4
7741440
×
75
f
n
-
25032
f
n
+
1
/
2
-
122530
f
n
+
1
d
.
d
-
107760
f
n
+
3
/
2
-
43080
f
n
+
2
-
3880
f
n
+
5
/
2
d
.
d
-
198
f
n
+
3
+
5
f
n
+
4
,
y
n
+
3
/
2
=
-
1
16
y
n
+
3
+
9
16
y
n
+
2
+
9
16
y
n
+
1
-
1
16
y
n
+
h
4
430080
×
f
n
+
260
f
n
+
1
/
2
+
2311
f
n
+
1
+
4936
f
n
+
3
/
2
d
.
d
+
2311
f
n
+
2
+
260
f
n
+
5
/
2
+
f
n
+
3
,
y
n
+
5
/
2
=
5
16
y
n
+
3
+
15
16
y
n
+
2
-
5
16
y
n
+
1
+
1
16
y
n
-
h
4
7741440
×
23
f
n
+
4680
f
n
+
1
/
2
+
41330
f
n
+
1
d
.
d
+
110000
f
n
+
3
/
2
+
120780
f
n
+
2
+
25832
f
n
+
5
/
2
d
d
-
250
f
n
+
3
+
5
f
n
+
4
.The general fourth order differential equations involve the first, second, and third derivatives. In order to generate the formula for the derivatives, the values ofϕ
j are substituted into
(10)
Y
′
t
=
∑
j
=
1
r
+
s
-
1
‍
j
ϕ
j
t
j
-
1
,
Y
′′
t
=
∑
j
=
2
r
+
s
-
1
‍
j
j
-
1
ϕ
j
t
j
-
2
,
Y
′′′
t
=
∑
j
=
3
r
+
s
-
1
‍
j
j
-
1
j
-
2
ϕ
j
t
j
-
3
.This is obtained by imposing that(11)
Y
′
t
=
1
h
∑
j
=
0
k
‍
α
j
′
y
n
+
j
+
h
4
∑
j
=
0
k
‍
β
j
′
f
n
+
j
+
∑
j
=
1
3
‍
β
ν
j
′
f
n
+
ν
j
d
d
d
×
∑
j
=
0
k
‍
β
j
′
f
n
+
j
+
∑
j
=
1
3
‍
β
ν
j
′
f
n
+
ν
j
,
(12)
Y
′′
t
=
1
h
2
∑
j
=
0
k
‍
α
j
′′
y
n
+
j
+
h
4
∑
j
=
0
k
‍
β
j
′′
f
n
+
j
+
∑
j
=
1
3
‍
β
′′
ν
j
f
n
+
ν
j
d
d
d
×
∑
j
=
0
k
‍
β
j
′′
f
n
+
j
+
∑
j
=
1
3
‍
β
ν
j
′′
f
n
+
ν
j
,
(13)
Y
′′′
t
=
1
h
3
∑
j
=
0
k
‍
α
j
′′′
y
n
+
j
+
h
4
∑
j
=
0
k
‍
β
j
′′′
f
n
+
j
+
∑
j
=
1
3
‍
β
ν
j
′′′
f
n
+
ν
j
d
d
d
×
∑
j
=
0
k
‍
β
j
′′′
f
n
+
j
+
∑
j
=
1
3
‍
β
ν
j
′′′
f
n
+
ν
j
.The formula for the first, second, and third derivatives is depicted in Tables1, 2, and 3, respectively.Table 1
Coefficientsα
i
′ and β
i
′ for the method (11) evaluated at t
n
+
i
/
2 for i
=
0,1
,
…
,
6 and t
n
+
4.
t
y
n
y
n
+
1
y
n
+
2
y
n
+
3
f
n
f
n
+
1
/
2
f
n
+
1
f
n
+
3
/
2
f
n
+
2
f
n
+
5
/
2
f
n
+
3
f
n
+
4
t
n
-
11
6
3
-
3
2
1
3
-
3397
3326400
-
146
3465
-
16201
166320
-
116
1485
-
3089
110880
-
26
7425
1
23760
-
1
665280
t
n
+
1
/
2
-
23
24
7
8
1
8
-
1
24
7
12165120
114727
13305600
636767
31933440
2461
295680
28903
6082560
-
127
1995840
179
1900800
-
593
255467520
t
n
+
1
-
1
3
-
1
2
1
-
1
6
127
9979200
113
34650
13613
498960
5827
155925
503
36960
569
311850
-
71
2494800
1
1425600
t
n
+
3
/
2
1
24
-
9
8
9
8
-
1
24
-
197
17031168
-
13
42240
-
4231
1064448
-
409
1900800
4079
946176
41
266112
481
10644480
-
409
425779200
t
n
+
2
1
6
-
1
1
2
1
3
13
3326400
-
89
51975
-
6913
498960
-
214
5775
-
9157
332640
-
491
155925
-
31
831600
1
1425600
t
n
+
5
/
2
1
24
-
1
8
-
7
8
23
24
-
2359
182476800
-
13
42240
-
17971
4561920
-
74749
7983360
-
271477
14192640
-
179503
19958400
23
285120
-
593
255467520
t
n
+
3
-
1
3
3
2
-
3
11
6
1
95040
113
34650
4721
166320
23
297
10859
110880
871
20790
893
831600
-
1
665280
t
n
+
4
-
11
6
7
-
19
2
13
3
2159
285120
-
146
3465
36541
99792
2956
155925
3697
3168
2918
31185
33047
71280
85741
9979200Table 2
Coefficientsα
i
′
′ and β
i
′
′ for the method (12) evaluated at t
n
+
i
/
2 for i
=
0,1
,
…
,
6 and t
n
+
4.
t
y
n
y
n
+
1
y
n
+
2
y
n
+
3
f
n
f
n
+
1
/
2
f
n
+
1
f
n
+
3
/
2
f
n
+
2
f
n
+
5
/
2
f
n
+
3
f
n
+
4
t
n
2
-
5
4
-
1
1597
100800
169
675
4439
15120
446
1575
1601
30240
37
1575
-
71
25200
23
302400
t
n
+
1
/
2
3
2
-
7
2
5
2
-
1
2
-
1973
3225600
107
19200
1933
15120
65911
604800
14927
322560
437
134400
439
1209600
-
13
1382400
t
n
+
1
1
-
2
1
0
1
12096
-
107
9450
-
185
3024
-
7
675
-
29
30240
1
1890
-
1
8400
1
302400
t
n
+
3
/
2
1
2
-
1
2
-
1
2
1
2
-
59
2419200
-
5753
1209600
-
4253
96768
-
13411
120960
-
4253
96768
-
5753
1209600
-
59
2419200
0
t
n
+
2
0
1
-
2
1
-
1
302400
0
1
5040
-
8
675
-
121
2016
-
8
675
1
5040
-
1
302400
t
n
+
5
/
2
-
1
2
5
2
-
7
2
3
2
109
3225600
5753
1209600
10399
241920
68459
604800
120527
967680
1223
172800
-
569
604800
13
1382400
t
n
+
3
-
1
4
-
5
2
-
47
302400
107
9450
401
5040
1177
4725
9683
30240
2251
9450
1399
75600
-
23
302400
t
n
+
4
-
2
7
-
8
3
10247
302400
-
169
675
1121
1008
-
146
105
18653
6048
-
1781
1575
106483
75600
143
2880Table 3
Coefficientsα
i
′
′
′ and β
i
′
′
′ for the method (13) evaluated at t
n
+
i
/
2 for i
=
0,1
,
…
,
6 and t
n
+
4.
t
y
n
y
n
+
1
y
n
+
2
y
n
+
3
f
n
f
n
+
1
/
2
f
n
+
1
f
n
+
3
/
2
f
n
+
2
f
n
+
5
/
2
f
n
+
3
f
n
+
4
t
n
-
1
3
-
3
1
-
33569
226800
-
2357
3150
-
275
4536
-
8747
14175
61
360
-
3317
28350
2507
113400
-
19
32400
t
n
+
1
/
2
-
1
3
-
3
1
139673
29030400
-
35453
201600
-
789907
1451520
-
134153
907200
-
9473
64512
26767
1814400
-
5101
1036800
3601
29030400
t
n
+
1
-
1
3
-
3
1
-
239
226800
73
3150
-
739
5670
-
4577
14175
-
61
1260
-
647
28350
34
14175
-
13
226800
t
n
+
3
/
2
-
1
3
-
3
1
14393
29030400
1027
201600
160589
1451520
1201
129600
-
40357
322560
2767
1814400
-
14107
7257600
1201
29030400
t
n
+
2
-
1
3
-
3
1
-
89
226800
43
3150
1553
22680
4213
14175
379
2520
-
131
4050
347
113400
-
13
226800
t
n
+
5
/
2
-
1
3
-
3
1
2399
4147200
1027
201600
30025
290304
184567
907200
161531
322560
355087
1814400
-
66427
7257600
3601
29030400
t
n
+
3
-
1
3
-
3
1
-
359
226800
73
3150
29
810
5023
14175
67
252
18553
28350
2389
14175
-
19
32400
t
n
+
4
-
1
3
-
3
1
21631
226800
-
2357
3150
61601
22680
-
70187
14175
17131
2520
-
126197
28350
45341
16200
55067
226800


## 3. Order and Stability Properties

Following the idea of Henrici [16] and Jator [2, 14], the linear difference operator associated with (3) is defined as
(14)
L
y
t
;
h
=
∑
j
=
0
k
‍
α
j
y
t
+
j
h
-
h
4
β
j
y
4
t
+
j
h
+
∑
j
=
1
3
‍
α
ν
j
y
t
+
ν
j
h
-
h
4
β
ν
j
y
4
t
+
ν
j
h
,
where y
(
t
) is an arbitrary function that is sufficiently differentiable. Expanding the test functions y
(
t
+
j
h
) and y
(
4
)
(
t
+
j
h
) about t and collecting the terms we obtain
(15)
L
y
(
x
)
;
h
=
C
0
y
(
x
)
+
C
1
h
y
′
(
x
)
+
⋯
+
C
q
h
q
y
(
q
)
(
x
)
+
⋯
whose coefficients C
q for q
=
0,1
,
… are constants and given as
(16)
C
0
=
∑
j
=
0
k
‍
α
j
+
∑
j
=
1
3
‍
α
ν
j
,
C
1
=
∑
j
=
1
k
‍
j
α
j
+
∑
j
=
1
3
‍
ν
j
α
ν
j
,
⋮
C
q
=
1
q
!
∑
j
=
1
k
‍
j
q
α
j
+
∑
j
=
1
3
‍
ν
j
q
α
ν
j
-
q
(
q
-
1
)
d
.
d
×
(
q
-
2
)
(
q
-
3
)
∑
j
=
1
k
‍
j
q
-
4
β
j
+
∑
j
=
1
3
‍
ν
j
q
-
4
β
ν
j
.
According to Jator [2], the linear multistep method is said to be of order p if
(17)
C
0
=
C
1
=
⋯
=
C
p
+
3
=
0
,
C
p
+
4
≠
0
.
The main method (8) and the additional methods (9) are the order eight methods with the error constants; C
12 are -
1
/
207360, -
13
/
679477248, -
1
/
7927234560, and 29
/
4756340736, respectively. With the order p
>
1, we stipulate the consistency of the method (see [2, 16]).In the sense of Jator [2], the hybrid methods (8)-(9) are normalized in block form to analyze the zero stability. The first characteristic polynomial is defined as
(18)
ρ
(
z
)
=
Det
z
A
0
-
A
1
=
z
6
(
z
-
1
)
with
(19)
A
(
0
)
=
7
×
7
Identity
matrix
,
A
(
1
)
=
0
-
5
4
0
-
5
2
0
-
10
3
5
16
0
22
3
0
44
3
0
176
9
-
11
6
0
1
4
0
1
2
0
2
3
-
1
16
0
-
8
0
-
16
0
-
64
3
2
0
-
1
4
0
-
1
2
0
-
2
3
1
16
0
4
0
8
0
32
3
-
1
0
4
0
8
0
32
3
-
1
.Since the roots of (18) satisfy |
z
i
|
≤
1 for i
=
1,2
,
…
,
7, the method is zero stable.


## 4. Numerical Experiment

The following problems are solved numerically to illustrate the efficiency of the block hybrid collocation method.Problem 1.
Consider the linear fourth order problem (see [2]):
(20)
y
(
4
)
=
y
′′′
+
y
′′
+
y
′
+
2
y
,
0
≤
t
≤
2
,
y
(
0
)
=
y
′
(
0
)
=
y
′′
(
0
)
=
0
,
y
′′′
(
0
)
=
30
,
and theoretical solution: y
(
t
)
=
2
e
2
t
-
5
e
-
t
+
3
cos
⁡
⁡
t
-
9
sin
⁡
t.Problem 2.
Consider the nonlinear fourth order problem (see [9]):
(21)
y
(
4
)
=
(
y
′
)
2
-
y
y
′′′
-
4
t
2
+
e
t
1
+
t
2
-
4
t
,
0
≤
t
≤
1
,
y
0
=
y
′
0
=
1
,
y
′′
0
=
3
,
y
′′′
(
0
)
=
1
,
and theoretical solution: y
(
t
)
=
t
2
+
e
t.The block hybrid collocation method is implemented together with theMathematica built-in packages, namely,  Solve and  FindRoot for the solution of linear and nonlinear problems, respectively.The performance comparison between block hybrid collocation method with the existing methods [2, 9] and the Adams Bashforth-Adams Moulton method is presented in Tables 4 and 5. The following notations are used in the tables:h: step size;
BHCM4: block hybrid collocation method;
Adams: Adams Bashforth-Adams Moulton method;
Awoyemi: multiderivative collocation method in Awoyemi [9];
Jator: finite difference method in Jator [2].Table 4
Numerical results for Problem1.
h
Method
Absolute error att
=
2
0.1
BHCM4
1.74 (−8)
Adams
2.11 (−3)
Jator
1.26 (−4)
0.05
BHCM4
8.45 (−11)
Adams
5.37 (−4)
Jator
1.91 (−6)
0.025
BHCM4
3.69 (−13)
Adams
5.09 (−5)
Jator
2.96 (−8)
0.02
BHCM4
7.11 (−14)
Adams
2.25 (−5)
Jator
8.65 (−9)Table 5
Numerical results for Problem2.
h
Method
Absolute error att
=
1
0.2
BHCM4
2.38 (−12)
Adams
5.01 (−7)
Awoyemi
5.84 (−4)
0.1
BHCM4
1.95 (−14)
Adams
2.44 (−6)
Awoyemi
9.26 (−5)Tables4 and 5 show the superiority of BHCM4 in terms of accuracy over the existing Adams method, Jator finite difference method [2], and Awoyemi multiderivative collocation method [9].


## 5. Application to Problem from Ship Dynamics [5–7]

The proposed method is also applied to solve a physical problem from ship dynamics. As stated by Wu et al. [5], when a sinusoidal wave of frequency Ω passes along a ship or offshore structure, the resultant fluid actions vary with time t. In a particular case study by Wu et al. [5], the fourth order problem is defined as
(22)
y
(
4
)
+
3
y
′′
+
y
2
+
ϵ
cos
⁡
Ω
t
=
0
,
t
>
0
,
which is subjected to the following initial conditions:
(23)
y
(
0
)
=
1
,
y
′
(
0
)
=
y
′′
(
0
)
=
y
′′′
(
0
)
=
0
,
where ϵ
=
0 for the existence of the theoretical solution, y
(
t
)
=
2
cos
⁡
t
-
cos
⁡
(
t
2
). The theoretical solution is undefined when ϵ
≠
0 (see [6]).In the literature, some numerical methods for solving fourth order ODEs have been extended to solve the problem from ship dynamics. Numerical investigation was presented in Twizell [6] and Cortell [7] concerning the fourth order ODEs (22) for the cases ϵ
=
0 and ϵ
=
1 with Ω
=
0.25
(
2
-
1
). Instead of solving the fourth order ODEs directly, Twizell [6] and Cortell [7] considered the conventional approach of reduction to system of first order ODEs. Twizell [6] developed a family of numerical methods with the global extrapolation to increase the order of the methods. On the other hand, Cortell [7] proposed the extension of the classical Runge-Kutta method.Table6 shows the comparison in terms of accuracy for y at the end point t
=
15. BHCM4 manages to achieve better accuracy compared to Adams Bashforth-Adams Moulton method, Twizell [6], and Cortell [7] when h
=
0.25 and h
=
0.1, respectively.Table 6
Performance comparison for Wu equation withϵ
=
0.
h
Method
Absolute error att
=
15
0.25
BHCM4
5.2 (−7)
Adams
4.9 (−3)
Twizell
1.9 (−4)
0.1
BHCM4
2.8 (−10)
Adams
8.4 (−5)
Cortell
3.7 (−5)Figure1 depicts the numerical solution for Wu equation (22) with ϵ
=
1 and Ω
=
0.25
(
2
-
1
) in the interval [
0,15
]. The solutions obtained by BHCM4 are in agreement with the observation of Cortell [7] andMathematica built-in package NDSolve.Figure 1
Response curve for Wu equation withϵ
=
1, Ω
=
0.25
(
2
-
1
).


## 6. Conclusion

As indicated in the numerical results, the block hybrid collocation method has significant improvement over the existing methods. Furthermore, it is applicable for the solution of physical problem from ship dynamics.As a conclusion, the block hybrid collocation method is proposed for the direct solution of general fourth order ODEs whereby it is implemented as self-starting method that generates the solution ofy at four main points and three off-step points concurrently.



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*Source: 561489-2015-02-22.xml*