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Research ArticleBlock Hybrid Collocation Method with Application toFourth Order Differential Equations

Lee Ken Yap1,2 and Fudziah Ismail2

1Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Setapak, 53300 Kuala Lumpur, Malaysia2Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM),43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to Lee Ken Yap; [email protected]

Received 10 July 2014; Revised 22 November 2014; Accepted 25 November 2014

Academic Editor: Maŕıa Isabel Herreros

Copyright © 2015 L. K. Yap and F. Ismail. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinarydifferential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main andadditional 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 themethod.Themethod is also applied to solve the fourth orderproblem from ship dynamics.

1. Introduction

Fourth order ordinary differential equations (ODEs) arisein several fields such as fluid dynamics (see [1]), beamtheory (see [2, 3]), electric circuits (see [4]), ship dynamics(see [5–7]), and neural networks (see [8]). Therefore, manytheoretical andnumerical studies dealingwith such equationshave appeared in the literature.

Here, we consider general fourth order ordinary differen-tial equations:

𝑦(4)

= 𝑓 (𝑡, 𝑦, 𝑦, 𝑦, 𝑦) (1)

with the initial conditions𝑦 (𝑎) = 𝑦0, 𝑦

(𝑎) = 𝑦

0,

𝑦(𝑎) = 𝑦

0 , 𝑦

(𝑎) = 𝑦

0 , 𝑡 ∈ [𝑎, 𝑏] .

(2)

Conventionally, fourth order problems (1) are reduced tosystem of first order ODEs and solved with the methodsavailable in the literature. Many investigators [2, 9, 10]remarked the drawback of this approach as it requiresheavier computational work and longer execution time.Thus,the direct approach on higher order ODEs has attractedconsiderable attention.

Recent developments have led to the implementation ofcollocation method for the direct solution of fourth orderODEs (1). Awoyemi [9] proposed a multiderivative colloca-tion method to obtain the approximation of 𝑦 at 𝑡𝑛+4.Moreover, Kayode [11, 12] developed collocation methods forthe approximation of 𝑦 at 𝑡𝑛+5 with the predictor of ordersfive and six, respectively. These schemes [9, 11, 12] are imple-mented in predictor-corrector mode with the employmentof Taylor series expansions for the computation of startingvalues. Jator [2] remarked that the implementation of theseschemes is more costly since the subroutines for incorporat-ing the starting values lead to lengthy computational time.Thus, some attempts have been made on the self-startingcollocation method which eliminates the requirement ofeither predictors or starting values from other methods. Jator[2] derived a collocation multistep method and used it togenerate a new self-starting finite difference method. On theother hand, Olabode and Alabi [13] developed a self-startingdirect block method for the approximation of 𝑦 at 𝑡𝑛+𝑗, 𝑗 =1, 2, 3, 4.

Here, we are going to derive a block hybrid collocationmethod 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

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 561489, 6 pageshttp://dx.doi.org/10.1155/2015/561489

2 Mathematical Problems in Engineering

collocation technique on basic polynomials to derive themain and additional methods which are combined and usedas block hybrid collocation method. This method generatesthe approximation of 𝑦 at four main points and three off-steppoints concurrently.

2. Derivation of Block HybridCollocation Methods

The hybrid collocation method that generates the approxi-mations to the general fourth order ODEs (1) is defined asfollows:

𝑘

∑

𝑗=0

𝛼𝑗𝑦𝑛+𝑗 +

3

∑

𝑗=1

𝛼]𝑗𝑦𝑛+]𝑗

= ℎ4(

𝑘

∑

𝑗=0

𝛽𝑗𝑓𝑛+𝑗 +

3

∑

𝑗=1

𝛽]𝑗𝑓𝑛+]𝑗) .

(3)

We approximate the solution by considering the interpolatingfunction

𝑌 (𝑡) =

𝑟+𝑠−1

∑

𝑗=0

𝜙𝑗𝑡𝑗, (4)

where 𝑡 ∈ [𝑎, 𝑏], 𝜙𝑗 are unknown coefficients to be deter-mined, 𝑟 is the number of interpolations for 4 ≤ 𝑟 ≤ 𝑘, and𝑠 is the number of distinct collocation points with 𝑠 > 0.The continuous approximation is constructed by imposingthe conditions as follows:

𝑌 (𝑡𝑛+𝑗) = 𝑦𝑛+𝑗, 𝑗 = 0, 1, 2, . . . , 𝑟 − 1, (5)

𝑌(4)

(𝑡𝑛+𝜇) = 𝑓𝑛+𝜇, 𝜇 = {𝑗, ]1, ]2, ]3} , 𝑗 = 0, 1, 2, . . . , 𝑘,(6)

where ]1, ]2, and ]3 are not integers. By considering 𝑟 = 4 and𝑠 = 8, we interpolate (5) at the points 𝑡𝑛, 𝑡𝑛+1, 𝑡𝑛+2, and 𝑡𝑛+3and collocate (6) at the points 𝑡𝑛, 𝑡𝑛+1/2, 𝑡𝑛+1, 𝑡𝑛+3/2, 𝑡𝑛+2, 𝑡𝑛+5/2,𝑡𝑛+3, and 𝑡𝑛+4.This leads to a systemof twelve equationswhichis solved byMathematica.The values of𝜙𝑗 are substituted into(4) to develop the multistep method:

𝑌 (𝑡) =

𝑘

∑

𝑗=0

𝛼𝑗𝑦𝑛+𝑗 + ℎ4(

𝑘

∑

𝑗=0

𝛽𝑗𝑓𝑛+𝑗 +

3

∑

𝑗=1

𝛽]𝑗𝑓𝑛+]𝑗) , (7)

where 𝛼𝑗, 𝛽𝑗, and 𝛽]𝑗 are constant coefficients. Hence, theblock hybrid collocation method can be derived as follows.

Main Method. Consider the following

𝑦𝑛+4 = 4𝑦𝑛+3 − 6𝑦𝑛+2 + 4𝑦𝑛+1 − 𝑦𝑛 +ℎ4

15120

× (19𝑓𝑛 + 1804𝑓𝑛+1 + 2560𝑓𝑛+3/2 + 6354𝑓𝑛+2

+ 2560𝑓𝑛+5/2 + 1804𝑓𝑛+3 + 19𝑓𝑛+4) .

(8)

Additional Method. Consider the following

𝑦𝑛+1/2 =1

16𝑦𝑛+3 −

5

16𝑦𝑛+2 +

15

16𝑦𝑛+1 +

5

16𝑦𝑛 +

ℎ4

7741440

× (75𝑓𝑛 − 25032𝑓𝑛+1/2 − 122530𝑓𝑛+1

− 107760𝑓𝑛+3/2 − 43080𝑓𝑛+2 − 3880𝑓𝑛+5/2

− 198𝑓𝑛+3 + 5𝑓𝑛+4) ,

𝑦𝑛+3/2 = −1

16𝑦𝑛+3 +

9

16𝑦𝑛+2 +

9

16𝑦𝑛+1 −

1

16𝑦𝑛 +

ℎ4

430080

× (𝑓𝑛 + 260𝑓𝑛+1/2 + 2311𝑓𝑛+1 + 4936𝑓𝑛+3/2

+ 2311𝑓𝑛+2 + 260𝑓𝑛+5/2 + 𝑓𝑛+3) ,

𝑦𝑛+5/2 =5

16𝑦𝑛+3 +

15

16𝑦𝑛+2 −

5

16𝑦𝑛+1 +

1

16𝑦𝑛 −

ℎ4

7741440

× (23𝑓𝑛 + 4680𝑓𝑛+1/2 + 41330𝑓𝑛+1

+ 110000𝑓𝑛+3/2 + 120780𝑓𝑛+2 + 25832𝑓𝑛+5/2

− 250𝑓𝑛+3 + 5𝑓𝑛+4) .

(9)

Thegeneral fourth order differential equations involve thefirst, second, and third derivatives. In order to generate theformula for the derivatives, the values of 𝜙𝑗 are substitutedinto

𝑌(𝑡) =

𝑟+𝑠−1

∑

𝑗=1

𝑗𝜙𝑗𝑡𝑗−1

,

𝑌(𝑡) =

𝑟+𝑠−1

∑

𝑗=2

𝑗 (𝑗 − 1) 𝜙𝑗𝑡𝑗−2

,

𝑌

(𝑡) =

𝑟+𝑠−1

∑

𝑗=3

𝑗 (𝑗 − 1) (𝑗 − 2) 𝜙𝑗𝑡𝑗−3

.

(10)

This is obtained by imposing that

𝑌(𝑡) =

1

ℎ(

𝑘

∑

𝑗=0

𝛼

𝑗𝑦𝑛+𝑗 + ℎ4

× (

𝑘

∑

𝑗=0

𝛽

𝑗𝑓𝑛+𝑗 +

3

∑

𝑗=1

𝛽

]𝑗𝑓𝑛+]𝑗)) ,

(11)

𝑌(𝑡) =

1

ℎ2(

𝑘

∑

𝑗=0

𝛼

𝑗 𝑦𝑛+𝑗 + ℎ4

× (

𝑘

∑

𝑗=0

𝛽

𝑗 𝑓𝑛+𝑗 +

3

∑

𝑗=1

𝛽

]𝑗𝑓𝑛+]𝑗)) ,

(12)

Mathematical Problems in Engineering 3

Table 1: Coefficients 𝛼𝑖 and 𝛽𝑖 for the method (11) evaluated at 𝑡𝑛+𝑖/2 for 𝑖 = 0, 1, . . . , 6 and 𝑡𝑛+4.

𝑡 𝑦𝑛 𝑦𝑛+1 𝑦𝑛+2 𝑦𝑛+3 𝑓𝑛 𝑓𝑛+1/2 𝑓𝑛+1 𝑓𝑛+3/2 𝑓𝑛+2 𝑓𝑛+5/2 𝑓𝑛+3 𝑓𝑛+4

𝑡𝑛 −11

63 −

3

2

1

3−

3397

3326400−

146

3465−

16201

166320−

116

1485−

3089

110880−

26

7425

1

23760−

1

665280

𝑡𝑛+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

𝑡𝑛+1 −1

3−1

21 −

1

6

127

9979200

113

34650

13613

498960

5827

155925

503

36960

569

311850−

71

2494800

1

1425600

𝑡𝑛+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

𝑡𝑛+2

1

6−1

1

2

1

3

13

3326400−

89

51975−

6913

498960−

214

5775−

9157

332640−

491

155925−

31

831600

1

1425600

𝑡𝑛+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

𝑡𝑛+3 −1

3

3

2−3

11

6

1

95040

113

34650

4721

166320

23

297

10859

110880

871

20790

893

831600−

1

665280

𝑡𝑛+4 −11

67 −

19

2

13

3

2159

285120−

146

3465

36541

99792

2956

155925

3697

3168

2918

31185

33047

71280

85741

9979200



𝑡𝑛 2 −5 4 −11597

100800

169

675

4439

15120

446

1575

1601

30240

37

1575−

71

25200

23

302400

𝑡𝑛+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

𝑡𝑛+1 1 −2 1 01

12096−

107

9450−

185

3024−

7

675−

29

30240

1

1890−

1

8400

1

302400

𝑡𝑛+3/2

1

2−1

2−1

2

1

2−

59

2419200−

5753

1209600−

4253

96768−

13411

120960−

4253

96768−

5753

1209600−

59

24192000

𝑡𝑛+2 0 1 −2 1 −1

3024000 1

5040−

8

675−

121

2016−

8

675

1

5040−

1

302400

𝑡𝑛+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

𝑡𝑛+3 −1 4 −5 2 −47

302400

107

9450

401

5040

1177

4725

9683

30240

2251

9450

1399

75600−

23

302400

𝑡𝑛+4 −2 7 −8 310247

302400−169

675

1121

1008−146

105

18653

6048−1781

1575

106483

75600

143

2880

𝑌

(𝑡) =1

ℎ3(

𝑘

∑

𝑗=0

𝛼

𝑗 𝑦𝑛+𝑗 + ℎ4

×(

𝑘

∑

𝑗=0

𝛽

𝑗 𝑓𝑛+𝑗 +

3

∑

𝑗=1

𝛽

]𝑗 𝑓𝑛+]𝑗)) .

(13)

The formula for the first, second, and third derivatives isdepicted in Tables 1, 2, and 3, respectively.

3. Order and Stability Properties

Following the idea of Henrici [16] and Jator [2, 14], the lineardifference operator associated with (3) is defined as

𝐿 [𝑦 (𝑡) ; ℎ] =

𝑘

∑

𝑗=0

[𝛼𝑗𝑦 (𝑡 + 𝑗ℎ) − ℎ4𝛽𝑗𝑦(4)

(𝑡 + 𝑗ℎ)]

+

3

∑

𝑗=1

[𝛼]𝑗𝑦 (𝑡 + ]𝑗ℎ) − ℎ4𝛽]𝑗𝑦(4)

(𝑡 + ]𝑗ℎ)] ,

(14)




𝑡𝑛 −1 3 −3 1 −33569

226800−2357

3150−

275

4536−

8747

14175

61

360−

3317

28350

2507

113400−

19

32400

𝑡𝑛+1/2 −1 3 −3 1139673

29030400−

35453

201600−

789907

1451520−134153

907200−

9473

64512

26767

1814400−

5101

1036800

3601

29030400

𝑡𝑛+1 −1 3 −3 1 −239

226800

73

3150−

739

5670−

4577

14175−

61

1260−

647

28350

34

14175−

13

226800

𝑡𝑛+3/2 −1 3 −3 114393

29030400

1027

201600

160589

1451520

1201

129600−

40357

322560

2767

1814400−

14107

7257600

1201

29030400

𝑡𝑛+2 −1 3 −3 1 −89

226800

43

3150

1553

22680

4213

14175

379

2520−

131

4050

347

113400−

13

226800

𝑡𝑛+5/2 −1 3 −3 12399

4147200

1027

201600

30025

290304

184567

907200

161531

322560

355087

1814400−

66427

7257600

3601

29030400

𝑡𝑛+3 −1 3 −3 1 −359

226800

73

3150

29

810

5023

14175

67

252

18553

28350

2389

14175−

19

32400

𝑡𝑛+4 −1 3 −3 121631

226800−2357

3150

61601

22680−70187

14175

17131

2520−126197

28350

45341

16200

55067

226800

where𝑦(𝑡) is an arbitrary function that is sufficiently differen-tiable. Expanding the test functions 𝑦(𝑡 + 𝑗ℎ) and 𝑦(4)(𝑡 + 𝑗ℎ)about 𝑡 and collecting the terms we obtain

𝐿 [𝑦 (𝑥) ; ℎ] = 𝐶0𝑦 (𝑥) + 𝐶1ℎ𝑦(𝑥) + ⋅ ⋅ ⋅ + 𝐶𝑞ℎ

𝑞𝑦(𝑞)

(𝑥) + ⋅ ⋅ ⋅

(15)

whose coefficients 𝐶𝑞 for 𝑞 = 0, 1, . . . are constants and givenas

𝐶0 =

𝑘

∑

𝑗=0

𝛼𝑗 +

3

∑

𝑗=1

𝛼]𝑗 ,

𝐶1 =

𝑘

∑

𝑗=1

𝑗𝛼𝑗 +

3

∑

𝑗=1

]𝑗𝛼]𝑗 ,

.

.

.

𝐶𝑞 =1

𝑞!

[

[

𝑘

∑

𝑗=1

𝑗𝑞𝛼𝑗 +

3

∑

𝑗=1

]𝑞𝑗𝛼]𝑗 − 𝑞 (𝑞 − 1)

× (𝑞 − 2) (𝑞 − 3)(

𝑘

∑

𝑗=1

𝑗𝑞−4

𝛽𝑗 +

3

∑

𝑗=1

]𝑞−4𝑗

𝛽]𝑗)]

]

.

(16)

According to Jator [2], the linear multistep method is said tobe of order 𝑝 if

𝐶0 = 𝐶1 = ⋅ ⋅ ⋅ = 𝐶𝑝+3 = 0, 𝐶𝑝+4 ̸= 0. (17)

The main method (8) and the additional methods (9)are the order eight methods with the error constants;𝐶12 are −1/207360, −13/679477248, −1/7927234560, and29/4756340736, respectively. With the order 𝑝 > 1, westipulate the consistency of the method (see [2, 16]).

In the sense of Jator [2], the hybrid methods (8)-(9) arenormalized in block form to analyze the zero stability. Thefirst characteristic polynomial is defined as

𝜌 (𝑧) = Det [𝑧𝐴(0) − 𝐴(1)] = 𝑧6 (𝑧 − 1) (18)

with

𝐴(0)

= 7 × 7 Identity matrix,

𝐴(1)

=

[[[[[[[[[[[[[[[[[[[[[[[[[

[

0 −5

40 −

5

20 −

10

3

5

16

022

30

44

30

176

9−11

6

01

40

1

20

2

3−

1

16

0 −8 0 −16 0 −64

32

0 −1

40 −

1

20 −

2

3

1

16

0 4 0 8 032

3−1

0 4 0 8 032

3−1

]]]]]]]]]]]]]]]]]]]]]]]]]

]

.

(19)

Since the roots of (18) satisfy |𝑧𝑖| ≤ 1 for 𝑖 = 1, 2, . . . , 7,the method is zero stable.

4. Numerical Experiment

The following problems are solved numerically to illustratethe efficiency of the block hybrid collocation method.

Mathematical Problems in Engineering 5

Table 4: Numerical results for Problem 1.

ℎ Method Absolute error at 𝑡 = 2

0.1BHCM4 1.74 (−8)Adams 2.11 (−3)Jator 1.26 (−4)

0.05BHCM4 8.45 (−11)Adams 5.37 (−4)Jator 1.91 (−6)

0.025BHCM4 3.69 (−13)Adams 5.09 (−5)Jator 2.96 (−8)

0.02BHCM4 7.11 (−14)Adams 2.25 (−5)Jator 8.65 (−9)

Problem 1. Consider the linear fourth order problem (see[2]):

𝑦(4)

= 𝑦

+ 𝑦+ 𝑦+ 2𝑦, 0 ≤ 𝑡 ≤ 2,

𝑦 (0) = 𝑦(0) = 𝑦

(0) = 0, 𝑦

(0) = 30,

(20)

and theoretical solution: 𝑦(𝑡) = 2𝑒2𝑡 − 5𝑒−𝑡 + 3 cos 𝑡 − 9 sin 𝑡.

Problem 2. Consider the nonlinear fourth order problem (see[9]):

𝑦(4)

= (𝑦)2− 𝑦𝑦

− 4𝑡2+ 𝑒𝑡(1 + 𝑡

2− 4𝑡) , 0 ≤ 𝑡 ≤ 1,

𝑦 (0) = 𝑦(0) = 1, 𝑦

(0) = 3, 𝑦

(0) = 1,

(21)

and theoretical solution: 𝑦(𝑡) = 𝑡2 + 𝑒𝑡.

The block hybrid collocation method is implementedtogether with the Mathematica built-in packages,namely, Solve and FindRoot for the solution of linearand nonlinear problems, respectively.

The performance comparison between block hybrid col-location method with the existing methods [2, 9] and theAdams Bashforth-Adams Moulton method is presented inTables 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 inAwoyemi [9];

Jator: finite difference method in Jator [2].

Tables 4 and 5 show the superiority of BHCM4 in terms ofaccuracy over the existing Adams method, Jator finite differ-ence method [2], and Awoyemi multiderivative collocationmethod [9].

Table 5: Numerical results for Problem 2.


0.2BHCM4 2.38 (−12)Adams 5.01 (−7)Awoyemi 5.84 (−4)

0.1BHCM4 1.95 (−14)Adams 2.44 (−6)Awoyemi 9.26 (−5)

Table 6: Performance comparison for Wu equation with 𝜖 = 0.


0.25BHCM4 5.2 (−7)Adams 4.9 (−3)Twizell 1.9 (−4)

0.1BHCM4 2.8 (−10)Adams 8.4 (−5)Cortell 3.7 (−5)

5. Application to Problem fromShip Dynamics [5–7]

The proposed method is also applied to solve a physicalproblem from ship dynamics. As stated by Wu et al. [5],when a sinusoidal wave of frequencyΩ passes along a ship oroffshore structure, the resultant fluid actions vary with time𝑡. In a particular case study by Wu et al. [5], the fourth orderproblem is defined as

𝑦(4)

+ 3𝑦+ 𝑦 (2 + 𝜖 cos (Ω𝑡)) = 0, 𝑡 > 0, (22)

which is subjected to the following initial conditions:

𝑦 (0) = 1, 𝑦(0) = 𝑦

(0) = 𝑦

(0) = 0, (23)

where 𝜖 = 0 for the existence of the theoretical solution,𝑦(𝑡) = 2 cos 𝑡−cos(𝑡√2).The theoretical solution is undefinedwhen 𝜖 ̸= 0 (see [6]).

In the literature, some numerical methods for solvingfourth order ODEs have been extended to solve the problemfrom ship dynamics. Numerical investigation was presentedin Twizell [6] and Cortell [7] concerning the fourth orderODEs (22) for the cases 𝜖 = 0 and 𝜖 = 1withΩ = 0.25(√2−1).Instead of solving the fourth order ODEs directly, Twizell[6] and Cortell [7] considered the conventional approach ofreduction to systemof first orderODEs. Twizell [6] developeda family of numerical methods with the global extrapolationto increase the order of the methods. On the other hand,Cortell [7] proposed the extension of the classical Runge-Kutta method.

Table 6 shows the comparison in terms of accuracy for 𝑦at the end point 𝑡 = 15. BHCM4 manages to achieve betteraccuracy compared to Adams Bashforth-Adams Moultonmethod, Twizell [6], and Cortell [7] when ℎ = 0.25 andℎ = 0.1, respectively.

Figure 1 depicts the numerical solution for Wu equation(22) with 𝜖 = 1 and Ω = 0.25(√2 − 1) in the interval


2 4 6 8 10 12 14

t

−40

−20

20

y(t)

NDSolve

BHCM4, h = 0.1Cortell, h = 0.1

Figure 1: Response curve for Wu equation with 𝜖 = 1, Ω =0.25(√2 − 1).

[0, 15]. The solutions obtained by BHCM4 are in agreementwith the observation of Cortell [7] andMathematica built-inpackage NDSolve.

6. Conclusion

As indicated in the numerical results, the block hybrid collo-cation method has significant improvement over the existingmethods. Furthermore, it is applicable for the solution ofphysical problem from ship dynamics.

As a conclusion, the block hybrid collocation methodis proposed for the direct solution of general fourth orderODEs whereby it is implemented as self-starting method thatgenerates the solution of 𝑦 at four main points and three off-step points concurrently.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

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[2] S. N. Jator, “Numerical integrators for fourth order initial andboundary value problems,” International Journal of Pure andApplied Mathematics, vol. 47, no. 4, pp. 563–576, 2008.

[3] O. Kelesoglu, “The solution of fourth order boundary valueproblem arising out of the beam-column theory usingAdomiandecomposition method,” Mathematical Problems in Engineer-ing, vol. 2014, Article ID 649471, 6 pages, 2014.

[4] A. Boutayeb and A. Chetouani, “A mini-review of numericalmethods for high-order problems,” International Journal ofComputer Mathematics, vol. 84, no. 4, pp. 563–579, 2007.

[5] X. J. Wu, Y. Wang, and W. G. Price, “Multiple resonances,responses, and parametric instabilities in offshore structures,”Journal of Ship Research, vol. 32, no. 4, pp. 285–296, 1988.

[6] E.H. Twizell, “A family of numericalmethods for the solution ofhigh-order general initial value problems,” Computer Methodsin Applied Mechanics and Engineering, vol. 67, no. 1, pp. 15–25,1988.

[7] R. Cortell, “Application of the fourth-order Runge-Kuttamethod for the solution of high-order general initial valueproblems,”Computers and Structures, vol. 49, no. 5, pp. 897–900,1993.

[8] A. Malek and R. Shekari Beidokhti, “Numerical solutionfor high order differential equations using a hybrid neu-ral network—optimization method,” Applied Mathematics andComputation, vol. 183, no. 1, pp. 260–271, 2006.

[9] D. O. Awoyemi, “Algorithmic collocation approach for directsolution of fourth-order initial-value problems of ordinarydifferential equations,” International Journal of ComputerMath-ematics, vol. 82, no. 3, pp. 321–329, 2005.

[10] N.Waeleh, Z. A. Majid, F. Ismail, andM. Suleiman, “Numericalsolution of higher order ordinary differential equations bydirect block code,” Journal of Mathematics and Statistics, vol. 8,no. 1, pp. 77–81, 2011.

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[13] B. T. Olabode and T. J. Alabi, “Direct block predictor-correctormethod for the solution of general fourth order ODEs,” Journalof Mathematics Research, vol. 5, no. 1, pp. 26–33, 2013.

[14] S. N. Jator, “Solving second order initial value problems by ahybrid multistep method without predictors,” Applied Mathe-matics and Computation, vol. 217, no. 8, pp. 4036–4046, 2010.

[15] L. K. Yap, F. Ismail, and N. Senu, “An accurate block hybrid col-locationmethod for third order ordinary differential equations,”Journal of Applied Mathematics, vol. 2014, Article ID 549597, 9pages, 2014.

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