Abstract and Applied Analysis

Volume 2014 (2014), Article ID 867095, 13 pages

http://dx.doi.org/10.1155/2014/867095

Research Article

Solving Nonstiff Higher-Order Ordinary Differential Equations Using 2-Point Block Method Directly

Hazizah Mohd Ijam,¹ Mohamed Suleiman,² Ahmad Fadly Nurullah Rasedee,² Norazak Senu,¹ Ali Ahmadian,1,2 and Soheil Salahshour³

¹Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

²Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

³Department of Computer Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

Received 18 July 2014; Accepted 23 August 2014; Published 17 September 2014

Academic Editor: Dumitru Baleanu

Copyright © 2014 Hazizah Mohd Ijam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We describe the development of a 2-point block backward difference method (2PBBD) for solving system of nonstiff higher-order ordinary differential equations (ODEs) directly. The method computes the approximate solutions at two points simultaneously within an equidistant block. The integration coefficients that are used in the method are obtained only once at the start of the integration. Numerical results are presented to compare the performances of the method developed with 1-point backward difference method (1PBD) and 2-point block divided difference method (2PBDD). The result indicated that, for finer step sizes, this method performs better than the other two methods, that is, 1PBD and 2PBDD.

1. Introduction

In this paper, we consider the system of th order ODEs of the form

with in the interval , where

For simplicity of discussion and without loss of generality, we consider the single equation

where

As shown in Figure 1, here the 2-point block method, the interval , is divided into series of blocks with each block containing two points; that is, and is the first block while and is the second block, where solutions to (3) are to be computed.

Figure 1: 2-point method.

Previous works on block method for solving (3) directly are given by Milne [1], Rosser [2], Shampine and Watts [3], and Chu and Hamilton [4]. According to Omar [5], both implicit and explicit block Adams methods in their divided difference form are developed for the solution of higher-order ODEs. Majid [6] has derived a code based on the variable step size and order of fully implicit block method to solve nonstiff higher-order ODEs directly. Ibrahim [7] has developed a new block backward differentiation formula method of variable step size for solving first- and second-order ODEs directly. Suleiman et al. [8] have introduced one-point backward difference methods for solving higher-order ODEs. Hence, this motivates us to extend the method to block method in solving nonstiff higher-order ODEs.

2. The Formulation of the Predict-Evaluate-Correct-Evaluate (PECE) Multistep Block Method in Its Backward Difference Form (MSBBD) for Nonstiff Higher-Order ODEs

The code developed will be using the PECE mode with constant stepsize. The predictor and corrector for first and second point will have the following form.

Predictor:

where is coefficient for predictor for and .

Corrector:

where is coefficient for corrector for and .

We also formulate the corrector in terms of the predictor. Both points and can be written as

We derived the formulation for both the predictor and corrector.

3. Derivation for Higher-Order Explicit Integration Coefficients

3.1. For the First Point

The derivation for up to third-order explicit integration coefficients for the first point has been given by Suleiman et al. [8].

3.2. For the Second Point

Integrating (3) once yields

Let be the interpolating polynomial which interpolates the values ; then

Approximating in (6) with and letting

gives

or

where

Define the generating function for the coefficient as follows:

Substituting in (14) into gives

which leads to

Equation (17) can be written as

or

Hence, the coefficients of are given by

Integrating (1) twice yields

Substituting with gives

The generating function of the coefficient is defined as follows:

Substituting (22) into above gives

Substituting into (24) yields

Equation (25) can be written as

or

Hence the coefficients of in relation to coefficients of the previous order are given by

By using the same process previously, we note that for integrating times yield

and, from (29), we get

Integrating times yield

or, in the backward difference formulation, given by

where

The generating function

Substituting (35) into above yields

As in (30), we now substitute in (37) giving

Equation (38) can be written as

or

Hence the coefficients of in relation to coefficients of the previous order are given by

4. Derivation for Higher-Order Implicit Integration Coefficients

4.1. For the First Point

The derivation for up to third-order implicit integration coefficients for the first point has been given by Suleiman et al. [8].

4.2. For the Second Point

Integrating (3) once yields

Let be the interpolating polynomial which interpolates the values ; then

As in the previous derivation, we choose

Replacing by yields

Simplify

where

Define the generating function for the coefficient as follows:

or

which leads to

For the case , the approximate solution of has the form

The coefficients are given by

where are the coefficients of the backward difference formulation of (54) which can be represented by

Define the generating function of the coefficient as follows:

Substituting (54) into above gives

Solving (57) with the substitution of (51) produces the relationship

By using the same process previously, we note that for integrating times yield

Integrating times yield

The coefficients are given by

where are the coefficients of the backward difference formulation of (62) which can be represented by

Define the generating function of the coefficient as follows:

Substituting (62) into above gives

Solving (65) with the substitution of (59) produces the relationship

5. The Relationship between the Explicit and Implicit Coefficients

5.1. For the First Point

Calculating the integration coefficients directly is time consuming when large numbers of integration are involved. An efficient technique of computing the coefficients is by formulating a recursive relationship between them. With this recursive relationship, we are able to obtain the implicit integration coefficient with minimal effort. The relationship between the explicit and implicit coefficients for the first point is already given by Suleiman et al. [8].

5.2. For the Second Point

For first-order coefficients,

It can be written as

By substituting

into (68), we have

Expanding the equation yields

This gives the recursive relationship

For second-order coefficient,

It can be written as

Substituting (70) into the equation above gives

or

Substituting (25) into (76) gives

Expanding the equation yields

This gives the recursive relationship

By using the same process previously, we note that, for -order coefficient, we have

which leads to a recursive relationship

For -order coefficient, we have

It can be written as

Substituting (80) into (83) gives

or

Substituting

into (85) leads to

Expanding the equation yields

which leads to a recursive relationship

Tables 1 and 2 are a few examples of the explicit and implicit integration coefficients.

Table 1: The explicit integration coefficients for from 0 to 6 (for ).

0
1
2
3
4
5
6

2
2
7/3
8/3
269/90
33/10
13613/3780

2
4/3
4/3
62/45
43/30
94/63
1466/945

4/3
2/3
3/5
26/45
359/630
179/315
16159/28350

Table 2: The implicit integration coefficients for from 0 to 6 (for ).

0
1
2
3
4
5
6

2

1/3
0
−1/90
−1/90
−8/945

2
/
2/3
4/90
1/90
1/315
1/1890

4/3

3/5
2/45
1/70
2/315
47/14175

6. Problem Tested

The problems shown in Table 3 are used to test the performance of the method.

Table 3: List of test problems.

Problem
Initial value
Interval

1

Exact solution:  

Source: artificial problem

2

Exact solution:  

Source: Suleiman [9]

3

Exact solution:

Source: Shampine and Gordon [10]

4

Exact solution:

Source: Suleiman [9]

5

Exact solution:

Source: Russell and Shampine [11]

7. Numerical Result

Tables 4, 5, 6, 7, and 8 give the numerical results for problems given in the previous section. The results for the 2PBBD are compared with those of 2PBDD and 1PBD according to Omar [5] and Suleiman et al. [8], respectively. Also given are graphs, where is plotted against and . The following notations are used in the tables:

 : step size, 2PBBD: 2-point block backward difference method, 2PBDD: 2-point block divided difference method, 1PBD: 1-point backward difference method, NS: total number of steps, MAXE: maximum error, TIME: total execution times (in microsecond).

Two sets of scaled graphs were plotted, namely, (i) against and (ii) against . For a particular abscissa, the lowest value of the ordinate is considered to be the more efficient at the abscissa considered. Hence, for the first set of graphs, that is, against , the method 2PBBD is better when , and loses out for value of . For the second set of graphs, as the time increases, the 2PBBD is the method of choice since it is lowest for all five sets of problems (see Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11). It gives us the impression of stability, where the errors grow most slowly compared with the other methods, 2PBDD and 1PBD.

Table 4: Numerical result for Problem 1.

Method
NS

Time

2PBBD
1000

6914

2PBDD
1000

9509

1PBD
2000

6912

2PBBD
10000

49146

2PBDD
10000

54025

1PBD
20000

50095

2PBBD
100000

195037

2PBDD
100000

256339

1PBD
200000

197800

2PBBD
1000000

1759047

2PBDD
1000000

2055758

1PBD
2000000

1638426

2PBBD
10000000

17500121

2PBDD
10000000

19917218

1PBD
20000000

14653209

2PBBD
100000000

176146062

2PBDD
100000000

199668362

1PBD
200000000

145626635

Table 5: Numerical result for Problem 2.

Method
NS

Time

2PBBD
3200

21516

2PBDD
3200

23350

1PBD
6400

22890

2PBBD
32000

86824

2PBDD
32000

111374

1PBD
64000

87426

2PBBD
320000

776833

2PBDD
320000

828900

1PBD
640000

725855

2PBBD
3200000

7769459

2PBDD
3200000

8208180

1PBD
6400000

6474481

2PBBD
32000000

77008100

2PBDD
32000000

81918617

1PBD
64000000

63364050

2PBBD
320000000

773213500

2PBDD
320000000

817874121

1PBD
640000000

616850084

Table 6: Numerical result for Problem 3.

Method
NS

Time

2PBBD
2513

24465

2PBDD
2513

24221

1PBD
5026

28522

2PBBD
25133

152099

2PBDD
25133

159389

1PBD
50265

139073

2PBBD
251328

1418931

2PBDD
251328

1366837

1PBD
502655

1219971

2PBBD
2513274

14315026

2PBDD
2513274

13192255

1PBD
5026548

11525418

2PBBD
25132742

141938590

2PBDD
25132742

130102816

1PBD
50265482

106896913

2PBBD
251327412

1412455700

2PBDD
251327412

1300828490

1PBD
502654824

1063913703

Table 7: Numerical result for Problem 4.

Method
NS

Time

2PBBD
2513

23909

2PBDD
2513

26684

1PBD
5026

31772

2PBBD
25133

121769

2PBDD
25133

131297

1PBD
50265

121536

2PBBD
251328

1102741

2PBDD
251328

1068239

1PBD
502655

1046930

2PBBD
2513274

10004344

2PBDD
2513274

9407995

1PBD
5026548

10087395

2PBBD
25132742

99147571

2PBDD
25132742

91521335

1PBD
50265482

93207026

2PBBD
251327412

988919951

2PBDD
251327412

911824508

1PBD
502654824

922126253

Table 8: Numerical result for Problem 5.

Method
NS

Time

2PBBD
2450

25345

2PBDD
2450

26172

1PBD
4900

27139

2PBBD
24500

106625

2PBDD
24500

116848

1PBD
49000

108650

2PBBD
245000

1021740

2PBDD
245000

996446

1PBD
490000

888813

2PBBD
2450000

9797882

2PBDD
2450000

8683164

1PBD
4900000

8211650

2PBBD
24500000

97582403

2PBDD
24500000

85958740

1PBD
49000000

80890328

2PBBD
245000000

978631746

2PBDD
245000000

844019109

1PBD
490000000

790057441

Figure 2: Graph of plotted against for Problem 1.

Figure 3: Graph of plotted against for Problem 1.

Figure 4: Graph of plotted against for Problem 2.

Figure 5: Graph of plotted against for Problem 2.

Figure 6: Graph of plotted against for Problem 3.

Figure 7: Graph of plotted against for Problem 3.

Figure 8: Graph of plotted against for Problem 4.

Figure 9: Graph of plotted against for Problem 4.

Figure 10: Graph of plotted against for Problem 5.

Figure 11: Graph of plotted against for Problem 5.

8. Conclusion

Of the 3 methods, 2PBBD is therefore preferred as a general code and should be included as a collection of methods, as a code for parallelization purposes, as an assembly of codes to be tested and studied, and as a code for solving ODEs.

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge that this research was supported by Universiti Putra Malaysia, GB-IBT Grant no. GP-IBT/2013/9410100.

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