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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

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.

867095.fig.001
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.

tab1
Table 1: The explicit integration coefficients for from 0 to 6 (for ).
tab2
Table 2: The implicit integration coefficients for from 0 to 6 (for ).

6. Problem Tested

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

tab3
Table 3: List of test problems.

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.

tab4
Table 4: Numerical result for Problem 1.
tab5
Table 5: Numerical result for Problem 2.
tab6
Table 6: Numerical result for Problem 3.
tab7
Table 7: Numerical result for Problem 4.
tab8
Table 8: Numerical result for Problem 5.
867095.fig.002
Figure 2: Graph of plotted against for Problem 1.
867095.fig.003
Figure 3: Graph of plotted against for Problem 1.
867095.fig.004
Figure 4: Graph of plotted against for Problem 2.
867095.fig.005
Figure 5: Graph of plotted against for Problem 2.
867095.fig.006
Figure 6: Graph of plotted against for Problem 3.
867095.fig.007
Figure 7: Graph of plotted against for Problem 3.
867095.fig.008
Figure 8: Graph of plotted against for Problem 4.
867095.fig.009
Figure 9: Graph of plotted against for Problem 4.
867095.fig.0010
Figure 10: Graph of plotted against for Problem 5.
867095.fig.0011
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.

References

  1. W. E. Milne, Numerical Solution of Differential Equations, John Wiley & Sons, New York, NY, USA, 1953. View at MathSciNet
  2. J. B. Rosser, “A Runge-Kutta for all seasons,” SIAM Review, vol. 9, no. 3, pp. 417–452, 1967. View at Publisher · View at Google Scholar · View at MathSciNet
  3. L. F. Shampine and H. A. Watts, “Block implicit one-step methods,” Mathematics of Computation, vol. 23, pp. 731–740, 1969. View at Publisher · View at Google Scholar
  4. M. T. Chu and H. Hamilton, “Parallel solution of ODE's by multi-block methods,” SIAM Journal on Scientific Computing, vol. 8, no. 3, pp. 342–353, 1987. View at Google Scholar
  5. Z. B. Omar, Parallel block method for solving higher order ordinary differential equations directly [Ph.D. thesis], Universiti Putra Malaysia, 1999.
  6. Z. B. Majid, Parallel block methods for solving ordinary differential equations [Ph.D. thesis], Universiti Putra Malaysia, 2004.
  7. Z. B. Ibrahim, Block multistep methods for solving ordinary differential equations [Ph.D. thesis], Universiti Putra Malaysia, 2006.
  8. M. Bin Suleiman, Z. B. Binti Ibrahim, and A. F. N. Bin Rasedee, “Solution of higher-order ODEs using backward difference method,” Mathematical Problems in Engineering, vol. 2011, Article ID 810324, 18 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. M. B. Suleiman, “Solving nonstiff higher order ODEs directly by the direct integration method,” Applied Mathematics and Computation, vol. 33, no. 3, pp. 197–219, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. L. F. Shampine and M. K. Gordon, Computer Solution of Ordinary Differential Equations, W. H. Freeman, San Francisco, Calif, USA, 1975.
  11. R. D. Russell and L. F. Shampine, “A collocation method for boundary value problems,” Numerische Mathematik, vol. 19, no. 1, pp. 1–28, 1972. View at Publisher · View at Google Scholar · View at Scopus