Development of Variable Stepsize Variable Order Block Method in Divided Difference Form for the Numerical Solution of Delay Differential Equations

Authors: Fuziyah Ishak, Mohamed B. Suleiman, Zanariah A. Majid, Khairil I. Othman

Abstract:

This paper considers the development of a two-point predictor-corrector block method for solving delay differential equations. The formulae are represented in divided difference form and the algorithm is implemented in variable stepsize variable order technique. The block method produces two new values at a single integration step. Numerical results are compared with existing methods and it is evident that the block method performs very well. Stability regions of the block method are also investigated.

Keywords: block method, delay differential equations, predictor-corrector, stability region, variable stepsize variable order.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1071037

References:

[1] A.N. Al-Mutib, Numerical Methods for Solving Delay Differential Equations. PhD. Thesis, University of Manchester, 1977.
[2] Z. Jackiewicz, "Variable-step variable-order algorithm for the numerical solution of neutral functional differential equations," Applied Numer. Math., vol. 4, pp. 317-329, 1987.
[3] Z. Jackiewicz and E. Lo, "The numerical solution of neutral functional differential equations by Adams predictor-corrector methods," Applied Numer. Math., vol. 8, pp. 477-491, 1991.
[4] L. F. Shampine and S.Thompson, "Solving DDEs in MATLAB," Applied Numer. Math., vol. 37, pp. 441-458, 2001.
[5] Z. Jackiewicz and E. Lo, "Numerical solution of neutral functional differential equations by Adams methods in divided difference form," J. of Comp. and Appl. Math.,vol. 189, pp.592-605, 2006.
[6] M. B. Suleiman and F.Ishak, "Numerical solution and stability of multistep method for solving delay differential equations (Accepted for publication)," Japan J. Indust. Appl. Math., published online 13 October 2010.
[7] L. G. Birta and O. Abou-Rabia, "Parallel block predictor-corrector methods for ODE's," IEEE Trans. on Computer, vol. c-36, pp. 299-311, 1987.
[8] A. Makroglou, "Block-by-block method for the numerical solution of Volterra delay integro-differential equations," Computing, vol. 30, pp. 49-62, 1983.
[9] L. F. Shampine and H. A. Watts, "Block implicit one-step methods," Math. Comp., vol. 23, pp. 731-170, 1969.
[10] B. P. Sommeijer, W. Couzy, and P. J. van der Houwen, "A-stable parallel block methods for ordinary and integro-differential equations," Applied Numer. Math., vol. 9, pp. 267-281, 1992.
[11] F. Ishak, M. Suleiman, and Z. Omar, "Two-point predictor-corrector block method for solving delay differential equations," MATEMATIKA, vol. 24, no. 2, pp. 131-140, 2008.
[12] V. K. Barwell, "Special stability problems for functional differential equations," BIT, vol. 15, pp. 130-135, 1975.
[13] L. F. Wiederholt, "Stability of multistep methods for delay differential equations," Math. Comp., vol. 30, pp. 283-290, 1976.
[14] A. N. Al-Mutib, "Stability properties of numerical methods for solving delay differential equations," J. Comp. Appl. Math., vol. 10, pp. 71-79, 1984.
[15] Z. Jackiewicz, "Asymptotic stability analysis of ╬© -methods for functional differential equations," Numer. Math., vol. 43, pp. 389-396, 1984.