Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31105
Fifth Order Variable Step Block Backward Differentiation Formulae for Solving Stiff ODEs

Authors: S.A.M. Yatim, Z.B. Ibrahim, K.I. Othman, F. Ismail


The implicit block methods based on the backward differentiation formulae (BDF) for the solution of stiff initial value problems (IVPs) using variable step size is derived. We construct a variable step size block methods which will store all the coefficients of the method with a simplified strategy in controlling the step size with the intention of optimizing the performance in terms of precision and computation time. The strategy involves constant, halving or increasing the step size by 1.9 times the previous step size. Decision of changing the step size is determined by the local truncation error (LTE). Numerical results are provided to support the enhancement of method applied.

Keywords: backward differentiation formulae, stiff ordinary differential equation, block backwarddifferentiation formulae, variablestep size

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1812


[1] L.G. Birta and O. Abou-Rabiaa, "Parallel block predictor-corrector methods for odes,",IEEE Transactions on Computers, vol. C-36(3), pp. 299-311, 1987.
[2] K. Burrage, "Efficient block predictor-corrector methods with a small number of corrections," J. of Comp. and App. Mat,. vol. 45, pp. 139-150, 1993.
[3] J.R. Cash, "On the integration of stiff systems of odes using extended backward differentiation formulae," Numer. Math,.vol. 34, pp. 235-246, 1980.
[4] J.R. Cash, "The integration of stiff initial value problems in odes using modified extended backward differentiation formulae," Comput. Math. Appl., vol. 9, pp. 645-660, 1983.
[5] M.T. Chu, and H. Hamilton, "Parallel solution of odes by multi-block methods," Siam J. Sci. Stat. Comput., vol. 8(1), pp. 342-353, 1987.
[6] S.O. Fatunla, "Block methods for second order odes," Intern. J. Computer Math., vol. 40, pp. 55-63, 1990.
[7] C.W. Gear, "Numerical initial value problems in ordinary differential equations," COMM. ACM., vol. 14, pp. 185-190, 1971.
[8] Z.B. Ibrahim, M.B. Suleiman and K.I. Othman, "Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations," European Journal of Scientific Research, vol. 21, no.3, pp. 508-520, 2008.
[9] Z.B. Ibrahim, K.I. Othman and M.B. Suleiman, "Variable stepsize block backward differentiation formula for solving stiff odes," Proceedings of World Congress on Engineering 2007, LONDON, U.K., vol. 2, pp. 785-789, 2007.
[10] Z.B. Ibrahim, M.B. Suleiman and K.I. Othman, "Implicit r-point block backward differentiation formula for solving first- order stiff odes," Applied Mathematics and Computation, vol. 186, pp. 558-565, 2007.
[11] Z.B. Ibrahim, "Block Multistep Methods For Solving Ordinary Differential Equations," Ph. D. Thesis, Universiti Putra Malaysia, Selangor, 2006.
[12] P. Kaps and G. Wanner, "A study of rosenbrock-type methods of high order," Numer. Math., vol. 38, pp. 279-298, 1981.
[13] J.D. Lambert, Numerical Methods for Ordinary Differential Equations: The Initial Value Problems, John Wiley & Sons, New York 1991.