BBDF-Alpha for solving stiff ordinary differential equations with oscillating solutions
Keywords:Block method, ordinary differential equations, stiff, oscillating
In this paper, the block backward differentiation α formulas (BBDF-α) is derived for solving first order stiff ordinary differential equations with oscillating solutions. The consistency and zero stability conditions are investigated to prove the convergence of the method. The stability region in the entire negative half plane shows that the derived method is A-stable for certain values of α. The implementation of the method using Newton iteration is also discussed. Several numerical experiments are conducted to demonstrate the performance of the method in terms of accuracy and computational time.
J. D. Lambert, Numerical methods for ordinary differential systems: The initial value problem, John Wiley and Sons: New York, USA, 1991.
J. Sunday, M. R. Odekunle, A. A. James and A. O. Adesanya, Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator, Brit. J. Math. Comput. Sci. 4(17) (2014), 2471-2481.
J. M. Franco, I. Gomez and L. Randez, SDIRK methods for stiff ODES with oscillating solutions, J. Comput. Appl. Math. 81 (1997), 197-209.
A. Tahmasbi, Numerical solutions for stiff ordinary differential equations systems, I. Math. Forum. 3 (2008), 703-711.
Z. B. Ibrahim, Block method for multistep formulas for solving ordinary differential equations. Universiti Putra Malaysia, Ph.D thesis, 2006.
Z. B. Ibrahim, M. Suleiman and K. I. Othman, Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations, Eur. J. Sci. Res. 21(3) (2008), 508-520.
Z. B. Ibrahim, M. B. Suleiman and K. I. Othman, Implicit r −point block backward differentiation formula for solving first order stiff ordinary differential equations, Appl. Math. Comput. 186 (2007), 558-565.
N. A. A. M. Nasir, Z. B. Ibrahim, K. I. Othman and M. Suleiman, Fifth order two-point block backward differentiation formulas for solving ordinary differential equations, Appl. Math. Sci. 5(71) (2011), 3505-3518.
N. A. A. M. Nasir, Z. B. Ibrahim, K. I. Othman and M. Suleiman, Numerical solution of first order stiff ordinary differential equations using fifth order block backward differentiation formulas, Sains Malays. 41 (2012), 489-492.
J. Sunday, Y. Skwame and M. R. Odekunle, A continuous block integrator for the solution of stiff and oscillatory differential equations, IOSR J. Math. 8(3) (2013), 75-80.
I. S. M. Zawawi, Z. B. Ibrahim and K. I. Othman, Derivation of diagonally implicit block backward differentiation formulas for solving stiff initial value problem, Math. Probl. Eng. (2015) 13 pages, Article ID 179231, http://dx.doi.org/10.1155/2015/179231.
I. S. M. Zawawi, Z. B. Ibrahim, F. Ismail and Z. A. Majid, Diagonally implicit block backward differentiation formulas for solving ordinary differential equations, Int. J. Math. Math. Sci. (2012), 8 pages, Article ID 767328, http://dx.doi.org/10.1155/2012/767328.
E. A. Celaya and J. J. Anza, BDF-α: A multistep method with numerical damping control, Universal J. Comput. Math. 1 (2013), 96-108.