Solving delay differential equations by the five-point one-step block method using Neville's interpolation

A five-point one-step block method based on the Newton backward divided difference formulae for the solution of first-order delay differential equations is derived. The proposed block method will approximate the solutions of initial value problems at five points simultaneously using variable step size. The approximation of the delay term is calculated using Neville's interpolation. The block method will be formulated in terms of linear multistep method, but the method is equivalent to one-step method. The order of the block method will be discussed. The P-stability and Q-stability regions of the block method using Neville's interpolation for the delay term are presented for a fixed step size. Numerical results are given to show the efficiency of the proposed method and compared with the existing method.

[1]  Simeon Ola Fatunla,et al.  Block methods for second order odes , 1991, Int. J. Comput. Math..

[2]  J. Barkley Rosser,et al.  A Runge-Kutta for all Seasons , 1967 .

[3]  Stability of implicit one-block methods for delay differential equations , 2001 .

[4]  Marino Zennaro,et al.  P-stability properties of Runge-Kutta methods for delay differential equations , 1986 .

[5]  P. Chartier.  L-stable parallel one-block methods for ordinary differential equations , 1994 .

[6]  Francesca Mazzia,et al.  Block-Boundary Value Methods for the Solution of Ordinary Differential Equations , 1999, SIAM J. Sci. Comput..

[7]  Raed Ali Al-Khasawneh,et al.  Numerical Treatment of Delay Differential Equations by Runge-Kutta Method Using Hermite Interpolation , 2002 .

[8]  Solving delay differential equations using coupled block method , 2011, 2011 Fourth International Conference on Modeling, Simulation and Applied Optimization.

[9]  David J. Evans,et al.  The Adomian decomposition method for solving delay differential equation , 2005, Int. J. Comput. Math..

[10]  Mohamed A. Ramadan,et al.  Numerical solution of system of first-order delay differential equations using polynomial spline functions , 2006, Int. J. Comput. Math..

[11]  Chengjian Zhang,et al.  Block boundary value methods for delay differential equations , 2010 .

[12]  V. Barwell,et al.  Special stability problems for functional differential equations , 1975 .

[13]  Hongjiong Tian,et al.  Continuous block implicit hybrid one-step methods for ordinary and delay differential equations , 2011 .

[14]  Zanariah Abdul Majid,et al.  3-point implicit block method for solving ordinary differential equations. , 2006 .

[15]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[16]  Fudziah Bt. Ismail,et al.  Comparison Of Interpolations Used In Solving Delay Differential Equations By Runge-Kutta Method , 2003, Int. J. Comput. Math..

[17]  A. N. Al-Mutib.  Stability properties of numerical methods for solving delay differential equations , 1984 .

[18]  M. Suleiman,et al.  Predictor-corrector block iteration method for solving ordinary differential equations , 2011 .

[19]  Hongjiong Tian,et al.  Continuous Block Theta-Methods for Ordinary and Delay Differential Equations , 2009, SIAM J. Sci. Comput..

[20]  H. J. Pesch,et al.  Numerical treatment of delay differential equations by Hermite Interpolation , 1981 .

[21]  H. A. Watts,et al.  Block Implicit One-Step Methods* , 1969 .