Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs
Science Journal of Applied Mathematics and Statistics
Volume 5, Issue 5, October 2017, Pages: 181-187
Received: Sep. 18, 2017;
Accepted: Oct. 8, 2017;
Published: Nov. 8, 2017
Views 1716 Downloads 96
Yohanna Sani Awari, Department of Mathematical Sciences, Taraba State University, Jalingo, Nigeria
A Modified Three Step Block Hybrid Extended Trapezoidal Multistep Method of Second Kind (BHETR2s) with two off-grid points, one at interpolation and another at collocation point yielding uniform order six (6, 6, 6, 6, 6)T for the Numerical Integration of initial value problems of stiff Ordinary Differential Equations was developed. The main method and additional equations were obtained from the same continuous formulation through interpolation and collocation procedures. The stability properties of the method was discussed and from the stability region obtained, the method is suitable for the solution Stiff Ordinary Differential Equations. Three numerical examples were considered to illustrate the efficiency and accuracy.
Yohanna Sani Awari,
Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs, Science Journal of Applied Mathematics and Statistics.
Vol. 5, No. 5,
2017, pp. 181-187.
Ajie I. J., Ikhile M. N. O and Onumanyi P. A Family of One-Block Implicit Multistep Backward Euler Type Methods American Journal of Computational and Applied Mathematics 2014, 4(2): 51-59.
Butcher, J. C. and Burrage, K. (1980). Non-Linear Stability of a General Class of Differential Equation Methods. BIT, 20: 185-203.
Carroll, J. “A Metrically Exponentially Fitted Scheme for the Numerical Solution of Stiff Initial Value Problems,” Computers & Mathematics with Applications, Vol. 26, 1993, pp. 57-64. doi:10.1016/0898-1221(93)90034-S
Dahlquist, G. (1963). A Special Stability Problem for Linear Multistep Methods. BIT., 3,27-43.
Fatunla, S. O. (1991). Block methods for second order IVP’s. International Journal of Computer Mathematics, 41, 55-63.
Fatunla, S. O. (1994). Higher Order Parallel Methods for Second Order ODE’s. Scientific Computing. Proceeding of fifth International Conference on Scientific Computing.
Gragg, W. B. and Stetter, H. J. Generalized Multistep Predictor-Corrector Methods. Journal of Association of Computing Machines, Vol. 11, No. 2, 1964, pp. 188-209.
Harier, E. and Wanner, G. “Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems,” Springer-Verlag, New York, 1996. doi:10.1007/978-3-642-05221-7.
Henrici, P. (1962). Discrete Variable Methods in ODE’s (1st Edition). Wiley and Sons Ltd., New York. pp. 285.
Hojjati, G., Rahimi, M. Y. and Hosseini, S. M. “An Adaptive Method for Numerical Solution of Stiff System of Ordinary Differential Equations,” Mathematics and Computers in Simulation, Vol. 66, No. 1, 2004, pp. 33-41. doi:10.1016/j.matcom.2004.02.019.
Hsiao, C. H. and Wang, W. J. “Haar Wavelet Approach to Non-Linear Stiff Systems,” Mathematics and Computer in Simulation, Vol. 57, No. 6, 2001, pp. 347-353. doi:10.1016/S0378-4754(01)00275-0.
Hsiao, C. H. “Haar Wavelet Approach to Linear Stiff Systems,” Mathematics and Computer in Simulation, Vol. 64, No. 1, 2004, pp. 561-567. doi:10.1016/j.matcom.2003.11.011.
James, A. A., Adesanya, A. O. and Joshua, S. Continuous block method for the solution of second order initial value problems of ordinary differential equations, Int. J. of pure and Appl. Math. 88(2013), 405-416.
James, A. A., Adesanya, A. O., Sunday, J., Yakubu, D. G. Half-Step Continuous Block Method for the Solutions of Modeled Problems of Ordinary Differential Equations, American Journal of Computational Mathematics, 2013, 3, 261-269.
Lopidus, L. and Schiesser, W. E. “Numerical Methods for Differential Systems,” Academic Press, New York, 1976.
Mickens, R. E. Non-Standard Finite Difference Models of Differential Equations. World Scientific, Singapore, 1994.
Rosser, J. B. A Runge-kutta Method for All Seasons, SIAM Review, Vol. 9, No. 3, 1967, pp. 417-452.
Sunday, J., Odekunleans, M. R. and Adesanya, A. O. Order Six Block Integrator for the Solution of First Order Ordinary Differential Equations. IJMS, Vol. 3, No. 1 2013, pp. 87-96.