Nonstandard finite difference method for ODEs for initial-value problems

Abstract

In this paper, a powerful recent non-standard finite different method by nonlocal approximation is improved. Also, compared standard finite difference method to this non-standard finite different method in terms of stability and accuracy. As a numerical example, Hybrid Selection & Genetics equation is considered as the candidate from class of first order ODEs with polynomial right-hand sides. Furthermore, results obtained from the non-standard finite different method and MATLAB ODE solvers (ode15s,ode23s) compared in terms of stability, accuracy, and execution time.

Keywords

• Nonstandard finite difference method,Nonlocal approximation,MATLAB ODE solvers

References

1. R. Anguelov, J.M.-S. Lubuma, Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simul. 61 (3-6) (2003) 465-475
2. R. Anguelov, J.M.-S. Lubuma, On the Mathematical Foundation of the Nonstandard Finite Difference Method, in: B. Fiedler,
3. K. Groger, J. Sprekels (Eds.), Proceedings of the International Conference on Differential Equations, EQUADIFF 99, World Scientific, Singapore, 2000, pp. 1401-1403
4. R. Anguelov, J.M.-S. Lubuma, On the the Nonstandard Finite Difference Method, Keynote address at the Annual Congress of the South African Mathematical Society, Pretoria, South Africa, 16-18 October 2000, Notices S.Afr. Math. Soc. 31 (3) 2000 143-152
5. R. Anguelov, J.M.-S. Lubuma, Contributions to the Mathematics of the Nonstandard Finite Difference Method and Applications, Num. Methods Partial Differential Equations 17 (5) (2001) 518-543
6. J.D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, Newyork, 1991
7. R.D. Richtmyer, K.W. Morton, Difference Methods for Initial Value Problems, Interscience, New York, 1967
8. R.E. Mickens (Ed.), Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2000.

9. R.E. Minkens, Nonstandard Finite Difference Schemes for Differential Equations, Journal of Difference Equations and Applications, 2002, 8:9, 823-847
10. Larson, R. (2006). Calculus: An Applied Approach 9E. Boston: Brooks.