Numerical comparison of nonstandard schemes for the Airy equation
About this article
Abstract
This paper considers the Airy ordinary differential equation (ODE) and different ways it can be discretized. We first consider a standard discretization using the central difference scheme. We then consider two difference schemes which were created using a nonstandard methodology. Finally, we compare the different schemes and how well they approximate solutions to the Airy ODE.
Keywords
- ODE
- Nonstandard Finite Difference Scheme
- Airy Equation
References
[1] L. Gr. Ixaru, M. Rizea, “Numerov method maximally adapted to the Schrodinger equation”, J. Comput. Phys., Vol. 73, (1987), 306-324.
[2] R. E. Mickens, “Nonstandard finite difference models of differential equations”, World Scientific Publishing Co. Pte. Ltd., (1994).
[3] R. E. Mickens, “Nonstandard finite difference schemes for differential equations”, Journal of Difference Equations and Applications, Vol. 8: 9, (2002), 823-847.
[4] R. E. Mickens, A. Smith, “Finite-difference models of ordinary differential equations: influence of denominator functions”, Journal of the Franklin Institute Vol. 327, (1990), 143-149.
[5] R. E. Mickens, “Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition”, SIAM J. Math. Anal. Vol. 33, (2002), 672-691.
[6] R. E. Mickens, Advances in the application of nonstandard finite difference schemes World Scientific Publishing Co. Pte. Ltd. (2005).
[7] R. Chen, Z. Xu and L. Sun, “Finite-difference scheme to solve Schrodinger equations”, Physical Review E, Vol. 47, (1993), 3799-3802.
[8] R. Yaghoubi, H. Saberi Najafi, “Comparison between standard and non-standard finite difference methods for solving first and second order ordinary differential equations”, International Journal of Applied Mathematical Research, Vol. 4, No. 2,(2015), 316-324.