Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method
International Journal of Theoretical and Applied Mathematics
Volume 3, Issue 1, February 2017, Pages: 30-35
Received: Sep. 26, 2016;
Accepted: Dec. 10, 2016;
Published: Jan. 12, 2017
Views 2563 Downloads 64
Oluwaseun Adeyeye, Department of Mathematics, Universiti Utara Malaysia, Kedah, Malaysia
Zurni Omar, Department of Mathematics, Universiti Utara Malaysia, Kedah, Malaysia
Numerov method is one of the most widely used algorithms in physics and engineering for solving second order ordinary differential equations. The numerical solution of this method has been improved by different authors by using different starting formulas but in recent years, there has been a dearth in that trend which informed the introduction of a two-step third-derivative block method in this paper to start Numerov method with the aim of getting better results than previous approaches. The selection of the steplength as two is to have a uniform basis for comparison with other existing two-step starting formula in literature. Although, the accuracy of the two-step method adopted in this article was enhanced by the introduction of higher derivative. Hence, this paper presents a two-step third-derivative block method which displayed better accuracy when adopted for starting Numerov method as shown in the numerical results. Thus, the third-derivative block method, as a starting formula, is seen to be quite suitable for starting Numerov method when applied to physical models.
Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method, International Journal of Theoretical and Applied Mathematics.
Vol. 3, No. 1,
2017, pp. 30-35.
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Adee, S. O., P. Onumanyi, U. W. Sirisena and Y. A. Yahaya. 2005. Note on starting the Numerov method more accurately by a hybrid formula of order four for an initial value problem. Journal of Computational and Applied Mathematics, 175 (2): 369-373. doi: 10.1016/j.cam.2004.06.016.
Ehle, B. L. 1968. High order A-stable methods for the numerical solution of systems of DE's. BIT Numerical Mathematics, 8 (4), 276-278. doi: 10.1007/BF01933437.
Enright, W. H. 1974. Second derivative multistep methods for stiff ordinary differential equations. SIAM Journal on Numerical Analysis, 11 (2), 321-331. doi: 10.1137/0711029.
Fatunla, S. O. 1988. Numerical methods for initial value problems in ordinary differential equations, Academic Press, New York.
Jator, S. N., and Li, J.. 2012. An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method. Numerical Algorithms, 59 (3), 333-346., doi: 10.1007/s11075-011-9492-3.
Gonzalez, J. Q. and D. Thompson. 1997. Getting started with Numerovs method. Computers in Physics, 11 (5): 514-515. http://dx.doi.org/10.1063/1.168593.
Konguetsof, A. 2010. A new two-step hybrid method for the numerical solution of the Schrodinger equation. Journal of Mathematical Chemistry, 47 (2): 871-890. doi: 10.1007/s10910-009-9606-5.
Lambert, J. D. Computational methods in ordinary differential equations, Wiley: London, 1973.
Mishra, B. N. and R. K. Mohanty. 2013. Single cell Numerov type discretization for 2D biharmonic and triharmonic equations on unequal mesh. Journal of Mathematical and Computational Science, 3 (1): 242-253. http://www.scik.org/index.php/jmcs/article/viewFile/774/308.
Mohanty, R. K. and R. Kumar. 2014. A novel numerical algorithm of Numerov Type for 2D quasi-linear elliptic boundary value problems. International Journal for Computational Methods in Engineering Science and Mechanics, 15 (6): 473-489. doi: 10.1080/15502287.2014.934488.
Norton, M. S. 2009. Numerov's Method for approximating solutions to Poisson's equation. https://www.siue.edu/_mnorton/Numerov.pdf (Accessed on October 29, 2015).
Obrechkoff, N. 1942. On mechanical quadrature (Bulgarian French summary). Spisanie Bulgar. Akad. Nauk, 65, 191-289.
Onumanyi, P., U. W. Sirisena and S. Adee. 2002. Some theoretical considerations of continuous linear multistep methods for . Bagale Journal of Pure and Applied Sciences, 2 (2): 1-5.
Sahi, R. K., Jator, S. N., & Khan, N. A. 2013. Continuous fourth derivative method for third order boundary value problems. International journal of pure and applied mathematics, 85 (5), 907-923. doi: 10.12732/ijpam.v85i5.9.
Simos, T. E. 2009. A new Numerov-type method for the numerical solution of the Schrodinger equation. Journal of Mathematical Chemistry, 46 (3): 981-1007. doi: 10.1007/s10910-009-9553-1.
Yusuph, Y. and P. Onumanyi. 2005. New multiple FDMs through multistep collocation for y = f (x, y). Proceedings of the National Mathematical Center, Abuja Nigeria.
Vigo-Aguiar, J. and H. Ramos. 2005. A variable-step Numerov method for the numerical solution of the Schrodinger equation. Journal of Mathematical Chemistry, 37 (3): 255-262. doi: 10.1007/s10910-004-1467-3.