Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method

Oluwaseun Adeyeye; Zurni Omar

doi:doi:10.11648/j.ijtam.20170301.15

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Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method

Oluwaseun Adeyeye, Zurni Omar

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1)

Received: 26 September 2016 Accepted: 10 December 2016 Published: 12 January 2017

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Abstract

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.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1)
DOI	10.11648/j.ijtam.20170301.15
Page(s)	30-35
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Numerov, Two-Step, Third-Derivative, Block Method, Second Order, Initial Value Problems

References

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[2]	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.
[3]	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.
[4]	Fatunla, S. O. 1988. Numerical methods for initial value problems in ordinary differential equations, Academic Press, New York.
[5]	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.
[6]	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.
[7]	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.
[8]	Lambert, J. D. Computational methods in ordinary differential equations, Wiley: London, 1973.
[9]	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.
[10]	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.
[11]	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).
[12]	Obrechkoff, N. 1942. On mechanical quadrature (Bulgarian French summary). Spisanie Bulgar. Akad. Nauk, 65, 191-289.
[13]	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.
[14]	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.
[15]	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.
[16]	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.
[17]	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.

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Oluwaseun Adeyeye, Zurni Omar. (2017). Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method. International Journal of Theoretical and Applied Mathematics, 3(1), 30-35. https://doi.org/10.11648/j.ijtam.20170301.15

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ACS Style

Oluwaseun Adeyeye; Zurni Omar. Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method. Int. J. Theor. Appl. Math. 2017, 3(1), 30-35. doi: 10.11648/j.ijtam.20170301.15

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AMA Style

Oluwaseun Adeyeye, Zurni Omar. Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method. Int J Theor Appl Math. 2017;3(1):30-35. doi: 10.11648/j.ijtam.20170301.15

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@article{10.11648/j.ijtam.20170301.15,
  author = {Oluwaseun Adeyeye and Zurni Omar},
  title = {Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {1},
  pages = {30-35},
  doi = {10.11648/j.ijtam.20170301.15},
  url = {https://doi.org/10.11648/j.ijtam.20170301.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170301.15},
  abstract = {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.},
 year = {2017}
}

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TY  - JOUR
T1  - Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method
AU  - Oluwaseun Adeyeye
AU  - Zurni Omar
Y1  - 2017/01/12
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170301.15
DO  - 10.11648/j.ijtam.20170301.15
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 30
EP  - 35
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170301.15
AB  - 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.
VL  - 3
IS  - 1
ER  -

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Author Information

Oluwaseun Adeyeye

Department of Mathematics, Universiti Utara Malaysia, Kedah, Malaysia
Zurni Omar

Department of Mathematics, Universiti Utara Malaysia, Kedah, Malaysia

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APA Style

Oluwaseun Adeyeye, Zurni Omar. (2017). Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method. International Journal of Theoretical and Applied Mathematics, 3(1), 30-35. https://doi.org/10.11648/j.ijtam.20170301.15

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ACS Style

Oluwaseun Adeyeye; Zurni Omar. Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method. Int. J. Theor. Appl. Math. 2017, 3(1), 30-35. doi: 10.11648/j.ijtam.20170301.15

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AMA Style

Oluwaseun Adeyeye, Zurni Omar. Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method. Int J Theor Appl Math. 2017;3(1):30-35. doi: 10.11648/j.ijtam.20170301.15

Copy | Download

@article{10.11648/j.ijtam.20170301.15,
  author = {Oluwaseun Adeyeye and Zurni Omar},
  title = {Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {1},
  pages = {30-35},
  doi = {10.11648/j.ijtam.20170301.15},
  url = {https://doi.org/10.11648/j.ijtam.20170301.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170301.15},
  abstract = {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.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Application of a Two-Step Third-Derivative Block Method for Starting Numerov Method
AU  - Oluwaseun Adeyeye
AU  - Zurni Omar
Y1  - 2017/01/12
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170301.15
DO  - 10.11648/j.ijtam.20170301.15
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 30
EP  - 35
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170301.15
AB  - 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.
VL  - 3
IS  - 1
ER  -

Copy | Download