Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs

Oluwasegun Micheal Ibrahim; Monday Ndidi Oziegbe Ikhile

doi:doi:10.11648/j.ajasr.20170303.11

| Peer-Reviewed

Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs

Oluwasegun Micheal Ibrahim, Monday Ndidi Oziegbe Ikhile

Published in American Journal of Applied Scientific Research (Volume 3, Issue 3)

Received: 29 June 2017 Accepted: 14 July 2017 Published: 31 October 2017

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.

Published in	American Journal of Applied Scientific Research (Volume 3, Issue 3)
DOI	10.11648/j.ajasr.20170303.11
Page(s)	21-27
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

Super-Implicit, Hybrid LMM, Stӧrmer-Cowell Method, P-stability

References

[1]	Cash J. R., (1981), High order P-stable Formulae for the Numerical Integration of Periodic Initial Value Problems. J. Numer. Maths., 37: 355-370.
[2]	Chawla M. M., and Neta B., (1986), Families of two-step fourth order P-stable methods for second order differential equations, J. Comp. and Appl. Maths 15: 213-223.
[3]	Dahlquist, G., (1978), On Accuracy and Unconditional Stability of the Linear Methods for Second Order Differential Equations, BIT, 18: 133- 136.
[4]	Fatunla S. O., (1985), One-leg Hybrid Formula for Second Order IVPs. Computers and Mathematics with Applications, 10: 329-333.
[5]	Fatunla S. O., (1988), Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Academic Press, Boston, USA.
[6]	Fatunla S. O., Ikhile M. N. O., and Otunta F. O., (1997), A Class of P-stable Linear Multistep Num. Methods. Int. J of Comp Math. 72: 1-13.
[7]	Fukushima T., (1998), “Symmetric multistep methods, revisited”, In Prec. of the 30 tu Syrup. on Celestial Mechanics, 4-6 March 1998, Hayama, Kanagawa, Japan (Edited by T. Fukushima, T. Ito, T. Fuse and H. Umehara): 229-247.
[8]	Fukushima T., (1999), Super-implicit multi-step methods, Proc. of the 31th Symp. on Celestial Mechanics, Kashima Space Research Center, Ibaraki, Japan (Edited by H. Umehara): 343-366.
[9]	Hairer E., (1979), Unconditionally Stable Methods for Second order Differential Equations, Numerical Maths., 32: 373 - 379.
[10]	Ibrahim O. M. (2016), High Order Symmetric Super-Implicit Hybrid LMM with Minimal Phase-Lag Error. M. Sc. Thesis, University of Benin, Nigeria.
[11]	Borwein J. M., and Skerritt M. P., (2012), An Introduction to Modern Mathematical Computing with Mathematica, Springer.
[12]	Jain M. K., Jain R. K., and Anantha Krishnaiah U., (1979), P-stable Methods for Periodic Initial Value Problems of Second Order Differential Equations. BIT 19: 347-355.
[13]	Lambert J. D., (1973), Numerical Methods for Ordinary Differential Systems: the Initial Value Problem. John Wiley & Sons Ltd.
[14]	Lambert J. D., and Watson I., (1976), Symmetric Multistep Methods For Periodic Initial Value Problems, J. Inst. Math. Appls., 18: 189-202.
[15]	Mehdizadeh Khalsaraei 1 M. and Molayi M., (2015) P-Stable Hybrid Super-Implicit Methods for Periodic Initial Value Problems. Journal of mathematics and computer science. 15: 129 – 136.
[16]	Neta B., (2005), P-stable Symmetric Super-Implicit Methods for Periodic Initial Value Problems, Comput. Math. Appl. 50: 701–705.
[17]	Neta B. (2007), P-Stable High-Order Super-Implicit and Obrechkoff Methods for Periodic Initial Value Problems, Computers and Mathematics with Applications 54: 117–126.
[18]	Otto S. R. and Denier J. P., (2005), An Introduction to Programming and Numerical methods in MATLAB, Springer-Verlag, London.
[19]	Simos T. E., (1993), A P-stable Complete in Phase Obrechkoff Trigonometric Fitted Method for Periodic Initial-value Problems, Prec. Royal See. London A, 441: 283-289.
[20]	Van Dooren R., (1974), Stabilization of Cowell’s classical finite difference methods for numerical integration. J. Comput. Phys. 16: 186-192.
[21]	Okuonghae R. I., and Ikhile M. N. O., (2014), A-Stable High Order Hybrid Linear Multistep Methods for Stiff Problems, J. Algor. Comp. Technol., 8 (4): 441-469.
[22]	Okuonghae R. I., and Ikhile M. N. O., (2015), Stiffly Stable Second Derivative Linear Multistep Methods with two Hybrid Points, Num. Analys. and Appl., 8 (3): 248-259.
[23]	Shoki A., and Saadat H., (2015), High Phase-Lag Order Trigonometrically Fitted Two-Step Obrechkoff Methods for the Numerical Solution of Periodic IVPs Num. Algorithm, 68 (2): 337-354.
[24]	Shoki A., (2017), A New High Order Implicit Four-Step Method with Vanished Phase-Lag and some of its Derivatives for the Numerical Solution of the Radial schrӧdinger equation, J. Modern Methods in Num. Maths., 8 (1-2): 1-16.
[25]	Ibrahim O. M., (2017), On the construction of high accuracy symmetric super-implicit hybrid formulas with phase-lag properties, Accepted for publication: Transaction J. of Nigeria Ass. of Mathematical Physics.

Cite This Article

Plain Text BibTeX RIS

APA Style

Oluwasegun Micheal Ibrahim, Monday Ndidi Oziegbe Ikhile. (2017). Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs. American Journal of Applied Scientific Research, 3(3), 21-27. https://doi.org/10.11648/j.ajasr.20170303.11

Copy | Download

ACS Style

Oluwasegun Micheal Ibrahim; Monday Ndidi Oziegbe Ikhile. Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs. Am. J. Appl. Sci. Res. 2017, 3(3), 21-27. doi: 10.11648/j.ajasr.20170303.11

Copy | Download

AMA Style

Oluwasegun Micheal Ibrahim, Monday Ndidi Oziegbe Ikhile. Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs. Am J Appl Sci Res. 2017;3(3):21-27. doi: 10.11648/j.ajasr.20170303.11

Copy | Download

@article{10.11648/j.ajasr.20170303.11,
  author = {Oluwasegun Micheal Ibrahim and Monday Ndidi Oziegbe Ikhile},
  title = {Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs},
  journal = {American Journal of Applied Scientific Research},
  volume = {3},
  number = {3},
  pages = {21-27},
  doi = {10.11648/j.ajasr.20170303.11},
  url = {https://doi.org/10.11648/j.ajasr.20170303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajasr.20170303.11},
  abstract = {The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs
AU  - Oluwasegun Micheal Ibrahim
AU  - Monday Ndidi Oziegbe Ikhile
Y1  - 2017/10/31
PY  - 2017
N1  - https://doi.org/10.11648/j.ajasr.20170303.11
DO  - 10.11648/j.ajasr.20170303.11
T2  - American Journal of Applied Scientific Research
JF  - American Journal of Applied Scientific Research
JO  - American Journal of Applied Scientific Research
SP  - 21
EP  - 27
PB  - Science Publishing Group
SN  - 2471-9730
UR  - https://doi.org/10.11648/j.ajasr.20170303.11
AB  - The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.
VL  - 3
IS  - 3
ER  -

Copy | Download

Author Information

Oluwasegun Micheal Ibrahim

Department of Mathematical Sciences, African Institute for Mathematical Sciences, Kigali City, Rwanda
Monday Ndidi Oziegbe Ikhile

Advance Research Laboratory, Department of Mathematics, University of Benin, Benin City, Nigeria

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Oluwasegun Micheal Ibrahim, Monday Ndidi Oziegbe Ikhile. (2017). Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs. American Journal of Applied Scientific Research, 3(3), 21-27. https://doi.org/10.11648/j.ajasr.20170303.11

Copy | Download

ACS Style

Oluwasegun Micheal Ibrahim; Monday Ndidi Oziegbe Ikhile. Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs. Am. J. Appl. Sci. Res. 2017, 3(3), 21-27. doi: 10.11648/j.ajasr.20170303.11

Copy | Download

AMA Style

Oluwasegun Micheal Ibrahim, Monday Ndidi Oziegbe Ikhile. Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs. Am J Appl Sci Res. 2017;3(3):21-27. doi: 10.11648/j.ajasr.20170303.11

Copy | Download

@article{10.11648/j.ajasr.20170303.11,
  author = {Oluwasegun Micheal Ibrahim and Monday Ndidi Oziegbe Ikhile},
  title = {Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs},
  journal = {American Journal of Applied Scientific Research},
  volume = {3},
  number = {3},
  pages = {21-27},
  doi = {10.11648/j.ajasr.20170303.11},
  url = {https://doi.org/10.11648/j.ajasr.20170303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajasr.20170303.11},
  abstract = {The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs
AU  - Oluwasegun Micheal Ibrahim
AU  - Monday Ndidi Oziegbe Ikhile
Y1  - 2017/10/31
PY  - 2017
N1  - https://doi.org/10.11648/j.ajasr.20170303.11
DO  - 10.11648/j.ajasr.20170303.11
T2  - American Journal of Applied Scientific Research
JF  - American Journal of Applied Scientific Research
JO  - American Journal of Applied Scientific Research
SP  - 21
EP  - 27
PB  - Science Publishing Group
SN  - 2471-9730
UR  - https://doi.org/10.11648/j.ajasr.20170303.11
AB  - The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.
VL  - 3
IS  - 3
ER  -

Copy | Download