Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs

Wenjuan Zhai; Bingzhen Chen

doi:doi:10.11648/j.sjams.20160402.19

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Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs

Wenjuan Zhai, Bingzhen Chen

Published in Science Journal of Applied Mathematics and Statistics (Volume 4, Issue 2)

Received: 12 April 2016 Published: 13 April 2016

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Abstract

In this paper, we derive the implicit exponentially fitted RKNd methods for solving oscillatory ODEs. The new methods integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from the set {exp(λt), exp(−λt)}, λ ∈ C, or equivalently when λ = iω, ω ∈ R. Numerical experiments are accompanied to show the efficiency and competence of the implicit exponentially fitted RKNd methods compared with implicit RKNd methods.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 4, Issue 2)
DOI	10.11648/j.sjams.20160402.19
Page(s)	74-80
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), 2016. Published by Science Publishing Group

Keywords

RKNd Method, Exponentially Fitted, Implicit, Stability, Efficiency, Oscillatory

References

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[4]	B. Zh. Chen and X. You. RKNd methods for solving initial value problems, Numer. Math. Sinica., 2010, 32(4): 399-412. (in Chinese).
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[10]	B. Paternoster, Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Appl. Numer. Math., 1998, 28: 401-412.
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[15]	L. Gr. Ixaru and G. Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht, 2004.
[16]	T. Monovasilis, Z. Kalogiratou, T E. Simos, Construction of Exponentially Fitted Symplectic Runge–Kutta–Nyström Methods from Partitioned Runge–Kutta Methods. Mediterranean Journal of Mathematics, 2014, 1618(1): 1-15.
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APA Style

Wenjuan Zhai, Bingzhen Chen. (2016). Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs. Science Journal of Applied Mathematics and Statistics, 4(2), 74-80. https://doi.org/10.11648/j.sjams.20160402.19

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ACS Style

Wenjuan Zhai; Bingzhen Chen. Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs. Sci. J. Appl. Math. Stat. 2016, 4(2), 74-80. doi: 10.11648/j.sjams.20160402.19

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AMA Style

Wenjuan Zhai, Bingzhen Chen. Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs. Sci J Appl Math Stat. 2016;4(2):74-80. doi: 10.11648/j.sjams.20160402.19

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@article{10.11648/j.sjams.20160402.19,
  author = {Wenjuan Zhai and Bingzhen Chen},
  title = {Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {4},
  number = {2},
  pages = {74-80},
  doi = {10.11648/j.sjams.20160402.19},
  url = {https://doi.org/10.11648/j.sjams.20160402.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20160402.19},
  abstract = {In this paper, we derive the implicit exponentially fitted RKNd methods for solving oscillatory ODEs. The new methods integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from the set {exp(λt), exp(−λt)}, λ ∈ C, or equivalently  when λ = iω, ω ∈ R. Numerical experiments are accompanied to show the efficiency and competence of the implicit exponentially fitted RKNd methods compared with implicit RKNd methods.},
 year = {2016}
}

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TY  - JOUR
T1  - Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs
AU  - Wenjuan Zhai
AU  - Bingzhen Chen
Y1  - 2016/04/13
PY  - 2016
N1  - https://doi.org/10.11648/j.sjams.20160402.19
DO  - 10.11648/j.sjams.20160402.19
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 74
EP  - 80
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20160402.19
AB  - In this paper, we derive the implicit exponentially fitted RKNd methods for solving oscillatory ODEs. The new methods integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from the set {exp(λt), exp(−λt)}, λ ∈ C, or equivalently  when λ = iω, ω ∈ R. Numerical experiments are accompanied to show the efficiency and competence of the implicit exponentially fitted RKNd methods compared with implicit RKNd methods.
VL  - 4
IS  - 2
ER  -

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Author Information

Wenjuan Zhai

Department of Mathematics, Beijing Jiaotong University Haibin College, Cangzhou, P. R. China
Bingzhen Chen

Department of Applied Mathematics, Beijing Jiaotong University, Beijing, P. R. China

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APA Style

Wenjuan Zhai, Bingzhen Chen. (2016). Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs. Science Journal of Applied Mathematics and Statistics, 4(2), 74-80. https://doi.org/10.11648/j.sjams.20160402.19

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ACS Style

Wenjuan Zhai; Bingzhen Chen. Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs. Sci. J. Appl. Math. Stat. 2016, 4(2), 74-80. doi: 10.11648/j.sjams.20160402.19

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AMA Style

Wenjuan Zhai, Bingzhen Chen. Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs. Sci J Appl Math Stat. 2016;4(2):74-80. doi: 10.11648/j.sjams.20160402.19

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@article{10.11648/j.sjams.20160402.19,
  author = {Wenjuan Zhai and Bingzhen Chen},
  title = {Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {4},
  number = {2},
  pages = {74-80},
  doi = {10.11648/j.sjams.20160402.19},
  url = {https://doi.org/10.11648/j.sjams.20160402.19},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20160402.19},
  abstract = {In this paper, we derive the implicit exponentially fitted RKNd methods for solving oscillatory ODEs. The new methods integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from the set {exp(λt), exp(−λt)}, λ ∈ C, or equivalently  when λ = iω, ω ∈ R. Numerical experiments are accompanied to show the efficiency and competence of the implicit exponentially fitted RKNd methods compared with implicit RKNd methods.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Implicit Exponentially Fitted RKNd Methods for Solving Oscillatory ODEs
AU  - Wenjuan Zhai
AU  - Bingzhen Chen
Y1  - 2016/04/13
PY  - 2016
N1  - https://doi.org/10.11648/j.sjams.20160402.19
DO  - 10.11648/j.sjams.20160402.19
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 74
EP  - 80
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20160402.19
AB  - In this paper, we derive the implicit exponentially fitted RKNd methods for solving oscillatory ODEs. The new methods integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from the set {exp(λt), exp(−λt)}, λ ∈ C, or equivalently  when λ = iω, ω ∈ R. Numerical experiments are accompanied to show the efficiency and competence of the implicit exponentially fitted RKNd methods compared with implicit RKNd methods.
VL  - 4
IS  - 2
ER  -

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