Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions

Gantulga Tsedendorj; Hiroshi Isshiki; Rinchinbazar Ravsal

doi:doi:10.11648/j.acm.s.2015040301.16

| Peer-Reviewed

Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions

Gantulga Tsedendorj, Hiroshi Isshiki, Rinchinbazar Ravsal

Published in Applied and Computational Mathematics (Volume 4, Issue 3-1)

Received: 17 April 2015 Accepted: 17 April 2015 Published: 12 May 2015

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we apply one-step GIRM to numerical evaluations of propagation of a single soliton, soliton-to-soliton interaction and soliton-to-bottom interaction. Firstly, in case of a single soliton, the bottom is considered to be constant in order to understand the behavior of the soliton propagation as it travels in the middle of the sea. Next, in case of soliton-to-bottom, we study behavior of a single soliton propagation when the bottom has different geometries. Finally, we evaluate interaction of two different i.e., big and small solitons. To carry out with the studies, we derive and implement GIRM to numerically solve the Korteweg-de Vries (KdV) equation. In order to verify the theory, numerical experiments are conducted and accurate approximate solutions are obtained in each case of the soliton interactions.

Published in	Applied and Computational Mathematics (Volume 4, Issue 3-1) This article belongs to the Special Issue Integral Representation Method and its Generalization
DOI	10.11648/j.acm.s.2015040301.16
Page(s)	78-86
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), 2015. Published by Science Publishing Group

Keywords

Korteweg-de Vries (KdV) equation, Single Soliton, Soliton-to-Soliton interaction, Soliton-to-Bottom interaction, Numerical Evaluation, Generalized Integral Representation Method (GIRM)

References

[1]	Russell J. S., "Report on Waves." Report of the 14th Meeting of the British Association for the Advancement of Science. London: John Murray, pp. 311-390, 1844.
[2]	Korteweg, D. J.; de Vries, G., "On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves." Philosophical Magazine. 39: 422–443, 1895
[3]	Demiray H., "Weakly nonlinear waves in water of variable depth: Variable-coefficient Korteweg-de Vries equation." Comput. Math. Appl. 60, 1747–1755, 2010.
[4]	Drazin P.G., Johnson R.S., "Solitons: An Introduction." 2nd ed. Cambridge University Press, 1989
[5]	Demiray H., "A note on the wave propagation in water of variable depth." Applied Mathematics and Computation 218, 2294–2299, 2011.
[6]	H. Isshiki, “From Integral Representation Method (IRM) to Generalized Integral Representation Method (GIRM),” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization, Vol. 4, No. 3-1, 2015, pp. 1-14. doi: 10.11648/j.acm.s.2015040301. 11
[7]	H. Isshiki, “Effects of Generalized Fundamental Solution (GFS) on Generalized Integral Representation Method (GIRM),” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization. Vol. 4, No. 3-1, 2015, pp. 40-51. doi: 10.11648/j.acm.s.2015 040301.13
[8]	H. Isshiki, T. Takiya, and H. Niizato, “Application of Generalized Integral representation (GIRM) Method to Fluid Dynamic Motion of Gas or Particles in Cosmic Space Driven by Gravitational Force,” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization. Vol. 4, No. 3-1, 2015, pp. 15-39. doi: 10.11648/j.acm.s.2015040301.12
[9]	H. Isshiki, “Application of the Generalized Integral Representation Method (GIRM) to Tidal Wave Propagation,” Applied and Computational Mathematics, Special Issue: Integral Representation Method and Its Generalization. Vol. 4, No. 3-1, 2015, pp. 52-58. doi: 10.11648/j.acm.s.20150403 01.14
[10]	H. Niizato, G. Tsedendorj, H. Isshiki. Implementation of One and Two-step Generalized Integral Representation Methods (GIRMs). Applied and Computational Mathematics, Special Issue: Integral Representation Method and its Generalization. Vol. 4, No. 3-1, 2015, pp. 59-77. doi: 10.11648/j.acm.s.2015 040301.1

Cite This Article

Plain Text BibTeX RIS

APA Style

Gantulga Tsedendorj, Hiroshi Isshiki, Rinchinbazar Ravsal. (2015). Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions. Applied and Computational Mathematics, 4(3-1), 78-86. https://doi.org/10.11648/j.acm.s.2015040301.16

Copy | Download

ACS Style

Gantulga Tsedendorj; Hiroshi Isshiki; Rinchinbazar Ravsal. Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions. Appl. Comput. Math. 2015, 4(3-1), 78-86. doi: 10.11648/j.acm.s.2015040301.16

Copy | Download

AMA Style

Gantulga Tsedendorj, Hiroshi Isshiki, Rinchinbazar Ravsal. Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions. Appl Comput Math. 2015;4(3-1):78-86. doi: 10.11648/j.acm.s.2015040301.16

Copy | Download

@article{10.11648/j.acm.s.2015040301.16,
  author = {Gantulga Tsedendorj and Hiroshi Isshiki and Rinchinbazar Ravsal},
  title = {Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3-1},
  pages = {78-86},
  doi = {10.11648/j.acm.s.2015040301.16},
  url = {https://doi.org/10.11648/j.acm.s.2015040301.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2015040301.16},
  abstract = {Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we apply one-step GIRM to numerical evaluations of propagation of a single soliton, soliton-to-soliton interaction and soliton-to-bottom interaction. Firstly, in case of a single soliton, the bottom is considered to be constant in order to understand the behavior of the soliton propagation as it travels in the middle of the sea. Next, in case of soliton-to-bottom, we study behavior of a single soliton propagation when the bottom has different geometries. Finally, we evaluate interaction of two different i.e., big and small solitons. To carry out with the studies, we derive and implement GIRM to numerically solve the Korteweg-de Vries (KdV) equation. In order to verify the theory, numerical experiments are conducted and accurate approximate solutions are obtained in each case of the soliton interactions.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions
AU  - Gantulga Tsedendorj
AU  - Hiroshi Isshiki
AU  - Rinchinbazar Ravsal
Y1  - 2015/05/12
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.s.2015040301.16
DO  - 10.11648/j.acm.s.2015040301.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 78
EP  - 86
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.s.2015040301.16
AB  - Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we apply one-step GIRM to numerical evaluations of propagation of a single soliton, soliton-to-soliton interaction and soliton-to-bottom interaction. Firstly, in case of a single soliton, the bottom is considered to be constant in order to understand the behavior of the soliton propagation as it travels in the middle of the sea. Next, in case of soliton-to-bottom, we study behavior of a single soliton propagation when the bottom has different geometries. Finally, we evaluate interaction of two different i.e., big and small solitons. To carry out with the studies, we derive and implement GIRM to numerically solve the Korteweg-de Vries (KdV) equation. In order to verify the theory, numerical experiments are conducted and accurate approximate solutions are obtained in each case of the soliton interactions.
VL  - 4
IS  - 3-1
ER  -

Copy | Download

Author Information

Gantulga Tsedendorj

Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia
Hiroshi Isshiki

IMA, Institute of Mathematical Analysis, Osaka, Japan
Rinchinbazar Ravsal

Department of Administration, National University of Mongolia, Ulaanbaatar, Mongolia

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Gantulga Tsedendorj, Hiroshi Isshiki, Rinchinbazar Ravsal. (2015). Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions. Applied and Computational Mathematics, 4(3-1), 78-86. https://doi.org/10.11648/j.acm.s.2015040301.16

Copy | Download

ACS Style

Gantulga Tsedendorj; Hiroshi Isshiki; Rinchinbazar Ravsal. Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions. Appl. Comput. Math. 2015, 4(3-1), 78-86. doi: 10.11648/j.acm.s.2015040301.16

Copy | Download

AMA Style

Gantulga Tsedendorj, Hiroshi Isshiki, Rinchinbazar Ravsal. Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions. Appl Comput Math. 2015;4(3-1):78-86. doi: 10.11648/j.acm.s.2015040301.16

Copy | Download

@article{10.11648/j.acm.s.2015040301.16,
  author = {Gantulga Tsedendorj and Hiroshi Isshiki and Rinchinbazar Ravsal},
  title = {Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3-1},
  pages = {78-86},
  doi = {10.11648/j.acm.s.2015040301.16},
  url = {https://doi.org/10.11648/j.acm.s.2015040301.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2015040301.16},
  abstract = {Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we apply one-step GIRM to numerical evaluations of propagation of a single soliton, soliton-to-soliton interaction and soliton-to-bottom interaction. Firstly, in case of a single soliton, the bottom is considered to be constant in order to understand the behavior of the soliton propagation as it travels in the middle of the sea. Next, in case of soliton-to-bottom, we study behavior of a single soliton propagation when the bottom has different geometries. Finally, we evaluate interaction of two different i.e., big and small solitons. To carry out with the studies, we derive and implement GIRM to numerically solve the Korteweg-de Vries (KdV) equation. In order to verify the theory, numerical experiments are conducted and accurate approximate solutions are obtained in each case of the soliton interactions.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Application of Generalized Integral Method (GIRM) to Numerical Evaluations of Soliton-to-Soliton and Soliton-to-Bottom Interactions
AU  - Gantulga Tsedendorj
AU  - Hiroshi Isshiki
AU  - Rinchinbazar Ravsal
Y1  - 2015/05/12
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.s.2015040301.16
DO  - 10.11648/j.acm.s.2015040301.16
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 78
EP  - 86
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.s.2015040301.16
AB  - Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we apply one-step GIRM to numerical evaluations of propagation of a single soliton, soliton-to-soliton interaction and soliton-to-bottom interaction. Firstly, in case of a single soliton, the bottom is considered to be constant in order to understand the behavior of the soliton propagation as it travels in the middle of the sea. Next, in case of soliton-to-bottom, we study behavior of a single soliton propagation when the bottom has different geometries. Finally, we evaluate interaction of two different i.e., big and small solitons. To carry out with the studies, we derive and implement GIRM to numerically solve the Korteweg-de Vries (KdV) equation. In order to verify the theory, numerical experiments are conducted and accurate approximate solutions are obtained in each case of the soliton interactions.
VL  - 4
IS  - 3-1
ER  -

Copy | Download