Research Article
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Gardner Denkleminin Trigonometrik Kuintik B-spline Kolokasyon Yöntemi ile Nümerik Çözümleri

Year 2018, Volume: 22 Issue: 6, 1576 - 1584, 01.12.2018
https://doi.org/10.16984/saufenbilder.342571

Abstract

Bu çalışmanın
amacı çeşitli disiplinlerde sıkça kullanılan Gardner denkleminin nümerik
çözümlerini elde etmektir. Bu amaç için geniş kararlılık bölgesine sahip
olmasından dolayı klasik Crank-Nicolson yöntemi ile zaman integrasyonu
yapılmıştır. Konum ayrıştırması ise trigonometrik quintik B-spline
fonksiyonları kullanılarak yapılmıştır. Bu yüzden Gardner denklemi beş bant
matris sistemine dönüştürülmüş ve Thomas algoritması uygulanmıştır.

References

  • [1] M. S. Ruderman, T. Talipova, E. Pelinovsky, "Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions", Journal of Plasma Physics, 74(05), 639-656, 2008.
  • [2] A. M. Kamchatnov, Y. H. Kuo, T. C. Lin, T. L. Horng, S. C., Gou, R., Clift, G.A. El, R. H. Grimshaw, "Undular bore theory for the Gardner equation", Physical Review E, 86(3), 036605, 2012.
  • [3] A. M. Kamchatnov, Y. H. Kuo, T. C. Lin, T. L. Horng, S. C., Gou, R. Clift, G.A. El, R. H. Grimshaw, "Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation" Journal of Fluid Mechanics, 736, 495-531, 2013.
  • [4] H. Nishiyama, T. Noi, "Conservative difference schemes for the numerical solution of the Gardner equation", Computational and Applied Mathematics, 35(1), 75-95, 2016.
  • [5] T. M. Rageh, G. Salem, F. A. El-Salam, "Restrictive Taylor Approximation for Gardner and KdV Equations" Int. J. Adv. Appl. Math. and Mech, 1(3), 1-10, 2014.
  • [6] A. Bekir, "On traveling wave solutions to combined KdV-mKdV equation and modified Burgers-KdV equation" Communications in Nonlinear Science and Numerical Simulation, 14(4), 1038-1042, 2009.
  • [7] Z. Fu, S. Liu, S. Liu, S, "New kinds of solutions to Gardner equation", Chaos, Solitons & Fractals, 20(2), 301-309, 2004.
  • [8] E. M. E. Zayed, M. A. M. Abdelaziz, "The Two-Variable (G'/G, 1/G)-Expansion Method for Solving the Nonlinear KdV-mKdV Equation", Mathematical Problems in Engineering, 2012, Article ID 725061, 1-14, 2012.
  • [9] E. V. Krishnan, H. Triki, M. Labidi, A. Biswas, "A study of shallow water waves with Gardner's equation", Nonlinear Dynamics, 66(4), 497-507, 2011.
  • [10] A. J. A. M. Jawad, "New Exact Solutions of Nonlinear Partial Differential Equations Using Tan-Cot Function Method", Studies in Mathematical sciences, 5(2), 13-25, 2012.
  • [11] N. M. Yagmurlu, O. Tasbozan, Y. Ucar, A. Esen, "Numerical Solutions of the CombinedKdV-MKdV Equation by a Quintic B-spline Collocation Method", Appl. Math. Inf. Sci. Lett., 4(1), 19-24, 2016.
  • [12] M. Abbas, A. A. Majid, A. I. M. Ismail, A. Rashid, (2014, May).” Numerical method using cubic trigonometric B-spline technique for nonclassical diffusion problems”, In Abstract and applied analysis, 2014.
  • [13] O. Ersoy, I. Dağ, "The Numerical Approach to the Fisher's Equation via Trigonometric Cubic B-spline Collocation Method", Communications in Numerical Analysis, 2017(2), 91-100, 2017.
  • [14] O. Ersoy, A. Korkmaz, I. Dağ, "Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves" Mediterranean Journal of Mathematics, 13, 4975-4994, 2016.
  • [15]. I. Dağ, O. Ersoy, " The exponential cubic B-spline algorithm for Fisher equation ", Chaos, Solitons and Fractals, 86, 101-106, 2016.
  • [16] S. G. Rubin, R. A. Graves, "Cubic spline approximation for problems in fluid mechanics", Nasa TR R-436,Washington, DC, 1975.
  • [17] A. Wazwaz, "Partial Differential Equations and Solitary Waves", Theory, Springer-Verlag Berlin Heidelberg, 2009.

Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method

Year 2018, Volume: 22 Issue: 6, 1576 - 1584, 01.12.2018
https://doi.org/10.16984/saufenbilder.342571

Abstract

The main purpose of this paper is to get the numerical
solutions of the Gardner equation which are widely used in various disciplines.
For this purpose, the time integration of the system is achieved by the
classical Crank-Nicolson method owing to its large stability region. Space
discretization is done by using the trigonometric quintic B-spline functions.
Thus the Gardner equation turns into a penta diagonoal matrix equation and the
Thomas algorithm is applied.

References

  • [1] M. S. Ruderman, T. Talipova, E. Pelinovsky, "Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions", Journal of Plasma Physics, 74(05), 639-656, 2008.
  • [2] A. M. Kamchatnov, Y. H. Kuo, T. C. Lin, T. L. Horng, S. C., Gou, R., Clift, G.A. El, R. H. Grimshaw, "Undular bore theory for the Gardner equation", Physical Review E, 86(3), 036605, 2012.
  • [3] A. M. Kamchatnov, Y. H. Kuo, T. C. Lin, T. L. Horng, S. C., Gou, R. Clift, G.A. El, R. H. Grimshaw, "Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation" Journal of Fluid Mechanics, 736, 495-531, 2013.
  • [4] H. Nishiyama, T. Noi, "Conservative difference schemes for the numerical solution of the Gardner equation", Computational and Applied Mathematics, 35(1), 75-95, 2016.
  • [5] T. M. Rageh, G. Salem, F. A. El-Salam, "Restrictive Taylor Approximation for Gardner and KdV Equations" Int. J. Adv. Appl. Math. and Mech, 1(3), 1-10, 2014.
  • [6] A. Bekir, "On traveling wave solutions to combined KdV-mKdV equation and modified Burgers-KdV equation" Communications in Nonlinear Science and Numerical Simulation, 14(4), 1038-1042, 2009.
  • [7] Z. Fu, S. Liu, S. Liu, S, "New kinds of solutions to Gardner equation", Chaos, Solitons & Fractals, 20(2), 301-309, 2004.
  • [8] E. M. E. Zayed, M. A. M. Abdelaziz, "The Two-Variable (G'/G, 1/G)-Expansion Method for Solving the Nonlinear KdV-mKdV Equation", Mathematical Problems in Engineering, 2012, Article ID 725061, 1-14, 2012.
  • [9] E. V. Krishnan, H. Triki, M. Labidi, A. Biswas, "A study of shallow water waves with Gardner's equation", Nonlinear Dynamics, 66(4), 497-507, 2011.
  • [10] A. J. A. M. Jawad, "New Exact Solutions of Nonlinear Partial Differential Equations Using Tan-Cot Function Method", Studies in Mathematical sciences, 5(2), 13-25, 2012.
  • [11] N. M. Yagmurlu, O. Tasbozan, Y. Ucar, A. Esen, "Numerical Solutions of the CombinedKdV-MKdV Equation by a Quintic B-spline Collocation Method", Appl. Math. Inf. Sci. Lett., 4(1), 19-24, 2016.
  • [12] M. Abbas, A. A. Majid, A. I. M. Ismail, A. Rashid, (2014, May).” Numerical method using cubic trigonometric B-spline technique for nonclassical diffusion problems”, In Abstract and applied analysis, 2014.
  • [13] O. Ersoy, I. Dağ, "The Numerical Approach to the Fisher's Equation via Trigonometric Cubic B-spline Collocation Method", Communications in Numerical Analysis, 2017(2), 91-100, 2017.
  • [14] O. Ersoy, A. Korkmaz, I. Dağ, "Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves" Mediterranean Journal of Mathematics, 13, 4975-4994, 2016.
  • [15]. I. Dağ, O. Ersoy, " The exponential cubic B-spline algorithm for Fisher equation ", Chaos, Solitons and Fractals, 86, 101-106, 2016.
  • [16] S. G. Rubin, R. A. Graves, "Cubic spline approximation for problems in fluid mechanics", Nasa TR R-436,Washington, DC, 1975.
  • [17] A. Wazwaz, "Partial Differential Equations and Solitary Waves", Theory, Springer-Verlag Berlin Heidelberg, 2009.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Özlem Ersoy Hepson

Publication Date December 1, 2018
Submission Date October 10, 2017
Acceptance Date March 11, 2018
Published in Issue Year 2018 Volume: 22 Issue: 6

Cite

APA Ersoy Hepson, Ö. (2018). Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method. Sakarya University Journal of Science, 22(6), 1576-1584. https://doi.org/10.16984/saufenbilder.342571
AMA Ersoy Hepson Ö. Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method. SAUJS. December 2018;22(6):1576-1584. doi:10.16984/saufenbilder.342571
Chicago Ersoy Hepson, Özlem. “Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-Spline Collocation Method”. Sakarya University Journal of Science 22, no. 6 (December 2018): 1576-84. https://doi.org/10.16984/saufenbilder.342571.
EndNote Ersoy Hepson Ö (December 1, 2018) Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method. Sakarya University Journal of Science 22 6 1576–1584.
IEEE Ö. Ersoy Hepson, “Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method”, SAUJS, vol. 22, no. 6, pp. 1576–1584, 2018, doi: 10.16984/saufenbilder.342571.
ISNAD Ersoy Hepson, Özlem. “Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-Spline Collocation Method”. Sakarya University Journal of Science 22/6 (December 2018), 1576-1584. https://doi.org/10.16984/saufenbilder.342571.
JAMA Ersoy Hepson Ö. Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method. SAUJS. 2018;22:1576–1584.
MLA Ersoy Hepson, Özlem. “Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-Spline Collocation Method”. Sakarya University Journal of Science, vol. 22, no. 6, 2018, pp. 1576-84, doi:10.16984/saufenbilder.342571.
Vancouver Ersoy Hepson Ö. Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method. SAUJS. 2018;22(6):1576-84.