Araştırma Makalesi
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Gardner Denkleminin Trigonometrik Kuintik B-spline Kolokasyon Yöntemi ile Nümerik Çözümleri

Yıl 2018, , 1576 - 1584, 01.12.2018
https://doi.org/10.16984/saufenbilder.342571

Öz

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.

Kaynakça

  • [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

Yıl 2018, , 1576 - 1584, 01.12.2018
https://doi.org/10.16984/saufenbilder.342571

Öz

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.

Kaynakça

  • [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.
Toplam 17 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Özlem Ersoy Hepson

Yayımlanma Tarihi 1 Aralık 2018
Gönderilme Tarihi 10 Ekim 2017
Kabul Tarihi 11 Mart 2018
Yayımlandığı Sayı Yıl 2018

Kaynak Göster

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. Aralık 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, sy. 6 (Aralık 2018): 1576-84. https://doi.org/10.16984/saufenbilder.342571.
EndNote Ersoy Hepson Ö (01 Aralık 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, c. 22, sy. 6, ss. 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 (Aralık 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, c. 22, sy. 6, 2018, ss. 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.

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