Araştırma Makalesi
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Yıl 2022, , 321 - 326, 29.06.2022

Murat Arı , Bahar Karaman , Yılmaz Dereli

https://doi.org/10.17776/csj.947289

Öz

Kaynakça

  • [1] Korteweg D., Vries G. D., On the change in form of long waves advancing in rectangular canal and on a new type of longstationary waves, Philos. Mag., 39 (1895) 422–443.
  • [2] Cui Y., Mao D. K., Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation, J. Comput. Phys., 227(1) (2007) 376–399.
  • [3] Zhu S. and Zhao J., The alternating segment explicit-implicit scheme for the dispersive equation, Appl. Math. Lett., 14(6) (2001) 57–662.
  • [4] Kudryashov N.A., On new travelling wave solutions of the KdV and the KdV-Burgers equations, Commun. Nonlinear Sci. Numer. Simul., 14 (2009) 1891-1900.
  • [5] Wazzan L. A., Modified tanh-coth method for solving the KdV and the KdV-Burgers equations, Commun. Nonlinear Sci Numer. Simul., 14 (2009) 443-450.
  • [6] Biswas A., Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients, Nonlinear Dyn., 58 (2009) 345-348.
  • [7] Wang G. W., Xu T. Z., Ebadi G., Johnson S., Strong A. J., A., Biswas A., Singular solitons, shock waves, and other solutions to potential KdV equation, Nonlinear Dyn., 76 (2014) 1059-1068.
  • [8] Dehghan M., Shokri A. A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dynamics, 50 (2007) 111-120.
  • [9] Vaneeva O. O., Papanicolaou N. C., Christou M. A., Sophocleous C., Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries, Commun. Nonlinear Sci. Numer. Simul., 19 (2014) 3074-3085.
  • [10] Rosenau P. A quasi-continuous description of a nonlinear transmission line, Phys. Scr., 34 (1986) 827-829.
  • [11] Rosenau P., Dynamics of dense discrete systems, Prog. Theor. Phys., 79 (1988) 1028-1042.
  • [12] Esfahani A., Solitary wave solutions for generalized Rosenau-KdV equation, Commun. Theor. Phys., 55(3) (2011) 396–398.
  • [13] Razborova P., Triki H., Biswas A. Perturbation of dispersive shallow water waves, Ocean Eng., 63 (2012) 1–7.
  • [14] Zuo J. M., Solitons and periodic solutions for the Rosenau-KdV and Rosenau-Kawahara equations, Appl. Math. Comput., 215(2) (2009) 835–840.
  • [15] Saha A., Topological 1-soliton solutions for the Generalized Rosenau-Kdv equation, Fundamental J. Math. Phys., 2(1) (2012) 19–23.
  • [16] Hu J., Xu Y., Hu B., Conservative linear difference scheme for Rosenau-KdV equation, Adv. Math. Phys., (2013) 423718.
  • [17] Zheng M., Zhou J., An Average Linear Difference Scheme for the Generalized Rosenau-KdV Equation, J. Appl. Math., (2014) 202793.
  • [18] Luo Y., Xu Y., Feng M., Conservative Difference Scheme for Generalized Rosenau-KdV Equation, Adv. Math. Phys., (2014) 986098 .
  • [19] Ak T., Dhawan S., Karakoç S. B. G., Bhowmik S. K., Raslan K. R., Numerical Study of Rosenau-KdV Equation Using Finite Element Method Based on Collocation Approach, Math. Model. Anal., 22(3) (2017) 373-388.
  • [20]Korkmaz B., Dereli Y., Numerical solution of the Rosenau-KdV-RLW equation by using RBFs Collocation method, Int. J. Mod. Phys. C, 27(10) (2016) 1650117.
  • [21] Karaman B., Dereli Y., Meshless Method Based on Radial Basis Functions for General Rosenau KdV-RLW Equation, Anadolu University Journal of Science and Technology B- Theoretical Sciences, 6(1) (2018) 45-54.
  • [22] Schaback R., The Meshless Kernel-Based Method of Lines for Solving Nonlinear Evolution Equations, Preprint, Göttingen, (2008).
  • [23]Rolland H.L. Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 176 (1971) 1905-1915.
  • [24]Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comp. Math., 4 (1995) 389-396.

Solving the Generalized Rosenau-KdV Equation by the Meshless Kernel-Based Method of Lines

Yıl 2022, , 321 - 326, 29.06.2022

Murat Arı , Bahar Karaman , Yılmaz Dereli

https://doi.org/10.17776/csj.947289

Öz

This current investigation consists of the numerical solutions of the Generalized Rosenau-KdV equation by using the meshless kernel-based method of lines, which is a truly meshless method. The governing equation is a nonlinear partial differential equation but the use of the method of lines leads to an ordinary differential equation. Thus, the partial differential equation is replaced by the ordinary differential equation. The numerical efficiency of the used technique is tested by different numerical examples. Numerical values of error norms and physical invariants are compared with known values in the literature. Moreover, Multiquadric, Gaussian, and Wendland’s compactly supported functions are used in computations. It is seen that the used truly meshless method in computations is very effective with high accuracy and reliability.

Anahtar Kelimeler

Generalized Rosenau-Kdv equation Meshless Kernel-based method of lines Radial basis function

Kaynakça

  • [1] Korteweg D., Vries G. D., On the change in form of long waves advancing in rectangular canal and on a new type of longstationary waves, Philos. Mag., 39 (1895) 422–443.
  • [2] Cui Y., Mao D. K., Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation, J. Comput. Phys., 227(1) (2007) 376–399.
  • [3] Zhu S. and Zhao J., The alternating segment explicit-implicit scheme for the dispersive equation, Appl. Math. Lett., 14(6) (2001) 57–662.
  • [4] Kudryashov N.A., On new travelling wave solutions of the KdV and the KdV-Burgers equations, Commun. Nonlinear Sci. Numer. Simul., 14 (2009) 1891-1900.
  • [5] Wazzan L. A., Modified tanh-coth method for solving the KdV and the KdV-Burgers equations, Commun. Nonlinear Sci Numer. Simul., 14 (2009) 443-450.
  • [6] Biswas A., Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients, Nonlinear Dyn., 58 (2009) 345-348.
  • [7] Wang G. W., Xu T. Z., Ebadi G., Johnson S., Strong A. J., A., Biswas A., Singular solitons, shock waves, and other solutions to potential KdV equation, Nonlinear Dyn., 76 (2014) 1059-1068.
  • [8] Dehghan M., Shokri A. A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dynamics, 50 (2007) 111-120.
  • [9] Vaneeva O. O., Papanicolaou N. C., Christou M. A., Sophocleous C., Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries, Commun. Nonlinear Sci. Numer. Simul., 19 (2014) 3074-3085.
  • [10] Rosenau P. A quasi-continuous description of a nonlinear transmission line, Phys. Scr., 34 (1986) 827-829.
  • [11] Rosenau P., Dynamics of dense discrete systems, Prog. Theor. Phys., 79 (1988) 1028-1042.
  • [12] Esfahani A., Solitary wave solutions for generalized Rosenau-KdV equation, Commun. Theor. Phys., 55(3) (2011) 396–398.
  • [13] Razborova P., Triki H., Biswas A. Perturbation of dispersive shallow water waves, Ocean Eng., 63 (2012) 1–7.
  • [14] Zuo J. M., Solitons and periodic solutions for the Rosenau-KdV and Rosenau-Kawahara equations, Appl. Math. Comput., 215(2) (2009) 835–840.
  • [15] Saha A., Topological 1-soliton solutions for the Generalized Rosenau-Kdv equation, Fundamental J. Math. Phys., 2(1) (2012) 19–23.
  • [16] Hu J., Xu Y., Hu B., Conservative linear difference scheme for Rosenau-KdV equation, Adv. Math. Phys., (2013) 423718.
  • [17] Zheng M., Zhou J., An Average Linear Difference Scheme for the Generalized Rosenau-KdV Equation, J. Appl. Math., (2014) 202793.
  • [18] Luo Y., Xu Y., Feng M., Conservative Difference Scheme for Generalized Rosenau-KdV Equation, Adv. Math. Phys., (2014) 986098 .
  • [19] Ak T., Dhawan S., Karakoç S. B. G., Bhowmik S. K., Raslan K. R., Numerical Study of Rosenau-KdV Equation Using Finite Element Method Based on Collocation Approach, Math. Model. Anal., 22(3) (2017) 373-388.
  • [20]Korkmaz B., Dereli Y., Numerical solution of the Rosenau-KdV-RLW equation by using RBFs Collocation method, Int. J. Mod. Phys. C, 27(10) (2016) 1650117.
  • [21] Karaman B., Dereli Y., Meshless Method Based on Radial Basis Functions for General Rosenau KdV-RLW Equation, Anadolu University Journal of Science and Technology B- Theoretical Sciences, 6(1) (2018) 45-54.
  • [22] Schaback R., The Meshless Kernel-Based Method of Lines for Solving Nonlinear Evolution Equations, Preprint, Göttingen, (2008).
  • [23]Rolland H.L. Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 176 (1971) 1905-1915.
  • [24]Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comp. Math., 4 (1995) 389-396.
Toplam 24 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Natural Sciences
Yazarlar

Murat Arı 0000-0002-4039-5970

Bahar Karaman 0000-0001-6631-8562

Yılmaz Dereli 0000-0003-0149-0542

Yayımlanma Tarihi 29 Haziran 2022
Gönderilme Tarihi 3 Haziran 2021
Kabul Tarihi 20 Mayıs 2022
Yayımlandığı Sayı Yıl 2022

Kaynak Göster

APA Arı, M., Karaman, B., & Dereli, Y. (2022). Solving the Generalized Rosenau-KdV Equation by the Meshless Kernel-Based Method of Lines. Cumhuriyet Science Journal, 43(2), 321-326. https://doi.org/10.17776/csj.947289