Research Article
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Year 2022, Volume: 43 Issue: 2, 321 - 326, 29.06.2022

Murat Arı , Bahar Karaman , Yılmaz Dereli

https://doi.org/10.17776/csj.947289
https://izlik.org/JA99NM66BU

Abstract

References

  • [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

Year 2022, Volume: 43 Issue: 2, 321 - 326, 29.06.2022

Murat Arı , Bahar Karaman , Yılmaz Dereli

https://doi.org/10.17776/csj.947289
https://izlik.org/JA99NM66BU

Abstract

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.

Keywords

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

References

  • [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.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Murat Arı 0000-0002-4039-5970

Bahar Karaman 0000-0001-6631-8562

Yılmaz Dereli 0000-0003-0149-0542

Submission Date June 3, 2021
Acceptance Date May 20, 2022
Publication Date June 29, 2022
DOI https://doi.org/10.17776/csj.947289
IZ https://izlik.org/JA99NM66BU
Published in Issue Year 2022 Volume: 43 Issue: 2

Cite

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

Cited By

An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains
Mathematics
https://doi.org/10.3390/math13071036

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