Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Numerical simulation of solitary waves of Rosenau–KdV equation by Crank–Nicolson meshless spectral interpolation method

  • 41 Accesses

Abstract

An efficient and accurate Crank–Nicolson meshless spectral radial point interpolation (CN-MSRPI) method is proposed for the numerical solution of nonlinear Rosenau–KdV equation. The proposed method uses meshless shape functions, owing Kronecker delta property, for approximation of spatial operator. Crank–Nicolson difference scheme is used for temporal operator approximation. Single solitary wave motion, interaction of double and triple solitary waves as well as generation of train of solitary waves from initial data are numerically simulated. Error analysis is made via computation of discrete \(L_{\infty }\), \(L_{2}\) and \(L_{\mathrm{rms}}\) error norms. Efficiency of the proposed numerical scheme is assessed via variation of number of nodes N and time step-size \(\tau \). Two invariant quantities correspond to mass and energy are computed using the proposed method for further validation. Stability of the proposed method is discussed and verified computationally. Comparison of obtained results made with exact and existing results in the literature revealed the proposed CN-MSRPI method superiority.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. 1.

    M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Form (Cambridge University Press, Cambridge, 1990)

  2. 2.

    M. Mirzazadeh, M. Ekici, A. Sonmezoglu et al., Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics. Eur. Phys. J. Plus 131, 166 (2016)

  3. 3.

    H. Bulut, T.A. Sulaiman, H.M. Baskonus, On the new soliton and optical wave structures to some nonlinear evolution equations. Eur. Phys. J. Plus 132, 459 (2017)

  4. 4.

    M. Boudoue-Hubert, N.A. Kudryashov, M. Justin et al., Exact traveling soliton solutions for the generalized Benjamin–Bona–Mahony equation. Eur. Phys. J. Plus 133, 108 (2018)

  5. 5.

    V.F. Morales-Delgado, J.F. Gómez-Aguilar, D. Baleanu, A new approach to exact optical soliton solutions for the nonlinear Schrödinger equation. Eur. Phys. J. Plus 133, 189 (2018)

  6. 6.

    D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary wave. Philos Mag. Ser. 5 39, 422–443 (1895)

  7. 7.

    M. Dehghan, A. Shokri, A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn. 50, 111–120 (2007)

  8. 8.

    S.A. El-Wakil, E.M. Abulwafa, M.A. Zahran, A.A. Mahmoud, Time-fractional KdV equation: formulation and solution using variational methods. Nonlinear Dyn. 65, 55–63 (2011)

  9. 9.

    L. Li, Y. Xie, S. Zhu, New exact solutions for a generalized KdV equation. Nonlinear Dyn. 92, 215–219 (2018)

  10. 10.

    F. Ferdous, M.G. Hafez, Nonlinear time fractional Korteweg–de–Vries equations for the interaction of wave phenomena in fluid-filled elastic tubes. Eur. Phys. J. Plus 133, 384 (2018)

  11. 11.

    P. Rosenau, Dynamics of dense discrete systems: high order effects. Prog. Theor. Phys. 79, 1028–1042 (1988)

  12. 12.

    J.-M. Zuo, Solitons and periodic solutions for the Rosenau–KdV and Rosenau–Kawahara equations. Appl. Math. Comput. 215, 835–840 (2009)

  13. 13.

    P. Razborova, A.H. Kara, A. Biswas, Additional conservation laws for Rosenau–KdV–RLW equation with power law nonlinearity by Lie symmetry. Nonlinear Dyn. 79, 743–748 (2015)

  14. 14.

    D. He, New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity. Nonlinear Dyn. 82, 1177–1190 (2015)

  15. 15.

    D. He, Exact solitary solution and a three-level linearly implicit conservative finite difference method for the generalized Rosenau–Kawahara–RLW equation with generalized Novikov type perturbation. Nonlinear Dyn. 85, 479–498 (2016)

  16. 16.

    M. Dehghan, J. Manafian, A. Saadatmandi, Application of semi-analytical methods for solving the Rosenau–Hyman equation arising in the pattern formation in liquid drops. Int. J. Numer. Methods Heat Fluid Flow 23(6), 777–790 (2012)

  17. 17.

    M. Abbaszadeh, M. Dehghan, The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis. Appl. Anal. 97(7), 1129–1153 (2018)

  18. 18.

    T. Ak, S.B.G. Karakoc, H. Triki, Numerical simulation for treatment of dispersive shallow water waves with Rosenau–KdV equation. Eur. Phys. J. Plus 131, 356 (2016)

  19. 19.

    T. Ak, S. Dhawan, S.B.G. Karakocc, S.K. Bhowmik, K.R. Raslan, Numerical study of Rosenau–Kdv equation using finite element method based on collocation approach. Math. Modell. Anal. 22(3), 373–388 (2017)

  20. 20.

    A. Esfahani, Solitary wave solutions for generalized Rosenau–KdV equation. Commun. Theor. Phys. 55, 396–398 (2011)

  21. 21.

    A. Saha, Topological 1-soliton solutions for the generalized Rosenau–Kdv equation. Fundam. J. Math. Phys. 2(1), 19–25 (2012)

  22. 22.

    B. Wongsaijai, K. Poochinapan, A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau–KdV equation and the Rosenau–RLW equation. Appl. Math. Comput. 245, 289–304 (2014)

  23. 23.

    J. Hu, Y. Xu, B. Hu, Conservative linear difference scheme for Rosenau–KdV equation. Adv. Math. Phys. (2013); Article ID 423718

  24. 24.

    S.B.G. Karakoca, T. Ak, Numerical simulation of dispersive shallow water waves with Rosenau–KdV equation. Int. J. Adv. Appl. Math. Mech. 33(3), 32–40 (2016)

  25. 25.

    S.B.G. Karakoca, T. Ak, Numerical solution of Rosenau–KdV equation using subdomain finite element method. New Trends Math. Sci. (NTMSCI) 4(1), 223–235 (2016)

  26. 26.

    M. Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71(1), 16–30 (2006)

  27. 27.

    M. Dehghan, A. Ghesmati, Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput. Phys. Commun. 181(4), 772–786 (2010)

  28. 28.

    M. Dehghan, V. Mohammadi, Two numerical meshless techniques based on radial basis functions (RBFs) and the method of generalized moving least squares (GMLS) for simulation of coupled Klein-Gordon-Schrödinger (KGS) equations. Comput. Math. Appl. 71, 892–921 (2016)

  29. 29.

    M. Dehghan, V. Mohammadi, A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge–Kutta method. Comput. Phys. Commun. 217, 23–34 (2017)

  30. 30.

    E. Shivanian, A new spectral meshless radial point interpolation (SMRPI) method: a well-behaved alternative to the meshless weak forms. Eng. Anal. Bound. Elem. 54, 1–12 (2015)

  31. 31.

    M. Hussain, S. Haq, A. Ghafoor, Meshless spectral method for solution of time-fractional coupled KdV equations. Appl. Math. Comput. 341, 321–334 (2019)

  32. 32.

    M. Hussain, S. Haq, Weighted meshless spectral method for the solution of multi-term time fractional advection-diffusion equation arising in heat and mass transfer. Int. J. Heat Mass. Transf. 129, 1305–1316 (2019)

  33. 33.

    S.A. Sarra, E.J. Kansa, Multiquadric radial basis function approximation methods for the numerical solution of partial differential equations. Adv. Comput. Mech. 2, 1–220 (2009)

  34. 34.

    P.P. Chinchapatnam, K. Djidjeli, P.B. Nair, Unsymmetric and symmetric meshless schemes for the unsteady convection-diffusion equation. Comput. Methods Appl. Mech. Eng. 195(19–22), 2432–2453 (2006)

  35. 35.

    B. Fornberg, C. Piret, On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere. J. Comput. Phys. 227, 2758–2780 (2008)

  36. 36.

    M. Uddin, H.U. Ali, A. Ali, I.A. Shah, Soliton Kernels for Solving PDEs. Int. J. Comput. Methods 13(2), 1640009 (2016). (14 pages)

  37. 37.

    R. Schaback, S. De-Marchi, Nonstandard kernels and their applications. Dolomit. Res. Notes Approx. (DRNA) 2, 16–43 (2009)

  38. 38.

    J. Rashidinia, M.N. Rasoulizadeh, Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation. Wave Motion (2019). https://doi.org/10.1016/j.wavemoti.2019.05.006

Download references

Author information

Correspondence to Manzoor Hussain.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix A

Appendix A

In order to show linearization process for the nonlinear term \(UU_{x}\) in Eq. (6) let us consider

$$\begin{aligned} \bigg [U(x)U_{x}(x)\bigg ]^{n+1}=[UU_{x}](t^{n}+\tau ,x) \end{aligned}$$
(16)

where \(\tau >0\) is the time step-size. By expanding R.H.S. of Eq. (16) using Taylor’s series about \(\tau \), one can write

$$\begin{aligned}{}[UU_{x}](x,t^{n}+\tau )= & {} UU_{x}(x,t^{n})+\tau \bigg [\partial _{t}(UU_{x})\bigg ](x,t^{n}) +\mathscr {O}(\tau ^2)\\= & {} UU_{x}(x,t^{n})+\tau \bigg [(\partial _{t}U)U_{x}+U(\partial _{t}U_{x})\bigg ](x,t^{n}) +\mathscr {O}(\tau ^2) \end{aligned}$$

Approximation of time derivative with finite differences accordingly the above equation gives us

$$\begin{aligned}{}[UU_{x}](x,t^{n}+\tau )= & {} UU_{x}(x,t^{n}) +\tau \bigg [\left( \frac{U^{n+1}-U^{n}}{\tau }+\mathscr {O}(\tau )\right) U^{n}_{x}\bigg ] \\&+\,\tau \bigg [\left( \frac{U_{x}^{n+1}-U_{x}^{n}}{\tau }+\mathscr {O}(\tau )\right) U^{n}\bigg ] +\mathscr {O}(\tau ^2) \end{aligned}$$

Simplification and re-arrangement then yield

$$\begin{aligned}{}[UU_{x}](x,t^{n}+\tau )= & {} UU_{x}(x,t^{n}) +\bigg [\left( U^{n+1}-U^{n}\right) U^{n}_{x}+\mathscr {O}(\tau ^2)\bigg ]\\&+\,\bigg [\left( U_{x}^{n+1}-U_{x}^{n}\right) U^{n}+\mathscr {O}(\tau ^2)\bigg ] +\mathscr {O}(\tau ^2)\\= & {} [U^{n}_{x}]U^{n+1}+[U^{n}]U_{x}^{n+1}-U^{n}U^{n}_{x}+\mathscr {O}(\tau ^2). \end{aligned}$$

Finally, omission of the \(\mathscr {O}(\tau ^2)\) error term gives us the linearized Eq. (6). It is to mentioned that the same process has been used by Rashidinia and Rasoulizadeh [38] for linearization.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hussain, M., Haq, S. Numerical simulation of solitary waves of Rosenau–KdV equation by Crank–Nicolson meshless spectral interpolation method. Eur. Phys. J. Plus 135, 98 (2020). https://doi.org/10.1140/epjp/s13360-020-00156-7

Download citation