Title: Chebyshev Rational Approximations for the Rosenau-KdV-RLW Equation on the Whole Line
Author(s): Mohammadreza Foroutan, Ali Ebadian
Pages: 1-15
Cite as:
Mohammadreza Foroutan, Ali Ebadian, Chebyshev Rational Approximations for the Rosenau-KdV-RLW Equation on the Whole Line, Int. J. Anal. Appl., 16 (1) (2018), 1-15.


In this paper, we consider the use of a modified Chebyshev rational approximations for the Rosenau-KdV-RLW equation on the whole line with initial-boundary values. It is shown that the proposed scheme leads to optimal error estimates. Furthermore, the stability and convergence of the proposed schemes are proved. The fully discrete Chebyshev pseudo-spectral scheme is constructed. Numerical results confirm well with the theoretical results. The idea and techniques presented in this paper will be useful to solve many other problems.

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  1. A. Biswas, Singular solitons, shock waves, and other solutions to potential KdV equation, Nonlinear Dyn. 58 (2009), 345–348. Google Scholar

  2. A. Biswas , Solitary waves for power-law regularized long-wave equation and R(m,n) equation, Nonlinear Dyn. 59 (2010), 423–426. Google Scholar

  3. A. Biswas, Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 3503–3506. Google Scholar

  4. A. Biswas, H. Triki, M. Labidi, Bright and dark solitons of the RosenauKawahara equation with power law nonlinearity, Phys. Wave Phenom. 19 (2011), 24–29. Google Scholar

  5. J.P. Boyd, Chebyshev and Fourier spectral methods, Second ed., Dover, New York, 2000. Google Scholar

  6. J.P. Boyd, Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys. 70 (1978), 63–88. Google Scholar

  7. J.P. Boyd, Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys. 69 (1978), 112–142. Google Scholar

  8. C.I. Christov, A complete orthogonal system of functions in L 2 (−∞,∞) spaces, SIAM J. Appl. Math. 42 (1982), 1337– 1344. Google Scholar

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

  10. D.Gottlieb, M.Y. Hussaini, S.Orszag, Theory and Applications of Spectral Methods in Spectral Methods for Partial Differential Equations edited by R. Voigt and D. Gottlieb and M.Y. Hussaini, SIAM, Philadelphia, 1984. Google Scholar

  11. B.Y. Guo, A class of difference schemes of two-dimensional viscous fluid flow, Acta Math. Sinica, 17 (1974), 242–258. Google Scholar

  12. B.Y. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comput. 68 (1999), 1067–1078. Google Scholar

  13. B.Y. Guo, Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, Contemp. Math. 163 (1994), 33–54. Google Scholar

  14. B.Y. Guo, J.Shen, On spectral approximations using modified Legendre rational functions: Approximation to the Korteweg-ele vries equation on the half line, Indiana university Mathematics journal, 50 (2001), 181–204. Google Scholar

  15. B.Y. Guo, J. Shen, Z.Q. Wang, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Int. J. Numer. Meth. Engng. 53 (2002), 65–84. Google Scholar

  16. B.Y. Guo, J. Shen, Z.Q. Wang, A rational approximation and its applications to differential equations on the half line, Journal of scientific computing, 15 (2000), 117–148. Google Scholar

  17. B.Y. Guo, Z.Q. Wang, Modified Chebyshev rational spectral method for the whole line, Discrete Contin. Dyn. Syst. Supplement Volume (2003), 365–374. Google Scholar

  18. M.R. Foroutan, A. Ebadian, S.Najafzadeh, The use of generalized Laguerre functions for solving the equation of magnetohydydinamic flow due to a stretching cylinder , SeMA Journal 73(4) (2016), 335–346. Google Scholar

  19. S.U. Islam, S. Haq, A. Ali, A meshfree method for the numerical solution of the RLW equation, J. Comput. Appl. Math. 223 (2009), 997–1012. Google Scholar

  20. Z.Q. Lv, M. Xue, Y.S. Wang, A new multi-symplectic scheme for the KdV equation, Chin. Phys. Lett. 28 (2011), 060205. Google Scholar

  21. K. Parand, M. Dehghan, A.R. Rezaei, S.M. Ghaderi, An approximation algorithm for the solution of the nonlinear LaneEmden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun. 181 (2010), 1096–1108. Google Scholar

  22. K. Parand, A. Taghavi, Rational scaled generalized Laguerre function collocation method for solving the Blasius equation, J. Comput. Appl. Math. 233(4) (2009), 980–989. Google Scholar

  23. D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 (1966), 321–330. Google Scholar

  24. J.L. Ramos, Explicit finite difference methods for the EW and RLW equations, Appl. Math. Comput. 179(2) (2006), 622–638. Google Scholar

  25. P. Razborova, B. Ahmed, A. Biswas, Solitons, shock waves and conservation laws of Rosenau-KdV-RLW equation with power law nonlinearity, Appl.Math. Inf. Sci.8(2) (2014), 485–491. Google Scholar

  26. P. Razborova, L. Moraru, A. Biswas, Perturbation of dispersive shallow water waves with Rosenau-KdV-RLW equation and power law nonlinearity, Rom. J. Phys. 59(7-8) (2014), 658–676. Google Scholar

  27. P. Razborova, A.H. Kara, A. Biswas, Additional conservation laws for Rosenau-KdV-RLW equation with power law nonlinearity by lie symmetry, Nonlinear Dyn. 179 (2015), 743–748. Google Scholar

  28. P. Razborova, H. Triki, A. Biswas, Perturbation of dispersive shallow water waves, cean Eng. 63 (2013), 1–7. Google Scholar

  29. P. Rosenau, A quasi-continuous description of a nonlinear transmission line , Phys. Scr. 34 (1986), 827–829. Google Scholar

  30. P. Rosenau, Dynamics of dense discrete systems, Prog. Theor. Phys. 79 (1988), 1028–1042. Google Scholar

  31. P. Sanchez, G. Ebadi, A. Mojaver, M. Mirzazadeh, M.Eslami, A. Biswas, Solitons and other solutions to perturbed Rosenau-KdV-RLW equation with power law nonlinearity, Acta Phys. Pol. A, 127 (2015), 1577–1586. Google Scholar

  32. J. Shen, T. Tang, L.L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, First edition 2010. Google Scholar

  33. T. Tajvidi, M. Razzaghi, M. Dehghan, Modified rational Legendre approach to laminar viscous flow over a semi-infinite, Caos, Solitons Fractals, 35 (2008), 59–66. Google Scholar

  34. Z.Q. Wang, B.Y. Guo, Modified Legendre rational spectral method for the whole line, J. Comput. Math. 22 (2004), 457–474. Google Scholar

  35. 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 (2014), 289–304. Google Scholar

  36. Z.Q. Zhang, H.P. Ma, A rational spectral method for the KdV equation on the half line, J. Comput. Appl. Math. 230 (2009), 614–625. Google Scholar