Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

NEW ANALYTIC APPROXIMATE SOLUTIONS TO THE GENERALIZED REGULARIZED LONG WAVE EQUATIONS

  • Bildik, Necdet (Department of Mathematics Faculty of Art and Sciences Manisa Celal Bayar University) ;
  • Deniz, Sinan (Department of Mathematics Faculty of Art and Sciences Manisa Celal Bayar University)
  • Received : 2017.03.13
  • Accepted : 2017.11.03
  • Published : 2018.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the new optimal perturbation iteration method has been applied to solve the generalized regularized long wave equation. Comparing the new analytic approximate solutions with the known exact solutions reveals that the proposed technique is extremely accurate and effective in solving nonlinear wave equations. We also show that,unlike many other methods in literature, this method converges rapidly to exact solutions at lower order of approximations.

Keywords

References

  1. Y. Aksoy and M. Pakdemirli, New perturbation-iteration solutions for Bratu-type equations, Computers & Math. Appl. 59 (2010), no. 8, 2802-2808. https://doi.org/10.1016/j.camwa.2010.01.050
  2. Y. Aksoy, M. Pakdemirli, S. Abbasbandy, and H. Boyaci, New perturbation-iteration solutions for nonlinear heat transfer equations, Int. J. Numer. Methods for Heat & Fluid Flow 22 (2012), no. 7, 814-828. https://doi.org/10.1108/09615531211255725
  3. Md. N. Alam and F. B. M. Belgacem, Application of the Novel (G/G)-expansion method to the regularized long wave equation, Waves, Wavelets and Fractals Advanced Analysis 1 (2015), 20-37.
  4. H. M. Baskonus, H. Bulut, and F. B. M. Belgacem, Analytical solutions for nonlinear long-short wave interaction systems with highly complex, J. Comput. Appl. Math. 312 (2017), 257-266. https://doi.org/10.1016/j.cam.2016.05.035
  5. A. Bekir, New solitons and periodic wave solutions for some nonlinear physical models by using the sine-cosine method, Phys. Scripta 77 (2008), no. 4, 045008. https://doi.org/10.1088/0031-8949/77/04/045008
  6. D. Bhardwaj and R. Shankar, A computational method for regularized long wave equation, Computers & Math. Appl. 40 (2000), no. 12, 1397-1404. https://doi.org/10.1016/S0898-1221(00)00248-0
  7. N. Bildik and D. Sinan, Comparative study between optimal homotopy asymptotic method and perturbation-iteration technique for different types of nonlinear equations, Iranian J. Sci. Techno. Transactions A: Science: 1-8.
  8. N. Bildik and D. Sinan, A new efficient method for solving delay differential equations and a comparison with other methods, The European Physical Journal Plus 132 (2017), no. 1, 51. https://doi.org/10.1140/epjp/i2017-11344-9
  9. J. L. Bona, W. R. McKinney, and J. M. Restrepo, Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation, J. Nonlinear Sci. 10 (2000), no. 6, 603-638. https://doi.org/10.1007/s003320010003
  10. J. L. Bona, W. G. Pritchard, and L. R. Scott, A comparison of solutions of two model equations for long waves, in Fluid dynamics in astrophysics and geophysics (Chicago, Ill., 1981), 235-267, Lectures in Appl. Math., 20, Amer. Math. Soc., Providence, RI, 1983.
  11. I. Dag and M. N. Ozer, Approximation of the RLW equation by the least square cubic B-spline finite element method, Appl. Math. Model. 25 (2001), 221-231. https://doi.org/10.1016/S0307-904X(00)00030-5
  12. M. Dehghan and R. Salehi, The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas, Comput. Phys. Comm. 182 (2011), no. 12, 2540-2549. https://doi.org/10.1016/j.cpc.2011.07.018
  13. S. Deniz and N. Bildik, A new analytical technique for solving Lane-Emden type equa-tions arising in astrophysics, Bull. Belg. Math. Soc. Simon Stevin 24 (2017), no. 2, 305-320.
  14. A. Dogan, Numerical solution of regularized long wave equation using Petrov-Galerkin method, Comm. Numer. Methods Engrg. 17 (2001), no. 7, 485-494. https://doi.org/10.1002/cnm.424
  15. A. Dogan, Numerical solution of RLW equation using linear finite elements within Galerkins method, Appl. Math. Model. 26 (2002), 771-783. https://doi.org/10.1016/S0307-904X(01)00084-1
  16. R. S. Dubey, F. B. M. Belgacem, and P. Goswami, Homotopy Perturbation Approximate Solutions for Bergman-Minimal Blood Glucose-Insulin Model, J. Fractal Geom. Nonlinear Anal. in Medicine and Biology (FGNAMB) 2 (2016), no. 3, 1-6.
  17. L. R. T. Gardner, G. A. Gardner, and I. Dag, A B-spline finite element method for the regularized long wave equation, Comm. Numer. Methods Engrg. 11 (1995), no. 1, 59-68. https://doi.org/10.1002/cnm.1640110109
  18. S. Iqbala, M. Idreesb, A. M. Siddiquic, and A. R. Ansarid, Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method, Appl. Math. Comput. 216 (2010), no. 10, 2898-2909. https://doi.org/10.1016/j.amc.2010.04.001
  19. H. Jafari and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition, Appl. Math. Comput. 180 (2006), no. 2, 488-497. https://doi.org/10.1016/j.amc.2005.12.031
  20. P. C. Jain and L. Iskandar, Numerical solutions of the regularized long-wave equation, Comput. Methods Appl. Mech. Engrg. 20 (1979), no. 2, 195-201. https://doi.org/10.1016/0045-7825(79)90017-3
  21. D. Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. Math. Comput. 149 (2004), no. 3, 833-841. https://doi.org/10.1016/S0096-3003(03)00189-9
  22. M. A. Khan, M. A. Akbar, and F. B. M. Belgacem, Solitary wave solutions for the boussinesq and fisher equations by the modified simple equation method, Math. Lett. 2 (2016), no. 1, 1-18.
  23. J. Lin, Z. Xie, and J. Zhou, High-order compact difference scheme for the regularized long wave equation, Comm. Numer. Methods Engrg. 23 (2007), no. 2, 135-156.
  24. V. Marinca and H. Nicolae, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, International Communications in Heat and Mass Transfer 35 (2008), no. 6, 710-715. https://doi.org/10.1016/j.icheatmasstransfer.2008.02.010
  25. D. Mirzaei and M. Dehghan, MLPG method for transient heat conduction problem with MLS as trial approximation in both time and space domains, CMES Comput. Model. Eng. Sci. 72 (2011), no. 3, 185-210.
  26. M. Mohammadi and R. Mokhtari, Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math. 235 (2011), no. 14, 4003-4014. https://doi.org/10.1016/j.cam.2011.02.012
  27. T. Ozis and D. Agirseven, He's homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients, Phys. Lett. A 372 (2008), no. 38, 5944-5950. https://doi.org/10.1016/j.physleta.2008.07.060
  28. D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 (1996), 321-330.
  29. M. M. Rashidi, G. Domairry, and S. Dinarvand, Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 14 (2009), no. 3, 708-717. https://doi.org/10.1016/j.cnsns.2007.09.015
  30. B. Saka and I. Dag, A collocation method for the numerical solution of the RLW equation using cubic B-spline basis, Arab. J. Sci. Eng. Sect. A Sci. 30 (2005), no. 1, 39-50.
  31. B. Saka and I. Dag, A numerical solution of the RLW equation by Galerkin method using quartic B-splines, Comm. Numer. Methods Engrg. 24 (2008), no. 11, 1339-1361.
  32. A. Shokri and M. Dehghan, A meshless method using the radial basis functions for numerical solution of the regularized long wave equation, Numer. Methods Partial Differential Equations 26 (2010), no. 4, 807-825. https://doi.org/10.1002/num.20457
  33. D. Sinan, Optimal Perturbation Iteration Method for Solving Nonlinear Heat Transfer Equations, J. Heat Transfer 139 (2017), no. 7, 074503. https://doi.org/10.1115/1.4036085
  34. D. Sinan and N. Bildik, Applications of optimal perturbation iteration method for solving nonlinear differential equations, AIP Conference Proceedings. Vol. 1798. No. 1. AIP Publishing, 2017.
  35. A. A. Soliman, Numerical simulation of the generalized regularized long wave equation by He's variational iteration method, Math. Comput. Simulation 70 (2005), no. 2, 119-124. https://doi.org/10.1016/j.matcom.2005.06.002
  36. K. A. Touchent and F. B. M. Belgacem, Nonlinear fractional partial differential equations systems solutions through a hybrid homotopy perturbation Sumudu transform method, Nonlinear Studies 22 (2015), no. 4, 591-600.