496
Views
9
CrossRef citations to date
0
Altmetric
Research Article

Exponential B-spline collocation solutions to the Gardner equation

, &
Pages 837-850 | Received 28 Sep 2018, Accepted 10 Mar 2019, Published online: 09 Apr 2019

References

  • E.J. Allen, J.A. Burns, and D.S. Gilliam, Numerical approximations of the dynamical system generated by Burgers' equation with Neumann-Dirichlet boundary conditions, ESAIM: Math. Model. Numer. Anal. 47(5) (2013), pp. 1465–1492.
  • D.C. Antonopoulou, Galerkin methods for a Schrödinger-type equation with a dynamical boundary condition in two dimensions, ESAIM: Math. Model. Numer. Anal. 49(4) (2015), pp. 1127–1156.
  • A. Aydin and B. Karasözen, Multisymplectic box schemes for the complex modified Korteweg-de Vries equation, J. Math. Phys. 51 (2010), pp. 083511.
  • W.K. Zahra, W.A. Ouf, and M.S. El-Azab, A robust uniform B-spline collocation method for solving the generalized PHI-four equation, Appl. Appl. Math. 11(1) (2016), p. 396.
  • B.K. Singh and M. Kumar, A numerical computation of a system of linear and nonlinear time dependent partial differential equations using reduced differential transform method, Int. J. Differ. Equ. 2016 (2016), pp. 1–9.
  • O. Nakoulima, N. Zahibo, E. Pelinovsky, T. Talipova, A. Slunyaev, and A. Kurkin, Analytical and numerical studies of the variable-coefficient Gardner equation, Appl. Math. Comput. 152 (2004), pp. 449–471.
  • M.S. Ruderman, T. Talipova, and E. Pelinovsky, Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions, J. Plasma Phys. 74(5) (2008), pp. 639–656.
  • A.M. Kamchatnov, Y.H. Kuo, T.C. Lin, T.L. Horng, S.C. Gou, R. Clift, G.A. El, and R.H.J. Grimshaw, Undular bore theory for the Gardner equation, Phys. Rev. E 86(3) (2012), pp. 036605.
  • R. Grimshaw, E. Pelinovsky, T. Taipova, and A. Sergeeva, Rogue internal waves in the ocean: long wave model, Euro. Phys. J. Spec. Topics 185(1) (2010), pp. 195–208.
  • A.M. Kamchatnov, Y.H. Kuo, T.C. Lin, T.L. Horng, S.C. Gou, R. Clift, G.A. El, and R.H.J. Grimshaw, Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation, J. Fluid Mech. 736 (2013), pp. 495–531.
  • A.V. Slyunyaev and E.N Pelinovski, Dynamics of large-amplitude solitons, J. Exp. Theor. Phys. 89(1) (1999), pp. 173–181.
  • H. Hu, M. Tan, and X. Hu, New interaction solutions to the combined KdV-mKdV equation from CTE method, J. Assoc. Arab Univ. Basic. Appl. Sci. 21 (2016), pp. 64–67.
  • W.-F. Yu, S.-Y. Lou, J. Yu, and H.-W. Hu, Interactions between solitons and cnoidal periodic waves of the Gardner equation, Chin. Phys. Lett. 31(7) (2014), pp. 070203.
  • A. Bekir, On traveling wave solutions to combined KdV-mKdV equation and modified Burgers-KdV equation, Commun. Nonlinear Sci. Numer. Simul. 14(4) (2009), pp. 1038–1042.
  • Z. Fu, S. Liu, and S. Liu, New kinds of solutions to Gardner equation, Chaos, Solitons Fractals 20(2) (2004), pp. 301–309.
  • H.L. Lü, X.Q. Liu, and L. Niu, A generalized (G′/G)-expansion method and its applications to nonlinear evolution equations, Appl. Math. Comput. 215(11) (2010), pp. 3811–3816.
  • H. Naher and F.A. Abdullah, Some new solutions of the combined KdV-MKdV equation by using the improved G/G-expansion method, World Appl. Sci. J. 16(11) (2012), pp. 1559–1570.
  • N. Taghizade and A. Neirameh, The solutions of TRLW and Gardner equations by-expansion method, Int. J. Nonlinear Sci. 9(3) (2010), pp. 305–310.
  • M.A. Akbar, N. Hj, and M. Ali, New solitary and periodic solutions of nonlinear evolution equation by exp-function method, In. World. Appl. Sci. J. 17(12) (2012), pp. 1603–1610.
  • A.M. Wazwaz, New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul. 12(8) (2007), pp. 1395–1404.
  • E.M.E. Zayed and M.A.M. Abdelaziz, The two-variable (G′/G,1/G)-expansion method for solving the nonlinear KdV-mKdV equation, Math. Prob. Eng. 2012 (2012), pp. 1–14.
  • E.V. Krishnan, H. Triki, M. Labidi, and A. Biswas, A study of shallow water waves with Gardner's equation, Nonlinear Dyn. 66(4) (2011), pp. 497–507.
  • Y.C. Guo and A. Biswas, Solitons and other solutions to Gardner equation by similarity Reduction, Roman. J. Phys. 60(7–8) (2015), pp. 961–970.
  • A.J.A.M. Jawad, New exact solutions of nonlinear partial differential equations using tan-cot function method, Stud. Math. Sci. 5(2) (2012), pp. 13–25.
  • S. Hamdi, B. Morse, B. Halphen, and W. Schiesser, Conservation laws and invariants of motion for nonlinear internal waves: part II, Natural Hazards 57(3) (2011), pp. 609–616.
  • H. Nishiyama and T. Noi, Conservative difference schemes for the numerical solution of the Gardner equation, Comput. Appl. Math. 35(1) (2016), pp. 75–95.
  • T.M. Rageh, G. Salem, and F.A. El-Salam, Restrictive Taylor approximation for Gardner and KdV equations, Int. J. Adv. Appl. Math. Mech. 1(3) (2014), pp. 1–10.
  • W.Q. Hu, Y.T. Gao, Z.Z. Lan, C.Q. Su, and Y.J. Feng, Lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients, Appl. Math. Model. 46 (2017), pp. 126–140.
  • T. Ak, H. Triki, S. Dhawan, and K.S. Erduran, Theoretical and numerical investigations on solitary wave solutions of Gardner equation, Eur. Phys. J. Plus 133 (2018), pp. 382.
  • O. Ersoy Hepson, A. Korkmaz, and I. Dag, Numerical solutions of the Gardner equation by extended form of the cubic B-splines, Pramana. J. Phys. 91(59) (2018), pp. 1–10.
  • B.J. McCartin, Theory of exponential splines, J. Approx. Theory 66 (1991), pp. 1–23.
  • O. Ersoy, A. Korkmaz, and I. Dag, Exponential B-Splines for numerical solutions to some boussinesq systems for water waves, Mediterr. J. Math. 13(6) (2016), pp. 4975–4994.
  • O. Ersoy, I. Dag, and N. Adar, Exponential twice continuously differentiable B-spline algorithm for Burgers'? equation, Ukrain. Math. J. 70(6) (2018), pp. 906–921.
  • A. Korkmaz, O. Ersoy, and I. Dag, Motion of patterns modeled by the gray-scott autocatalysis system in one dimension, MATCH Commun. Math. Comput. Chem. 77 (2017), pp. 507–526.
  • S.G. Rubin and R.A. Graves, Cubic spline approximation for problems in fluid mechanics, Nasa TR R-436, Washington, DC, 1975.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.