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B-SPLINE COLLOCATION METHOD FOR NUMERICAL SOLUTION OF NONLINEAR KAWAHARA AND MODIFIED KAWAHARA EQUATIONS

Year 2017, Volume: 7 Issue: 2, 0 - 2, 01.12.2017

Abstract

In this paper, a collocation method is applied for solving the Kawahara and modi ed Kawahara equations. For the spatial discretization, we use the sextic B-spline collocation SBSC method on uniform meshes, nite di erence scheme is employed for the time discretization. The stability analysis of the collocation methods are examined by the Von Neumann approach. Numerical results demonstrate the eciency and accuracy of the proposed methods.

References

  • Kawahara,T, (1972), Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33, pp.260
  • Hunter,J.K. and Scheule,J., (1988), Existence of perturbed solitary wave solution to a model equation for water waves, Physica D., 32, pp.253-268.
  • Wazwaz,A.M., (2003), Compacton solutions of the Kawahara-type nonlinear dispersive equation, Appl. Math. Comput., 145, pp.133-150.
  • Zhang,D., (2005), Doubly periodic solutions of the modified Kawahara equations, Chaos Solitons Fractals, 25, pp.1155-1160.
  • Shuangping,T. and Shuangbin,C., (2002), Existence and uniqueness of solutions to nonlinear Kawa- hara equations, Chinese Ann. Math. Ser. A, 23, 2, pp.221-228.
  • Khan,Y. and Faraz,N., (2010), A new approach to differential difference equations, J. Adv. Res. Differ. Equ., 2, 2, pp.1-12.
  • Khan,Y. and Wu,Q., (2011), Homotopy perturbation transform method for nonlinear equations using
  • He’s polynomials, Computers and Mathematics with Applications, 61, 8, pp.1963-1967.
  • Yuan,J.M., Shen,J., and Wu,J., (2008), A Dual-Petrov-Galerkin method for the Kawahara-type equa- tions, J. Sci. Comput, 34, pp.48-63.
  • Zarebnia,M. and Jalili,S., (2011), A numerical solution to a modified Kawahara equation, J. of Ad- vanced Research in Differential Equ., 3, pp.65-76.
  • De Boor,C., (1978), A practical guide to splines, Springer-Verlag, New York.
  • Prenter,P.M, (1985), Splines and Variational Methods, Wiley, New York, NY.
  • Rubin,S.G. and Graves,R.A., (1975), Cubic spline approximation for problems in fluid mechanics
  • Nasa, Washington, D. C., Nasa TR R-436, pp.253-268. Korkmaz,A. and Daˇg,I., (2009), Crank-Nicolson differential quadrature algorithms for the Kawahara equation, Chaos Solutions and Fractals, Vol.42, No.1, pp.65-73.
  • Malik,R.P., (1997), On fifth order KdV-type equation, solv-int/9710010, JINR, Dubna, October 1997.
  • Bibi,N., Tirmizi,S.I.A., and Haq,S., (2011), Meshless method of lines for numerical solution of Kawa- hara type equation, Applied Mathematics, 2, pp.608-618, doi:10.4236/am.2011.25081.
Year 2017, Volume: 7 Issue: 2, 0 - 2, 01.12.2017

Abstract

References

  • Kawahara,T, (1972), Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33, pp.260
  • Hunter,J.K. and Scheule,J., (1988), Existence of perturbed solitary wave solution to a model equation for water waves, Physica D., 32, pp.253-268.
  • Wazwaz,A.M., (2003), Compacton solutions of the Kawahara-type nonlinear dispersive equation, Appl. Math. Comput., 145, pp.133-150.
  • Zhang,D., (2005), Doubly periodic solutions of the modified Kawahara equations, Chaos Solitons Fractals, 25, pp.1155-1160.
  • Shuangping,T. and Shuangbin,C., (2002), Existence and uniqueness of solutions to nonlinear Kawa- hara equations, Chinese Ann. Math. Ser. A, 23, 2, pp.221-228.
  • Khan,Y. and Faraz,N., (2010), A new approach to differential difference equations, J. Adv. Res. Differ. Equ., 2, 2, pp.1-12.
  • Khan,Y. and Wu,Q., (2011), Homotopy perturbation transform method for nonlinear equations using
  • He’s polynomials, Computers and Mathematics with Applications, 61, 8, pp.1963-1967.
  • Yuan,J.M., Shen,J., and Wu,J., (2008), A Dual-Petrov-Galerkin method for the Kawahara-type equa- tions, J. Sci. Comput, 34, pp.48-63.
  • Zarebnia,M. and Jalili,S., (2011), A numerical solution to a modified Kawahara equation, J. of Ad- vanced Research in Differential Equ., 3, pp.65-76.
  • De Boor,C., (1978), A practical guide to splines, Springer-Verlag, New York.
  • Prenter,P.M, (1985), Splines and Variational Methods, Wiley, New York, NY.
  • Rubin,S.G. and Graves,R.A., (1975), Cubic spline approximation for problems in fluid mechanics
  • Nasa, Washington, D. C., Nasa TR R-436, pp.253-268. Korkmaz,A. and Daˇg,I., (2009), Crank-Nicolson differential quadrature algorithms for the Kawahara equation, Chaos Solutions and Fractals, Vol.42, No.1, pp.65-73.
  • Malik,R.P., (1997), On fifth order KdV-type equation, solv-int/9710010, JINR, Dubna, October 1997.
  • Bibi,N., Tirmizi,S.I.A., and Haq,S., (2011), Meshless method of lines for numerical solution of Kawa- hara type equation, Applied Mathematics, 2, pp.608-618, doi:10.4236/am.2011.25081.
There are 16 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Amir Saboor Bagherzadeh This is me

Publication Date December 1, 2017
Published in Issue Year 2017 Volume: 7 Issue: 2

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