Advertisement

Improved wavelets based technique for nonlinear partial differential equations

  • Muhammad Asad Iqbal
  • Muhammad Shakeel
  • Ayyaz Ali
  • Syed Tauseef Mohyud-Din
Article
  • 113 Downloads

Abstract

One of the most challenging task now a days for engineers and scientists is finding solutions of nonlinear Partial Differential Equations (PDEs) which frequently arise in many engineering and physical phenomena’s. Encouraged by the ongoing research, a new technique is proposed in this article for obtaining more accurate results of nonlinear PDEs. Shifted Legendre wavelets and Picard’s Iteration Technique are used in the proposed technique. To test the significance of the proposed technique, nonlinear Gardner equation is considered and solved. The proposed technique provides very accurate results over a wider interval because of the use of the shifted polynomials. The results obtained are also compared with the results of Variational Iteration Method and the supremacy of the proposed method is established.

Keywords

Shifted Legendre wavelets Picard’s iteration Nonlinear problems Exact solutions 

Notes

Acknowledgements

The authors are highly grateful to the unknown referees for their valuable comments.

Compliance with ethical standards

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. Ali, A., Iqbal, M.A., Mohyud-Din, S.T.: Chebyshev wavelets method for boundary value problems. Sci. Res. Essays 8(46), 2235–2241 (2013)Google Scholar
  2. Babolian, E., Zadeh, F.F.: Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Appl. Math. Comput. 188, 417–426 (2007)MathSciNetMATHGoogle Scholar
  3. Bekir, A., Aksoy, E., Guner, O.: Optical soliton solutions of the long-short-wave interaction system. J. Nonlinear Optic. Phys. Mater. 22(2), 1350015 (2013)ADSCrossRefGoogle Scholar
  4. Bekir, A., Guner, O., Bilgil, H.: Optical soliton solutions for the variable coefficient modified Kawahara equation. Opt. Int J. Light Electron Opt. 126(20), 2518–2522 (2015). doi: 10.1016/j.ijleo.2015.06.051 CrossRefGoogle Scholar
  5. Cattani, C., Kudreyko, A.: Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind. Appl. Math. Comput. 215, 4164–4171 (2010)MathSciNetMATHGoogle Scholar
  6. Guner, O., Bekir, A.: Optical solitons for nonlinear coupled Klein-Gordon equations. Optoelectron. Adv. Mater. Rap. Commun. 9(3–4), 332–343 (2015)Google Scholar
  7. Guner, O., Bekir, A.: Bright and dark soliton solutions for some nonlinear fractional differential equations. Chin. Phys. B 25(3), 30203 (2016). doi: 10.1088/1674-1056/25/3/030203 CrossRefGoogle Scholar
  8. Islam, S., Zuhra, S., Idrees, M., Ullah, H., Shah, I.A., Zaman, A.: Application of Optimal Homotopy Asymptotic Method to Benjamin-Bona-Mahony and Sawada-Kotera Equations. World Appl. Sci. J. 31(11), 1945–1951 (2014)Google Scholar
  9. Mohammadi, F., Hosseini, M.M., Mohyud-din, S.T.: Legendre wavelet Galerkin method for solving ordinary differential equations with non analytical solution. Int. J. Syst. Sci. 42(4), 579–585 (2011)CrossRefMATHGoogle Scholar
  10. Noor, M.A., Mohyud-Din, S.T.: Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials. Int. J. Nonlinear Sci. Numer Simul. 9(2), 141–157 (2008)CrossRefMATHGoogle Scholar
  11. Saeed, U., Rehman, M., Iqbal, M.A.: Haar Wavelet-Picard technique for fractional order nonlinear initial and boundary value problems. Sci. Res. Essays 9(12), 571–580 (2014)CrossRefGoogle Scholar
  12. Wazwaz, A.M.: Approximate solutions to boundary value problems of higher order by the modified decomposition method. Comput. Math Appl. 40, 679–691 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer, Nonlinear physical science (2009)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Muhammad Asad Iqbal
    • 1
  • Muhammad Shakeel
    • 1
  • Ayyaz Ali
    • 2
  • Syed Tauseef Mohyud-Din
    • 3
  1. 1.Department of MathematicsMohiUd Din Islamic UniversityNerian SharifPakistan
  2. 2.Department of MathematicsInstitute of Southern PunjabMultanPakistan
  3. 3.Department of Mathematics, Faculty of SciencesHITEC UniversityTaxilaPakistan

Personalised recommendations