Advertisement

Journal of Mathematical Chemistry

, Volume 48, Issue 4, pp 1044–1061 | Cite as

Haar wavelet method for solving some nonlinear Parabolic equations

  • G. Hariharan
  • K. Kannan
Original Paper

Abstract

Wavelet transform or wavelet analysis is a recently developed mathematical tool in applied mathematics. In this paper, we develop an accurate and efficient Haar transform or Haar wavelet method for some of the well-known nonlinear parabolic partial differential equations. The equations include the Nowell-whitehead equation, Cahn-Allen equation, FitzHugh-Nagumo equation, Fisher’s equation, Burger’s equation and the Burgers-Fisher equation. The proposed scheme can be used to a wide class of nonlinear equations. The power of this manageable method is confirmed. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.

Keywords

Nonlinear parabolic equations Haar wavelet method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wadati M.: Introduction to solitons. Pramana: J. Phys. 57(5–6), 841–847 (2001)CrossRefGoogle Scholar
  2. 2.
    Wadati M.: The modified Korteweg-de veries equation. J. Phys. Soc. Jpn. 34, 1289–1296 (1973)CrossRefGoogle Scholar
  3. 3.
    Ablowitz M., Segur H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)Google Scholar
  4. 4.
    Hirota R.: Direct Methods in Soliton Theory. Springer, Berlin (1980)Google Scholar
  5. 5.
    Malfliet W., Hereman W.: The tanh method I: exact solutions of nonlinear evolution and wave equations. Physica Scr. 54, 563 (1996)CrossRefGoogle Scholar
  6. 6.
    Wazwaz A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131 (2007)CrossRefGoogle Scholar
  7. 7.
    Yusufoğlu E., Bekir A.: Exact solutions of coupled nonlinear evolution equations. Chaos, Solitons Fractals 37(3), 842 (2008)CrossRefGoogle Scholar
  8. 8.
    Wazwaz A.M.: A sine–cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499 (2004)CrossRefGoogle Scholar
  9. 9.
    Fan E., Zhang H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403 (1998)CrossRefGoogle Scholar
  10. 10.
    Zhang S.: Application of exp-function method to a KdV equation with variable coefficients. Phys. Lett. A 365, 448 (2007)CrossRefGoogle Scholar
  11. 11.
    Wazwaz A.M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 154(3), 713 (2004)CrossRefGoogle Scholar
  12. 12.
    Wazwaz A.M.: An analytical study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 154, 609–620 (2004)CrossRefGoogle Scholar
  13. 13.
    Wazwaz A.M.: The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. J. Appl. Math. Comput. 188, 1467 (2007)CrossRefGoogle Scholar
  14. 14.
    Lax P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 62, 467–490 (1968)CrossRefGoogle Scholar
  15. 15.
    Chen C.F., Hsiao C.H.: Haar wavelet method for solving lumped and distributed-parameter systems. IEEE Proc. Pt. D 144(1), 87–94 (1997)Google Scholar
  16. 16.
    Lepik U.: Numerical solution of evolution equations by the Haar wavelet method. J. Appl. Math. Comput. 185, 695–704 (2007)CrossRefGoogle Scholar
  17. 17.
    Lepik U.: Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul. 68, 127–143 (2005)CrossRefGoogle Scholar
  18. 18.
    Lepik U.: Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci. Phys. Math. 56(1), 28–46 (2007)Google Scholar
  19. 19.
    Hariharan G., Kannan K., Sharma K.R.: Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 211, 284–292 (2009)CrossRefGoogle Scholar
  20. 20.
    Rosu H.C., Cornejo-Pe’rez O.: Super symmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71, 1–13 (2005)CrossRefGoogle Scholar
  21. 21.
    Brazhnik P., Tyson J.: On traveling wave solutions of Fisher’s equation in two spatial dimensions. SIAM J. Appl. Math. 60(2), 371–391 (1999)Google Scholar
  22. 22.
    Monsour M.B.A.: Travelling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation. Appl. Math. Model. 32, 240–247 (2008)CrossRefGoogle Scholar
  23. 23.
    Olmos D., Shizgal B.D.: A pseudospectral method of solution of Fisher’s equation. J. Comput. Appl. Math. 193, 219–242 (2006)CrossRefGoogle Scholar
  24. 24.
    Wazwaz A.M.: Analytical study on Burgers, Fisher, Huxley equations and combined forms of these equations. J. Appl. Math. Comput. 195, 754–761 (2008)CrossRefGoogle Scholar
  25. 25.
    Rajendran L., Senthamarai R.: Traveling-wave solution of non-linear coupled reaction-diffusion equation arising in mathematical chemistry. J. Math. Chem. 46, 550–561 (2009)CrossRefGoogle Scholar
  26. 26.
    Kolmogorov A., Petrovsky I., Piskunov N.: Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Moscow Bull. Univ. Math. 1, 1–25 (1937)Google Scholar
  27. 27.
    Cattani C.: Haar wavelet spline. J. Interdisciplinary Math. 4, 35–47 (2001)Google Scholar
  28. 28.
    Hariharan G., Kannan K., Sharma K.R.: Haar wavelet in estimating depth profile of soil temperature. Appl. Math. Comput. 210, 119–125 (2009)CrossRefGoogle Scholar
  29. 29.
    Hsiao C.H., Wang W.J.: Haar wavelet approach to nonlinear stiff systems. Math. Comput. Simul. 57, 347–353 (2001)CrossRefGoogle Scholar
  30. 30.
    Karpovsky M.G.: Finite Orthogonal Series in the Design of Digital Devices. Wiley, New York (1976)Google Scholar
  31. 31.
    L.A. Zalmonzon, Fourier, Walsh, and Haar Transforms and Their Applications in Control, Communication and other Fields (Nauka, Moscow, 1989) (in Russian)Google Scholar
  32. 32.
    Haar A.: Zur theorie der orthogonalen Funktionsysteme. Math. Annal 69, 331–371 (1910)CrossRefGoogle Scholar
  33. 33.
    Fitzhugh R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsSASTRA UniversityTirumalaisamudramIndia

Personalised recommendations