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Haar wavelet method for solving some nonlinear Parabolic equations

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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.

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References

  1. Wadati M.: Introduction to solitons. Pramana: J. Phys. 57(5–6), 841–847 (2001)

    Article  Google Scholar 

  2. Wadati M.: The modified Korteweg-de veries equation. J. Phys. Soc. Jpn. 34, 1289–1296 (1973)

    Article  Google Scholar 

  3. Ablowitz M., Segur H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)

    Google Scholar 

  4. Hirota R.: Direct Methods in Soliton Theory. Springer, Berlin (1980)

    Google Scholar 

  5. Malfliet W., Hereman W.: The tanh method I: exact solutions of nonlinear evolution and wave equations. Physica Scr. 54, 563 (1996)

    Article  CAS  Google Scholar 

  6. Wazwaz A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131 (2007)

    Article  Google Scholar 

  7. Yusufoğlu E., Bekir A.: Exact solutions of coupled nonlinear evolution equations. Chaos, Solitons Fractals 37(3), 842 (2008)

    Article  Google Scholar 

  8. Wazwaz A.M.: A sine–cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499 (2004)

    Article  Google Scholar 

  9. Fan E., Zhang H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403 (1998)

    Article  CAS  Google Scholar 

  10. Zhang S.: Application of exp-function method to a KdV equation with variable coefficients. Phys. Lett. A 365, 448 (2007)

    Article  CAS  Google Scholar 

  11. Wazwaz A.M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 154(3), 713 (2004)

    Article  Google Scholar 

  12. Wazwaz A.M.: An analytical study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 154, 609–620 (2004)

    Article  Google Scholar 

  13. Wazwaz A.M.: The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. J. Appl. Math. Comput. 188, 1467 (2007)

    Article  Google Scholar 

  14. Lax P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 62, 467–490 (1968)

    Article  Google Scholar 

  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. Lepik U.: Numerical solution of evolution equations by the Haar wavelet method. J. Appl. Math. Comput. 185, 695–704 (2007)

    Article  Google Scholar 

  17. Lepik U.: Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul. 68, 127–143 (2005)

    Article  Google Scholar 

  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. Hariharan G., Kannan K., Sharma K.R.: Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 211, 284–292 (2009)

    Article  Google Scholar 

  20. Rosu H.C., Cornejo-Pe’rez O.: Super symmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71, 1–13 (2005)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  23. Olmos D., Shizgal B.D.: A pseudospectral method of solution of Fisher’s equation. J. Comput. Appl. Math. 193, 219–242 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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. Cattani C.: Haar wavelet spline. J. Interdisciplinary Math. 4, 35–47 (2001)

    Google Scholar 

  28. Hariharan G., Kannan K., Sharma K.R.: Haar wavelet in estimating depth profile of soil temperature. Appl. Math. Comput. 210, 119–125 (2009)

    Article  Google Scholar 

  29. Hsiao C.H., Wang W.J.: Haar wavelet approach to nonlinear stiff systems. Math. Comput. Simul. 57, 347–353 (2001)

    Article  Google Scholar 

  30. Karpovsky M.G.: Finite Orthogonal Series in the Design of Digital Devices. Wiley, New York (1976)

    Google Scholar 

  31. L.A. Zalmonzon, Fourier, Walsh, and Haar Transforms and Their Applications in Control, Communication and other Fields (Nauka, Moscow, 1989) (in Russian)

  32. Haar A.: Zur theorie der orthogonalen Funktionsysteme. Math. Annal 69, 331–371 (1910)

    Article  Google Scholar 

  33. Fitzhugh R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Article  CAS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, SASTRA University, Tirumalaisamudram, Tamilnadu, 613402, India

    G. Hariharan & K. Kannan

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  1. G. Hariharan
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  2. K. Kannan
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Correspondence to G. Hariharan.

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Hariharan, G., Kannan, K. Haar wavelet method for solving some nonlinear Parabolic equations. J Math Chem 48, 1044–1061 (2010). https://doi.org/10.1007/s10910-010-9724-0

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  • DOI: https://doi.org/10.1007/s10910-010-9724-0

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