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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Wadati M.: Introduction to solitons. Pramana: J. Phys. 57(5–6), 841–847 (2001)
Wadati M.: The modified Korteweg-de veries equation. J. Phys. Soc. Jpn. 34, 1289–1296 (1973)
Ablowitz M., Segur H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Hirota R.: Direct Methods in Soliton Theory. Springer, Berlin (1980)
Malfliet W., Hereman W.: The tanh method I: exact solutions of nonlinear evolution and wave equations. Physica Scr. 54, 563 (1996)
Wazwaz A.M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Appl. Math. Comput. 187, 1131 (2007)
Yusufoğlu E., Bekir A.: Exact solutions of coupled nonlinear evolution equations. Chaos, Solitons Fractals 37(3), 842 (2008)
Wazwaz A.M.: A sine–cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499 (2004)
Fan E., Zhang H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403 (1998)
Zhang S.: Application of exp-function method to a KdV equation with variable coefficients. Phys. Lett. A 365, 448 (2007)
Wazwaz A.M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 154(3), 713 (2004)
Wazwaz A.M.: An analytical study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 154, 609–620 (2004)
Wazwaz A.M.: The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. J. Appl. Math. Comput. 188, 1467 (2007)
Lax P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 62, 467–490 (1968)
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)
Lepik U.: Numerical solution of evolution equations by the Haar wavelet method. J. Appl. Math. Comput. 185, 695–704 (2007)
Lepik U.: Numerical solution of differential equations using Haar wavelets. Math. Comput. Simul. 68, 127–143 (2005)
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)
Hariharan G., Kannan K., Sharma K.R.: Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 211, 284–292 (2009)
Rosu H.C., Cornejo-Pe’rez O.: Super symmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71, 1–13 (2005)
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)
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)
Olmos D., Shizgal B.D.: A pseudospectral method of solution of Fisher’s equation. J. Comput. Appl. Math. 193, 219–242 (2006)
Wazwaz A.M.: Analytical study on Burgers, Fisher, Huxley equations and combined forms of these equations. J. Appl. Math. Comput. 195, 754–761 (2008)
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)
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)
Cattani C.: Haar wavelet spline. J. Interdisciplinary Math. 4, 35–47 (2001)
Hariharan G., Kannan K., Sharma K.R.: Haar wavelet in estimating depth profile of soil temperature. Appl. Math. Comput. 210, 119–125 (2009)
Hsiao C.H., Wang W.J.: Haar wavelet approach to nonlinear stiff systems. Math. Comput. Simul. 57, 347–353 (2001)
Karpovsky M.G.: Finite Orthogonal Series in the Design of Digital Devices. Wiley, New York (1976)
L.A. Zalmonzon, Fourier, Walsh, and Haar Transforms and Their Applications in Control, Communication and other Fields (Nauka, Moscow, 1989) (in Russian)
Haar A.: Zur theorie der orthogonalen Funktionsysteme. Math. Annal 69, 331–371 (1910)
Fitzhugh R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-010-9724-0