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Fractals, Wavelets, and their Applications pp 501–508Cite as

A New Wavelet-Based Hybrid Method for Fisher Type Equation

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 92)

Abstract

In this paper, we have introduced a new wavelet-based hybrid method for solving the Fisher’s type equations. To the best of our knowledge, until now there is no rigorous wavelet solution has been addressed for the Fisher’s equations. With the help of wavelets operational matrices, the Fisher’s equations are converted into a system of algebraic equations. Some numerical examples are presented to demonstrate the validity and applicability of the method.

Keywords

  • Fisher’s equation
  • Operational matrices
  • Legendre wavelets
  • Homotopy analysis method
  • Haar wavelets

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References

  1. Al-Khaled, K.: Numerical study of Fisher’s reaction-diffusion equation by the sinc-collocation method. J. Comput. Appl. Math. 13, 245–255 (2001)

    MathSciNet  CrossRef  Google Scholar 

  2. Carey, G.F., Shen, Y.: Least-squares finite element approximationn of Fisher’s reaction-diffusion equation. Numer. Meth. Part. Differ. Equat. 175–186 (1995)

    Google Scholar 

  3. Hariharan, G.: The homotopy analysis method applied to the Kolmogorov-Petrovskii-Piskunov (KPP) and fractional KPP equations. J. Math. Chem. 51, 992–1000 (2013)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Hariharan, G., Kannan, K., Sharma, K.: Haar wavelet in estimating the depth profile of soil temperature. Appl. Math. Comput. 210, 119–225 (2009a)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Hariharan, G., Kannan, K.: Haar wavelet method for solving nonlinear parabolic equations. J. Math. Chem. 48, 1044–1061 (2010a)

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Hariharan, G., Kannan, K.: A comparative study of a Haar wavelet method and a restrictive Taylor’s series method for solving convection-diffusion equations. Int. J. Comput. Meth. Eng. Sci. Mech. 11(4), 173–184 (2010b)

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Hariharan, G., Rajaraman, R.: A new coupled wavele-based method applied to the nonlinear reaction-diffusion equation arising in mathematical chemistry. J. Math. Chem. 51, 2386–2400 (2013)

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 30, 700–708 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M., Mohammadi, F.: Wavelet collocation method for solving multiorder fractional differential equations. J. Appl. Math. 2012, Article ID 163821 (2012)

    Google Scholar 

  11. Jafari, H., Soleymanivaraki, M., Firoozjaee, M.A.: Legendre wavelets for solving fractional differential equations. J. Appl. Math. 4(27), 65–70 (2011)

    Google Scholar 

  12. Khan, N.A., Khan, N.-U., Ara, A., Jamil, M.: Approximate analytical solution of fractional reaction-diffussion equations. J. Kind Saud Univ. Sci. 24, 111–118 (2012)

    CrossRef  Google Scholar 

  13. Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. CRC Press/Chapman and Hall, Boca Raton (2004)

    Google Scholar 

  14. Maleknejad, K., Sohrabi, S.: Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets. Appl. Math. Comput. 186, 836–843 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Matinfar, M., Ghanbari, M.: Solving the Fisher’s equations by means of variational iteration method. Int. J. Contemp. Math. Sci. 4(7), 343–348 (2009a)

    MathSciNet  MATH  Google Scholar 

  16. Matinfar, M., Ghanbari, M.: Homotopy perturbation method for the Fisher’s equation and its generalized. Int. J. Nonlinear Sci. 8(4), 448–455 (2009b)

    MathSciNet  Google Scholar 

  17. Matinfar, M., Bahar, S.R., Ghasemi, M.: Solving the generalized Fisher’s equation by the differential transform method. J. Appl. Math. Inform. 30(3–4), 555–560 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Mittal, R.C., Jiwari, R.: Numerical study of Fisher’s equation by using differential quadrature method. Int. J. Inform. Syst. Sci. 5(1), 143–160 (2008)

    MathSciNet  Google Scholar 

  19. Mohammadi, F., Hosseini, M.M.: A new Legendre wavelet operational matrix of derivative and its applications in solving singular ordinary differential equations. J. Franklin Inst. 348, 1787–1796 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Parsian, H.: Two dimension Legendre wavelets and operational matrices of integration. Acta Math. Academiae Paedagogicae Nyireghaziens 21, 101–106 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Razzaghi, M., Yousefi, S.: The Legendre wavelets direct method for variational problems. Math. Comput. Simulat. 53, 185–192 (2000)

    MathSciNet  CrossRef  Google Scholar 

  23. Razzaghi, M., Yousefi, S.: The Legendre wavelets operational matrix of integration. Int. J. Syst. Sci. 32, 495–502 (2001)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

  25. Yang, Y.: Solving a nonlinear multi-order fractional differential equation using legendre psedu-spectral method. Appl. Math. 4, 113–118 (2013)

    CrossRef  Google Scholar 

  26. Yildirem, K., Ibis, B., Bayram, M.: New solutions of the nonlinear Fisher type equations by the reduced differential transform. Nonlinear Sci. Lett. A. 3(1), 29–36 (2012)

    Google Scholar 

  27. Yin, F., Song, J., Lu, F.: A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein-Gordan equations. Math. Meth. Appl. Sci. 2013 (Press)

    Google Scholar 

  28. Yousefi, S.A.: Legendre wavelets method for solving differential equations of Lane-Emden type. App. Math. Comput. 181, 1417–1442 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  29. Zhou, X.W.: Exp-function method for solving Fisher’s equation. J. Phys. Conf. Ser. 96 (2008)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, School of Humanities and Sciences SASTRA University, Thanjavur, Tamilnadu, 613401, India

    R. Rajaram & G. Hariharan

Authors
  1. R. Rajaram
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  2. G. Hariharan
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Corresponding author

Correspondence to R. Rajaram .

Editor information

Editors and Affiliations

  1. Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Mecklenburg-Vorpomm., Germany

    Christoph Bandt

  2. Mathematical Sciences Institute, Australian National University, Canberra, Australia

    Michael Barnsley

  3. Math Department, Boston University, Boston, Massachusetts, USA

    Robert Devaney

  4. Mathematical Institute, University of St Andrews, St Andrews, Fife, United Kingdom

    Kenneth J. Falconer

  5. Department of Mathematics and Statistics, University of Hyderabad, Hyderabad, India

    V. Kannan

  6. Department of Basic Sciences & Humanities, Rajagiri School of Engineering & Technology, Kerala, India

    Vinod Kumar P.B.

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Rajaram, R., Hariharan, G. (2014). A New Wavelet-Based Hybrid Method for Fisher Type Equation. In: Bandt, C., Barnsley, M., Devaney, R., Falconer, K., Kannan, V., Kumar P.B., V. (eds) Fractals, Wavelets, and their Applications. Springer Proceedings in Mathematics & Statistics, vol 92. Springer, Cham. https://doi.org/10.1007/978-3-319-08105-2_35

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