Skip to main content

2014 | OriginalPaper | Buchkapitel

A New Wavelet-Based Hybrid Method for Fisher Type Equation

verfasst von : R. Rajaram, G. Hariharan

Erschienen in: Fractals, Wavelets, and their Applications

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Al-Khaled, K.: Numerical study of Fisher’s reaction-diffusion equation by the sinc-collocation method. J. Comput. Appl. Math. 13, 245–255 (2001)MathSciNetCrossRef Al-Khaled, K.: Numerical study of Fisher’s reaction-diffusion equation by the sinc-collocation method. J. Comput. Appl. Math. 13, 245–255 (2001)MathSciNetCrossRef
2.
Zurück zum Zitat Carey, G.F., Shen, Y.: Least-squares finite element approximationn of Fisher’s reaction-diffusion equation. Numer. Meth. Part. Differ. Equat. 175–186 (1995) Carey, G.F., Shen, Y.: Least-squares finite element approximationn of Fisher’s reaction-diffusion equation. Numer. Meth. Part. Differ. Equat. 175–186 (1995)
3.
Zurück zum Zitat Hariharan, G.: The homotopy analysis method applied to the Kolmogorov-Petrovskii-Piskunov (KPP) and fractional KPP equations. J. Math. Chem. 51, 992–1000 (2013)MathSciNetCrossRefMATH Hariharan, G.: The homotopy analysis method applied to the Kolmogorov-Petrovskii-Piskunov (KPP) and fractional KPP equations. J. Math. Chem. 51, 992–1000 (2013)MathSciNetCrossRefMATH
4.
Zurück zum Zitat Hariharan, G., Kannan, K., Sharma, K.: Haar wavelet in estimating the depth profile of soil temperature. Appl. Math. Comput. 210, 119–225 (2009a)MathSciNetCrossRefMATH Hariharan, G., Kannan, K., Sharma, K.: Haar wavelet in estimating the depth profile of soil temperature. Appl. Math. Comput. 210, 119–225 (2009a)MathSciNetCrossRefMATH
5.
6.
Zurück zum Zitat Hariharan, G., Kannan, K.: Haar wavelet method for solving nonlinear parabolic equations. J. Math. Chem. 48, 1044–1061 (2010a)MathSciNetCrossRefMATH Hariharan, G., Kannan, K.: Haar wavelet method for solving nonlinear parabolic equations. J. Math. Chem. 48, 1044–1061 (2010a)MathSciNetCrossRefMATH
7.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
8.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
10.
Zurück zum Zitat 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) 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)
11.
Zurück zum Zitat Jafari, H., Soleymanivaraki, M., Firoozjaee, M.A.: Legendre wavelets for solving fractional differential equations. J. Appl. Math. 4(27), 65–70 (2011) Jafari, H., Soleymanivaraki, M., Firoozjaee, M.A.: Legendre wavelets for solving fractional differential equations. J. Appl. Math. 4(27), 65–70 (2011)
12.
Zurück zum Zitat 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 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
13.
Zurück zum Zitat Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. CRC Press/Chapman and Hall, Boca Raton (2004) Liao, S.J.: Beyond Perturbation: Introduction to Homotopy Analysis Method. CRC Press/Chapman and Hall, Boca Raton (2004)
14.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
15.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
16.
Zurück zum Zitat 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 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
17.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
18.
Zurück zum Zitat 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 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
19.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
20.
21.
Zurück zum Zitat Parsian, H.: Two dimension Legendre wavelets and operational matrices of integration. Acta Math. Academiae Paedagogicae Nyireghaziens 21, 101–106 (2005)MathSciNetMATH Parsian, H.: Two dimension Legendre wavelets and operational matrices of integration. Acta Math. Academiae Paedagogicae Nyireghaziens 21, 101–106 (2005)MathSciNetMATH
22.
Zurück zum Zitat Razzaghi, M., Yousefi, S.: The Legendre wavelets direct method for variational problems. Math. Comput. Simulat. 53, 185–192 (2000)MathSciNetCrossRef Razzaghi, M., Yousefi, S.: The Legendre wavelets direct method for variational problems. Math. Comput. Simulat. 53, 185–192 (2000)MathSciNetCrossRef
23.
24.
Zurück zum Zitat Wazwaz, A.M., Gorguis, A.: An analytical study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 154, 609–620 (2004)MathSciNetCrossRefMATH Wazwaz, A.M., Gorguis, A.: An analytical study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput. 154, 609–620 (2004)MathSciNetCrossRefMATH
25.
Zurück zum Zitat Yang, Y.: Solving a nonlinear multi-order fractional differential equation using legendre psedu-spectral method. Appl. Math. 4, 113–118 (2013)CrossRef Yang, Y.: Solving a nonlinear multi-order fractional differential equation using legendre psedu-spectral method. Appl. Math. 4, 113–118 (2013)CrossRef
26.
Zurück zum Zitat 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) 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)
27.
Zurück zum Zitat 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) 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)
28.
Zurück zum Zitat Yousefi, S.A.: Legendre wavelets method for solving differential equations of Lane-Emden type. App. Math. Comput. 181, 1417–1442 (2006)MathSciNetCrossRefMATH Yousefi, S.A.: Legendre wavelets method for solving differential equations of Lane-Emden type. App. Math. Comput. 181, 1417–1442 (2006)MathSciNetCrossRefMATH
29.
Zurück zum Zitat Zhou, X.W.: Exp-function method for solving Fisher’s equation. J. Phys. Conf. Ser. 96 (2008) Zhou, X.W.: Exp-function method for solving Fisher’s equation. J. Phys. Conf. Ser. 96 (2008)
Metadaten
Titel
A New Wavelet-Based Hybrid Method for Fisher Type Equation
verfasst von
R. Rajaram
G. Hariharan
Copyright-Jahr
2014
DOI
https://doi.org/10.1007/978-3-319-08105-2_35