Cardinal B-Spline Wavelet Based Numerical Method for the Solution of Generalized Burgers–Huxley Equation

  • S. C. Shiralashetti
  • S. Kumbinarasaiah
Original Paper


In this paper, Cardinal B-spline wavelet numerical method is developed for the solution of Generalized Burgers–Huxley(GBH) Equation. It is a new approach for the accurate numerical solution of the GBH equation with the initial and boundary conditions using Cardinal B-spline wavelets. This method is based on the truncated Cardinal B-spline wavelet expansions is used to convert the initial and boundary value problems into system of algebraic equations which can be efficiently solved by suitable solvers. Numerical results of the GBH equation with initial and boundary conditions shows the efficiency and accuracy of the present method.


Cardinal B-spline wavelet Linear and nonlinear Burgers–Huxley equations Collocation method 

Mathematics Subject Classification

35-XX 42C40 65L60 



It is a pleasure to thank the University Grants Commission (UGC), Govt. of India for the financial support under UGC-SAP DRS-III for 2016-2021:F.510/3/DRS-III/2016(SAP-I) Dated: 29th Feb. 2016.


  1. 1.
    Lepik, U.: Solving pdes with the aid of two-dimensional haar wavelets. Comput. Math. Appl. 61, 1873–1879 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burden, R.L., Faires, J.D.: Numerical Analysis. PWS Publishing Company, Boston (1993)zbMATHGoogle Scholar
  3. 3.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)zbMATHGoogle Scholar
  4. 4.
    Machado, J.A.T., Babaei, A., Moghaddam, B.P.: Highly accurate scheme for the Cauchy Problem of the generalized Burgers–Huxley equation. Acta Polytech. Hungarica 13(6), 183–195 (2016)Google Scholar
  5. 5.
    Satsuma, J., Ablowitz, M., Fuchssteiner, B., Kruskal, M.: Topics in Soliton Theory and Exactly Solvable Nonlinear Equations. World Scientific, The Singapore City (1987)Google Scholar
  6. 6.
    Rosu, H.C., Perez, O.C.: Super symmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71, 1–13 (2005)CrossRefGoogle Scholar
  7. 7.
    Wang, X.Y.: Nerve propagation and wall in liquid crystals. Phys. Lett. A 112(8), 402–406 (1985)CrossRefGoogle Scholar
  8. 8.
    Pap, E.: Pseudo-analysis approach to nonlinear partial differential equations. Acta Polyt. Hungarica 5(1), 31–45 (2008)MathSciNetGoogle Scholar
  9. 9.
    Wang, G., Liu, X., Zhang, Ying-yuan: New explicit solutions of the generalized Burger’s–Huxley equation. Vietnam J. Math. 41, 161–166 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Inan, B.: Finite difference methods for the generalized Huxley and Burgers–Huxley equations. Kuwait J. Sci. 44(3), 20–27 (2017)Google Scholar
  11. 11.
    Inan, B., Bahadir, A.R.: An explicit exponential finite difference method for the Burger’s equation. Eur. Int. J. Sci. Technol. 2(10), 61–72 (2013)Google Scholar
  12. 12.
    Inan, B., Bahadir, A.R.: Numerical solution of the one-dimensional Burgers equation: implicit and fully implicit exponential finite difference methods. Pramana J. Phys. 81(4), 547–556 (2013)CrossRefGoogle Scholar
  13. 13.
    Khattak, A.J.: A computational meshless method for the generalized Burgers–Huxley equation. Appl. Math. Model. 33, 3718–3729 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ismail, H.N.A., Raslan, K., Abd Rabboh, A.A.: Adomian decomposition method for Burgers–Huxley and Burgers–Fisher equations. Appl. Math. Comput. 159, 291–301 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Abdulghafor, M., Rozbayani, A., Al-Amr, M.O.: Discrete adomian decomposition method for solving Burgers–Huxley equation. Int. J. Contemp. Math. Sci. 8(13), 623–631 (2013)MathSciNetGoogle Scholar
  16. 16.
    Mittala, R.C., Tripathi, A.: Numerical solutions of generalized Burger’s–Fisher and generalized Burger’s–Huxley equations using collocation of cubic B-splines. Int. J. Comput. Math. 92(5), 1053–1077 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    El-Kady, M., El-Sayed, S.M., Fathy, H.E.: Development of Galerkin method for solving the generalized Burger’s–Huxley equation. Math. Probl. Eng. 2013, 1–9 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Celik, I.: Chebyshev wavelet collocation method for solving generalized Burgers–Huxley equation. Math. Methods Appl. Sci. 39(3), 341–613 (2015)MathSciNetGoogle Scholar
  19. 19.
    Wang, Q., Guo, X., Lu, L.: On the new solutions of the generalized Burger’s–Huxley equation. Math. Methods Appl. Sci. 40, 2034–2041 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Macias-Diaz, J.E., Gonzalez, A.E.: Some remarks on an exact and dynamically consistent scheme for the Burger’s-Huxley equation in higher dimensions. Adv. Differ. Equ. (2015). MathSciNetGoogle Scholar
  21. 21.
    Singh, B.K., Arora, G., Singh, M.K.: A numerical scheme for the generalized Burgers–Huxley equation. J. Egypt. Math. Soc. 24(4), 629–637 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rogers, C., Shadwich, W.F.: Bcklund Transformations and Their Application. Academic Press, New York (1982)Google Scholar
  23. 23.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  24. 24.
    Adomian, G.: A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135, 501–544 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Abbasbandy, S., Shirzadi, A.: The first integral method for modified Benjamin–Bona–Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 15, 1759–1764 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hirota, R.: Exact solution of the Korteweg-de Vries Equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)CrossRefzbMATHGoogle Scholar
  27. 27.
    Liao, S.J.: An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169, 854–865 (2005)MathSciNetzbMATHGoogle Scholar
  28. 28.
    He, J.H.: Asymptotology by homotopy perturbation method. Appl. Math. Comput. 156, 591–596 (2004)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ganji, Z.Z., Ganji, D.D., Bararnia, H.: Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method. Appl. Math. Model. 33, 1836–1841 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shiralashetti, S.C., Angadi, L.M., Deshi, A.B., Kantli, M.H.: Haar wavelet method for the numerical solution of Benjamin–Bona–Mahony equations. J. Inf. Comput. Sci. 11, 136–145 (2016)zbMATHGoogle Scholar
  31. 31.
    Lahmar, N.A., Belhamitib, O., Bahric, S.M.: A new Legendre wavelets decomposition method for solving PDEs. Malaya J. Mat. 1(1), 72–81 (2014)Google Scholar
  32. 32.
    Heydari, M.H., Hooshmandasl, M.R., Maalek Ghaini, F.M.: A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type. Appl. Math. Model. 38, 1597–1606 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Bachman, G., Narici, L., Beckenstein, E.: Fourier and Wavelet Analysis. Springer, Berlin (1991)zbMATHGoogle Scholar
  34. 34.
    Chui, C.K.: Introduction to Wavelets (Volume 1). Academic press, New York (1992)zbMATHGoogle Scholar
  35. 35.
    Neves, A.J., Pereira, T.P.: Convolutions power of a characteristic function (2012). arXiv:1204.4860v1 [math.NA]
  36. 36.
    Chui, C.K.: An analysis of cordinal spline wavelets. J. Approx. Theory 72, 54–68 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKarnatak UniversityDharwadIndia

Personalised recommendations