A numerical algorithm based on a new kind of tension B-spline function for solving Burgers-Huxley equation

  • N. AliniaEmail author
  • M. Zarebnia
Original Paper


In this paper, a numerical algorithm based on a new kind of tension B-spline, named hyperbolic-trigonometric tension B-spline method, is applied for solving Burgers-Huxley equation. This method is generated over the space span {sin(tt),cos(tt),sinh(tt),cosh(tt),1,t,...,tn-?5},n =?5, where t is the tension parameter. Properties of it are the same in most of the properties of the usual polynomial B-splines and benefit from some other advantages, as well. Therefore, in this paper, we apply three methods consisting of trigonometric method, hyperbolic tension B-spline method, and our new hyperbolic-trigonometric tension B-spline method, to solve Burgers-Huxley equation. The convergence analysis is discussed. Then, we use some numerical examples to illustrate the accuracy and implementation of the proposed algorithm.


Burgers-Huxley equation Tension B-spline Hyperbolic-trigonometric Collocation method Numerical algorithm 


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  1. 1.
    Satsuma, J.: Topics in soliton theory and exactly solvable nonlinear equations. In: Ablowitz, M., Fuchssteiner, B., Kruskal, M. (eds.) , pp 255–-262. World Scientific, Singapore (1987)Google Scholar
  2. 2.
    Wang, X.Y., Zhu, Z.S., Lu, Y.K.: Solitary wave solutions of the generalized BurgersHuxley equation. J. Phys. A: Math. Gen. 23, 271–274 (1990)CrossRefGoogle Scholar
  3. 3.
    Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)CrossRefGoogle Scholar
  4. 4.
    Fitzhugh, R.: Mathematical models of excitation and propagation in nerve. In: Schwan, H. P. (ed.) Biological Engineering, pp 1–85. McGraw-Hill, New york (1969)Google Scholar
  5. 5.
    Ismail, H.N.A., Raslan, K., Rabboh, A.A.A.: Adomian decomposition method for BurgersHuxley and BurgersFisher equations. Appl. Math. Comput. 159, 291–301 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Javidi, M.: A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method. Appl. Math. Comput. 178, 338–344 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Deng, X.: Traveling wave solutions for the generalized BurgersHuxley equation. Appl. Math. Comput. 204, 733–737 (2008)MathSciNetGoogle Scholar
  8. 8.
    Dehghan, M., Saray, B.N., Lakestani, M.: Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized BurgersHuxley equation. Math. Comput. Model. 55, 11291142 (2012)zbMATHGoogle Scholar
  9. 9.
    Zhou, S., Cheng, X.: A linearly semi-implicit compact scheme for the BurgersHuxley equation. Int. J. Comput. Math. 88, 795804 (2011)Google Scholar
  10. 10.
    Gupta, V., Kadalbajoo, M.K.: A singular perturbation approach to solve BurgersHuxley equation via monotone finite difference scheme on layer adaptive mesh. Commun. Nonlinear Sci. Numer. Simul. 16, 18251844 (2011)Google Scholar
  11. 11.
    Mohanty, R.K., Dai, W., Liu, D.: Operator compact method of accuracy two in time and four in space for the solution of time dependent Burgers-Huxley equation. Numer. Algor. 70(3), 591605 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Molabahramia, A., Khani, F.: The homotopy analysis method to solve the Burgers-Huxley equation. Nonlinear Anal. Real World Appl.10(2), 589600 (2009)MathSciNetGoogle Scholar
  13. 13.
    Wazwaz, A.M.: Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations. Appl. Math. Comput. 195(2), 754761 (2008)MathSciNetGoogle Scholar
  14. 14.
    Batiha, B., Noorani, M.S.M., Hashim, I.: Application of variational iteration method to the generalized Burgers-Huxley equation. Chaos Solitons Fractals 36(3), 660663 (2008)CrossRefGoogle Scholar
  15. 15.
    Batiha, B., Noorani, M.S.M., Hashim, I.: Numerical simulation of the generalized Huxley equation by Hes variational iteration method. Appl. Math. Comput. 186(2), 13221325 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Efimova, O.Y., Kudryashov, N.A.: Exact solutions of the Burgers-Huxley equation. J. Appl. Math. Mech. 68(3), 413420 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zhang, J.W.: C-curves, an extension of cubic curves. Comput. Aided Geom. Design 13, 199–217 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, J.W.: Two different forms of C-B-splines. Comput. Aided Geom. Design 14, 31–41 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Koch, P.E., Lyche, T.: Construction of Exponential Tension B-splines of Arbitrary Order, pp 255–258. Academic Press, New York (1991)zbMATHGoogle Scholar
  20. 20.
    Lu, Y.G., Wang, G.Z., Yang, X.N.: Uniform hyperbolic polynomial B-spline curves. Comput. Aided Geom. Design 19, 379–393 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mainar, E., Peña, J.M.: A basis of C-Bezier splines with optimal properties. Comput. Aided Geom. Design 19, 161–175 (2002)CrossRefGoogle Scholar
  22. 22.
    Chen, Q.Y., Wang, G.Z.: A class of Bezier-like curves. Comput. Aided Geom. Design 20, 29–39 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang, G.Z., Chen, Q.Y., Zhou, M.H.: NUAT B-spline curves. Comput. Aided Geom. Design 21, 193–205 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jena, M.K., Shunmugaraj, P., Das, P.C.: A subdivision algorithm for trigonometric spline. Comput. Aided Geom. Design 19, 71–88 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jena, M.K., Shunmugaraj, P., Das, P.C.: A subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes. Comput. Aided Geom. Design 20, 61–77 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, Y.J., Wang, G.Z.: Two kinds of B-basis of the algebraic hyperbolic space. J. Zhejiang Univ. Sci. 6, 750–759 (2005)CrossRefGoogle Scholar
  27. 27.
    Xu, G., Wang, G.: AHT Bezier curves and NUAHT B-spline curves. J. Comput. Sci. Technol. 22, 597–607 (2007)CrossRefGoogle Scholar
  28. 28.
    Wang, G., Fang, M.: Unified and extended form of three types of splines. J. Comput. Appl. Math. 216, 498–508 (2008)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Jianzhong, W., Daren, H.: On quartic and quintic interpolation splines and their optimal error bounds. Appl. Numer. Math. 11, 1130–1141 (1982)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wasim, I., Abbas, M., Amin, M.: Hybrid B-Spline collocation method for solving the generalized Burgers-Fisher and Burgers-Huxley equations. Math. Probl. Eng. 2018, 1–18 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hammad, D.A., El-Azab, M.S.: 2N order compact finite difference scheme with collocation method for solving the generalized Burgers-Huxley and Burgers-Fisher equations. Appl. Math. Comput. 258, 296–311 (2015)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Inan, B., Bahadir, A.R.: Numerical solutions of the generalized Burgers-Huxley equation by implicit exponential finite difference method. J. Appl. Math. Stat. Inform. 11, 57–67 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Chen, J.: An effcient multiscale Runge-Kutta Galerkin method for generalized Burgers-Huxley equation. Appl. Math. Sci. 11(30), 1467–1479 (2017)Google Scholar
  34. 34.
    Mittala, R.C., Tripathia, A.: Numerical solutions of generalized BurgersFisher and generalized BurgersHuxley equations using collocation of cubic B-splines. Int. J. Comput. Math. 92(5), 1053–1077 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Bukhari, M., Arshad, M., Batool, S., Saqlain, S.M.: Numerical solution of generalized Burgers-Huxley equation using local radial basis functions. International Journal of Advanced and Applied Sciences 4(5), 1–11 (2017)CrossRefGoogle Scholar
  36. 36.
    Singh, B.K., Arora, G., Batool, S., Singh, M.K.: A numerical scheme for the generalized BurgersHuxley equation. Journal of the Egyptian Mathematical Society 24, 629–637 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Bratsos, A.G.: A fourth order improved numerical scheme for the generalized BurgersHuxley equation. Am. J. Comput. Math. 1(3), 152–158 (2011)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Mohaghegh ArdabiliArdabilIran

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