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


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.

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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)

  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)

    Article  Google 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)

    Article  Google 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)

  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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Deng, X.: Traveling wave solutions for the generalized BurgersHuxley equation. Appl. Math. Comput. 204, 733–737 (2008)

    MathSciNet  Google 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)

    MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Google 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)

    MathSciNet  Google 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)

    Article  Google 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)

    MathSciNet  Google Scholar 

  16. 16.

    Efimova, O.Y., Kudryashov, N.A.: Exact solutions of the Burgers-Huxley equation. J. Appl. Math. Mech. 68(3), 413420 (2004)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Zhang, J.W.: C-curves, an extension of cubic curves. Comput. Aided Geom. Design 13, 199–217 (1996)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zhang, J.W.: Two different forms of C-B-splines. Comput. Aided Geom. Design 14, 31–41 (1997)

    MathSciNet  Article  Google 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)

    Google 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)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  22. 22.

    Chen, Q.Y., Wang, G.Z.: A class of Bezier-like curves. Comput. Aided Geom. Design 20, 29–39 (2003)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Wang, G.Z., Chen, Q.Y., Zhou, M.H.: NUAT B-spline curves. Comput. Aided Geom. Design 21, 193–205 (2004)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  27. 27.

    Xu, G., Wang, G.: AHT Bezier curves and NUAHT B-spline curves. J. Comput. Sci. Technol. 22, 597–607 (2007)

    Article  Google Scholar 

  28. 28.

    Wang, G., Fang, M.: Unified and extended form of three types of splines. J. Comput. Appl. Math. 216, 498–508 (2008)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

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Alinia, N., Zarebnia, M. A numerical algorithm based on a new kind of tension B-spline function for solving Burgers-Huxley equation. Numer Algor 82, 1121–1142 (2019). https://doi.org/10.1007/s11075-018-0646-4

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