10.1007/s40096-022-00461-5

Numerical study of nonlinear generalized Burgers–Huxley equation by multiquadric quasi-interpolation and pseudospectral method

  • Mahbubeh Rahimi1,
  • Hojatollah Adibi*1,
  • Majid Amirfakhrian2,
  1. Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, IR
  2. Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, IR Department of Computer Science, University of Calgary, Calgary, CA

Published in Issue 2022-03-11

How to Cite

Rahimi, M., Adibi, H., & Amirfakhrian, M. (2022). Numerical study of nonlinear generalized Burgers–Huxley equation by multiquadric quasi-interpolation and pseudospectral method. Mathematical Sciences, 17(4 (December 2023). https://doi.org/10.1007/s40096-022-00461-5
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Abstract

Abstract This paper develops an efficient numerical meshless method to solve the nonlinear generalized Burgers–Huxley equation (NGB-HE). The proposed method approximates the unknown solution in the two stages. First, the θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} -weighted finite difference technique is adopted to discretize the temporal dimension. Second, a combination of the multiquadric quasi-interpolation and pseudospectral (denoted by MQQI-PS) is constructed to approximate the spatial derivatives. In addition, a cross-validation technique is used to find the shape parameter value. Finally, numerical results are illustrated to show the accuracy and efficiency of the MQQI-PS method.

Keywords

  • Nonlinear generalized Burgers–Huxley equation,
  • Multiquadric quasi-interpolation method,
  • Pseudospectral method,
  • θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}-Method

References

  1. Wang et al. (1990) Solitary wave solutions of the generalised Burgers-Huxley equation 23(3) (pp. 271-274) https://doi.org/10.1088/0305-4470/23/3/011
  2. FitzHugh, R.: Mathematical models of excitation and propagation in nerve. In: Biological Engineering. Schwan, H.P.(eds), New York: McGraw Hill (1969)
  3. Bratsos (2010) A fourth-order numerical scheme for solving the modified Burgers equation 60(5) (pp. 1393-1400)
  4. Çelik (2012) Haar Wavelet method for solving generalized Burgers-Huxley equation 18(1) (pp. 25-37)
  5. Javidi and Golbabai (2009) A new domain decomposition algorithm for generalized Burger’s-Huxley equation based on Chebyshev polynomials and preconditioning 39(2) (pp. 849-857) https://doi.org/10.1016/j.chaos.2007.01.099
  6. Dehghan et al. (2012) Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers-Huxley equation 55(3–4) (pp. 1129-1142) https://doi.org/10.1016/j.mcm.2011.09.037
  7. Mohammadi (2013) B-spline collocation algorithm for numerical solution of the generalized Burger’s-Huxley equation 29(4) (pp. 1173-1191) https://doi.org/10.1002/num.21750
  8. Mohan and Khan (2021) On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies 26(7) (pp. 3943-3988)
  9. Zhong et al. (2021) The modified high-order Haar wavelet scheme with Runge-Kutta method in the generalized Burgers-Fisher equation and the generalized Burgers-Huxley equation 35(24) https://doi.org/10.1142/S0217984921504194
  10. Çiçek and Korkut (2021) Numerical solution of generalized Burgers-Huxley equation by Lie-Trotter splitting method 14(1) (pp. 90-102) https://doi.org/10.1134/S1995423921010080
  11. Wang (2021) Variational principle and approximate solution for the generalized Burgers-Huxley equation with fractal derivative 29(2) https://doi.org/10.1142/S0218348X21500444
  12. Shukla, S., Kumar, M.: Error analysis and numerical solution of Burgers-Huxley equation using 3-scale Haar wavelets. Eng. Comput. (2020).
  13. https://doi.org/10.1007/s00366-020-01037-4
  14. Alinia and Zarebnia (2019) A numerical algorithm based on a new kind of tension B-spline function for solving Burgers-Huxley equation 82(4) (pp. 1121-1142) https://doi.org/10.1007/s11075-018-0646-4
  15. Abbasbandy et al. (2012) Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function 36(12) (pp. 1811-1818) https://doi.org/10.1016/j.enganabound.2012.06.011
  16. Abbasbandy, S., Azarnavid, B., Hashim, I., Alsaedi, A.: Approximation of backward heat conduction problem using Gaussian radial basis functions. U.P.B. Sci. Bull., Series A.
  17. 76
  18. (4), 67–76 (2014)
  19. Chen et al. (2014) Springer https://doi.org/10.1007/978-3-642-39572-7
  20. Kansa et al. (2009) Numerical simulation of two-dimensional combustion using mesh-free methods 33(7) (pp. 940-950) https://doi.org/10.1016/j.enganabound.2009.02.008
  21. Assari and Dehghan (2019) A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations (pp. 249-265)
  22. Assari (2019) Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations 98(11) (pp. 2064-2084) https://doi.org/10.1080/00036811.2018.1448073
  23. Assari and Dehghan (2018) Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method (pp. 137-158) https://doi.org/10.1016/j.apnum.2017.09.002
  24. Assari (2019) The numerical solution of Fredholm-Hammerstein integral equations by combining the collocation method and radial basis functions 33(3) (pp. 667-682) https://doi.org/10.2298/FIL1903667A
  25. Xiao et al. (2011) Applying multiquadric quasi-interpolation to solve KdV equation 31(2) (pp. 191-201)
  26. Nikan, O., Avazzadeh, Z., Machado, J.T.: Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport. Commun. Nonlinear Sci. Numer. Simul.
  27. 99
  28. , 105755 (2021)
  29. Powell, M.J.D.: Univariate multiquadric approximation: Reproduction of linear polynomials. In: Multivariate Approximation and Interpolation. Haussman, W. and Jetter, K.(eds), Birkhäuser Verlag, Basel (1990)
  30. Beatson and Powell (1992) Univariate multiquadric approximation: Quasi-interpolation to scattered data (pp. 275-288) https://doi.org/10.1007/BF01279020
  31. Wu and Schaback (1994) Shape preserving properties and convergence of univariate multiquadric quasi-interpolation 10(4) (pp. 441-446) https://doi.org/10.1007/BF02016334
  32. Jiang et al. (2011) High accuracy multiquadric quasi-interpolation 35(5) (pp. 2185-2195) https://doi.org/10.1016/j.apm.2010.11.022
  33. Singh et al. (2016) A numerical scheme for the generalized Burgers-Huxley equation 24(4) (pp. 629-637) https://doi.org/10.1016/j.joems.2015.11.003
  34. 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.
  35. 2013
  36. (2013). 10.1155/2013/165492
  37. Hardy (1971) Multiquadric equations of topography and other irregular surfaces (pp. 1905-1915) https://doi.org/10.1029/JB076i008p01905
  38. Madych and Nelson (1990) Multivariate interpolation and conditionally positive definite functions (pp. 211-230) https://doi.org/10.1090/S0025-5718-1990-0993931-7
  39. Sarboland, M., Aminataei, A.: On the numerical solution of one-dimensional nonlinear nonhomogeneous Burgers’ equation. J. Appl. Math.
  40. 2014
  41. (2014). 10.1155/2014/598432
  42. Rashidinia et al. (2010) Numerical solution of the nonlinear Klein-Gordon equation 233(8) (pp. 1866-1878) https://doi.org/10.1016/j.cam.2009.09.023
  43. Fasshauer et al. (2005) RBF collocation methods as Pseudospectral methods (pp. 47-56) WIT Press
  44. Wendland (2005) Cambridge University Press
  45. Schaback (2007) Convergence of unsymmetric kernel-based meshless collocation methods 45(1) (pp. 333-351) https://doi.org/10.1137/050633366
  46. Emamjomeh, M., Abbasbandy, S., Rostamy, D.: Quasi interpolation of radial basis functions-pseudospectral method for solving nonlinear Klein–Gordon and sine-Gordon equations. Iranian J. Numer. Anal. Optim.
  47. 10
  48. (1), 81–106 (2020). 10.22067/IJNAO.V10I1.75129
  49. Twizell (1984) Ellis Horwood Limited
  50. Islam et al. (2009) A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations 33(3) (pp. 399-409) https://doi.org/10.1016/j.enganabound.2008.06.005
  51. Lee, M.B., Yoon, J.: Sampling inequalities for infinitely smooth radial basis functions and its application to error estimates. Appl. Math. Lett
  52. Fasshauer and Zhang (2007) On choosing optimal shape parameters for RBF approximation (pp. 345-368) https://doi.org/10.1007/s11075-007-9072-8
  53. Rippa (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation (pp. 193-210) https://doi.org/10.1023/A:1018975909870