735
Views
50
CrossRef citations to date
0
Altmetric
Section B

Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

&
Pages 1053-1077 | Received 20 Jan 2014, Accepted 25 Apr 2014, Published online: 04 Jun 2014

References

  • İ. Çelik, Haar wavelet method for solving generalized Burgers-Huxley equation, Arab J. Math. Sci. 18 (2011), pp. 25–37. Available at http://dx.doi.org/10.1016/j.ajmsc.2011.08.003. doi: 10.1016/j.ajmsc.2011.08.003
  • Z. Chen, A. Gumel, and R. Mickens, Nonstandard discretizations of the generalized Nagumo reaction-diffusion equation, Numer. Methods Partial Differential Equations 19(3) (2003), pp. 363–379. doi: 10.1002/num.10048
  • S. Coen, M. Tlidi, P. Emplit, and M. Haelterman, Convection versus dispersion in optical bistability, Phys. Rev. Lett. 83(12) (1999), pp. 2328–2331. doi: 10.1103/PhysRevLett.83.2328
  • M. Dehghan and F. Fakhar-Izadi, Pseudospectral methods for nagumo equation, Internat. J. Numer. Methods Biomed. Eng. 27(4) (2011), pp. 553–561. doi: 10.1002/cnm.1319
  • M. Dehghan, J.M. Heris, and A. Saadatmandi, Application of semi-analytic methods for the Fitzhugh–Nagumo equation, which models the transmission of nerve impulses, Math. Methods Appl. Sci. 33(11) (2010), pp. 1384–1398.
  • M. Dehghan, B.N. Saray, and M. Lakestani, 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, Math. Comput. Modelling 55 (2011), pp. 1129–1142. Available at http://dx.doi.org/10.1016/j.mcm.2011.09.037. doi: 10.1016/j.mcm.2011.09.037
  • Y. Duan, L. Kong, and R. Zhang, A lattice Boltzmann model for the generalized Burgers–Huxley equation, Phys. A 391(3) (2012), pp. 625–632. doi: 10.1016/j.physa.2011.08.034
  • A. Golbabai and M. Javidi, A spectral domain decomposition approach for the generalized Burger's–Fisher equation, Chaos Solitons Fractals 39(1) (2009), pp. 385–392. doi: 10.1016/j.chaos.2007.04.013
  • S. Gottlieb, On high order strong stability preserving Runge–Kutta and multi step time discretizations, J. Sci. Comput. 25(1) (2005), pp. 105–128. doi: 10.1007/s10915-004-4635-5
  • S. Haq, A. Hussain, and M. Uddin, On the numerical solution of nonlinear Burgers’-type equations using meshless method of lines, Appl. Math. Comput. 218 (2011), pp. 6280–6290. Available at http://dx.doi.org/10.1016/j.amc.2011.11.106. doi: 10.1016/j.amc.2011.11.106
  • H.N.A. Ismail, K. Raslan, and A.A. Abd Rabboh, Adomian decomposition method for Burger's–Huxley and Burger's–Fisher equations, Appl. Math. Comput. 159(1) (2004), pp. 291–301. doi: 10.1016/j.amc.2003.10.050
  • M. Javidi, A numerical solution of the generalized Burger's–Huxley equation by spectral collocation method, Appl. Math. Comput. 178(2) (2006), pp. 338–344. doi: 10.1016/j.amc.2005.11.051
  • M. Javidi and A. Golbabai, A new domain decomposition algorithm for generalized Burger's–Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fractals 39(2) (2009), pp. 849–857. doi: 10.1016/j.chaos.2007.01.099
  • A. Kaushik, Pointwise uniformly convergent numerical treatment for the non-stationary Burger–Huxley equation using grid equidistribution, Int. J. Comput. Math. 84(10) (2007), pp. 1527–1546. doi: 10.1080/00207160701314505
  • A.J. Khattak, A computational meshless method for the generalized Burger's–Huxley equation, Appl. Math. Modelling 33(9) (2009), pp. 3718–3729. doi: 10.1016/j.apm.2008.12.010
  • A.N. Kolmogorov, I.G. Petrovskii, and N.S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos Univ. 1(7) (1937), pp. 1–26.
  • R.C. Mittal and R.K. Jain, Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method, Appl. Math. Comput. 218 (2012), pp. 7839–7855. Available at http://dx.doi.org/10.1016/j.amc.2012.01.059. doi: 10.1016/j.amc.2012.01.059
  • R.C. Mittal and R. Jiwari, Numerical study of Burger–Huxley equation by differential quadrature method, Int. J. Appl. Math. Mech. 5 (2009), pp. 1–9.
  • R. Mohammadi, B-spline collocation algorithm for numerical solution of the generalized Burger's–Huxley equation, Numer. Methods Partial Differential Equations 29(4) (2013), pp. 1173–1191. doi: 10.1002/num.21750
  • J.D. Murray, Mathematical Biology: I. An Introduction, Vol. 2, Springer, New York, NY, 2002.
  • E.J. Parkes and B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Comm. 98(3) (1996), pp. 288–300. doi: 10.1016/0010-4655(96)00104-X
  • J. Riordan, C.R. Doering, and D. Ben-Avraham, Fluctuations and stability of Fisher waves, Phys. Rev. Lett. 75(3) (1995), pp. 565–568. doi: 10.1103/PhysRevLett.75.565
  • M. Sari, G. Gürarslan, and İ. Dağ, A compact finite difference method for the solution of the generalized Burgers–Fisher equation, Numer. Methods Partial Differential Equations 26(1) (2010), pp. 125–134. 10.1002/num.20421.
  • J. Satsuma, Exact Solutions of Burgers Equation with Reaction TermsTopics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Sci Pub, Singapore, 1986, pp. 255–262.
  • J. Satsuma, M. Ablowitz, B. Fuchssteiner, and M. Kruskal, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore City, 1987.
  • A.C. Scott, Neurophysics, Wiley, New York, 1977.
  • K. Sepehrnoori and G. Carey, Numerical integration of semidiscrete evolution systems, Comput. Methods Appl. Mech. Engrg. 27(1) (1981), pp. 45–61. doi: 10.1016/0045-7825(81)90167-5
  • G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, Oxford, NY, 1985.
  • S. Tomasiello, Numerical solutions of the Burgers–Huxley equation by the IDQ method, Int. J. Comput. Math. 87(1) (2010), pp. 129–140. doi: 10.1080/00207160801968762
  • X. Wang, Nerve propagation and wall in liquid crystals, Phys. Lett. A 112(8) (1985), pp. 402–406. doi: 10.1016/0375-9601(85)90411-6
  • X.Y. Wang, Z.S. Zhu, and Y.K. Lu, Solitary wave solutions of the generalised Burgers–Huxley equation, J. Phys. A. 23(3) (1999), pp. 271–274. doi: 10.1088/0305-4470/23/3/011
  • A.M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations, Appl. Math. Comput. 169(1) (2005), pp. 321–338. doi: 10.1016/j.amc.2004.09.054
  • G.B. Whitman, Linear and Nonlinear Waves, Wiley and Sons, New York, 1974.
  • R. Zhang, X. Yu, and G. Zhao, The local discontinuous Galerkin method for Burger's–Huxley and Burger's–Fisher equations, Appl. Math. Comput. 218 (2012), pp. 8773–8778. Available at http://dx.doi.org/10.1016/j.amc.2012.02.035. doi: 10.1016/j.amc.2012.02.035
  • T. Zhao, C. Li, Z. Zang, and Y. Wu, Chebyshev–Legendre pseudo-spectral method for the generalised Burgers–Fisher equation, Appl. Math. Modelling 36(3) (2012), pp. 1046–1056. Available at http://dx.doi.org/10.1016/j.apm.2011.07.059. doi: 10.1016/j.apm.2011.07.059
  • C.G. Zhu and W.S. Kang, Numerical solution of Burgers–Fisher equation by cubic B-spline quasi-interpolation, Appl. Math. Comput. 216(9) (2010), pp. 2679–2686. doi: 10.1016/j.amc.2010.03.113

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.