255
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

Numerical solutions of strongly non-linear generalized Burgers–Fisher equation via meshfree spectral technique

ORCID Icon & ORCID Icon
Pages 1727-1748 | Received 23 Oct 2019, Accepted 28 Sep 2020, Published online: 19 Nov 2020

References

  • G.P. Boswell and F.A. Davidson, Diffusion fronts in enzyme-catalysed reactions, J. Eng. Math 59(2) (2007), pp. 157–169.
  • P. Brazhnik and J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math 60(2) (1999), pp. 371–391.
  • R.L. Burden and J.D. Faires, Numerical Analysis, 9th ed., Brooks/Cole Cengage Learning, Boston, 2011.
  • F. Cavarretta and G. Naldi, Mathematical study of a nonlinear neuron model with active dendrites, AIMS Math. 4(3) (2019), pp. 831–846.
  • V. Chandraker, A. Awasthi, and S. Jayaraj, Numerical treatment of Burger-Fisher equation, Procedia Tech. 25 (2016), pp. 1217–1225.
  • S. Coen, M. Tlidi, P. Emplit, and M. Haelterman, Convection versus dispersion in optical bistability, Phys. Rev. Lett. 83(12) (1999), pp. 2328–2331.
  • D.A. Di Pietro, A. Ern, and L. Formaggia, Numerical Methods for PDEs: State of the Art Techniques, SEMA SIMAI Springer Series, Springer-Nature, Switzerland AG, 2018.
  • B. Ermentrout and D. Terman, Mathematical Foundations of Neuroscience, Springer-Verlag, New York, 2010.
  • G.E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific, River Edge, NJ , 2007.
  • X. Feng, R. Glowinski, and M. Neilan, Recent developments in numerical methods for fully nonlinear second order partial differential equations, SIAM Rev. 55(2) (2013), pp. 205–267.
  • R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), pp. 355–369.
  • J. Geiser, Discretization methods with embedded analytical solutions for convection-diffusion dispersion-reaction equations and applications, J. Eng. Math. 57(1) (2007), pp. 79–98.
  • P. Graben, C. Zhou, M. Thiel, and J. Kurths, Foundations of neurophysics, Lectures in supercomputational neuroscience: Dynamics in complex brain networks, Springer, Berlin, 2008.
  • S. Haq and M. Hussain, Selection of shape parameter in radial basis functions for solution of time-fractional Black-Scholes models, Appl. Math. Comput. 335 (2018), pp. 248–263.
  • S. Haq and M. Hussain, The meshless Kansa method for time-fractional higher order partial differential equations with constant and variable coefficients, RACSAM 113(3) (2019), pp. 1935–1954.
  • S. Haq, M. Hussain, and A. Ghafoor, A computational study of variable coefficients fractional advection-diffusion-reaction equations via implicit meshless spectral algorithm, Eng. Comput. (2019), pp. 1–21. doi:10.1007/s00366-019-00760-x.
  • S. Haq and M. Uddin, Numerical solution of nonlinear Fisher's equations using collocation method radial basis functions, in Proceedings of 2nd International Symposium on Frontiers of Computational Sciences, M. Abid and H.A. Wajid eds., Islamabad, Pakistan, 2012.
  • 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.
  • E.J. Kansa, Multiquadrics – A scattered data approximation scheme with application to computation fluid dynamics. II. solutions to hyperbolic, parabolic, and elliptic partial differential equations, Comput. Math. Appl. 19 (1990), pp. 149–158.
  • T. Kawahara and M. Tanaka, Interactions of traveling fronts an, exact solution of a nonlinear diffusion equation, Phys. Lett. A 97 (1983), pp. 311–314.
  • A. Khaliq, J. Ku, and Q. Sheng, Recent advances in numerical methods for systems of partial differential equations, J. Appl. Comput. Math. 299 (2016), pp. 1–256.
  • J.S. Kim and F.A. Williams, Extinction of diffusion flames with nonunity Lewis numbers, J. Eng. Math. 31(2–3) (1997), pp. 101–118.
  • J.R. King and R.D. O'Dea, Pushed and pulled fronts in a discrete reaction-diffusion equation, J. Eng. Math. 102(1) (2017), pp. 89–116.
  • 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.
  • G.R. Liu and T.Y. Gu, An Introduction to Meshfree Methods and Their Programming, Berlin, Springer Press, 2005.
  • J.H. Merkin and D.J. Needham, Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system, J. Eng. Math. 23(4) (1989), pp. 343–356.
  • C.A. Micchelli, Interpolation of scattered data: distance matrix and conditionally positive definite functions, Construct. Approx 2 (1986), pp. 11–22.
  • R.C. Mittal and G. Arora, Efficient numerical solution of Fisher's equation by using B-spline method, Int. J. Comput. Math. 87(1) (2009), pp. 3039–3051.
  • R.C. Mittal and S. Kumar, Numerical study of Fisher's equation by wavelet Galerkin method, Int. J. Comput. Math. 83(3) (2006), pp. 287–298.
  • R.C. Mittal and R. Rohila, A study of one dimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method, J. Math. Chem. 55(2) (2017), pp. 673–695.
  • R.C. Mittal and A. Tripathi, Numerical solutions of generalized Burgers-Fisher and generalized Burgers-Huxley equations using collocation of cubic B-splines, Int. J. Comput. Math. 92(5) (2015), pp. 1053–1077.
  • R. Mohammadi, Spline solution of the generalized Burgers'-Fisher equation, Appl. Anal. 91(12) (2012), pp. 2189–2215.
  • J.D. Murray, Mathematical Biology: I. An Introduction, Springer, New York NY, 2002.
  • J. Rashidinia and M.N. Rasoulizadeh, Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation, Wave Motion 90 (2019), pp. 152–167.
  • J. Riordan, C.R. Doering, and D. Ben-Avraham, Fluctuations and stability of Fisher waves, Phys. Rev. Lett. 75(3) (1995), pp. 565–568.
  • V. Sangwan and B. Kaur, An exponentially fitted numerical technique for singularly perturbed Burgers-Fisher equation on a layer adapted mesh, Int. J. Comput. Math. 96(7) (2019), pp. 1502–1513.
  • A.C. Scott, Neurophysics, Wiley, New York, 1977.
  • Q. Sheng, Y. Tang, B.A. Wade, and Y. Wang, Recent trends in highly accurate and structure-preserving numerical methods for partial differential equations, Int. J. Comput. Math. 95(1) (2018), pp. 1–2.
  • J. Sherratt, On the transition from initial data traveling waves in the Fisher-KPP equation, Dyn. Stab. Syst. 13(2) (1998), pp. 167–174.
  • E. Shivanian, A new spectral meshless radial point interpolation (SMRPI) method: a well-behaved alternative to the meshless weak forms, Eng. Anal. Bound. Elem. 54 (2015), pp. 1–12.
  • E. Tadmor, A review of numerical methods for nonlinear partial differential equations, Bull. Amer. Math. Soc. (New Series) 49(4) (2012), pp. 507–554.
  • X. Wang, Nerve propagation and wall in liquid crystals, Phys. Lett. A 112(8) (1985), pp. 402–406.
  • X.Y. Wang, Exact and explicit solitray wave solutions for the generalized Fisher equation, Phy. Lett. A 131(4–5) (1988), pp. 277–279.
  • T.-J. Wang, Generalized Laguerre spectral method for Fisher's equation on a semi-infinite interval, Int. J. Comput. Math. 92(5) (2015), pp. 1039–1052.
  • 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.
  • T. Zhao, C. Li, Z. Zang, and Y. Wu, Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation, Appl. Math. Modell. 36(3) (2012), pp. 1046–1056.
  • 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.

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.