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Articles

Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

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Pages 989-1004 | Received 08 Nov 2013, Accepted 16 Dec 2013, Published online: 13 Mar 2014

References

  • G.Abramson, A.R.Bishop, and V.M.Kenkre, Effects of transport memory and nonlinear damping in a generalized Fisher's equation, Phys. Rev. E64 (2001), p. 066615.
  • R.Anguelov, J.M.S.Lubuma, and M.Shillor, Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems, Discrete Contin. Dyn. Syst.61 (2009), pp. 34–43.
  • R.Anguelov, J.M.S.Lubuma, and M.Shillor, Topological dynamic consistency of non-standard finite difference schemes for dynamical systems, J. Differ. Equ. Appl.17 (2011), pp. 1769–1791.
  • H.J.Eberl and L.Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, Electron. J. Differ. Equ.15 (2007), pp. 77–95.
  • H.J.Eberl, D.F.Parker, and M.C.M.van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med.3 (2001), pp. 161–176.
  • M.A.Efendiev, H.J.Eberl, and S.V.Zelik, Existence and longtime behavior of solutions of a nonlinear reaction–diffusion system arising in the modeling of biofilms, RIMS Kokyuroko1258 (2002), pp. 49–71.
  • R.A.Fisher, The wave of advance of advantageous genes, Ann. Eugenics7 (1937), pp. 355–369.
  • D.Furihata, Finite difference schemes that inherit energy conservation or dissipation property, J. Comput. Phys.156 (1999), pp. 181–205.
  • D.Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math.134 (2001), pp. 37–57.
  • S.M.Garba, A.B.Gumel, and J.M.S.Lubuma, Dynamically-consistent non-standard finite difference method for an epidemic model, Math. Comput. Model.53 (2011), pp. 131–150.
  • A.L.Hodgkin and A.F.Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol.117 (1952), pp. 500–544.
  • A.Kolmogorov, I.Petrovsky, and N.Piscounov, Étude de l'équations de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. Univ. Moskou Ser. Int.1A (1937), pp. 1–25.
  • J.E.Macías-Díaz, A numerical method with properties of consistency in the energy domain for a class of dissipative nonlinear wave equations with applications to a Dirichlet boundary-value problem, Z. Angew. Math.88 (2008), pp. 828–846.
  • J.E.Macías-Díaz, Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping, Comput. Phys. Commun.182 (2011), pp. 2471–2478.
  • J.E.Macías-Díaz, A Mickens-type monotone discretization for bounded travelling-wave solutions of a Burgers–Fisher partial differential equation, J. Differ. Equ. Appl.19 (2013), pp. 1907–1920.
  • J.E.Macias-Diaz, S.Jerez-Galiano, and A.Puri, Positivity-preserving methods for a linearised Fisher–KPP equation with consistency properties in the energy domain, J. Differ. Equ. Appl.16 (2010), pp. 389–402.
  • J.E.Macías-Díaz and A.Puri, A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein-Gordon equation, Numer. Meth. Part. Differ. Equ.21 (2005), pp. 998–1015.
  • J.E.Macías-Díaz and A.Puri, An energy-based computational method in the analysis of the transmission of energy in a chain of coupled oscillators, J. Comput. Appl. Math.214 (2008), pp. 393–405.
  • J.E.Macías-Díaz and A.Puri, An explicit positivity-preserving finite-difference scheme for the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation, Appl. Math. Comput.218 (2012), pp. 5829–5837.
  • J.E.Macías-Díaz, J.Ruiz-Ramírez, and J.Villa, The numerical solution of a generalized Burgers–Huxley equation through a conditionally bounded and symmetry-preserving method, Comput. Math. Appl.61 (2011), pp. 3330–3342.
  • R.E.Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comput. Appl. Math.110 (1999), pp. 181–185.
  • R.E.Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl.8 (2002), pp. 823–847.
  • R.E.Mickens, A nonstandard finite-difference scheme for the Lotka–Volterra system, Appl. Numer. Math.45 (2003), pp. 309–314.
  • R.E.Mickens, Dynamic consistency: A fundamental principle for constructing nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl.11 (2005), pp. 645–653.
  • R.E.Mickens, A note on a non-standard finite difference scheme for the Reluga model, J. Differ. Equ. Appl.16 (2010), pp. 1501–1504.
  • R.E.Mickens, A SIR-model with square-root dynamics: An NSFD scheme, J. Differ. Equ. Appl.16 (2010), pp. 209–216.
  • R.E.Mickens and P.M.Jordan, A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Meth. Part. Differ. Equ.20 (2004), pp. 639–649.
  • R.E.Mickens and P.M.Jordan, A new positivity-preserving nonstandard finite difference scheme for the DWE, Numer. Meth. Part. Differ. Equ.21 (2005), pp. 976–985.
  • R.E.Mickens and T.M.Washington, A note on exact finite difference schemes for the differential equations satisfied by the Jacobi cosine and sine functions, J. Differ. Equ. Appl.19 (2012), pp. 1–6.
  • A.D.Polyanin, and V.F.Zaitsev, Handbook of Nonlinear Partial Differential Equations, 1st ed., Chapman & Hall CRC Press, Boca Raton, FL, 2004.
  • L.I.W.Roeger and G.Lahodny, Jr, Dynamically consistent discrete Lotka–Volterra competition systems, J. Differ. Equ. Appl.19 (2013), pp. 191–200.
  • L.I.W.Roeger and R.E.Mickens, Exact finite difference and non-standard finite difference schemes for, J. Differ. Equ. Appl.18 (2012), pp. 1511–1517.
  • J.Ruiz-Ramírez and J.E.Macías-Díaz, A finite-difference scheme to approximate non-negative and bounded solutions of a FitzHugh–Nagumo equation, Int. J. Comput. Math.88 (2011), pp. 3186–3201.
  • M.Sari, G.Gürarslan, and İ.Dağ, A compact finite difference method for the solution of the generalized Burgers–Fisher equation, Numer. Meth. Part. Differ. Equ.26 (2010), pp. 125–134.
  • M.Tatari, B.Sepehrian, and M.Alibakhshi, New implementation of radial basis functions for solving Burgers–Fisher equation, Numer. Meth. Part. Differ. Equ.28 (2011), pp. 248–262.
  • X.Y.Wang, Exact and explicit solitary wave solutions for the generalised Fisher equation, Phys. Lett. A.131 (1988), pp. 277–279.
  • X.Y.Wang, Z.S.Zhu, and Y.K.Lu, Solitary wave solutions of the generalised Burgers–Huxley equation, J. Phys. A: Math. Gen.23 (1990), p. 271.
  • G.Wu, Uniformly constructing soliton solutions and periodic solutions to Burgers–Fisher equation, Comput. Math. Appl.58 (2009), pp. 2355–2357.

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