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Numerical solutions of FitzHugh–Nagumo equation by exact finite-difference and NSFD schemes

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Abstract

In this paper, numerical solution of the FitzHugh–Nagumo (FN) equation is presented based on the nonstandard finite-difference (NSFD) scheme. At first, two exact finite-difference schemes for FN equation are obtained. Moreover, two NSFD schemes are presented for this equation. The positivity, boundedness, and local truncation error of the schemes are discussed. The numerical results obtained by the NSFD schemes are compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD schemes.

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References

  • Arenas AJ, Gonzalez-Parra G, Chen-Charpentier BM (2010) A nonstandard numerical scheme of predictor-corrector type for epidemic models. Comput Math Appl 59(12):3740–3749

    Article  MathSciNet  MATH  Google Scholar 

  • Batiha B, Noorani MSM, Hashim I (2007) Numerical simulation of the generalized Huxley equation by He’s variational iteration method. Appl Math Comput 186(2):1322–1325

    MathSciNet  MATH  Google Scholar 

  • Batiha B, Noorani MSM, Hashim I (2008) Application of variational iteration method to the generalized Burgers–Huxley equation. Chaos Solitons Fractals 36(3):660–663

    Article  MATH  Google Scholar 

  • Bhrawy AH (2013) A Jacobi–Gauss–Lobatto collocation method for solving generalized Fitzhugh–Nagumo equation with time-dependent coefcients. Appl Math Comput 222:255–264

    MathSciNet  MATH  Google Scholar 

  • Bhrawy AH (2014) An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl Math Comput 247:30–46

    MathSciNet  MATH  Google Scholar 

  • Bhrawy AH (2016) A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer Algorithm 1–23: doi:10.1007/s11075-015-0087-2

  • Bhrawy AH (2016) A space–time collocation scheme for modified anomalous subdiffusion and nonlinear superdiffusion equations. Eur Phys J Plus 131(12):1–20

    Google Scholar 

  • Bhrawy AH, Doha EH, Ezz-Eldien SS, Abdelkawy MA (2016) A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations. Calcolo 53:1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Biazar J, Mohammadi F (2010) Application of differential transform method to the generalized Burgers–Huxley equation. Appl Appl Math 2:1726–1740

    MathSciNet  MATH  Google Scholar 

  • Erdogan U, Ozis T (2011) A smart nonstandard finite difference scheme for second order nonlinear boundary value problems. J Comput Phys 230(17):6464–6474

    Article  MathSciNet  MATH  Google Scholar 

  • Fitzhugh R (1961) Impulse and physiological states in models of nerve membrane. Biophys J 1(6):445–466

    Article  Google Scholar 

  • Gonzalez-Parra G, Arenas AJ, Chen-Charpentier BM (2010) Combination of nonstandard schemes and Richardsons extrapolation to improve the numerical solution of population models. Math Comput Model 52(7–8):1030–1036

    Article  MathSciNet  MATH  Google Scholar 

  • Hariharan G, Kannan K (2010) Haar wavelet method for solving FitzHugh–Nagumo equation. World Acad Sci Eng Technol 67(43):560–574

    Google Scholar 

  • Hashim I, Noorani MSM, Batiha B (2006) A note on the Adomian decomposition method for the generalized Huxley equation. Appl Math Comput 181(2):1439–1445

    MathSciNet  MATH  Google Scholar 

  • Li H, Guo Y (2006) New exact solutions to the Fitzhugh–Nagumo equation. Appl Math Comput 180(2):524–528

    MathSciNet  MATH  Google Scholar 

  • Mickens RE (2005) Advances in the applications of nonstandard finite difference schemes. Wiley-Interscience, Singapore

    Book  MATH  Google Scholar 

  • Mickens RE (2007) Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numer Methods Partial Differ Equ 23(3):672–691

    Article  MathSciNet  MATH  Google Scholar 

  • Murray JD (2003) Mathematical biology I, II, 3rd edn. Springer, Berlin

    Google Scholar 

  • Nagumo JS, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061–2071

    Article  Google Scholar 

  • Nucci MC (1992) The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh–Nagumo equation. Phys Lett A 164(1):49–56

    Article  MathSciNet  Google Scholar 

  • Roeger LIW (2008) Exact finite-difference schemes for two dimensional linear systems with constant coefficients. J Comput Appl Math 219:102–109

    Article  MathSciNet  MATH  Google Scholar 

  • Roeger LIW (2008) Exact nonstandard finite-difference methods for a linear system the case of centers. J Differ Equ Appl 14(4):381–389

    Article  MathSciNet  MATH  Google Scholar 

  • Roeger LIW, Mickens RE (2007) Exact finite-difference schemes for first order differential equations having three distinct fixed-points. J Differ Equ Appl 13(12):1179–1185

    Article  MathSciNet  MATH  Google Scholar 

  • Rucker S (2003) Exact finite difference scheme for an advectionreaction equation. J Differ Equ Appl 9(11):1007–1013

    Article  MATH  Google Scholar 

  • Shih M, Momoniat E, Mahomed FM (2005) Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh–Nagumo equation. J Math Phys 46:0235031

    Article  MathSciNet  MATH  Google Scholar 

  • Wazwaz AM (2009) Partial differential equations and solitary waves theory–higher education press. Springer, New York

    Book  Google Scholar 

  • Wazwaz AM, Gorguis A (2004) An analytic study of Fisher’s equation by using adomian decomposition method. Appl Math Comput 154(3):609–620

    MathSciNet  MATH  Google Scholar 

  • Whitehead JA, Newell AC (1969) Finite bandwidth, nite amplitude convection. J Fluid Mech 38(2):279–303

    Article  MathSciNet  MATH  Google Scholar 

  • Zibaei S, Namjoo M (2014) A NSFD scheme for Lotka–Volterra food web model. Iran J Sci Technol Trans A Sci 38:399–414

    MathSciNet  Google Scholar 

  • Zibaei S, Namjoo M (2015) A nonstandard finite difference scheme for solving fractional-order model of HIV-1 infection of CD \(4^+\) T-cells. Iran J Math Chem 6(2):145–160

    MATH  Google Scholar 

  • Zibaei S, Namjoo M (2016) A nonstandard finite difference scheme for solving three-species food chain with fractional-order Lotka-Volterra model. Iran J Numer Anal Optim 6(1):53–78

    MATH  Google Scholar 

  • Zibaei S, Zeinadini M, Namjoo M (2016) Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes. J Differ Equ Appl 1–16: doi:10.1080/10236198.2016.1173687

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Authors and Affiliations

  1. Department of Mathematics, School of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

    Mehran Namjoo

  2. Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Iran

    Sadegh Zibaei

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  1. Mehran Namjoo
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  2. Sadegh Zibaei
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Correspondence to Mehran Namjoo.

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Communicated by Eduardo Souza de Cursi.

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Namjoo, M., Zibaei, S. Numerical solutions of FitzHugh–Nagumo equation by exact finite-difference and NSFD schemes. Comp. Appl. Math. 37, 1395–1411 (2018). https://doi.org/10.1007/s40314-016-0406-9

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  • DOI: https://doi.org/10.1007/s40314-016-0406-9

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