Research Article
BibTex RIS Cite
Year 2019, , 180 - 187, 26.12.2019
https://doi.org/10.32323/ujma.561873

Abstract

References

  • [1] A. C. Scott, Neunstor propagation on a tunnel diode loaded transmission line, Proceedings of IEEE 51 (1963), 240-249.
  • [2] A. H. Bhrawy, A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222 (2013), 255-264.
  • [3] D. E. Jackson, Error estimates for the semidiscrete Galerkin approximations of the Fitzhugh-Nagumo equations, Appl. Math. Comput., 50 (1992), 93-114.
  • [4] D. G. Aronson, H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 30 (1978), 33-76.
  • [5] F. Wu, D. Li, J. Wen, J. Duan, Stability and convergence of compact finite difference method for parabolic problems with delay, Appl. Math. and Comp., 322 (2018), 129-139.
  • [6] H. Li, Y. Guo, New exact solutions to the Fitzhugh-Nagumo equation, Appl. Math. Comput., 180 (2006), 524-528.
  • [7] H. Triki, A.-M. Wazwaz, On soliton solutions for the Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Model., 37 (2013) 3821-3828.
  • [8] J. Nagumo, S. Yoshizawa, S. Arimoto, Bistable trunsmission lines, Transactions on IEEE Circuit Theory, 12 (1965) 400-412.
  • [9] M. Dehghan, J. M. Heris, A. Saadatmandi, Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses, Math. Methods Appl. Sci., (2010)
  • [10] M. Shih, E. Momoniat, F. M. Mahomed, Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation, J. Math. Phys., 46 (2005), (023503).
  • [11] M. C. Nucci, P. A. Clarkson, The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh- Nagumo equation, Phys Lett. A, 164 (1992), 49-56.
  • [12] P. G. Dlamini and M. Khumalo, A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations, Open Math., 15 (2017), 1450-1462.
  • [13] R. A. Van Gorder, A variational formulation of the Ngumo reaction-diffusion equation and the Nagumo telegraph equation, Nonlinear Anal. Real World Appl., 11 (2010), 2957-2962.
  • [14] RK. Mohanty, D. Weizhong, L. Donn, Operator compact method of accuracy two time in time and four in space for the solution of time dependent Burgers-Huxley equation, Numer Algor, 70 (2015), 591-605.
  • [15] S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Appl. Math. Model., 32 (2008), 2706-2714.
  • [16] SK. Lele, Compact finite difference schemes with Spectral-like Resolution, Journal of Computational Physics, 103 (1992), 16-42.
  • [17] T. Wang, J. Jiang, H. Wang, W. Xu, An efficient and conservative compact finite difference scheme for the coupled Gross-Pitaevskii equations describing spin-1 Bose-Einstein condensate, Appl. Math. and Comp., 323 (2018), 164-181.
  • [18] T. Wu, R. Xu, An optimal compact sixth-order finite difference scheme for the Helmholtz equation, Comp. Math. and Appl., (2018).

Compact Finite Differences Method for FitzHugh-Nagumo Equation

Year 2019, , 180 - 187, 26.12.2019
https://doi.org/10.32323/ujma.561873

Abstract

In this paper, we developed the compact finite differences method to find approximate solutions for the FitzHugh-Nagumo (F-N) equations. To the best of our knowledge, until now there is no compact finite difference solutions have been reported for the FitzHugh-Nagumo equation arising in gene propagation and model. We have given numerical example to demonstrate the validity and applicability.

References

  • [1] A. C. Scott, Neunstor propagation on a tunnel diode loaded transmission line, Proceedings of IEEE 51 (1963), 240-249.
  • [2] A. H. Bhrawy, A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222 (2013), 255-264.
  • [3] D. E. Jackson, Error estimates for the semidiscrete Galerkin approximations of the Fitzhugh-Nagumo equations, Appl. Math. Comput., 50 (1992), 93-114.
  • [4] D. G. Aronson, H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 30 (1978), 33-76.
  • [5] F. Wu, D. Li, J. Wen, J. Duan, Stability and convergence of compact finite difference method for parabolic problems with delay, Appl. Math. and Comp., 322 (2018), 129-139.
  • [6] H. Li, Y. Guo, New exact solutions to the Fitzhugh-Nagumo equation, Appl. Math. Comput., 180 (2006), 524-528.
  • [7] H. Triki, A.-M. Wazwaz, On soliton solutions for the Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Model., 37 (2013) 3821-3828.
  • [8] J. Nagumo, S. Yoshizawa, S. Arimoto, Bistable trunsmission lines, Transactions on IEEE Circuit Theory, 12 (1965) 400-412.
  • [9] M. Dehghan, J. M. Heris, A. Saadatmandi, Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses, Math. Methods Appl. Sci., (2010)
  • [10] M. Shih, E. Momoniat, F. M. Mahomed, Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation, J. Math. Phys., 46 (2005), (023503).
  • [11] M. C. Nucci, P. A. Clarkson, The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh- Nagumo equation, Phys Lett. A, 164 (1992), 49-56.
  • [12] P. G. Dlamini and M. Khumalo, A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations, Open Math., 15 (2017), 1450-1462.
  • [13] R. A. Van Gorder, A variational formulation of the Ngumo reaction-diffusion equation and the Nagumo telegraph equation, Nonlinear Anal. Real World Appl., 11 (2010), 2957-2962.
  • [14] RK. Mohanty, D. Weizhong, L. Donn, Operator compact method of accuracy two time in time and four in space for the solution of time dependent Burgers-Huxley equation, Numer Algor, 70 (2015), 591-605.
  • [15] S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Appl. Math. Model., 32 (2008), 2706-2714.
  • [16] SK. Lele, Compact finite difference schemes with Spectral-like Resolution, Journal of Computational Physics, 103 (1992), 16-42.
  • [17] T. Wang, J. Jiang, H. Wang, W. Xu, An efficient and conservative compact finite difference scheme for the coupled Gross-Pitaevskii equations describing spin-1 Bose-Einstein condensate, Appl. Math. and Comp., 323 (2018), 164-181.
  • [18] T. Wu, R. Xu, An optimal compact sixth-order finite difference scheme for the Helmholtz equation, Comp. Math. and Appl., (2018).
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Canan Akkoyunlu 0000-0002-0999-6997

Publication Date December 26, 2019
Submission Date May 8, 2019
Acceptance Date October 15, 2019
Published in Issue Year 2019

Cite

APA Akkoyunlu, C. (2019). Compact Finite Differences Method for FitzHugh-Nagumo Equation. Universal Journal of Mathematics and Applications, 2(4), 180-187. https://doi.org/10.32323/ujma.561873
AMA Akkoyunlu C. Compact Finite Differences Method for FitzHugh-Nagumo Equation. Univ. J. Math. Appl. December 2019;2(4):180-187. doi:10.32323/ujma.561873
Chicago Akkoyunlu, Canan. “Compact Finite Differences Method for FitzHugh-Nagumo Equation”. Universal Journal of Mathematics and Applications 2, no. 4 (December 2019): 180-87. https://doi.org/10.32323/ujma.561873.
EndNote Akkoyunlu C (December 1, 2019) Compact Finite Differences Method for FitzHugh-Nagumo Equation. Universal Journal of Mathematics and Applications 2 4 180–187.
IEEE C. Akkoyunlu, “Compact Finite Differences Method for FitzHugh-Nagumo Equation”, Univ. J. Math. Appl., vol. 2, no. 4, pp. 180–187, 2019, doi: 10.32323/ujma.561873.
ISNAD Akkoyunlu, Canan. “Compact Finite Differences Method for FitzHugh-Nagumo Equation”. Universal Journal of Mathematics and Applications 2/4 (December 2019), 180-187. https://doi.org/10.32323/ujma.561873.
JAMA Akkoyunlu C. Compact Finite Differences Method for FitzHugh-Nagumo Equation. Univ. J. Math. Appl. 2019;2:180–187.
MLA Akkoyunlu, Canan. “Compact Finite Differences Method for FitzHugh-Nagumo Equation”. Universal Journal of Mathematics and Applications, vol. 2, no. 4, 2019, pp. 180-7, doi:10.32323/ujma.561873.
Vancouver Akkoyunlu C. Compact Finite Differences Method for FitzHugh-Nagumo Equation. Univ. J. Math. Appl. 2019;2(4):180-7.

 23181

Universal Journal of Mathematics and Applications 

29207              

Creative Commons License  The published articles in UJMA are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.