[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,
Volume: 2 Issue: 4, 180 - 187, 26.12.2019
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.
[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).
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.