Pseudospectral methods for Nagumo equation

International Journal for Numerical Methods in Biomedical Engineering

Volumn 27, Issue 4, 2011, Pages 553-561

  Dehghan, Mehdi ^a   Fakhar Izadi, Farhad ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Benjamin Ono equation; Discrete Fourier series; Leapfrog; Nagumo equation; Pseudospectral; Rational Chebyshev functions

Indexed keywords

BENJAMIN-ONO EQUATION; CHEBYSHEV FUNCTIONS; DISCRETE FOURIER SERIES; LEAPFROG; NAGUMO EQUATION; PSEUDOSPECTRAL;

FOURIER ANALYSIS; HARMONIC ANALYSIS; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; RATIONAL FUNCTIONS; RUNGE KUTTA METHODS;

FOURIER SERIES;

EID: 79953035899     PISSN: 20407939     EISSN: 20407947     Source Type: Journal    
DOI: 10.1002/cnm.1319     Document Type: Article

References (32)

1. Fitzhugh R. Impulse and physiological states in models of nerve membrane. Biophysics Journal 1961; 1:445-466.
2. Nagumo JS, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE 1962; 50:2061-2070.
3. Aronson DG, Weinberger HF. Multidimensional nonlinear diffusion arising in population genetics. Advances in Mathematics 1978; 30:33-76.
4. Scott AC. Neunstor propagation on a tunnel diode loaded transmission line. Proceedings of IEEE 1963; 51:240-249.
5. Nagumo J, Yoshizawa S, Arimoto S. Bistable transmission lines. Transactions on IEEE Circuit Theory 1965; 12:400-412.
6. Pickard WF. On the propagation of the nervous impulse down medullated and unmedullated fibers. Journal of Theoretical Biology 1966; 11:30-45.
7. Buratti RJ, Lindgren AG. Neuristor wave forms and stability by the linear approximation. Proceedings of IEEE 1968; 56:1392-1393.
8. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 1952; 117:500-544.
9. Ramos JI. Exponential methods for one-dimensional reaction-diffusion equations. Applied Mathematics and Computation 2005; 170:380-398.
10. Ogawa S, Lécot C. A quasi-random walk method for one-dimensional reaction-diffusion equations. Mathematics and Computers in Simulation 2003; 62:487-494.
11. Chen Z, Gumel AB, Mickens RE. Nonstandard discretizations of the generalized Nagumo reaction-diffusion equation. Numerical Methods for Partial Differential Equations 2003; 19:363-379.
12. Dehghan M. Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Mathematics and Computers in Simulation 2006; 71:16-30.
13. Van Gorder Robert A, Vajravelu K. Analytical and numerical solutions of the density dependent diffusion Nagumo equation. Physics Letters A 2008; 372:5152-5158.
14. Abdusalam HA. Analytic and approximate solutions for Nagumo telegraph reaction diffusion equation. Applied Mathematics and Computation 2004; 157:515-522.
15. Kaushik A, Sharma MD. Numerical analysis of a mathematical model for propagation of an electrical pulse in a neuron. Numerical Methods for Partial Differential Equation 2008; 24:1055-1079.
16. Khalifa Ahmed K, Farea Hussain A. Numerical treatment of the nerve conduction equation. International Journal of Computational Mathematics 2000; 76(2):149-158.
17. Rogers JM, McCulloch AD. A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Transactions on Biomedical Engineering 1994; 41:743-757.
18. Shih M, Momoniat E, Mahomed FM. Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation. Journal of Mathematical Physics 46: 2005; 023503(1-10).
19. Jackson DE. Error estimates for the semidiscrete Galerkin approximations of the Fitzhugh-Nagumo equations. Applied Mathematics and Computation 1992; 50:93-114.
20. Olmos D, Shizgal BD. Pseudospectral method of solution of the Fitzhugh-Nagumo equation. Mathematics and Computers in Simulation 2009; 79:2258-2278.
21. Barkley D. A model for fast computer simulation of excitable media. Physica D 1991; 49:61-70.
22. Chen Z, Guo B. Analytic solutions of the Nagumo equation. IMA Journal of Applied Mathematics 1992; 48:107-115.
23. Kawahara T, Tanaka M. Interaction of travelling fronts: an exact solution of a nonlinear diffusion equation. Physics Letters A 1983; 97:311-314.
24. James RL, Weideman JAC. Pseudospectral methods for the Benjamin-Ono equation. Advances in Computer Methods for Partial Differential Equations 1992; VII:371-377.
25. Canuto C, Hussaini MY, Quarteroni A, Zang TA. Spectral Methods in Fluid Dynamics. Springer: New York, 1988.
26. Boyd JP. Chebyshev and Fourier Spectral Methods (2nd edn). Dover: Mineola, New York, 2001.
27. Boyd JP. Spectral method using rational basis functions on an infinite interval. Journal of Computational Physics 1987; 69:112-142.
28. Wang Z-Q, Guo B-Y. A rational approximation and its applications to nonlinear partial differential equations on the whole line. Journal of Mathematical Analysis and Application 2002; 274:374-403.
29. Dehghan M, Shamsi M. Numerical solution of two-dimensional parabolic equation subject to nonstandard boundary specifications using the pseudospectral Legendre method. Numerical Methods for Partial Differential Equations 2006; 22:1255-1266.
30. Dehghan M, Saadatmandi A. A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification. Computers and Mathematics with Applications 2006; 52:933-940.
31. Dehghan M, Ramezani M. Composite spectral method for solution of the diffusion equation with specification of energy. Numerical Methods for Partial Differential Equations 2008; 24:950-959.
32. Dehghan M, Shakeri F. The numerical solution of the second Painleve equation. Numerical Methods for Partial Differential Equations 2009; 25:1238-1259.