Numerical Approach of a Nonlinear Forward-backward Equation

AUTHOR(S):

TITLE

Numerical Approach of a Nonlinear Forward-backward Equation

PDF BIBTEX

ABSTRACT

This paper contains a computational approximation for the solution of a forward-backward differential equation that models nerve conduction in a myelinated axon. We look for a solution of an equation defined in R, which tends to known values at ±∞. Extending the approach introduced in [5, 10, 6] for linear case, a numerical method for the solution of problem, is adapted to non linear case using a continuation method 1.

KEYWORDS

Mixed-type functional differential equations, Non linear Boundary Value Problem, Nerve Conduction, Collocation, Continuation Method, Numerical Approximation, Method of Steps.

REFERENCES

[1] K.A. Abell, C.E. Elmer, A.R. Humphries, E.S. VanVleck, Computation of mixed type functional differential boundary value problems, SIADS - Siam J App Dyn Syst 4, 3, 2005, pp. 755-781. [1] K.A. Abell, C.E. Elmer, A.R. Humphries, E.S. VanVleck, Computation of mixed type functional differential boundary value problems, SIADS - Siam J App Dyn Syst 4, 3, 2005, pp. 755-781.

[2] J. Bell, Behaviour of some models of myelinated axons, IMA J Math App in Med and Biol 1, 2, 1984, pp. 149-167.

[3] H. Chi, J. Bell, B. Hassard, Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction, J Math Biol 24, 1986, pp. 583-601.

[4] V. Iakovleva and C. Vanegas, On the Solution of differential equations withe delayed and advanced arguments, Elect. J Diff Eq Conf 13, 2005, pp. 57-63.

[5] P.M. Lima, M.F. Teodoro, N.J. Ford, P.M. Lumb, Analytical and Numerical Investigation of Mixed Type Functional Differential Equations, J Comp App Math 234, 9, 2010, pp. 2826-2837.

[6] P.M. Lima, M.F. Teodoro, N.J. Ford, P.M. Lumb, Finite Element Solution of a Linear Mixed-Type Functional Differential Equation, Num Alg 55, 2, 2010, pp. 301-320.

[7] P.M. Lima, M.F. Teodoro, N.J. Ford, P.M. Lumb, in: S. Pinelas et al. (Ed.) Analysis and Computational Approximation of a Forward-Backward Equation Arising in Nerve Conduction, ICCEAD 2011, Springer Proc Math Stat NY 2013 47, pp. 475-485.

[8] L.S. Pontryagin, R.V. Gamkreledze, E.F. Mischenko, The Mathematical Theory of Optimal Process, Interscience, New York, 1962.

[9] A. Rustichini, Functional differential equations of mixed type: The linear autonomous case, J Dyn Diff Eq 1, 2, 1989, pp. 121–143.

[10] M.F. Teodoro, P.M. Lima, N.J. Ford, P.M. Lumb, New approach to the numerical solution of forwardbackward equations, Front Math China 4, 1, 2009, pp. 155-168.

[11] M.F. Teodoro, P.M. Lima, N.J. Ford, P.M. Lumb, Numerical modelling of a functional differential equation with deviating arguments using a collocation method, In: T. E. Simos (Ed.), ICNAAM 2008, AIP Proc, Melville-NY, 2008 48, pp. 533-537.

Cite this paper

M. Filomena Teodoro. (2016) Numerical Approach of a Nonlinear Forward-backward Equation. International Journal of Mathematical and Computational Methods, 1, 75-78