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Keywords:

  • Hodgkin Huxley equation;
  • electrical pulses;
  • singular perturbation;
  • fitted mesh;
  • uniform convergence;
  • quasi-linearization

Abstract

This article is devoted to the study of a mathematical model arising in the mathematical modeling of pulse propagation in nerve fibers. A widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations. When considered as part of a particular initial boundary value problem the equation models the electrical activity in a neuron. A small perturbation parameter ε is introduced to the highest order derivative term. The parameter if decreased, speeds up the fast variables of the model equations whereas it does not affect the slow variables. In order to formally reduce the problem to a discussion of the moment of fronts and backs we take the limit ε [RIGHTWARDS ARROW] 0. This limit is singular and is therefore the solution tends to a slowly moving solution of the limiting equation. This leads to the boundary layers located in the neighborhoods of the boundary of the domain where the solution has very steep gradient. Most of the classical methods are incapable of providing helpful information about this limiting solution. To this effort a parameter robust numerical method is constructed on a piecewise uniform fitted mesh. The method consists of standard upwind finite difference operator. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives. A parameter uniform error estimate for the numerical scheme so constructed is established in the maximum norm. It is then proven that the numerical method is unconditionally stable and provides a solution that converges to the solution of the differential equation. A set of numerical experiment is carried out in support of the predicted theory, which validates computationally the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008