An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

Pius W.M. Chin, Claude R.B. Moutsinga*, Khadijo R. Adem

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to analyze because of the presence of the nonlinearity term, which is always very difficult to approximate. This paper aims to design a reliable scheme that consists of a combination of the nonstandard finite difference in time method, the Galerkin method and the compactness methods in space variables. This method is used to show that the solution of the problem exists uniquely. The a priori estimate from the existence process is applied to the scheme to show that the numerical solution from the scheme converges optimally in the (Formula presented.) as well as the (Formula presented.) norms. We proceed to show that the scheme preserves the decaying properties of the exact solution. Numerical experiments are introduced with a chosen example to validate the proposed theory.

Original languageEnglish
Article number867
JournalMathematics
Volume12
Issue number6
DOIs
Publication statusPublished - Mar 2024

Keywords

  • Galerkin method
  • Huxley equations
  • nonlinear equation
  • nonstandard finite difference method
  • optimal rate of convergence

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