TY - JOUR

T1 - An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

AU - Chin, Pius W.M.

AU - Moutsinga, Claude R.B.

AU - Adem, Khadijo R.

N1 - Publisher Copyright:
© 2024 by the authors.

PY - 2024/3

Y1 - 2024/3

N2 - 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.

AB - 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.

KW - Galerkin method

KW - Huxley equations

KW - nonlinear equation

KW - nonstandard finite difference method

KW - optimal rate of convergence

UR - http://www.scopus.com/inward/record.url?scp=85188942389&partnerID=8YFLogxK

U2 - 10.3390/math12060867

DO - 10.3390/math12060867

M3 - Article

AN - SCOPUS:85188942389

SN - 2227-7390

VL - 12

JO - Mathematics

JF - Mathematics

IS - 6

M1 - 867

ER -