An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation
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 becaus...
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MDPI AG
2024-03-01
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| author | Pius W. M. Chin Claude R. B. Moutsinga Khadijo R. Adem |
| author_facet | Pius W. M. Chin Claude R. B. Moutsinga Khadijo R. Adem |
| author_sort | Pius W. M. Chin |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> as well as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> 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. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T18:02:21Z |
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| spelling | doaj.art-342836990dcd4b71a40993c3c901830b2024-03-27T13:53:09ZengMDPI AGMathematics2227-73902024-03-0112686710.3390/math12060867An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley EquationPius W. M. Chin0Claude R. B. Moutsinga1Khadijo R. Adem2Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, Ga-Rankuwa, Pretoria 0204, South AfricaDepartment of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, Ga-Rankuwa, Pretoria 0204, South AfricaDepartment of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, Ga-Rankuwa, Pretoria 0204, South AfricaThe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula> as well as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> 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.https://www.mdpi.com/2227-7390/12/6/867Huxley equationsnonlinear equationnonstandard finite difference methodGalerkin methodoptimal rate of convergence |
| spellingShingle | Pius W. M. Chin Claude R. B. Moutsinga Khadijo R. Adem An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation Mathematics Huxley equations nonlinear equation nonstandard finite difference method Galerkin method optimal rate of convergence |
| title | An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation |
| title_full | An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation |
| title_fullStr | An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation |
| title_full_unstemmed | An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation |
| title_short | An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation |
| title_sort | analysis of the nonstandard finite difference and galerkin methods applied to the huxley equation |
| topic | Huxley equations nonlinear equation nonstandard finite difference method Galerkin method optimal rate of convergence |
| url | https://www.mdpi.com/2227-7390/12/6/867 |
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