A Fractional PDE-Based Model for Nerve Impulse Transport Solved Using a Conforming Virtual Element Method: Application to Prosthetic Implants
The main objective of this study is to present a fundamental mathematical model for nerve impulse transport, based on the underlying physical phenomena, with a straightforward application in describing the functionality of prosthetic devices. The governing equation of the resultant model is a two-di...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/6/398 |
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| Summary: | The main objective of this study is to present a fundamental mathematical model for nerve impulse transport, based on the underlying physical phenomena, with a straightforward application in describing the functionality of prosthetic devices. The governing equation of the resultant model is a two-dimensional nonlinear partial differential equation with a time-fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. novel and effective numerical approach for solving this fractional-order problem is constructed based on the virtual element method. Three basic technical building blocks form the basis of our methodology: the regularity theory related to nonlinearity, discrete maximal regularity, and a fractional variant of the Grünwald–Letnikov approximation. By utilizing these components, along with the energy projection operator, a fully discrete virtual element scheme is formulated in such a way that it ensures stability and consistency. We establish the uniqueness and existence of the approximate solution. Numerical findings confirm the convergence in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>–norm and <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>–norm on both uniform square and regular Voronoi meshes, confirming the effectiveness of the proposed model and method, and their potential to support the efficient design of sensory prosthetics. |
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| ISSN: | 2075-1680 |