A Fractional PDE-Based Model for Nerve Impulse Transport Solved Using a Conforming Virtual Element Method: Application to Prosthetic Implants - Zaffar Mehdi Dar; Chandru Muthusamy; Higinio Ramos (Axioms, 2025)
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 <semantics>α∈(0,1)</semantics>. 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 <semantics>L2</semantics>–norm and <semantics>H1</semantics>–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.