A Petrov–Galerkin Finite Element Method for the space time fractional Fitzhugh–Nagumo equation

Abstract

The Classical Nagumo equation is a non-linear reaction diffusion equation which is modelled to analyse the transmission of nerve impulses. Its Fractional Order in the Riemann–Liouville sense simplifies the model knowing that the Fractional Calculus of arbitrary order handles better real life problems than the classical calculus. We here present a Petrov–Galerkin Finite Element Method, perturbed by the newly developed Mamadu–Njoseh Orthogonal Polynomials for the solution of this model. This work aims at determining the compatibility of the Mamadu–Njoseh polynomials as basis function for the Petrov–Galerkin Finite Element Method, and obtaining an approximate solution for the FitzHugh–Nagumo Equation combined with the Riemann–Liouville fractional calculus. Our result compared with that found in literature showed that our method converges better with minimal error to the exact solution.