Numerical Computation of Fitzhugh-Nagumo Equation: A Novel Galerkin Finite Element Approach

Abstract

The key objective of this research paper is to find the numerical solution of the famous FitzHugh-Nagumo equation. The numerical scheme used here is the Galerkin finite element method (GFEM) in a simple and convenient way. Because the advantages of using GFEM are that it can be used directly without any linearization or any other restrictive assumption, it uses shape functions instead of trial functions, and it gives a polynomial at each point instead of value, so it can be used to find value at any point within the domain. First we derive the detail formulation of GFEM for this nonlinear parabolic partial differential equation. Then we solve the FitzHugh-Nagumo equation for various values of. Later, we solve another renowned Newell-Whitehead equation for the verification of the consistency of this algorithm. The results are depicted both graphically and numerically. All results are compared with the analytical solutions to show the convergence of the proposed algorithm. Those results demonstrate that our proposed algorithm works efficiently and gives a very good agreement with the exact solution. This can be applied for solving any nonlinear parabolic partial differential equation (PDE).