Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions

Abstract

The generalization of the classical FitzHugh–Nagumo model provides a more accurate description of the physical phenomena of neurons by incorporating both nonlinearity and fractional derivatives. In this article, we present a numerical method for solving the time-fractional FitzHugh–Nagumo equation (TFFNE) in the sense of the Atangana–Baleanu fractional derivative using B-spline functions. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while θ\theta -weighted scheme is used to discretize the space directions. The efficiency of the scheme is demonstrated through numerical results and rate of convergence. The convergence order and error norms are studied at different values of the noninteger parameter, temporal directions, and spatial directions. Finally, the effectiveness of the proposed methodology is examined through the analysis of three applications.

Keywords

• computational physics
• nonlinear time-fractional fitzhugh–nagumo equation
• atangana–baleanu fractional derivative
• cubic b-spline functions
• finite difference formulation
• convergence and stability