Unconditionally stable high-order scheme for the fractional generalized Burgers-Huxley equation

This study presents a high-order numerical scheme for solving the generalized fractional-order Burgers-Huxley equation, a nonlinear evolutionary PDE combining integer-order spatial derivatives with temporal fractional derivatives. The model captures essential features of reaction-diffusion-convection systems and neural pulse propagation dynamics. We develop a fourth-order compact finite difference method for spatial discretization coupled with a Grünwald-Letnikov approximation for the fractional time derivative. The resulting nonlinear system is solved using an efficient iterative algorithm. Rigorous analysis demonstrates that the proposed method is unconditionally stable and achieves O(τ³ + h⁴) convergence, where τ and h represent temporal and spatial step sizes, respectively. Numerical experiments confirm the theoretical results and illustrate the scheme's accuracy through multiple test cases, demonstrating its effectiveness in handling this class of fractional PDEs.