A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients - Aniley, Worku Tilahun; Duressa, Gemechis File (Frontiers in Applied Mathematics and Statistics, 2025)
This study presents an exponentially fitted finite-difference scheme for addressing singularly perturbed convection–diffusion problems involving the time-fractional derivative. The Caputo fractional derivative defines the time-fractional derivative. Then, the implicit finite-difference method is used to discretize the temporal variable in a uniform mesh discretization. To manage the effect of the perturbation parameter on the solution profile, an exponentially fitted factor is introduced into the resulting system of ordinary differential equations. Finally, on a uniform spatial domain discretization, an exponentially fitted scheme is developed using the Numerov finite-difference approach. The ε-uniform of the proposed scheme is rigorously demonstrated, confirming that it is uniformly convergent with a convergence order of O((Δt)2−α+M−1). The validity of the proposed method is illustrated through model examples. The numerical results match the theoretical predictions and demonstrate that the proposed method is more accurate than some recent existing methods.