A novel exponentially fitted finite difference method for time-fractional singularly perturbed convection-diffusion problems with variable coefficients

Worku Tilahun Aniley ^1,2* Gemechis File Duressa ²

• ¹ PhD Candidate, Jimma, Ethiopia
• ² Jimma University, Jimma, Oromia, Ethiopia

This study presents an exponentially fitted finite difference scheme for addressing singularly perturbed convection-diffusion problems involving time-fractional derivative. The Caputo fractional derivative is used to define the time-fractional derivative. Then, implicit finite difference method is used to discretize the temporal variable on 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((∆t) 2-α + M -1 ). The validity of the proposed method is illustrated through model examples, with the numerical results matching theoretical predictions and showing that the proposed method is more accurate than some recent existing methods.