Exponentially Fitted Finite Difference Method for Time-Fractional Singularly Perturbed Convection-Diffusion Problems

Abstract

In this thesis, uniformly convergent numerical method is presented for time-fractional singularly per turbed parabolic convection-diffusion problem with variable coefficients. The time-fractional deriva tive is considered in the Caputo sense and Crank-Nicholson technique is applied to discretize the temporal variable uniformly. Then, a uniformly convergent numerical method is developed along the spatial direction by applying a finite difference method. To regulate the effect of the singular perturbation parameter on the solution behavior, artificial viscosity σ(x,ε) is introduced. The ε uniform convergence of the proposed method rigorously proved and shown to be accurate of order O (∆t)2+ h2 ε+h . The applicability of the proposed method is validated by using model examples. The numerical results obtained are in agreement with the theoretical expectations and the proposed method is more accurate than some existing method in the literature.