Exact difference approach on the Shishkin mesh for solving time fractional singularly perturbed parabolic PDE

A novel approach has been introduced to address time-fractional singularly perturbed parabolic partial differential equations. This method utilizes the L1-Caputo finite difference technique to approximate the fractional derivative term and employs an exact difference scheme for spatial derivative approximation on a Shishkin mesh. Conventional numerical methods in FDM, FEM and Collocation methods relying on uniform meshes often fail to provide accurate solutions due to the presence of boundary layers. The proposed method overcomes this limitation by ensuring the discrete maximum principle, stability bounds, and uniform convergence while effectively resolving boundary layers. Numerical experiments have confirmed the effectiveness of the scheme across various perturbation parameter values and mesh sizes.