A novel fitted mesh scheme involving Caputo–Fabrizio approach for singularly perturbed fractional-order differential equations with large negative shift

This study focuses on the formulation and analysis of a novel fitted mesh method for solving singularly perturbed time-fractional partial differential equations (SPTPDEs) with large negative shift. These equations pose significant challenges due to the combined effects of fractional-order derivative in time, small perturbation parameter, and negative shift term that lead to sharp boundary layers and potential numerical instability. In this work, the non-locality effect of the fractional–order is treated applying the Caputo–Fabrizio approach. The influence of the small perturbation parameter and large shift argument are controlled by formulating a fitted-mesh method utilizing the Crank–Nicolson approach in time and central difference method with Shishkin-type piece-wise uniform meshes in space. By these coupled techniques, the abruptly varying nature of the solution in the layer regions can be resolved effectively. We investigate and prove that the proposed numerical scheme is stable and convergent of order two in time, and order two with logarithmic factor in space. Numerical experiments are conducted to validate the theoretical findings and showcase the scheme’s efficiency and accuracy in handling the intricacies of the problem. The convergence analysis and numerical results show that the formulated scheme is a reliable tool to treat the considered class of time-fractional singularly perturbed differential equations consisting of large negative shift.