Uniformly convergent numerical method for two parameter singularly perturbed parabolic differential equations with discontinuous initial condition

This paper investigates a class of singularly perturbed problems with two small parameters having a jump discontinuity in the initial condition. To elucidate the nature of the singularity stemming from this discontinuity, a specific singular function is identified. By subtracting this singular function from the solution of the problem, a remainder function is obtained. This remainder function is then numerically estimated using an appropriate piece-wise uniform mesh, taking into consideration the relation between the two perturbation parameters. Through rigorous error analysis, the numerical scheme employed is demonstrated to be parameter-uniform. The numerical results obtained in this research align well with the theoretical point-wise error bounds, affirming the robustness and accuracy of the proposed method.