Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem

The parabolic convection-diffusion-reaction problem is examined in this work, where the diffusion and convection terms are multiplied by two small parameters, respectively. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method on uniform mesh is utilized to discretize the time variables in the first step. Two-point Gaussian quadrature rule is used for further discretizing these semi-discrete problems in space, and the second order interpolation of the first derivatives is utilized. The fitting factor's value, which accounts for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by two examples, which yielded more accurate results than some other methods found in the literatures.