A second-order finite difference method for singularly perturbed nonlinear delay parabolic problems with periodic initial-boundary conditions

In this paper, a numerical study for the singularly perturbed nonlinear delay parabolic problems with boundary conditions are made. A finite difference method based on the mesh with adaptive points are proposed. The method employs interpolating quadrature rules containing integral remainder terms with linear basis functions ensuring a second-order accuracy rate on an adaptive mesh. The proposed method exhibits second-order convergence in the space variable, and first-order in the time variable, regardless of the perturbation parameter. Stability analysis is provided, and numerical experiments corroborate the theoretical findings. The results demonstrate the effectiveness of the proposed approach in accurately solving singularly perturbed nonlinear delay parabolic problems with boundary value conditions.