Adaptive temporal mesh equidistribution for studying the singularly perturbed delay-in-time parabolic equations

This article intends to contribute a time-dependent mesh method to analyze the singularly perturbed delay parabolic partial differential equation exhibiting the boundary layer instabilities. In engineering control systems, past population effects in ecological models, stress responses in viscoelastic materials, and past pressures or velocities in complex fluid flows are all modeled by delayed partial differential equations. In the vicinity of the boundary layer, where the velocity field exerts a major influence, the solution exhibits sharp gradients. Because these gradients are steep, to obtain accurate solution a numerical method with dynamic mesh is obliged. To compute these singularities more accurately, we imposed adaptive moving mesh method that is a time-dependent self-adaptive method by imposing the center differences in the spatial direction and implicit Euler in the temporal direction. A stability analysis for the proposed scheme is performed using the energy technique. Numerical experiments are performed with various examples, and the approximated results support theoretical findings. Six numerical examples are presented with their errors and convergence orders.