Abstract This study explores time‐dependent singularly perturbed equations, which are progressed by boundary layers and steep gradients that are often challenging to resolve accurately using conventional numerical methods. These singularities arise in various physical systems—such as heat transfer, fluid dynamics, and chemical reactions—where rapid spatial or temporal displacement makes numerical simulation difficult. To resolve these phenomena adequately, we develop a backward‐time integration scheme combined with a weighted essentially nonoscillatory and central‐space difference discretization. The adaptation of an r‐adaptive equidistributed mesh method is particularly advantageous for responding to the stiffness associated with boundary layer regions. This adaptive technique serves as a robust framework for temporal integration and ensures second‐order spatial convergence even in the presence of sharp gradients. Offering improved visualization and physical interpretation of the underlying processes through numerical experiments demonstrates that the proposed method is not only stable but also efficiently resolves the complex structure of the boundary layers.
Sultan et al. (Thu,) studied this question.