TURK 2025

This study addresses the solution of a first-order partial differential equation containing a small parameter that leads to boundary layer behavior, rendering classical solution methods ineffective. To tackle this, we employ asymptotic expansion techniques enhanced by boundary layer corrections. The problem involves an initial-boundary value formulation where the differential operator contains a small coefficient multiplying the time derivative. As this parameter approaches zero, sharp gradients develop near the domain boundaries. To regularize the problem, we introduce new stretched variables that separate the boundary layer dynamics from the smooth part of the solution. The solution is then expressed as a series expansion in powers of the small parameter, and each term is obtained by solving simplified subproblems that are well-posed and compatible with the original initial and boundary conditions. Special functions are used to accurately capture the rapid variations near the boundaries. The proposed method offers a systematic way to approximate solutions of singularly perturbed problems and provides insight into their structure, with potential applications in various fields where multiscale phenomena are present.