A Semi-Analytical Approximation for a Coupled System of Singularly Perturbed Convection–Diffusion Equations with Shifts

This paper presents a semi-analytical approach to solve a coupled system of singularly perturbed differential equations with shifts of mixed type. The solution to the problem manifests a distinctive multiscale nature, characterized by localized narrow regions where the solution undergoes exponential changes. Beyond these regions, the solutions exhibit smooth variations. We employ a factorization approach to address intricate multiscale characteristics, splitting the given coupled system into two explicit systems. The first, a degenerate system, captures the smooth solution outside boundary layers through initial value problems. The second, addressing solutions within boundary layers, utilizes stretching transformations to form a set of boundary value problems. Even though this factorization seems straightforward, the solutions obtained from these simplified systems capture the essential characteristics of the given system. The Runge–Kutta method is employed to solve the degenerate system of initial value problems, while the system of boundary value problems is solved analytically using asymptotic expansions. We establish the stability and consistency of the proposed method. The method converges uniformly with higher-order accuracy. The proposed method is easy to implement, does not require an adaptive mesh generation procedure to sustain approximation, and the estimates do not involve logarithmic terms. Numerical results and illustrations are presented for four model problems to underscore the effectiveness and potential of the presented approach.