Spline-based approximation for two-parameter singularly perturbed systems with large time delay with applications in science and engineering

A numerical method, used for solving two-parameter singularly perturbed systems with large time delays which are commonly encountered in a variety of scientific and engineering applications, is constructed and analyzed in this work. To attain high precision and stability, the suggested approach combines the cubic spline interpolation with the Crank–Nicolson method. The two-parameter nature of the problem introduces significant challenges due to the presence of boundary layers and the interaction of small perturbation parameters with large time delays. The technique successfully captures the abrupt changes and steep slopes present in such systems by using cubic splines. According to theoretical analysis, the suggested scheme significantly outperforms current techniques by achieving second-order convergence in both spatial and temporal variables. The theoretical conclusions are corroborated in practice by numerical experiments, which show the method’s robustness and its efficiency. Discussions of applications in fluid dynamics, heat transport and control systems reflect clearly the approach’s applicability in real-world scenarios. The findings show that the Crank–Nicolson method together with the cubic spline approach is an effective technique for precisely resolving two-parameter singularly perturbed systems with significant time delays, providing a solid foundation for practical scientific and engineering problems.