An effective numerical approach for solving a system of singularly perturbed differential–difference equations in biology and physiology

This study aims to analyze a system of time-dependent singularly perturbed differential–difference equations characterized by small shifts, particularly relevant in neuroscience. We employ Taylor series expansions for approximation to manage the equations’ delay and advance parameters. This method allows for a detailed examination of the complex dynamics, ensuring accuracy and feasibility. To discretize the problem, we use the Crank–Nicolson finite difference method in the time direction on a uniform mesh, combined with a Shishkin-type mesh and cubic $B$-spline collocation method in the spatial direction. This integrated approach leverages the strengths of each discretization technique in their respective dimensions, ensuring a robust and highly precise numerical solution. We thoroughly investigate the convergence of our proposed method, demonstrating its nearly second-order accuracy. Numerical experiments on two examples confirm its efficiency and effectiveness in practical applications. Furthermore, this approach is highly adaptable and can be implemented seamlessly in any programming language.