Numerical approximation for singularly perturbed differential equations exhibiting significant positive shift arising in neuronal activity

This article introduces an approximation method for solving singularly perturbed differential equations with a large positive shift parameter, a problem that has received limited attention in existing research. A key challenge in this context is to estimate the solution in [Formula: see text] by using a function in the range [Formula: see text]. The proposed numerical scheme is implemented using the Gauss elimination method in MATLAB R2022a and exhibits a linear convergence rate on a uniform grid. Both theoretical and numerical analyses confirm the scheme's accuracy and reliability. To demonstrate its effectiveness, several examples are presented in tables and charts, validating the theoretical results. Compared to existing approaches, our method delivers improved accuracy with significantly lower error rates for the tested examples.