Rational spectral collocation method for two classes of singularly perturbed reaction-diffusion equations with non-smooth data

X. Mai, W. Bian, L. Liu, H. Chen

For two classes of singularly perturbed reaction-diffusion problems with discontinuous data, this paper presents a novel high-precision numerical method. Firstly, a barycentric rational spectral collocation method with hybrid sinh transformation is designed for each problem class. The original second-kind Chebyshev points are transformed twice via the sinh function, enabling them to cluster near the boundary layers, interior layers, and discontinuity points of the domain. Subsequently, a nonlinear unconstrained optimization problem is formulated to determine the boundary layer widths in the sinh transformation. Finally, an Enhanced Arctic Puffin Optimization (EAPO) algorithm is constructed herein to solve this optimization problem, with its performance verified through 12 test functions of CEC2022. Additionally, multiple numerical experiments on singularly perturbed reaction-diffusion equations with non-smooth data are conducted, and the results demonstrate that the proposed method exhibits excellent performance in both computational efficiency and accuracy. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 19(2) (2026), pp. 29-52