Abstract
This article examines periodic Sobolev reports with a singular deviation, which causes significant difficulties in numerical approximation due to the presence of sharp or boundary layers. A stable quantitative method for the effective solution of such problems in the Bakhvalov lattice, a special grid for the deviant action of the solution, is proposed. Singularly perturbed periodic Sobolev problems create significant difficulties in numerical approximation due to the presence of sharp layers or boundary layers. Our proposed reliable numerical method for efficiently solving such problems on the Bakhvalov grid, a specialized grid, is designed to account for the singular behavior of the solution. First, an asymptotic analysis of the exact solution is performed. Then a finite difference scheme is created by applying quadrature interpolation rules to an adaptive network. The stability and convergence of the presented algorithm in a discrete maximum norm is analyzed. The results show that the proposed approach provides an accurate approximation of the solution for singular problems while maintaining computational efficiency.