Numerical Modeling of a Boundary Value Problem for a Singularly Perturbed Differential Equation with Two Boundary Layers Using the Spectral-Grid Method

This paper proposes a spectral-grid method based on Chebyshev polynomials of the first kind for the numerical solution of second-order singularly perturbed boundary value problems containing two boundary layers. The proposed method possesses several important advantages, including high numerical accuracy, computational efficiency in terms of the number of arithmetic operations, reduced memory requirements, accurate localization and resolution of boundary layers, and applicability to singularly perturbed boundary value problems containing one, two, or multiple boundary layers. In the proposed approach, the computational domain is partitioned into several grid elements, and the solution on each element is approximated by a truncated series of Chebyshev polynomials. Continuity conditions for the solution and its derivatives are imposed at the interfaces between adjacent elements, resulting in a system of algebraic equations for the unknown expansion coefficients. The principal advantage of the method lies in its ability to accurately localize boundary layers by appropriately selecting the lengths of the grid elements and the degrees of the approximation polynomials. Numerical experiments for a wide range of perturbation parameters are presented in the form of tables and graphical illustrations and are compared with existing results available in the literature. The obtained results demonstrate that the proposed spectral-grid method provides highly accurate numerical solutions even for very small values of the perturbation parameter while significantly reducing the maximum absolute error. The convergence of the proposed method has been theoretically established, and its convergence rate has been analyzed. The numerical results confirm the accuracy, computational efficiency, robustness, and reliability of the proposed method for solving singularly perturbed boundary value problems.