Numerical Modeling of the Boundary Value Problem for a two-Parameter Singularly Perturbed Differential Equation using the Spectral-Grid Method

Abstract

The solutions of boundary value problems associated with two-parameter singularly perturbed differential equations are well known to exhibit the formation of two distinct boundary layers, typically occurring near the endpoints of the domain. The presence of these narrow regions, characterized by steep gradients in the solution, poses significant challenges for classical numerical techniques. As a result, standard finite difference, finite element, or low-order discretization methods often fail to capture the layer behavior accurately unless excessively fine meshes are employed, leading to increased computational cost and reduced efficiency. To address these difficulties, the present work proposes an efficient spectral-grid method for the numerical solution of boundary value problems involving two-parameter singular perturbations. The core idea of the proposed approach is to combine the high accuracy of spectral methods with a carefully designed grid that adapts to the boundary layer structure of the solution. By employing this spectral-grid framework, the original boundary value problem is transformed into an equivalent system of algebraic equations, which can be solved efficiently using standard linear algebra techniques. Extensive numerical experiments are carried out to assess the performance of the proposed method. The computed results are systematically compared with those available in the existing literature. These comparisons clearly demonstrate that the spectral-grid method developed in this study achieves superior accuracy while maintaining a relatively low computational cost.