MESHLESS APPROACHES FOR THE NUMERICAL RESOLUTION OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH LAYERED SOLUTION STRUCTURES

Abstract

To overcome the limitations of traditional mesh-based methods that struggle to capture steep
gradients in Singularly Perturbed Boundary Value Problems (SPBVPs) with sharp boundary
and interior layer structures without costly and complex mesh refinement. We investigate point
interpolation and boundary-type meshless techniques that bypass mesh generation and flexibly
adapt to complex layer phenomena by systematically decomposing solutions into regular and
singular components, validated by numerical experiments with error and convergence analysis
and reproducible Python implementations. The framework ensures computational stability,
strong layer-resolving capabilities, and parameter-uniform convergence across all perturbation
regimes, with numerical experiments validating the accuracy, efficiency, and robustness of the
methods. Presents Meshless point interpolation and boundary-type techniques with solution
decomposition to resolve complex layer phenomena in SPBVPs without mesh generation.