a numerical approach for singular perturbation problems with an interior layer using an adaptive spline

An adaptive spline is used in this work to deal with singularly perturbed boundary value problems with layers in the interior region. to evaluate the layer behavior in the solution, a different technique on a uniform mesh is designed by replacing the first-order derivatives with nonstandard differences in the adaptive cubic spline. a tridiagonal solver is used to solve the tridiagonal system of the difference scheme. the fourth-order convergence of the approach is established. the validity of the suggested computational method is demonstrated through numerical experiments, which are compared to other methods in the literature. layer profile is depicted in graphs.