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  • E-ISSN:

    2454-9584

    P-ISSN

    2454-8111

    Impact Factor 2024

    6.713

    Impact Factor 2023

    6.464

  • E-ISSN:

    2454-9584

    P-ISSN

    2454-8111

    Impact Factor 2024

    6.713

    Impact Factor 2023

    6.464

  • E-ISSN:

    2454-9584

    P-ISSN

    2454-8111

    Impact Factor 2024

    6.713

    Impact Factor 2023

    6.464

INTERNATIONAL JOURNAL OF INVENTIONS IN ENGINEERING & SCIENCE TECHNOLOGY

International Peer Reviewed (Refereed), Open Access Research Journal
(By Aryavart International University, India)

Paper Details

A Stabilized Adaptive Spectral Collocation Method for Singularly Perturbed Nonlinear Boundary Value Problems

Jwan Majid Khalaf

Continuous Education Center, University of Technology, Baghdad, Iraq

192 - 213 Vol. 12, Issue 1, Jan-Dec, 2026
Receiving Date: 2026-02-11;    Acceptance Date: 2026-03-09;    Publication Date: 2026-03-30
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http://doi.org/10.37648/ijiest.v12i01.012

Abstract

In this paper, we present a stabilized adaptive spectral collocation method. It is designed to solve singularly perturbed nonlinear boundary value problems. These problems are stiff and generate narrow boundary layers. Reduced perturbation parameters cause problems for conventional numerical discretization methods. The?presented strategy couples a selective stabilization approach together with an adaptive node refinement procedure, allowing for stability and accuracy in stiff limit cases. Stabilization is imposed at the level of the discrete operators to dampen undesirable high-frequency oscillations (spurious waves) and enhance spectral conditioning,?while adaptivity rearranges collocation nodes in an efficient way so that layer-dominated problems can be solved accurately. We?conduct a rigorous stability analysis, which shows uniform discrete stability with respect to the perturbation parameter, and convergence analysis showing exponential accuracy inpresence of adaptive refinement. Wide-spread numerical tests on benchmark nonlinear problems demonstrate that the method is able to provide high-accuracy solutions with much fewer number of degrees-of-freedom than those required?by both classical spectral-collocation and low-order methods. Results additionally demonstrate the reliability of our approach for very small?perturbations parameters and a mild sensitivity to stabilisation and adaptivity parameters. In general, the stabilization of the adaptation spectral?collocation yields an efficient numerical tool for solving singularly perturbed nonlinear boundary value problems and paves way for potential extensions to more complicated multi-scale systems.

Keywords: Singular perturbations; Spectral collocation; Adaptive refinement; Numerical stabilization; Boundary layers; Nonlinear boundary value problems

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