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<p>This paper investigates a system of singularly perturbed turning point problems subject to Robin-type boundary conditions, that exhibit sharp boundary and interior layers that pose significant challenges for standard numerical methods. To effectively handle these difficulties, a trigonometric cubic B-spline (TCBS) collocation approach is proposed for obtaining accurate numerical approximations. The smoothness and flexibility of the TCBS basis functions enable the method to capture the rapid variations associated with turning point behavior. A rigorous theoretical framework is developed to analyze the stability and convergence of the scheme, and it is shown that the method achieves parameter-uniform convergence of order <InlineEquation ID="IEq1"><EquationSource Format="TEX">\(\mathcal {O}(\mathcal {N}^{-2}(\ln N)^3)\)</EquationSource></InlineEquation>, which corresponds to second-order accuracy up to a logarithmic factor. The performance of the method is demonstrated through a numerical example, where the computed results are compared with existing results available in the literature. In addition, the computational efficiency of the proposed method is evaluated through the reported CPU time. The numerical results confirm the reliability, accuracy, and effectiveness of the present approach.</p>

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Parameter-uniform numerical method for a coupled system of singularly perturbed turning point problems with Robin boundary conditions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月22日

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  • Chaturya Karanam,
  • Prashanth Maroju

摘要

This paper investigates a system of singularly perturbed turning point problems subject to Robin-type boundary conditions, that exhibit sharp boundary and interior layers that pose significant challenges for standard numerical methods. To effectively handle these difficulties, a trigonometric cubic B-spline (TCBS) collocation approach is proposed for obtaining accurate numerical approximations. The smoothness and flexibility of the TCBS basis functions enable the method to capture the rapid variations associated with turning point behavior. A rigorous theoretical framework is developed to analyze the stability and convergence of the scheme, and it is shown that the method achieves parameter-uniform convergence of order \(\mathcal {O}(\mathcal {N}^{-2}(\ln N)^3)\), which corresponds to second-order accuracy up to a logarithmic factor. The performance of the method is demonstrated through a numerical example, where the computed results are compared with existing results available in the literature. In addition, the computational efficiency of the proposed method is evaluated through the reported CPU time. The numerical results confirm the reliability, accuracy, and effectiveness of the present approach.