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<p>The study offers a substantive contribution to the numerical analysis of quasilinear singularly perturbed parabolic reaction-diffusion equations, which pose considerable analytical and computational challenges due to the interplay between nonlinearities and small perturbation parameters that induce sharp boundary layers. The significance of the work lies in the formulation of a robust and high-order numerical scheme, integrating the extrapolated Crank–Nicolson method for temporal discretization with orthogonal spline collocation on a Shishkin mesh for spatial resolution. The methodological framework is designed to ensure parameter-uniform accuracy, addressing the limitations of conventional techniques in resolving steep gradients. The principal aim is to establish a theoretically sound and computationally efficient approach, substantiated through rigorous convergence analysis and comprehensive numerical experimentation. This establishes the proposed method as a versatile and reliable tool for the simulation of complex physical phenomena in applied mathematics, physics, and engineering disciplines.</p>

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The computational methods based on OSC for quasilinear parabolic singularly perturbed problems

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Jewel Howlader,
  • Kapil K. Sharma

摘要

The study offers a substantive contribution to the numerical analysis of quasilinear singularly perturbed parabolic reaction-diffusion equations, which pose considerable analytical and computational challenges due to the interplay between nonlinearities and small perturbation parameters that induce sharp boundary layers. The significance of the work lies in the formulation of a robust and high-order numerical scheme, integrating the extrapolated Crank–Nicolson method for temporal discretization with orthogonal spline collocation on a Shishkin mesh for spatial resolution. The methodological framework is designed to ensure parameter-uniform accuracy, addressing the limitations of conventional techniques in resolving steep gradients. The principal aim is to establish a theoretically sound and computationally efficient approach, substantiated through rigorous convergence analysis and comprehensive numerical experimentation. This establishes the proposed method as a versatile and reliable tool for the simulation of complex physical phenomena in applied mathematics, physics, and engineering disciplines.