Scholarlink

<p>This paper presents numerical methods based on non-polynomial spline functions for solving singularly perturbed parabolic partial differential equations (SPPPDEs) with small shift terms. These equations frequently arise in applications such as neural signal transmission and tumor growth modeling, where delay effects and sharp gradients pose significant computational challenges. In the numerical schemes proposed in this paper, Taylor series expansion is used for handling the shift terms, backward Euler method is used for discretizing the time domain, and a generalized Shishkin mesh for spatial discretization. This combination ensures accurate resolution of boundary and interior layers. Representative numerical experiments demonstrate the methods’ second-order convergence in both time and space, with uniform accuracy across a range of perturbation parameters (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\epsilon\)</EquationSource> </InlineEquation>). Compared to traditional finite difference methods and spline-based methods, the proposed approaches have shown an improved <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\epsilon\)</EquationSource> </InlineEquation>-uniform convergence of the solution. The maximum pointwise errors, graphs of solutions of the test problems and the error graphs further validate the method’s effectiveness in capturing layer behavior and delay effects.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Numerical Methods based on Non-Polynomial Splines for Solving Singularly Perturbed Parabolic Partial Differential Equations with Small Shifts

  • 期刊论文

  • 发表日期：

    2026年3月19日

查看全文
  • T. Prathap,
  • R. Nageshwar Rao

摘要

This paper presents numerical methods based on non-polynomial spline functions for solving singularly perturbed parabolic partial differential equations (SPPPDEs) with small shift terms. These equations frequently arise in applications such as neural signal transmission and tumor growth modeling, where delay effects and sharp gradients pose significant computational challenges. In the numerical schemes proposed in this paper, Taylor series expansion is used for handling the shift terms, backward Euler method is used for discretizing the time domain, and a generalized Shishkin mesh for spatial discretization. This combination ensures accurate resolution of boundary and interior layers. Representative numerical experiments demonstrate the methods’ second-order convergence in both time and space, with uniform accuracy across a range of perturbation parameters ( \(\epsilon\) ). Compared to traditional finite difference methods and spline-based methods, the proposed approaches have shown an improved \(\epsilon\) -uniform convergence of the solution. The maximum pointwise errors, graphs of solutions of the test problems and the error graphs further validate the method’s effectiveness in capturing layer behavior and delay effects.