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<p>In this paper, we are interested in the efficient numerical resolution of one dimensional two-parameter singularly perturbed systems; for simplicity, we only give the theoretically details corresponding to the simplest case of systems with two equations. The diffusion parameters are distinct and can have a very different value; on the other hand, the convection parameter is the same for both equations. Finally, we assume that a large time delays term appears in the partial differential equation. So, the exact solution has overlapping boundary layers at both end points of the spatial interval, when the magnitude of the diffusion parameters is very different; the behavior of the boundary layers depends on the value and the ratio between the diffusion and the convection parameters. To approximate the exact solution of the continuous problem, we construct a numerical method, which combines the Crank-Nicolson method to discretize in time, which is constructed on a uniform mesh, and a type of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{B}\)</EquationSource> </InlineEquation>-splines to dscretize in space, which are defined on a special nonuniform Shishkin mesh. We prove that the fully discrete scheme is a uniformly convergent method, having second order in time and almost second order in space. From a practical point of view, higher order numerical methods are convenient because they permit to obtain good numerical approximations with a small increase of the computational cost. To corroborate in practice the good properties of the numerical method, some test problems are solved; from the numerical results obtained for these examples, clearly follows both the efficiency and the order of uniform convergence of the numerical method, in agreement with the theoretical results.</p>

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An effcient spline method to solve two-parameter 1D parabolic singularly perturbed systems with time delay and different diffusion parameters

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月19日

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  • Parvin Kumari,
  • Carmelo Clavero

摘要

In this paper, we are interested in the efficient numerical resolution of one dimensional two-parameter singularly perturbed systems; for simplicity, we only give the theoretically details corresponding to the simplest case of systems with two equations. The diffusion parameters are distinct and can have a very different value; on the other hand, the convection parameter is the same for both equations. Finally, we assume that a large time delays term appears in the partial differential equation. So, the exact solution has overlapping boundary layers at both end points of the spatial interval, when the magnitude of the diffusion parameters is very different; the behavior of the boundary layers depends on the value and the ratio between the diffusion and the convection parameters. To approximate the exact solution of the continuous problem, we construct a numerical method, which combines the Crank-Nicolson method to discretize in time, which is constructed on a uniform mesh, and a type of \(\mathcal{B}\) -splines to dscretize in space, which are defined on a special nonuniform Shishkin mesh. We prove that the fully discrete scheme is a uniformly convergent method, having second order in time and almost second order in space. From a practical point of view, higher order numerical methods are convenient because they permit to obtain good numerical approximations with a small increase of the computational cost. To corroborate in practice the good properties of the numerical method, some test problems are solved; from the numerical results obtained for these examples, clearly follows both the efficiency and the order of uniform convergence of the numerical method, in agreement with the theoretical results.