Fitted-mesh cubic spline methods for layer-dominated singularly perturbed parabolic problems with non-smooth data

Document Type : Research Article

Authors

Shegaye Lema Cheru ¹

Gemechis File Duressa ²

Tariku Birabasa Mekonnen ¹

¹ Department of Mathematics, College of Natural and Computational Sciences, Wollega University, Ethiopia.

² Department of Mathematics, College of Natural and Computational Sciences, Jimma University, Ethiopia.

10.22067/ijnao.2026.96617.1784

Abstract

This paper introduces a numerical framework for addressing a two-parameter singularly perturbed parabolic problem characterized by discontinuities in both the convection coefficients and the source terms. The presence of these discontinuities alongside small perturbation parameters creates distinct boundary and interior layers, which present considerable obstacles to obtaining precise numerical approximations. To address these difficulties, a layer-adapted (Shishkin-type) spatial mesh is constructed a priori, providing enhanced resolution in regions where steep solution gradients occur. The suggested method uses the implicit Euler scheme for time integration and a cubic spline in tension for spatial discretization on a Shishkin-type mesh. A thorough analysis shows that the method achieves first-order in time and second-order up to a logarithmic factor in space when using the maximum norm, regardless of how big the perturbation parameters are. The method is especially good for modeling real-world processes like heat and mass transfer in layered media, pollutant dispersion in mixed environments, and viscous flow through composite channels, where there are often discontinuities and strong layer behavior.

Keywords

Singular perturbations

Two parameters

Discontinuous data

Uniform convergence

Boundary layer

Interior layer

Subjects

• Numerical methods for differential equations and its applications

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