Estimates for singularly perturbed parabolic PDEs with spatial delay

Pradyut Hazarika; Avijit Das

doi:10.3934/mfc.2026010

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2026, Volume 12: 139-157. Doi: 10.3934/mfc.2026010

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Estimates for singularly perturbed parabolic PDEs with spatial delay

Pradyut Hazarika and
Avijit Das^,

Department of Mathematics, National Institute of Technology Silchar, Assam 788010, India

^*Corresponding author: Avijit Das
^*Corresponding author: Avijit Das

Received: December 30, 2025

Revised: February 18, 2026

Published online: March 16, 2026

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Abstract

In this article, the analytical bounds of the solution as well as its derivatives of the singularly perturbed parabolic delay differential equation are analyzed. The solution exhibits both a boundary layer and an interior layer due to the existence of perturbation and delay effect on the equation. Consequently, the solution is decomposed into regular and singular components, and bounds are established for both components. For the regular component, bounds for mixed derivatives are obtained directly, while for the singular component, bounds for spatial, temporal, and mixed derivatives are presented separately. The results simultaneously illustrate the effect of the small perturbation parameter and the space delay parameter itself on the layers while providing a theoretical basis for accurate numerical approximation.

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Mathematics Subject Classification: Primary: 65N30, 65N15, 35J35; Secondary: 35J40.

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Figure 1. Numerical solution $u(x, t)$ over the domain $0 \le x \le 2, \; 0 \le t \le 1$ when $ \epsilon = 10^{-2} $

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Figure 2. Numerical solution $u(x, t)$ over the domain $0 \le x \le 2, \; 0 \le t \le 1$ when $ \epsilon = 1 $

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