Study On Singularly Perturbed Integro-Differential Parabolic Equation With Space Delay
Date
2025
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Publisher
Department of Mathematics, NITS
Abstract
This thesis presents a comprehensive numerical study of a class of singularly perturbed delay differential equations (SP-DDEs), which arise frequently in applications across science and engineering. The presence of both a small perturbation parameter and a delay term introduces significant challenges in analysis and computation. To address these, the solution is decomposed into regular and singular components, facilitating both theoretical and numerical treatment. Parameter-uniform bounds for these components and their derivatives are rigorously estab lished. A finite element framework is developed, beginning with the weak formulation of the problem and leading to a fully discretized numerical scheme. The associated bilinear form is shown to be coercive and continuous, thereby ensuring the stability and well-posedness of the method. A piecewise-uniform Shishkin mesh is employed to effectively resolve boundary and interior layers, enabling accurate approximations even in regions with steep gradients. The theoretical results are supported by numerical experiments, which confirm the parameter-uniform convergence and demonstrate the efficiency of the proposed method in capturing the complex layer behavior induced by singular perturbation and delay.