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Parameter-robust numerical scheme for time-dependent singularly perturbed reaction-diffusion problem with large delay. (English) Zbl 1448.65090

Summary: This work presents the development of numerical scheme for second-order time-dependent singularly perturbed reaction-diffusion problem with large delay in the undifferentiated term. These types of problems arise frequently in many areas of science and engineering that take into consideration the effect of present situation as well as the past history of the physical system. As the characteristics of the reduced problem \((\varepsilon =0)\) corresponding to the original singularly perturbed problem considered here are parallel to the boundary of the domain this implies, parabolic layers exhibit in the solution. In this paper, we initiate the study of parabolic layers together with interior layers in the solution of singularly perturbed parabolic partial differential-difference equations due to propagation of singularity. Proposed numerical scheme comprised of finite difference scheme and piecewise uniform Shishkin mesh. The method is shown to be accurate of order \((M^{-1}+N^{-2}(\ln N)^2)\), where \(M\) and \(N\) are the number of mesh elements in time and spatial direction, respectively. Proposed numerical scheme is proved to be parameter uniform convergent in the maximum norm. Numerical experiments have been performed to show the existence of interior layer due to large state-dependent delay argument in the reaction term and to confirm the predicted theory.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K67 Singular parabolic equations
35B25 Singular perturbations in context of PDEs
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