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The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of 'Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions' (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

This article focuses on the investigation of two-dimensional elliptic singularly perturbed problems that incorporate positive and negative shifts, the solution of this class of problems may demonstrate regular/parabolic/degenerate or interior boundary layers. The goal of this article is to establish the development of numerical techniques for two-dimensional elliptic singularly perturbed problems with positive and negative shifts having regular boundary layers. The three numerical schemes are proposed to estimate the solution of this class of problems based on the fitted operator and fitted mesh finite-difference methods. The fitted operator finite difference method is analysed for convergence. The effect of shift terms on the solution behaviour is demonstrated through numerical experiments. The paper concludes by providing several numerical results that demonstrate the performance of proposed numerical schemes. [ABSTRACT FROM AUTHOR]

This paper continues the authors's study of robust numerical scheme for generalized Stein's model of neuronal variability, which is a singularly perturbed parabolic partial differential–difference equation with general values of shift arguments. The work on this class of problems is initiated in the papers [K. Bansal and K.K. Sharma, Parameter-robust numerical scheme for time dependent singularly perturbed reaction- diffusion problem with large delay, Numer. Funct. Anal. Optim. 39(2) (2018), pp. 127–154] for unit shift argument and in [K. Bansal and K.K. Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection- diffusion-reaction problems with general shift arguments, Numer. Algorithms 75(1) (2017), pp. 113–145] for general shift arguments. In the present paper, this work is further extended to the development of numerical scheme based on Mickens techniques, interpolation and θ scheme. The advantage of this work over the previous research is that it deals with the problem having general values of the shift arguments with higher order of convergence and without any constraints on the number of intervals. [ABSTRACT FROM AUTHOR]

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 (𝜀 = 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, whereMandNare 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. [ABSTRACT FROM AUTHOR]

In this article we propose an efficient numerical scheme based on a Shishkin mesh for a class of singularly perturbed parabolic convection-diffusion problems with boundary turning point and retarded arguments. The solution of the considered problem exhibit a boundary layer on the left side of the domain. The continuous problem is semidiscretized by means of backward Euler finite difference method in time to get a system of ordinary differential equations at each time level. This system of differential equations is discretized by using the standard upwind finite difference scheme on a nonuniform mesh of Shishkin type. It has been shown theoretically that the numerical solution generated by the method converges uniformly to the solution of the continuous problem with respect to the singular perturbation parameter. Numerical experiments supporting the theoretical results are given. [ABSTRACT FROM AUTHOR]