Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data

Parameter-uniform approximation on equidistributed meshes for singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions

This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of $N_{r} \times N_{t}$
 elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

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Finite difference scheme for singularly perturbed reaction diffusion problem of partial delay differential equation with nonlocal boundary condition

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be - uniformly convergent of , where and denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter . Some numerical results are carried out to support the theory.

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A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

In this paper, we consider a boundary value problem for a singularly perturbed delay differential equation of reaction–diffusion type. A fitted operator finite difference scheme based on Numerov’s method is constructed. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

A Novel Method for Singularly Perturbed Delay Differential Equations of Reaction-Diffusion Type

A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition

An iterative numerical method for singularly perturbed reaction-diffusion equations with negative shift

In this paper, a numerical method based on an iterative scheme is proposed for a singularly perturbed weakly coupled system of delay differential equations of convection–diffusion type. In this method the solution of the delay problem is obtained as the limit of the solutions to a sequence of the non-delay problems. The numerical solutions of the non-delay problems are obtained by applying existing finite difference scheme for the non-delay weakly coupled system of singularly perturbed equations. An error estimate is derived by using maximum principle and in supremum norm. Numerical results illustrating the theory are also included.

An Iterative Numerical Method for a Weakly Coupled System of Singularly Perturbed Convection–Diffusion Equations with Negative Shifts

A numerical method based on an iterative scheme is proposed for a system of singularly perturbed differential equations of reaction-diffusion type with negative shift term. In this method the solution of the delay problem is obtained as the limit of the solutions to a sequence of the non-delay problems. Then non-delay problems are solved by applying available finite difference scheme and finite element method in the literature. An error estimate in supremum norm is derived. Numerical experiments are carried out.

P. Avudai Selvi

Papers

A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition

An iterative numerical method for singularly perturbed reaction-diffusion equations with negative shift

An Iterative Numerical Method for a Weakly Coupled System of Singularly Perturbed Convection–Diffusion Equations with Negative Shifts

A Numerical Method for a System of Singularly Perturbed Differential Equations of Reaction-Diffusion Type with Negative Shift