scholarly journals Solving Singularly Perturbed Differential-Difference Equations with Dual Layer using Fourth Order Numerical Method

2019 ◽  
Vol 9 (2) ◽  
pp. 3770-3773
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Numerical Method ◽  
Fourth Order ◽  
Difference Equations ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Layer Behavior ◽  
Maximum Absolute Errors

In this paper, we presented a fourth-order numerical method to solve SPDDE with the dual-layer. The answer to the problem shows dual-layer behavior. A fourth-order finite difference plan on a uniform mesh is developed. The result of the delay and also advance parameters on the boundary layer(s) has likewise been evaluated as well as represented in charts. The applicability of the planned plan is actually confirmed through executing it on model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.

scholarly journals Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Habtamu Garoma Debela ◽  
Solomon Bati Kejela ◽  
Ayana Deressa Negassa
Keyword(s):  
Numerical Method ◽  
Difference Equations ◽  
Oscillatory Behavior ◽  
Singularly Perturbed ◽  
Stability And Convergence ◽  
Uniform Mesh ◽  
Small Shift ◽  
Exponentially Fitted ◽  
Maximum Absolute Errors ◽  
The Stability

This paper presents a numerical method to solve singularly perturbed differential-difference equations. The solution of this problem exhibits layer or oscillatory behavior depending on the sign of the sum of the coefficients in reaction terms. A fourth-order exponentially fitted numerical scheme on uniform mesh is developed. The stability and convergence of the proposed method have been established. The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on four model examples. Maximum absolute errors in comparison with the other numerical experiments are tabulated to illustrate the proposed method.

Fitted Mesh Method for a Class of Singularly Perturbed Differential-Difference Equations

2015 ◽  
Vol 8 (4) ◽  
pp. 496-514 ◽  
Author(s):  
Devendra Kumar
Keyword(s):  
Boundary Layer ◽  
Small Parameter ◽  
Uniform Convergence ◽  
Difference Equations ◽  
Numerical Study ◽  
Second Order ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Small Parameters ◽  
Spline Basis

AbstractThis paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter ε and the shifts depend on the small parameter ε has been considered. The fitted-mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm. The stability and parameter-uniform convergence analysis of the proposed method have been discussed. The method has been shown to have almost second-order parameter-uniform convergence. The effect of small parameters on the boundary layer has also been discussed. To demonstrate the performance of the proposed scheme, several numerical experiments have been carried out.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

A robust numerical method for a coupled system of singularly perturbed parabolic delay problems

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mukesh Kumar ◽  
Joginder Singh ◽  
Sunil Kumar ◽  
Aakansha Aakansha
Keyword(s):  
Numerical Method ◽  
A Priori ◽  
Coupled System ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
A Priori Bounds ◽  
Central Difference ◽  
Spatial Direction ◽  
Content Type ◽  
Shishkin Meshes

Purpose The purpose of this paper is to design and analyze a robust numerical method for a coupled system of singularly perturbed parabolic delay partial differential equations (PDEs). Design/methodology/approach Some a priori bounds on the regular and layer parts of the solution and their derivatives are derived. Based on these a priori bounds, appropriate layer adapted meshes of Shishkin and generalized Shishkin types are defined in the spatial direction. After that, the problem is discretized using an implicit Euler scheme on a uniform mesh in the time direction and the central difference scheme on layer adapted meshes of Shishkin and generalized Shishkin types in the spatial direction. Findings The method is proved to be robust convergent of almost second-order in space and first-order in time. Numerical results are presented to support the theoretical error bounds. Originality/value A coupled system of singularly perturbed parabolic delay PDEs is considered and some a priori bounds are derived. A numerical method is developed for the problem, where appropriate layer adapted Shishkin and generalized Shishkin meshes are considered. Error analysis of the method is given for both Shishkin and generalized Shishkin meshes.

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