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.

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.

Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Musa Cakir ◽  
Baransel Gunes
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Difference Scheme ◽  
Singularly Perturbed ◽  
Stability And Convergence ◽  
Uniform Mesh ◽  
Numerical Examples ◽  
Exponentially Fitted ◽  
The Stability ◽  
Exponentially Fitted Difference Scheme

Abstract In this study, singularly perturbed mixed integro-differential equations (SPMIDEs) are taken into account. First, the asymptotic behavior of the solution is investigated. Then, by using interpolating quadrature rules and an exponential basis function, the finite difference scheme is constructed on a uniform mesh. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Some numerical examples are solved, and numerical outcomes are obtained.

scholarly journals Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Meku Ayalew ◽  
Gashu Gadisa Kiltu ◽  
Gemechis File Duressa
Keyword(s):  
Numerical Methods ◽  
Difference Equations ◽  
Numerical Scheme ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Maximum Absolute Errors

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.

scholarly journals Robust numerical method for singularly perturbed differential equations having both large and small delay

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Habtamu Garoma Debela
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Large Delay ◽  
Content Type ◽  
Small Delay ◽  
Numerical Experimentation ◽  
The Stability

PurposeThe purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential equations having both small and large delay.Design/methodology/approachThis study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation.FindingsThe stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points.Originality/valueIn this paper, the authors consider a new governing problem having both small delay on convection term and large delay. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered.

An Efficient Galerkin BEM to Compute High Acoustic Eigenfrequencies

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Mario Durán ◽  
Jean-Claude Nédélec ◽  
Sebastián Ossandón
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
High Frequency ◽  
High Accuracy ◽  
Integral Representations ◽  
Stability And Convergence ◽  
Symmetric Matrices ◽  
Frequency Interval ◽  
Numerical Treatment ◽  
The Stability

An efficient numerical method, using integral equations, is developed to calculate precisely the acoustic eigenfrequencies and their associated eigenvectors, located in a given high frequency interval. It is currently known that the real symmetric matrices are well adapted to numerical treatment. However, we show that this is not the case when using integral representations to determine with high accuracy the spectrum of elliptic, and other related operators. Functions are evaluated only in the boundary of the domain, so very fine discretizations may be chosen to obtain high eigenfrequencies. We discuss the stability and convergence of the proposed method. Finally we show some examples.

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.

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.

scholarly journals Exponentially Fitted Initial Value Technique for Singularly Perturbed Differential-Difference Equations

2015 ◽  
Vol 127 ◽  
pp. 424-431 ◽  
Author(s):  
Lakshmi Sirisha Ch. ◽  
Y.N. Reddy
Keyword(s):  
Difference Equations ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Exponentially Fitted

scholarly journals A numerical method using Laplace-like transform and variational theory for solving time-fractional nonlinear partial differential equations with proportional delay

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Alemu Senbeta Bekela ◽  
Melisew Tefera Belachew ◽  
Getinet Alemayehu Wole
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Nonlinear Partial Differential Equations ◽  
Stability And Convergence ◽  
Test Problems ◽  
Proportional Delay ◽  
Variational Theory ◽  
Partial Differential ◽  
The Stability

Abstract Time-fractional nonlinear partial differential equations (TFNPDEs) with proportional delay are commonly used for modeling real-world phenomena like earthquake, volcanic eruption, and brain tumor dynamics. These problems are quite challenging, and the transcendental nature of the delay makes them even more difficult. Hence, the development of efficient numerical methods is open for research. In this paper, we use the concepts of Laplace-like transform and variational theory to develop a new numerical method for solving TFNPDEs with proportional delay. The stability and convergence of the method are analyzed in the Banach sense. The efficiency of the proposed method is demonstrated by solving some test problems. The numerical results show that the proposed method performs much better than some recently developed methods and enables us to obtain more accurate solutions.

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