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 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.

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.   

scholarly journals Note on the Sensitivity of Simulated Solutions of the Nonlinear Singularly Perturbed Dynamical System on the Used Different Rewriting into a System of First Order Differential Equations

2016 ◽  
Vol 24 (39) ◽  
pp. 51-55
Author(s):  
Vladimír Liška ◽  
Zuzana Šútova ◽  
Dušan Pavliak
Keyword(s):  
Dynamical System ◽  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Numerical Scheme ◽  
Singularly Perturbed ◽  
First Order ◽  
First Order Differential Equations

Abstract In this paper we analyze the sensitivity of solutions to a nonlinear singularly perturbed dynamical system based on different rewriting into a System of the First Order Differential Equations to a numerical scheme. Numerical simulations of the solutions use numerical methods implemented in MATLAB.

Application of a Hybrid Method to the Solution of the Nonlinear Burgers’ Equation

2003 ◽  
Vol 70 (6) ◽  
pp. 926-929 ◽  
Author(s):  
Bor-Lih Kuo ◽  
Chao-Kuang Chen
Keyword(s):  
Reynolds Number ◽  
Numerical Methods ◽  
Finite Difference ◽  
Hybrid Method ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Techniques ◽  
Good Agreement

This paper presents the use of a hybrid method which combines differential transformation and finite difference approximation techniques in the solution of the nonlinear Burgers’ equation for various values of Reynolds number including high values. In order to demonstrate the accuracy and validity of the proposed method, it is used to solve several examples of Burgers’ equation, with each example having different initial conditions and boundary conditions. It is found that the results obtained are in good agreement with the analytical solutions, and that the results are more accurate than those provided by other approximate numerical methods.

Laplace-Transform Finite-Difference and Quasistationary Solution Method for Water-Injection/Falloff Tests

SPE Journal ◽  
2013 ◽  
Vol 19 (03) ◽  
pp. 398-409 ◽  
Author(s):  
Azeb D. Habte ◽  
Mustafa Onur
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Water Injection ◽  
Time Discretization ◽  
Finite Difference Approximation ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Phase Flow ◽  
Slightly Compressible ◽  
Oil Water

Summary In this work, we present a method for efficiently and accurately simulating the pressure-transient behavior of oil/water flow associated with water-injection/falloff tests. The method uses the Laplace-transform finite-difference (LTFD) method coupled with the well-known Buckley-Leverett frontal-advance formula to solve the radial diffusivity equation describing slightly compressible oil/water two-phase flow. The method is semianalytical in time, and as a result, the issue of time discretization in the finite-difference approximation method is eliminated. Thus, stability and convergence problems caused by time discretization are avoided. Two approaches are presented and compared in terms of accuracy for simulating the tests with multiple-rate injection and falloff periods: One is based on solving the initial-boundary-value (IVB) problem with the initial condition attained from the end of the previous flow period, and the other is based on the conventional superposition on the basis of the single-phase flow of a slightly compressible fluid. The former is shown to always provide a more accurate and efficient solution. The method is quite general in that it allows one to incorporate the effect of wellbore storage and thick-skin and finite outer-boundary conditions. The accuracy of the method was evaluated by considering various synthetic test cases with favorable and unfavorable mobility ratios and by comparing the pressure and pressure-derivative signatures with a commercial black-oil simulator, and an excellent agreement was seen.

Solving Fully Three-Dimensional Microscale Dual Phase Lag Problem Using Mixed-Collocation, Finite Difference Discretization

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Alaeddin Malek ◽  
Zahra Kalateh Bojdi ◽  
Parisa Nuri Niled Golbarg
Keyword(s):  
Finite Difference ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Finite Difference Approximation ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Phase Lag ◽  
Dual Phase ◽  
Linear System Of Equations ◽  
Finite Difference Discretization

In the present work, we investigate laser heating of nanoscale thin-films irradiated in three dimensions using the dual phase lag (DPL) model. A numerical solution based on mixed-collocation, finite difference method has been employed to solve the DPL heat conduction equation. Direct substitution in the model transforms the differential equation into a linear system of equations in which related system is solved directly without preconditioning. Consistency, stability, and convergence of the proposed method based on a mixed-collocation, finite difference approximation are proved, and numerical results are presented. The general form of matrices and their corresponding eigenvalues are presented.

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