scholarly journals Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Worku Tilahun Aniley ◽  
Gemechis File Duressa
Keyword(s):  
Time Delay ◽  
Numerical Test ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Parabolic Differential Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Derivative Term ◽  
The Stability ◽  
The Right

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

scholarly journals An adaptive mesh method for time dependent singularly perturbed differential-difference equations

2019 ◽  
Vol 8 (1) ◽  
pp. 328-339
Author(s):  
P. Pramod Chakravarthy ◽  
Kamalesh Kumar
Keyword(s):  
Prior Information ◽  
Grid Method ◽  
Adaptive Mesh ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Mesh Method

Abstract In this paper, a time dependent singularly perturbed differential-difference convection-diffusion equation is solved numerically by using an adaptive grid method. Similar boundary value problems arise in computational neuroscience in determination of the behaviour of a neuron to random synaptic inputs. The mesh is constructed adaptively by using the concept of entorpy function. In the proposed scheme, prior information of the width and position of the layers are not required. The method is independent of perturbation parameter ε and gives us an oscillation free solution, without any user introduced parameters. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

scholarly journals Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Gemechis File Duressa
Keyword(s):  
Boundary Layer ◽  
Hybrid Method ◽  
Singularly Perturbed ◽  
Nonoscillatory Solutions ◽  
Convection Diffusion ◽  
Layer Region ◽  
The Stability ◽  
Regular Region ◽  
The Right ◽  
Diffusion Problems

This paper deals with numerical treatment of nonstationary singularly perturbed delay convection-diffusion problems. The solution of the considered problem exhibits boundary layer on the right side of the spatial domain. To approximate the term with the delay, Taylor’s series approximation is used. The resulting time-dependent singularly perturbed convection-diffusion problems are solved using Crank-Nicolson method for temporal discretization and hybrid method for spatial discretization. The hybrid method is designed using mid-point upwind in regular region with central finite difference in boundary layer region on piecewise uniform Shishkin mesh. Numerical examples are used to validate the theoretical findings and analysis of the proposed scheme. The present method gives accurate and nonoscillatory solutions in regular and boundary layer regions of the solution domain. The stability and the uniform convergence of the scheme are proved. The scheme converges uniformly with almost second-order rate of convergence.

scholarly journals Fitted numerical scheme for singularly perturbed convection-diffusion reaction problems involving delays

2021 ◽  
pp. 6-6
Author(s):  
Mesfin Woldaregay ◽  
Worku Aniley ◽  
Gemechis Duressa
Keyword(s):  
Numerical Scheme ◽  
Numerical Test ◽  
Singularly Perturbed ◽  
Diffusion Reaction ◽  
Convection Diffusion ◽  
Solution Methods ◽  
Solution Bound ◽  
The Stability ◽  
Delay Differential ◽  
Exponential Boundary Layer

This paper deals with solution methods for singularly perturbed delay differential equations having delay on the convection and reaction terms. The considered problem exhibits an exponential boundary layer on the left or right side of the domain. The terms with the delay are treated using Taylor?s series approximation and the resulting singularly perturbed boundary value problem is solved using a specially designed exponentially finite difference method. The stability of the scheme is analysed and investigated using a comparison principle and solution bound. The formulated scheme converges uniformly with linear order of convergence. The theoretical findings are validated using three numerical test examples.

scholarly journals Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Gemechis File Duressa
Keyword(s):  
Difference Equations ◽  
Numerical Test ◽  
Perturbation Parameter ◽  
Richardson Extrapolation ◽  
Singularly Perturbed ◽  
Derivative Term ◽  
Uniformly Convergent ◽  
Solution Bound ◽  
Derivatives Of ◽  
Fitted Operator

This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter ε taking arbitrary values in the interval 0,1 . For small values of ε , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test examples for different values of ε and mesh number N .

scholarly journals Robust numerical method for singularly perturbed parabolic differential equations with negative shifts

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2383-2401
Author(s):  
Mesfin Woldaregay ◽  
Gemechis Duressa
Keyword(s):  
Differential Equations ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Spatial Discretization ◽  
Parabolic Differential Equations ◽  
First Order ◽  
Temporal Discretization ◽  
Solution Bound ◽  
The Stability

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having delay on the zeroth and first order derivative terms. The solution of the considered problem exhibits boundary layer behaviour as the perturbation parameter tends to zero. The equation is solved using ?-method in temporal discretization and exponentially fitted finite difference method in spatial discretization. The stability of the scheme is proved by using solution bound technique by constructing barrier functions. The parameter uniform convergence analysis of the scheme is carried out and it is shown to be accurate of order O(N-2/N-1+c?+(?t)2) for the case ?= 1/2, where N is the number of mesh points in spatial discretization and ?t is the mesh size in temporal discretization. Numerical examples are considered for validating the theoretical analysis of the scheme.

scholarly journals Novel finite point approach for solving time-fractional convection-dominated diffusion equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xiaomin Liu ◽  
Muhammad Abbas ◽  
Honghong Yang ◽  
Xinqiang Qin ◽  
Tahir Nazir
Keyword(s):  
Variable Coefficients ◽  
Numerical Dispersion ◽  
Discrete Equation ◽  
Finite Point ◽  
Spatial Direction ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Tailored Finite Point Method ◽  
L1 Approximation ◽  
The Stability

AbstractIn this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.

Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Subal Ranjan Sahu ◽  
Jugal Mohapatra
Keyword(s):  
Time Delay ◽  
Numerical Study ◽  
Singularly Perturbed ◽  
Parabolic Differential Equations ◽  
Small Space ◽  
First Order ◽  
Taylor Series Approximation ◽  
Backward Euler ◽  
Differential Difference Equation ◽  
Negative Space

Abstract A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov–Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

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