scholarly journals Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Gemechis File Duressa
Keyword(s):  
Singularly Perturbed ◽  
Discrete Maximum Principle ◽  
Spatial Discretization ◽  
Parabolic Pdes ◽  
Temporal Discretization ◽  
Taylor Series Approximation ◽  
Uniformly Convergent ◽  
Solution Bound ◽  
The Right ◽  
Convergence Of The Scheme

In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence O N − 1 + Δ t 2 , where N is number of subintervals in spatial discretization and Δ t is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.

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 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 .

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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