scholarly journals Numerical study of singularly perturbed differential–difference equation arising in the modeling of neuronal variability

2012 ◽  
Vol 63 (1) ◽  
pp. 118-132 ◽  
Author(s):  
Pratima Rai ◽  
Kapil K. Sharma
Keyword(s):  
Difference Equation ◽  
Numerical Study ◽  
Singularly Perturbed ◽  
Differential Difference Equation

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.

Spline approximation method for singularly perturbed differential-difference equation on nonuniform grids

2020 ◽  
pp. 2150005
Author(s):  
P. Mushahary ◽  
S. R. Sahu ◽  
J. Mohapatra
Keyword(s):  
Difference Equation ◽  
Spline Approximation ◽  
Perturbation Parameter ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Nonuniform Grids ◽  
Fine Mesh ◽  
Differential Difference Equation ◽  
Perturbed Differential Equation

In this paper, a second-order singularly perturbed differential-difference equation involving mixed shifts is considered. At first, through Taylor series approximation, the original model is reduced to an equivalent singularly perturbed differential equation. Then, the model is treated by using the hybrid finite difference scheme on different types of layer adapted meshes like Shishkin mesh, Bakhvalov–Shishkin mesh and Vulanović mesh. Here, the hybrid scheme consists of a cubic spline approximation in the fine mesh region and a midpoint upwind scheme in the coarse mesh region. The error analysis is carried out and it is shown that the proposed scheme is of second-order convergence irrespective of the perturbation parameter. To display the efficacy and accuracy of the proposed scheme, some numerical experiments are presented which support the theoretical results.

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