A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations☆
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TL;DR: In this article, a numerical study is made for solving a class of time-dependent singularly perturbed convection-diffusion problems with retarded terms which often arise in computational neuroscience.
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About: This article is published in Applied Mathematical Modelling. The article was published on 01 Jun 2011. and is currently open access. The article focuses on the topics: Upwind scheme & Finite difference method.
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Citations
Numerical Treatment for the Class of Time Dependent Singularly Perturbed Parabolic Problems with General Shift Arguments
TL;DR: In this article, two numerical schemes for solving a class of time dependent singularly perturbed parabolic convection-diffusion problems with general shift arguments in the reaction term were proposed.
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Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments
Komal Bansal,Kapil K. Sharma +1 more
TL;DR: It is proved that the scheme is unconditionally stable and parameter uniform convergent for bigger shift arguments as well as for small shift arguments, and the performance of the method is corroborated by numerical examples.
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A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay
Devendra Kumar,Parvin Kumari +1 more
TL;DR: In this paper, a numerical scheme for a class of singularly perturbed parabolic partial differential equation with the time delay on a rectangular domain in the x-t plane is constructed.
48
Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay
TL;DR: In this paper, the authors presented a numerical scheme for second-order time-dependent singularly perturbed reaction-diffusion problem with large delay in the undifferentiated term.
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Extended cubic B-spline collocation method for singularly perturbed parabolic differential-difference equation arising in computational neuroscience.
TL;DR: A parameter uniform numerical method is presented for solving singularly perturbed parabolic differential-difference equations with small shift arguments in the reaction terms arising in computational neuroscience and is shown to be accurate of order by preserving an ε-uniform convergence.
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References
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Introduction to theoretical neurobiology
Henry C. Tuckwell
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TL;DR: Tuckwell as discussed by the authors describes the basic properties of an electrically active nerve cell and develops mathematical theories for the way neurons respond to the various stimuli they receive, including the Lapicque model, linear cable theory, and time-dependent solutions of the cable equations.
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A theoretical analysis of neuronal variability.
TL;DR: The assumptions of the model are relaxed and the effects of such experimentally found phenomena as relative refractory and supernormal periods, adaptation, potentiation, and rhythmic slow potentials are discussed.
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Some models of neuronal variability.
TL;DR: The pattern of nerve action potentials produced by unit permeability changes (quantal inputs) occurring at random is considered analytically and by computer simulation methods to approximate the variability of interspike intervals.
516
Analysis of some difference approximations for a singular perturbation problem without turning points
R. Bruce Kellogg,Alice Tsan +1 more
TL;DR: Some three point difference schemes are considered for a singular perturbation problem without turning points in this article, and bounds for the discretization error are obtained which are uniformly valid for all h and e > 0.