Fitted Numerical Methods for Singularly Perturbed One-Dimensional Parabolic Partial Differential Equations with Small Shifts Arising in the Modelling of Neuronal Variability

In this paper, we presented exponentially fitted finite difference methods for a class of time dependent singularly perturbed one-dimensional convection diffusion problems with small shifts. Similar boundary value problems arise in computational neuroscience in determination of the behavior of a neu...

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Bibliographic Details
Published in:Differential equations and dynamical systems 2019-01, Vol.27 (1-3), p.1-18
Main Authors: Nageshwar Rao, R., Pramod Chakravarthy, P.
Format: Article
Language:English
Subjects:
Boundary value problems
Computer Science
Convection
Convergence
Difference equations
Engineering
Finite difference method
Finite element method
Mathematical models
Mathematics
Mathematics and Statistics
Numerical methods
Original Research
Parameters
Partial differential equations
Taylor series
Time dependence
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Summary:In this paper, we presented exponentially fitted finite difference methods for a class of time dependent singularly perturbed one-dimensional convection diffusion problems with small shifts. Similar boundary value problems arise in computational neuroscience in determination of the behavior of a neuron to random synaptic inputs. When the shift parameters are smaller than the perturbation parameter, the shifted terms are expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable and is convergent with order O ( Δ t + h 2 ) where Δ t and h respectively the time and space step-sizes. When the shift parameters are larger than the perturbation parameter a special type of mesh is used for the space variable so that the shifts lie on the nodal points and an exponentially fitted scheme is developed. This scheme is also unconditionally stable. By means of two examples, it is shown that the proposed methods provide uniformly convergent solutions with respect to the perturbation parameter. On the basis of the numerical results, it is concluded that the present methods offer significant advantage for the linear singularly perturbed partial differential difference equations.
ISSN:0971-3514
0974-6870
DOI:10.1007/s12591-017-0363-9