R. Nageshwar Rao

A finite difference method for singularly perturbed differential-difference equations arising from a model of neuronal variability

Abstract In this paper a finite difference method is presented for singularly perturbed differential-difference equations with small shifts of mixed type (i.e., terms containing both negative shift and positive shift). Similar boundary value problems are associated with expected first exit time problems of the membrane potential in the models for the neuron. To handle the negative and positive shift terms, we construct a special type of mesh, so that the terms containing shift lie on nodal points after discretization. The proposed finite difference method works nicely when the shift parameters are smaller or bigger to perturbation parameter. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameters on the boundary layer behavior or oscillatory behavior of the solution of the problem.

Numerical Solution of Second Order Singularly Perturbed Delay Differential Equations via Cubic Spline in Tension

This paper deals with the singularly perturbed boundary value problem for a second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. A difference scheme on a uniform mesh is accomplished by the method based on cubic spline in tension. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results.

An Exponentially Fitted Spline Method for Second-Order Singularly Perturbed Delay Differential Equations

This paper deals with the singularly perturbed boundary value problem for the second-order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. A fitted numerical scheme has been developed to solve the boundary value problem. The difference scheme which is shown to converge to the continuous solution uniformly with respect to the perturbation parameter is illustrated with numerical results.

A numerical scheme for singularly perturbed reaction-diffusion problems with a negative shift via numerov method

In this paper, we consider a boundary value problem for a singularly perturbed delay differential equation of reaction-diffusion type. We construct an exponentially fitted numerical method using Numerov finite difference scheme, which resolves not only the boundary layers but also the interior layers arising from the delay term. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

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 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

A modified Numerov method for solving singularly perturbed differential–difference equations arising in science and engineering

O(\Delta t+h^{2})

A Fourth Order Finite Difference Method for Singularly Perturbed Differential-Difference Equations

O(Δt+h2) where