P. Pramod Chakravarthy

Method of reduction of order for solving singularly perturbed two-point boundary value problems

In this paper, a method of reduction of order is proposed for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. It is distinguished by the following fact: the original singularly perturbed boundary value problem is replaced by a pair of initial value problems. Classical fourth order Runge-Kutta method is used to solve these initial value problems. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory.

An initial-value approach for solving singularly perturbed two-point boundary value problems

In this paper an initial-value approach is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. This approach is based on the boundary layer behavior of the solution. The method is distinguished by the following fact: The given singularly perturbed two-point boundary value problem is replaced by three first order initial-value problems. Several linear and non-linear problems are solved to demonstrate the applicability of the method. It is observed that the present method approximates the exact solution very well.

An exponentially fitted finite difference method for singular perturbation problems

In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. A fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the system. The stability of the algorithm is investigated. Several linear and nonlinear problems are solved to demonstrate the applicability of the method. It is observed that the present method approximates the exact solution very well.

Numerical patching method for singularly perturbed two-point boundary value problems using cubic splines

A numerical patching method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. The method is distinguished by the following fact: The original singularly perturbed two-point boundary value problem is divided into two problems, namely inner and outer region problems. The terminal boundary condition is obtained from the solution of the reduced problem. Using general stretching transformation, a modified inner region problem is constructed. Then, both inner region problem and outer region problems are solved as two-point boundary value problems by employing cubic splines. Several linear and non-linear problems are solved to demonstrate the applicability of the method.

A seventh order numerical method for singular perturbation problems

In this paper, a seventh order numerical method is presented for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. The two-point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a seventh order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two-point boundary value problem is obtained from the theory of singular perturbations. It is used in the seventh order compact difference scheme to get a two term recurrence relation and is solved. Several linear and nonlinear singular perturbation problems have been solved and the numerical results are presented to support the theory.

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 adaptive mesh method for time dependent singularly perturbed differential-difference equations

Abstract In this paper, a time dependent singularly perturbed differential-difference convection-diffusion equation is solved numerically by using an adaptive grid method. Similar boundary value problems arise in computational neuroscience in determination of the behaviour of a neuron to random synaptic inputs. The mesh is constructed adaptively by using the concept of entorpy function. In the proposed scheme, prior information of the width and position of the layers are not required. The method is independent of perturbation parameter ε and gives us an oscillation free solution, without any user introduced parameters. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

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.