Numerical integration of singularly perturbed delay differential equations using exponential integrating factor

In this paper, we proposed a numerical integration technique with exponential integrating factor for the solution of singularly perturbed differential-difference equations with negative shift, namely the delay differential equation, with layer behaviour. First, the negative shift in the differentiated term is approximated by Taylor\u27s series, provided the shift is of (o(varepsilon )). Subsequently, the delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. An exponential integrating factor is introduced into the first order delay equation. Then Trapezoidal rule, along with linear interpolation, has been employed to get a three term recurrence relation. The resulting tri-diagonal system is solved by Thomas algorithm. The proposed technique is implemented on model examples, for different values of delay parameter, $delta$ and perturbation parameter, $varepsilon$. Maximum absolute errors are tabulated and compared to validate the technique. Convergence of the proposed method has also been discussed

Numerical integration of singularly perturbed delay differential equations using exponential integrating factor

In this paper, we proposed a numerical integration technique with exponential integrating factor for the solution of singularly perturbed differential-difference equations with negative shift, namely the delay differential equation, with layer behaviour. First, the negative shift in the differentiated term is approximated by Taylor\u27s series, provided the shift is of (o(varepsilon )). Subsequently, the delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. An exponential integrating factor is introduced into the first order delay equation. Then Trapezoidal rule, along with linear interpolation, has been employed to get a three term recurrence relation. The resulting tri-diagonal system is solved by Thomas algorithm. The proposed technique is implemented on model examples, for different values of delay parameter, $delta$ and perturbation parameter, $varepsilon$. Maximum absolute errors are tabulated and compared to validate the technique. Convergence of the proposed method has also been discussed

An Initial Value Technique using Exponentially Fitted Non Standard Finite Difference Method for Singularly Perturbed Differential-Difference Equations

In this paper, an exponentially fitted non standard finite difference method is proposed to solve singularly perturbed differential-difference equations with boundary layer on left and right sides of the interval. In this method, the original second order differential difference equation is replaced by an asymptotically equivalent singularly perturbed problem and in turn the problem is replaced by an asymptotically equivalent first order problem. This initial value problem is solve by using exponential fitting with non standard finite differences. To validate the applicability of the method, several model examples have been solved by taking different values for the delay parameter δ , advanced parameter η and the perturbation parameter ε . Comparison of the results is shown to justify the method. The effect of the small shifts on the boundary layer solutions has been investigated and presented in figures. The convergence of the scheme has also been investigated

Hybrid Algorithm for Singularly Perturbed Delay Parabolic Partial Differential Equations

This study aims at constructing a numerical scheme for solving singularly perturbed parabolic delay differential equations. Taylor’s series expansion is applied to approximate the shift term. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is an ε−uniformly convergent accuracy of order one. Some test examples are considered to testify the theoretical investigations

A seventh order numerical method for singular perturbed differential-difference equations with negative shift

In this paper, a seventh order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been used for delay. Such problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, we first use Taylor approximation to tackle terms containing small shifts which converts into a singularly perturbed boundary value problem. This 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 employed for the first order system and solved by using the boundary conditions. Several numerical examples are solved and compared with exact solution. We also present least square errors, maximum errors and observed that the present method approximates the exact solution very well

Fitted non-polynomial spline method for singularly perturbed differential difference equations with integral boundary condition

The aim of this paper is to present fitted non-polynomial spline method for singularly perturbed differential-difference equations with integral boundary condition. The stability and uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε and mesh size, h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence and it is observed that the present method is more accurate and uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result and it also improves the results of the methods existing in the literature

Relaxation oscillations in a class of delay-differential equations.

We study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics. Under certain circumstances the model exhibits self-sustained oscillations, with periods which can be significantly longer than the basic cell cycle time. We show that the long periods in the oscillations occur when the cell generation rate is small, and we provide an asymptotic analysis of the model in this case. This analysis bears a close resemblance to the analysis of relaxation oscillators (such as the Van der Pol oscillator), except that in our case the slow manifold is infinite dimensional. Despite this, a fairly complete analysis of the problem is possible