Abstract
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.
Highlights
The boundary value problems for singularly perturbed differential-difference equations arise in various practical problems in biomechanics and physics such as in variation problems in control theory and depolarization in Stein’s model
We have considered numerical results for these test examples to show the effect of small shifts on boundary layer solution of the problem
From the numerical experiments presented here, we observe as δ increases, the thickness of the boundary layer decreases and maximum error decreases as the grid size h decreases, which shows the convergence of the computed solution to the exact solution
Summary
The boundary value problems for singularly perturbed differential-difference equations arise in various practical problems in biomechanics and physics such as in variation problems in control theory and depolarization in Stein’s model. In [5] M.K. Kadalbajoo and K.K. Sharma presented a numerical method to solve boundary value problems for singularly perturbed differential difference equations of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift. Sharma presented non-standard finite difference methods for second order, linear, singularly perturbed differential-difference equations. The objective of this paper is to describe the seventh order numerical method to the boundary value problems for singularly perturbed differential difference equations with negative shift. In this method, we approximate the shifted term by Taylor series and apply a difference scheme, provided shifts are of o(ε). It is observed that the present method approximates the exact solution very well
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