No 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.

In this paper, an exponential fitted method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point via deviating argument. The original second order boundary value problem is transformed to first order differential equation with a small deviating argument. This problem is solved efficiently by using exponential fitting and discrete invariant imbedding method. Maximum absolute errors of several standard examples are presented to support the method.

An extended second order finite difference method on a variable mesh is proposed for the solution of a singularly perturbed boundary value problem. A discrete equation is achieved on the non uniform mesh by extending the first and second order derivatives to the higher order finite differences. This equation is solved efficiently using a tridiagonal solver. The proposed method is analysed for convergence, and second order convergence is derived. Model examples are solved by the proposed scheme and compared with available methods in the literature to uphold the method.