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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.

A numerical approach is suggested for the layer behaviour differential-difference equations with small and large delays in the differentiated term. Using the non-polynomial spline, the numerical scheme is derived. The discretization equation is constructed using the first order derivative continuity at non-polynomial spline internal mesh points. A fitting parameter is introduced into the scheme with the help of the singular perturbation theory to minimize the error in the solution. The maximum errors in the solution are tabulated to verify the competence of the numerical method relative to the other methods in literature. We also focused on the impact of large delays on the layer behaviour or oscillatory behaviour of solutions using a special mesh with and without fitting parameter in the proposed scheme.Graphs show the effect of the fitting parameter on the solution layer.

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In this paper, we descend a variable mesh finite difference scheme based on non polynomial spline approximation for the solution of singular perturbation problems with twin boundary layers. We develop the discretization equation for the problem using the condition of continuity for the first order derivatives of the variable mesh non polynomial spline at the interior nodes. The discrete invariant imbedding algorithm is utilized to solve the tridiagonal system obtained by the method. Endeavor examples are illustrated and maximum absolute errors in comparison to the other methods in the literature are shown to vindicate the method.

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In this paper, a quadrature technique is employed for the solution of singularly perturbed delay differential equation. A first-order neutral type delay differential equation is achieved, which is asymptotically equivalent to the given singularly perturbed delay differential equation. Then Gaussian quadrature two-point formula is implemented on the first order equation to get a tridiagonal. Thomas algorithm is used to solve the resulting tri-diagonal system. The proposed method is implemented on model example, for different value of delay parameter and perturbation parameter. Maximum absolute errors are tabulated with a comparison to authorize the method. Theoretical convergence of the method is discussed. The layer behaviour is discussed using the graphical histrionics.