A boundary-value problem for singularly perturbed second order differential-difference equation with both the negative(delay) and positive(advance) shifts is considered and a new exponentially fitted three term finite difference scheme is proposed for the numerical approximation of its solution. An approximate version of the considered problem is constructed by the use of Taylor’s series expansion procedure first, and then, a new three term recurrence relationship is derived by applying the finite difference approximation techniques on it. Furthermore, an exponential fitting factor is introduced in the derived new scheme using the theory of singular perturbations and an efficient ’discrete invariant imbedding algorithm’ is used to solve the resulting tridiagonal system of equations. Convergence of the method is analyzed. Numerical experiments performed on several test examples; presentation of the computational results in terms of maximum absolute errors and the comparison of the results with some other method results, show the applicability and efficiency of the proposed scheme. Graphs plotted for the solution with varying shifts show the effect of small shifts on the boundary layer behavior of the solution. Both the theoretical and numerical analysis of the method reveals that the method is able to produce uniformly convergent solutions with quadratic convergence rate.

研究了同时具有负（时滞）和正（先行）位移的奇摄动二阶微分方程的边值问题，并提出了一种新的指数拟合三项有限差分方案，用于其解的数值逼近。首先通过使用泰勒级数展开过程来构造所考虑问题的近似版本，然后通过对其应用有限差分近似技术来推导新的三项递归关系。此外，使用奇异摄动理论在导出的新方案中引入了指数拟合因子，并使用有效的“离散不变嵌入算法”来求解所得的三对角方程组。分析了该方法的收敛性。在几个测试示例上进行了数值实验；以最大绝对误差表示计算结果，并将计算结果与其他方法的结果进行比较，证明了该方案的适用性和效率。针对具有变化位移的解绘制的图形显示了小位移对溶液边界层行为的影响。该方法的理论和数值分析均表明，该方法能够产生具有二次收敛速率的均匀收敛解。针对具有变化位移的解绘制的图形显示了小位移对溶液边界层行为的影响。该方法的理论和数值分析均表明，该方法能够产生具有二次收敛速率的均匀收敛解。针对具有变化位移的解绘制的图形显示了小位移对溶液边界层行为的影响。该方法的理论和数值分析均表明，该方法能够产生具有二次收敛速率的均匀收敛解。