ε-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay

Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are ε-uniformly convergent.