A Novel Fitted Method for a Class of Singularly Perturbed Differential-Difference Equations with Small Delay Exhibiting Twin Layer or Oscillatory Behaviour

Abstract

A new exponentially fitted three term method is developed for the numerical treatment of a class of linear second order singularly perturbed differential-difference equations (SPDDEs) which involves the small delay in un-differentiated term. The solution of such equations with the interval and boundary conditions exhibits twin layer or oscillatory behaviour. The method uses the Taylor’s series expansion for constructing an equivalent valid version of the original problem first and then, to derive a new three term finite difference recurrence relationship/scheme. The non-uniformity in the solution is resolved by the introduction of a suitable fitting parameter in the derived new scheme. Finally the resulting system of algebraic equations is solved by the well known “discrete invariant algorithm.” Method is analyzed for the stability and convergence, and the theory is illustrated by solving several test example problems. Computational results are tabulated and compared to show the applicability, accuracy and efficiency of the method. Theory and computation show that the method is able to approximate the solution very well with second order convergence rate.