SINGULARLY PERTURBED ORDINARY DIFFERENTIAL EQUATIONS WITH TIME DELAY AND ADVANCED SHIFTS

Abstract

Differential equations play a fundamental role in modelling real-world phenomena in science and engineering. The class of problems called singularly perturbed differential problems had a major role in setting up the foundations of fluid dynamics, control theleads to turning-point. The presence of sharp boundaries or interior layers can be observed in the solutions due to the multiplication of the highest order derivative by a small perturbation parameter called ε. The introduction of other parameters, such as delay, advanced, or a combination of both, makes the problem harder to solve. Furthermore, the vanishing of the convection term leads to turning-point problems and interior layers, making the problem more challenging. In this thesis, we study a class of singularly perturbed differential-difference equations with mixed delay and advance. We study two cases with respect to delays and advances of order o(ε) and O(ε). Numerical results and applications are presented to confirm the theoretical analysis and also show how the delay and advance terms affect the position of the interior layer, while the rate of convergence does not depend on the perturbation and shift parameters.