Numerical integration of singularly perturbed delay differential equations using exponential integrating factor

Abstract

In this paper, we proposed a numerical integration technique with exponential integrating factor for the solution of singularly perturbed differential-difference equations with negative shift, namely the delay differential equation, with layer behaviour. First, the negative shift in the differentiated term is approximated by Taylor's series, provided the shift is of $o(\varepsilon )$. Subsequently, the delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. An exponential integrating factor is introduced into the first order delay equation. Then Trapezoidal rule, along with linear interpolation, has been employed to get a three term recurrence relation. The resulting tri-diagonal system is solved by Thomas algorithm. The proposed technique is implemented on model examples, for different values of delay parameter, $\delta $ and perturbation parameter, $\varepsilon $. Maximum absolute errors are tabulated and compared to validate the technique. Convergence of the proposed method has also been discussed.