The solution for singularly perturbed differential-difference equation with boundary layers at both ends by a numerical integration method

Abstract

This paper presents a numerical integration method for the solution of ‘Singularly Perturbed Differential-Difference Equations' having dual layers. It is well known that when we use already existing numerical methods to solve such problems, we get oscillatory or unsatisfactory results unless we take a very small step size, which is time-consuming and costly. To get a numerical solution for such a problem, first, the delay and advanced parameters present in the SPDDE are approximated by Taylor’s series to get an equivalent ‘Singularly Perturbed Differential Equation' of second order. Second, an asymptotically equivalent first-order differential equation is obtained from SPDE using Taylor’s transformation. Composite Simpson’s 1/3 rule is implemented to get a three-term recurrence relation. The Thomas algorithm is applied to get the solution of the tri-diagonal system of equations. Several model examples are tested and it was found that the numerical solution approximates the available/exact solution very well.

KEYWORDS:

• Delay parameter
• advance parameter
• differential-difference equations
• dual layer
• numerical integration
• boundary layer

Mathematics Subject Classification:

Acknowledgements

The authors are grateful to the anonymous referees. All authors equally contributed to this work and read and approved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Declarations of interest

All authors equally contributed to this work.