A collocation method for singularly perturbed differential-difference turning point problems exhibiting boundary/interior layers

ABSTRACT

In this article, a collocation method for the problems where the second-order derivative is multiplied by a small perturbation parameter ϵ, the coefficient of the convection term vanishes at a point within the domain of interest, and the shift δ is of $o\left(\epsilon \right)$ is proposed. Similar boundary value problems are encountered while simulating several real-life processes, for instance, first exit time problem in the modelling of neuronal variability. The presence of turning point results into twin boundary layers or an interior layer in the solution of the problem under consideration. A rigorous analysis is carried out and it has been shown theoretically that the numerical solution generated by the method converges uniformly to the solution of the continuous problem with respect to the singular perturbation parameter. The effect of the small shift on the boundary/interior layer(s) has also been observed. Several numerical examples are presented to support the theoretical analysis developed in this article.

KEYWORDS:

• Singular perturbation
• fitted-mesh method
• differential-difference equation
• collocation method
• boundary layer
• turning point
• interior layer

AMS SUBJECT CLASSIFICATIONS:

• 39A10
• 39A70
• 34B05
• 34E20
• 34K10
• 34K06
• 34K28

Acknowledgments

The author would like to express thanks to the anonyms referees for their valuable comments/suggestions which have improved the presentation of this work.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Devendra Kumar  http://orcid.org/0000-0002-4832-8481