Higher Order Uniformly Convergent Numerical Algorithm for Time-Dependent Singularly Perturbed Differential-Difference Equations

  • V. P. RameshEmail author
  • B. Priyanga
Original Research


This paper presents a higher order uniformly convergent discretization for second order singularly perturbed parabolic differential equation with delay and advance terms. The retarded terms are approximated by the Taylor series which give rise to a nearby singularly perturbed parabolic differential equation. A hybrid numerical algorithm based on implicit Euler scheme is proposed to discretize the time variable and a combined finite difference scheme made out of modified upwind and central difference schemes to discretize the spatial variable on a piecewise uniform mesh of Shishkin type in space and a uniform mesh in time. The existence and uniqueness of a solution for the proposed hybrid algorithm is analyzed. It is proved that the algorithm is \(\epsilon \)-uniformly convergent of almost second order in space, \(O(N^{-2} (\ln N)^2)\) and first order in time, \(O(M^{-1})\). A few numerical experiments supporting the results are presented. The efficiency of the proposed hybrid algorithm is demonstrated by comparing upwind and modified upwind algorithm.


Singularly perturbed problem Differential difference equation Parabolic equation Shishkin mesh Hybrid algorithm Neuronal model 

Mathematics Subject Classification

65M60 35K20 35K67 



We thank the referees for all their comments and for the suggestions to improve the presentation.


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Copyright information

© Foundation for Scientific Research and Technological Innovation 2019

Authors and Affiliations

  1. 1.Department of MathematicsCentral University of Tamil NaduThiruvarurIndia
  2. 2.Department of MathematicsMathematics and Analytics Center of Excellence (MACoE), Central University of Tamil NaduThiruvarurIndia

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