A numerical scheme for singularly perturbed delay differential equations of convection-diffusion type on an adaptive grid

    Pramod Chakravarthy Podila Affiliation
    ; Trun Gupta Affiliation
    ; Nageshwar Rao Affiliation


In this paper, an adaptive mesh strategy is presented for solving singularly perturbed delay differential equation of convection-diffusion type using second order central finite difference scheme. Layer adaptive meshes are generated via an entropy production operator. The details of the location and width of the layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

Keyword : singular perturbation, entropy like variable, delay differential equation, layer adaptive meshes, central finite difference scheme, convection-diffusion problem

How to Cite
Podila, P., Gupta, T., & Rao, N. (2018). A numerical scheme for singularly perturbed delay differential equations of convection-diffusion type on an adaptive grid. Mathematical Modelling and Analysis, 23(4), 686-698.
Published in Issue
Oct 9, 2018
Abstract Views
PDF Downloads
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.


G.M. Amiraliyev and E. Cimen. Numerical method for a singularly perturbed convection-diffusion problem with delay. Appl. Math. Comp.,216:2351–2359,2010

G.M. Amiraliyev and F. Erdogan. Uniform numerical method for singularly perturbed delay differential equations. Comput. Math. Applicat., 53:1251–1259., 2007.

I.G. Amiraliyeva, F. Erdogan and G.M. Amiraliyev. A uniform numerical method for dealing with a singularly perturbed delay initial value problem. Appl. Math.L., 23:1221–1225, 2010

N.S. Bakhvalov. On the optimization of the methods for solving boundary value problems in the presence of boundary layers. Zh. Vychisl. Mater. Fiz., 9:841–859, 1969.

K. Bansal, P. Rai and K.K. Sharma. Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments.Diff. Equ. Dyn. Sys., 25(2):327–346, 2017

K. Bansal and K.K. Sharma. Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments. Num. Algor., 75(1):113–145, 2017.

A. Bellen and M. Zennaro. Numerical methods for delay differential Equations. Oxford Science Publications, New York, 2003.

P.P. Chakravarthy, S.D. Kumar and R.N. Rao. Numerical solution of second order singularly perturbed delay differential equations via cubic spline in tension. Int. J. Appl. Comput. Math, 3(3):1703–1717, 2016.

Y. Chen and J. Wu. The asymptotic shapes of periodic solutions of a singular delay differential system. J. Diff. Equ., 169:614–632, 2001.

R.V. Culshaw and S. Ruan. A delay differential equation model of hiv infection of CD4 + T-cells. Math. Biosci.,165(1):27–39, 2000.

M.W. Derstine, H.M. Gibbs, F.A. Hopf and D.L. Kaplan. Bifurcation gap in a hybrid optical system. Phys. Rev. A, 26:3720–3722, 1982.

R.D. Driver.Ordinary and delay differnential equations. Springer, New York, 1977.

F. Erdogan and G.M. Amiraliyev. Fitted finite difference method for singularly perturbed delay differential equations.Numer. Algor., 59:131–145, 2012.

P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Singularly perturbed convection-diffusion problems with boundary and weak interior layers. J. Comput. Appl. Math.,166:133–151, 2004.

P.A. Farrell, J.J.H. Miller, E. O’Riordan and G.I. Shishkin. Singularly perturbed differential equations with discontinuous source terms. In J.J.H. Miller G.I. Shishkin and L. Vulkov (Eds.), Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, pp. 23–32, New York,USA, 2000. Nova Science Publishers. Inc.

E.C. Gartland. Graded-mesh difference schemes for singularly perturbed two-point boundary value problems.Math. comput., 51(184):631–657, 1988.

V.Y. Glizer. Asymptotic solution of a boundary-value problem for linear singularly-perturbed functional differential equations arising in optimal control theory. J. Opt. Th. and Applicat., 106:49–85, 2000.

V.Y. Glizer. Block wise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution.J. Math. Anal. Applicat., 278:409–433, 2003.

M. Kot. Elements of Mathematical Ecology. Cambridge University Press, 2001.

Y. Kuang. Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York, 1993.

V. Kumar and B. Srinivasan. An adaptive mesh strategy for singularly perturbed convection diffusion problems. Appl. Math. Modell., 39:2081–2091, 2015.

C.G. Lange and R.M. Miura. Singular perturbation analysis of boundary-value problems for differential-difference equations. SIAM J. Appl. Math., 42:502–531, 1982.

R.J. Leveque. Finite volume methods for hyperbolic problems. Cambridge University Press, 2002.

T. Linb. Layer-adapted meshes for convection diffusion problems. Comput. Meth-ods Appl. Mech. Eng., 192:1061–1105, 2003.

V.D. Liseikin. The use of special transformations in the numerical solution of boundary value problems. Comput. Math. Math. Phys.,30:43–53, 1990.

A. Longtin and G.J. Milton. Complex oscillations in the human pupil light reflex with mixed and delayed feedback.Math. Biosci., 90:183–199, 1988.

M.C. Mackey and L. Glass. Oscillation and chaos in physiological control systems. Science, 197:287–289, 1977.

J.M. Mahaffy, J. Belair and M.C. Mackey. Hematopoietic model with moving boundary condition and state dependent delay: application in erythropoiesis. J. Theoret. Biol.,190:135–146, 1998.

J.Mallet-Paret and R.D.Nussbaum. A differential-delay equations arising in optics and physiology. SIAM J. Math. Anal., 20:249–292, 1989.

H. Mayer, K.S. Zaenker and U. an der Heiden. A basic mathematical model of immune response. Chaos, 5:155–161, 1995.

J.J.H. Miller, E.O. Riordan and I.G. Shishkin. Fitted numerical methods for singular perturbation problems. Word Scientific, Singapore, 1996.

J.D. Murray.Mathematical Biology. An Introduction, Springer, 2002.

P.W. Nelson and A.S. Perelson. Mathematical analysis of delay differential equation models of HIV–1 infection. Math. Biosci.,179:73–94, 2002.

V. Subburayan and N. Ramanujam. An initial value technique for singularly perturbed convection - diffusion problems with a negative shift. J. Opt. Th. Applicat., 158:234–250, 2013.

R. Vulanovic. On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat.,13:187–201, 1983.

R. Vulanovic. Mesh construction for discretization of singularly perturbed Boundary value problems. Ph.d thesis, University of Novi sad, 1986.

W.J. Wilbur and J. Rinzel. An analysis of Stein’s model for stochastic neuronal excitation. Biol. Cyber., 45:107–114, 1982.