Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 701–722

# Singularly perturbed delay differential equations of convection–diffusion type with integral boundary condition

• E. Sekar
• A. Tamilselvan
Original Research

## Abstract

In this paper we consider a class of singularly perturbed delay differential equations of convection diffusion type with integral boundary condition. A finite difference scheme with an appropriate piecewise Shishkin type mesh is suggested to solve the problem. We prove that the method is of almost first order convergent. An error estimate is derived in the discrete norm. Numerical experiments support our theoretical results.

## Keywords

Singularly perturbed problems Delay differential equation Finite difference scheme Shishkin mesh Integral boundary condition

## Mathematics Subject Classification

65L11 65L12 65L20

## References

1. 1.
Amiraliyev, G.M., Amiraliyev, I.G., Kudu, M.: A numerical treatment for singularly perturbed differential equations with integral boundary condition. Appl. Math. Comput. 185, 574–582 (2007)
2. 2.
Bahuguna, D., Dabas, J.: Existence and uniqueness of a solution to a semilinear partial delay differential equation with an integral condition. Nonlinear Dyn. Syst. Theory 8(1), 7–19 (2008)
3. 3.
Bahuguna, D., Abbas, S., Dabas, J.: Partial functional differential equation with an integral condition and applications to population dynamics. Nonlinear Anal. 69, 2623–2635 (2008)
4. 4.
Boucherif, A.: Second order boundary value problems with integral boundary condition. Nonlinear Anal. 70(1), 368–379 (2009)
5. 5.
Cakir, M., Amiraliyev, G.M.: A finite difference method for the singularly perturbed problem with nonlocal boundary condition. Appl. Math. Comput. 160, 539–549 (2005)
6. 6.
Cannon, J.R.: The solution of the heat equation subject to the specification of energy. Q. Appl. Math. 21, 155–160 (1963)
7. 7.
Cen, Z., Cai, X.: A Second Order Upwind Difference Scheme for a Singularly Perturbed Problem with Integral Boundary Condition in Netural Network, pp. 175–181. Springer, Berlin (2007)Google Scholar
8. 8.
Choi, Y.S., Chan, K.-Y.: A parabolic equation with nonlocal boundary conditions arising from electrochemistry. Nonlinear Anal. Theory Methods Appl. 18(4), 317–331 (1992)
9. 9.
Culshaw, R.V., Ruan, S.: A delay differential equation model of HIV infection of $$CD4^+$$ T-cells. Math. Biosci. 165, 27–39 (2000)
10. 10.
Day, W.A.: Parabolic equations and thermodynamics. Q. Appl. Math. 50, 523–533 (1992)
11. 11.
Els’gol’ts, E.L.: Qualitative Methods in Mathematical Analysis in: Translations of Mathematical Monographs, vol. 12. American Mathematical Society, Providence (1964)Google Scholar
12. 12.
Feng, M., Ji, D., Weigao, G.: Positive solutions for a class of boundary value problem with integral boundary conditions in banach spaces. J. Comput. Appl. Math. 222, 351–363 (2008)
13. 13.
Glizer, V.Y.: Asymptotic analysis and Solution of a finite-horizon $$H_{\propto }$$ control problem for singularly perturbed linear systems with small state delay. J. Optim. Theory Appl. 117, 295–325 (2003)
14. 14.
Kadalbajoo, M.K., Kumar, D.: Fitted mesh B-spline collocation method for singularly perturbed differential equations with small delay. Appl. Math. Comput. 204, 90–98 (2008)
15. 15.
Kadalbajoo, M.K., Sharma, K.K.: Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations. Comput. Appl. Math. 24(2), 151–172 (2005)
16. 16.
Kadalbajoo, M.K., Sharma, K.K.: Parameter-uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior. Electron. Trans. Numer. Anal. 23, 180–201 (2006)
17. 17.
Kudu, M., Amiraliyev, G.: Finite difference method for a singularly perturbed differential equations with integral boundary condition. Int. J. Math. Comput. 26, 72–79 (2015)
18. 18.
Lange, C.G., Miura, R.M.: Singularly perturbation analysis of boundary-value problems for differential-difference equations. SIAM J. Appl. Math. 42(3), 502–530 (1982)
19. 19.
Li, H., Sun, F.: Existence of solutions for integral boundary value problems of second order ordinary differential equations. Bound. Value Probl. 1, 147 (2012)
20. 20.
Longtin, A., Milton, J.: Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Math. Biosci. 90, 183–199 (1988)
21. 21.
Mahendran, R., Subburayan, V.: Fitted finite difference method for third order singularly perturbed delay differential equations of convection diffusion type. Int. J. Comput. Methods 15(1), 1840007 (2018)Google Scholar
22. 22.
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific Publishing Co., London (1996)
23. 23.
Nicaise, S., Xenophontos, C.: Robust approximation of singularly perturbed delay differential equations by the hp finite element method. Comput. Methods Appl. Math. 13(1), 21–37 (2013)
24. 24.
Tang, Z.Q., Geng, F.Z.: Fitted reproducing kernel method for singularly perturbed delay initial value problems. Appl. Math. Comput. 284, 169–174 (2016)
25. 25.
Zarin, H.: On discontinuous Galerkin finite element method for singularly perturbed delay differential equations. Appl. Math. Lett. 38, 27–32 (2014)
26. 26.
Zhang, L., Xie, F.: Singularly perturbed first order differential equations with integral boundary condition. J. Shanghai Univ. (Eng) 13, 20–22 (2009)