DOI QR코드

DOI QR Code

SOLVING SECOND ORDER SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR VIA INITIAL VALUE METHOD

  • GEBEYAW, WONDWOSEN (Department of Mathematics, College of Natural and Computational Sciences, Dilla University) ;
  • ANDARGIE, AWOKE (Department of Mathematics, College of Sciences, Bahir Dar University) ;
  • ADAMU, GETACHEW (Department of Mathematics, College of Sciences, Bahir Dar University)
  • Received : 2017.11.16
  • Accepted : 2018.03.18
  • Published : 2018.05.30

Abstract

In this paper, an initial value method for solving a class of singularly perturbed delay differential equations with layer behavior is proposed. In this approach, first the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay using Taylor series expansion. Then from the modified problem, two explicit Initial Value Problems which are independent of the perturbation parameter, ${\varepsilon}$, are produced: the reduced problem and boundary layer correction problem. Finally, these problems are solved analytically and combined to give an approximate asymptotic solution to the original problem. To demonstrate the efficiency and applicability of the proposed method three linear and one nonlinear test problems are considered. The effect of the delay on the layer behavior of the solution is also examined. It is observed that for very small ${\varepsilon}$ the present method approximates the exact solution very well.

Keywords

References

  1. A. Andargie, Y.N. Reddy, An asymptotic-Fitted method for solving singularly perturbed delay differential equation, J. of App. math. 3 (2012), 895-902. https://doi.org/10.4236/am.2012.38132
  2. S. Cengizci, An asymptotic-numerical hybrid method for solving singularly perturbed linear delay differential equations, Int. Jou. of DEs. 20 (2017), Article ID 7269450, 8 pages.
  3. M.W. Derstine, H.M. Gibbs, F.A. Hopf, D.L. Kaplan, Bifurcation gap in a hybrid optical system, Phys. Rev. A 26 (1982), 3720-3722. https://doi.org/10.1103/PhysRevA.26.3720
  4. E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
  5. G. File, Y.N. Reddy, A non-asymptotic method for solving singularly perturbed delay differential equations, J. of App. Maths. And Info. 32 (2014), 39-53.
  6. G. File, Y.N. Reddy, computational method for solving singularly perturbed delay differential equations wit negative shift, Int. J. of App. sci. and engine. 11 (2013), 101-103.
  7. G. File, Y.N. Reddy, Numerical integration of a class of singularly perturbed delay differential equations with small shift, Int. J. of DEs. 2012, Article ID 572723, 12 pages.
  8. G. File, Y.N. Reddy, Terminal boundary-value technique for solving singularly perturbed delay differential equations, J. of Taibah Univ. for sci. 8 (2014), 289-300. https://doi.org/10.1016/j.jtusci.2014.01.006
  9. V.Y. Glizer, Asymptotic analysis and solution of a finite-horizon $H_{\infty}$ control problem for singularly-perturbed linear systems with small state delay, J. Optim. Theory Appl. 117 (2003), 295-325. https://doi.org/10.1023/A:1023631706975
  10. D.D. Joseph, L. Preziosi, Heat waves, Rev. Mod. Phys. 61 (1989), 41-73. https://doi.org/10.1103/RevModPhys.61.41
  11. M.K. Kadalbajoo, D. Kumar, A computational method for singularly perturbed nonlinear differential-difference equations with small shift, J. Appl. Math. Modl. 34 (2010), 2584-2596. https://doi.org/10.1016/j.apm.2009.11.021
  12. M.K. Kadalbajoo , K.C. Patidal, K.K. Sharma,${\varepsilon}$-uniform convergent fitted method for the numerical solution of the problems arising from singularly perturbed general DDEs, App. Math. and Comp. 182 (2006), 119-139. https://doi.org/10.1016/j.amc.2006.03.043
  13. M.K. Kadalbajoo and K.K. Sharma, A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations, App. Math. and Comp. 197 (2008), 692-707. https://doi.org/10.1016/j.amc.2007.08.089
  14. A. Kanshik, V. Kumar, and A.K. Vashishth, An efficient mixed asymptotic-numerical scheme for singularly perturbed convection diffusion problems, App. Math. and Comp. 218 (2012), 8645-8658. https://doi.org/10.1016/j.amc.2012.02.026
  15. C.G. Lange and R.M. Miura, singular perturbation analysis of boundary value problems for differential difference equations, SIAM J. on App. math. 45 (1985), 680-707.
  16. C.G. Lange and R.M. Miura, singular perturbation analysis of boundary value problems for differential difference equations, V. small shift with layer behavior, SIAM J. on App. math. 54(1994), 249-272. https://doi.org/10.1137/S0036139992228120
  17. X. Liao, Hopf and resonant co-dimension two bifurcations in vander Pol equation with two time delays, Chaos Soliton Fract. 23 (2005), 857-871. https://doi.org/10.1016/j.chaos.2004.05.048
  18. A. Longtin, J. Milton, Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci. 90 (1988), 183-199. https://doi.org/10.1016/0025-5564(88)90064-8
  19. M.C. Mackey, L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977), 287-289. https://doi.org/10.1126/science.267326
  20. A.H. Nayfe , Introduction to perturbation methods, Wiley, New York, 1981.
  21. D.Y. Tzou, Macro-to-micro scale heat transfer, Taylor and Francis, Washington, DC, 1997.
  22. M. Wazewska-Czyzewska, A. Lasota, Mathematical models of the red cell system, Mat. Stos. 6 (1976), 25-40.
  23. E.R. El-Zahar, and Sabel M.M. El-Leabeir, A new method for solving singularly perturbed boundary value problems, App. Math. Inf. Sci. 7(2013), No. 3, pp. 927-938. https://doi.org/10.12785/amis/070310