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Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 179–206 | Cite as

A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay

  • Devendra KumarEmail author
  • Parvin Kumari
Original Research

Abstract

A numerical scheme for a class of singularly perturbed parabolic partial differential equation with the time delay on a rectangular domain in the xt plane is constructed. The presence of the perturbation parameter in the second-order space derivative gives rise to parabolic boundary layer(s) on one (or both) of the lateral side(s) of the rectangle. Thus the classical numerical methods on the uniform mesh are inadequate and fail to give good accuracy and results in large oscillations as the perturbation parameter approaches zero. To overcome this drawback a numerical method comprising the Crank–Nicolson finite difference method consisting of a midpoint upwind finite difference scheme on a fitted piecewise-uniform mesh of \(N\times M\) elements condensing in the boundary layer region is constructed. A priori explicit bounds on the solution of the problem and its derivatives which are useful for the error analysis of the numerical method are established. To establish the parameter-uniform convergence of the proposed method an extensive amount of analysis is carried out. It is shown that the proposed difference scheme is second-order accurate in the temporal direction and the first-order (up to a logarithmic factor) accurate in the spatial direction. To validate the theoretical results, the method is applied to two test problems. The performance of the method is demonstrated by calculating the maximum absolute errors and experimental orders of convergence. Since the exact solutions of the test problems are not known, the maximum absolute errors are obtained by using double mesh principle. The numerical results show that the proposed method is simply applicable, accurate, efficient and robust.

Keywords

Delay partial differential equations Time delay Singular perturbation Piecewise-uniform mesh Parabolic boundary layers Parameter-uniform convergence 

Mathematics Subject Classification

35K10 65M06 65M12 65M15 65N06 65N12 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments that improved the quality of this paper.

References

  1. 1.
    Adomian, G., Rach, R.: Nonlinear stochastic differential delay equations. J. Math. Anal. Appl. 91, 94–101 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ansari, A.R., Bakr, S.A., Shishkin, G.I.: A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. J. Comput. Appl. Math. 205, 552–566 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asl, F.M., Ulsoy, A.G.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Control 125, 215–223 (2003)CrossRefGoogle Scholar
  4. 4.
    Aziz, I., Amin, R.: Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet. Appl. Math. Model. 40, 10286–10299 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baranowski, J.: Legendre polynomial approximations of time delay systems. In: Proceeding of 12th International PhD Workshop, vol. 1, pp. 15–20 (2010)Google Scholar
  6. 6.
    Bashier, E.B.M., Patidar, K.C.: A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Appl. Math. Comput. 217, 4728–4739 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Campbell, S.A., Edwards, R., Van Den Driessche, P.: Delayed coupling between two neural network loops. SIAM J. Appl. Math. 65, 316–335 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ciupe, M.S., Bivort, B.L., Bortz, D.M., Nelson, P.W.: Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models. Math. Biosci. 200, 1–27 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cooke, K.L., Kuang, Y., Li, B.: Analysis of an antiviral immune response model with time delays. Can. Appl. Math. Q. 6, 321–354 (1998)zbMATHGoogle Scholar
  10. 10.
    Cooke, K.L., van den Driessche, P., Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39, 332–354 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Das, A., Natesan, S.: Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh. Appl. Math. Comput. 271, 168–186 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Epstein, I.R.: Delay effects and differential delay equations in chemical kinetics. Int. Rev. Phys. Chem. 11, 135–160 (1992)CrossRefGoogle Scholar
  13. 13.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman & Hall, London (2000)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gowrisankar, S., Natesan, S.: \(\epsilon \)-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations. Int. J. Comput. Math. 94, 902–921 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  16. 16.
    Van Harten, A., Schumacher, J.M.: On a Class of Partial Functional Differential Equations Arising in Feedback Control Theory. North Holland, Amsterdam (1978)zbMATHGoogle Scholar
  17. 17.
    Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: The use of defect correction for the solution of parabolic singular perturbation problems. Z. Angew. Math. Mech. 76, 59–74 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hemker, P.W., Shishkin, G.I., Shishkina, L.P.: \(\epsilon \)-Uniform schemes with high-order time-accuracy for parabolic singular perturbation problems. IMA J. Numer. Anal. 20, 99–121 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kadalbajoo, M.K., Awasthi, A.: A parameter uniform difference scheme for singularly perturbed parabolic problem in one space dimension. Appl. Math. Comput. 183, 42–60 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kadalbajoo, M.K., Kumar, D.: A computational method for singularly perturbed nonlinear differential-difference equations with small shift. Appl. Math. Model. 34, 2584–2596 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kaushik, A., Sharma, M.D.: Numerical analysis of a mathematical model for propagation of an electrical pulse in a neuron. Numer. Methods Partial Differ. Equ. 27, 1–18 (2008)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning point. Math. Comp. 32, 1025–1039 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)zbMATHGoogle Scholar
  24. 24.
    Kumar, D., Kadalbajoo, M.K.: A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations. Appl. Math. Model. 35, 2805–2819 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kumar, D., Kadalbajoo, M.K.: A parameter uniform method for singularly perturbed differential-difference equations with small shifts. J. Numer. Math. 21, 1–22 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kuramoto, Y., Yamada, T.: Turbulent state in chemical reactions. Prog. Theor. Phys. 56, 679–681 (1976)CrossRefGoogle Scholar
  27. 27.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)CrossRefGoogle Scholar
  28. 28.
    McCartin, B.J.: Discretization of the semiconductor device equations. In: Miller, J.J.H. (ed.) New Problems and New Solutions for Device and Process Modelling. Boole Press, Dublin (1985)Google Scholar
  29. 29.
    Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore (1996)CrossRefzbMATHGoogle Scholar
  30. 30.
    Murray, J.D.: Mathematical Biology. I. An Introduction, 3rd edn. Springer, New York (2002)zbMATHGoogle Scholar
  31. 31.
    Nelson, P.W., Murray, J.D., Perelson, A.S.: A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163, 201–215 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nelson, P.W., Perelson, A.S.: Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179, 73–94 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion and Flow Problems. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  34. 34.
    Samarskii, A.A., Vabishchevich, P.N.: Computational Heat Transfer. Wiley, New York (1995)Google Scholar
  35. 35.
    Shishkin, G.I.: Approximation of solutions of singularly perturbed boundary value problems with a parabolic boundary layer. USSR Comput. Math. Math. Phys. 29, 1–10 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Smolen, P., Baxter, D.A., Byrne, J.H.: A reduced model clarifies the role of feedback loops and time delays in the drosophila circadian oscillator. Biophys. J. 83, 2349–2359 (2002)CrossRefGoogle Scholar
  37. 37.
    Symko, R.M., Glass, L.: Spatial switching in chemical reactions with heterogeneous catalysts. J. Chem. Phys. 60, 835–841 (1974)CrossRefGoogle Scholar
  38. 38.
    Takahashi, Y., Rabins, M.J., Auslander, D.M.: Control and Dynamic Systems. Addison Wesley, Boston (1970)zbMATHGoogle Scholar
  39. 39.
    Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Nauka, Moscow (1972)zbMATHGoogle Scholar
  40. 40.
    Vielle, B., Chauvet, G.: Delay equation analysis of human respiratory stability. Math. Biosci. 47, 105–122 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Villasana, M., Radunskaya, A.: A delay differential equation model for tumor growth. J. Math. Biol. 47, 270–294 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wang, P.K.C.: Asymptotic stability of a time-delayed diffusion system. J. Appl. Mech. 30, 500–504 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wang, X.T.: Numerical solution of delay systems containing inverse time by hybrid functions. Appl. Math. Comput. 173, 535–546 (2006)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Wang, Y., Tian, D., Li, Z.: Numerical method for singularly perturbed delay parabolic partial differential equations. Thermal Science 21, 1595–1599 (2017)CrossRefGoogle Scholar
  45. 45.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  46. 46.
    Zhao, T.: Global periodic-solutions for a differential delay system modeling a microbial population in the chemostat. J. Math. Anal. Appl. 193, 329–352 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of MathematicsBirla Institute of Technology and SciencePilaniIndia

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