Advertisement

Numerical approximation for a class of singularly perturbed delay differential equations with boundary and interior layer(s)

  • Pratima Rai
  • Kapil K. SharmaEmail author
Original Paper
  • 49 Downloads

Abstract

This paper is devoted to the study of singularly perturbed delay differential equations with or without a turning point. The solution of the considered class of problems may exhibit boundary or interior layer(s) due to the presence of the perturbation parameter, the turning point, and the delay term. Some a priori estimates are derived on the solution and its derivatives. To solve the problem numerically, a finite difference scheme on piecewise uniform Shishkin mesh along with interpolation to tackle the delay term is proposed. The solution is decomposed into regular and singular components to establish parameter uniform error estimate. It is shown that the proposed scheme converges to the solution of the continuous problem uniformly with respect to the singular perturbation parameter. The numerical experiments corroborate the theoretical findings.

Keywords

Delay differential equations Singular perturbation Turning point Interior layer Interpolation Shishkin mesh 

Mathematics Subject Classification (2010)

34K26 34K28 65L10 65L12 65L20 

Notes

References

  1. 1.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’ Riordon, E., Shishkin, G.I.: Robust computational techniques for boundary layers. Chapman & hall/ CRC, Boca Raton (2000)CrossRefGoogle Scholar
  2. 2.
    Geng, F.Z., Qian, S.P.: Modified reproducing kernel method for singularly perturbed boundary value problems with a delay. Appl. Math. Model. 39, 5592–5597 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kadalbajoo, M.K., Sharma, K.K.: Numerical analysis of boundary value problem for singularly perturbed delay differential equations with layer behaviour. Appl. Math. Comput. 151(1), 11–28 (2004)zbMATHGoogle Scholar
  4. 4.
    Kadalbajoo, M.K., Sharma, K.K.: ε-uniform fitted mesh method for singularly perturbed differential-difference equations: Mixed type of shifts with layer behavior. Int. J. Comput. Math. 81, 49–62 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kadalbajoo, M.K., Sharma, K.K.: Numerical treatment of a mathematical model arising from a model of neuronal variability. J. Math. Anal. Appl. 307, 606–627 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kadalbajoo, M.K., Sharma, K.K.: A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations. Appl. Math. Comput. 197, 692–707 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary value problems for differential-difference equations. SIAM J. Appl. Math. 42, 502–531 (1982)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary value problems for differential-difference equations. III. Turning point problems. SIAM J. Appl. Math. 45, 708–734 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nicaise, S., Xenophontos, C.: Robust approximation of singularly perturbed delay differential equation by the hp finite element method. Comput. Meth. Appl. Meth. 13, 21–37 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rai, P., Sharma, K.K.: Numerical analysis of singularly perturbed delay differential turning point problem. Appl. Math. Comput. 218, 3483–3498 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Rai, P., Sharma, K.K.: Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability. Comp. Math. Appl. 63, 118–132 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rai, P., Sharma, K.K.: Parameter uniform numerical method for singularly perturbed differential-difference equations with interior layer. Int. J. Comput. Math. 88(16), 3416–3435 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rai, P., Sharma, K.K.: Fitted numerical method for singularly perturbed delay differential turning point problems exhibiting boundary layers. Int. J. Comp. Math. 89(7), 944–961 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rao, R.N., Chakravarthy, P.P.: A finite difference method for singularly perturbed differential difference equations with layer and oscillatory behaviour. Appl. Math. Model. 37, 5743–5755 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Subburayan, V., Ramanujam, N.: Asymptotic initial value technique for singularly perturbed convection diffusion delay problem with boundary and weak interior layers. Appl. Math. Lett. 25, 2272–2278 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tang, Z.Q., Geng, F.Z.: Fitted reproducing kernel method for singularly perturbed delay initial value problem. Appl. Math. Comput. 284, 169–174 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zarin, H.: On discontinuous Galerkin finite element method for singularly perturbed delay differential equations. Appl. Math. Lett. 38, 27–32 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Selvi, P., Ramanujam, N.: An iterative numerical method for singularly perturbed reaction diffusion equations with negative shift. J. Comput. Appl. Math. 296, 10–23 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia
  2. 2.Department of MathematicsSouth Asian University, Akbar BhavanChanakypuri, New DelhiIndia

Personalised recommendations