64
Views
0
CrossRef citations to date
0
Altmetric
Articles

A higher order compact numerical approach for singularly perturbed parabolic problem with retarded term

, &
Pages 530-552 | Received 27 Jun 2022, Accepted 23 Dec 2023, Published online: 03 Jan 2024

References

  • A.R. Ansari, S. Bakr, and G.I. Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205(1) (2007), pp. 552–566. DOI:10.1016/j.cam.2006.05.032
  • G. Babu and K. Bansal, A high order robust numerical scheme for singularly perturbed delay parabolic convection diffusion problems, J. Appl. Math. Comput. 68 (2022), pp. 363–389. DOI:10.1007/s12190-021-01512-1
  • E.B. Bashier and K.C. Patidar, A fitted numerical method for a system of partial delay differential equations, Comput. Math. with Appl. 61(6) (2011), pp. 1475–1492. DOI:10.1016/j.camwa.2010.11.010
  • E.B.M. Bashier and K.C. Patidar, An almost second order fitted mesh numerical method for a singularly perturbed delay parabolic partial differential equation, Neural. Parallel Sci. Comput. 18(2) (2010), pp. 137–154.
  • E.B.M. Bashier and K.C. Patidar, A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation, Appl. Math. Comput. 217(9) (2011), pp. 4728–4739. DOI:10.1016/j.amc.2010.11.028
  • E. Cimen, Numerical solution of a boundary value problem including both delay and boundary layer, Math. Model. Anal. 23(4) (2018), pp. 568–581. DOI:10.3846/mma.2018.034
  • E. Cimen and G.M. Amiraliyev, Uniform convergence method for a delay differential problem with layer behaviour, Mediterr. J. Math. 16 (2019), pp. 1–15. DOI:10.1007/s00009-019-1335-9
  • E. Cimen and M. Cakir, Convergence analysis of finite difference method for singularly perturbed nonlocal differential-difference problem, Miskolc Math. Notes. 19(2) (2018), pp. 795–812. DOI:10.18514/MMN.2018.2302
  • C. Clavero and J.L. Gracia, A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction–diffusion problems, Appl. Math. Comput. 218(9) (2012), pp. 5067–5080. DOI:10.1016/j.amc.2011.10.072
  • A. Das and S. Natesan, Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection–diffusion problems on Shishkin mesh, Appl. Math. Comput. 271 (2015), pp. 168–186. DOI:10.1016/j.amc.2015.08.137
  • P. Das, A higher order difference method for singularly perturbed parabolic partial differential equations, J. Differ. Equ. Appl. 24(3) (2018), pp. 452–477. DOI:10.1080/10236198.2017.1420792
  • S. Gowrisankar and S. Natesan, A robust numerical scheme for singularly perturbed delay parabolic initial-boundary-value problems on equidistributed grids, Electron. Trans. Numer. Anal. 41(2014), pp. 376–395.
  • S. Gowrisankar and S. Natesan, ε-uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations, Int. J. Comput. Math. 94(5) (2017), pp. 902–921. DOI:10.1080/00207160.2016.1154948
  • M.K. Kadalbajoo and A. Awasthi, A parameter uniform difference scheme for singularly perturbed parabolic problem in one space dimension, Appl. Math. Comput. 183(1) (2006), pp. 42–60. DOI:10.1016/j.amc.2006.05.023
  • M.K. Kadalbajoo and K.K. Sharma, Numerical analysis of boundary-value problems for singularly-perturbed differential-difference equations with small shifts of mixed type, J. Optim. Theory. Appl. 115(1) (2002), pp. 145–163. DOI:10.1023/A:1019681130824
  • A. Kaushik, Error estimates for a class of partial functional differential equation with small dissipation, Appl. Math. Comput. 226(2014), pp. 250–257. DOI:10.1016/j.amc.2013.10.040
  • A. Kaushik, K.K. Sharma, and M. Sharma, A parameter uniform difference scheme for parabolic partial differential equation with a retarded argument, Appl. Math. Model. 34(12) (2010), pp. 4232–4242. DOI:10.1016/j.apm.2010.04.020
  • A. Kaushik and M. Sharma, A robust numerical approach for singularly perturbed time delayed parabolic partial differential equations, Comput. Math. Model. 23(1) (2012), pp. 96–106. DOI:10.1007/s10598-012-9122-5
  • D. Kumar, A parameter-uniform scheme for the parabolic singularly perturbed problem with a delay in time, Numer. Methods. Partial. Differ. Equ. 37(1) (2021), pp. 626–642. DOI:10.1002/num.22544
  • D. Kumar and P. Kumari, A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay, J. Appl. Math. Comput. 59(1-2) (2019), pp. 179–206. DOI:10.1007/s12190-018-1174-z
  • D. Kumar and P. Kumari, A parameter-uniform scheme for singularly perturbed partial differential equations with a time lag, Numer. Methods. Partial. Differ. Equ. 36(4) (2020), pp. 868–886. DOI:10.1002/num.22455
  • K. Kumar, P.C. Podila, P. Das, and H. Ramos, A graded mesh refinement approach for boundary layer originated singularly perturbed time-delayed parabolic convection diffusion problems, Math. Methods. Appl. Sci. 44(16) (2021), pp. 12332–12350. DOI:10.1002/mma.7358
  • M. Kumar, J. Singh, and S. Kumar, et al. A robust numerical method for a coupled system of singularly perturbed parabolic delay problems, Eng. Comput. (2020). DOI:10.1108/EC-04-2020-0191
  • P.M.M. Kumar and A.R. Kanth, Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension spline, Comput. Appl. Math. 39(1) (2020), pp. 1–19. DOI:10.1007/s40314-020-01278-5
  • S. Kumar and M. Kumar, High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay, Comput. Math. with Appl. 68(10) (2014), pp. 1355–1367. DOI:10.1016/j.camwa.2014.09.004
  • C.G. Lange and R.M. Miura, Singular perturbation analysis of boundary value problems for differential-difference equations, SIAM J. Appl. Math. 42(3) (1982), pp. 502–531. DOI:10.1137/0142036
  • T. Linß, Robust convergence of a compact fourth-order finite difference scheme for reaction–diffusion problems, Numer. Math. 111(2) (2008), pp. 239–249. DOI:10.1007/s00211-008-0184-4
  • J.J.H. Miller, E. O'Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific Publishing Company Inc, River Edge, NJ, 1996.
  • N. Negero and G. Duressa, An efficient numerical approach for singularly perturbed parabolic convection–diffusion problems with large time-lag, J. Math. Model. 10(2) (2022), pp. 173–110. DOI:10.22124/JMM.2021.19608.1682
  • N.T. Negero and G.F. Duressa, An exponentially fitted spline method for singularly perturbed parabolic convection–diffusion problems with large time delay, Tamkang J. Math. (2022). https://journals.math.tku.edu.tw/index.php/TKJM/article/view/3983.
  • P. Rai and S. Yadav, Robust numerical schemes for singularly perturbed delay parabolic convection-diffusion problems with degenerate coefficient, Int. J. Comput. Math. 98(1) (2021), pp. 195–221. DOI:10.1080/00207160.2020.1737030
  • H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection–Diffusion–Reaction and Flow Problems. Vol. 24. Springer Science & Business Media, Berlin Heidelberg, 2008.
  • J. Singh, S. Kumar, and M. Kumar, A domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay, Numer. Methods. Partial. Differ. Equ. 34(5) (2018), pp. 1849–1866. DOI:10.1002/num.22256
  • S. Yadav and P. Rai, A higher order numerical scheme for singularly perturbed parabolic turning point problems exhibiting twin boundary layers, Appl. Math. Comput. 376(2020), pp. 125095. DOI:10.1016/j.amc.2020.125095
  • S. Yadav and P. Rai, A higher order scheme for singularly perturbed delay parabolic turning point problem, Eng. Comput. (2020). DOI:10.1108/EC-03-2020-0172

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.