Skip to main content
SpringerLink
Log in

A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this study, we consider the time-delayed singularly perturbed parabolic PDEs (SPPPDEs). We know that the classical finite difference scheme will not produce good results for singular perturbation problems on a uniform mesh. Here, we propose a new stable finite difference (NSFD) scheme, which produces good results on a uniform mesh and also on an adaptive mesh. The NSFD scheme is constructed based on the stability of the analytical solution. Results are compared with the results available in the literature and observed that the proposed method is efficient over the existing methods for solving SPPPDEs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Ansari AR, Bakr SA, Shishkin GI (2007) A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. J Comput Appl Math 205:552–566

    Article  MathSciNet  Google Scholar 

  • Arino O, Hbid ML, Ait Dads E (2006) Delay differential equations and applications. Springer, Berlin

    Book  Google Scholar 

  • Bashier EBM, Patidar KC (2011) A novel fitter operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Appl Math Comput 217(9):4728–4739

    MathSciNet  MATH  Google Scholar 

  • Cheng O, Jia-qi M (2005) The nonlinear singularly perturbed problems for predator-prey reaction diffusion equation. J Biomath 20(2):135–141

    MathSciNet  MATH  Google Scholar 

  • Das A, Natesan S (2015) Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh. Appl Math Comput 271:168–186

    MathSciNet  MATH  Google Scholar 

  • Das A, Natesan S (2018) Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations. Int J Comput Math 95:490–510

    Article  MathSciNet  Google Scholar 

  • Doolan ER, Miller JJH, Schilders WHA (1980) Uniform numerical methods for problems with initial and boundary layers. Boole Press, Dublin

    MATH  Google Scholar 

  • Gowrisankar S, Natesan S (2017) \(\varepsilon \)-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations. Int J Comput Math 94:902–921

    Article  MathSciNet  Google Scholar 

  • Kellogg RB, Tsan A (1978) Analysis of some difference approximations for a singular perturbation problem without turning points. Math Comput 32(144):1025–1039

    Article  MathSciNet  Google Scholar 

  • Kumar D, Kumari P (2019) A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay. J Appl Math Comput 59:179–206

    Article  MathSciNet  Google Scholar 

  • Lange CG, Miura RM (1982) Singular perturbation analysis of boundary-value problems for differential-difference equations. SIAM J Appl Math 42:502–531

    Article  MathSciNet  Google Scholar 

  • Miller JJH, Riordan EO, Shishkin IG (1996) Fitted numerical methods for singular perturbation problems. Word Scientific, Singapore

    Book  Google Scholar 

  • Murray JD (2001) Mathematical biology I: an introduction, 3rd edn. Springer, Berlin

    Google Scholar 

  • Musila M, Lansky P (1991) Generalized ’s model for anatomically complex neurons. BioSystems 25(3):179–191

    Article  Google Scholar 

  • Peiraviminaei A, Ghoreishi F (2014) Numerical solutions based on Chebyshev collocation method for singularly perturbed delay parabolic PDEs. Math Methods Appl Sci 37(14):2112–2119

    Article  MathSciNet  Google Scholar 

  • Roos HG, Stynes M, Tobiska L (2008) Numerical methods for singularly perturbed differential equations. Springer, Berlin

    MATH  Google Scholar 

  • Salama AA, AI-Amery DG (2017) A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations. Int J Comput Math 94(12):2520–2546

    Article  MathSciNet  Google Scholar 

  • Shishkin GI (1989) Approximation of solutions of singularly perturbed boundary value problems with a parabolic boundary layer. USSR Comput Math Math Phys 29(4):1–10

    Article  MathSciNet  Google Scholar 

  • Wang K, Wong YS (2014) Pollution-free finite difference schemes for non-homogeneous Helmholtz equation. Int J Numer Anal Model 11:787–815

    MathSciNet  Google Scholar 

  • Wang K, Wong YS, Deng J (2015) Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates. Commun Comput Phys 17:779–807

    Article  MathSciNet  Google Scholar 

  • Wu J (1996) Theory and applications of partial functional differential equations. Springer, New York

    Book  Google Scholar 

  • Xuefei H, Wang K (2018) New finite difference methods for singularly perturbed convection-diffusion equations. Taiwan J Math 22(4):949–978

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, 440010, India

    Pramod Chakravarthy Podila & Kamalesh Kumar

Authors
  1. Pramod Chakravarthy Podila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kamalesh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pramod Chakravarthy Podila.

Additional information

Communicated by Frederic Valentin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Podila, P.C., Kumar, K. A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs. Comp. Appl. Math. 39, 140 (2020). https://doi.org/10.1007/s40314-020-01170-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01170-2

Keywords

Mathematics Subject Classification

Search

Navigation