Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A numerical technique for solving singularly perturbed two-point boundary value problems

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

Abstract

In this article, we first convert a second order singularly perturbed boundary value problem (SPBVP) into a pair of initial value problems, which are solved later using exponential time differencing (ETD) Runge–Kutta methods. The stability analysis of the proposed scheme is addressed. Some linear and non-linear problems have been solved to study the applicability of the proposed method.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.

References

  • Andargie A, Reddy UN (2007a) Fitted fourth-order tridiagonal finite difference method for singular perturbation problems. Appl Math Comput 192:90–100

    MathSciNet  Google Scholar 

  • Andargie A, Reddy YN (2007b) An exponentially fitted special second-order finite difference method for solving singular perturbation problems. Appl Math Comput 190:1767–1782

    MathSciNet  Google Scholar 

  • Ashi HA, Cummings LJ, Matthews PC (2010) Exponential time differencing methods: stability analysis and application to the nonlinear Schrodinger equation. Int J Numer Methods Appl 4(2):99–128

    Google Scholar 

  • Aziz T, Khan A (2002) A spline method for second-order singularly perturbed boundary value problems. J Comput Appl Math 147:445–452

    Article  MathSciNet  Google Scholar 

  • Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers. McGraw-Hill, New York

    Google Scholar 

  • Cox SM, Matthews PC (2002) Exponential time differencing for stiff systems. J Comput Phys 176:430–455

    Article  MathSciNet  Google Scholar 

  • de la Hoz F, Vadillo F (2008) An exponential time differencing method for the nonlinear Schrödinger equation. Comput Phys Commun 179:449–456

    Article  Google Scholar 

  • Du Q, Zhu W (2004) Stability analysis and application of exponential time differencing scheme. J Comput Math 22:200–209

    MathSciNet  Google Scholar 

  • Du Q, Zhu W (2005) Analysis and applications of the exponential time differencing schemes and their contour integration modifications. BIT Numer Math 45:307–328

    Article  MathSciNet  Google Scholar 

  • Gasparo MG, Macconi M (1998) New initial value method for singularly perturbed boundary value problems. J Optim Theory Appl 63:213–224

    Article  MathSciNet  Google Scholar 

  • Johnson RS (2005) Singular perturbation theory. Mathematical and analytical techniques with applications to engineering. Springer, Berlin

    Google Scholar 

  • Jung CY, Nguyen TB (2015) Semi-analytical time differencing methods for stiff problems. J Sci Comput 63:355–373

    Article  MathSciNet  Google Scholar 

  • Kadalbajoo MK, Reddy UN (1987) Initial-value technique for a class of nonlinear singular perturbation problems. J Optim Theory Appl 53:395–406

    Article  MathSciNet  Google Scholar 

  • Kaushik A, Kumar V, Vashishth AK (2012) An efficient mixed asymptotic-numerical scheme for singularly perturbed convection diffusion problems. Appl Math Comput 218:8645–8658

    MathSciNet  Google Scholar 

  • Kumar V, Srinivasan B (2015) An adaptive mesh strategy for singularly perturbed convection diffusion problems. Appl Math Model 39:2081–2091

    Article  MathSciNet  Google Scholar 

  • Kumar V, Kaushik A, Vashishth AK (2012) An efficient mixed asymptotic-numerical scheme for singularly perturbed convection diffusion problems. Appl Math Comput 218:8645–8658

    MathSciNet  Google Scholar 

  • Munyakazi JB, Kehinde OO (2022) A new parameter uniform discretization of semilinear singularly perturbed problems. Mathematics 10(13):2254

    Article  Google Scholar 

  • Natesan S, Ramanujam N (1998) Initial value technique for singularly perturbed boundary value problems for second order ordinary differential equations arising in chemical reactor theory. J Optim Theory Appl 97:455–470

    Article  MathSciNet  Google Scholar 

  • Nayfeh AH (1981) Introduction to perturbation techniques. Wiley, New York

    Google Scholar 

  • O’Malley RE (1974) Introduction to singular perturbations. Academic Press, New York

    Google Scholar 

  • Reddy YN, Chakravarthy PP (2003) Method of reduction of order for solving singularly perturbed two-point boundary value problems. Appl Math Comput 136:27–45

    MathSciNet  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Visvesvaraya National Institute of Technology, Nagpur, India

    Pramod Chakravarthy Podila & Rahul Mishra

  2. Universidad de Salamanca, Salamanca, Spain

    Higinio Ramos

Authors
  1. Pramod Chakravarthy Podila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rahul Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Higinio Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pramod Chakravarthy Podila.

Ethics declarations

Conflict of interest

Both authors have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Podila, P.C., Mishra, R. & Ramos, H. A numerical technique for solving singularly perturbed two-point boundary value problems. Comp. Appl. Math. 43, 366 (2024). https://doi.org/10.1007/s40314-024-02880-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-024-02880-7

Keywords

Mathematics Subject Classification

Search

Navigation