[go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Second-order a priori and a posteriori error estimations for integral boundary value problems of nonlinear singularly perturbed parameterized form

  • Research
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this work, we present the a priori and a posteriori error analysis of a hybrid difference scheme for integral boundary value problems of nonlinear singularly perturbed parameterized form. The discretization for the nonlinear parameterized equation constitutes a hybrid difference scheme which is based on a suitable combination of the trapezoidal scheme and the backward difference scheme. Further, we employ the composite trapezoidal scheme for the discretization of the nonlocal boundary condition. A priori error estimation is provided for the proposed hybrid scheme, which leads to second-order uniform convergence on various a priori defined meshes. Moreover, a detailed a posteriori error analysis is carried out for the present hybrid scheme which provides a proper discretization of the error equidistribution at each partition. Numerical results strongly validate the theoretical findings for nonlinear problems with integral boundary conditions.

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

Algorithm 1
Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Availability of Supporting Data

(not applicable)

References

  1. Jankowski, T.: Application of the numerical-analytic method to systems of differential equations with parameter. Ukr. Math. J. 54(4), 671–683 (2002)

    MathSciNet  Google Scholar 

  2. Samoilenko, A.M., Martynyuk, S.V.: Justification of a numerical-analytic method of successive approximations for problems with integral boundary conditions. Ukr. Math. J. 43(9), 1150–1157 (1991)

    MathSciNet  Google Scholar 

  3. Srivastava, H. M., Nain, A. K., Vats, R. K., Das, P.: A theoretical study of the fractional-order p-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam-Hyers stability, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 117(160) (2023). https://doi.org/10.1007/s13398-023-01488-6

  4. O’Malley, R. E.: Singular perturbation methods for ordinary differential equations, vol. 89, Springer (1991)

  5. Iragi, B.C., Munyakazi, J.B.: A uniformly convergent numerical method for a singularly perturbed Volterra integro-differential equation. Int. J. Comput. Math. 97(4), 759–771 (2020)

    MathSciNet  Google Scholar 

  6. Mbroh, N.A., Munyakazi, J.B.: A robust numerical scheme for singularly perturbed parabolic reaction-diffusion problems via the method of lines. Int. J. Comput. Math. 99(6), 1139–1158 (2022)

    MathSciNet  Google Scholar 

  7. Munyakazi, J.B.: A uniformly convergent nonstandard finite difference scheme for a system of convection–diffusion equations. Comput. Appl. Math. 34, 1153–1165 (2015)

    MathSciNet  Google Scholar 

  8. Mbroh, N.A., Munyakazi, J.B.: A fitted operator finite difference method of lines for singularly perturbed parabolic convection–diffusion problems. Math. Comput. Simul. 165, 156–171 (2019)

    MathSciNet  Google Scholar 

  9. Durmaz, M.E., Amiraliyev, G.M.: A robust numerical method for a singularly perturbed Fredholm integro-differential equation. Mediterr. J. Math. 18, 24 (2021)

    MathSciNet  Google Scholar 

  10. Cimen, E., Amiraliyev, G.M.: Uniform convergence method for a delay differential problem with layer behaviour. Mediterr. J. Math. 16, 57 (2019)

    MathSciNet  Google Scholar 

  11. Sahoo, S.K., Gupta, V.: Higher order robust numerical computation for singularly perturbed problem involving discontinuous convective and source term. Math. Methods Appl. Sci. 45(8), 4876–4898 (2022)

    MathSciNet  Google Scholar 

  12. Gupta, V., Sahoo, S.K., Dubey, R.K.: Robust higher order finite difference scheme for singularly perturbed turning point problem with two outflow boundary layers. Comput. Appl. Math. 40, 179 (2021)

    MathSciNet  Google Scholar 

  13. Shiromani, R., Shanthi, V., Das, P.: A higher order hybrid-numerical approximation for a class of singularly perturbed two-dimensional convection-diffusion elliptic problem with non-smooth convection and source terms. Comput. Math. Appl. 142, 9–30 (2023)

    MathSciNet  Google Scholar 

  14. Kudu, M.: A parameter uniform difference scheme for the parameterized singularly perturbed problem with integral boundary condition. Adv. Difference Equ. 2018, 170 (2018)

    MathSciNet  Google Scholar 

  15. Kudu, M., Amirali, I., Amiraliyev, G.M.: Uniform numerical approximation for parameter dependent singularly perturbed problem with integral boundary condition. Miskolc Math. Notes 19(1), 337–353 (2018)

    MathSciNet  Google Scholar 

  16. Kumar, S., Kumar, S.: Sumit, A posteriori error estimation for quasilinear singularly perturbed problems with integral boundary condition. Numer. Algorithm. 89, 791–809 (2022)

    Google Scholar 

  17. Mortimer, R.G.: Physical chemistry. Academic Press, London (2008)

    Google Scholar 

  18. Murray, J.D.: Mathematical biology I. An introduction, Springer, New York (2002)

    Google Scholar 

  19. Kokotović, P., Khalil, H. K., O’Reilly, J.: Singular perturbation methods in control: analysis and design, SIAM (1999)

  20. Amiraliyev, G.M., Duru, H.: A note on a parameterized singular perturbation problem. J. Comput. Appl. Math. 182(1), 233–242 (2005)

    MathSciNet  Google Scholar 

  21. Saini, S., Das, P., Kumar, S.: Computational cost reduction for coupled system of multiple scale reaction diffusion problems with mixed type boundary conditions having boundary layers, Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales. Serie A. Matematicas (2023). https://doi.org/10.1007/s13398-023-01397-8

    Article  Google Scholar 

  22. Amiraliyev, G.M., Kudu, M., Duru, H.: Uniform difference method for a parameterized singular perturbation problem. Appl. Math. Comput. 175(1), 89–100 (2006)

    MathSciNet  Google Scholar 

  23. Kumar, S., Kumar, M.: A second order uniformly convergent numerical scheme for parameterized singularly perturbed delay differential problems. Numer. Algorithm. 76(2), 349–360 (2017)

    MathSciNet  Google Scholar 

  24. Das, P.: Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J. Comput. Appl. Math. 290, 16–25 (2015)

    MathSciNet  Google Scholar 

  25. Das, P.: An a posteriori based convergence analysis for a nonlinear singularlyperturbed system of delay differential equations on an adaptive mesh. Numer. Algorithm. 81, 465–487 (2019)

    Google Scholar 

  26. Cen, Z.: A second-order difference scheme for a parameterized singular perturbation problem. J. Comput. Appl. Math. 221(1), 174–182 (2008)

    MathSciNet  Google Scholar 

  27. Wang, Y., Chen, S., Wu, X.: A rational spectral collocation method for solving a class of parameterized singular perturbation problems. J. Comput. Appl. Math. 233(10), 2652–2660 (2010)

    MathSciNet  Google Scholar 

  28. Xie, F., Wang, J., Zhang, W., He, M.: A novel method for a class of parameterized singularly perturbed boundary value problems. J. Comput. Appl. Math. 213(1), 258–267 (2008)

    MathSciNet  Google Scholar 

  29. Kudu, M., Amirali, I., Amiraliyev, G.M.: A fitted second-order difference method for a parameterized problem with integral boundary condition exhibiting initial layer. Mediterr. J. Math. 18, 106 (2021)

    MathSciNet  Google Scholar 

  30. Kudu, M., Amirali, I., Amiraliyev, G.M.: A second order accurate method for a parameterized singularly perturbed problem with integral boundary condition. J. Comput. Appl. Math. 404, 113894 (2022)

    MathSciNet  Google Scholar 

  31. Satra, S., Mohapatra, J., Das, P., Choudhuri, D.: Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis. Comput. Math. Appl. (2023). https://doi.org/10.1016/j.camwa.2023.09.008

    Article  MathSciNet  Google Scholar 

  32. Linß, T.: Layer-adapted meshes for reaction-convection-diffusion problems, Springer (2009)

  33. de Boor, C.: Good approximation by splines with variable knots. In: Spline Functions and Approximation Theory, pp. 57–72. Springer (1973)

  34. Huang, J., Cen, Z., Xu, A.: An improved a posteriori error estimation for a parameterized singular perturbation problem. Appl. Math. Lett. 114, 106912 (2021)

    MathSciNet  Google Scholar 

  35. Atkinson, K. E.: An introduction to numerical analysis, John wiley & sons (2008)

  36. Farrell, P., Hegarty, A., Miller, J. J., O’Riordan, E., Shishkin, G. I.: Robust computational techniques for boundary layers, CRC Press (2000)

Download references

Acknowledgements

The author Sunil Kumar would like to thank the Science and Engineering Research Board (SERB), Government of India, for giving the research support grant CRG/2023/003228 for the present work. The author Pratibhamoy Das wanted to acknowledge the support of the Science and Engineering Research Board, Government of India, through the Project No. MTR/2021/000797 for carrying out the present work. The authors gratefully acknowledge the valuable comments and suggestions from the anonymous referees.

Funding

Sunil Kumar was supported by the Science and Engineering Research Board, Government of India, through the Project No. CRG/2023/003228. Pratibhamoy Das was supported by the Science and Engineering Research Board, Government of India, through the Project No. MTR/2021/000797

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, India

    Shashikant Kumar & Sunil Kumar

  2. Department of Mathematics, Indian Institute of Technology, Patna, India

    Pratibhamoy Das

Authors
  1. Shashikant Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sunil Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pratibhamoy Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors have equivalently contributed to the work.

Corresponding author

Correspondence to Sunil Kumar.

Ethics declarations

Ethical Approval

(not applicable)

Competing Interests

The authors declare no competing interests.

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

Kumar, S., Kumar, S. & Das, P. Second-order a priori and a posteriori error estimations for integral boundary value problems of nonlinear singularly perturbed parameterized form. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01918-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11075-024-01918-5

Keywords

Mathematics Subject Classification (2010)