Existence results and numerical method for solving a fourth-order nonlinear integro-differential equation | Numerical Algorithms Skip to main content
Springer Nature Link
Log in

Existence results and numerical method for solving a fourth-order nonlinear integro-differential equation

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we consider a boundary value problem (BVP) for a fourth-order nonlinear integro-differential equation. By reducing the problem to an operator equation, we establish the existence and uniqueness of the solution and construct a numerical method for solving it. We prove that the method is of second-order accuracy and obtain an estimate for the total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical 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
¥17,985 /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 includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Aruchunan, E., Wu, Y., Wiwatanapataphee, B., Jitsangiam, P.: A new variant of arithmetic mean iterative method for Fourth order integro-differential equations solution. In: 2015 3rd International Conference on Artificial Intelligence, Modelling and Simulation (AIMS), Kota Kinabalu, pp. 82–87. https://doi.org/10.1109/AIMS.2015.24 (2015)

  2. Chen, J., Huang, Y., Rong, H., Wu, T., Zeng, T.: A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation. J. Comput. Appl. Math. 290, 633–640 (2015)

    Article  MathSciNet  Google Scholar 

  3. Chen, J., He, M., Zeng, T.: A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation II: Efficient algorithm for the discrete linear system. J. Vis. Commun. Image R 58, 112–118 (2019)

    Article  Google Scholar 

  4. Dang, Q.A., Ngo, T.K.Q.: Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term. Nonlinear Anal. Real World Appl. 36, 56–68 (2017)

    Article  MathSciNet  Google Scholar 

  5. Dang, Q.A., Dang, Q.L., Ngo, T.K.Q.: A novel efficient method for nonlinear boundary value problems. Numer. Algor. 76, 427–439 (2017)

    Article  MathSciNet  Google Scholar 

  6. Dang, Q.A., Nguyen, T.H.: The unique solvability and approximation of BVP for a nonlinear fourth order kirchhoff type equation, east asian. J. Appl. Math. 8(2), 323–335 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Dang, Q.A., Nguyen, T.H.: Existence results and iterative method for solving a nonlinear biharmonic equation of Kirchhoff type. Comput. Math. Appl. 76, 11–22 (2018)

    Article  MathSciNet  Google Scholar 

  8. Dang, Q.A., Dang, Q.L.: A simple efficient method for solving sixth-order nonlinear boundary value problems, Comp. Appl. Math 37(1). https://doi.org/10.1007/s40314-018-0643-1 (2018)

  9. Dang, Q.A., Nguyen, T.H.: Existence results and numerical method for a fourth order nonlinear problem. Int. J. Appl. Comput. Math 4, 148 (2018). https://doi.org/10.1007/s40819-018-0584-9

    Article  MathSciNet  MATH  Google Scholar 

  10. Dang, Q.A., Vu, T.L.: Iterative method for solving a nonlinear fourth order boundary value problem. Computers and Mathematics with Applications 60, 112–121 (2010)

    Article  MathSciNet  Google Scholar 

  11. Dascioglu, A., Sezer, M.: A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form. Int. J. Comput. Math. 84(4), 527–539 (2007)

    Article  MathSciNet  Google Scholar 

  12. Lakestania, M., Dehghanb, M.: Numerical solution of fourth-order integro-differential equations using Chebyshev Cardinal Functions. Int. J. Comput. Math. 87, 1389–1394 (2010)

    Article  MathSciNet  Google Scholar 

  13. Singh, R., Wazwaz, A.M.: Numerical solutions of fourth-order Volterra integro-differential equations by the Green’s function and decomposition method. Math. Sci. 10, 159–166 (2016)

    Article  MathSciNet  Google Scholar 

  14. Sweilam, N.H.: Fourth order integro-differential equations using variational iteration method. Comput. Math. Appl. 54, 1086–1091 (2007)

    Article  MathSciNet  Google Scholar 

  15. Tahernezhad, T., Jalilian, R.: Exponential spline for the numerical solutions of linear Fredholm integro-differential equations. Adv. Difference Equ. 2020, 141 (2020)

    Article  MathSciNet  Google Scholar 

  16. Yulan, W., Chaolu, T., Jing, P.: New algorithm for second-order boundary value problems of integro-differential equation. J. Comput. Appl. Math. 229, 1–6 (2009)

    Article  MathSciNet  Google Scholar 

  17. Wang, J.: Monotone iterative technique for nonlinear fourth order integro-differential equations arXiv:2003.04697 [math.CA] (2020)

  18. Zhuang, Q., Ren, Q.: Numerical approximation of a nonlinear fourth-order integro-differential equation by spectral method. Appl. Math. Comput. 232, 775–783 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Karman, T.V., Biot, M.A.: Mathematical Methods in Engineering: an Introduction to the Mathematical Treatment of Engineering Problems. McGraw Hill, New York (1940)

    MATH  Google Scholar 

  20. Pugsley, A.: The Theory of Suspension Bridge. Edward Arnold Pub. Ltd , London (1968)

    Google Scholar 

  21. Itzykson, C., Zube, J.B.: Quantum Field Theory. Dover Publications, INC, Mineola (2005)

    Google Scholar 

Download references

Funding

This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 102.01-2021.03.

Author information

Authors and Affiliations

  1. Institute of Information Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    Dang Quang Long

  2. Centre for Informatics and Computing, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    Dang Quang A

Authors
  1. Dang Quang Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dang Quang A
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dang Quang A.

Additional information

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

Long, D.Q., A, D.Q. Existence results and numerical method for solving a fourth-order nonlinear integro-differential equation. Numer Algor 90, 563–576 (2022). https://doi.org/10.1007/s11075-021-01198-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01198-3

Keywords

Search

Navigation