Skip to main content
SpringerLink
Log in

Solving the Dirichlet problem for fully fourth order nonlinear differential equation

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this paper we study the existence and uniqueness of a solution and propose an iterative method for solving a beam problem which is described by the fully fourth order equation

$$\begin{aligned} u^{(4)}(x)=f(x,u(x),u'(x),u''(x),u'''(x)), \quad a< x < b \end{aligned}$$

associated with the Dirichlet boundary conditions. This problem was well studied by Agarwal by the reduction of it to a nonlinear operator equation for the unknown function u(x). Here we propose a novel approach by the reduction of the problem to an operator equation for the nonlinear term \(\varphi (x)=f(x,u(x),u'(x),u''(x),u'''(x))\). Under some easily verified conditions on the function f in a specified bounded domain, we prove the existence, uniqueness and positivity of a solution and the convergence of an iterative method for finding it. Some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method. The advantages of the proposed approach to the problem over the well-known approach of Agarwal is shown on examples.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Agarwal, R.P., Chow, Y.M.: Iterative methods for a fourth order boundary value problem. J. Comput. Appl. Math. 10, 203–217 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, R.P.: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986)

    Book  MATH  Google Scholar 

  3. Bai, Z.: The upper and lower solution method for some fourth-order boundary value problems. Nonlinear Anal. 67, 1704–1709 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. 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  MATH  Google Scholar 

  6. Dang, Q.A., Ngo, T.K.Q.: New fixed point approach for a fully nonlinear fourth order boundary value problem. Bol. Soc. Parana. Math. 36(4), 209–223 (2018). https://doi.org/10.5269/bspm.v36i4.33584

    Article  MathSciNet  MATH  Google Scholar 

  7. Doedel, E.J.: Finite difference collocation methods for nonlinear two point boundary value problems. SIAM J. Numer. Anal. 16, 173–185 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Du, J., Cui, M.: Constructive proof of existence for a class of fourth-order nonlinear BVPs. Comput. Math. Appl. 59, 903–911 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ertürk, V.S., Momani, S.: Comparing numerical methods for solving fourth-order boundary value problems. Appl. Math. Comput. 188, 1963–1968 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Feng, H., Ji, D., Ge, W.: Existence and uniqueness of solutions for a fourth-order boundary value problem. Nonlinear Anal. 70, 3561–3566 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, Y., Liang, Q.: Existence results for a Fourth-order boundary value problem. J. Funct. Spaces Appl. (2013) (Article ID 641617)

  12. Li, Y.: Existence of positive solutions for the cantilever beam equations with fully nonlinear terms. Nonlinear Anal. Real World Appl. 27, 221–237 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Melnikov, Yuri A., Melnikov, Max Y.: Greens Functions: Construction and Applications. De Gruyter, Berlin (2012)

    Book  MATH  Google Scholar 

  14. Minhós, F., Gyulov, T., Santos, A.I.: Lower and upper solutions for a fully nonlinear beam equation. Nonlinear Anal. 71, 281–292 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mohanty, R.K.: A fourth-order finite difference method for the general one-dimensional nonlinear biharmonic problems of first kind. J. Comput. Appl. Math. 114, 275–290 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Noor, M.A., Mohyud-Din, S.T.: A efficient method for fourth-order boundary value problems. Comput. Math. Appl. 54, 1101–1111 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pei, M., Chang, S.K.: Existence of solutions for a fully nonlinear fourth-order two-point boundary value problem. J. Appl. Math. Comput. 37, 287–295 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers for their helpful comments and suggestions for improving the quality of the paper. The first author, Quang A. Dang, is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the Grant number 102.01-2017.306, and the second author, Thanh Huong Nguyen, is supported by Thai Nguyen University under the project ƉH2016-TN07-02.

Author information

Authors and Affiliations

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

    Quang A Dang

  2. College of Sciences, Thainguyen University, Thainguyen, Vietnam

    Thanh Huong Nguyen

Authors
  1. Quang A Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thanh Huong Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quang A Dang.

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

Dang, Q.A., Nguyen, T.H. Solving the Dirichlet problem for fully fourth order nonlinear differential equation. Afr. Mat. 30, 623–641 (2019). https://doi.org/10.1007/s13370-019-00671-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-019-00671-6

Keywords

Mathematics Subject Classification

Search

Navigation