Advertisement

Numerical Algorithms

, Volume 76, Issue 2, pp 427–439 | Cite as

A novel efficient method for nonlinear boundary value problems

  • Dang Quang AEmail author
  • Dang Quang Long
  • Ngo Thi Kim Quy
Original Paper

Abstract

In this paper, we propose a novel efficient method for a fourth-order nonlinear boundary value problem which models a statistically bending elastic beam. Differently from other authors, we reduce the problem to an operator equation for the right-hand side function. Under some easily verified conditions on this function in a specified bounded domain, we prove the contraction of the operator. This guarantees the existence and uniqueness of a solution of the problem and the convergence of an iterative method for finding it. The positivity of the solution and the monotony of iterations are also considered. We show that the examples of some other authors satisfy our conditions; therefore, they have a unique solution, while these authors only could prove the existence of a solution. Numerical experiments on these and other examples show the fast convergence of the iterative method.

Keywords

Elastic beam equation Existence and uniqueness of solution Positivity of solution Iterative method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aftabizadeh, A.R.: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116, 415–426 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alves, E., Ma, T.F., Pelicer, M.L.: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. Nonlin. Anal. 71, 3834–3841 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amster, P., Cárdenas Alzate, P.P.: A shooting method for a nonlinear beam equation. Nonlin. Anal. 68, 2072–2078 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bai, Z., Ge, W., Wang, Y.: The method of lower and upper solutions for some fourth-order equations. J. Inequal. Pure Appl. Math. 5(1), Art. 13 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dang, Q.A.: Iterative method for solving the Neumann boundary value problem for biharmonic type equation. J. Comput. Appl. Math. 196, 634–643 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li, Y.: A monotone iterative technique for solving the bending elastic beam equations. Appl. Math. Comput. 217, 2200–2208 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ma, R., Zhang, J., Fu, S.: The method of lower and upper solutions for fourth-order two-point boundary value problems. J. Math. Anal. Appl. 215, 415–422 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ma, T.F., da Silva, J.: Iterative solutions for a beam equation with nonlinear boundary conditions of third order. Appl. Math. Comput. 159, 11–18 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ma, T.F.: Existence results and numerical solutions for a beam equation with nonlinear boundary conditions. Appl. Numer. Math. 47, 189–196 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pao, C.V.: Numerical methods for fourth order nonlinear elliptic boundary value problems. Numer Methods Partial Differ. Equ. 17, 347–368 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer (1984)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Centre for Informatics and ComputingVASTCau GiayVietnam
  2. 2.Institute of Information TechnologyVASTCau GiayVietnam
  3. 3.Thai Nguyen University of Economics and Business AdministrationThai NguyenVietnam

Personalised recommendations