10.1186/2251-7456-6-1

Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions

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  • Fazhan Geng*1,
  1. Department of Mathematics, Changshu Institute of Technology, 215500, CN

Published in Issue 2012-05-18

How to Cite

Geng, F. (2012). Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions. Mathematical Sciences, 6(1). https://doi.org/10.1186/2251-7456-6-1
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Abstract

Abstract Purpose This paper investigates an analytical approximate solution of a fourth-order differential equation with nonlinear boundary conditions modeling beams on elastic foundations using iterative reproducing kernel method. Methods The solution obtained using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve the problems since there is no method of obtaining a reproducing kernel satisfying nonlinear boundary conditions. The aim of this paper is to fill this gap. Results Several illustrative examples are given to demonstrate the effectiveness of the present method. Conclusions Results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the beam equation with third-order nonlinear boundary conditions.

Keywords

  • Iterative reproducing kernel method,
  • Beam equation,
  • Fourth-order boundary value problem,
  • Nonlinear boundary conditions
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References

  1. Agarwal and Chow (1984) Iterative methods for a fourth order boundary value problem https://doi.org/10.1016/0377-0427(84)90058-X
  2. Feireisl (1993) Non-zero time periodic solutions to an equation of Petrovsky type with nonlinear boundary conditions: slow oscillations of beams on elastic bearings
  3. Humphreys (1997) Numerical mountain pass solutions of a suspension bridge equation https://doi.org/10.1016/S0362-546X(96)00020-X
  4. Ma and Silva (2004) Iterative solutions for a beam equation with nonlinear boundary conditions of third order https://doi.org/10.1016/j.amc.2003.08.088
  5. Geng (2009) A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems https://doi.org/10.1016/j.amc.2009.02.053
  6. Geng and Cui (2007) Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space https://doi.org/10.1016/j.amc.2007.03.016
  7. Cui and Lin (2009) Nova Science Pub Inc
  8. Berlinet and Thomas-Agnan (2004) Kluwer Academic Publishers https://doi.org/10.1007/978-1-4419-9096-9
  9. Daniel (2003) Springer
  10. Cui and Geng (2007) Solving singular two-point boundary value problem in reproducing kernel space https://doi.org/10.1016/j.cam.2006.04.037
  11. Geng and Cui (2007) Solving a nonlinear system of second order boundary value problems https://doi.org/10.1016/j.jmaa.2006.05.011
  12. Du and Cui (2009) Constructive approximation of solution for fourth-order nonlinear boundary value problems https://doi.org/10.1002/mma.1064
  13. Yao and Lin (2009) Solving singular boundary-value problems of higher even-order https://doi.org/10.1016/j.cam.2008.02.010
  14. Li and Geng (2008) Solving a class of singular boundary value problems in the reproducing kernel Hilbert space
  15. Geng and Shen (2010) Solving a Volterra integral equation with weakly singular kernel in the reproducing kernel space
  16. Geng (2009) Analytic approximations of solutions to systems of ordinary differential equations with variable coefficients
  17. Li and Cui (2003) The exact solution for solving a class nonlinear operator equations in the reproducing kernel space https://doi.org/10.1016/S0096-3003(02)00370-3