### Abstract

In this paper we develop further the Galerkin technique based on the so-called beam functions with application to nonlinear problems. We make use of the formulas expressing a product of two beam functions into a series with respect to the system. First we prove that the overall convergence rate for a fourth-order linear b.v.p is algebraic fifth order, provided that the derivatives of the sought function up to fifth order exist. It is then shown that the inclusion of a quadratic nonlinear term in the equation does not degrade the fifth-order convergence. We validate our findings on a model problem which possesses analytical solution in the linear case. The agreement between the beam-Galerkin solution and the analytical solution for the linear problem is better than 10^{-12} for 200 terms. We also show that the error introduced by the expansion of the nonlinear term is lesser than 10 ^{-9}. The beam-Galerkin method outperforms finite differences due to its superior accuracy whilst its advantage over the Chebyshev-tau method is attributed to the smaller condition number of the matrices involved in the former.

Original language | English |
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Title of host publication | APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS'33 |

Subtitle of host publication | 33rd International Conference |

Pages | 119-126 |

Number of pages | 8 |

Volume | 946 |

DOIs | |

Publication status | Published - 2007 |

Event | 33rd International Conference on Applications of Mathematics in Engineering and Economics - Sozopol, Bulgaria Duration: 8 Jun 2007 → 14 Jun 2007 |

### Other

Other | 33rd International Conference on Applications of Mathematics in Engineering and Economics |
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Country | Bulgaria |

City | Sozopol |

Period | 8/06/07 → 14/06/07 |

### Keywords

- Beam functions
- Fourth-order boundary value problems
- Galerkin spectral method

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## Cite this

*APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS'33: 33rd International Conference*(Vol. 946, pp. 119-126) https://doi.org/10.1063/1.2806046