On the beam functions spectral expansions for fourth-order boundary value problems

N. C. Papanicolaou, C. I. Christov

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

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 languageEnglish
Title of host publicationAPPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS'33
Subtitle of host publication33rd International Conference
Pages119-126
Number of pages8
Volume946
DOIs
Publication statusPublished - 2007
Event33rd International Conference on Applications of Mathematics in Engineering and Economics - Sozopol, Bulgaria
Duration: 8 Jun 200714 Jun 2007

Other

Other33rd International Conference on Applications of Mathematics in Engineering and Economics
CountryBulgaria
CitySozopol
Period8/06/0714/06/07

Keywords

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

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  • Cite this

    Papanicolaou, N. C., & Christov, C. I. (2007). On the beam functions spectral expansions for fourth-order boundary value problems. In APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS'33: 33rd International Conference (Vol. 946, pp. 119-126) https://doi.org/10.1063/1.2806046