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URN etd-0630108-234657
Author Daniel Ting-chun
Author's Email Address No Public.
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Department Applied Mathematics
Year 2007
Semester 2
Degree Master
Type of Document
Language English
Title The Trefftz Method using Fundamental Solutions for
Biharmonic Equations
Date of Defense 2008-05-29
Page Count 71
Keyword
  • stability analysis
  • greedy adaptive techniques
  • error analysis
  • singularity problems
  • particular solutions
  • biharmonic equations
  • Almansi
  • fundamental solutions
  • Abstract In this thesis, the analysis of the method of fundamental solution(MFS) is expanded for biharmonic equations. The bounds of errors are derived for the traditional and the Almansi's approaches in bounded simply-connected domains. The exponential and the polynomial convergence rates are obtained from highly and finite smooth solutions, respectively. Also the bounds of condition number are derived for the disk domains, to show the exponential growth rates. The analysis in this thesis is the first time to provide the rigor analysis of the CTM for biharmonic equations, and the intrinsic nature of accuracy and stability is similar to that of Laplace's equation.
    Numerical experiment are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions(MPS) is always superior to MFS, supported by numerical examples. However, if such singular particular solutions near the singular points can not be found, the local refinement of collocation nodes and the greedy adaptive techniques can be used. It seems that the greedy adaptive techniques may provide a better solution for singularity problems. Beside, the numerical solutions by Almansi's approaches are slightly better in accuracy and stability than those by the traditional FS. Hence, the MFS with Almansi's approaches is recommended, due to the simple analysis, which can be obtained directly from the analysis of MFS for Laplace's equation.
    Advisory Committee
  • Tzon-Tzer Lu - chair
  • Cheng-Sheng Chien - co-chair
  • Hung-Tsai Huang - co-chair
  • Chien-Sen Huang - co-chair
  • Zi-Cai Li - advisor
  • Files
  • etd-0630108-234657.pdf
  • indicate access worldwide
    Date of Submission 2008-06-30

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