Title page for etd-0614106-030140


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URN etd-0614106-030140
Author Cheng-Feng Lee
Author's Email Address No Public.
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Department Applied Mathematics
Year 2005
Semester 2
Degree Master
Type of Document
Language English
Title High precision computations of multiquadric collocation method for partial differential equations
Date of Defense 2006-06-07
Page Count 37
Keyword
  • meshless method
  • multiquadric collocation method
  • condition number
  • exponential convergence
  • arbitrary precision computation
  • error estimate
  • optimal shape factor
  • Abstract Multiquadric collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. More amazingly, there are two ways to reduce the error: the traditional way of refining the grid, and the unexpected way of simply increasing the value of shape constant $c$ contained in the multiquadric basis function, $sqrt{r^2 + c^2}$. The latter is accomplished without increasing computational cost. It has been speculated that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $c
    ightarrow infty$. The ability to obtain infinitely accurate solution is limited only by the roundoff error induced instability of matrix solution with large condition number. Using the arbitrary precision computation capability of {it Mathematica}, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented in this paper. A formula for a finite, optimal $c$ value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal $c$ formula can be obtained. These results are supported by numerical examples.
    Advisory Committee
  • Tzon-Tzer Lu - chair
  • Leevan Ling - co-chair
  • Zi-Cai Li - co-chair
  • ALLEN T. L. HORNG - co-chair
  • Chien-Sen Huang - advisor
  • Files
  • etd-0614106-030140.pdf
  • indicate in-campus access in a year and off_campus not accessible
    Date of Submission 2006-06-14

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