research-article

Approximate rational solutions torational ODEs defined on discrete differentiable curves

Authors:

Hongbo Li,

Ruiyong Sun,

Shoubin Yao,

Ge LiAuthors Info & Claims

ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation

Pages 217 - 224

https://doi.org/10.1145/1993886.1993921

Published: 08 June 2011 Publication History

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Abstract

In this paper, a new concept is proposed for discrete differential geometry: discrete n-differentiable curve, which is a tangent n-jet on a sequence of space points. A complete method is proposed to solve ODEs of the form n^(m)=F(r, r', ..., r⁽ⁿ⁾, n, n', ..., n^(m-1), u)/G(r, r', ..., r⁽ⁿ⁾, n, n', ..., n^(m-1), u), where F, G are respectively vector-valued and scalar-valued polynomials, where r is a discrete curve obtained by sampling along an unknown smooth curve parametrized by u, and where n is the vector field to be computed along the curve. Our Maple-13 program outputs an approximate rational solution with the highest order of approximation for given data and neighborhood size.

The method is used to compute rotation minimizing frames of space curves in CAGD. For one-step backward-forward chasing, a 6th-order approximate rational solution is found, and 6 is guaranteed to be the highest order of approximation by rational functions. The theoretical order of approximation is also supported by numerical experiments.

References

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Y. Chang, G. Corliss. ATOMFT: solving ODEs and DAEs using Taylor series. Comput. Math. Appl. 28: 209--233, 1994.

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R. Feng, X. Gao. A polynomial time algorithm for finding rational general solutions of first-order autonomous ODEs. J. Symb. Comput. 41}(7): 739--762, 2006.

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N. Shawagfeh, D. Kaya. Comparing numerical methods for the solutions of systems of ODEs. Appl. Math. Letters 17: 323--328, 2004.

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P. Kim and P.J. Olver. Geometric integration via multi-space. Regular and Chaotic Dynamics 9 (2004) 213--226.

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W. Wang, B. Jutter, D. Zheng, Y. Liu. Computation of rotation minimizing frames. ACM Trans. Graphics 27, 2008.

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Index Terms

Approximate rational solutions torational ODEs defined on discrete differentiable curves
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations

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Published In

ISSAC '11: Proceedings of the 36th international symposium on Symbolic and algebraic computation

June 2011

372 pages

ISBN:9781450306751

DOI:10.1145/1993886

General Chair:
Éric Schost
The University of Western Ontario, Canada
,
Program Chair:
Ioannis Z. Emiris
University of Athens, Greece

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 June 2011

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ISSAC '11

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SIGSAM

ISSAC '11: International Symposium on Symbolic and Algebraic Computation

June 8 - 11, 2011

California, San Jose, USA

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Overall Acceptance Rate 395 of 838 submissions, 47%

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