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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic solution of a small parametered $ 2$-D integral equation arising from a contact problem of elasticity based on the solution of a $ 2$-D integral equation


Author: Tian Quan Yun
Journal: Proc. Amer. Math. Soc. 123 (1995), 1221-1227
MSC: Primary 73T05
MathSciNet review: 1231307
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Abstract: Asymptotic solution of a 2-D integral equation of constant kernel with small parameter $ \varepsilon $,

$\displaystyle \int_0^\pi {\int_{ - \infty }^\infty p } \,dsd\psi + \varepsilon r\int_0^\pi {\int_{ - \infty }^\infty p } \,ds\cos \psi d\psi = G(r),$

which occurs in a more exact form of Hertz's contact problem in elasticity, is presented in this paper based on the solution of a 2-D integral equation

$\displaystyle \int_0^\pi {\int_{ - \infty }^\infty} pdsd\psi = F(r)$

with constant kernel, and the unknown function $ p = p(s,\psi ) = p(t,\phi )$ is subjected to the following two constraints:

\begin{displaymath}\begin{array}{*{20}{c}} {p(t,\phi ) = p(t)\quad \forall \phi ... ...phi ) \notin E = \{ (t,\phi )\vert t \leq a\} } \\ \end{array} \end{displaymath}

where $ (s,\psi )$ are local polar coordinates with origin at $ M(r,0)$, with $ (r,0)$ measured by global polar coordinates $ (t,\phi )$ with origin at $ O(0,0)$. A more exact solution of Hertz's contact problem is found as an example.

References [Enhancements On Off] (What's this?)

  • [1] S. P. Timoshenko and J. N. Goodier, Theory of elasticity, 3rd ed., McGraw-Hill, New York, 1970, pp. 411-412.
  • [2] Tian Quan Yun, The exact integral equation of Hertz’s contact problem, Appl. Math. Mech. 12 (1991), no. 2, 165–169 (Chinese, with English summary); English transl., Appl. Math. Mech. (English Ed.) 12 (1991), no. 2, 181–185. MR 1104095 (92c:73125), http://dx.doi.org/10.1007/BF02016536
  • [3] Gabor T. Herman, Image reconstruction from projections, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. The fundamentals of computerized tomography; Computer Science and Applied Mathematics. MR 630896 (83e:92008)
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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1231307-4
PII: S 0002-9939(1995)1231307-4
Keywords: Hertz's contact problem, Radon transform, Abel integral equation, asymptotic expansion
Article copyright: © Copyright 1995 American Mathematical Society