Mathematics of Computation

On the highly accurate summation of certain series occurring in plate contact problems

Author(s): D. A. MacDonald.
Journal: Math. Comp. 66 (1997), 1619-1627.
MSC (1991): Primary 65B10; Secondary 35Q80, 35J25
Retrieve article in: PDF
This article is available free of charge

Abstract: The infinite series $R_p = \sum _{k=1}^\infty {(2 k - 1)}^{- p} \, x^{2 k - 1}, 0 <1-x\ll 1 , p = 2 \quad \text {or} 3 \, ,$ and the related series

$\begin{equation*}\begin {split} C(x,b,2)&=\sum _{k=1}^\infty {(2k-1)}^{-2} \cosh (2k-1)x/\cosh (2k-1)b,\quad 0 <1-x/b \ll 1, S(x,b,3)&=\sum _{k=1}^\infty {(2k-1)}^{-3} \sinh (2k-1)x/\cosh (2k-1)b, \end {split} \end{equation*}$

are of interest in problems concerning contact between plates and unilateral supports. This article will re-examine a previously published result of Baratella and Gabutti for $R_p$

, and will present new, rapidly convergent, series for $C(x,b,2)$

and

1.: M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover Publications Inc., New York, (1966). MR 34:8606
2.: P. Baratella and B. Gabutti, Uniform approximation on unit disk of series arising in plate contact problems, Jl of Comp. & Appl. Math. 62 (1995) 181-196. MR 96h:65003
3.: J. Boersma and J.P. Dempsey, Numerical evaluation of Legendre's Chi-function, Math. Comp. 59 (1992) 157-163. MR 92k:65008
4.: J.P. Dempsey , L.M. Keer, N.B. Patel and M.L. Glasser, Contact between plates and unilateral supports, ASME J Appl. Mech 51 (1984) 324-328.
5.: K.M. Dempsey, D Liu, and J.P. Dempsey, Plana's summation formula for $\sum ^\infty _{m=1,3,\dotsc }m^{-2} \sin (m\alpha ),m^{-3}\cos (m\alpha ),m^{-2}A^m,m^{-3}A^m$ , Math. Comp. 55 (1990) 693-704. MR 91b:65003
6.: M. Filonenko-Borodich, Theory of Elasticity, Mir Publishers, Moscow (1968).
7.: W. Gautschi, On certain slowly convergent series occurring in plate contact problems , Math. Comp. 57, (1991) 325-338. MR 91j:40002
8.: E.T. Whittaker and G.N. Watson, A course of modern analysis, C.U.P., Cambridge (1927).

Retrieve articles in Mathematics of Computation with MSC (1991): 65B10, 35Q80, 35J25

D. A. MacDonald
Affiliation: Department of Mathematical Sciences, P.O. Box 147, The University, Liverpool L69 3BX, United Kingdom
Email: sx10@liv.uk

DOI: 10.1090/S0025-5718-97-00869-7
PII: S 0025-5718(97)00869-7
Keywords: Slowly convergent series, boundary value problems.
Received by editor(s): July 10, 1996
Copyright of article: Copyright 1997, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS