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Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations

Chandler-Wilde, Simon N. and Peplow, Andrew T. LU (1995) In Journal of Integral Equations and Applications 7(3). p.303-327
Abstract

We consider second kind integral equations of the form x(s) - ʃΩ k(s, t)x(t) dt = y(s) (abbreviated x - Kx = y), in which Ω is some unbounded subset of Rn Let Xp denote the weighted space of functions x continuous on Ω and satisfying x(s) = O(|s| -p), s → ∞. We show that if the kernel k(s, t) decays like | s - t | -q as | s - t | → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∊ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I - K)-1 ∊ B(X0), then (I - K)-1 ∊ B(Xp) for 0 ≤ p < q, and (I - K)-1 ∊ B(Xq) if further conditions on k hold. Thus, if... (More)

We consider second kind integral equations of the form x(s) - ʃΩ k(s, t)x(t) dt = y(s) (abbreviated x - Kx = y), in which Ω is some unbounded subset of Rn Let Xp denote the weighted space of functions x continuous on Ω and satisfying x(s) = O(|s| -p), s → ∞. We show that if the kernel k(s, t) decays like | s - t | -q as | s - t | → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∊ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I - K)-1 ∊ B(X0), then (I - K)-1 ∊ B(Xp) for 0 ≤ p < q, and (I - K)-1 ∊ B(Xq) if further conditions on k hold. Thus, if k(s, t) = 0(| s - t | -q), | s - t | → ∞, and y(s) = O(| s | -p), s → ∞, the asymptotic behavior of the solution x may be estimated as x(s) = O(| s | -r), | s | → ∞, r:= min(p, q). The case when k(s, t) = k(s - t), so that the equation is of Wiener-Hopf type, receives especial attention. Conditions, in terms of the symbol of I - K, for I - K to be invertible or Fredholm on Xp are established for certain cases (Ω a half-space or cone). A boundary integral equation, which models three-dimensional acoustic propagation above flat ground, absorbing apart from an infinite rigid strip, illustrates the practical application and sharpness of the above results. This integral equation models, in particular, road traffic noise propagation along an infinite road surface surrounded by absorbing ground. We prove that the sound propagating along the rigid road surface eventually decays with distance at the same rate as sound propagating above the absorbing ground.

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published
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Journal of Integral Equations and Applications
volume
7
issue
3
pages
25 pages
publisher
Rocky Mountain Mathematics Consortium
external identifiers
  • scopus:0000261879
ISSN
0897-3962
DOI
10.1216/jiea/1181075881
language
English
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4b0b7cee-5190-4eef-aaee-07203aff0727
date added to LUP
2021-02-15 19:59:35
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@article{4b0b7cee-5190-4eef-aaee-07203aff0727,
  abstract     = {<p>We consider second kind integral equations of the form x(s) - ʃ<sub>Ω</sub> k(s, t)x(t) dt = y(s) (abbreviated x - Kx = y), in which Ω is some unbounded subset of R<sup>n</sup> Let X<sub>p</sub> denote the weighted space of functions x continuous on Ω and satisfying x(s) = O(|s| <sup>-p</sup>), s → ∞. We show that if the kernel k(s, t) decays like | s - t | <sup>-q</sup> as | s - t | → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∊ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I - K)<sup>-1</sup> ∊ B(X<sub>0</sub>), then (I - K)<sup>-1</sup> ∊ B(X<sub>p</sub>) for 0 ≤ p &lt; q, and (I - K)<sup>-1</sup> ∊ B(X<sub>q</sub>) if further conditions on k hold. Thus, if k(s, t) = 0(| s - t | <sup>-q</sup>), | s - t | → ∞, and y(s) = O(| s | <sup>-p</sup>), s → ∞, the asymptotic behavior of the solution x may be estimated as x(s) = O(| s | <sup>-r</sup>), | s | → ∞, r:= min(p, q). The case when k(s, t) = k(s - t), so that the equation is of Wiener-Hopf type, receives especial attention. Conditions, in terms of the symbol of I - K, for I - K to be invertible or Fredholm on Xp are established for certain cases (Ω a half-space or cone). A boundary integral equation, which models three-dimensional acoustic propagation above flat ground, absorbing apart from an infinite rigid strip, illustrates the practical application and sharpness of the above results. This integral equation models, in particular, road traffic noise propagation along an infinite road surface surrounded by absorbing ground. We prove that the sound propagating along the rigid road surface eventually decays with distance at the same rate as sound propagating above the absorbing ground.</p>},
  author       = {Chandler-Wilde, Simon N. and Peplow, Andrew T.},
  issn         = {0897-3962},
  language     = {eng},
  number       = {3},
  pages        = {303--327},
  publisher    = {Rocky Mountain Mathematics Consortium},
  series       = {Journal of Integral Equations and Applications},
  title        = {Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations},
  url          = {http://dx.doi.org/10.1216/jiea/1181075881},
  doi          = {10.1216/jiea/1181075881},
  volume       = {7},
  year         = {1995},
}