Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations
(1995) In Journal of Integral Equations and Applications 7(3). p.303327 Abstract
We consider second kind integral equations of the form x(s)  ʃ_{Ω} k(s, t)x(t) dt = y(s) (abbreviated x  Kx = y), in which Ω is some unbounded subset of R^{n} Let X_{p} 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(X_{0}), then (I  K)^{1} ∊ B(X_{p}) for 0 ≤ p < q, and (I  K)^{1} ∊ B(X_{q}) if further conditions on k hold. Thus, if... (More)
We consider second kind integral equations of the form x(s)  ʃ_{Ω} k(s, t)x(t) dt = y(s) (abbreviated x  Kx = y), in which Ω is some unbounded subset of R^{n} Let X_{p} 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(X_{0}), then (I  K)^{1} ∊ B(X_{p}) for 0 ≤ p < q, and (I  K)^{1} ∊ B(X_{q}) 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 WienerHopf 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 halfspace or cone). A boundary integral equation, which models threedimensional 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.
(Less)
 author
 ChandlerWilde, Simon N. and Peplow, Andrew T. ^{LU}
 publishing date
 1995
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Journal of Integral Equations and Applications
 volume
 7
 issue
 3
 pages
 25 pages
 publisher
 Rocky Mountain Mathematics Consortium
 external identifiers

 scopus:0000261879
 ISSN
 08973962
 DOI
 10.1216/jiea/1181075881
 language
 English
 LU publication?
 no
 id
 4b0b7cee51904eefaaee07203aff0727
 date added to LUP
 20210215 19:59:35
 date last changed
 20210331 10:54:01
@article{4b0b7cee51904eefaaee07203aff0727, 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 < 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 WienerHopf 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 halfspace or cone). A boundary integral equation, which models threedimensional 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 = {ChandlerWilde, Simon N. and Peplow, Andrew T.}, issn = {08973962}, language = {eng}, number = {3}, pages = {303327}, 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}, }