×

Hypersingular integrals: How smooth must the density be? (English) Zbl 0846.65070

This is a very interesting paper that examines the conditions on the density \(f(t)\) for the hypersingular integrals \[ \int^B_A {f(t)\over (t- x)^n} dx,\qquad n= 1,2,\dots \] to exist. It is well known that it is sufficient that \(f(t)\) has a Hölder-continuous first derivative. This paper is concerned with finding weaker conditions and it is established that it is sufficient for \(n= 2\) (this is a Hadamard finite-part integral) that the even part of \(f\) has a Hölder-continuous first derivative. A similar condition is found for \(n= 1\) (a Cauchy principal value). The non-trivial consequences of these results are discussed, particularly with regard to collocation at a point \(x\) between two boundary elements.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Monegato, J. Comput. Appl. Math. 50 pp 9– (1994) · Zbl 0818.65016
[2] Toh, Int. J. Solids Struct. 31 pp 2299– (1994) · Zbl 0946.74584
[3] Martin, Proc. Roy. Soc. A 421 pp 341– (1989) · Zbl 0674.73071
[4] and , ’Hypersingular boundary integral equations: their occurrence, interpretation, regularization and computation’, in and (eds.), Developments in Boundary Element Methods, Elsevier, Amsterdam, 1992, pp. 207-252.
[5] ’Direct evaluation of hypersingular integrals in 2D BEM’, in (ed.), Numerical Techniques for Boundary Element Methods, Vieweg, Braunschweig, 1992, pp. 23-34. · Zbl 0743.65013
[6] Krishnasamy, Comput. Mech. 9 pp 267– (1992) · Zbl 0755.65108
[7] Guimaräes, Eng. Anal. Boundary Elements 13 pp 353– (1994)
[8] Gray, Math. Models Meth. Appl. Sci. 4 pp 179– (1994) · Zbl 0801.73057
[9] Huang, Int. J. numer. methods eng. 37 pp 2041– (1994) · Zbl 0832.73076
[10] Holzer, Comm. numer. methods eng. 9 pp 219– (1993) · Zbl 0781.65091
[11] ’Hypersingular integrals and the Helmholtz equation in two dimensions’, in et al. (eds.), Advances in Boundary Element Methods in Japan and USA, Computational Mechanics Publications, Southampton, 1990, pp. 379-388.
[12] Singular Integral Equations, Noordhoff, Groningen, 1953.
[13] Mathematics for the Physical Sciences, Hermann, Paris, 1966. · Zbl 0151.34001
[14] and , ’Numerical approximation of the solution for a model 2-D hypersingular integral equation’, in et al. (eds.), Computational Engineering with Boundary Elements, Computational Mechanics Publications, Southampton, 1990, pp. 85-99.
[15] Potential Theory, Ungar, New York, 1967.
[16] Guiggiani, Eng. Anal. Boundary Elements 13 pp 169– (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.