Direct Boundary Equation Formulation

Gavin R. Thomson () and Christian Constanda ()
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Gavin R. Thomson: A.C.C.A.
Christian Constanda: The University of Tulsa, Department of Mathematical and Computer Sciences

Chapter Chapter 7 in Stationary Oscillations of Elastic Plates, 2011, pp 87-102 from Springer

Abstract: Abstract To overcome the difficulties that arose in the indirect method—that is, the ill-posed nature of certain integral equations even though the boundary value problems they represent have at most one solution—in this chapter we do not assume a priori that the solutions of the boundary value problems take the form of specific layer potentials. Instead, using the Somigliana representation formulas (4.36) and (4.37), we apply the so-called direct method to obtain an integral equation of the second kind for an unknown density. These equations alone do not, however, prevent the nonuniqueness of solution since they involve the same boundary integral operators encountered in the indirect method. An equation of the first kind is also derived for each problem, and we show that, in the case of (Dω-) and (Nω-), the pair of second-kind and first-kind integral equations constructed in this way has a unique solution irrespective of the oscillation frequency ω.

Date: 2011
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DOI: 10.1007/978-0-8176-8241-5_7

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