A Note on Transforming a Plane Strain First-Kind Fredholm Integral Equation into an Equivalent Second-Kind Equation

Abstract

Methods to convert Fredholm integral equations of the first kind into equivalent Fredholm integral equations of the second kind are used to study issues of existence and uniqueness of solutions. For some examples applied to plane strain problems, see (Constanda, Proc. Amer. Math. Soc. 123:3385–3396, 1995) and (Constanda, Direct and Indirect Boundary Integral Equations. Chapman & Hall/CRC, Boca Raton-London-New York-Washington, DC, 2000, Sec. 2.12). In this paper, another technique to convert the Fredholm integral equation of the first kind that arises in a direct boundary integral formulation for the plane strain Dirichlet problem into an equivalent Fredholm integral equation of the second kind is developed. The technique presented in this paper generalizes work of Y. Yan and I.H. Sloan that was done for the scalar Laplace equation (Yan and Sloan, J. Integral Equations Appl. 1:549–579, 1988) to the plane strain system of displacement equations.