The authors consider integral equations of the first kind \(\int_ D R(x,y)h(y)dy = f(x)\), \(x\in \overline D\subset R^ r\). Here, \(D\) is a bounded domain with a smooth boundary, and \(R\) is in the class of kernels of positive rational functions of self-adjoint elliptic operators. The right-hand side \(f\) is assumed to be in a Sobolev space whose order depends on the order of the self-adjoint elliptic operator and of the degrees of the polynomials involved in the rational function mentioned above.
It is shown that the inverse of the integral operator is an isomorphism between that Sobolev space and its dual, so that the integral equation of the first kind is well-posed if considered between these spaces. A numerical method based on a decomposition of the solution into a smooth part and a “singular part” (which contains derivatives of the \(\delta\)- distribution concentrated on the boundary of D) and on a least-squares- approach is presented and its behaviour exemplified by numerical examples.