We extend the main result of {KKL11b} to the case of more general weighted singular integral operators with two shifts of the form \[ (aI-b U_\alpha )P_\gamma ^++(cI-dU_\beta )P_\gamma ^-, \] acting on the space $L^p(\mathbb{R} _+)$, $1\lt p\lt \infty $, where \[ P_\gamma ^\pm =(I\pm S_\gamma )/2 \] are operators associated with the weighted Cauchy singular integral operator $S_\gamma $, given by \[ (S_\gamma f)(t)=\frac {1}{\pi i}{\int _{\mathbb{R} _+}} \bigg (\frac {t}{\tau }\bigg )^\gamma \frac {f(\tau )}{\tau -t}\,d\tau \] with $\gamma \in \mathbb{C} $ satisfying $0\lt 1/p+\Re \gamma \lt 1$, and $U_\alpha ,U_\beta $ are the isometric shift operators given by \[ U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha ), \qquad U_\beta f=(\beta ')^{1/p}(f\circ \beta ), \] generated by diffeomorphisms $\alpha ,\beta $ of $\mathbb{R} _+$ onto itself having only two fixed points at the endpoints $0$ and $\infty $, under the assumptions that the coefficients $a,b,c,d$ and the derivatives $\alpha ',\beta '$ of the shifts are bounded and continuous on $\mathbb{R} _+$ and admit discontinuities of slowly oscillating type at $0$ and $\infty $.

Two high order methods are constructed and analyzed for a class of Fredholm integral equations of the second kind with kernels that may have weak boundary and diagonal singularities. The proposed methods are based on improving the boundary behavior of the exact solution with the help of a change of variables, and on central part interpolation by polynomial splines on the uniform grid. A detailed error analysis for the proposed numerical schemes is given. This includes, in particular, error bounds under various types of assumptions on the equation, and shows that the proposed central part collocation approach has accuracy and numerical stability advantages compared with standard piecewise polynomial collocation methods, including the collocation at Chebyshev knots.

We present a regularization strategy that leads to well-conditioned boundary integral equation formulations of Helm\-holtz equations with impedance boundary conditions in two-dimensional Lipschitz domains. We consider both the case of classical impedance boundary conditions, as well as that of transmission impedance conditions wherein the impedances are certain coercive operators. The latter type of problem is instrumental in the speed up of the convergence of Domain Decomposition Methods for Helmholtz problems. Our regularized formulations use as unknowns the Dirichlet traces of the solution on the boundary of the domain. Taking advantage of the increased regularity of the unknowns in our formulations, we show through a variety of numerical results that a graded-mesh based Nystr\"om discretization of these regularized formulations leads to efficient and accurate solutions of interior and exterior Helmholtz problems with impedance boundary conditions.