Quadrature formula for normal derivative of double layer potential

Looking for a solution to the Dirichlet and Neumann boundary value problems for the Helmholtz equation in the form of a combination of simple and double layer potentials, the considered boundary value problems are reduced to a curvilinear integral equation depending on the operators generated by the simple and double layer potentials and by their normal derivative. It is known that the latter operators are weakly singular integral ones. However, a counterexample constructed by Lyapunov shows that for the double layer potential with continuous density, the derivative, generally speaking, does not exist, that is, the operator generated by the normal derivative of the double layer potential is a singular integral operator.
Since in many cases it is impossible to find exact solutions to integral equations, it is of interest to study an approximate solution of the obtained integral equations. In its turn, in order to find an approximate solution, it is necessary, first of all, to construct quadrature formulas for the simple and double layer potentials of the and for their normal derivatives. In this work we prove the existence theorem for the normal derivative of the double layer potential and we provide a formula for its calculation. In addition, we develop a new method for constructing a quadrature formula for a singular curvilinear integral and on the base of this we construct a quadrature formula for the normal derivative of the double layer potential and we estimate the error.