Constructive method for solving some classes of hypersingular integral equations of the second kind

The paper studies approximate methods for solving hypersingular integral equations obtained from the Neumann external boundary value problem and from the external boundary value problem with the impedance condition for the Helmholtz equation in two-dimensional space. It should be pointed out that these hypersingular integral equations involve an operator generated by the normal derivative of the double layer potential. A counterexample built by A.M. Lyapunov shows that the normal derivative forAa double layer potential with continuous density, generally speaking, does not exist, i.e., the operator generated by the normal derivative of the double layer potential is not defined in the space of continuous functions. Using the regularization method, the considered hypersingular integral equations of the external Neumann boundary value problem and the external boundary value problem with the impedance condition for the Helmholtz equation are reduced to weakly singular integral equations. Having constructed quadrature formulas for one class of curvilinear integrals, the integral equations under consideration are replaced by a system of algebraic equations. Then, using G.M. Vainikko's theorem on convergence for linear operator equations, we prove that the resulting systems of algebraic equations are uniquely solvable, and the solutions to the system of algebraic equations converge to the value of the exact solution of the considered hypersingular integral equations at the reference points, and the rate of convergence of the method is indicated.