On the numerical solution of the Neumann boundary value problem for the Helmholtz equation using the method of hypersingular integral equations

A numerical method for solving a boundary hypersingular integral equation arising from the Neumann boundary value problem for the Helmholtz equation is proposed. The proposed numerical method is based on the explicit separation of the hypersingular main part in the kernel of the integral equation. After discretization, this boundary integral equation is reduced to a system of linear algebraic equations. The coefficients of this system are represented as the sums of hypersingular and weakly singular integrals. The hypersingular integrals are understood in the sense of the finite Hadamard value and are calculated analytically. A number of quadrature formulas for the weakly singular integrals are developed using the smoothing procedures for singularity. The proposed numerical scheme is tested on the basis of the following model examples: a hypersingular integral equation on a sphere and the problems of diffraction of acoustic waves on inelastic spheres and discs. The numerical solutions obtained are compared with existing analytical and numerical data.