On the approximate solution of a class of weakly singular integral equations

The work is devoted to the study of the solution of one class of weakly singular surface integral equations of the second kind. First, a Lyapunov surface is partitioned into “regular” elementary parts, and then a cubature formula for one class of weakly singular surface integrals is constructed at the control points. Using the constructed cubature formula, the considered integral equation is replaced by a system of algebraic equations. As a result, under the additional conditions imposed on the kernel of the integral, it is proved that the considered integral equation and the resulting system of algebraic equations have unique solutions, and the solution of the system of algebraic equations converges to the value of the solution of the integral equation at the control points. Moreover, using these results, we substantiate the collocation method for various integral equations of the external Dirichlet boundary-value problem for the Helmholtz equation.