About numerical solution of some integral equations of the first kind. — I. optimal approximations

Modeling of electrostatic fields at the environments with different characters lead to necessity of solution of the various boundary value problems for the Laplacian in R³ in the case of closed and tired boundary surfaces. Integral equations method allows to avoid the direct solving or significantly to simplify such process for the series of boundary value problems for the Laplacian [1]. The bilateral Dirichlet problem at the Hilbert space the normal derivative elements of which has the jump through boundary surface or the Neumann problem at the Hilbert space the elements of which has the jump through boundary surface such problems includes. Solution of these problems we obtain by means of the simple and double layer potentials by substituting instead of corresponding potential densities the values of difference of the boundary conditions. Solution of bilateral Dirichlet and Neumann problems at the Hilbert space elements of which as their normal derivatives has the jump through boundary surface by means of the sum of simple and double layer potentials reduce to solving only one integral equation of the first kind for simple layer potential in the case of Dirichlet problem and integral equation of the first kind for double layer potential in the case of Neumann problem [1].