The effectiveness of the indirect method of near-boundary elements (as a variant of the method of boundary integral equations) for constructing numerical solutions of direct and inverse problems of potential theory in a limited piecewise homogeneous object of arbitrary shape whose components are in ideal contact is substantiated. The integral representation of the solution of the direct problem is constructed using the fundamental solution of the Laplace equation for the plane. To find the intensities of unknown sources introduced in the near-boundary elements, the collocation technique was used, i.e. the boundary conditions of the first and second kinds and the conditions of ideal contact are satisfied in the middle of each boundary element. After solving the resulting system of linear algebraic equations, the unknown potential in the medium and inclusions and the flow through their boundaries are found, taking into account that the components are considered as completely independent domains. Based on the nature of the potential change or its derivative, the initial approximations for the conductivity of the inclusions, their centers of mass, orientation, and size are determined. To solve the inverse problem, an algorithm for recognizing the main physical and geometric characteristics of inclusions based on excess data of the potential or flow at the boundary of the object was built. It consists of two cascades of iterations: in the first of them, the location of inhomogeneities and their approximate sizes is determined, in the second one, it is specified their shape and orientation on the plane. A computational experiment was conducted for the problem of electrical exploration using a constant artificial field and the resistance method, in particular, electroprofiling.
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