Purpose. In order to adequately describe real processes that characterize the spread in the crust harmonic electromagnetic field (EMF) excited by artificial sources , unlike classical models (homogeneous and piecewise homogeneous half-space), a locally inhomogeneous half-space (its electrical characteristics depend on coordinate only within the local area) is considered. Taking into account depending on the coordinates of conductivity, permeability and permittivity (so-called geometric heterogeneity) we get linear boundary problems of mathematical physics with variable coefficients, which mainly solved by a combination of analytical and computational methods. Methodology. To find solutions to these problems the numerically-analytical approach based on the combination of integral equations method (IEM) with extraction of the operator that describes the influence of the local area geometric heterogeneity is constructed. Taking into account the advantages of the IEM in a homogeneous infinite medium, we make discretization only in the local area and find the unknown EMF components in the grid nodes after their interpolation in the element of discretization. Results. A half-space containing the local area with an arbitrary curved boundary is considered. Its electrical characteristics are continuous functions of the coordinates. To find the component of the electric field (EF) mathematical model of the problem, composed of the Helmholtz equations system and zero boundary conditions on the free surface of the half-space, is built. The right side of the system describes the effect of local heterogeneity and contains unknown EF strength vector components. Using special fundamental solution of the Helmholtz equation that automatically satisfies the boundary condition, integral representations (IR) of solutions of equations initial problem conditions are written. They are used for constructing a system of linear equations formed as a result of satisfaction coincidence unknown EF strength vector components calculated using the integral representations with the values in the grid nodes of the local area. After solving this system using IR of solution and their derivatives of the coordinates the vector components of the electric and magnetic fields at an arbitrary point of a half-space are calculated. Originality. Without the introduction of electric or magnetic potentials the numerically-analytical solution of the problem of established oscillations of EMF in a locally homogeneous half-space is constructed. Dependencies on three Cartesian coordinates all its electrical characteristics are included. The effectiveness of a combination of methods of integral equations and weighted residuals for solving this problem is justified. Practical significance. Built discrete-continual models take into account the impact separate and mutual dependence on the coordinates conductivity, permeability and permittivity on the EMF distribution. This allows you to explore the effect of conductive charges, induced polarization and magnetic polarization (magnetization) in the process.
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