Synthesized field method in inverse problems of magnetotelluric sounding

A new method is proposed and investigated for solving the inverse three-dimensional problem of magnetotelluric sounding. The method uses the tangential electric field on the Earth's surface, which is synthesized from the known impedance.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 115–121.     

An inverse problem for a system with a moving boundary

The problem of estimating the right-hand side of a nonlinear parabolic equation is considered. A finite-step algorithm based on the model positional control method and the finite element method is proposed. The algorithm is robust to informational noise and computational errors. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 23–33.     

Continuous regularized variable-metric proximal minimization method

The article considers the variable-metric continuous proximal method for the convex programming problem. The convergence of the method is proved. The regularized version of the method is proposed for the case where the objective function and the feasible set are defined inexactly. A regularizing operator is constructed. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 39–47.     

Axisymmetric excitation of a dielectric torus: Numerical solution

The article considers the mathematical model of a lens antenna in the shape of a homogeneous dielectric torus excited by an axial magnetic dipole. The boundary-value problem for an elliptical differential equation is posed and reduced by the fundamental solution method to an integral equation of 2nd kind over the torus section. A numerical algorithm is proposed, and the numerical results are compared with experimental findings. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 109–115.     

Method of successive approximations in the nonlinear theory of viscoelasticity

A method of successive approximations, a generalization of the Il'yushin method of elastic solutions, is proposed for solving problems of the nonlinear theory of elasticity in which the stress-strain relation is given in the form of a time operator Frechet-differentiable in a neighborhood of zero. The nonlinear relaxation kernels are found from the given nonlinear creep kernels for the principal quadratic theory of elasticity. These relations make it possible to formulate the boundary value problem for this theory. By way of illustration the problem of the pressure exerted on a space by a sphere is examined within the framework of the developed theory. The question of the convergence of the method is discussed in relation to the quadratic theory of visco-elasticity.Presented at the Third All-Union Conference on Theoretical and Applied Mechanics, Moscow (January, 1968).Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 236–242, March–April, 1969.     

Mathematical modeling of multioscillator antennas located above a real Earth

A method is proposed for calculating the current distribution and the main electrical characteristics of arbitrarily oriented multioscillator antennas above a plane Earth. The boundary-value problem is reduced to a system of Gallen integral equations, which is solved by Galerkin's method with self-regularization. The saddle-point method is used to evaluate the Sommerfeld integrals for the far-zone field. Numerical results are presented.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 77–85.     

Inverse sounding problem using stabilization of the magnetic field of a loop

We consider the inverse sounding problem for a stratified medium using the variable magnetic field of a loop. A uniqueness theorem and a theorem ensuring stable determination of the integral conductivity of the medium are proved. An algorithm is proposed for the solution of the inverse problem based on a transformation from the time domain to the frequency domain. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 87–95.     

High-frequency approximation of the green's function for the Helmholtz equation in an axial nonhomogeneous medium

A method is proposed for constructing the field of a point source in an axial two-dimensionally nonhomogeneous medium. The problem is reduced to an integral equation of the 2nd kind. If the frequency ω of the point source is sufficiently high, the equation can be solved by the method of successive approximations. The Neumann series converges in the usual sense for ω>>1 and it is an asymptotic series for ω→∞. The Neumann series also converges in the case of weak dependence of the medium on the coordinate φ. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 104–108.     

Galerkin method for the linear-radiator integral equation

The article examines the general Galerkin method scheme for the singular integral equation in the problem of radiation from a finite-thickness linear radiator. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 150–158.     

An inverse problem of atmospheric radar sounding

The paper considers the problem of determining the velocities of targets in the atmosphere given a set of Doppler spectra produced by continuous Doppler systems. The problem of determining the velocities of particles moving in a closed trajectory (a circle) is examined. A uniqueness theorem is proved for the reconstruction of velocities from two spectra. An algorithm is proposed for determining the sought function, and a numerical experiment is conducted to establish its efficiency.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 162–167.     

Analysis of the iterative method of solution of integral diffraction equations

The article investigates the convergence of an iterative method of solution of the integral equation for the diffraction problem of an electromagnetic field in a nonabsorbing medium. The convergence region of the iterative process is determined for the diffraction of a plane wave on a dielectric cylinder in a homogeneous space.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 112–114.     

Separation of variables as a means of solving the mixed boundary-value problem of viscoelasticity

The quasistatic mixed boundary-value problem is considered for a viscoelastic body with constant Poisson's ratio. A method of reducing viscoelastic problems to the solution of a series of uniform elastic problems is described. The convergence of the method is proved. The method is illustrated with reference to the solution of a specific viscoelastic problem. Tabulated influence functions of the Abel type are used in the solutions.Moscow M. V. Lomonosov State University. Translated from Mekhanika Polimerov, No. 4, pp. 588–596, July–August, 1969.     

Inverse coefficient problem for a nonlinear differential equation and an iterative solution method

The article considers the determination of the solution-dependent coefficient of a nonautonomous ordinary differetial equation with a parameter. Reduction of the inverse problem to a nonlinear operator equation is used to prove existence and uniqueness theorems and to propose an iterative solution method. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 5–17.     

High-frequency asymptotic method for investigating the problem of excitation of a thin dielectric coating of an ideally conducting cylinder (H-polarization)

An incomplete Galerkin projection method is constructed and proved as a generalization of the Sommerfeld method for the problem of plane diffraction of the field of a point source on an ideally conducting circular cylinder coated by a thin homogeneous dielectric layer with an arbitrary boundary, in the H-polarization case. An explicit asymptotic solution is obtained in the form of a generalized Watson series. Specific eesonance phenomena are investigated for a boundary layer with corner points. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 132–149.     

Method of solving quasistatic boundary value problems of viscoelasticity for media with time-varying properties

It is shown that, if certain conditions are satisfied, the solution of a viscoelastic mixed boundary value problem can be obtained from the solution of the corresponding elastic problem by substituting time operators for the body and surface forces or given displacements. A similar assertion is also proved for media with time-varying properties. The applicability of the method to the theory of shallow flexible viscoelastic shells is demonstrated.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, No. 6, pp. 987–993, November–December, 1969.     

Equilibrium routing strategies in data transmission networks

Search for optimal routing (in the game-theoretical sense) in a packet data transmission network is considered. The proposed method reduces the general problem to a particular problem of routing in ring networks. A fast parallel (distributed) algorithm is constructed, which may be used in network-level data transmission protocols. Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 107–123, 1998.     

Inverse problem of electromagnetic sounding of the shape of a conducting body

The inverse problem of electromagnetic sounding of the shape of a conducting body is considered in the framework of the two-dimensional model with E-polarization, assuming a cylindrical body of an arbitrary cross section embedded in a layered medium. The integral equations are obtained for the case of an ideally conducting body and a body of finite conductivity. The linearization method is applied to obtain an iterative method that finds a correction to the initial shape. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 96–103.     

Method of elastic solutions of problems of thermoviscoelasticity

The quasi-static mixed boundary value problem is considered for a viscoelastic body with time-dependent properties and a singular nondegenerate relaxation kernel. The existence of a time-temperature (or similar) analogy is not assumed. A method of reducing the viscoelastic problem to the solution of a series of uniform elastic problems is proposed and its convergence proved. The method is illustrated with reference to examples of solutions of specific viscoelastic problems.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 4, pp. 603–614, July–August, 1970.     

Mathematical modeling of vertical electrical sounding of quasilayered media

The direct solution of the vertical electrical sounding problem is considered for three-dimensional quasilayered media. An efficient fast algorithm is obtained for computing the apparent resistivity curves in the two-dimensional case. A PC implementation of the algorithm is proposed. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 123–131.     

Direct problem of calculating the magnetic dipole field in a quasilayered medium

A method is developed for calculating the electromagnetic field of a magnetic dipole in a quasilayered two-dimensional medium. The quasi-three-dimensional problem is reduced to a two-dimensional problem for the Fourier-transformed electromagnetic field. An equivalent system of integral equations on the layer boundaries is obtained. This research was partially supported by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010) and the Interuniversity Scientific Program “Russian Universities: Basic Research.” Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 94–110, 1998.