Том 29, № 3 (2018)

Unique solvability theorems are proved for the inverse two-dimensional sounding problem in a nonhomogeneous conducting half-space with different primary-field polarizations.

We construct the complete trajectory of a meteorite from observations on the visible (“bright”) section of the trajectory. Mathematically, this is a minimization problem for a functional defined as the deviation of the observed trajectory from the trajectory calculated using the system of meteor physics equations. This differential system has been solved numerically, and the minimization problem has been solved by the Nelder–Mead method. The solution makes it possible to determine the physical parameters of the body entering the Earth’s atmosphere and to complete its trajectory on the unobservable (“dark”) section until its collision with the Earth’s surface.

A new method is considered for the calculation of traveling-wave characteristics in a layered medium. It is mainly based on the transfer-matrix method [2–10] and the generalized reflection-transmission coefficient (GRTC) method [11–14] for the calculation of characteristics. Introducing an impedance tensor of rank 2, Ẑ (z), in the boundary-value stress problem and utilizing in this way the boundary conditions and the Lame equation, we obtain a system of differential equations which is solved by the fourth-order Runge-Kutta method. Specifying a model of the layered medium in terms of known constants, we find the traveling-wave characteristic γ as a function of frequency ω . Given γ, we easily calculate the dispersion curves, which closely fit the GRTC method. Compared with the GRTC method, the impedance-tensor method fully solves the dispersion curve of the normal-mode traveling waves, and it is applicable to both a homogeneous planar Earth model and a model with a slow layer.

We describe the application of an explicit homogeneous conservative quasi-acoustic scheme to numerical solution of one-dimensional shallow-water equations with an uneven bottom. The scheme performs linear reconstruction of the numerical solution within a single numerical cell and partitions the linear reconstruction into small-perturbation horizontal layers. The quasi-acoustic scheme correctly reproduces the physical solution in the neighborhood of the sonic point without invoking artificial regularizers or tuning parameters. The scheme is verified on a number of test and prototype problems.

In this paper, the numerical solution of multidimensional hyperbolic problems is discussed with the quantized tensor train (QTT)-approximation methods. Three schemes are proposed. First, an improved implicit time iteration scheme is presented by using the two-site density matrix renormalization group (DMRG) algorithm to solve a linear system at each time step. Second, the time is considered as an independent dimension, and a discretization of the whole differential equation is introduced with all spatial and time dimensions connected in one big global linear system. Then the problem is solved in the QTT-format. The third scheme is to solve the global system by splitting the global time interval into several subintervals. The numerical experiments, with these three schemes applied to the wave equation, show that the complexity of the first scheme is linear while that of the second and third schemes is log-linear in both time and spatial grid points.

We consider singularly perturbed problems of convection-diffusion-reaction type which involve two small parameters. A new discrete cubic spline method is developed for the solution of this problem on a Shishkin mesh. A convergence analysis is given and the method is shown to be almost second-order uniformly convergent with respect to the perturbation parameters ????_d and ????_c. Numerical results are presented to validate the theoretical results as well as the robustness of the method.

An iterative method is proposed and investigated for the solution of an integral equation with a kernel with a logarithmic singularity in an arbitrary definition domain.