Vol 40, No 4 (2019)

In this paper, we present the results of investigation of the production model that extends Houthakker—Johansen approach of distribution of capacities over technologies and takes into account financial characteristics of manufacturer that works in conditions of current assets deficit and Poisson process of demand arrivals. The model is formalized in form of Bellman equation. The solution of Bellman equation defines the value of a company in dependence on parameters of manufacturing process and economic environment (interest rates, price, cost, etc.). The closed form expression for the solution of Bellman equation and calculation of average characteristics of production activity in model terms allow developing the methodology of calculation and analysis of company’s value that bases on official company reporting data and economic environment parameters. The developed method we apply to compute and analyze the economic conditions of functioning of FCA Company (Fiat Chrysler Automobiles, Italy). We calculate average debt in terms of the model and analyze the dynamic of average debt of FCA. We present the results of modified model investigation that reflects the crisis management strategy of business owner and takes into account the debt load and how it affects the company’s value. The condition of uniqueness for the solution of Bellman equation in modified model is identification of coefficient of debt accounting due to the bankruptcy procedure. In terms of the model, we calculate a closed form of critical debt value that corresponds to the bankruptcy bound.

Ulrasound transducers with phased arrays are used in multiple areas, including medical and technical applications. The numerical modeling of the composite ultrasound requires high spatial and temporal resolution with a full viscoelastic anisotropic material model. The grid-characteristic method that can provide it is demanding in terms of calculation time. A problem of optimal adjusting of the phased array requires a fast and effective numerical method. The retracing method for acoustics that is used for ultrasound and seismic problems, improved with a wavefront construction method was implemented. A technique of modeling the ultrasound of composites with good accuracy and in a reasonable time is proposed.

This paper investigates an inverse non-stationary problem of the restoration of the spatial law of a homogeneous isotropic Timoshenko beam of finite length. Hinge support conditions are used as boundary conditions. Initial conditions are assumed to be zero. It is assumed that of the beam’s ends is fitted with sensors which in the course of corresponding experiment register the amount of deflection of the beam at the sensor points. The method of the solution of a direct problem is based on the principle of superposition where the deflection of the beam is associated with the space load the beam is exposed to, by means of an integral operator by the spatial coordinate and time. The kernel of such operator is so called influence function. This function is a fundamental solution of a system of differential equations of motion of the study beam. The construction of such solution represents a separate problem. The influence function is found by means of Laplace time transformation and expansion into Fourier series in a system of the problem’s eigenfunctions. The solution of the inverse problem at the first stage reduces to a system of algebraic equations for vector operator whose components are time convolutions of the coefficients of expansion series for an influence function with the desired coefficients of expansion of the load in a Fourier series. At the same time, the components of the vector of the rights parts are time dependencies registered by the sensors. The resulting system is ill-conditioned [1]. The second stage serves to resolve independent Volterra integral equations of the first kind for the desired coefficients of Fourier serials for the load.

In the paper, we propose an efficient method based on the use of a bicubic Hermite finite element coupled with collocation for the reaction-diffusion equation with a variable coefficient. This enables one to reduce the dimension of the system of equations in comparison with the standard finite element scheme. Numerical experiments confirm a theoretical convergence estimate and demonstrate the advantage of the proposed method.

We solve finite-difference approximations of a linear-quadratic optimal control problem governed by Dirichlet boundary value problem with fractional time derivative. The state equation of the problem is approximated using locally one-dimensional difference schemes. The stability estimates of discrete state equations necessary for studying the convergence of iterative solution methods for the constructed discrete optimal control problems are proved. The rate of convergence of the proposed iterative method is obtained and the optimal iterative parameter is found. The results of numerical tests for a model problem are presented.

The aim of this article is representing of mathematical modeling results in spatial exploration seismology problems, in particular the problem of seismic waves propagation in fractured layers with fractures of different dimensions. The modeling was produced by grid-characteristic method with use of irregular computational meshes (triangle in 2D-case and tetrahedral in 3D-case). The use of such numerical method takes into consideration the characteristic physical properties of describing processes and give us possibility to construct correct algorithms on boundaries and contact boundaries in integrational domain. By the use of unstructured meshes we can produce computations of wave processes in any areas of different complex shapes.

A numerical method is presented for solving economic problems formulated in the Mean Field Game (MFG) form. The mean-field equilibrium is described by the coupled system of two parabolic partial differential equations: the Fokker-Planck-Kolmogorov equation and the Hamilton-Jacobi-Bellman one. The description is focused on the discrete approximation of these equations which accurately transfers the properties of MFG from the differential level to the discrete one. This approach results in an efficient algorithm for finding the corresponding grid control function. Contrary to other difference schemes, here the semi-Lagrangian approximation is applied which improves properties of a discrete problem. This implies the faster convergence of an iterative algorithm for the monotone minimization of the cost functional even with non-quadratic and non-symmetric contribution of control.