Mathematical simulation of acoustic wave refraction near a caustic

Issues related to the computation of wave fields in an acoustic medium near caustics are considered. A boundary condition on a caustic is established, and the Green’s function of a boundary value problem for the general case of a varying speed of sound is constructed. For this purpose, an auxiliary Goursat problem is considered and a system of its particular solutions is constructed using hyper-geometric functions. A Volterra integral equation for the Green’s function is obtained, and an algorithm for its expansion with respect to smoothness is described. A finite difference scheme approximating the solution of the differential problem with an unbounded coefficient is proposed. Numerical results are presented.     

Mathematical modeling of infiltration metasomatism in a porous medium

We construct a mathematical model describing the processes of dissolution and redeposition of minerals in a medium with a nonhomogeneous distribution of acidity. The dynamics of extraction of a mineral from a leaching solutions is investigated. We show that filtration of solutions through reduced acidity regions induces deposition, increasing the concentration of the target mineral in the solid phase; in high pH regions, on the other hand, the mineral dissolves. The stratum may retain certain reserves of the target mineral after leaching depending on the size of the reduced pH region and its proximity to the extraction borehole. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 5–17, 2007.     

Mathematical description of artificial sense-of-touch systems

The paper proposes a mathematical formalism for describing artificial sense-of-touch systems. Mathematical models for obtaining, processing, and interpreting tactile information are provided. Problems of encoding and reproducing tactile information are formulated and algorithms for solving these problems are proposed. The problem of interpreting tactile information is considered, and the corresponding simple mathematical model is studied. Within the framework of this simple model, an exact solution of the interpretation problem is obtained for the case of finite deformations, and the insufficiency of the linear method (Hooke’s law) for describing problems of interpreting tactile data is shown. The presentation of mathematical models for the theory of artificial sense-of-touch systems in this paper is the first such detailed presentation in the Russian literature. These models are of interest to mathematicians, mechanicians, physicians, and engineers who construct or use artificial sense-of-touch systems.     

Mathematical description of the properties of muscle tissue

Theoretical research on the problem of describing the properties of muscle tissue is briefly reviewed. The principal approaches to the problem are indicated, their shortcomings are noted, and the requirements that must be satisfied by an up-to-date continuum model are formulated. The procedure for constructing such a model is described, together with the principal and secondary hypotheses, and the conclusions that follow from a comparison of theory and experiment are noted.     

Magnetohydrodynamic flow of a rarefied gas near an accelerated porous plate

An investigation is made of the flow of an electrically conducting rarefied gas due to the time-varying motion of an infinite porous plate, the gas being permeated by a transverse magnetic field. The suction is taken to be a constant and the magnetic lines of force are taken to be fixed relative to the fluid. The effects of magnetic field, rarefaction parameter, suction parameter are shown by means of some tables. The expressions of the skin friction for the two particular cases have also been obtained.     

Mathematical description of the pulsations of the third ventricle

A mechanical vibratory system consisting of two masses connected with each other and with the fixed supports by three springs is used to study the pulsations of the third ventricle of the brain. The frequency equation of this system is examined and solved for various relations between the stiffnesses of the connecting springs. The values obtained for the principal vibration frequencies are used to describe the vibrations of the walls of the third ventricle. As a result, the frequency and base line characteristics of the vibrations of the walls of the third ventricle in normal and pathological states are established.     

Mathematical modelling of a quasi-linear feedback system

This paper develops a method for system identification of a two input-one output quasi-linear system modelling the dynamics of a non-linear element of a feedback system. The identification method, which may be performed in either the time domain or the frequency domain, is designed to use normal operating records without the need to introduce special test inputs.     

Mathematical modeling of flows through inelastic porous solids

We propose a framework, based on classical mixture theory, to describe the isothermal flow of an incompressible fluid through a deformable inelastic porous solid. The modeling of the behavior of the inelastic solid takes into account changes in the elastic response due to evolution in the microstructure of the material. We apply the model to a compression layer problem. The mathematical problem generated by the model is a free boundary problem.     

Mathematical modelling of a diesel common-rail system

In diesel common-rail systems, the exact knowledge of the injection pressure is important to accurately control the injected diesel mass and thus the combustion process. This paper focuses on the mathematical modelling of the hydraulic and mechanical components of a common-rail system in order to describe the dynamics of the diesel rail pressure. Based on this model, an average model is derived to reduce the model complexity and to allow for a fast calculation of the mass flow into the rail for different crank shaft revolution speeds and openings of the fuel metering unit. The main purpose of this average model is to serve as a basis for a model-based (non-linear) controller design. The stationary accuracy of the models is validated by means of measurement data.     

Mathematical analysis of a two-shuttle transportation system

Summary A mathematical analysis of a two-shuttle system, using car ferries as the example, is formulated as an interdependent queueing system. The main features of the model include a fixed ferry travel time, a single dock on each side, stochastic (Poisson) car arrivals and infinitely large parking lots. Equations are derived to describe the behavior of the system, namely: equations of the number of cars waiting at both ferry docks, times for the two ferries to reach the next docking at appropriate clock times, number of cars on and boarding the ferries, and waiting times of the ferries.

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Zusammenfassung Die mathematische Analyse eines zweifachen Pendlersystems mit Autofähren wird als unabhängiges Warteschlangensystem formuliert. Das Modell enthält konstante Reisezeiten der Fähren, ein Dock an jedem Ufer, stochastische (Poisson) Autoankünfte und einen beliebig großen Parkraum. Es werden Gleichungen abgeleitet, die das Verhalten des Systems beschreiben: Gleichungen für die Zahl der wartenden Autos an beiden Docks, Zeiten für die beiden Fähren, um das nächste Dock zur entsprechenden Zeit zu erreichen, Zahl der geladenen Autos und Wartezeiten der Fähren.

     

Mathematical description of surface effects in deformable solid bodies

We consider the general balance equations for the interface surface of two media and the interface line of three media, and we analyze the generalized Laplace and Young equations and study the balance equations for surface dislocations and disclinations. We establish the relations between the densities and flows of surface defects at the junction of three grain boundaries.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 42–47.     

Hydromagnetic flow near an oscillating porous flat plate

A solution in the closed form has been obtained for the flow of a viscous incompressible fluid of small electrical conductivity near an infinite insulated porous flat plate oscillating harmonically in its own plane in the presence of a transverse magnetic field of uniform strength fixed relative to the fluid. Small uniform suction has been imposed along the surface of the plate. This is a generalisation of the result of Ong and Nicholls for the hydromagnetic flow near an oscillating solid flat plate.