Global weak solutions to a class of compressible non-Newtonian fluids with vacuum

The global weak solution of an initial-boundary value problem for a compressible non-Newtonian fluid is studied in a three-dimensional bounded domain. By the techniques of artificial pressure, a solution to the initial-boundary value problem is constructed through an approximation scheme and a weak convergence method. The existence of a global weak solution to the three-dimensional compressible non-Newtonian fluid with vacuum and large data is established.     

Solution of the stationary problem of heat conduction of laminated anisotropic inhomogeneous plates by the method of initial functions

The problem of stationary heat conduction of laminated plates of constant and variable thickness is formulated in the three-dimensional statement. We reduce the three-dimensional problem to a twodimensional one by the method of initial functions. For plates with layers of variable thickness, a system of resolving equations with variable coefficients is obtained. The obtained two-dimensional boundary-value problems are analyzed. For plates with homogeneous layers of constant thickness, we construct a solution in an analytic form. It is shown that this solution coincides with a solution obtained by the method of separation of variables.     

三维种群生态动力系统的参数识别

研究具有反应一扩散现象的三维种群生态动力系统的参数识别问题，依该系统正问题解的性质，建立了参数识别的数学模型；论证了系统正问题解关于待识别参数的连续依赖性与参数识别问题最优解的存在性。     

Design of continuous strain-type sensors for systems of structural mechanics

In this paper, we present a general approach for designing continuous strain-type sensors to measure specific kinematical entities in systems of structural mechanics. We first discuss the case of three-dimensional bodies in the geometrically linear regime, for which the solution of the sensor design problem is found from a problem-oriented application of a work-based integral formulation to an auxiliary problem of statics; yet, the resulting sensor works for static and dynamic problems. The deduction of solutions for the sensor design problem on a structural mechanics level is straightforward, because the three-dimensional solution is work-based. Hence, for any theory of structural mechanics, for which a corresponding work-based integral formulation exists, we can easily find a solution of the sensor design problem. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.     

三个不动中心问题的一个三维解

本文对三个不动中心问题的平面解作进一步发展，通过对三个不动中心在四维空间中的定义得到三个不动中心问题的三维解．     

Numerical solution of the three-dimensional Dirichlet problem for inhomogeneous media by the method of integral equations

We consider the three-dimensional Dirichlet problem for equations of elliptic type in inhomogeneous media. The problem can be reduced to a system of loaded Fredholm integral equations of the second kind over the volume. We prove the uniqueness of a classical solution of the problem. We suggest a numerical solution algorithm of iterative type. An example of the numerical solution of the problem is considered, and the convergence of the iterative procedure is demonstrated numerically.     

A review of methods for computer-aided reconstruction of solid models of objects by their orthographic views

The paper contains a review of methods for computer-aided reconstruction of three-dimensional solid models of objects by their orthographic views. This problem arises when passing from engineering 2D-drawings to spatial models of objects in systems of three-dimensional solid modeling. In spite that there exists a large variety of such systems, none of them provides a computer-aided solution of this problem. This is due to the fact that the existing methods for solution are imperfect and require further elaboration. In this paper, we review the existing approaches and compare their application domains.     

On the one problem of wave diagnostic

This work is devoted to the development of methods and algorithms of the solution of a nonlinear three-dimensional inverse problem of wave diagnostics of heterogeneities in homogeneous environments in approximation of Helmholtz equation. By virtue of computational complexity of the problem, its solution is possible only with use of a supercomputer. The problem consists in seeking the unknown factor in the equation with partial derivatives. The problem is transformed to a system of operator equations with respect to an unknown factor and a wave field. Effective algorithms of its solution with the use of supercomputers having a parallel architecture as a result.     

Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor–Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf–sup condition and optimal a priori error estimates.     

Asymptotic splitting in the three-dimensional problem of linear elasticity for elongated bodies with a structure

The equilibrium of linearly-elastic elongated bodies (rods) with an extremely arbitrary geometry and structure subjected to the effects of force and heat is considered. Owing to the presence of a small parameter-the relative thickness-this is a singularly perturbed problem. The asymptotic analysis involves splitting the three-dimensional problem into one- and two-dimensional problems. The one-dimensional problem gives the same result as the classical theory, even when the material is structurally heterogeneous and anisotropic, which invalidates the conventional hypotheses of applied theories. The two-dimensional problems yield not only the parameters of the one-dimensional model, but also a complete solution of the three-dimensional problem. The algorithm used to split the three-dimensional problem is implemented on a computer. It is sometimes more effective than the conventional finite-element, boundary-element and difference methods in the case of elongated bodies.     

A three-dimensional analog of the Tricomi problem for a parabolic-hyperbolic equation

For a parabolic-hyperbolic equation, we study the three-dimensional analog of the Tricomi problem with a noncharacteritic plane on which the type of the equation changes. The uniqueness of the solution to the problem is proved by the method of a priori estimates, and the existence of a solution is reduced to the existence of a solution to a Volterra integral equation of the second kind.     

Justification of the two-dimensional equations of a linearly elastic shallow shell

We consider a problem in three-dimensional linearized elasticity, posed over a shell with a specific geometry, subjected to general loadings, and clamped on a portion of its lateral surface. We show that, as the thickness of the shell goes to zero, the solution of the three-dimensional problem converges to the solution of two-dimensional shallow shell equations. This approach, which provides in particular a mathematical definition of “shallowness”, clearly delineates conditions under which a three-dimensional problem may be deemed asymptotically equivalent to a two-dimensional shallow shell problem.     

Two-Dimensional and Three-Dimensional Thermal Structures in a Medium with Nonlinear Thermal Conductivity

The article considers self-similar solutions of the nonlinear heat equation with a three-dimensional source that evolve in a blow-up setting. The self-similar problem is a boundary-value problem for a nonlinear equation of elliptical type that has a nonunique solution. We investigate the eigenfunction spectrum of the self-similar problem in two- and three-dimensional space. The problem is solved on a grid by Newton’s iteration method. The implementation of Newton’s method requires analysis of a linearized equation and construction of initial approximations. The eigenfunctions are continued in a parameter. Structures of various symmetry are obtained. New types of multidimensional structures are observed: these are multiply connected three-dimensional heat localization regions.__________Translated from Prikladnaya Matematika i Informatika, No. 17, pp. 84–111, 2004.     

Three-stage heuristic algorithm for three-dimensional irregular packing problem

This paper studies a new practical problem which can be decomposed into three three-dimensional packing problems: three-dimensional irregular packing with variable-size cartons problem, three-dimensional variable-size bin packing problem, and the single container loading problem. Since the three sub-problems are NP-hard, searching a good solution becomes more difficult. In this paper, mathematical models of each sub-problem are developed and three-stage heuristic algorithms are proposed to solve this new problem. Experiments are conducted with random instances generated by real-life case. Computational results indicate that the proposed algorithm is efficient and can yield satisfactory results.     

Solvability of the stationary heat convection problem for a high-viscosity fluid

We study the stationary heat convection problem in the Boussinesq approximation. We derive a priori estimates for its solution. We prove existence and uniqueness theorems for a weak solution of the problem and analyze the smoothness of a weak solution for raised smoothness of the problem data. We consider the two- and three-dimensional cases.     

Inverse problems for stationary Navier-Stokes systems

An inverse problem for a nonlinear equation in a Hilbert space is considered in which the right-hand side that is a linear combination of given functionals is found from given values of these functionals on the solution. Sufficient conditions for the existence of a solution are established, and the solution set is shown to be homeomorphic to a finite-dimensional compact set. A boundary inverse problem for the three-dimensional thermal convection equations for a viscous incompressible fluid and an inverse magnetohydrodynamics problem are considered as applications.     

Global solution to the incompressible flow of liquid crystals

The initial–boundary value problem for the three-dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution must be equal to the unique strong solution with the same data.     

Numerical solution of the three-dimensional stationary diffraction problem for elastic waves

Three-dimensional problems concerning the propagation of stationary elastic oscillations in media with three-dimensional inclusions are solved numerically. By applying potential theory methods, the original problem is stated as a system of two singular vector integral equations for the unknown internal and external densities of auxiliary sources of waves. An approximate solution of the original problem is obtained by approximating the integral equations by a system of linear algebraic equations, which is then solved numerically. The underlying algorithm has the property of self-regularization, due to which a numerical solution is found without using cumbersome regularizing algorithms. Results of test computations and numerical experiments are presented that characterize the capabilities of this approach as applied to the diffraction of elastic waves in three-dimensional settings.     

A Certain Inversion Problem for the Ray Transform with Incomplete Data

The problem is considered of reconstructing a function in three-dimensional space from its integrals over a family of straight lines that are generators of cones, i.e., an inversion problem for the ray transform with incomplete data. This problem is connected with an auxiliary problem of analytic continuation and is strongly ill-posed. A uniqueness theorem and conditional stability estimates for a solution are obtained.