Reconstruction of a bounded solution of a linear functional equation

We consider the problem of finding a bounded solution of a linear finite-difference equation with continuous time. We present conditions for the existence of such a solution and algorithms for finding the solution in real time under various additional assumptions.     

On a convective flow of a binary mixture in a vertical layer

Under study is an invariant solution of the equations of thermal diffusive convection which describes a stationary process of a binary mixture flow in a vertical layer under the action of the pressure gradient and the buoyancy force that depends nonlinearly on temperature and concentration. Some general properties of this solution are established and an existence theorem is proved. Analysis of the numerical solution of the problem is carried out in the cases of a power-law and exponential dependence of the buoyancy force on its argument.     

The compression of a thin strip of material in a superplasticity state

The viscoplastic flow of a thin strip of material in a superplasticity state between rigid, converging parallel planes (an analogue of Prandtl's problem) is investigated. An analytical quadrature solution of the problem is constructed, asymptotically precise in the same sense as Prandtl's solution. Special cases are considered where the solution (including an approximate solution) is written out fully. The effects of superplasticity are determined.     

A concept of a robust solution of a multicriterial linear programming problem

A new concept of a robust solution of a multicriterial linear programming problem is proposed. The robust solution is understood here as the best starting point, prepared while the preferences of the decision maker with respect to the criteria are still unknown, for the adaptation of the solution to the preferences of the decision maker, once they are finally known. The objective is the total cost of the initial preparation and of the later potential adaptation of the solution. In the starting robust solution the decision variables may have interval values. The problem can be solved by means of the simplex algorithm. A numerical example illustrates the approach.     

On fractional powers of a pair of matrices and a plate-beam problem

In this paper we consider the transversal deflections of a dynamically-coupled Von Kármán system consisting of a plate which has a beam attached to its one edge. The problem is considered in the form of a non-linear evolution problem in a product space. We show the existence of a unique local solution by following a fractional powers approach to first construct a “weak” solution in a larger space. Regularity properties for this solution yield a unique local strong solution for the original boundary-value problem. This approach entails the introduction of fractional powers of a pair of matrices.     

Determining dimension of the solution component that contains a computed zero of a polynomial system

When the Jacobian of a computed numerical solution of a polynomial system in Cn allows very small singular values, the solution could be isolated with a multiple multiplicity or may belong to a solution component with positive dimension. The algorithm constructed in this article intends to differentiate those cases by determining the dimension of the solution component M in which the solution lies. Of particular interest is the case when dim(M)=0, then the solution is of course isolated. While the proposed algorithm is experimental, it has been tested successfully on the class of problems with the solution in question belonging to a reduced component. Numerical results are provided to illustrate the accuracy of the algorithm.     

Asymptotic properties of a solution of a boundary-value problem

The asymptotic behavior of the solution of a boundary-value problem for the equation u_txx+ u_x =f when the time tends to infinity is investigated. It is proved that the time mean of the solution tends to a stationary solution everywhere except in a boundary region at the left end of the interval.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 273–284, September, 1970.     

Behavior of the solution of a parabolic equationon a characteristic

We consider the solution of a linear second-order parabolic equation with one spatial variable and a zero right side. We prove that since the solution decreases quite rapidly in the spatial variable as it approaches a particular point, it vanishes on the part of the characteristic joining the point to the boundary of the region in which the solution is defined.Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 257–262, September, 1972.     

On the decay of a solution of a nonuniformly parabolic equation

For a uniformly parabolic second-order equation with lower-order terms in an unbounded domain, we obtain an upper bound for the decay rate of the solution of the mixed problem with alternating boundary conditions of the first and third types. We prove that the bound is sharp in the case of an equation without lower-order terms in a wide class of domains of revolution. In addition, we show that a solution of a nonuniformly parabolic equation can decay much more rapidly than a solution of a uniformly parabolic equation.     

The motion of a screw dislocation in a wedge-shaped region

The problem of antiplane deformation for a wedge-shaped region containing a uniformly moving screw dislocation is considered. A general solution of the problem is obtained using Laplace and Kontorovich-Lebedev integral transformations. It is shown during the solution that the method is suitable for a wedge apex angle greater than n. The limiting case, when the wedge-shaped region is a half-plane, is also considered. For this case the solution can be simplified considerably.     

Creeping flow of a sphere nearby a cylinder

We present a three-dimensional solution of a sphere nearby an infinite cylinder at low Reynolds number. We utilize the Lamb’s general solution based on spherical harmonics and develop a framework based on cylindrical harmonics to solve the flow field around the sphere and outside the cylinder, respectively. The solution is solved semi-analytically by considering geometrical parameters, including sphere radius, sphere velocity, separation distance and cylinder radius. The drag force coefficients of the sphere which are dependent on the distance between the cylinder surface and the sphere, as well as the velocity contours in the vicinity of the sphere, are analyzed. We also provide an analytical formula to calculate the drag force. The analytical formula has good quantitative agreement with the semi-analytical solution when the radius of the cylinder is smaller than the sphere. Such analysis can give insights into the details of the complex interaction between the sphere and cylinder.     

Vibrations of a beam-string complex system under a moving force

In this paper, the dynamic response of an Euler-Bernoulli beam and string system traversed by a constant moving force is considered. The force is moving with a constant velocity on the top beam. The complex system is finite, simply supported, parallel one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a beam-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case the aperiodic part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations. The dynamic influence lines of complex systems may be used for the analysis the complex models of moving load. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions of a dissipative diffusion equation in a Hilbert space

Two types of dissipation of a diffusion process on an abstract space are discussed. The main results are the bound, critical length, and extension of solutions of a dissipative Riccati equation on a Hilbert space. The bound result enables the local solution to be extended to a global solution. The analysis of critical length provides computational and theoretical information about the maximal interval of existence of solutions. It is shown that no bounded solution can be extended beyond the critical length.     

Solvability of a free-boundary problem for a heterogeneous elastic body

We consider the free-boundary problem for a stationary elastic system. We prove that the problem has a classical solution if the initial data are close to the stationary solution.     

On the Existence and Uniqueness of a Solution of the Bellman Equation in a Model of Operation of a Manufacturing Company with Regard to the Debt Load

We study the existence and uniqueness of a solution of the Bellman equation formalizing the operation of a manufacturing company under conditions of irregular demand and debt load. The class of functions where this equation has a unique solution is determined. The choice of a specific solution is justified.     

On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation

We study a nonlocal boundary value problem for a degenerating pseudoparabolic third-order equation of the general form. For the solution of the problem, we obtain a priori estimates in differential and difference form, which imply the stability of the solution with respect to the initial data and right-hand side on a layer as well as the convergence of the solution of the difference problem to the solution of the differential problem.     

Differential properties of a generalized solution to a hyperbolic system of first-order differential equations

We study some questions of the qualitative theory of differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations whose right-hand sides contain some discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. In particular, its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts the common approach which uses the reduction of a system of two first-order equations to a single second-order equation.     

An Estimate for a Fundamental Solution of a Parabolic Equation with Drift on a Riemannian Manifold

We construct a fundamental solution for a parabolic equation with drift on a Riemannian manifold of nonpositive curvature. We obtain some estimates for this fundamental solution that depend on the conditions on the drift field.     

Solution of a regularized problem for a stationary magnetic field in a nonhomogeneous conducting medium by a finite element method

The use of a vector finite element method for solving a regularized stationary magnetic field problem, which is formulated in terms of a vector magnetic potential, is validated. A generalized solutionis approximated using first-order Nedelec vector elements of the second kind on tetrahedrons. The existence and uniqueness of the solution to a discrete regularized problem and its convergence to the generalized solution for the case of an inhomogeneous (in the electromagnetic properties) domain are justified. Some issues of the numerical solution to the discrete regularized problem are discussed. Approaches to optimize the algorithms are shown on a series of numerical experiments.     

Asymptotics of a Solution to the Quasiliniear Neumann Problem in a Neighborhood of a Conical Point

The existence of a small solution to the quasilinear Neumann problem is established in a H\"older class with separated asymptotics. The lower-order asymptotic terms are constructed. The homogeneity exponents of these terms turn out to be dependent on the value of the solution at a conical point. Bibliography: 33 titles.