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In this paper we study a new problem for a parabolic-hyperbolic equation with fractional order Caputo operator in rectangular domain. There are many works devoted to study problems for the second order mixed parabolic-hyperbolic and elliptic-hyperbolic type equations in rectangular domains with two gluing conditions with respect to second argument and with boundary value conditions on all borders of the domain. In studying the unique solvability of this problem, it becomes necessary to specify an additional condition on the hyperbolic boundary of the domain. For this reason, the considering problem became unresolved in an arbitrary rectangular domain. In this paper, we were able to remove this restriction by setting three gluing conditions for the second argument.

The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann–Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin’s method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev ’s attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates.A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann–Dirichle boundary condition.

The paper presents an efficient numerical technique for 3D-modeling of seismic stability of large buried structures. This technique allows considering the subsoil-structure contact interaction, gravity field effects, the inhomogeneous structure of soil and the varieties location of the earthquake hypocenter. As part of this technique is substantiated sizing of the soil-structure computational domain and continuum model for hard and soft soil foundations describing. The numerical technique for determining the kinematic conditions at the lower boundary of the computational domain from the experimental accelerations at the soil surface is given, offered special non-reflecting waves boundary conditions. The methods described above and algorithms for seismic resistance solving have been implemented in certified software package “Dynamica-3”, parallelization of the algorithm has been held according to the spatial domain decomposition principle. The developed numerical technique allows correctly posing the problem of seismic vibrations of buried structure, reducing computing costs and increasing the efficiency of numerical studies of Earthquake Engineering. Through this, the multiple recalculation of the task with different action scenarios generated from experimental seismograms by probabilistic methods became technically possible. The results of these calculations allow reflecting the experience of many earthquakes, which increases reliability of the estimates. The paper presents the model calculations results of seismic stability of large-sized buried structure—the mutual vertical and horizontal displacement of soil and building walls—which can then be used to assess strength of adjacent underground pipelines. A number of numerical experiments on seismic vibrations of nearby large-sized structures have been carried out to assess the mutual influence of structures on their behavior in an earthquake. It is established that the influence of nearby large-sized objects on seismic vibrations of structures differs for different buildings.

The transient interaction of a rigid convex indenter with elastic half-plane boundary is investigated in this paper. At the initial time point, the rigid indenter and half-plane have some distance between them. Indenter moves towards the half-plane on arbitrary law of motion. The indenter and half-plane interaction is considered in two stages: contactless interaction, which includes only adhesive forces, and the stage with mechanical contact between indenter and half-plane’s boundary. On the second stage, the adhesive force also takes place. In this paper the first stage of interaction is considered. To obtain the normal displacement of the half-plane, the superposition principle is used. For this method, the Lamb problem solution is considered as a Green function. However, the adhesive force function’s support is unknown. In this article the original numerical and analytical algorithm is constructed and realized, the solution for considered problem is obtained. Graphical results for half-plane boundary normal displacements space-time distribution are presented. Also graphical results for adhesive force support boundary position and velocity on time dependence are shown.

Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.

This paper is devoted to the study of so-called finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries and quasi-isometries. We obtain several criteria for the homeomorphic extension to the boundary of finitely bi-Lipschitz homeomorphisms f between domains in ℝn, n ≥ 2, whose outer dilatations KO(x, f) satisfy the integral constraints $$\int {\Phi (K_O^{n - 1} (x,f))dm(x) < \infty } $$ with an increasing convex function Φ: [0,∞] → [0,∞]. Note that the integral conditions on the function Φ (obtained in the paper) are not only sufficient, but also necessary for the continuous extension of f to the boundary.

We study the canonical and boundary representa� tions on a sphere Ω with an action of the generalized Lorentz group, following the extended interpretation suggested by Molchanov in [10, 11]. In this case, the sphere is not a homogeneous space, the action of the group G is not transitive, and the representations are not unitary. The manifold Ω is the flag space of the overgroup G*. The fundamental role in the theory is played by the Berezin transform (which is the basic object in Berezin quantization), that is, the operator intertwining canonical representations with “conju� gate” parameters. In this paper, we present an decomposition of the Berezin forms and construct harmonic analysis on a pair hyperboloids. The constructions and methods of the paper are of independent interest.

The present paper deals with finding an optimal control of vibrations of a spherical layer, where the control is given by conditions of the third kind on the boundary. We continue and develop the results obtained by V.A. Il’in, E.I. Moiseev, and their students. We consider a more general control optimality criterion than in the previous papers.

We seek the optimal boundary control of vibrations of a spherical layer in the spherically symmetric case. This paper continues the series of papers by V.A. Il’in and his students and, unlike the previous papers, uses a more general control optimality criterion. We obtain closed formulas for the controls; these formulas are consistent with Il’in’s earlier results.

This paper analyzes the stability of a linear autonomous nonconservative system with an even number of degrees of freedom in the presence of potential, gyroscopic, dissipative, and nonconservative positional forces. It is well known that, when applied separately, dissipative and nonconservative positional forces destroy gyroscopic stabilization [1, 3]. However, their combination can make a system asymptotically stable. It is found that the complexity of the choice of such a combination is associated with a Whitney umbrella singularity existing on the boundary of the gyroscopic stabilization domain of the nonconservative system. In this paper, an approximation to the boundary of the asymptotic stability domain near the singularity is explicitly found and an analytical estimate of the critical gyroscopic parameter is obtained. As an example, we analyze the stability of Hauger’s gyropendulum under the action of a follower torque. 1. Consider an autonomous nonconservative system of the form

2 [0 ,l ]a ndL2[0 ,l ], respectively. The pair {ϕ(x) ,ψ (x)} can be treated as an element of the Hilbert space H = W 1 2 [0 ,l ] × L2[0 ,l ]. Along with the pair {ϕ(x) ,ψ (x)}, which specifies the initial state of the system, we consider an arbitrary pair {ϕ1(x) ,ψ 1(x)} in the space H . The paper [1] gives necessary and sufficient conditions for the pair {ϕ1(x) ,ψ 1(x)} to be a state of the system at time T and for the existence of a boundary control μ(t )i nW 1 2 [0 ,T ] (an close-form expression is given for it) bringing the vibration process from the state {u(x, 0) = ϕ(x) ,u t(x, 0) = ψ(x)} to the state {u(x, T )= ϕ1(x) ,u t(x, T )= ψ1(x)}. In this connection, we face the natural problem on the existence and close-form representation of a boundary control μ∗(t) applied to the left end and, in the case of failure of the above-mentioned necessary and sufficient conditions, bringing the vibration process from a state {u(x, 0) = ϕ(x) ,u t(x, 0) = ψ(x)} to a state {u(x, T )= ϕ∗(x) ,u t(x, T )= ψ∗(x)} that has the least deviation from the a priori unattainable state {ϕ1(x) ,ψ 1(x)} in the norm of the space H . This problem is considered in the present paper for 0

The present paper is a continuation of [1] and uses the notation and definitions introduced there. In [1], the authors also presented the properties of transformation operators, studied new function spaces, introduced the notion of σ-trace, and proved the direct and inverse trace theorems. In the present paper, we pose a singular boundary value problem in a domain with a single corner (singular) point. We prove some auxiliary results. The main result is given by the theorem on the unique solvability of the corresponding boundary value problem.

This paper studies spectral properties of ordinary differential operators of even order 2n, n > 1, with nonsmooth coefficients in the differential operation, in particular, with nonsmooth coefficient multiplying the (2n – 1)th derivative. We use the Il’in’s approach [1] to studying spectral properties of operators irre� spective of the particular form of the boundary condi� tions. We investigate the dependence of estimates for the rate of local convergence of spectral decomposi� tions on the distance from an interior compact set K to the boundary ∂G in the case where the normalized Fourier coefficients αkfk have asymptotics O () , where ν = const > 0 and λk is the spectral parameter (see Theorem 1) and in the case where the coefficients have asymptotics O (l n –β λ k ), β = const (see Theo� rem 2). We begin with a brief review of related results. Paper [2] studies the rate of convergence of biorthog� onal expansions of functions in systems of root func�

The paper is devoted to the study of conditional bounds for the expectation of the maximum of independent identically distributed standardized random variables for which the values of the skewness and kurtosis coefficients are known. With the aid of Holder’s inequality, an upper bound (in the form of a lower bound for a certain expression with parameters) is obtained and a criterion for the reachability of this estimate is formulated. A lower bound for the upper boundary of the expectation of the maximum is also found. A simpler and rougher upper bound is given in explicit form.

An ideal triangulation of a compact $ 3 $ -manifold with nonempty boundary is known to be minimal if and only if the triangulation contains the minimum number of edges among all ideal triangulations of the manifold. Therefore, every ideal one-edge triangulation (i.e., an ideal singular triangulation with exactly one edge) is minimal. Vesnin, Turaev, and Fominykh showed that an ideal two-edge triangulation is minimal if no $ 3 $ – $ 2 $ Pachner move can be applied. In this paper we show that each of the so-called poor ideal three-edge triangulations is minimal. We exploit this property to construct minimal ideal triangulations for an infinite family of hyperbolic $ 3 $ -manifolds with totally geodesic boundary.

The paper deals with the problem of recovering solutions of a generalized Cauchy–Riemann system in a multidimensional spatial domain from their values on a piece of the boundary of this domain, i.e., an approximate solution of this problem based on the Carleman–Yarmukhamedov matrix method is constructed.

An advantage of the discontinuous Galerkin method is the ability to describe complex contact surfaces by using unstructured computational grids. However, the use of the discontinuous Galerkin method requires significant computational resources, including at the preprocessing stage. The grid-characteristic method on structured regular computational grids saves computational resources, but problems arise when taking into account complex inhomogeneities, including surface topography. Therefore, it is of interest to combine the grid-characteristic method on structured computational grids with the discontinuous Galerkin method, to which this paper is devoted. The work describes in detail the Galerkin method and the description of the contact boundary between the subdomains of the integration domain in which the grid-characteristic method and the discontinuous Galerkin method are used. The corresponding calculation algorithm is discussed. Examples are given on the calculation of the propagation of seismic waves from the hypocenter of an earthquake to the Earth’s surface, taking into account the surface topography.

For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity $d(E)/d(E,\partial\Omega)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $\Omega$ has the hyperbolic distance $h_\Omega(z,w),$ we consider the infimum $\kappa(\Omega)$ of the quantity $h_\Omega(E)/\log(1+d(E)/d(E,\partial\Omega))$ over compact subsets $E$ of $\Omega$ with at least two points, where $h_\Omega(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $\mathbb{H}$. Our main results claim that $\kappa(\Omega)$ is positive if and only if the boundary of $\Omega$ is uniformly perfect and that the inequality $\kappa(\Omega)\leq\kappa(\mathbb{H})$ holds for all $\Omega,$ where equality holds precisely when $\Omega$ is convex., Comment: 21 pages, 2 figures

The paper focuses on the problem of normal surface waves in an open metal-dielectric inhomogeneous waveguide of arbitrary cross-section with losses. The medium filling the waveguide is characterized by a isotropic inhomogeneous complex permittivity. The setting is reduced to a boundary eigenvalue problem for longitudinal components of the electromagnetic field in Sobolev spaces. Variational formulation of the problem is considered in terms of the analysis of operator-functions. A discreteness of the set of the sought-for eigenvalues is proved.

In this paper, we study the existence, uniqueness and boundary behavior of positive boundary blow-up solutions to the quasilinear system $$\Delta_{\infty}u=a(x)u^{p}v^{q}$$ , $$\Delta_{\infty}v=b(x)u^{r}v^{s}$$ in a smooth bounded domain $$\Omega\subset R^{N}$$ , with the explosive boundary condition $$u=v=+\infty$$ on $$\partial\Omega$$ , where the operator $$\Delta_{\infty}$$ is the $$\infty$$ -Laplacian, the positive weight functions $$a(x)$$ , $$b(x)$$ are Holder continuous in $$\Omega$$ , and the exponents verify $$p$$ , $$s > 3$$ , $$q$$ , $$r>0$$ , and $$(p-3)(s-3) > qr$$ .

The paper deals with the study of dynamical systems admitting compact attractors, which describe the behavior of trajectories of the system at infinity and play an important role in the qualitative theory of stability of motion. The properties of trajectories of a dynamical system in the attraction domain of the attractors and on the boundary are established by using concepts such as elliptic and weakly elliptic sets, as well as the pseudo-stability and pseudo-prolongation properties.

In previous works ([4–8]), the author studied the general approach to solving mixed boundary, spectral, and initial-boundary transmission problems. In this paper, this approach is applied to the mixed boundary transmission problem of the linear theory of elasticity. On the basis of the corresponding Green formulas the solution of the problem can be represented as the sum of the solutions of auxiliary problems containing a given function only in one place. As a result, four auxiliary problems are obtained. We find their weak solutions using Green’s formulas. And then we conclude that the solution of the original problem is the sum of the auxiliary solutions.

We prove new integral inequalities for real-valued test functions defined on subdomains of the Euclidean space. Namely, we obtain several new Hardy-type inequalities that contain the scalar product of gradients of test functions and of the gradient of the distance function from the boundary of an open subset of the Euclidean space. Our method of proof is based on interior and exterior approximations of a given domain by sequences of simplest domains and has two important ingredients. The first one is approximations of a given domain by elementary domains that admit a special partition. The second ingredient is presented in this paper by Theorems 1 and 2 about convergence everywhere of the sequences of distance functions from boundary of approximating elementary domains as well as about convergence almost everywhere for their gradients. In the proofs we also use some basic theorems due to Rademacher, Hardy, Motzkin and Hadwiger.

The present paper considers the determination of the two-phase region boundary within a simplified version of the wide-range equations of state of water and steam by Nigmatulin–Bolotnova (EOS NB) in the case the determination is based on the temperature-dependence of the saturated vapor pressure. This version of EOS NB can be used when the temperature and pressure ranges under investigation do not include the area of abnormal features of water at temperatures close to freezing ones (lower than $$10^{o}C$$ ) and when some errors in adiabatic speed of sound in a nearly-critical zone are acceptable. In particular, it is applied to numerically simulating dynamics of cavitation bubbles under their strong collapse. It is shown that in determining the two-phase region boundary within the simplified version of EOS NB on the basis of the temperature-dependencies of the saturated vapor pressure, a number of problems can occur. A simple modification of one of the known temperature-dependencies of the saturated vapor pressure, free of such problems, is presented.

The paper is devoted to studying limit behavior for a solution of model elliptic pseudo-differential equation with some integral boundary condition in 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev–Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

This paper is devoted to the study of the asymptotics of roots of a sequence of Bernstein polynomials approximating a piecewise linear function. This sequence arises in the construction of modified compactly supported wavelets that, in contrast to classical Daubechies wavelets, preserve localization with the growth of smoothness. It is proved that the limiting curve for roots is the boundary of the domain of convergence of the Bernstein polynomials on the complex plane.

In this paper lower bounds for entire functions of exponential type and regular growth, zero sets of which have zero condensation indices, are obtained. In this case, the exceptional set consists of pairwise disjoint disks centered at zeroes. Sufficient conditions for radii of these circles are indicated. We also obtain a result on representation of analytic functions in the closure of a bounded convex domain (as well as analytic functions in domain and continuous up to the boundary) by series of exponential monomials. This result extends the classical result of A.F. Leont’ev to the case of multiple zero set of entire function. The obtained result is applied to the problem on distribution of singular points of a sum of series of exponential monomials at the boundary of its convergence domain.

The paper concerns the development of a constructive method of solving one of the main boundary value problems of Riemann problem type in the class of metaanalytic functions of the second type in the case when a unit circle serves as a boundary

Together with the classical plane billiards, topological billiards can be considered, where the motion occurs on a locally flat surface obtained by isometrically gluing together several plane domains along their boundaries, which are arcs of confocal quadrics. A point moves inside each of the domains along straight line segments; when it reaches the boundary of a domain, it passes to another domain. Previously, the author gave a Liouville classification of all topological billiards obtained by gluing along convex boundaries. In the present paper, all topological integrable billiards obtained by gluing along convex or nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics are classified. For some of such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules W*) for Liouville equivalence are calculated.

The main subject of this paper is the study of a general linear boundary-value problem with Drazin or right Drazin (respectively, left Drazin) invertible operators corresponding to initial boundary operators. The obtained results are then employed to solve a Schro¨ dinger equation.

This paper concerns the extension of univalent conditions known for a ring domain to the case of a half-plane. We study the possibilities of applying Rahman and Brickman methods to functions analytic in the upper half-plane. Some generalizations of the functions convex in one direction are obtained. We also investigate the curvelinear half-planes, the boundary of which can be attained by a family of shift curves, and demonstrate the efficiency of the quasiconformal extension method. The subclasses of domains, whose boundaries are quasiconformal curves, are identified. Some sufficient conditions for univalence of functions analytic in these domains are established.

For an algebra A, denote by V A (n) the dimension of the vector space spanned by the monomials whose length does not exceed n. Let T A (n) = V A (n) − V A (n − 1). An algebra is said to be boundary if T A (n) − n < const. In the paper, the normal bases are described for algebras of slow growth or for boundary algebras. Let L be a factor language over a finite alphabet A. The growth function T L (n) is the number of subwords of length n in L. We also describe the factor languages such that T L (n) ≤ n + const.

We consider the Cauchy problem with spatially localized initial data for the twodimensional wave equation degenerating on the boundary of the domain. This problem arises, in particular, in the theory of tsunami wave run-up on a shallow beach. Earlier, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi developed a method for constructing asymptotic solutions of this problem. The method is based on a modified Maslov canonical operator and on characteristics (trajectories) unbounded in the momentum variables; such characteristics are nonstandard from the viewpoint of the theory of partial differential equations. In a neighborhood of the velocity degeneration line, which is a caustic of a special form, the canonical operator is defined via the Hankel transform, which arises when applying Fock’s quantization procedure to the canonical transformation regularizing the above-mentioned nonstandard characteristics in a neighborhood of the velocity degeneration line (the boundary of the domain). It is shown in the present paper that the restriction of the asymptotic solutions to the boundary is determined by the standard canonical operator, which simplifies the asymptotic formulas for the solution on the boundary dramatically; for initial perturbations of special form, the solutions can be expressed via simple algebraic functions.

The boundary control of vibrations of a plane membrane is considered. A constraint is imposed on the absolute value of the control function. The goal of the control is to drive the membrane to rest. The proof technique used in this paper can be applied to a membrane of any dimension, but the two-dimensional case is considered for simplicity and illustrative purposes.

The paper is concerned with the interpolation of a function of two variables with large gradients in the boundary layers. An underlying function is assumed to be the sum of the regular componentwith derivatives bounded up to some order and two boundary layer components, the latter are known up to a multiplicative factor. Such a representation is typical for the solution of a singular perturbed elliptic problem. A two-dimensional interpolation formula exact on the boundary layer components is put forward. The formula has an arbitrary number of nodes in each direction. An error estimate is obtained which is uniform on the gradients of the underlying function in boundary layers. Results of numerical experiments are provided.

This paper deals with the boundary-value problem for a nonlinear elliptic equation containing a small parameter multiplying the derivatives and degenerating into a finite equation as the small parameter tends to zero. The existence theorem for the solution with a boundary layer and its Lyapunov stability are proved.

The paper is devoted to the two dimensional Ericksen–Leslie system describing the nematodynamics of liquid crystals. The moment of inertia of molecules is supposed to be strictly positive. The existence of the solution was proved in the case of a periodic domain and in the case of a bounded domain. In the last case the media is supposed to adhere to the solid surface the director vector field describing orientation of the molecules is constant in the neighbourhood of the boundary. The uniqueness of the strong solution was proved in both cases. Also we prove the propagation of director disturbance has finite speed. This fact shows the difference between the model under consideration and other models with zero moment of inertia of the molecules. The estimate of the speed of propagation depending on physical properties of the liquid crystal and the flow is obtained.

In this paper, we study the existence of weak solutions for the boundary-value problem $$\Delta _\gamma u + g(x,u) = 0in\Omega u = u_0 on\partial \Omega ,$$ (1) where Ω is a bounded domain with smooth boundary in ℝN (N ≥ 2) and Δγ is a subelliptic operator of the type $$\Delta _\gamma u = \sum\limits_{j = 1}^N {\partial _{x_j } (\gamma _j^2 \partial _{x_j } u)} ,\partial _{x_j } u = \frac{{\partial u}} {{\partial x_j }},\gamma = (\gamma _1 ,\gamma _2 ,...,\gamma _N ) $$ . We use the sub-super solution and variational methods.

In this paper, we study a posteriori error estimates of finite element methods for boundary parabolic optimal control problems. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for the boundary parabolic optimal control problems.

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ⊂ ℝn, n ≥ 3, where 0 ≤ V (x) ∈ L1(Ω), 0 ≤ ϕ(x) ∈ L1(∂Ω) and ϕ(x) is continuous on the boundary ∂Ω. It is proved that there exists a constant C*(n) = (n − 2)2/4 such that if V0(x) = (c + λ1)|x|−2, then, for 0 ≤ c ≤ C*(n) and V(x) ≤ V0(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ϕ(x) ∈ L1(∂Ω); for c > C*(n) and V(x) ≥ V0(x) in Ω, this problem has no nonnegative solutions if ϕ(x) > 0.

Rational approximations to the function zα, α ∈ ℝ \ ℤ, were studied by Newman, Gonchar, Bulanov, Vyacheslavov, Andersson, Stahl, and others. The present paper deals with the order of best rational approximations to this function in a domain with zero external angle and vertex at the point z = 0. In particular, the obtained results show that the conditions imposed on the boundary of the domain in the Jackson-type inequality proved by the author in 2001 for the best rational approximations in Smirnov spaces cannot be weakened significantly.

In the present paper, we consider a boundary control by displacements at one endpoint of an inhomogeneous rod that has two parts of different densities and elasticities with fixed other endpoint for the case in which the times of passage of the wave through each of these inhomogeneity parts coincide. We find a closed analytic form of the boundary control by displacements bringing the stick from the original rest state into a given terminal state specified by given terminal displacement and velocity.

151 Recently, the control theory of thermal and hydro� dynamic fields in continuous media has been inten� sively developed. The mathematical description of such problems includes three components: a goal, the control mechanisms used to achieve this goal, and constraints imposed on the state and controls of the system under study. The constraints are specified by the equations of the used continuous medium model together with boundary conditions set on the bound� ary of the domain, while the desired goal is achieved by minimizing a certain cost functional. Control problems and inverse extremum problems for stationary models of heat and mass transfer have been theoretically studied, for example, in [1–8], where the solvability of optimization problems was analyzed and optimality systems describing necessary extremum conditions were derived and examined. In [6–8], based on an analysis of an optimality system, sufficient conditions for the uniqueness and stability of solutions to control problems were established in spe� cial cases corresponding to purely hydrodynamic or temperature cost functionals and controls. The con� trol of viscous flows by applying heat sources and the control of thermal processes by applying hydrody� namic sources have been investigated to a lesser degree. At the same time, computations show that effective mechanisms for the control of viscous flows can be found in this direction. The goal of this paper is to study such control prob� lems for the Oberbeck–Boussinesq thermal convec� tion model. First, we formulate the general control problem for this model and present a theorem provid� ing sufficient conditions on the initial data under which the solution of the control problem is unique and stable with respect to small perturbations in the cost functional and the boundary function involved in the Dirichlet boundary condition for velocity. Next, we describe a numerical solution algorithm for the control problem based on Newton’s method and dis� cuss numerical results. They confirm that the algo� rithm is highly effective in a wide range of the basic parameters of the boundary value problem.

It is well known that the numerical solution of initial‐boundary value problems for the nonlinear Navier‐ Stokes equations based on Newton’s method faces two practical problems: inverting the Frechet derivative of the corresponding nonlinear mapping and choosing an initial approximation from a sufficiently small neighborhood of the desired solution. The goal of this paper is to develop new effective approaches to coping with these problems. In a bounded domain Ω ⊂ � 3 with boundary ∂Ω ∈ C 2 , we consider the initial‐boundary value problem for the nonlinear nonstationary Navier‐Stokes equations

In the present paper, in terms of a generalized solution of the wave equation, we perform an exhaustive study of the problem on the boundary control by an elastic force u x (0, t) = µ(t) at one endpoint x = 0 of a string in the presence of a model nonlocal boundary condition of one of four types relating (with the sign “+” or “−”) the values of the displacement u(x, t) or its derivative u x (x, t) at the boundary point x = l of the string to their values at some interior point $$ \mathop x\limits^ \circ $$ of the string (0 < $$ \mathop x\limits^ \circ $$ < l). We prove necessary and sufficient conditions for the existence of such boundary controls. Under these conditions, we optimize the controls by minimizing the boundary energy integral and then write out the optimal boundary controls in closed analytic form.

In this paper, using mathematical simulation, we establish that extraordinary light transmission arises not only in a hole but also in other inhomogeneities present in metallic films. It is shown that this effect is most pronounced at those wavelengths for which the scattering cross section in the spectral domain is maximal. Moreover, the effect depends considerably on the material of the film. The mechanism of this phenomenon is established using simulation results. Consider a mathematical model of an inhomogeneity in a metallic film deposited on a glass prism. Suppose that the entire space is divided into three parts: air (domain D 0 ), the film ( D f ), and the glass prism ( D 1 ). Let the film and the air be separated by the plane Ξ 0 ; and the prism and the film, by the plane Ξ 1 . A local cylindrical inhomogeneity occupying the domain D i with a smooth boundary ∂ D i lies entirely inside the metallic film of thickness d . Assume that the symmetry axis of the inhomogeneity is aligned with the outward normal to Ξ 1 . We introduce a rectangular coordinate system such that its origin lies in Ξ 1 ( z = 0) and the Oz axis is directed along the inhomogeneity’s axis of symmetry. As an external excitation, we use a linearly polar

Suppose that λ is an arbitrary positive function from C[0, 1) such that λ(r) → ∞ as r → 1 − 0 and satisfying some growth regularity conditions, A(λ) is the set of all holomorphic functions f in the unit disk for which ln|f(z)| ≤ c · λ(|z|), |z| < 1. In this paper, we establish that there exists a function f ∈ A(λ) with root set {z k } =1 +∞ such that the sequence {|z k |} =1 +∞ is the uniqueness set for the class A(λ).

In the present paper, we exhaustively solve the problem of boundary control by the displacement u(0, t) = µ(t) at the end x = 0 of the string in the presence of a model nonlocal boundary condition of one of four types relating the values of the displacement u(x, t) or its derivative u x (x, t) at the boundary point x = l of the string to their values at some interior point $$ \mathop x\limits^ \circ $$ .

We obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as limits of integrals over the Cartesian powers of the domain; the integrands are elementary functions depending on the geometric characteristics of the manifold, the coefficients of the equation, and the initial data. It is natural to call such representations Feynman formulas. Besides, we obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as functional integrals with respect to Weizsacker-Smolyanov surface measures and the restriction of the Wiener measure to the set of trajectories in the domain; such a restriction of the measure corresponds to Brownian motion in a domain with absorbing boundary. In the proof, we use Chernoff’s theorem and asymptotic estimates obtained in the papers of Smolyanov, Weizsacker, and their coauthors.

We establish that the optimal boundary controls of string vibrations found in the papers published in Differentsial’nye Uravneniya, 2007, vol. 43, nos. 10–12 and 2008, vol. 44, no. 1 preserve their form if an arbitrary point of the string is taken as the point where the initial and terminal conditions are coordinated.