# Showing papers in "Lobachevskii Journal of Mathematics in 2020"

••

TL;DR: In this article, weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for a parabolic differential equation is studied under initial and boundary conditions.

Abstract: The questions of weakly generalized solvability of a nonlinear inverse problem in nonlinear optimal control of thermal processes for a parabolic differential equation are studied. The parabolic equation is considered under initial and boundary conditions. To determine the recovery function, a nonlocal integral condition is specified. Moreover, the recovery function nonlinearly enters into the differential equation. Is applied the method of variable separation based on the search for a solution to the mixed inverse problem in the form of a Fourier series. It is assumed that the recovery function and nonlinear term of the given differential equation are also expressed as a Fourier series. For fixed values of the control function, the unique solvability of the inverse problem is proved by the method of compressive mappings. The quality functional has a nonlinear form. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to a complicated functional-integral equation, the process of solving which consists of solving separately taken two nonlinear functional and nonlinear integral equations. Nonlinear functional and integral equations are solved by the method of successive approximations. Formulas are obtained for the approximate calculation of the state function of the controlled process, the recovery function, and the optimal control function. Is proved the absolutely and uniformly convergence of the obtained Fourier series.

39 citations

••

TL;DR: In this article, a family of discrete Schrodinger operators is considered in the lattice and the existence of eigenvalues outside the essential spectrum and their dependence on the parameters of the operator is derived.

Abstract: We consider a family of the discrete Schrodinger operators
$$H_{\lambda\mu}$$
, depending on parameters, in the
$$3$$
-dimensional lattice,
$$\mathbb{Z}^{3}$$
with a non-local potential constructed via the Dirac delta function and the shift operator. The existence of eigenvalues outside the essential spectrum and their dependence on the parameters of the operator are explicitly derived. The threshold eigenvalue is proven to be absorbed into the essential spectrum and it turns into an embedded eigenvalue at the left intercept of a particular parabola, and the threshold resonance at the other points of the parabola.

32 citations

••

TL;DR: In this article, a Boussinesq type nonlinear partial integro-differential equation of the fourth order with a degenerate kernel, integral form conditions, spectral parameters and reflecting argument is considered.

Abstract: In the three-dimensional domain a Boussinesq type nonlinear partial integro-differential equation of the fourth order with a degenerate kernel, integral form conditions, spectral parameters and reflecting argument is considered. The solution of this partial integro-differential equation is studied in the class of generality functions. The method of separation of variables and the method of a degenerate kernels are used. Using these methods, the nonlocal boundary value problem is integrated as a countable system of ordinary differential equations. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving countable system of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed and we obtained the countable system of nonlinear integral equations for each of five cases. To establish the unique solvability of this countable system of nonlinear integral equations we use the method of successive approximations and the method of compressing mappings. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution of the boundary value problem with respect to given functions in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed as Fourier series. For other cases, the absence of nontrivial solutions of the problem is proved. The corresponding theorems are formulated.

31 citations

••

TL;DR: In this article, an analogue of the Gellerstedt problem for a loaded parabolic-hyperbolic equation of the third order in an infinite three-dimensional domain is studied, and sufficient conditions are found such that all differentiation operations are legal.

Abstract: In this paper, we study an analogue of the Gellerstedt problem for a loaded parabolic-hyperbolic equation of the third order in an infinite three-dimensional domain. The main method to study this Gellerstedt problem is the Fourier transform. Based on the Fourier transform, we reduce the considering problem to a planar analogue of the Gellerstedt spectral problem with a spectral parameter. The uniqueness of the solution of this problem is proved by the new extreme principle for loaded third-order equations of the mixed type. The existence of a regular solution of the Gellerstedt spectral problem is proved by the method of integral equations. In addition, the asymptotic behavior of the solution of the Gellerstedt spectral problem is studied for large values of the spectral parameter. Sufficient conditions are found such that all differentiation operations are legal in this work.

28 citations

••

TL;DR: In this article, the authors derived a set of quantum channels attainable in the completely positive divisible phase covariant dynamics and showed that this set coincides with the set of channels available in semigroup phase covariants, which can still exhibit non-monotonicity of populations for any initial state.

Abstract: Phase covariant qubit dynamics describes an evolution of a two-level system under simultaneous action of pure dephasing, energy dissipation, and energy gain with time-dependent rates
$$\gamma_{z}(t)$$
,
$$\gamma_{-}(t)$$
, and
$$\gamma_{+}(t)$$
, respectively. Non-negative rates correspond to completely positive divisible dynamics, which can still exhibit such peculiarities as non-monotonicity of populations for any initial state. We find a set of quantum channels attainable in the completely positive divisible phase covariant dynamics and show that this set coincides with the set of channels attainable in semigroup phase covariant dynamics. We also construct new examples of eternally indivisible dynamics with
$$\gamma_{z}(t)<0$$
for all
$$t>0$$
that is neither unital nor commutative. Using the quantum Sinkhorn theorem, we for the first time derive a restriction on the decoherence rates under which the dynamics is positive divisible, namely,
$$\gamma_{\pm}(t)\geq 0$$
,
$$\sqrt{\gamma_{+}(t)\gamma_{-}(t)}+2\gamma_{z}(t)>0$$
. Finally, we consider phase covariant convolution master equations and find a class of admissible memory kernels that guarantee complete positivity of the dynamical map.

23 citations

••

TL;DR: In this article, the authors studied the steady adiabatic filtration of real gases by Legendrian surfaces in 5-dimensional thermodynamic contact space and showed the relation between phase transitions and singularities of projection of the Legendrian surface on the plane of intensive variables.

Abstract: Steady adiabatic filtration of real gases is studied. Thermodynamical states of real gases are presented by Legendrian surfaces in 5-dimensional thermodynamical contact space. The relation between phase transitions and singularities of projection of the Legendrian surfaces on the plane of intensive variables is shown. The constructive method of finding solutions of the Dirichlet filtration problem together with analysis of critical phenomena is presented. Case of van der Waals gas is discussed in details.

23 citations

••

TL;DR: An open two-level quantum system evolving under coherent and incoherent piecewise constant controls constrained in their magnitude and variations is considered, which combines the approach of $$k$$ nearest neighbors and training a multi-layer perceptron neural network to predict suboptimal final times and controls.

Abstract: This work considers an open two-level quantum system evolving under coherent and incoherent piecewise constant controls constrained in their magnitude and variations. The control goal is to steer an initial pure density matrix into a given target density matrix in a minimal time. A machine learning algorithm was developed, which combines the approach of $$k$$
nearest neighbors and training a multi-layer perceptron neural network, to predict suboptimal final times and controls. For 18 sets of initial pure states with different size (between 10 and 200) training datasets were constructed. The numerical results are described, including the analysis of the dependence of the quality of the machine learning algorithm on the size of the training set.

20 citations

••

TL;DR: It is established that associative steganography retains the property of provable (computational) stability in this case as well and recommended for its use to protect the text characteristics of objects.

Abstract: The case of analysis of associatively protected cartographic scenes is considered. Protection of objects and their coordinates is achieved masking binary matrices of their code symbols. The set of inverse mask matrices is the recognition key. This allows such protection to be attributed to associative steganography. A message is considered to be unconditionally steadfast if it is statistically indistinguishable from a random sequence. Therefore, the study of its steadfastness is carried out using statistical tests of randomness NIST. If a pseudo-random sequence successfully passes the test of all 15 tests, then it is considered random (‘‘white’’). If there is a failure at least on one test, then it is considered ‘‘black’’. But in the case of the application of the basic masking algorithm, this cannot help the disclosure of the stegomessage. The effect of masking redundancy introduced with the aim of improving noise immunity on the durability of mapping objects to the effects of various attacks is considered. It is established that associative steganography retains the property of provable (computational) stability in this case as well. Recommendations for its use to protect the text characteristics of objects are given.

18 citations

••

TL;DR: For a mixed-type equation of the second kind with a singular coefficient by the spectral expansions uniqueness criterion, the solution was constructed as the sum of a Fourier-Bessel series.

Abstract: For a mixed-type equation of the second kind with a singular coefficient by the spectral expansions uniqueness criterion is installed for solving the first boundary-value problem. The solution was constructed as the sum of a Fourier–Bessel series. In justifying the uniform convergence appeared the problem of small denominators. In connection with this evaluation is set on a small separation from zero denominator with the corresponding asymptotic behavior that allowed to justify the convergence of the series in the class of regular solutions.

18 citations

••

TL;DR: The equivalence of using the contact condition of complete adhesion with the absence of an interface in the case of the same elastic and acoustic properties is proved in this paper.

Abstract: This work is devoted to the modification of the grid-characteristic numerical method. We propose using joint structured regular and curved computational grids to describe the complex geometric shape of objects and save computing resources. In subdomains, where possible, we propose use structured regular computational grids conformal with structured curved grids. Moreover, we use different calculation algorithms in subdomains with structured regular grids and with curved grids. The elastic and acoustic wave equations are also jointly solved in various subdomains of the integration domain for a more accurate description of the simulated model. We use the corresponding contact conditions at the boundaries between subdomains with different types of computational grids, different solved systems of equations, and different elastic and acoustic parameters of the media. We have proved in this paper the equivalence of using the contact condition of complete adhesion with the absence of an interface in the case of the same elastic and acoustic properties. The proposed modification of the grid-characteristic method was tested on the problems of studying the earthquake resistance of a bridge over a river and a bridge over a highway.

16 citations

••

TL;DR: In this paper, a mathematical model is proposed that describes the process of pumping gas into a porous medium initially filled with methane and water, where non-ideal gas and non-isothermal effects during its filtering are taken into account.

Abstract: Mathematical model is proposed that describes the process of
pumping gas into a porous medium initially filled with methane
and water. Non-ideal gas and non-isothermal effects during its
filtration are taken into account. The hydrate formation process
is assumed to be equilibrium. Methodology for solving the system
of equations of the mathematical model is constructed. The
methodology was tested, in the one-dimensional axisymmetric case
the calculation results were compared with a self-similar solution
for a perfect gas, which showed a good qualitative and
quantitative agreement.

••

TL;DR: In this paper, the Rogosinski inequality for analytic functions was considered for harmonic mappings of the form of the harmonic form, where the analytic part of the power series is bounded by the coefficients of the derivatives.

Abstract: In this paper we first consider another version of the Rogosinski inequality for analytic functions $$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$$
in the unit disk $$|z|<1$$
, in which we replace the coefficients $$a_{n}$$
$$(n=0,1,\ldots,N)$$
of the power series by the derivatives $$f^{(n)}(z)/n!$$
$$(n=0,1,\ldots,N)$$
. Secondly, we obtain improved versions of the classical Bohr inequality and Bohr’s inequality for the harmonic mappings of the form $$f=h+\overline{g}$$
, where the analytic part $$h$$
is bounded by $$1$$
and that $$|g^{\prime}(z)|\leq k|h^{\prime}(z)|$$
in $$|z|<1$$
and for some $$k\in[0,1]$$
.

••

TL;DR: In this article, the authors define a new class of confluent hypergeometric functions of several variables, study their properties and determine the system of hypergeometrical equations that these functions satisfy.

Abstract: An investigation of applied problems related to heat conduction and dynamics, electromagnetic oscillations and aerodynamics, quantum mechanics and potential theory leads to the study of various hypergeometric functions. The great success of the theory of hypergeometric functions in one variable has stimulated the development of corresponding theory in two and more variables. In the theory of hypergeometric functions, an increase in a number of variables will always be accompanied by a complication in the study of the function of several variables. Therefore, the decomposition formulas that allow us to represent the hypergeometric function of several variables through an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitate the process of studying the properties of multidimensional functions. In the literature, hypergeometric functions are divided into two types: complete and confluent. In all respects, confluent hypergeometric functions including the decomposition formulas, have been little studied in comparison with other types of hypergeometric functions, especially when the dimension of the variables exceeds two. In this paper we define a new class of confluent hypergeometric functions of several variables, study their properties and determine the system of hypergeometric equations that these functions satisfy, because all fundamental solutions of the generalized Helmholtz equation with singular coefficients are written out through one new introduced confluent hypergeometric function of several variables. Using the decomposition formulas which are established here the order of the singularity of the found fundamental solutions of the elliptic equation which mentioned above is determined.

••

TL;DR: In this paper, the authors presented a modification based on a local linear reconstruction of the solution of the Rusanov scheme for the hydrodynamic equations for weak shock waves without any external piecewise polynomial reconstruction.

Abstract: The results of numerical modeling of white dwarf mergers on massive parallel supercomputers using a AVX-512 technique are presented. A hydrodynamic model of white dwarfs closed by a star equation of state and supplemented by a Poisson equation for the gravitational potential is constructed. This paper presents a modification based on a local linear reconstruction of the solution of the Rusanov scheme for the hydrodynamic equations. This reconstruction makes it possible to considerably decrease the numerical dissipation of the scheme for weak shock waves without any external piecewise polynomial reconstruction. The scheme is efficient for unstructured grids, when it is difficult to construct a piecewise polynomial solution, and also in parallel implementations of structured nested or adaptive grids, when the costs of interprocess interactions increase significantly. As input data, piecewise constant values of the physical variables in the left and right cells of a discontinuity are used. The smoothness of the solution is measured by the discrepancy between the maximum left and right eigenvalues. This discrepancy is used for a local piecewise polynomial reconstruction in the left and right cells. Then the solutions are integrated along the characteristics taking into account the piecewise linear representation of the physical variables. A performance of 234 gigaflops and 33-fold speedup are obtained on two Intel Skylake processors on the cluster NKS-1P of the Siberian Supercomputer Center ICM & MG SB RAS.

••

TL;DR: This note describes the degree spectra of domains, the set of Turing degrees that compute an isomorphic copy of the structure that is not computably presentable, and discusses similar notions for topological structures.

Abstract: The investigation of computability in topological structures develops in some aspects similar to the investigation of computability in algebraic structures. If a countable algebraic structure is not computably presentable then its ‘‘degree of non-computability’’ is measured by the so called degree spectrum, i.e. the set of Turing degrees that compute an isomorphic copy of the structure. In this note we initiate a discussion of similar notions for topological structures, in particular we describe the degree spectra of domains.

••

TL;DR: In this article, a numerical investigation of the fracture connectivity effect on attenuation of seismic waves propagating in fractured porous fluid-saturated media is presented, where an algorithm for statistical modeling is designed to generate fracture systems with prescribed percolation length.

Abstract: We present a numerical investigation of the fracture connectivity effect on attenuation of seismic waves propagating in fractured porous fluid-saturated media. We design an algorithm for statistical modeling to generate fracture systems with prescribed percolation length. Generated statistical realizations of the fractured systems are then analyzed to evaluate the fracture-cluster length-scale. After that for all statistical realizations we simulated wave propagation observing formation of the wave-induced fluid flows. We show that fracture-to-background fluid flows are secretive to the branch size. Thus, in the case of permeable background, seismic attenuation is affected by the branch length; i.e., attenuation increases with the increase of the branches length. If the permeability of the background material is low, no fracture-to-background wave-induced fluid flows appear, whereas strong fracture-to-fracture fluid flows may take place. However, fracture-to-fracture fluid flows are local and depend only on the parameters of the individual fractures and their intersections. As a result, the effect of the fracture-to-fracture fluid flows on seismic attenuation is relatively low, even smaller than the attenuation due to scattering.

••

Ghent University

^{1}TL;DR: In this paper, the authors construct special solutions for a certain class of degenerating differential equations of parabolic type of a high order, expressed in terms of hypergeometric functions of one variable.

Abstract: In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this investigation, we construct special solutions for a certain class of degenerating differential equations of parabolic type of a high order. These special solutions are expressed in terms of hypergeometric functions of one variable.

••

TL;DR: In this paper, a Frankl-type problem with integral conjugating condition for a mixed type equation consisting of sub-diffusion and wave equations is investigated. And the existence of formulated problem has been proved using energy integrals and the method of integral equations, imposing certain conditions on given data.

Abstract: In this work, we investigate Frankl-type problem with integral conjugating condition for a mixed type equation consisting of sub-diffusion and wave equations. A uniqueness and the existence of formulated problem have been proved using energy integrals and the method of integral equations, imposing certain conditions on given data.

••

TL;DR: A numerical algorithm for the self-consistent simulations of surface water and sediment dynamics, based on the original Lagrangian-Eulerian CSPH-TVD approach, taking into account the physical factors essential for the understanding of the shallow water and surface soil layer motions.

Abstract: In this paper, we describe a numerical algorithm for the self-consistent simulations of surface water and sediment dynamics. The method is based on the original Lagrangian-Eulerian CSPH-TVD approach for solving the Saint-Venant and Exner equations, taking into account the physical factors essential for the understanding of the shallow water and surface soil layer motions, including complex terrain structure and its evolution due to sediment transport. Additional Exner equation for sediment transport has been used for the numerical CSPH-TVD scheme stability criteria definition. By using OpenMP-CUDA and GPUDirect technologies for hybrid computing systems (supercomputers) with several graphic coprocessors (GPUs) interacting with each other via the PCI-E/NVLINK interface we also develop a parallel numerical algorithm for the CSPH-TVD method. The developed parallel version of the algorithm demonstrates high efficiency for various configurations of Nvidia Tesla CPU + GPU computing systems. In particular, maximal speed up is 1800 for a system with four C2070 GPUs compare to the serial version for the CPU. The calculation time on the GPU V100 (Volta architecture) is reduced by 95 times compared to the GPU C2070 (Fermi architecture).

••

TL;DR: In this article, a complete solution to the problem of joint analysis of the properties of algorithms and features with the architecture of computing systems is presented, which leads to a huge gap between real and peak performance indicators.

Abstract: The described project is aimed at a complete solution to the problem of joint analysis of the properties of algorithms and features with the architecture of computing systems. This problem arose in the mid-70s of the last century, and over time, its importance in the practice of using computer systems is constantly growing. The main reason is a significant complication of the architecture of computers, which determines a strong dependence of the efficiency of their work on the properties of algorithms and programs. Exactly this dependence leads in practice to a huge gap between real and peak performance indicators, which is typical for all classes of computers from mobile devices to supercomputers of the highest performance range. It is this dependence that leads to a decrease in the quality of work of supercomputer centers and a drop in the efficiency of computer systems below a fraction of a percent. And at the same time, the fundamental nature of the problem itself determines two important facts. First, it is characteristic of all computer systems and centers of the world without exception. Second, practically all scientific groups of the world in all science areas conducting research using high-performance computing systems face this problem.

••

TL;DR: In this paper, the existence and uniqueness of the three-dimensional static boundary-value problems in the framework of gradient-incomplete strain-gradient elasticity was considered and the mathematical properties of weak solutions were discussed.

Abstract: In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

••

TL;DR: In this article, the authors considered a three-dimensional elliptic equation with two singular coefficients, for which a nonlocal problem is studied in a semi-infinite parallelepiped.

Abstract: We consider a three-dimensional elliptic equation with two singular coefficients, for which a nonlocal problem is studied in a semi-infinite parallelepiped. The study of the problem is carried out using the method of separation of Fourier variables and spectral analysis. For the problem posed, using the Fourier method, two one-dimensional spectral problems are obtained. Based on the completeness property of the systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution to the problem is constructed in the form of the sum of a double Fourier series with respect to trigonometric and Bessel functions. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates are obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

••

TL;DR: In this paper, self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order are expressed in terms of hypergeometric functions, and they are shown to be linearly independent.

Abstract: When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this study, we construct self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order. These self-similar solutions are expressed in terms of hypergeometric functions.

••

TL;DR: In this article, the authors prove existence and uniqueness of a nonlocal boundary-value problem in the Sobolev space for the loaded Chaplygin's equation under some conditions on coefficients.

Abstract: Boundary-value problems for loaded equations in the plane in the case of loaded parts consist of traces of an unknown solution or its first normal derivatives, are well studied. The multidimensional loaded differential equations are relatively less investigated. Moveover, when the loaded part consists of not only the traces of the solution or its first normal derivatives, but also second derivatives of the solutions, the classical methods are not effective. Therefore, in this paper, we propose a method which overcomes these difficulties. Under some conditions on coefficients of the loaded multidimensional Chaplygin’s equation, we prove existence and uniqueness of a solution of a nonlocal boundary-value problem in the Sobolev space $$W_{2}^{3}(Q)$$.

••

TL;DR: In this article, the authors considered the task of generalized discrimination between symmetric coherent states and constructed an operation which enlarges the information content of the states with fixed failure probability, and applied this transformation to develop a zero-error eavesdropping strategy for quantum cryptography on symmetric coherence states.

Abstract: Symmetric coherent states are of interest in quantum cryptography, since for such states there is an upper bound for unambiguous state discrimination (USD) probability, which is used to resist USD attack. But it is not completely clear what an eavesdropper can do for shorter channel length, when USD attack in not available. We consider the task of generalized discrimination between symmetric coherent states and construct an operation which enlarges the information content of the states with fixed failure probability. We apply this transformation to develop a zero-error eavesdropping strategy for quantum cryptography on symmetric coherent states.

••

TL;DR: The main result of as discussed by the authors is that the inverse statement is not true for many root classes of groups and the proof of this result is based on the criterion for the fundamental group of a graph of isomorphic groups to be residually a δ-group, which is of independent interest.

Abstract: Let
$$\mathcal{C}$$
be a root class of groups and
$$\mathcal{\pi}_{1}(\mathcal{G})$$
be the fundamental group of a graph
$$\mathcal{G}$$
of groups. We prove that if
$$\mathcal{G}$$
has a finite number of edges and there exists a homomorphism of
$$\mathcal{\pi}_{1}(\mathcal{G})$$
onto a group of
$$\mathcal{C}$$
acting injectively on all the edge subgroups, then
$$\mathcal{\pi}_{1}(\mathcal{G})$$
is residually a
$$\mathcal{C}$$
-group. The main result of the paper is that the inverse statement is not true for many root classes of groups. The proof of this result is based on the criterion for the fundamental group of a graph of isomorphic groups to be residually a
$$\mathcal{C}$$
-group, which is of independent interest.

••

TL;DR: In this article, the authors studied 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.

Abstract: 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.

••

TL;DR: In this article, the authors studied the asymptotic behavior of the solutions of singularly perturbed three boundary value problems on an interval, where the singularities of the problem are that the small parameter is found at the highest derivative of the unknown function and the corresponding unperturbed first-order differential equation has higher order singular point at the left end of the segment.

Abstract: The article studies the asymptotic behavior of the solutions of a singularly perturbed three boundary value problems on an interval. The object of the study is a linear inhomogeneous ordinary differential equation of the second order with a small parameter with the highest derivative of the unknown function. The singularities of the problem are that the small parameter is found at the highest derivative of the unknown function and the corresponding unperturbed first-order differential equation has higher order an irregular singular point at the left end of the segment. At the ends of the segment, boundary conditions are imposed. Three problems are considered, in one Dirichlet problem, in two Neumann problem and in the three Roben problem. Asymptotic expansions of problems are constructed by the classical method of Vishik–Lyusternik–Vasilyeva–Imanaliev boundary functions. However, this method cannot be applied directly, since the external solution has a singularity. We first remove this singularity from the external solution, then apply the method of boundary functions. The constructed asymptotic expansions are substantiated using the maximum principle, i.e. estimates for the residual functions are obtained.

••

TL;DR: In this article, the authors constructed a matrix differential tensor operator of cofactors to the matrix differential metric operator of the micropolar theory of elasticity for any inhomogeneous anisotropic materials.

Abstract: The motion equations of the micropolar theory of elasticity in
displacements and rotations represented by the matrix differential
tensor-operator for any inhomogeneous anisotropic materials. As a
particular case the micropolar isotropic homogeneous materials
with a center of symmetry is considered. In this case, the matrix
differential tensor-operator of cofactors to the matrix
differential tensor-operator of the motion equations is
constructed. This constructed operator makes possible to decompose
the equations. The equations are obtained separately with respect
to the displacement and rotation vectors. Decomposed equations
also obtained for a reduced medium. In this case, the equation
with respect to the displacement vector is the same as the
equation of the classical theory, and the equation with respect to
the rotation vector has a similar form. In addition, in the
absence of volume loads, the equations of the reduced medium do
not depend on the properties of the material. This suggests that
these equations can be used to identify the material constants of
this medium. The cases under which the static boundary conditions
are easily split are revealed. From the decomposed equations of
the micropolar theories of elasticity, the corresponding
decomposed equations of the static (quasistatic) problem of the
theories of single-layer and multi-layer prismatic bodies of
constant thickness in displacements and rotations are obtained.
From the last systems of equations the equations in the moments of
unknown vector functions with respect to any systems of orthogonal
polynomials are derived. As a particular case, we obtain a system
of equations of the eighth approximation in moments with respect
to the system of Legendre polynomials, which decomposes into two
systems. One of them is the system with respect to the even order
moments of the unknown vector function, and the other system is
the system with respect to odd order moments of the same
functions. Based on the obtained operator of cofactors to the
operator of any of these systems we get a high order (the order of
the system depends on the order of approximation) elliptic type
equation for each moment of the unknown vector function, which
characteristic roots are easily found. Using Vekua’s method for
solving such equations [66], we can obtain their analytical
solution. Note also that the analytic method with the use of the
orthogonal polynomial systems (Legendre and Chebyshev) in
constructing the one-layer [2, 3, 7, 10, 15, 17, 18, 20–22, 63,
68, 69] and multilayer [4–6, 13, 60, 61] thin body theory was
also applied by other authors. In this direction the authors had
published the papers [24–31, 33–37, 41–45, 51–53], and others
with the application of Legendre and Chebyshev polynomial systems.
These expansions can be successfully used in constructing any thin
body theory. Despite this, classical theories are far from
perfect, and micropolar theories and theories of another rheology
are very far from perfect.

••

TL;DR: In this article, symmetry and corresponding fields of differential invariants of the inviscid flows on a curve are given, and their dependence on thermodynamic states of media is studied.

Abstract: Symmetries and the corresponding fields of differential invariants of the inviscid flows on a curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given.