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We study Sturm–Liouville differential operators on the time scales consisting of finitely many isolated points and closed intervals. In the author’s previous paper, it was established that such operators are uniquely determined by the spectral characteristics of all classical types. In the present paper, an algorithm for their recovery based on the method of spectral mappings is obtained. We also prove that the eigenvalues of two Sturm–Liouville boundary-value problems on time scales with one common boundary condition alternate.

This article explores and develops opportunities Fourier method of separation of variables for the study of the asymptotic behavior of harmonic functions on noncompact Riemannian manifolds of a special form. These manifolds generalize spherically symmetric manifold and are called model ones in a series of works. In the first part of the paper, an estimate of the eigenfunctions of the Laplace operator is obtained on compact Riemannian manifolds $$S$$ in the norm $$C^{m}(S)$$ . In the second part of the paper, the conditions for the unique solvability of the Dirichlet problem for harmonic functions on model manifolds with smooth boundary data at ‘‘infinity’’ are found. It was shown that the solution of this boundary value problem converges to the boundary data in the $$C^{1}$$ -norm.

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.

It is well known that developing methods for solving Dirichlet problems is important and relevant for various areas of mathematical physics related to the Laplace equation, the Helmholtz equation, the Stokes equation, the Maxwell equation, the Dirac equation, and others. The author in previous papers studied the solvability of Dirichlet boundary value problems of the first and second orders in quaternionic analysis. In the present paper, we study a higher-order Dirichlet boundary value problem associated with the two-dimensional Helmholtz equation with complex potential. The existence and uniqueness of a solution to the Dirichlet boundary value problem in the two-dimensional case is proved and an appropriate representation formula for the solution of this problem is found. Most Dirichlet problems are solved for the case in three variables. Note that the case of two variables is not a simple consequence of the three-dimensional case. To solve the problem, we use the method of orthogonal decomposition of the quaternion-valued Sobolev space. This orthogonal decomposition of the space is also a tool for the study of many elliptic boundary value problems that arise in various areas of mathematics and mathematical physics. An orthogonal decomposition of the quaternion-valued Sobolev space with respect to the high-order Dirac operator is also obtained in this paper.

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.

We consider an initial-boundary value problem for the heat equation with an inhomogeneous initial condition and boundary conditions. One of the boundary conditions contains a mixed derivative. When solving this problem by the method of separation of variables, a spectral problem arises. A system of eigenfunctions of this spectral problem and a biorthogonally conjugate system, are constructed explicitly. Also we obtain an asymptotic formula for the eigenvalues. In this paper we formulate theorems on the properties of the system of eigenfunctions of the spectral problem and a theorem about representing the solution of the initial initial-boundary value problem in the form of a Fourier series in the system of eigenfunctions. Thus, the existence of a solution is shown if the initial condition belongs to the Holder class. However, it has been shown that the solution is not unique. We show that additional condition guarantees the uniqueness of the solution. The unique solution of this problem is also obtained in the article.

In this paper, we study the third boundary value problem for the convection-diffusion equation with a time fractional derivative and variable coefficients in a multidimensional domain. For an approximate solution in a rectangular parallelepiped of the problem set, in the same area, the third boundary value problem for a differential equation with a small parameter is considered. An a priori estimate is obtained from which it follows the convergence of the solution of the differential problem with a small parameter to the solution of the original problem for small values of the parameter. For a problem with a small parameter, a locally one-dimensional difference scheme by A. A. Samarski is constructed. Using the maximum principle for solving the difference problem, an a priori estimate is obtained in the grid norm $$C$$ , which expresses the stability of the locally one-dimensional difference scheme. The uniform convergence of the locally one-dimensional scheme is proved for $$0

In this paper, we consider the questions of the unique solvability of a boundary value problem for a third-order partial integro-differential equation with a degenerate kernel and multiple characteristics. An explicit solution of the boundary value problem is constructed. In this case, a combination of three methods was used: the method for constructing Green’s function, the method of Fourier series and the Fredholm method for the degenerate kernel.

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 purpose of this article is to demonstrate with a simple example the complexities that arise when considering boundary-value problems for differential-difference equations. In this paper we discuss in detail smoothness issues of solutions to the Dirichlet problem for linear differential-difference equation with integer shifts. After analysis of this example we consider more general statement of the problem and certain results concerning the solvability and smoothness of solutions.

In this paper, we investigate a problem on small motions of a system of bodies partially filled with ideal fluids under the action of an elastic damping device. The initial boundary value problem is reduced to the Cauchy problem for a first-order differential operator equation on a Hilbert space. Properties of the resulting operator matrices, which are coefficients of the equation, are studied. Theorems on solvability of the Cauchy problem and the initial boundary value problem are proven.

A class of quadratic irregular pencils of the second order ordinary differential operators with constant coefficients is considered in this paper. It is assumed that the boundary conditions are splitting and the characteristic equation has positive roots. Both amounts of two-fold expansions in the series in terms of eigenfunctions of such pencils and necessary and sufficient conditions for convergence of these expansions to the decomposed vector-valued function are found. For this class of irregular pencils, the question of a two-fold expansion in terms of eigenfunctions was not considered earlier.

In this paper, we investigate the global structure of positive solutions of $$\begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0 0$$ is a parameter, $$h\in C[0,1]$$ , $$f\in C[0,\infty)$$ and $$f(s)>0$$ for $$s>0$$ . We show that the problem has three positive solutions suggesting suitable conditions on the nonlinearity. Furthermore, we also establish the existence of infinitely many positive solutions. The proof is based on the bifurcation method.

The paper deals with a family of second-order partial differential equations on the complex plane. The coefficient of the unknown function depends on a natural parameter $$n$$ . Solutions of the equations are called generalized analytic functions of order $$n$$ . In a simply connected domain $$T^{+}$$ , we investigate Neumann boundary value problem for generalized analytic functions of order $$n$$ . If $$T^{+}$$ is a disk, we suggest an explicit method for solving the problem in the class of first-order generalized analytic functions. We establish that the solvability of Neumann problem significantly depends on the value of radius of the disk.

In this paper we consider of a multidimensional loaded mixed type equation of the second order with some seminonlocal conditions on the coefficients. The existence and uniqueness of solution of the seminonlocal boundary value problem of second kind is proved in the Sobolev space $$W\ _{2}^{3}(Q)$$ . In proof of the theorems are used the methods of ‘‘ $$\varepsilon$$ -regularizations’’, a priori estimate and successive approximations.

In this paper we study a boundary value problem with the Poincare–Tricomi condition for a degenerate partial differential equation of elliptic-hyperbolic type of the second kind. In the hyperbolic part of a degenerate mixed differential equation of the second kind the line of degeneracy is a characteristic. For this type of differential equations a class of generalized solutions is introduced in the characteristic triangle. Using the properties of generalized solutions, the modified Cauchy and Dirichlet problems are studied. The solutions of these problems are found in the convenient form for further investigations. A new method has been developed for a differential equation of mixed type of the second kind, based on energy integrals. Using this method, the uniqueness of the considering problem is proved. The existence of a solution of the considering problem reduces to investigation of a singular integral equation and the unique solvability of this problem is proved by the Carleman–Vekua regularization method.

The paper deals with the study of solvability to geometrically nonlinear boundary value problem for elastic inhomogeneous isotropic shallow shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation relative to deflection of shell in Sobolev space. Solvability of equation is proved with the use of contracting mappings principle.

We prove a theorem on the equivalence of a scalar Riemann boundary value problem with a shift-reflection on the real axis and a matrix Riemann boundary value problem without the shift. A relationship between the boundary value problem in question and a boundary value problem with orientation-preserving shift-rotation on the unit circle is established. The operator identities constructed by the present authors in their preceding papers and permitting one to eliminate the shift-reflection from the integral equation corresponding to the boundary value problem are the main research tool.

The present paper is devoted to the study of the asymptotic behavior of the optimal control for the boundary value problem in an e-periodically perforated domain with linear Robin-type boundary condition, when the period of the structure tends to zero, and the problem parameters, diameter of perforations and adsorption coefficient, take critical values.

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.

In this paper we present analytical solutions for the cantilever beam bending problems obtained in the non-classical electroelasticity theory with strain and electric field gradient effects. We show that considered model allows to provide the refined analysis for the electric field distribution around the supproted end of the cantilever taking into account the extended number of boundary conditions, which cannot be captured in classical models.

In this paper we study a spectral problem with spectral parameter in the boundary condition. This problem arises when solving boundary value problems for mixed type equations using the spectral method. We analyze the system of root functions of this problem and proof two theorems about basis properties of this system. The biortogonal system is constructed and uniform convergence of spectral expansions is studied.

In the paper, the solvability problems of an nonhomogeneous boundary value problem in the first quadrant for a fractionally loaded heat equation are studied. Feature of this problem is that, firstly, the loaded term is presented in the form of the Caputo fractional derivative with respect to the spatial variable, secondly, the order of the derivative in the loaded term is less than the order of the differential part and, thirdly, the point of load is moving. The problem is reduced to the Volterra integral equation of the second kind, the kernel of which contains the generalized hypergeometric series. The kernel of the obtained integral equation is estimated and it is shown that the kernel of the equation has a weak singularity (under certain restrictions on the load), this is the basis for the statement that the loaded term in the equation is a weak perturbation of its differential part. In addition, the limiting cases of the order of the fractional derivative are considered. It is proved that there is continuity in the order of the fractional derivative.

One approach to designing turbomachine blades is to use the theory of inverse boundary-value problems of aerohydrodynamics. The basis of the method for solving the inverse problem for a profile lattice located on an axisymmetric stream surface in a layer of variable thickness is the solution of the problem for a flat lattice. In this paper, we consider a number of issues arising in the numerical implementation of the solution of this problem for dense lattices, and we propose methods for solving them and thereby increase the accuracy of numerical calculations.

In earlier research, the double- and simple layer potentials have been successfully applied in solving boundary value problems for two-dimensional elliptic equations. Despite the fact that all fundamental solutions of a three-dimensional elliptic equation with one, two and three singular coefficients were known, the potential theory was not constructed in any case of a singularity. Here, in this paper, our goal is to construct a potential theory corresponding to the three-dimensional elliptic equation with one singular coefficient. We used some properties of Gaussian hypergeometric function to prove the limiting theorems, while deriving integral equations concerning the denseness of potentials.

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.

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.

We consider the Cauchy problem for a wave equation with variable velocity on the simplest decorated graph obtained by gluing a ray to ℝ3, with initial conditions localized on the ray. For the wave operator to be self-adjoint, we impose certain boundary conditions at the gluing point. This paper describes the asymptotic expansion of the solution of the problem under consideration for arbitrary boundary conditions at the gluing point under the assumption that the velocity on ℝ3 is radially symmetric. Also we study the distribution of the energy of the wave as the small parameter tends to zero, which depends on the boundary conditions.

The main subjects of the present paper are the Goursat and Darboux boundary-value problems for hyperbolic systems with two independent variables. We show that obtained by T.V. Chekmarev in terms of successive approximations formulas for solution of the Goursat problem can be built also by the Riemann method, work out an analog of the Riemann–Hadamard method for the system, and introduce its Riemann–Hadamard matrix. We solve the Darboux problem in terms of the introduced matrix.

In this paper we are interested in Riesz basisness of root functions of the non-selfadjoint a discontinuous Sturm–Liouville operator with periodic boundary condition which are not strong regular and with conjugate conditions. Here we assume that the potentials are complex valued and continuously differentiable functions. One of conjugate conditions have different finite one-sided limits at point zero. In order to prove Riesz basisness of root functions, we firstly obtain asymptotic expressions of fundamental solutions. Putting these solutions into characteristic determinant, we get asymptotic formulas of eigenvalues by means of Rouche theorem. Asymptotic formulas of eigenfunctions acquired by obtained relation and fundamental solutions. By the aid of asymptotic formulas of eigenfunctions and Bessel properties of eigenfunctions we prove the basisness of the root functions of the boundary value problem. We also prove the Riesz basisness of root functions of the same operator with antiperiodic boundary conditions and with same conjugate conditions.

In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation $${u_{tt}} - M(||\nabla u||_2^2)\Delta u + \int_0^t {h(t - s)\Delta u(s)ds + a|{u_t}{|^{m - 2}}{u_t} = |u{|^{p - 2}}u} $$ with initial conditions and acoustic boundary conditions. We show that, depending on the properties of convolution kernel h at infinity, the energy of the solution decays exponentially or polynomially as t → + ∞. Our approach is based on integral inequalities and multiplier techniques. Instead of using a Lyapunov-type technique for some perturbed energy, we concentrate on the original energy, showing that it satisfies a nonlinear integral inequality which, in turn, yields the final decay estimate.

The main results of the paper are related to the study of differential operators of the form $$Ly = {y^{\left( n \right)}}\left( { - x} \right) + \sum\limits_{k = 1}^n {pk\left( x \right){y^{\left( {n - k} \right)}}\left( { - x} \right) + } \sum\limits_{k = 1}^n {{q_k}\left( x \right){y^{\left( {n - k} \right)}}} \left( x \right),\,x \in \left[ { - 1,1} \right],$$ with boundary conditions of general form concentrated at the endpoints of a closed interval. Two equivalent definitions of the regularity of boundary conditions for the operator L are given, and a theorem on the unconditional basis property with brackets of the generalized eigenfunctions of the operator L in the case of regular boundary conditions is proved.

A non-singular solution of the gradient elastic fracture mechanics for the cracks of Modes I and II is constructed in this paper. Previously, a similar problem was solved only for cracks of Mode III. A generalized theory of elasticity is used, in which the governing equation in displacements is represented as a product of the Lame operator and the Helmholtz operator, and the classical boundary value problem for the total stress is completely distinguished in the problems of crack mechanics. As a result, the determination of local stress fields reduces to solving the Helmholtz equations with the known right-hand side of the equations. The Papkovich-Neuber representation in a complex form is used to construct a solution of the mechanics of cracks. We used also the method of radial multipliers, which allows us to construct fundamental solutions of the Helmholtz equations corresponding to analytical functions with a fractional exponent and, as a result, to find solutions that compensate for the singularities.

This paper investigates an inverse non-stationary problem of the restoration of the spatial law of a homogeneous isotropic Timoshenko beam of finite length. Hinge support conditions are used as boundary conditions. Initial conditions are assumed to be zero. It is assumed that of the beam’s ends is fitted with sensors which in the course of corresponding experiment register the amount of deflection of the beam at the sensor points. The method of the solution of a direct problem is based on the principle of superposition where the deflection of the beam is associated with the space load the beam is exposed to, by means of an integral operator by the spatial coordinate and time. The kernel of such operator is so called influence function. This function is a fundamental solution of a system of differential equations of motion of the study beam. The construction of such solution represents a separate problem. The influence function is found by means of Laplace time transformation and expansion into Fourier series in a system of the problem’s eigenfunctions. The solution of the inverse problem at the first stage reduces to a system of algebraic equations for vector operator whose components are time convolutions of the coefficients of expansion series for an influence function with the desired coefficients of expansion of the load in a Fourier series. At the same time, the components of the vector of the rights parts are time dependencies registered by the sensors. The resulting system is ill-conditioned [1]. The second stage serves to resolve independent Volterra integral equations of the first kind for the desired coefficients of Fourier serials for the load.

A formulation of a boundary value problem to find natural frequencies of an inhomogeneous beam in the framework of the Euler–Bernoulli hypotheses are represented. Questions related to various classical variational formulations for a spectral problem arising in the beam theory are discussed. Particularities of the application of the Hamiltonian principles to boundary-value problems are considered. The method of integro-differential relations, which is an alternative to the classical variational approaches is discussed. Various bilateral energy quality estimates for approximate solutions that follow from the method of integro-differential relations are proposed. In the final part of the paper advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed based on a numerical example.

In this paper are considered the questions of unique solvability and redefinitions of a nonlocal inverse problem for the Fredholm integro-differential equation of the second order with degenerate kernel, integral condition, and spectral parameter. Calculations of the value of the spectral parameter are reduced to the solve of trigonometric equations. Systems of algebraic equations are obtained. The singularities that arose in determining arbitrary constants are studied. A criterion for unique solvability of the problem is established and the corresponding theorem is proved.

In the paper, the numerical modeling of a supersonic flow around a wedge by viscous heat-conducting gas is considered. A numerical algorithm is proposed for the initial boundary-value problem for the Navier–Stokes equations. These equations are modified and amplified by new boundary conditions to provide the conservation law for the full energy: kinetic and inner. Then the combination of the Lagrangian approximation for the transfer operators and the conforming finite element method for other terms provides an efficient algorithm. Particular attention has been given to the approximation providing the conservation laws for mass and full energy at discrete level. Test calculations have been performed for a wide range of Mach and Reynolds numbers.

The purpose of this paper is to establish new sufficient conditions for the univalence of analytical functions. We consider the case when the functions are defined in the domains belonging to one of the Rahmanov classes. Our study is based upon an examination of the properties of some generalizations of spiral domains using the quasiconformal extension method. Possible applications of the obtained results to the solution of the strong problem of univalence in inverse boundary value problems are shown.

The paper deals with the study of solvability of geometrically nonlinear boundary value problem for elastic shallow isotropic inhomogeneous shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation whose solvability is proved with the use of contracting mappings principle.

We considered 6-fold linear equation in the class of analytic functions outside the equilateral triangle and vanishing at infinity. We proposed a method of equivalent regularization, using the theory of boundary value problem of Carleman. Also, the paper is dedicated to applications to the problem of moments for entire functions of exponential type. In particular, we construct a system of functions biorthogonal whit piecewise-quasipolynomial weight to some power system on the three rays. The indicator of such functions is a piecewise-trigonometric function, with a period of 2π/3.

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

In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data.

In this paper we apply spectral method to the Gellerstedt problem for Lavrent’ev-Bitsadze equation in a half-strip. We consider Frankl matching condition at the type change line. Using the Darboux solution formulae for mixed type equationinthe hyperbolic subdomainwe reduce problem to an auxiliary problem for the Laplace operator.

The aim of the paper is to show, using the one-dimensional problem as an example, what is to be expected and what should be pursued when studying the multidimensional case. The one-dimensional case has been chosen as a model, because here the problem admits an explicit solution permitting one to follow the phase transformation process.

In the paper we consider the problem of searching for coexisting modes in a nonlinear boundary value problem with a delay from population dynamics. For this we construct the asymptotic of spatially homogeneous cycle using the normal forms method and research the dependence of its stability on the diffusion parameter. Then we find coexisting attractors of the problem using numerical methods. Numerical experiment required an application of massively parallel computing systems and adaptation of solutions search algorithms to them. Based on the numerical analysis we come to the conclusion of the existence in the boundary value problem of solutions of two types. The first type has a simple spatial distribution and inherits the properties of a homogeneous solution. The second called the mode of self-organization is more complex distributed in space and is much more preferred in terms of population dynamics.

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.

The paper proposes an analogue of Vishik–Lyusternik–Vasileva–Imanalieva boundary functions method for constructing a uniform asymptotic expansion of solutions to many band (or with an intermediate boundary layers) bisingularly problems. By means of this method we construct the uniform asymptotic expansion for the solution to the three-band bisingular Dirichlet problem for second order ordinary differential equation on the interval. By the maximum principle we justify formal asymptotic expansion of the solution, that is, an estimate for the error term is established.

This paper studies interconnections between holomorphic vector bundles on compact Riemann surfaces and the solution of the homogeneous conjugation boundary value problem for analytic functions on the one hand, and cohomology and the solution of the inhomogeneous problem on the other. We establish that constructing the general solution to the homogeneous problem with arbitrary coefficients in the boundary conditions is equivalent to classifying holomorphic vector bundles. Solving the inhomogeneous problem is equivalent to checking the solvability of 1-cocycles with coefficients in the sheaf of sections of a bundle; in particular, the solvability conditions in the inhomogeneous problem determine obstructions to the solvability of 1-cocycles, i.e. the first cohomology group. Using this connection, we can apply the methods of boundary value problems to vector bundles. The results enable us to elucidate the role of boundary value problems in the general theory of Riemann surfaces.