# Vol 21, No 4 (2017)

**Year:**2017**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1226

### On a nonlocal boundary-value problem with constant coefficients for a multidimensional mixed type equation

#### Abstract

In this paper the unique solvability and smoothness of generalized solution of a nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind in Sobolev spaces $W_{2}^{l }(Q)$, ($2\le l $ is integer number), have been proved. First, the unique solvability of the generalized solution from space $W_{2}^{2 }(Q)$ has been studied. Further, the uniqueness of the generalized solution of nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation was proved by a priory estimates. For the proof of the existence of the generalized solution, we used method of `“ε-regularization” together with Galerkin method. Precisely, first, we study regular solvability of the nonlocal boundary value problem for the multidimensional mixed type equation by functional analysis methods, i.e. we obtained necessary a priory estimates for the considered problems. Using these estimates we solve composite type equation, then by the theorem on weak compactness, we pass to the limit and deduce to the multidimensional mixed type equation of the first kind. At the end, smoothness of the generalized solution of the considered problems has been discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):597-610

597-610

### On a spectral problem for a system of differential equations of mixed type

#### Abstract

This article explores the spectral characteristics of the differential operator generated by the boundary problem for linear systems of differential equations of mixed type. The simplest example of a classical system of equations that falls within the field of our consideration is the system of equations of mixed type: $$D_tu_1-signD_xu_2-\varepsilon u_2=f_1, \quad D_tu_2+D_xu_1+\varepsilon u_1=f_2$$elliptic for $t>0$ and hyperbolic for $t<0$.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):611-632

611-632

### Yang-Mills equations on conformally connected torsion-free 4-manifolds with different signatures

#### Abstract

In this paper we study spaces of conformal torsion-free connection of dimension 4 whose connection matrix satisfies the Yang-Mills equations. Here we generalize and strengthen the results obtained by us in previous articles, where the angular metric of these spaces had Minkowski signature. The generalization is that here we investigate the spaces of all possible metric signatures, and the enhancement is due to the fact that additional attention is paid to calculating the curvature matrix and establishing the properties of its components. It is shown that the Yang-Mills equations on 4-manifolds of conformal torsion-free connection for an arbitrary signature of the angular metric are reduced to Einstein's equations, Maxwell's equations and the equality of the Bach tensor of the angular metric and the energy-momentum tensor of the skew-symmetric charge tensor. It is proved that if the Weyl tensor is zero, the Yang-Mills equations have only self-dual or anti-self-dual solutions, i.e the curvature matrix of a conformal connection consists of self-dual or anti-self-dual external 2-forms. With the Minkowski signature (anti)self-dual external 2-forms can only be zero. The components of the curvature matrix are calculated in the case when the angular metric of an arbitrary signature is Einstein, and the connection satisfies the Yang-Mills equations. In the Euclidean and pseudo-Euclidean 4-spaces we give some particular self-dual and anti-self-dual solutions of the Maxwell equations, to which all the Yang-Mills equations are reduced in this case.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):633-650

633-650

### A boundary value problem for a third order hyperbolic equation with degeneration of order inside the domain

#### Abstract

In this paper we study the boundary value problem for a degenerating third order equation of hyperbolic type in a mixed domain. The equation under consideration in the positive part of the domain coincides with the Hallaire equation, which is a pseudoparabolic type equation. Moreover, in the negative part of the domain it coincides with a degenerating hyperbolic equation of the first kind, the particular case of the Bitsadze-Lykov equation. The existence and uniqueness theorem for the solution is proved. The uniqueness of the solution to the problem is proved with the Tricomi method. Using the functional relationships of the positive and negative parts of the domain on the degeneration line, we arrive at the convolution type Volterra integral equation of the 2nd kind with respect to the desired solution by a derivative trace. With the Laplace transform method, we obtain the solution of the integral equation in its explicit form. At last, the solution to the problem under study is written out explicitly as the solution of the second boundary-value problem in the positive part of the domain for the Hallaire equation and as the solution to the Cauchy problem in the negative part of the domain for a degenerate hyperbolic equation of the first kind.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):651-664

651-664

### The Dirichlet problem for a three-dimensional equation of mixed type with three singular coefficients

#### Abstract

We study the Dirichlet problem in a parallelepiped for a three-dimensional equation of mixed type with three singular coefficients. Separation of variables with Fourier series and spectral analysis are used to investigate this problem. Two one-dimensional spectral problems are obtained for the possed problem using the Fourier method. On the basis of the completeness property of the eigenfunction systems of these problems, the uniqueness theorem is proved. The solution of the problem is constructed as the sum of a double Fourier-Bessel series. In justification of the uniform convergence of the series constructed, asymptotic estimates of the Bessel functions of the real and imaginary argument are used. On their basis, estimates are obtained for each member of the series. The estimates obtained made it possible to prove the convergence of the series and its derivatives up to the second order inclusive, and also the existence theorem in the class of regular solutions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):665-683

665-683

### Torsion of a growing shaft

#### Abstract

The torsion of a shaft by rigid disks is considered. The shaft has the form of circular cylinder. Two rigid disks are attached to its end faces. The process of continuous growth of such shaft under the influence of twisting torques applied to the disks is studied. Dual series equations which reflect the mathematical content of the problem at the different stages of the growing process are derived and solved. Results of the numerical analysis and singularities of the qualitative mechanical behaviour of the fundamental characteristics are discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):684-698

684-698

### Complex bending and initial destruction of hybrid timber beams

#### Abstract

A mathematical model of the deformation of hybrid timber beams has been developed. By hybrid we mean bars, formed by rigid connection (gluing) on certain contact surfaces of a set of layers of different forms of crosssections and different types of timber. In general, the bars are in conditions of complex bending with stretching-compression. The physical non-linearity of timber, as well as the different tensile and compression resistance, is taken into account. In the general case, the problem reduces either to solving a system of three nonlinear algebraic equations of the third degree with respect to generalized deformations of the cross section or to a system of three nonlinear ordinary differential equations with respect to the components of the displacement vector of the points of the axis of the rod. To solve the obtained algebraic equations the Newton method is used, the solution of the differential equations is performed using the Galerkin type method. An analytical approximation of the experimental tension-compression diagrams of timber along the fibers in the form of polynomials of the second and third degree is proposed. The coefficients of the approximating functions are determined in two ways: using the least squares method with the experimental deformation diagrams; by imposing certain requirements on the diagrams, using the basic mechanical characteristics of the timber (maximum stresses and deformations, moduli of elasticity). Numerical values of the approximation coefficients for 15 different types of timber are given. The above examples of calculations of hybrid timber beams have shown the possibility of the emergence of hidden mechanisms of destruction, as well as the strong influence of the rearrangement of layer materials on the stress-strain state of the structure. The method developed in the article for the calculation of hybrid rodshaped timber structures offers great opportunities for solving optimization problems in the design, and allows rational use of various types of timber.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):699-716

699-716

### Digital processing of the results of optoelectronic measurements. The photoelasticity method and its application for determination of coeﬃcients of the multiparameter asymptotic Williams expansion of the stress ﬁeld

#### Abstract

The study is aimed at experimental and computational determination of the coefficients in crack tip asymptotic expansions for a wide class of specimens under mixed mode loading conditions. Multiparametric presentation of the stress ﬁled near the crack tips for a wide class of specimens is given. Theoretical, experimental and computational results obtained in this research show that the isochromatic fringes in the vicinity of the crack tip require to keep the higher order stress terms in the asymptotic expansion of the stress ﬁeld around the crack tip since the contribution of the higher order stress terms (besides the stress intensity factors and the T-stress) is not negligible in the crack tip stress ﬁeld. One can see that the higher-order terms of the asymptotic expansion are important when the stress distribution has to be known also farther from the crack tip and it is necessary to extend the domain of validity of the Williams solution. It is shown that at large distances from the crack tips the effect of the higher order terms of the Williams series expansion becomes more considerable. The knowledge of more terms of the stress asymptotic expansions will allow us to approximate the stress ﬁeld near the crack tips with high accuracy.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):717-735

717-735

### A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation

#### Abstract

In this paper a new exact solution of an overdetermined system of Oberbeck-Boussinesq equations that describes a stationary shear flow of a viscous incompressible fluid in an infinite layer is under study. The given exact solution is a generalization of the Ostroumov-Birich class for a layered unidirectional flow. In the proposed solution, the horizontal velocities depend only on the transverse coordinate z. The temperature field and the pressure field are three-dimensional. In contradistinction to the Ostroumov-Birich solution, in the solution presented in the paper the horizontal temperature gradients are linear functions of the z coordinate. This structure of the exact solution allows us to find a nontrivial solution of the Oberbeck-Boussinesq equations by means of the identity zero of the incompressibility equation. This exact solution is suitable for investigating large-scale flows of a viscous incompressible fluid by quasi-two-dimensional equations. Convective fluid motion is caused by the setting of tangential stresses on the free boundary of the layer. Inhomogeneous thermal sources are given on both boundaries. The pressure in the fluid at the upper boundary coincides with the atmospheric pressure. The paper focuses on the study of temperature and pressure fields, which are described by polynomials of three variables. The features of the distribution of the temperature and pressure profiles, which are polynomials of the seventh and eighth degree, respectively, are discussed in detail. To analyze the properties of temperature and pressure, algebraic methods are used to study the number of roots on a segment. It is shown that the background temperature and the background pressure are nonmonotonic functions. The temperature field is stratified into zones that form the thermocline and the thermal boundary layer near the boundaries of the fluid layer. Investigation of the properties of the pressure field showed that it is stratified into one, two or three zones relative to the reference value (atmospheric pressure).

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):736-751

736-751

### The Cauchy problem for a system of the hyperbolic differential equations of the n-th order with the nonmultiple characteristics

#### Abstract

In the paper the Cauchy problem is considered for the hyperbolic differential equation of the n-th order with the nonmultiple characteristics. The regular solution of the Cauchy problem for the hyperbolic differential equation of the n-th order with the nonmultiple characteristics is considered. In the paper the solution of the Cauchy problem for the system of the hyperbolic differential equations of the n-th order with the nonmultiple characteristics is considered. The existence and uniqueness theorem for the regular solution of the Cauchy problem for the system of the hyperbolic differential equations of the n-th order with the nonmultiple characteristics is considered as the result of the research.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):752-759

752-759

### Dynamic stability of heated geometrically irregular plates on the basis of the Reisner model

#### Abstract

On the basis of the continuum model of geometrically irregular plate the problem of dynamic stability has been solved. The Reissner type model is considered. The heated plate with ribs is subjected to periodic temporary coordinate of the tangential forces. For the tangential forces a nonhomogeneous boundary problem of membrane thermoelasticity in displacements is solved. The system of singular equations of dynamic stability recorded through the function of the deflection and additional functions. The additional functions characterize the law of change of stresses in vertical planes dependent variables x and y. The solution is reduced to the Mathieu equation. The characteristics of the Mathieu equation represented by terms in classical theory of plates and contain corrections of temperature, transverse shear and ribs. Three areas of dynamic stability of the thermoelastic system are determined. Quantitative analysis has been carried out. Dependence of the configuration of the areas of dynamic stability on temperature, shear deformation in vertical planes and relative height of ribs is presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):760-772

760-772

### Forced axisymmetric oscillations of circular multilayer bimorph plates

#### Abstract

A method for calculating circular multilayer bimorph plates is presented and new analytical solutions of axisymmetric dynamic problems of direct and inverse piezoelectric effects are obtained. The cases of hinged and rigid fixing of the outer contour of the structure are considered. To investigate related linear problems, a mathematical apparatus is used in the form of a method of finite integral transformations. The constructed calculated relationships allow us to substantiate the constructive solutions of piezoceramic transducers. Built design ratio allows to prove the constructive solutions of multilayer piezoelectric ceramic transducers, namely, to choose the geometrical dimensions and physical characteristics of the materials used, define the dimensions of a split circular electrode, allowing most effectively to convert the external electrical stimulation into mechanical vibrations at various frequency. In addition, it is possible to perform stress-strain state, the nature of the change of the electric field and frequency range of the axisymmetric vibrations of the considered systems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(4):773-785

773-785