# Vol 25, No 2 (2021)

## Differential Equations and Mathematical Physics

### The Riemann method for equations with a dominant partial derivative (A Review)

Mironov A.N., Mironova L., Yakovleva J.O.

#### Abstract

This review article is devoted to a class of linear equations with a dominant (leading) partial derivative of the form $$(D+M)u=f$$, where $$Du$$ is a mixed partial derivative, and $$M$$ is a linear differential operator containing the derivatives of the function $$u$$ obtained from $$D$$ by discarding at least one differentiation. We can point out the structural similarity of such linear equations with linear ordinary differential equations. We present the Riemann method for linear equations with a dominant partial derivative, which is a natural generalization of the well-known Riemann method for a second-order hyperbolic equation with two independent variables.

The article deals with the main provisions of the theory developed for the equation with the dominant partial derivative of the general form, allowing the interested reader to apply the obtained results to the task that interests him. The definition of the Riemann function as a solution of the Volterra integral equation is given. The main differential identity is discussed, and the process of obtaining a formula for solving the Cauchy problem in terms of the Riemann function by integrating the specified identity over the corresponding domain in $$n$$-dimensional space is demonstrated. An example of constructing a solution to the Cauchy problem for the third-order equation is given.

The Riemann method is described below for a fairly wide class of linear systems of hyperbolic equations (including those with multiple characteristics). This method is ideologically very close to the Riemann method for linear equations with a dominant partial derivative.

Applications of the Riemann method to the study of new problems for partial differential equations are discussed. In particular, using the Riemann method, the correctness of new boundary value problems for factorized hyperbolic equations is proved, the solvability of integral equations with partial integrals is investigated, and a certain modification of the Riemann method allows us to develop the Riemann--Hadamard method for Darboux problems. The explicit representation of solutions of hyperbolic systems in terms of the Riemann matrix allows us to study new boundary value problems, in particular, problems with the assignment of normal derivatives of the desired functions on the characteristics, problems with conditions on the entire boundary of the domain, and Darboux problems.

The Riemann method described here for linear equations with a dominant partial derivative is obviously transferred to matrix equations. In this regard, some cases are indicated when the Riemann matrix is constructed explicitly (in terms of hypergeometric functions) for such matrix equations.

The paper provides a review of the literature, briefly describes the history of the development of this direction in Russia and in foreign countries.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):207-240
207-240

### The first boundary value problem in a rectangular domain for a differential equation with the Bessel operator and the Riemann–Liouville partial derivative

Khushtova F.G.

#### Abstract

The paper is devoted to the first boundary-value problem in a rectangular domain for a differential equation with the singular Bessel operator acting with respect to a spatial variable and the Riemann–Liouville fractional differentiation operator acting with respect to a time variable. An explicit representation of the solution is constructed. The uniqueness of the solution is proved in the class of functions satisfying the Hölder condition with respect to the time variable. When the order of the fractional derivative is equal to unity, and the Bessel operator has no singularity, the studied equation coincides with the heat equation and the obtained results coincide with well-known corresponding classical results.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):241-256
241-256

### Potentials for a three-dimensional elliptic equation with one singular coefficient and their application

Ergashev T.G.

#### Abstract

A potential theory for a three-dimensional elliptic equation with one singular coefficient is considered. Double- and simple-layer potentials with unknown density are introduced, which are expressed in terms of the fundamental solution of the mentioned elliptic equation. When studying these potentials, the properties of the Gaussian hypergeometric function are used.

Theorems are proved on the limiting values of the introduced potentials and their conormal derivatives, which make it possible to equivalently reduce boundary value problems for singular elliptic equations to an integral equation of the second kind, to which the Fredholm theory is applicable.

The Holmgren problem is solved for a three-dimensional elliptic equation with one singular coefficient in the domain bounded $$x=0$$ by the coordinate plane and the Lyapunov surface for $$x>0$$ as an application of the stated theory. The uniqueness of the solution to the stated problem is proved by the well-known $$abc$$ method, and existence is proved by the method of the Green's function, the regular part of which is sought in the form of the double-layer potential with an unknown density. The solution to the Holmgren problem is found in a form convenient for further research.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):257-285
257-285

## Mechanics of Solids

### Experimental studies of high-temperature creep of titanium alloy VT6 under conditions of a complex stress state under the influence of an aggressive medium

Igumnov L.A., Kazakov D.A., Shishulin D.N., Modin I.A., Zhegalov D.V.

#### Abstract

The results of experimental studies of high-temperature creep and long-term strength under conditions of a uniaxial and complex stress-strain state are presented. Tests for uniaxial tension, torsion and their combined action.

The tests were carried out on laboratory tubular specimens made of VT6 material at a temperature of 600$$\,^\circ$$C as delivered and under conditions of exposure to an aggressive environment. An aggressive environment was created by preliminary hydrogenation of laboratory samples with different hydrogen-ion concentration by mass $$\mathrm{C}_m$$ equal to 0.15 % and 0.3 %.

Experimental information for the construction of material parameters and scalar functions of a thermal creep model with isotropic-kinematic hardening is presented. This information is obtained from basic experiments to determine: the initial radius of the zero level creep surface; fans of creep curves at different levels of specified stresses, with obtaining the characteristics of the third section on the creep diagram, which precedes the failure of the sample at a fixed temperature at a given time interval; torsional creep curves up to the moment of loss of stability in the working part of the specimen. Based on the results of tests for uniaxial loading, two levels of stress intensity were selected, with different combinations of which experiments were carried out under conditions of complex loading.

The results of experimental studies of high-temperature creep and long-term strength under several different programs of isothermal loading under conditions of a complex stress-strain state are presented. Investigations are carried out for specimens made of VT6 alloy at delivery condition, under conditions of exposure to an aggressive environment. The obtained experimental information makes it possible to determine the necessary material parameters and to verify the used mathematical model of thermal creep.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):286-302
286-302

### The equilibrium state of a hollow transversally-isotropic thick-walled sphere, which is fixed on the external surfaces and is subject to a uniform internal lateral pressure and weight forces

Fukalov A.A., Zaitsev A.V., Sokolkin Y.V., Bayandin Y.V.

#### Abstract

We use the decomposition of the components of the displacement vector along the hoop and radial coordinates in series in Legendre polynomials and generalized power series to obtain an exact analytical solution to the equilibrium problem of a thick-walled transversely isotropic centrally symmetric hollow body, which is rigidly fixed on the external surface and is subject to a uniform internal pressure and weight forces.

As an example of using the obtained analytical solution, we analyzed the influence of weight forces on distribution of independent invariants of the stress tensor in the cross section of a heavy reinforced concrete sphere, which internal surface is free from pressure. Based on the multicriteria approach describing various loss of strength mechanisms (from tension or compression in the radial and hoop direction and interlayer shear), we found the regions of a heavy reinforced concrete sphere, in which damage can be initiated.

A qualitative and quantitative comparison of the stress fields at the points of the cross-sections of the thick-walled heavy spheres with the results of the numerical solution of the same problem in the axisymmetric and 3D formulations in the FEM packages ANSYS 13.0 and ABAQUS 6.11 is carried out. Both packages demonstrated the minimum deviation of the numerically determined values of the stress invariants from the exact analytical solution in the axisymmetric formulation. Also the difference with a comparable error in the 3D setting was found. In the latter case, the presentation of the FEM results for stresses and strains in the component form led to an unexpected result, i.e. significant errors in comparison with the exact analytical solution. To eliminate the errors found in determining the stress-strain state, which are caused only by features of determining the spherical coordinate system in the FEM packages ANSYS 13.0 and ABAQUS 6.11, it is necessary to use the presentation of the results obtained in the invariant form.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):303-319
303-319

### Solution of the coupled nonstationary problem of thermoelasticity for a rigidly fixed multilayer circular plate by the finite integral transformations method

Shlyakhin D.A., Kusaeva Z.M.

#### Abstract

A new closed solution of an axisymmetric non-stationary problem is constructed for a rigidly fixed round layered plate in the case of temperature changes on its upper front surface (boundary conditions of the 1st kind) and a given convective heat exchange of the lower front surface with the environment (boundary conditions of the 3rd kind).

The mathematical formulation of the problem under consideration includes linear equations of equilibrium and thermal conductivity (classical theory) in a spatial setting, under the assumption that their inertial elastic characteristics can be ignored when analyzing the operation of the structure under study.

When constructing a general solution to a non-stationary problem described by a system of linear coupled non-self-adjoint partial differential equations, the mathematical apparatus for separating variables in the form of finite integral Fourier–Bessel transformations and generalized biorthogonal transformation (CIP) is used. A special feature of the solution construction is the use of a CIP based on a multicomponent relation of eigenvector functions of two homogeneous boundary value problems, with the use of a conjugate operator that allows solving non-self-adjoint linear problems of mathematical physics. This transformation is the most effective method for studying such boundary value problems.

The calculated relations make it possible to determine the stress-strain state and the nature of the distribution of the temperature field in a rigid round multilayer plate at an arbitrary time and radial coordinate of external temperature influence. In addition, the numerical results of the calculation allow us to analyze the coupling effect of thermoelastic fields, which leads to a significant increase in normal stresses compared to solving similar problems in an unrelated setting.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):320-342
320-342

### A refined model of viscoelastic-plastic deformation of flexible spatially-reinforced cylindrical shells

Yankovskii A.P.

#### Abstract

A model of viscoelastic-plastic deformation of flexible circular cylindrical shells with spatial reinforcement structures is developed. The instant plastic behavior of the materials of the composition is determined by flow theory with isotropic hardening. The viscoelastic deformation of the components of the composition is described by the equations of the Maxwell–Boltzmann body model. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The relations used make it possible to calculate with varying degrees of accuracy the residual displacements of the points of the construction and the residual deformed state of the components of the composition. In this case, a possible weak resistance of the reinforced shell to transverse shear is simulated. In the first approximation, the equations used, the initial and boundary conditions, are reduced to the formulas of the nonclassical Ambardzumyan theory.

The numerical solution of the formulated initial boundary-value problem is constructed according to the explicit “cross” scheme. Elastoplastic and viscoelastic-plastic dynamic deformation of thin fiberglass shells under the influence of internal pressure of an explosive type is investigated. Two reinforcement structures are considered:

1) orthogonal reinforcement in the longitudinal and circumferential directions;
2) spatial reinforcement in four directions.

It is shown that even for relatively thin composite shells the Ambardzumyan theory is unacceptable to obtain adequate results of calculations of their viscoelastic-plastic dynamic deformation. It has been demonstrated that a calculation according to the theory of elastoplastic deformation of reinforced structures does not allow even an approximate estimate of the residual states of composite shells after their dynamic loading. It is shown that even for a relatively thin and long cylindrical shell, the replacement of the traditional “flat”-cross-reinforcement structure with a spatial structure can significantly reduce the residual strain of the binder material. In cases of relatively thick and especially short shells, the positive effect of such a replacement of the reinforcement structures is manifested to a much greater extent.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):343-364
343-364

## Mathematical Modeling, Numerical Methods and Software Complexes

### Horizontal-axis wind turbine weathervane yaw differential error

Solomin E.V., Terekhin A.A., Martyanov A.S., Kovalyov A.A., Ismagilov D.R., Miroshnichenko A.A., Yang Y., Ryavkin G.N.

#### Abstract

The paper deals with the problem of orientation of the traditional horizontal-axis wind turbine (HAWT) when changing the direction, strength and speed of the wind.

When the wind direction changes, the active swept area of the rotor, which is a circle when the rotation axis and the incoming air flow vector are collinear, decreases and takes the form of an ellipse. This, in turn, leads to a decrease in the electricity volume generation.

Weathervane or rumba-anemometer is a device for registering the speed and direction of wind flow. When the wind direction changes, the device transmits a corresponding signal to the Control System, which in turn generates the command to turn the HAWT rotor on the wind. However, when the wind flow is passing between the rotating blades, the direction of distorted wake is changing, causing eddy formation. As a result the device gives out the initially incorrect data about the direction and velocity of the air flow. Furthermore, when adjusting the position of the rotor (yawing), the collinearity of the rotation axis and the vector of the incoming flow is not achieved, the swept area remains mostly elliptical, and the power generated is proportionally reduced. In accordance with the relevancy of the said problem, the goal of the study was to calculate numerical values of the wake deflection angle in various modes, using the three-dimensional modeling in Ansys CFX software package. The obtained information can be used then to develop the algorithm for eliminating this error.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):365-380
365-380

## Short Communications

### Exact boundaries for the analytical approximate solution of a class of first-order nonlinear differential equations in the real domain

Orlov V.N., Kovalchuk O.A.

#### Abstract

The paper gives a solution to one of the problems of the analytical approximate method for one class first order nonlinear differential equations with moving singular points in the real domain. The considered equation in the general case is not solvable in quadratures and has movable singular points of the algebraic type. This circumstance requires the solution of a number of mathematical problems.

Previously, the authors have solved the problem of the influence of a moving point perturbation on the analytical approximate solution. This solution was based on the classical approach and, at the same time, the area of application of the analytic approximate solution shrank in comparison with the area obtained in the proved theorem of existence and uniqueness of the solution.

Therefore, the paper proposes a new research technology based on the elements of differential calculus. This approach allows to obtain exact boundaries for an approximate analytical solution in the vicinity of a moving singular point.

New a priori estimates are obtained for the analytical approximate solution of the considered class of equations well in accordance with the known ones for the common area of action. These results complement the previously obtained ones, with the scope of the analytical approximate solution in the vicinity of the movable singular point being significantly expanded.

These estimates are consistent with the theoretical positions, as evidenced by the experiments carried out with a non-linear differential equation having the exact solution. A technology for optimizing a priori error estimates using a posteriori estimates is provided. The series with negative fractional powers are used.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):381-392
381-392

### Quantum evolution in terms of mechanical motion

Samarin A.Y.

#### Abstract

Quantum tunneling is considered from the point of view of local realism. It is concluded that a quantum object tunneling through a potential barrier cannot be interpreted as a point-like particle because such an interpretation generates a contradiction with the impossibility of faster-than-light motion. Such a contradiction does not arise if a quantum object is considered as a continuous medium formed by the fields of matter. The dynamics law of the mechanical motion of these matter fields is derived from the quantum evolution law in the path integral form. The analysis of tunneling shows that this dynamics law has a form of the principle of least action on a complex time variable. The approach used here is not only a physical interpretation of quantum tunneling consistent with special relativity but is also applicable to the description of a wide range of quantum phenomena for which traditional research methods are impracticable.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(2):393-401
393-401