Vol 25, No 2 (2021)

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.

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.

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.

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.

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.

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.