Vol 28, No 4 (2024)
- Year: 2024
- Articles: 11
- URL: https://journals.eco-vector.com/1991-8615/issue/view/9391
Differential Equations and Mathematical Physics
Some necessary and some sufficient conditions for local extrema of polynomials and power series in two variables
Abstract
This study extends the author’s previous works establishing necessary and sufficient conditions for a local extremum at a stationary point of a polynomial or an absolutely convergent power series in its neighborhood. It is known that in the one-dimensional case, the necessary and sufficient conditions for an extremum coincide, forming a single criterion.
The next stage of analysis focuses on the two-dimensional case, which constitutes the subject of the present research. Verification of extremum conditions in this case reduces to algorithmically feasible procedures: computing real roots of univariate polynomials and solving a series of practically implementable auxiliary problems.
An algorithm based on these procedures is proposed. For situations where its applicability is limited, a method of substituting polynomials with undetermined
coefficients is developed. Building on this method, an algorithm is constructed to unambiguously verify the presence of a local minimum at a stationary point for polynomials representable as a sum of two $A$-quasihomogeneous forms, where $A$ is a two-dimensional vector with natural components.



Problem of optimal dynamic measurement with multiplicative effects in spaces of differentiable "noises"
Abstract
The article deals with a model of optimal dynamic measurement with multiplicative influence, considered as an optimal control problem for a nonstationary Leontief-type system. The existence of a solution to this problem in a stochastic formulation is established. The main objective is to find a recoverable signal (control action) that brings the system state as close as possible to the observed indicators, given the presence of an additional input process modeling noise. Solutions to the system must be sought in spaces of random processes. To achieve this, the optimal control problem in spaces of
differentiable "noises" is preliminarily analyzed. The linearity of the transducer model, described by a non-stationary Leontief-type system, allows the original system to be decomposed into deterministic and stochastic subsystems. Based on the results regarding the solvability of optimal control problems for each subsystem, a solution to the original problem is obtained.
The first part of the article presents the solvability conditions for a stochastic non-stationary Leontief-type system. The second part explores the optimal control problem in the stochastic case and derives estimates for the minimized functionals using results previously obtained for the deterministic counterpart. In conclusion, an algorithm for studying the problem of optimal dynamic measurement with multiplicative influence in spaces of "noises" is presented.



Optimization of the error in exponential-trigonometric interpolation formula
Abstract
In engineering geodesy, point clouds obtained through area measurement methods, such as terrestrial laser scanning or photogrammetry, need to be approximated by a curve or surface that can be described by using a continuous mathematical function, often employing splines and optimal interpolation formulas.
This work is devoted to the construction of an optimal interpolation formula that is exact for exponential-trigonometric functions in a Hilbert space. The optimal interpolation formula is obtained by minimizing the norm of the error functional with respect to the coefficients. The article proves the existence and uniqueness of the optimal interpolation formula and provides explicit analytical expressions for the optimal coefficients of the interpolation formula. Using the constructed optimal interpolation formula, specific functions were interpolated, and a comparison was made with known results from other authors.



Mechanics of Solids
Model of bending of an orthotropic cantilever beam of Bernoulli-Euler under the action of unsteady thermomechanodiffusion loads
Abstract
The paper examines the interaction of mechanical, temperature and diffusion fields during unsteady bending of a cantilevered beam. The mathematical formulation of the problem includes a system of equations of non-stationary bending vibrations of a Bernoulli-Euler beam taking into account heat and mass transfer, which is obtained from the general model of thermomechanical diffusion for continuum using the generalized principle of virtual displacements. It is assumed that the speed of propagation of thermal and diffusion disturbances is finite. Using the example of a cantilevered three-component beam made of an alloy of zinc, copper and aluminum, under the influence of a non-stationary load applied to the free end, the interaction of mechanical, temperature and diffusion fields was studied.



The influence of anisotropy and strength-differential effect on the design of equi-strength rotating disk of variable thickness
Abstract
The work is devoted to the calculation of the geometry of an equi-strength annular disk taking into account the anisotropy and strength differential effect. The disk is under centrifugal forces and tractions on the inner and outer surfaces. The problem statement is based on the anisotropic elasticity theory and the plane stress assumption. General quadratic failure criterion is used, the only requirement for which is ellipticity. In particular cases, the used condition is reduced to many known strength criteria (Tsai–Wu, Hill, Drucker–Prager, von Mises, etc.).
The governing system of equations consists of the compatibility equation, the equilibrium equation and the condition of constant equivalent stress. This condition is satisfied by a trigonometric substitution and an introduced auxiliary function. The two remaining equations are solved sequentially in an implicit form, in which the auxiliary function is treated as independent variable. The found analytical solution allows to construct the geometry of the disk (profile and inner radius of the disk) of equal strength, and also to determine the distribution of stresses in it. It is established that the solution may not exist and be non-unique. In particular cases, the solution is reduced to solutions for many known failure criteria, as well as to the classical solution of Rabotnov. Comparison of calculations obtained for the Tsai–Wu and von Mises criteria showed that anisotropy and different strengths under tension and compression can have a significant effect on the geometry of a disk of equal strength and the stress state in it.



Stochastic superelastic properties of materials with phase transformations
Abstract
The study is devoted to the impact of stochastic isothermal phase transformations in an unstable material on its superelastic hardening.
A stochastic differential equation is derived to describe the dynamics of nucleation, growth of the new phase volume, and its interaction with the parent phase, depending on the level of irreversible structural deformations.
Macroscopic constitutive relations are established for the unstable material, incorporating the stochastic nature of phase transformations and their dependence on structural deformations. Effective elastic moduli of the material are calculated based on these relations.
Stochastic differential equations for direct and reverse phase transitions are formulated.
Numerical simulations demonstrate strong agreement with experimental data, validating the proposed model.



Antiplane axisymmetric elastic-plastic shear in an isotropic hardening material
Abstract
The paper presents an analytical solution to the problem of axisymmetric antiplane shear. The deformable material is enclosed between two cylindrical surfaces, one of which is fixed, and the other moves along the generatrix. This problem models a shear-off testing scheme. We use a geometrically nonlinear formulation of the elastic-plastic problem, taking the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts. The elastic properties of the specimen are described by the Mooney–Rivlin hyperelastic model. We consider an isotropic hardening material with the hardening law that is an arbitrary monotonic function of the accumulated plastic strain. The Tresca yield condition is utilized. The original nonlinear coupled system of partial differential equations is reduced to ordinary linear differential equations, the solution of which requires the calculation of definite integrals. The resulting solution includes deformation in the elastic range, the initiation of plastic flow, propagation of the plastic deformation region, and subsequent intensive plastic flow. The solution is illustrated with examples of materials with linear hardening, quadratic hardening, and Voce-type hardening with saturation. For these examples, ''force – displacement'' relationships, the distribution of accumulated plastic strain over the sample cross-section, and data on the distortion of material fibers, which were located in the radial direction before deformation, are
presented.



Mathematical Modeling, Numerical Methods and Software Complexes
Exact solution to the velocity field description for Couette–Poiseulle flows of binary liquids
Abstract
Exact solution of the Oberbeck–Boussinesq equations for describing steady flows of binary Poiseuille-type fluids is proposed and studied. The fluid motion is considered in the infinite horizontal layer. Shear flows are described by overdetermined system of equations. Nontrivial exact solution for the Oberbeck–Boussinesq system exists in the class of velocities with two vector components and depends only on the transverse coordinate. This structure of the velocity vector coordinates ensures naturally the fulfillment of the continuity equation as an ''extra'' equation. The pressure field, the temperature field, and the concentration field of the dissolved substance are described by linear functions of horizontal (longitudinal) coordinates with coefficients that functionally depend on the third coordinate. Fluid layer, as it is shown, can have two points where the velocity becomes zero. In this case, the spiral flow is realized (the hodograph of the velocity vector has a turning point).



Mathematical modeling of gas oscillations in a methane pyrolysis reactor
Abstract
A mathematical model of gas oscillations induced by external harmonic loading has been developed, taking into account spatiotemporal nonlocality. The model is based on the equilibrium (motion) equation and a modified Hooke’s law, which incorporates relaxation terms accounting for the mean free path and time of microparticles (electrons, atoms, molecules, ions, etc.).
Numerical studies of the model have shown that resonance occurs when the natural frequency of gas oscillations coincides with the frequency of the external load. This resonance is characterized by a sharp increase in the amplitude of oscillations, which is limited by the gas friction coefficient. When the frequency of the external load is close to the natural frequency of gas oscillations, bifurcation-flutter oscillations (beats) are observed, accompanied by periodic increases and decreases in the oscillation amplitude at each point of the spatial variable. In this case, the gas oscillations exhibit an infinite
number of amplitudes and frequencies.
Periodic variations in gas displacement and pressure, ranging from zero to a certain maximum value and propagating along the length of the methane pyrolysis reactor, contribute to the cleaning of its internal surfaces from loose carbon deposits. The carbon removed from the reactor walls accumulates in the lower part between two gas-tight shut-off valves, allowing for its removal without interrupting the pyrolysis process. This model can be useful for optimizing reactor cleaning processes and improving the efficiency of methane pyrolysis.



Short Communications
A common fixed-point result via a supplemental function with an application
Abstract
In this paper, we prove a novel common fixed-point theorem for two commuting mappings. This assertion is proved using the measure of noncompactness in Banach spaces. Moreover, an application is given to demonstrate the usability of the obtained results.



The Riemann matrix for some systems of the differential hyperbolic-type equations of the high order
Abstract
Solutions to some boundary value problems for systems of hyperbolic partial differential equations can be constructed explicitly in terms of the Riemann matrix. In this regard, the question of explicitly constructing the Riemann matrix for high-order hyperbolic systems of equations is relevant.
We consider a system of third-order hyperbolic partial differential equations with three independent variables. For the specified system, the Riemann matrix is constructed as a solution to a special Goursat problem. Furthermore, the Riemann matrix satisfies a Volterra integral equation. The Riemann matrix is expressed explicitly in terms of a hypergeometric function of a matrix argument. Similarly, a system of fourth-order hyperbolic partial differential equations with four independent variables is considered. These results are generalized for a system of hyperbolic partial differential equations of order $n$ that does not contain derivatives of order less than $n$.


