Vol 23, No 4 (2019)

Articles
Second boundary-value problem for the generalized Aller–Lykov equation
Kerefov M.A., Gekkieva S.K.
Abstract

The equations that describe a new type of wave motion arise in the course of mathematical modeling for continuous media with memory. This refers to differential equations of fractional order, which form the basis for most mathematical models describing a wide class of physical and chemical processes in media with fractal geometry. The paper presents a qualitatively new equation of moisture transfer, which is a generalization of the Aller–Lykov equation, by introducing the concept of the fractal rate of change in humidity clarifying the presence of flows affecting the potential of humidity. We have studied the second boundary value problem for the Aller–Lykov equation with the fractional Riemann–Liouville derivative. The existence of a solution to the problem has been proved by the Fourier method. To prove the uniqueness of the solution we have obtained an a priori estimate, in terms of a fractional Riemann–Liouville using the energy inequality method.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):607-621
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Dirichlet problem for mixed type equation with characteristic degeneration
Sabitova Y.K.
Abstract
For a mixed elliptic-hyperbolic type equation with characteristic degeneration, the first boundary value problem in a rectangular region is investigated. The criterion for the uniqueness of the solution of the problem is established. Earlier, in proving the uniqueness of solutions of boundary value problems for equations of mixed type, the extremum principle or the method of integral identities was used. The uniqueness of the solution to this problem is established on the basis of the completeness of the system of eigenfunctions of the corresponding one-dimensional spectral problem. The solution of the problem is constructed as a sum of a series in the system of eigenfunctions. When we proved the convergence of the obtained series, the problem of small denominators of a more complicated structure than in other known works arose. These denominators contain a parameter depending on the lengths of the sides of the rectangle in the hyperbolic part of the domain and the exponent of the degree of degeneration. In this connection, estimates are established about separation from zero with the corresponding asymptotics, in cases where this parameter is a natural, rational and algebraic irrational number of degree two. If this parameter is not an algebraic irrational number of degree two, then the solution of the problem as a sum of a series does not exist. Using the obtained estimates, the uniform convergence of the constructed series in the class of regular solutions is justified under certain sufficient conditions with respect to the boundary functions. The stability of the solution of the problem with respect to the boundary functions in the norms of the space of summable functions and in the space of continuous functions is also proved.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):622-645
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On a differential constraint in the continuum theory of growing solids
Murashkin E.V., Radayev Y.N.
Abstract
The present paper is devoted to the problem of boundary conditions formulation for asymmetric problems in the mechanics of growing solids (MGS). The boundary conditions on the propagating growing surface (PGS) is the fundamental problem of this branch of mechanics. Results from the algebra of rational invariants are used for deriving constitutive equations on PGS. Geometrically and mechanically consistent differential constraints are obtained on PGS. Those are valid for a wide range of materials and metamaterials. A number of constitutive equations on PGS of different complexity levels are proposed. The boundary conditions simultaneously can be treated as differential constraints within the frameworks of variational formulations. The differential constraints imply an experimental identification of constitutive functions. For this reason, the obtained results furnish a general ground in applied problems of the MGS.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):646-656
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Orthotropic strip with central semi-infinite crack under arbitrary loads applied far apart from the crack tip
Ustinov K.B., Lisovenko D.S., Chentsov A.V.
Abstract
The exact analytical solution has been obtained for a problem of orthotropic strip with central semi-infinite crack loaded normally with self-balanced system of forces applied far enough from the crack tip to be considered as applied at infinity. The general solution is expressed as a superposition of solutions for two modes of loading: (i) symmetrically applied moments; (ii) symmetrically applied transverse forces with compensating moments. The exact expressions for stress intensity factor (SIF) have been obtained. Due to symmetry only the opening mode of SIF is present for each case of loading. For both cases of loading the stress states are determined by two dimensionless parameters composed by four elastic constants. Expression for SIF for the case of loading with symmetrically applied moments is obtained in terms of elementary functions and coincides with the elementary solution due to beam theory. Expression for SIF for the case of loading with symmetrically applied transverse forces with compensating moments has been obtained in terms of one function of one of the parameters expressed as a single integral, multiplied by a power function of the second parameter. The solution for this case demonstrated good agreement with the existing numerical solution for the range of parameters, for which the latter had been obtained. The obtained solution covers all possible range of parameters.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):657-670
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Analysis of the bulk creep influence on stress-strain curves under tensile loadings at constant rates and on Poisson's ratio evolution based on the linear viscoelasticity theory
Khokhlov A.V.
Abstract
The Boltzmann–Volterra linear constitutive equation for isotropic non-aging viscoelastic materials is studied analytically in order to elucidate its abilities to provide a qualitative simulation of rheological effects related to different behavior types of lateral strain and the Poisson's ratio (i.e. lateral contraction ratio) observed in uni-axial tests under tension or compression at constant stress rate. The viscoelasticity equation is controlled by two material functions of a positive real argument (that is shear and bulk creep compliances); they are implied to be positive, differentiable, increasing and convex up functions. General properties of the volumetric, longitudinal and lateral strain-time curves, stress-strain curves and the Poisson's ratio evolution in time generated by the viscoelasticity relation (with an arbitrary shear and bulk creep functions) are examined, their dependence on stress rate and on qualitative characteristics of two creep functions are analyzed, conditions for their monotonicity and convexity or for existence of extrema, inflection points and sign changes are studied. Taking into account compressibility and volumetric creep (governed by a time-dependent bulk creep function) is proved to affect strongly the qualitative behavior of lateral strain and the Poisson's ratio. In particular, it is proved that the linear theory can reproduce increasing, decreasing or non-monotone and convex up or down dependencies of lateral strain and Poisson's ratio on time under tension or compression at constant stress rate, it can provide existence of minimum, maximum or inflection points and sign changes from minus to plus and vice versa. It is shown, that the Poisson's ratio at any moment of time is confined in the interval from $-1$ to 0.5 and the restriction on creep compliancies providing negative values of the Poisson's ratio is derived. Criteria for the Poisson's ratio increase or decrease and for extrema existence are obtained. The analysis revealed the set of characteristic features of the theoretic volumetric, axial and lateral strain-time curves, stress-strain curves families and the Poisson's ratio dependence on time which are convenient to check in tensile tests at constant stress rates and should be employed as indicators of the linear viscoelasticity theory applicability (or non-applicability) for simulation of a material behavior before identification. The specific properties of the two models are considered based on the assumption that the Poisson's ratio is time-independent or the assumption that bulk creep function is constant which neglects bulk creep and simulates purely elastic volumetric strain dependence on a mean stress. This assumptions reduce the number of material function to the single one and one scalar parameter and are commonly (and very often) used for simplification of viscoelasticity problems solutions. A number of restrictions and additional applicability indicators are found for this models. In particular, it is proved that elastic volumetric deformation assumption does not cut the overall range of the Poisson's ratio values and does not demolish the Boltzmann–Volterra relation ability to describe non-monotonicity and sign changes of lateral strain and to produce negative values of the Poisson's ratio, but neglecting bulk creep restricts this ability significantly and reduces drastically the variety of possible behavior modes of lateral strain-time curves and the Poisson's ratio evolution and so contracts applicability field of the model. The model with constant bulk compliance generates only convex-up lateral strain-time curves which can not have minima or inflection points and can change sign from minus to plus only and the Poisson's ratio is increasing convex-up function of time (without any extrema or inflection points which are possible in general case) and can not change sign from positive to negative.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):671-704
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Modeling of the extracellular information field influence in dynamics of the formation and development risks of a cancer tumor
Artemova O.I., Krevchik V.D., Semenov M.B.
Abstract
The dynamic nonlinear 2D model of the extracellular information field influence in the dynamics of risks of the cancer tumor formation and development has been considered. Physical properties of the extracellular matrix, availability of nutrients, oxygen concentration, pH of the extracellular matrix, interaction with stromal cells, and etc. are considered as the main external parameters forming the informational metabolic potential. Within the framework of the constructed 2D analytical model, it has been shown that microinteraction through the extracellular matrix of emerging cancer cells through a dynamic informational metabolic profile significantly influences the risk dynamics of the formation and development of a cancer tumor. It is shown that, depending on the structure of the 2D informational metabolic profile, a number of characteristic nonlinear features such as 2D bifurcations, beats, chaos, imposed on integral dynamic curves resembling by the Gompertz function, describing the probable risks of the formation and development of a cancerous tumor, are appeared. A comparison of the results of our analytical model under consideration with the results of the modeling of other authors on the consideration of chaotic and bifurcation dynamics in the “tumor–immune cluster–virus” system has been made. As a result of the quantitative estimations carried out within framework of the proposed theoretical model, we can formulate a method for assessing the risks of developing malignant neoplasms, characterized in that subfebrile temperature, caspase level, colposcopic Raid index, which determine the threshold for the formation of malignant neoplasms, and identified as the risk factors.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):705-723
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Active adaptation of a distributed multi-sensor filtering system
Semushin I.V., Tsyganova J.V.
Abstract
A multi-sensor filtering system is characterized mathematically as a result of the solution to the problem of synthesizing the multi-dimensional discrete system of filtering a single signal from heterogeneous data sources.The stationary problem statement has three variants of its solution: by Kolmogorov–Wiener, Kalman covariance, and Kalman information forms.In the body of the paper, we actualize a problem of these solutions under uncertainty conditions.Aimed at the Active Principle of Adaptation, we have found a method to form an instrumental performance index to substitute the inaccessible original performance index (filtering error mean square) by that criterion functional we created. This substitution makes it possible to apply for system adaptation all apparatus and tools of practical optimization methods, first of all, the gradient and Newton-like methods.
Our findings follow:
– Stretching one-step prediction and measurement update operations are wise to perform at the Decision Making Center; computation operations aimed to minimize the instrumental performance index are to be done in this place, too.
– Uncompounded procedures of adaptive data scaling are advisable to complete at the sensors' location in the network.
– Adaptation algorithms may be implemented based for filter structures taken in different forms: Kolmogorov–Wiener, Kalman covariance, or Kalman information forms.
– Computational operations for minimizing the instrumental performance index would be beneficial to develop as versions to implement the modern practical optimization methods of different levels of complexity.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):724-743
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Mathematical modeling and prediction of the effectiveness of surgical treatment in surgery of the spine and pelvic complex
Kossovich L.Y., Kharlamov A.V., Lysunkina Y.V., Shulga A.E.
Abstract
Based on the study of the literature on the quality assessment of operativetreatment in reconstructive surgery of the spine and pelvic complex, it can beconcluded that, as a rule, multiple linear or logistic regression, adecision tree, is used to predict the quality of operative treatment. Neuralnetworks are less commonly used.
Forecasting is performed on the basis of a comparison of the pre- andpostoperative condition of the patient, assessed according to variousordinal and quantitative scales as a result of interviewing the patient.
With a relatively small number of analyzed cases of the disease (severaltens or hundreds) and a small number of indicators (no more than two orthree dozen), the use of neural networks seems premature for two reasons: asmall amount of data allows analyzing them with classical methods ofmathematical statistics, and identifying dependencies on a given stagerequires constant “manual” intervention, taking into account informationfrom the subject area.
The application of statistical analysis methods to data on the treatment ofchronic injuries showed the presence of standard problems for medical data.This is the presentation of the initial information in nominal or ordinalscales, the subjective nature of some indicators, as well as theinterdependence of the presented characteristics, which reduces the qualityof research.
The search for the objective function that characterizes the quality ofsurgical treatment has shown the ambiguity of solving this problem even fora highly specialized situation.
The identification of objectively present relationships also revealed alarge number of problems, especially related to the choice of the type ofsurgical treatment, which is largely determined by the experience of thesurgeon.
Based on the study, it was proposed to build a model for predicting thequality of surgical treatment, based on expert assessments in the form of aforecast tree with recommended surgical treatment options and a statisticalforecast based on the available experience. It is assumed that the modelwill be dynamic with feedback and be able to self-update.
To predict the quality of surgical treatment in reconstructive surgery ofthe spine and pelvic complex, it is advisable to use a forecast tree, which allows usto recommend the type of surgery for a specific case of injury or diseaseand calculate the predicted values of quality of life indicators.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):744-755
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On the electrostatic field in expansion dynamics of gas bubbles
Museibli P.T.
Abstract
The work is devoted to the study of the dynamics of the formation ofbubbles in a gas–liquid system taking into account the potential difference.The electrical conductivity of the fluid is determined depending on theconcentration of the electrolyte and, accordingly, the electrostatic field thatoccurs when the fluid flows. The effect of the electrostatic field on the bubbleformation dynamics has shown that the radius of the gas bubbles and thedynamics of its expansion, formed by the pressure difference, can be regulatedby the potential difference parameter.
Depending on the electrolytic concentration,the electric conductivity of the liquid and, accordingly, the electrostatic fieldarising from friction in fluid are determined. The effect of the electrostaticfield on the dynamics of the bubble formation has shown that the radius ofgas bubbles and expansion dynamics formed by the pressure drop can beregulated by the potential difference parameter. It is presented that one of themain factors affecting the flow of two-phase fluids is the nature of the liquidphase and the concentration of electrolyte added. The results of regulation ofthe bubble formation dynamics in the gas–liquid system via the electrostaticfield and a number of physical parameters can be applied in the oil and gasindustry, chemical processes, biomechanics.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):756-763
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Non-helical exact solutions to the Euler equations for swirling axisymmetric fluid flows
Prosviryakov E.Y.
Abstract
Swirling axisymmetric stationary flows of an ideal incompressible fluid are considered within the framework of the Euler equations. A number of new exact solutions to the Euler equations are presented, where, as distinct from the known Gromeka–Beltrami solutions, vorticity is noncollinear with velocity. One of the obtained solutions corresponds to the flow inside a closed volume, with the nonpermeability condition fulfilled at its boundary, the vector lines of vorticity being coiled on revolution surfaces homeomorphic to a torus.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):764-770
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On a proplem for generalized Boussinesq–Love equation
Zhegalov V.I.
Abstract
For a fourth-order equation with two independent variables a variant of the Goursat problem with data on two intersecting characteristics is considered. It includes not only the construction of the desired function, but also the coefficients of the equation. Thus, we are talking about the inverse problem of determining the coefficients of the equation. The method of construction of conditions providing allocation of infinite number of sets of this type equations is offered, for which the problem under consideration is solvable in quadratures. Instead of introducing additional boundary conditions, restrictions on the structure of the equation are proposed, related to the possibilities of its factorization.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):771-776
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Modeling of stress state of a perforated cement sheath in a well with hydraulic fracture
Kireev T.F., Bulgakova G.T.
Abstract
Modeling of stress state of a perforated cement sheath in a well with hydraulic fracture is performed. The incompressible fluid flow model is used to calculate the pore pressure of a fluid. The linear-elastic body model and finite volume method with multipoint stress approximation are used to calculate the stress state of the cement sheath and production casing. The numerical model was verified by comparing the calculation results with a calculation in the Fenics open-source computing platform. It is shown that the maximum value of von Mises stress falls on the perforation zone at the junction of the cement sheath and the production casing. The presence of a hydraulic fracture can reduce the stress of the cement sheath.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):777-788
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The Bitsadze–Samarskii problem for some characteristically loaded hyperbolic-parabolic equation
Khubiev K.U.
Abstract
The paper considers a characteristically loaded equation of a mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity part of the domain. In the hyperbolic part of the domain, we have a loaded one-velocity transport equation, known in mathematical biology as the Mac Kendrick Von Forester equation, in the parabolic part we have a loaded diffusion equation. The purpose of the paper is to study the uniqueness and existence of the solution of the nonlocal inner boundary value problem with Bitsadze-Samarskii type boundary conditions and the continuous conjugation conditions in the parabolic domain; the hyperbolic domain is exempt from the boundary conditions.
The problem under investigation is reduced to a non-local problem for an ordinary second-order differential equation with respect to the trace of the unknown function in the line of the type changing. The existence and uniqueness theorem for the solution of the problem has been proved; the solution is written out explicitly in the hyperbolic part of the domain. In the parabolic part, the problem under study is reduced to the Volterra integral equation of the second kind, and the solution representation has been found.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):789-796
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The solution of equations of ideal gas that describes Galileo invariant motion with helical level lines, with the collapse in the helix
Yulmukhametova Y.V.
Abstract
We consider the equations of ideal gas dynamics in a cylindrical coordinate system with the arbitrary equation of state and one two-dimensional subalgebra from the optimum system of an 11-dimensional Lie algebra of differentiation operators of the first order. The basis of the subalgebra operators consists of the operator of Galilean transfer and the operator of movement on spiral lines. Invariants of operators set representation: type of speed, density and entropy. After substitution of the solution representation into the equations of gas dynamics the assumption of the linear relation of a radial component of speed and spatial coordinate is entered. Transformations of equivalence which are allowed by a set of equations of gas dynamics after substitution of the solution representation are written down. For the state equation of polytropic gas all four solutions depending on an isentropic exponent are found. For each case the equations of world lines of gas particles motion are written down. The transition Jacobian from Eulerian variables to Lagrangian is found. The instants of collapse of gas particles are determined by value of the Jacobian. As a result the solutions describe movement on straight lines from a helicoid surface. Movements of the particles on equiangular spirals lying on a paraboloid and on hyperbolic spirals, lying on a cone.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2019;23(4):797-808
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