# Vol 23, No 1 (2019)

**Year:**2019**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1969

### Dirichlet problem for the mixed type equation with two degeneration lines in a half-strip

#### Abstract

In this article, the first boundary problem for the mixed type equation with two degeneration lines at a half-strip in the class of the regular and limited in infinity decisions is discussed. The criterion of uniqueness for the stated problem was formulated by the methods of a spectral analysis.The solution of a problem is constructed in the form of a series on eigenfunctions of the corresponding one-dimensional eigenvalues problem. At justification of the uniform convergence of the constructed series there was a problem of small denominators. The estimation for the separation from zero of a small denominator with the corresponding asymptotics was provided in connestion with mentioned problem in the present paper.This assessment at some sufficient conditions on boundary function allowed to prove convergence of the constructed series in a class of the regular solutions of this equation. In difference from other works of similar subject is the criterion of uniqueness and existence of the solution of the stated problem to be proved at all positive values of the parameters entering the equation, not necessarily equal. Such fact is an important consequence of the received result that the constructed solution everywhere in the considered area is the solution of the equation. Therefore the line of change-type of the equation as a special one is eliminated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):7-19

7-19

### Boundary value problem for mixed-compound equation with fractional derivative, functional delay and advance

#### Abstract

We study the Tricomi problem for the functional-differential mixed-compound equation $LQu(x,y)=0$ in the class of twice continuously differentiable solutions. Here $L$ is a differential-difference operator of mixed parabolic-elliptic type with Riemann–Liouville fractional derivative and linear shift by $y$. The $Q$ operator includes multiple functional delays and advances $a_1(x)$ and $a_2(x)$ by $x$. The functional shifts $a_1(x)$ and $a_2(x)$ are the orientation preserving mutually inverse diffeomorphisms. The integration domain is $D=D^+\cup D^-\cup I$. The “parabolicity” domain $D^+$ is the set of $(x,y)$ such that $x_00$. The ellipticity domain is $D^-=D_0^-\cup D_1^-\cup D_2^-$, where $D_k^-$ is the set of $(x,y)$ such that $x_k

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):20-36

20-36

### The vortex filament dynamics: New viewpoint on the problems of energy and effective mass

#### Abstract

The paper is devoted to the dynamics of a zero thickness infinite vortex filament in the local induction approximation. The filament is asymptotically considered as a straight line defined by the certain vector ${\boldsymbol{b}}_3 \in E_3$.

We also investigate the possibility of interpretation of such object as a planar “quaziparticle”. The configuration space for some “collective coordinates” for such object is the plane $E_2 \perp {\boldsymbol{b}}_3$.

The “quaziparticle” has a certain number of the internal degrees of the freedom. The Hamiltonian description of the filament is constructed in terms of the variables allowing the natural classification into “external” and “internal” groups.

The external variables (coordinates and momenta of a planar structureless particle) and the internal ones (the variables for the continuous Heisenberg spin chain) are entangled by the constraints. Because of these constraints, the constructed theory is non-trivial. The space symmetry group of the system was constructed by two stages: the contraction $ SO(3) \to E(2)$ and the subsequent extension $E(2) \times T \to \tilde{\mathcal G}_2$. The group $E(2)$ is the group of the plane motion for the plane $E_2 \perp {\boldsymbol{b}}_3$, symbol $T$ denotes the group of time translations and the group $\tilde{\mathcal G}_2$ is the central extended Galilei group for the plane mentioned above.

The appearance of the Galilei group makes it possible to introduce the invariant Cazimir functions for the Lee algebras for this group and to formulate the new approach for the problem of the energy of the infinite vortex filament with zero thickness. The formula for the tensor of the inverse effective mass of the constructed system is also being deduced. It is demonstrated that the suggested theory can be interpreted as a model of the planar vortex particle having an infinite number of internal degrees of freedom.

We also investigate the possibility of interpretation of such object as a planar “quaziparticle”. The configuration space for some “collective coordinates” for such object is the plane $E_2 \perp {\boldsymbol{b}}_3$.

The “quaziparticle” has a certain number of the internal degrees of the freedom. The Hamiltonian description of the filament is constructed in terms of the variables allowing the natural classification into “external” and “internal” groups.

The external variables (coordinates and momenta of a planar structureless particle) and the internal ones (the variables for the continuous Heisenberg spin chain) are entangled by the constraints. Because of these constraints, the constructed theory is non-trivial. The space symmetry group of the system was constructed by two stages: the contraction $ SO(3) \to E(2)$ and the subsequent extension $E(2) \times T \to \tilde{\mathcal G}_2$. The group $E(2)$ is the group of the plane motion for the plane $E_2 \perp {\boldsymbol{b}}_3$, symbol $T$ denotes the group of time translations and the group $\tilde{\mathcal G}_2$ is the central extended Galilei group for the plane mentioned above.

The appearance of the Galilei group makes it possible to introduce the invariant Cazimir functions for the Lee algebras for this group and to formulate the new approach for the problem of the energy of the infinite vortex filament with zero thickness. The formula for the tensor of the inverse effective mass of the constructed system is also being deduced. It is demonstrated that the suggested theory can be interpreted as a model of the planar vortex particle having an infinite number of internal degrees of freedom.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):37-48

37-48

### Analytical and experimental determination of the specific penetration resistance. Description of the facial and rear weakening free-surface effect

#### Abstract

In the present paper, the main force parameter is estimated under the static tests. This parameter is the specific penetration resistance and it is generally believed that the parameter is a constant in case of static penetration. However, a number of experimental and analytical data illustrate varying the specific penetration resistance according to the current depth of penetration. It is noted that the weakening free-surface effect decreases the specific penetration resistance near the facial or rear edge. Consequently, the relevance of the topic is emphasized by the influence of the facial and rear weakening free-surface effect on the penetration parameters detected in the experimental studies and engineering calculations.

The refined approximation of the specific penetration resistance presented in this paper is taking account of the penetration of the sharp indenter into the plate of middle thickness within the framework of the viscous crater formation and the facial and rear weakening free-surface effect. Also this article contains data processing technique.

For carrying out the tests a number of experimental samples were made. It is plates of different thicknesses, it must be emphasized that test sample materials are technical plasticine, plumbum and Wood's metal. It should also be noted that for the static tests three cone-nose indenters were made. Indenter sizes: the diameter of the cylindrical part is 7 mm in all cases and the lengths of the conical nose are 3.2 mm, 5.6 mm, and 8.4 mm. The test were carryed out on a testing machine Zwick/Roell Z-250. The key parameters derived from the experiment are the specific penetration resistance of the deep layers, the friction coefficient and the parameters of the weakening free-surface effect.

The results obtained in the experiment lead to the approximation of the resistance force from more general parameters. These parameters are the specific penetration resistance of the deep layers and the friction coefficient of a sample, geometric parameters of indenter and plate. An approximation error does not exceed 25 % for the technical plasticine, 16 % for the Wood's metal, and 25 % for the plumbum. These errors are given for “sharp” (the length of the cone-nose is 8.4 mm) and “middle” (the length of the cone-nose is 5.6 mm) indenter because of a problem has been in depth considered in the investigation. This problem is that penetration of the “blunt” indenter is not follow to condition of viscous crater formation. Therefore, different versions should be used to describe the penetration process (for example, plugging mechanism).

It is proposed in penetration models for the estimation of the penetration resistance force of sharp indenters into the plate of middle thickness.

The refined approximation of the specific penetration resistance presented in this paper is taking account of the penetration of the sharp indenter into the plate of middle thickness within the framework of the viscous crater formation and the facial and rear weakening free-surface effect. Also this article contains data processing technique.

For carrying out the tests a number of experimental samples were made. It is plates of different thicknesses, it must be emphasized that test sample materials are technical plasticine, plumbum and Wood's metal. It should also be noted that for the static tests three cone-nose indenters were made. Indenter sizes: the diameter of the cylindrical part is 7 mm in all cases and the lengths of the conical nose are 3.2 mm, 5.6 mm, and 8.4 mm. The test were carryed out on a testing machine Zwick/Roell Z-250. The key parameters derived from the experiment are the specific penetration resistance of the deep layers, the friction coefficient and the parameters of the weakening free-surface effect.

The results obtained in the experiment lead to the approximation of the resistance force from more general parameters. These parameters are the specific penetration resistance of the deep layers and the friction coefficient of a sample, geometric parameters of indenter and plate. An approximation error does not exceed 25 % for the technical plasticine, 16 % for the Wood's metal, and 25 % for the plumbum. These errors are given for “sharp” (the length of the cone-nose is 8.4 mm) and “middle” (the length of the cone-nose is 5.6 mm) indenter because of a problem has been in depth considered in the investigation. This problem is that penetration of the “blunt” indenter is not follow to condition of viscous crater formation. Therefore, different versions should be used to describe the penetration process (for example, plugging mechanism).

It is proposed in penetration models for the estimation of the penetration resistance force of sharp indenters into the plate of middle thickness.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):49-68

49-68

### Simulation of metal creep in nonstationary complex stress state

#### Abstract

The simulation of the results of metal testing under creep conditions at the nonstationary complex stress state is considered. As an example, we consider the experimental data obtained by a group of Japanese scientists for testing tubular samples of stainless steel under the temperature of $650 ^{\circ}$C. The following article presents the test results for four different loading programs. These loading programs are various combinations of piecewise constant dependencies of tangential and normal stresses on time. The presented data was simulated using the hardening theory and the flow theory; two material constants used are determined from the condition of the minimum relative integral discrepancy between the experimental and theoretical values of the corresponding creep deformations. A comparison was made of the results of the simulation carried out with the results of the simulation of the same experimental data, carried out by other researchers using other theories. In these theories, a large number of material characteristics are used: from three to nine constants and additionally one material function. The advantage of the hardening theory and flow theory with only two material constants each compared to the other theories has been shown.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):69-89

69-89

### Modeling of elastoplastic behavior of flexible spatially reinforced plates under refined theory of bending

#### Abstract

On the basis of the time-steps algorithm the structural model is constructed for elastic-plastic deformation of bended plates with spatial reinforcement structures. The inelastic behavior of the composition phase materials is described by equations of the theory of plastic flow with isotropic hardening. The possible weakened resistance of the reinforced plates to the transverse shear is taken into account on the basis of the refined theory, from which the relations of the Reddy theory are obtained in the first approximation. The geometric nonlinearity of the problem is considered in the Karman approximation. The solution of the formulated initial boundary value problems is based on an explicit numerical “cross” scheme. The dynamic inelastic deformation of spatially- and flat-cross-reinforced metal-composite and fiberglass flexible plates of different relative thickness is investigated in the case of the load caused by an air blast wave. It is demonstrated that for relatively thick fiberglass plates, the replacement of the flat-cross reinforcement structure by the spatial structure with the preservation of the total fiber consumption leads to a decrease in the structural flexibility in the transverse direction by almost 1.5 times, as well as to a decrease of the maximum of intensity of deformation in the binder by half. For relatively thin both fiberglass and metal-composite plates, the replacement of flat-cross 2D reinforcement structure with 3D and 4D spatial structures does not lead to a noticeable decrease in their deflections, but allows to reduce the intensity of deformations in the binder by 10 % or more. It is shown that the widely used non-classical Reddy theory does not allow obtaining reliable results of calculations of the elastic-plastic dynamic behavior of the bended plates, both with plane and spatial reinforcement structures, even with a small relative thickness of the structures and weak anisotropy of the composition.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):90-112

90-112

### A dual active set algorithm for optimal sparse convex regression

#### Abstract

The shape-constrained problems in statistics have attracted much attention in recent decades. One of them is the task of finding the best fitting monotone regression. The problem of constructing monotone regression (also called isotonic regression) is to find best fitted non-decreasing vector to a given vector. Convex regression is the extension of monotone regression to the case of $2$-monotonicity (or convexity). Both isotone and convex regression have applications in many fields, including the non-parametric mathematical statistics and the empirical data smoothing. The paper proposes an iterative algorithm for constructing a sparse convex regression, i.e. for finding a convex vector $z\in \mathbb{R}^n$ with the lowest square error of approximation to a given vector $y\in \mathbb{R}^n$ (not necessarily convex). The problem can be rewritten in the form of a convex programming problem with linear constraints. Using the Karush–Kuhn–Tucker optimality conditions it is proved that optimal points should lie on a piecewise linear function. It is proved that the proposed dual active-set algorithm for convex regression has polynomial complexity and obtains the optimal solution (the Karush–Kuhn–Tucker conditions are fulfilled).

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):113-130

113-130

### Tumor growth and mathematical modeling of system processes

#### Abstract

The paper deals with applying mathematical modeling to study tumor growth process and optimizing cancer treatment. A structured review of the studies devoted to this problem is given. The role of the cell life cycle in understanding the tumor growth and the mechanisms of cancer treatment is discussed. It is important that modern cancer treatment methods, in particular, chemotherapy and radiation therapy, affect both normal and tumor cells in certain stages of the life cycle and do not influence on cells in other stages. Cell life cycle description is given as well as the mechanisms that maintain and restore normal density of the cell population. A graph of cell life cycle stages and transitions is demonstrated. Dynamic mathematical model of proliferative homeostasis in the cell population is proposed, which takes into account the heterogeneity of cell populations by life cycle stages. The model is a system of differential equations with delays. The stationary state of the model is investigated, which allows to determine the parameters values for the normal cell population. The results of a numeric experiment is obtained, which is focused on the process of cell population density recovery after mass death of cells. As the experiment shows, after cell death, the densities of cells in different life cycle stages are restored to normal values, which corresponds to the concepts of proliferative homeostasis in cell populations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):131-151

131-151

### Comparison of various mathematical models on the example of solving the equations of the movement of large planets and the Moon

#### Abstract

In this paper, we study the accuracy of solving various differential equations describing the motion of large planets, the Moon and Sun. On the time interval from 31 years BC to 3969 AD, the numerical integration of Newtonian relativistic differential equations and equations obtained on the basis of the interaction of the surrounding space with moving material bodies was carried out. The range of applicability of the considered differential equations for the investigated objects is revealed. By comparing of the coordinates of the Moon, found by solving various differential equations and the DE405 data bank, it is shown that the greatest accuracy in the elements of the orbits of large planets is achieved by solving differential equations obtained on the basis of the interaction of the surrounding space with moving material bodies. The solution of relativistic equations provides high accuracy of the orbit elements for Mercury and the outer planets throughout the integration interval. However, for the remaining inner planets and the Moon, the accuracy of the orbital elements obtained by solving relativistic equations is comparable to the accuracy obtained by solving Newton equations. It is believed that the use of the harmonic coordinate system is justified only for Mercury from the point of view of the velocity of the secular longitude displacement of its perihelion, but for other internal planets (the Venus, Earth & Moon, and Mars) the velocities of secular displacements of the longitude of the perihelion's are overstated. It is shown that the solution of differential equations obtained on the basis of the interaction of the surrounding space with moving material bodies ensures a high accuracy of obtaining orbital elements for all objects under consideration on the time interval under study.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):152-185

152-185

### The Goursat-type problem for a hyperbolic equation and system of third order hyperbolic equations

#### Abstract

In the first part of this study, the well-posed Goursat-type problem is considered for the hyperbolic differential equation of the third order with non-multiple characteristics. The example illustrating the non-well-posed Goursat-type problem for the hyperbolic differential equation of the third order is discussed. The regular solution of the Goursat-type problem for the hyperbolic differential equation of the third order with the non-multiple characteristics is obtained in an explicit form.

In the second part, the well-posed Goursat-type problem is considered for a system of the hyperbolic differential equations of the third order. The regular solution of the Goursat-type problem for this system is also obtained in an explicit form.

The theorems for the Hadamard's well-posedness of Goursat-type problem for the hyperbolic differential equation and for a system of the hyperbolic differential equations is formulated as the result of the research.

In the second part, the well-posed Goursat-type problem is considered for a system of the hyperbolic differential equations of the third order. The regular solution of the Goursat-type problem for this system is also obtained in an explicit form.

The theorems for the Hadamard's well-posedness of Goursat-type problem for the hyperbolic differential equation and for a system of the hyperbolic differential equations is formulated as the result of the research.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):186-194

186-194

### Exact analytical solution for the stationary two-dimensional heat conduction problem with a heat source

#### Abstract

The exact analytic solution for the stationary two-dimensional heat conduction problem with a heat source for an infinite square bar was obtained. It was based on the Bubnov–Galyorkin orthogonal method using trigonometric systems of coordinate functions. The infinite system of ordinary differential equations obtained by the Bubnov–Galyorkin method is divided and reduced by the orthogonality property of trigonometric coordinate functions to the solution of a generalized equation which provides the exact analytical solution in a simple form, i.e. in the form of an infinite series. In view of the symmetry of the problem, only a quarter of the cross-section of the bar is considered for the boundary conditions of the adiabatic wall (the absence of heat transfer) along the cut lines, which allows (in contrast to the well-known classical exact analytical solution) to significantly simplify the process of the solution and the final equation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(1):195-203

195-203