# Vol 18, No 3 (2014)

**Year:**2014**Articles:**15**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1245

### A nonlocal problem for mixed type equation with singular coefficient in domain with half-strip as hyperbolic part

#### Abstract

A nonlocal problem for mixed type equation with a singular coefficient and the spectral parameter is formulated in the ﬁeld, which hyperbolic part is vertical half-strip and elliptic part is rectangle. The nonlocal condition of problem combines the values of required function on the right and left boundaries of half-stripe and rectangle. The only requirement on the unknown function in the change type line is continuity. To research the given problem we apply the spectral method. The uniqueness and existence of a solution are proved. The solution is constructed as biortogonal series. Coefficients of this series should require special ODE systems, solved in the paper. The uniform convergence of the series is proved with the restrictions on problem conditions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):7-20

7-20

### A criterion for the unique solvability of the Dirichlet spectral problem in a cylindrical domain for multidimensional hyperbolic equations with wave operator

#### Abstract

We consider the Dirichlet spectral problem with the homogeneous boundary conditions in a cylindrical domain of Euclidean space for multidimensional hyperbolic equation with wave operator. We construct the solution as an expansion in multidimensional spherical functions; prove the existence and uniqueness theorems. The obtained conditions of the problem unique solvability essentially depend on the “height” of the cylinder.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):21-30

21-30

### On the determination of the unknown coefficients of the highest derivatives in a linear elliptic equation

#### Abstract

Inverse problems on restoration of coefficients to the differential equations with partial derivatives are of interest in many applied researches. These problems lead to necessity of the approached decision of inverse problems for the equations of mathematical physics which are incorrect in classical sense. In the article the existence, uniqueness and stability of the solution of the given inversion problem for the elliptic equation are proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):31-43

31-43

### On the solvability of boundary value problem for mixed-type equation with a singular coefficient

#### Abstract

In this paper we study a problem with conditions on the inner characteristic and on some parts of the degeneration line for mixed type equation with singular coefficient in unbounded domain. We prove the uniqueness of solution of the mentioned problem with the help of the extremum principle. The proof of the existence of solution is based on the theory of singular integral equations and Fredholm integral equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):44-56

44-56

### A cases of solvability of the integral equation in quadratures

#### Abstract

We consider Volterra equation with two-variable, commonly encountered in the theory of elasticity. The purpose is to ﬁnd new variants of sufficient conditions for it’s solvability in explicit calculation. The reduction principle of the original equation, ﬁrst, to Goursat problem for differential equation of third order, and after that to two problems solving consecutively for equations of the ﬁrst and second order is devised. One of these problems can be solved by direct equation integration, and the other’s solution can be written through Riemann function for which variants of its explicit construction are found. Seven variants of conditions for mentioned calculation were obtained in terms of coefficients of the original equation. Considering that there are four variants of factorization of equation of third order involved into the reasoning, virtually there are 28 variants of conditions for original equation solvability in quadratures noted in this article.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):57-65

57-65

### Contact problem of torsion of a multilayer base with elastic connections between layers

#### Abstract

Torsion of a multilayer base with elastic connections between layers by cylindrical punch with a ﬂat sole was considered. The Hankel integral transform of the ﬁrst order and the method of the compliance functions were used. Two auxiliary functions connected with transformants of tangential stresses and displacements points of the upper boundary of the layer were introduced for each layer. The components of the stress-strain state were represented as a linear combination of these functions. The recurrent formulas binding auxiliary functions of neighboring layers were built based on conditions of joint deformation of neighboring layers. The compliance functions were introduced. The recurrent formulas binding the compliance functions of neighboring layers were built. The problem was resolved into the integral equation. The kernel of the integral equation contains Sonine-Weber integral. The approximate solution of equation was found by the method of mechanical quadratures. Mechanical effects for one-layer and two-layer bases were obtained. Elastic connections between the layers of base led to a reduction of contact stresses in comparison with the case of full contact. Decrease of the shear modulus of one of the layers of a two-layer base was reduced to decrease of the contact stresses, as in the cause of ideal contact and when elastic connections are between the layers of the base.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):66-78

66-78

### Comparative analysis of the approximate analytical and finite element solutions for misaligned tube

#### Abstract

The boundary value problem of steady-state creep for thick-walled misaligned tube under internal pressure was considered. The approximate analytical solution of this problem by method of small parameter including the second approach is under construction. The solution for the state of plane deformation is constructed. The hypothesis of incompressibility of material for creep strain is used. As a small parameter the misalignment of the centers of the inner and outer radii of the tube is used. The main attention to the convergence of the resulting analytical solution considering the second approximation and assessment of its error is paid. It is noted that the convergence problem is solved only for boundary value problems in the theory of elasticity. Therefore the error assessment in the problem is solved on the basis of a comparison of the approximate analytical solution with the numerical solution constructed on the ﬁnite element method, for some special cases. Considering the symmetry of the problem, the ﬁnite element model was built for the half tube. The number of ﬁnite elements is about 18,000. Considering the symmetry of the problem the second half of the tube is replaced by boundary conditions. Analysis of analytical and numerical solutions is executed depending on the steady-state creep nonlinearity parameter and misaligned parameter that is ratio of the misalignment of the centers of the outer and inner diameter to the outer radius. It is shown that the error of deviation of the approximate analytical solution in the second approximation from numerical solution until the misalignment value of the centers of the inner and outer diameters of 0.1 for the tubes with small exponent of the steady-state creep (3 to 8) is not more than 9 %, and error to 8 % for the tubes with a large exponent of the steady-state creep nonlinearity is observed in the misaligned parameter to 0.06. Results of computations are presented in tabular form and in the form of graphs. Recommendations for the use of the constructed approximate analytical solution in applied problems are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):79-93

79-93

### The optimal location of the polygonal internal supports to the circular rigid-plastic plates

#### Abstract

The general solution of a problem of the limit behavior and dynamic bend is obtained for the perfect rigid-plastic circular plates, hinge supported on immobile polygonal contour, located inside the plate. The plate is subjected to short-term dynamic load of explosive type with high intensity, uniformly distributed over the surface. It is shown that there are several mechanisms of limit and dynamic deformation of plates depending on the location of the support contour. The simple analytic expressions are obtained for the limit load and maximum ﬁnal deﬂection of plates. The optimal location of support and the number of sides of the polygonal contour are determined, at which the plate has maximum limit load. Numerical examples are given. Keywords: rigid-plastic plate, circular plate, internal polygonal support, explosive load, limit load, ﬁnal deﬂection, optimal location of support.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):94-105

94-105

### On the creep theory for the strain-hardening materials

#### Abstract

The deformation process of the medium with full strains equaled to the sum of elastic and creep strains is considered. Elastic strains are described by Hook’s law while creep strains velocities are functions of stress components and some structural parameters, velocities of their changing are described by Rabotnov’s kinetic equations. It is assumed that Drucker’s stability postulate in big, formulated for materials with time-depending behaviour, is valid for creep strains. The inversion of depends between stresses and strains as well as uniqueness of the solution of boundary value problems are discussed. A special case of aforementioned creep equations for a strengthening material, when the strengthening parameter is the value of speciﬁc scattered creep energy, is considered. The suﬃcient conditions for Drucker’s stability postulate fulﬁllment in big are determined for this case, the reasons in favor of necessity of these conditions are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):106-117

106-117

### The investigation of the motion of planets, the Moon, and the Sun based on a new principle of interaction

#### Abstract

A new principle of interaction of the surrounding space with material bodies is investigated. Under the surrounding space we understand the physical vacuum, whose properties are currently still in the formative stage. Gravity is the result of the interaction of the physical vacuum with moving material bodies. It is assumed that the movement of material objects leads to a change in the density of the surrounding space, i.e. areas which density is signiﬁcantly less than the density of the environment are forming. Gravity is explained by the properties of compression space relative the motion of material bodies. The diﬀerential equations of motion of n material bodies are received. It should be noted that the system of diﬀerential equations does not contain the masses and forces of interaction between bodies explicitly. The elements of orbits of the large planets are calculated in the interval of time (1600-2200 years). The results of calculation are compared with elements of orbits founded on data of coordinates and of velocities DE405/LE405. It is shown that the coordinates and the elements of orbits of the large planets, the Moon and the Sun obtained with help of new method are in satisfactory agreement with the coordinates DE405/LE405. Based on the studies, the following conclusions are made: the differential equations of motion satisfactorily describe the motion of the major planets in the time interval of 600 years; these equations are signiﬁcantly simpler than the differential equations taking into account the relativistic eﬀects, moreover, outlay of machine time is more than twice smaller the latter’s.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):118-131

118-131

### A method of extended normal equations for tikhonov’s regulatization problems with differentiation operator

#### Abstract

This article is devoted to a new method of ill-conditioned linear algebraic systems solving with the help of differentiation operator. These problems appear while solving the ﬁrst kind integral Fredholm equations. The most difficult thing about this method is that differential operator discrete analogue matrix is rank deﬁciency matrix. The generalized singular value decomposition methods are used to solve those problems. The approach has high computational complexity. This also leads to additional computational error. Our method is based on the original regularized problem transformation into equivalent augmented regularized normal equation system using differential operator discrete analogue. The problem of spectrum matrix investigation of augmented regularized normal equation system with rank deﬁciency differential operator discrete analogue matrix is very relevant nowadays. Accurate eigenvalue spectrum research for this problem is impossible. That is why we estimated spectrum matrix bounds. Our estimation is based on a wellknown Courant-Fisher theorem. It is shown that estimated spectrum matrix bounds are rather accurate. The comparison between the proposed method and standard method based on the solving of normal system of equations is done. As shown in the paper, the condition number of normal method matrix is bigger than the condition number of augmented normal equations method matrix. In conclusion test problems description is given which proves our theoretical background.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):132-142

132-142

### Estimation of the order of the matrix method approximation of numerical integration of boundary-value problems for the second order inhomogeneous linear ordinary differential equations

#### Abstract

Using the ﬁrst three terms of Taylor expansion of the required function in the approximate derivative by ﬁnite differences leads to the second order approximation of the traditional numerical quadrature method of boundary value problems for linear ordinary second order differential equations with variable coefficients. The paper shows previously proposed numerical quadrature method using tools of matrix calculus where the approximate derivative by ﬁnite differences was not used. Agreeing to above method the arbitrary number of terms of Taylor expansion for the required solution may be used when compiling the difference equation system. When using the three ﬁrst terms of expansion the difference equation system coincided with the traditional system. The estimation of residuals and the order of approximation depending on the number of the used terms of Taylor expansion is given. It is theoretically shown that for the boundary value problem with boundary conditions of the ﬁrst kind the approximation method order increases in direct proportion with the increasing in the number of members used in Taylor series expansion only for odd values of this number. For even values of this number the order of approximation coincides with the order of approximation for the number less by unit of the odd values. For boundary value problems with boundary conditions of the second and third kinds the order of approximation was directly proportional to the number of used terms in the Taylor series expansion of the required solution of the problem regardless of evenness. In these cases the order of approximation of the boundary points and therefore the whole problem turned out to be one unit less than the order for the inner points of the grid for the interval of integration. The method of approximation order increase at the boundary points up to the approximation order in the inner points of the grid is presented. The theoretical conclusions are conﬁrmed by a numerical experiment for a boundary value problem with boundary conditions of the ﬁrst and third kinds.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):143-160

143-160

### On the one property of the free components concerning to the sum of equal powers

#### Abstract

The given paper contains the proof of that the number of combinatorial arrangements coincides with free components of the sums of equal powers with the natural bases and parameters in the presence of the simple equality connecting elements of these arrangements. In the proof the modiﬁed exposition of the components participating in formation of the sum of equal powers is used. This exposition becomes simpler and led to an aspect of product of binomial factors. Other variants of construction of corresponding product of binomial factors do not exist here. The received proof allows both to represent number of arrangements in the form of product, and to apply at this representation summation elements. Thus, the number of arrangements supposes characteristic expression not only in the form of product of its elements.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):161-168

161-168

### The Templet language preprocessor: a programming tool for process-per-message modeling

#### Abstract

Motivation: A large number of applications can be described as a set of processes that exchange messages. Traditionally the process-per-message model is used in the form of a specialized language or a run-time library for general purpose language. The ﬁrst approach lacks implementation simplicity, while the second approach is difficult in use. We propose a new method that comprises domain-speciﬁc language called Templet for code markup, a general purpose language, and the preprocessor. Our approach is free from disadvantages mentioned above. Method: A code of a program is divided into blocks. Block boundaries are indicated by comments. The entire code structure is deﬁned in Templet language so it can be checked out automatically before compilation. Description of Channels: A channel deﬁnes a message exchange protocol between two interconnected processes. We provide channel syntax in the form of Extended Backus-Naur Formalism (EBNF). The informational structure of the channel is described with Entity-Relation diagram (ER). Description of Processes: A process deﬁnes the algorithm for message processing. Information structure of the process is shown in conjunction with the syntax. EBNF and ER models are also used in the process speciﬁcation. Syntax rules are illustrated with the fork-join code sample. Preprocessor structure and work scheme: We present the algorithm and the structure of the preprocessor. Subsystems discussed are: syntax analyzer; semantics analyzer; internal database; inference mechanism; and code generator. The method for estimation of workload of manual coding is presented. It shows the diminishment of workload in 20 times comparing with manual coding. Discussion: The preprocessor is used for skeleton programming as a part of web-service for automated parallel programming. Its advantages and features are discussed in comparison with parallelization with markup technique; general-purpose macro processor; parallel programming language, metaprogramming; and model-driven development. This paper is an extended version of a PIT 2014 paper [1].

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):169-182

169-182

### Development and application of the computational model for skeleton solutions. Case study - using “bag-of-task” for hrbf neural network learning

#### Abstract

The article proposes a solution to the problem of mapping an algorithm from the ﬁeld of Computational Mathematics on the target computing environment. The solution is based on a formal method for constructing parallel skeletons. The method comprises a speciﬁcation of concurrency with the directed graphs and a formula for interpretation of dynamic behavior of such graphs. This interpretation is based on Temporal Logic of Actions approach proposed by Leslie Lamport. To illustrate the use of the method the “bag-oftasks” parallel skeleton is discussed hereinafter. We present graphically basic skeleton operations with the proposed computational model. After that we specify a learning algorithm of hyper-radial basis function neural network in the terms of skeleton operations as a case study. This made it possible to parallelize the leaning algorithm and map it on desired computing environments with predeﬁned run-time libraries. Computational experiments conﬁrming that our approach does not reduce the performance of the resulting programs are presented. The approach is suitable for researchers not familiar with parallel computing. It helps to get a reliable and effective supercomputer application both for SMP and distributed architectures.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(3):183-195

183-195