## Vol 19, No 4 (2015)

**Year:**2015**Articles:**15**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1216

Articles

Boundary value problems for matrix Euler-Poisson-Darboux equation with data on a characteristic

#### Abstract

We consider the system of $n$ partial differential equations in matrix notation (the system of Euler-Poisson-Darboux equations). For the system we formulate the Cauchy-Goursat and Darboux problems for the case when the eigenvalues of the coefficient matrix lie in $(0; 1/2)$. The coefficient matrix is reduced to the Jordan form, which allows to separate the system to the $r$ independent systems, one for each Jordan cell. The coefficient matrix in the obtained systems has the only one eigenvalue in the considered interval. For a system of equations having the only coefficient matrix in form of Jordan cell, which is the diagonal or triangular matrix, we can construct the solution using the properties of matrix functions. We form the Riemann-Hadamard matrices for each of $r$ systems using the Riemann matrix for the considered system, constructed before. That allow to ﬁnd out the solutions of the Cauchy-Goursat and Darboux problems for each system of matrix partial differential equations. The solutions of the original problems are represented in form of the direct sum of the solutions of systems for Jordan cells. The correctness theorem for the obtained solutions is formulated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):603-612

On the solution of the convolution equation with a sum-difference kernel

#### Abstract

The paper deals with the integral equations of the second kind with a sumdifference kernel. These equations describe a series of physical processes in a medium with a reﬂective boundary. It has noted some difficulties at applying the methods of harmonic analysis, mechanical quadrature, and other approaches to approximate solution of such equations. The kernel average method is developed for numerical-analytical solution of considered equation in non singular case. The kernel average method has some similarity with known strip method. It was applied for solution of Wiener-Hopf integral equation in earlier work of the author. The kernel average method reduces the initial equation to the linear algebraic system with Toeplitz-plus-Hankel matrix. An estimate for accuracy is obtained in the various functional spaces. In the case of large dimension of the obtained algebraic system the known methods of linear algebra are not efficient. The proposed method for solving this system essentially uses convolution structure of the system. It combines the method of non-linear factorization equations and discrete analogue of the special factorization method developed earlier by the author to the integral equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):613-623

The oscillator's model with broken symmetry

#### Abstract

The equations of the oscillator motion are considered. The exact solutions are given in the form of exponents with an additional parameter that characterizes the asymmetry of the oscillations. It is shown that these equations are the special case of the Hill’s equation. The equations for the three types of exponents, including having the property of unitarity are obtained. Lagrangians and Hamiltonians are found for these equations. It is proved that all the equations are associated by canonical transformations and essentially are the same single equation, expressed in different generalized coordinates and momenta. Moreover, the solutions of linear homogeneous equations of the same type are both solutions of inhomogeneous linear equations of another one. A quantization possibility of such systems is discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):624-633

The Dirichlet problem for mixed type equation with two lines of degeneracy in a rectangular area

#### Abstract

We study the ﬁrst boundary value problem for the elliptic-hyperbolic type equation with two perpendicular lines of change of type and spectral parameter. We prove the existence and uniqueness of the solution. In the proof of the uniqueness of solution we use the completeness of biorthogonal system in space $L_2$ . When building a solution as the sum of a series there is a problem of small denominators. We obtained estimates of the denominators of the separation from zero.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):634-649

The width of the higgs boson decay into two photons in framework of nmssm with CP violation: one loop approximation

#### Abstract

The article considers the problem of the extension of standard model physics of elementary particles. We focus on nonminimal supersymmetric models. Nowadays this model is the most appropriate compared with other developed supersymmetric theories. We assume the violation of CP invariance in the Higgs sector. We ﬁnd a local minimum of the Higgs potential for calculating the physical states of Higgs bosons. This procedure must be performed in order to obtain a stable state of vacuum. Next, calculations of the decay widths of the Higgs boson into two photons there are in this work. We use the method of quantum-ﬁeld perturbation theory with Feynman diagrams. The results of calculations are presented graphically. We show, that the CP violating phase affects the value of the decay widths. Also there are dependences of the masses of the Higgs bosons from the phase of CP violation: tree-level approximation. Some model parameters are not determined in the experiment, and concluded in a range of values that gives the possibility to consider various scenarios of multiple Higgs bosons. In particular, several Higgs bosons may be existing with mass less than 125 GeV in case of CP violation. This can be explained by the absence of a speciﬁc CP state of this particle.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):650-657

On the uniqueness of kernel determination in the integro-differential equation of parabolic type

#### Abstract

We study the problem of determining the kernel of the integral term in the one-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of the kernel. Next, assuming that there are two solutions $ k_1 (x, t) $ and $ k_2 (x, t), $ integro-differential equations, Cauchy and additional conditions for the difference of solutions of the Cauchy problem corresponding to the functions $ k_1 (x, t), $ $ k_2 (x, t)$ are obtained. Further research is being conducted for the difference $k_1 (x, t) - k_2 (x, t) $ of solutions of the problem and using the techniques of integral equations estimates it is shown that if the unknown kernel $ k (x, t) $ can be represented as $ k_j (x, t) = \sum_ {i = 0} ^ N a_i (x) b_i (t)$, $ j = 1, 2, $ then $ k_1 (x, t ) \equiv k_2 (x, t). $ Thus, the theorem on the uniqueness of the solution of the problem is proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):658-666

An inverse problem for two-dimensional equations of finding the thermal conductivity of the initial distribution

#### Abstract

The inverse problem of ﬁnding the initial distribution has been studied on the basis of formulas for the solution of the ﬁrst initial-boundary value problem for the inhomogeneous two-dimentional heat equation. The uniqueness of the solution of the direct initial-boundary value problem has proved with the completeness of the eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. The existence theorem for solving direct initial boundary value problem has been proved. Inverse problem has been investigated on the basis of the solution of direct problem, a criterion for the uniqueness of the inverse problem of ﬁnding the initial distribution has been proved. The existence of the inverse problem solution has been equivalently reduced to Fredholm integral equation of the ﬁrst kind.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):667-679

On a class of vector fields

#### Abstract

It is shown that a simple postulate ``The displacement field of the vacuum is a normalized electric field'', is equivalent to three parametric representation of the displacement field of the vacuum: $$ u(x;t) = P(x) \cos k(x)t + Q(x) \sin k(x)t. $$ Here $t$ is time; $k(x)$ -- frequency vibrations at the point of three-dimensional Euclidean space; $P(x), Q(x)$ -- a pair of stationary orthonormal vector fields; $(k,P, Q)$ -- parameter list of the displacement field. In this case, the normalization factor has dimension $T^{-2}$. The speed of the displacement field $$ v(x;t) = \frac{\partial u(x;t)}{\partial t} = k(x)(Q(x) \cos k(x)t - P(x) \sin k(x)t). $$ The electric field corresponding to this distribution of the displacement field of vacuum, is given by the formula $$ E(x;t) = -\frac{\partial v(x;t)}{\partial t} = k^2(x)u(x;t). $$ Moreover, the magnetic induction $$ B(x;t) = \mathop{\mathrm{rot }} v(x; t). $$ These constructions are used in the determination of local and global solutions of Maxwell's equations describing the dynamics of electromagnetic fields.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):680-696

A similar for ∆1 problem for the second order hyperbolic equation in the 3D euclidean space

#### Abstract

The second-order hyperbolic type equation is considered in the 3D Euclidean space. Boundary value problem is posed in the inﬁnite cylindrical region bounded by the characteristic surfaces of this equation with data on the related characteristic surfaces of the equation and with conditions mates on the internal non-descriptive plane. The solution is also assumed to be zero when z → ∞ with derivative by variable z. By the Fourier transform method the problem reduced to the corresponding planar problem ∆1 for hyperbolic equation, which in characteristic coordinates is the generalized Euler-Darboux equation with a negative parameter. Authors obtained estimates of the plane problem solution and its partial derivatives up to the second order inclusive. This, in turn, provided an opportunity to impose the conditions to given boundary functions ensuring the existence of a classical solution of the problem in the form of the Fourier transform.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):697-709

On the usage of special functions of two variables for studying of orthogonal polynomials of two variables

#### Abstract

It is shown that the second order partial diﬀerential equations system deﬁned by author is the most general system. It is possible to get all systems, solutions of which are hypergeometric functions of two variables from a Horn list and biorthogonal systems of Hermite and Appel polynomials. In this case the main apparatus of biorthogonal polynomials of two variables study is special functions of two variables. The resulting system of hypergeometric type allows us to use uniﬁed approach for the construction of biorthogonal systems of polynomials. All possible singular curves of the studied system are set. The existence of regular solutions is set by Frobenius-Latysheva method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):710-721

Fundamental solution of the model equation of anomalous diffusion of fractional order

#### Abstract

Fundamental solution of the model equation of anomalous diffusion with Riemann-Liouville operator is constructed. Using the properties of the integral transformation with Wright function in kernel, we give estimates for the fundamental solution. When the considered equation transformes into the diffusion equation of fractional order, constructed fundamental solution goes into the corresponding fundamental solution of the diffusion equation of fractional order. General solution of the model equation of anomalous diffusion of fractional order is constructed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):722-735

An inverse problem for a nonlinear Fredholm integro-differential equation of fourth order with degenerate kernel

#### Abstract

We consider the questions of one value solvability of the inverse problem for a nonlinear partial Fredholm type integro-differential equation of the fourth order with degenerate kernel. The method of degenerate kernel is developed for the case of inverse problem for the considering partial Fredholm type integro-differential equation of the fourth order. After denoting the Fredholm type integro-differential equation is reduced to a system of integral equations. By the aid of differentiating the system of integral equations reduced to the system of differential equations. When a certain imposed condition is fulfilled, the system of differential equations is changed to the system of algebraic equations. For the regular values of spectral parameterthe system of algebraic equations is solved by the Kramer metod. Using the given additional condition the nonlinear Volterra type integral equation of second kind with respect to main unknowing function and the nonlinear Volterra special type integral equation of first kind with respect to restore function are obtained. We use the method of successive approximations combined with the method of compressing maps. Further the restore function is deﬁned. This paper developes the theory of Fredholm integro-differential equations with degenerate kernel.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):736-749

An aeroelastic stability of the circular cylindrical shells containing a flowing fluid

#### Abstract

The paper is concerned with the analysis of the panel flutter of circular cylindrical shells containing an ideal compressible liquid and subjected to the external supersonic gas flow. The aerodynamic pressure is calculated based on the quasi-static aerodynamic theory. The behavior of the liquid is described in the framework of the potential theory. Using the Bubnov- Galerkin method, the corresponding wave equation together with the impermeability condition and specified boundary conditions are transformed into the system of equations. The classical shell theory based on the Kirchhoff-Love hypotheses and the principle of virtual displacements are used as the mathematical framework for the elastic structure dynamic problem. A solution to the problem is searched for by a semi-analytical version of the finite element method and involves the calculation of the complex eigenvalues of the coupled system of equations using the Muller-based iterative algorithm. The reliability of the obtained numerical solution of the aeroelastic and hydroelastic stability problem has been estimated by comparing it with the available theoretical data. For shells with different dimensions and variants of boundary conditions the numerical experiments have been performed to estimate the influence of velocity of the internal liquid flow on the value of static pressure in the unperturbed gas flow, which is taken as a variable parameter. It has been found that a growth of liquid velocity causes a change in the flutter type of stability loss. It has been shown that with increase of linear dimensions of the shell the stabilizing effect of the internal liquid flow extending the boundaries of aeroelastic stability changes to the destabilizing effect. Specific values of geometrical dimensions determining the variation in the character of dynamic behavior of the system depend on the prescribed combination of boundary conditions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):750-767

Mathematical modeling of deformation of reinforced femur during prolonged static loads

#### Abstract

A two-layer mathematical model of a human femur neck reinforced implants of different design for modeling stress-strain state which occurs during a surgical procedure to prevent femur neck fractures by the forced introduction of metallic implants is proposed. Engineered implant designs are provided. Methods and software for geometric modeling of femur embedded with the implants are developed. New boundary value problems to evaluate kinetics in creep conditions of the stress-strain state of reinforced and non-reinforced femoral neck during prolonged static loads corresponding to human foot traffic are formulated. Effective elastic properties of cortical and cancellous bone, power and kinematic boundary value problems. A phenomenological creep model for compact bone tissue is constructed. The technique of identifying the parameters is developed. A check of its adequacy to experimental data is carried out. Based on the finite element method the numerical method for solving the provided boundary value problems at macro level of continuum mechanics is developed. A lot of variative calculations allowed developing recommendations for the rational positioning of the implant in order to minimize stress concentrations. The performed analysis showed that there is a significant relaxation of stresses in the most loaded areas due to creep. Relaxation is more intense in reinforced femoral neck than in the unreinforced. Thus the tension in the most loaded femoral neck area due to creep is reduced by 49 % with respect to the intensity of the initial time of loading for femur which is reinforced by the spoke-spoke-type implant when loading duration is 1 year under natural loads corresponding to human foot traffic. It was found that the time component (long-term fixed load) does not impair the positive effect of reducing the stress concentration due to a femoral neck reinforcement which is a positive fact from the medical practice point of view.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):768-784

Tooling software for development and execution support of scientific computing applications in cluster systems

#### Abstract

Rationale: Many diﬀerent tools exist for development of scientiﬁc computing applications. Most of them are focused on the process of writing software code, but often there is a need for applications that organize the computation process and support team development. The article describes application development speciﬁcs in the ﬁeld of science-oriented computing and highlights individual issues in the development of such software. Classiﬁcation of task management systems: The systems are classiﬁed by means of computing process organization and the layer of hardware abstraction. Templet development tools: The tools for application development considered in the article include parallel programming libraries, a task running and monitoring service and the monitoring subsystem for SSAU cluster. Close interaction between these tools enables eﬀective teamwork for scientiﬁc application development. Applied problems solved by Templet tools: Tooling is used to solve practical issues in the ﬁeld of modeling multi-dimensional dynamic systems behavior. The article demonstrates an approach that splits application development into system-level and applied development layers. Conclusion: The article concludes about the use of design techniques and the beneﬁts provided by software development tools.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(4):785-798