# Vol 23, No 2 (2017)

**Year:**2017**Articles:**6**URL:**https://journals.ssau.ru/est/issue/view/248

## Full Issue

## Статьи

### A PROBLEM ON VIBRATION OF A BAR WITH UNKNOWN BOUNDARY CONDITION ON A PART OF THE BOUNDARY

#### Abstract

In this paper, we study an inverse problem for hyperbolic equation. This problem arises when we consider vibration of a nonhomogeneous bar if one endpoint is fixed by spring but behavior of the other is unknown and is the subject to find. Overdetermination is given in the form of integral with respect to spacial variable. Unique solvability of this problem is proved under some conditions on data. The proof is based on a priori estimates in Sobolev space.

**Vestnik of Samara University. Natural Science Series**. 2017;23(2):7-14

### MACKAY FUNCTIONS AND EXACT CUTTING IN SPACES OF MODULAR FORMS

#### Abstract

#### In the article we consider structure problems in the theory of modular forms. The phenomenon of the exact cutting for the spaces Sk(Γ0(N), χ), where χ is a quadratic character with the condition χ(−1) = = (−1)k . We prove that for the levels N ̸= 3, 17, 19 the cutting function is a multiplicative eta–product of an integral weight. In the article we give the table of the cutting functions. We prove that the space of an cutting function is one–dimensional. Dimensions of the spaces are calculated by the Cohen-Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

**Vestnik of Samara University. Natural Science Series**. 2017;23(2):15-25

### ABOUT ONE NONLOCAL TASK FOR THE EQUATION OF THE FOURTH ORDER WITH THE DOMINATING MIXED DERIVATIVE

#### Abstract

In the article the nonlocal task for the model equation from the dominating mixed derivative of the fourth order is considered. The unique solubility of an objective in which two of four conditions are nonlocal is proved and represent integrals both on a space variable, and on time variable. For the proof the new method based on equivalence of an objective and set of equations of the second order is offered.

**Vestnik of Samara University. Natural Science Series**. 2017;23(2):26-31

### ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE

#### Abstract

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

**Vestnik of Samara University. Natural Science Series**. 2017;23(2):32-43

### QUAZILINEARIZATION METHOD FOR THE SOLUTION TO THE PROBLEM OF PLATE WITH THE CENTRAL CIRCULAR HOLE UNDER CREEP REGIME

#### Abstract

The approximation solution of the problem for an infinite plate with the circular hole under creep regime is obtained by the quazilinearization method. Four approximations of the solution of the nonlinear problems are found. It is shown that with increasing of the number of approximations the solution converges to the limit numerical solution. It is worth to note that the tangential stress reaches its maximum value not at the circular hole but at the internal point of the plate. It is also shown that quazilinearization method is an effective method for nonlinear problems.

**Vestnik of Samara University. Natural Science Series**. 2017;23(2):44-50

### THE MAPPINGS OF VAN DER POL — DYUFFING GENERATOR IN DISCRETE TIME

#### Abstract

In the work transition to discrete time in the equation of movement of van der Pol – Dyuffing generator is described. The transition purpose—to create mappings of the generator as subjects of the theory of nonlinear oscillations (nonlinear dynamics) in discrete time. The method of sampling is based on the use of counting of the pulse characteristic of an oscillatory contour as the sampling series for a signal in a self-oscillating ring ”active nonlinearity – the resonator – feedback”. The choice of the consecutive scheme of excitement of a contour allows to receive the iterated displays in the form of recurrent formulas. Two equivalent forms of discrete displays of the generator of van der Pol – Dyuffing—complex and valid are presented. In approximation of method of slow-changing amplitudes it is confirmed that the created discrete mappings have dynamic properties of an analog prototype. Also within the numerical experiment it is shown that in case of the high power of generation the effect of changing of frequencies of harmonicas of the generated discrete signal significantly influence dynamics of the self-oscillators. In particular, in the discrete generator of van der Pol – Dyuffing the chaotic self-oscillations are observed.

**Vestnik of Samara University. Natural Science Series**. 2017;23(2):51-59