# Vol 26, No 1 (2020)

**Year:**2020**Articles:**6**URL:**https://journals.ssau.ru/est/issue/view/437

## Articles

### ON THE NOETHER THEORY OF TWO-DIMENSIONAL SINGULAR OPERATORS AND APPLICATIONS TO BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF FOURTH-ORDER ELLIPTIC EQUATIONS

#### Abstract

It is known that the general theory of multidimensional singular integral operators over the entire space Em was constructed by S. G. Mikhlin. It is shown that in the two-dimensional case, if the operator symbol does not turn into zero, then the Fredholm theory holds. As for operators over a bounded domain, in this case the boundary of the domain significantly affects the solvability of such operator equations. In this paper we consider two-dimensional singular operators with continuous coefficients over a bounded domain. Such operators are widely used in many problems of the theory of partial differential equations. In this regard, it would be interesting to find criteria of Noetherity of such operators as explicit conditions for its coefficients. Depending on the 2m+ 1 connected components, necessary and sufficient conditions of Noetherity for such operators are obtained and a formula for the evaluation of the index is given. The results are applied to the Dirichlet problem for general fourth-order elliptic systems.

**Vestnik of Samara University. Natural Science Series**. 2020;26(1):7-13

### ON THE SOLUTION OF SOME HIGHER-ORDER INTEGRO-DIFFERENTIAL EQUATIONS OF SPECIAL FORM

#### Abstract

The article is devoted to the solution of boundary value problems for higher-order linear integro-differential equations of Fredholm type with differential and integral operators encompassing powers of an ideal bijective linear differential operator whose inverse is known explicitly. The conditions for existence and uniqueness of solutions are derived and the solutions are delivered in closed form. The approach is based on the view that an integro-differential operator is a perturbed differential operator. The results obtained are of both theoretical and practical importance. The method is elucidated by solving two illustrative problems.

**Vestnik of Samara University. Natural Science Series**. 2020;26(1):14-22

### SEMIGROUPS OF BINARY OPERATIONS AND MAGMA-BASED CRYPTOGRAPHY

#### Abstract

In this article, algebras of binary operations as a special case of finitary homogeneous relations algebras are investigated. The tools of our study are based on unary and associative binary operations acting on the set of ternary relations. These operations are generated by the converse operation and the left-composition of binary relations. Using these tools, we are going to define special kinds of ternary relations that correspond to functions, injections, right- and left-total binary relations. Then we obtain criteria for these properties in terms of ordered semigroups. Note, that there is an embedding of the semigroup of quasigroups operations in the semigroup of magmas operation and further in the semigroup of ternary relations. This is similar to embedding the semigroup of bijections in the semigroup of functions and then in the semigroup of binary relations. Taking a binary operation as the generator of a cyclic semigroup, we can apply an exponential squaring method for the fast computation of its positive integer powers. Given that this is the main method of public key cryptography, we are adapting the Diffie-Hellman-Merkle key exchange algorithm for magmas as a result.

**Vestnik of Samara University. Natural Science Series**. 2020;26(1):23-51

### PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 4. THE CASE OF EXACT TRAJECTORIAL MEASUREMENTS

#### Abstract

Proposed work is the fourth in the cycle, therefore, the diagnostic problem is formulated for the case of exact trajectorial measurements, the diagnostic theorem is stated and proved, and two diagnostic algorithms that follow from this theorem are presented. Techniques for an a priori counting of constants, which should be stored in a program for the computer-aided diagnostics whenever the first diagnostic algorithm is used, and other algorithmic parameters are considered. If the second algorithm is applied, the constants should not be stored; this algorithm is based on the search for the minimum value of the diagnostic functional among the values of this functional that were obtained in the process of diagnostics for the a priori chosen set of reference malfunctions. Various extensions of the diagnostic theorem are considered, namely, the problem of whether the diagnostic algorithms thus obtained are applicable when the dimension of the diagnostic vector being used is lower than that of the state vector or when the uninterrupted express-diagnostics with no checking surface is carried out, the problem of selecting the “minimum” diagnostic time, the diagnostics of malfunctions occurring in the neighborhoods of reference non-degenerate malfunctions and not envisaged in the a priori list. We consider other functionals solving the diagnostic problem. Finally, we state the extended diagnostic problem that is solved by using the proposed algorithms.

**Vestnik of Samara University. Natural Science Series**. 2020;26(1):52-68

### EXTRACTION OF FRACTURE MECHANICS PARAMETERS FROM FEM ANALYSIS: ALGORITHMS AND PROCEDURES

#### Abstract

The paper describes new algorithms and procedures proposed for determining fracture mechanics parameters from finite element analysis using the over deterministic method. The multi-parameter crack tip stress field description is used. The algorithms and procedures based on multi-parameter stress field representations in series form are shown to be a powerful tool for reliable and accurate parameter determination. The technique is aimed at the determination of coefficients of the Williams series expansion from finite element analysis and is based on the over deterministic approach. The methodology is illustrated and applied to several cases of cracked specimens. Examples are presented for crack-tip fields recorded using digital photoelasticity. The results of finite element analysis are compared with the digital photoelasticity experiments. The results are in good agreement. The principal stresses obtained from finite element method are in good agreement with the isochromatic fringe patterns obtained by the photoelasticity method.

Explanation has been made for giving guidance to a user on how best to approach implementation of the method from a practical standpoint.

**Vestnik of Samara University. Natural Science Series**. 2020;26(1):69-77

### REFINED ANALYSIS OF STEADY STATE CREEP OF A ROTATING DISK AND A PLATE WITH A CENTRAL HOLE UNDER UNIFORM TENSILE LOAD BY THE QUAZILINEARIZATION METHOD

#### Abstract

Refined analysis of the steady state creep of a rotating disk and a plate with a central hole under uniform tensile load by the quasilinearization method is presented. It is shown that the high values of the creep exponent in power law constitutive equations require more iterations in the framework of the quasilinearization method in each problem. 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 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**. 2020;26(1):78-94