scholarly journals On one approach to the solution of miscellaneous problems of the theory of elasticity

2021 ◽  
Vol 57 (3) ◽  
pp. 263-273
Author(s):  
V. V. Amel’kin ◽  
M. N. Vasilevich ◽  
L. A. Khvostchinskaya
Keyword(s):  
Theory Of Elasticity ◽  
Singular Points ◽  
Cauchy Type ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Piecewise Constant ◽  
Analytical Functions ◽  
Cauchy Type Integrals ◽  
Upper Half Plane ◽  
Product Of Matrices

Herein, a miscellaneous contact problem of the theory of elasticity in the upper half-plane is considered. The boundary is a real semi-axis separated into four parts, on each of which the boundary conditions are set for the real or imaginary part of two desired analytical functions. Using new unknown functions, the problem is reduced to an inhomogeneous Riemann boundary value problem with a piecewise constant 2 × 2 matrix and four singular points. A differential equation of the Fuchs class with four singular points is constructed, the residue matrices of which are found by the logarithm method of the product of matrices. The single solution of the problem is represented in terms of Cauchy-type integrals when the solvability condition is met.

The Riemann boundary value problem in the class of Cauchy type integrals with densities of grand variable exponent Lebesgue spaces

2016 ◽  
Vol 23 (4) ◽  
pp. 551-558 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi ◽  
Vakhtang Paatashvili
Keyword(s):  
Boundary Value Problem ◽  
Variable Exponent ◽  
Boundary Value ◽  
Function Spaces ◽  
Cauchy Type ◽  
Riemann Boundary Value Problem ◽  
Lebesgue Spaces ◽  
Standard Type ◽  
Riemann Boundary ◽  
Variable Exponent Lebesgue Spaces

AbstractThe present paper deals with the Riemann boundary value problem for analytic functions in the framework of the new function spaces introduced by the first two authors, the so-called grand variable exponent Lebesgue spaces which unify two non-standard type function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

scholarly journals ABOUT THE APPROXIMATE SOLUTION OF THE USUAL AND GENERALIZED HILBERT BOUNDARY VALUE PROBLEMS FOR ANALYTICAL FUNCTIONS

2000 ◽  
Vol 5 (1) ◽  
pp. 119-126
Author(s):  
V. R. Kristalinskii
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Cauchy Type ◽  
Analytical Functions ◽  
Method Of Solution ◽  
Cauchy Type Integrals ◽  
Hilbert Boundary Value Problem

In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the formproposed by the author. All methods are implemented with the Mathcad and Maple.

scholarly journals The Riemann boundary value problem related to the time-harmonic Maxwell equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pei Yang ◽  
Liping Wang ◽  
Zuoliang Xu
Keyword(s):  
Boundary Value Problem ◽  
Maxwell Equations ◽  
Boundary Value ◽  
Uniform Boundedness ◽  
Matrix Operator ◽  
Cauchy Type ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Time Harmonic ◽  
Definition Of

AbstractIn this paper, we first give the definition of Teodorescu operator related to the $\mathcal{N}$ N matrix operator and discuss a series of properties of this operator, such as uniform boundedness, Hölder continuity and so on. Then we propose the Riemann boundary value problem related to the $\mathcal{N}$ N matrix operator. Finally, using the intimate relationship of the corresponding Cauchy-type integral between the $\mathcal{N}$ N matrix operator and the time-harmonic Maxwell equations, we investigate the Riemann boundary value problem related to the time-harmonic Maxwell equations and obtain the integral representation of the solution.

Thermal mechanical coupling analysis of two-dimensional decagonal quasicrystals with elastic elliptical inclusion

2021 ◽  
pp. 108128652110387
Author(s):  
Yuan-Yuan Ma ◽  
Xue-Fen Zhao ◽  
Ting Zhai ◽  
Sheng-Hu Ding
Keyword(s):  
Flow Direction ◽  
Elastic Field ◽  
Cauchy Type ◽  
Two Dimensional ◽  
Elliptic Inclusion ◽  
Riemann Boundary Value Problem ◽  
Decagonal Quasicrystal ◽  
Riemann Boundary ◽  
Linear Temperature ◽  
Mechanical Coupling

In this paper, the thermal mechanical coupling problem of an infinite two-dimensional decagonal quasicrystal matrix containing elastic elliptic inclusion is studied under remote uniform loading and linear temperature variation. Combining with the theory of the sectional holomorphic function, conformal transformation, singularity analysis, Cauchy-type integral and Riemann boundary value problem, the analytic relations among the sectional functions are obtained, and the problem is transformed into a basic complex potential function equation. The closed form solutions of the temperature field and thermo-elastic field in the matrix and inclusion are obtained. The solutions demonstrate that the uniform temperature and remote uniform stresses will induce an internal uniform stress field. Numerical examples show the effects of the thermal conductivity coefficient ratio, the heat flow direction angle and the elastic modulus on the interface stresses. The results provide a valuable reference for the design and application of reinforced quasicrystal materials.

scholarly journals Hypersingular integro-differential equations with power factors in coefficients

2019 ◽  
pp. 48-56 ◽  
Author(s):  
Andrei P. Shilin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Arbitrary Order ◽  
Closed Curve ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Solvability Conditions ◽  
Analytical Functions ◽  
Linear Differential

The linear hypersingular integro-differential equation of arbitrary order on a closed curve located on the complex plane is considered. A scheme is proposed to study this equation in the case when its coefficients have some particular structure. This scheme providers for the use of generalized Sokhotsky formulas, the solution of the Riemann boundary value problem and the solution in the class of analytical functions of linear differential equations. According to this scheme, the equations are explicitly solved, the coefficients of which contain power factors, so that along with the Riemann problem the arising differential equations are constructively solved. Solvability conditions, solution formulas, examples are given.

DESIGN METHOD FOR MULTILAYER LININGS OF PARALLEL UNDERSEA TUNNELS CONSTRUCTED WITH SEQUENCE DRIVING

2019 ◽  
Vol 4 (1) ◽  
pp. 195-206
Author(s):  
I.Y. Voronina ◽  
◽  
A.S. Sammal ◽  
N.V. Shelepov ◽  
◽  
...  
Keyword(s):  
Stress State ◽  
Design Method ◽  
Cauchy Type ◽  
Underground Structures ◽  
Calculation Algorithm ◽  
Load Bearing Capacity ◽  
Analytical Design ◽  
Analytical Functions ◽  
Cauchy Type Integrals ◽  
Complete Calculation

The analytical design methodfor multilayer non-circular linings of complexes mutually influencing underwater tunnels with the sequence of their driving is proposed. The method is based on the solution of the corresponding elasticity theory plane problem based on the application of the analytical functions of complex variable apparatus, conform mapping and properties of Cauchy-type integrals. The method is implemented in the form of a complete calculation algorithm and the original computer program. Specific results of design illustrating the effect of the sequence of construction of a complex consisting of three underwater tunnels on the stress state and load - bearing capacity of multilayer underground structures are presented.

The use of Riemann problems in solving a class of transcendental equations

1973 ◽  
Vol 73 (1) ◽  
pp. 111-118 ◽  
Author(s):  
E. E. Burniston ◽  
C. E. Siewert
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Explicit Solutions ◽  
Riemann Boundary Value Problem ◽  
Riemann Problems ◽  
Riemann Boundary ◽  
Canonical Solution ◽  
Transcendental Equations ◽  
Explicit Expressions

AbstractA method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

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