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

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.

A solution of the hypersingular integro-differential equation with determinants of the Vronsky type

In this paper, we consider a new hypersingular integro-differential equation of arbitrary order on a closed curve located in the complex plane. The integrals in the equation are understood in the sense of the finite Hadamard part. The equation refers to linear integro-differential equations with variable coefficients of a particular form. A characteristic feature of the equation is its representation with the help of determinants close to the Vronsky ones. The method of analytical continuation, properties of determinants, and generalized Sokhotsky formulas are used for the study. The equation reduces to the Riemann boundary value problem of a jump in a certain class of functions. If the Riemann boundary problem turns out to be solvable, then one should solve linear inhomogeneous differential equations in the class of analytic functions in the domains of the complex plane. The analysis of the obtained solutions in an infinitely distant point is not evident. The study has a complete look. The conditions for the solvability of the original equation are explicitly written out. When they are fulfilled, the solution is explicitly written, to which an example is given.

A note on the solvability of homogeneous Riemann boundary problem with infinity index

In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.

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

In 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

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.