Abstract
In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.
Highlights
Analytical and generalized analytical functions are widely used in modeling various physical and mechanical processes
The determination of the hydrodynamic pressure on the side surfaces of the shell in the presence of a surface load is reduced to the Riemann-Hilbert problem of the theory of generalized analytical functions [3], p. 491
The boundary value problems for generalized analytical functions are the apparatus for solving mechanical problems, and the methods developed for the boundary value problems of the theory of generalized analytical functions can be used to solve many nonlinear problems of the general bending problem
Summary
Analytical and generalized analytical functions are widely used in modeling various physical and mechanical processes (for example [1, 2]). The theory of generalized analytic functions has deep applications to mechanical problems of infinitesimal surface bends [3] and problems of stress state of membrane shell theory [3, 4]. We can interpret any infinitesimal bending of the surface as a certain state of stress equilibrium of the shell This state of the shell can be described by solving a homogeneous equation. We reduce this equation to the Hilbert boundary value problem for analytic functions. The determination of the hydrodynamic pressure on the side surfaces of the shell in the presence of a surface load is reduced to the Riemann-Hilbert problem of the theory of generalized analytical functions [3], p. The boundary value problems for generalized analytical functions are the apparatus for solving mechanical problems, and the methods developed for the boundary value problems of the theory of generalized analytical functions can be used to solve many nonlinear problems of the general bending problem
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