Riemann Boundary Value Problem Research Articles

The Solvability and Explicit Solutions of Singular Integral–Differential Equations with Reflection

This article deals with a classes of singular integral–differential equations with convolution kernel and reflection. By means of the theory of boundary value problems of analytic functions and the theory of Fourier analysis, such equations can be transformed into Riemann boundary value problems (i.e., Riemann–Hilbert problems) with nodes and reflection. For such problems, we propose a novel method different from classical one, by which the explicit solutions and the conditions of solvability are obtained.

A higher dimensional Marcinkiewicz exponent and the Riemann boundary value problems for polymonogenic functions on fractals domains

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space Rn+1,n≥2 for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.

Solvability of Riemann Boundary Value Problems and Applications to Approximative Properties of Perturbed Exponential System in Orlicz Spaces

Solvability of Riemann Boundary Value Problems and Applications to Approximative Properties of Perturbed Exponential System in Orlicz Spaces

Riemann boundary value problems on the Archimedean spiral

This paper studies the Riemann boundary value problems on the Archimedean spiral. We characterize the functions which are integrable on the Archimedean spiral. We also study the asymptotic behaviors of Cauchy-type integral and Cauchy principal value integral on Archimedean spiral at the origin and infinity. At the end, we discuss the Riemann boundary value problems for sectionally holomorphic functions with the Archimedean spiral as their jump curve and obtain the singular integral representations of the solutions if solvable.

Hypersingular Integro-Differential Equation Containing Polynomials and Their Derivatives in Coefficients

A new linear integro-differential equation is considered on a closed curve located on the complex plane. The integrals in the equation are understood in the sense of a finite part, according to Hadamard. The order of the equation can be higher than one. The coefficients of the equation have a special structure. A characteristic of the coefficients is that they contain two arbitrary polynomials and their derivatives. Solvability conditions are explicitly stated. Whenever they are satisfied, an exact analytical solution is given. Generalized Sokhotsky formulas, the theory of Riemann boundary value problems, methods for solving linear differential equations, and the properties of analytic functions of a complex variable are used. An example is given.

The Riemann problem in a half-plane for generalized analytic functions with a supersingular point on the contour of the boundary condition

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on the real axis for a generalized equation Cauchy–Riemann with supersingular coefficients. To solve the problem, it was necessary to derive a structural formula for the general solution of this equation and to investigate the solvability of the Riemann boundary value problem of the theory of analytic functions with an infinite index due to the power-order vorticity point.

Inverse scattering problem for a third-order operator with local potential

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem.

Invertibility and Spectrum of the Riemann Boundary Value Problem Operator in a Countably Normed Space of Smooth Functions on a Circle

Invertibility and Spectrum of the Riemann Boundary Value Problem Operator in a Countably Normed Space of Smooth Functions on a Circle

Riemann boundary value problem on a spiral

ABSTRACT The Riemann boundary value problem is a classic problem of complex analysis. It has numerous applications in hydrodynamics and aerodynamics. It is well-studied in the case of rectifiable curves, but results for non-rectifiable curves are relatively new. Here we apply a new approach to this problem on such curves to spirals, including those with infinite length.

A Class of Inverse Boundary Value Problems for (λ, 1) Bi-Analytic Functions

In this paper, a class of inverse boundary value problems for (λ, 1) bi-analytic functions is given. Using the method of Riemann boundary value problem for analytic functions, the conditions of solvability and the expression of the solutions for the inverse problems are obtained.

Inverse scattering problem for a third-order operator with non-local potential

Direct and inverse scattering problem for an operator with non-local potential is solved in the paper. The method is based on the Riemann boundary value problem on a bundle of three straight lines. Description of scattering problem data is given.

Riemann Boundary Value Problems on the Curve of Parabola

Riemann Boundary Value Problems on the Curve of Parabola

A hypersingular integro-differential equation with linear functions in coefficients

In this paper, we investigated a new linear integro-differential equation of arbitrary order given on the closed curve located on a complex plane. The coefficients of the equation are variables and have a special form. The characteristic feature is the presence of linear functions in the coefficients. The equation is reduced to the consecutive solution of a Riemann boundary value problem on an initial curve and two linear differential equations. Differential equations are solved for analytic functions in areas into which the initial curve separates a complex plane. The corresponding fundamental systems of solutions are found, after that the arbitrary-constant variation method is applied. To achieve the analyticity of the obtained solutions the restrictions are imposed. All the arising conditions of resolvability of the input equation are written down explicitly, and if they are carried out then the solution is written in an explicit form. We represent the example demonstrating the existence of the cases when all conditions of resolvability are satisfied.

On solvability of homogeneous Riemann boundary value problems in Hardy-Orlicz classes

This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary value problems with piecewise continuous coefficients are considered in these classes. New characteristic of Orlicz space is defined which depends on whether the power function belongs to this space or not. Relationship between this characteristic and Boyd indices of Orlicz space is established. The concept of canonical solution of homogeneous problem is defined, which depends on the argument of the coefficient. In terms of the above characteristic, a condition on the jumps of the argument is found which is sufficient for solvability of these problems, and, in case of solvability, a general solution is constructed. It is established the basicity of the parts of exponential system in Hardy-Orlicz classes.

Hypersingular integro-differential equation with recurrent relations in coefficients

A new hypersingular integro-differential equation is considered on a closed curve located on the complex plane. The equation refers to linear equations with variable coefficients of a special kind. A characteristic feature is the presence of constant multipliers in the coefficients, given by some recurrent relations. The equation is first reduced to solving the Riemann boundary value problem on the original curve. A class of functions is established for solving the Riemann problem, after which this problem is solved. Next, it is necessary to solve two linear differential equations of arbitrary order for analytical functions in two different regions of the complex plane. The corresponding fundamental systems of solutions are found, after which the method of variation of arbitrary constants is used for the solution. Restrictions are imposed on the obtained solutions of differential equations in order to achieve their analyticity. As a result, all the resulting solvability conditions of the original equation are written explicitly. The solution of the original equation after solving the differential equations can be written explicitly. Solved the example.

On a Riemann boundary value problem in the space of p-summable functions with infinite index

Abstract. The paper considers the Riemann boundary value problem in the half-plane in the space Lp(ρ), where weight function ρ(x) has infinite number of zeros. A necessary and sufficient condition is obtained for the normal solvability and Noetherianness of the considered problem. If the problem is solvable, solutions are represented in an explicit form.

On a Riemann boundary value problem for weighted spaces in the half-plane

The paper considers the Riemann boundary value problem in the half-plane in the class of functions that are $C(\rho)$-continuous with respect to the weight $\rho(x)$, when the weight function has infinite number of zeros. Necessary and sufficient conditions for solvability of the problem are established. If the problem is solvable, solutions are represented in an explicit form.

Closed-form solutions for several classes of singular integral equations with convolution and Cauchy operator

The main goal of this paper is to study the existence of solutions for several new classes of singular convolution integral equations containing the Cauchy operator in the normal type case. To investigate the solutions of such equations, we establish the Noethericity theory of solvability. By means of the properties of complex Fourier transforms, we transform these equations into Riemann boundary value problems with nodes. The analytic solutions and conditions of solvability are obtained via using Riemann–Hilbert approach. Moreover, we also discuss the asymptotic property of solutions near nodes. Thus, this paper generalizes the theories of complex analysis, functional analysis, integral equations and the classical Riemann boundary value problems.

Riemann boundary value problem on non-rectifiable curves with torsion points

The paper is dealing with the Riemann boundary value problem under assumption that the boundary and the boundary data under consideration have singularities at the same points. The singularities of the boundary are the points of strong torsion, and the boundary data have at these points discontinuities of various kinds.

Periodic Riemann boundary value problem

In this paper, we discuss the Riemann boundary value problems for the analytic functions with period in the situation where the solution can be unbounded at . When the jump curve is a closed contour, by using the tangent transformation , we transform the singularities on the z plane to the singularities on the ζ plane. When the jump curve is the real axis, by using the transformation , we transform the singularities on the z plane to the singularities 0 and ∞ on the ω plane. Thus, the original periodic Riemann boundary value problems are reduced to the classical Riemann boundary value problems whose solutions and the corresponding solvability conditions are well known.