Riemann-Hilbert Boundary Value Problem Research Articles

New inversion, convolution and Titchmarsh's theorems for the half-Hilbert transform

While exploiting the generalized Parseval equality for the Mellin transform, we derive the reciprocal inverse operator in the weighted L2-space related to the Hilbert transform on the nonnegative half-axis. Moreover, employing the convolution method, which is based on the Mellin–Barnes integrals, we prove the corresponding convolution and Titchmarsh's theorems for the half-Hilbert transform. Some applications to the solvability of a new class of singular integral equations are demonstrated. Our technique does not require the use of methods of the Riemann–Hilbert boundary value problems for analytic functions. The same approach is applied recently to invert the half-Hartley transform and to establish its convolution theorem.

A Riemann-Hilbert boundary value problem in a bounded sector

In this article a Riemann-Hilbert boundary value problem in a bounded sector domain with angle is considered. A plane parqueting-reflection technique is used. Boundary behaviours of resulting integral operators are discussed in detail. Then we study the Riemann-Hilbert boundary value problem, with an arbitrary index, for both homogeneous and inhomogeneous Cauchy–Riemann equations. Expressions of solutions and the condition of solvability are explicitly obtained.

Solution of a singular homogeneous Hilbert boundary-value problem for analytic function in multiply connected circular domain

We offer a new approach for solving the homogeneous Riemann-Hilbert boundary-value problem for analytic function in multiply connected circular domains. The approach is based on determination of analytic function in terms of known boundary values of its argument in a special case.

Solvability of the Riemann–Hilbert boundary value problem with a two-side curling at infinity of order less than 1

We consider the Hilbert boundary value problem for the upper half-plane in the situation where its coefficients have countably many discontinuities of jump type and two-side curling at infinity. We obtain the general solution and describe completely its solvability in a special class of functions for the case where the index of the problem has power singularity of order lesser than 1.

Riemann-hilbert characterization for main bessel polynomials with varying large negative parameters

In this article, we establish the Bessel polynomials with varying large negative parameters and discuss their orthogonality based on the generalized Bessel polynomials. By using the Riemann-Hilbert boundary value problem on the positive real axis, we get the Riemann-Hilbert characterization of the main Bessel polynomials with varying large negative parameters.

A New Well-Posed Version of Riemann–Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Domains

In this article, we first propose the Riemann–Hilbert problem for uniformly elliptic complex equations of first order in multiply connected domains and its new well-posed-ness with continuous solutions. Then we obtain a priori estimates of solutions of the modified Riemann–Hilbert problem, and derive the solvability results of the modified and original Riemann–Hilbert problems. Finally we shall verify the equivalence of two kinds of well-posed-ness for the Riemann–Hilbert boundary value problem.

DIFFRACTION FROM A GRATING ON A CHIRAL MEDIUM: APPLICATION OF ANALYTICAL REGULARIZATION METHOD

Theoretical results on the electromagnetic wave difiraction from a periodic strip grating placed on a chiral medium are obtained. Analytical Regularization Method based on the solution to the vector Riemann-Hilbert boundary value problem was used to get robust numerical results in the resonant domain, where direct solution methods typically fail. It was shown that in the case of normal incidence of linearly polarized wave the cross-polarized fleld appears in the re∞ected fleld. For elliptically polarized incident wave the difiraction character essentially depends on the polarization direction of the incident wave. These difiraction peculiarities are more pronounced in the resonant domain. In∞uence of the dichroism caused by chiral medium losses is thoroughly studied. The combination of a chiral medium and a grating can be efiectively used for a frequency and polarization selection and for a mode conversion.

Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations

Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.

The discontinuous Riemann-Hilbert problem for elliptic complex equations of first order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann-Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

On some constructive methods for the matrix Riemann–Hilbert boundary-value problem

In this paper we consider the relations between the Riemann–Hilbert monodromy problem and the matrix Riemann–Hilbert boundary-value problem with piecewise continuous coefficient and construct the so-called canonical matrix for the boundary-value problem for a piecewise continuous matrix-function. The formula for the calculation of the index is also obtained.

К решению неоднородной краевой задачи Гильберта для аналитической функции в многосвязной круговой области в особом случае

К решению неоднородной краевой задачи Гильберта для аналитической функции в многосвязной круговой области в особом случае

Approximate Method of Riemann-Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Unbounded Domains

In this article, we discuss the approximate method of solving the Riemann-Hilbert boundary value problem for nonlinear uniformly elliptic complex equation of first order (0.1) with the boundary conditions (0.2) in a multiply connected unbounded domain D, the above boundary value problem will be called Problem A. If the complex Equation (0.1) satisfies the conditions similar to Condition C of (1.1), and the boundary condition (0.2) satisfies the conditions similar to (1.5), then we can obtain approximate solutions of the boundary value problems (0.1) and (0.2). Moreover the error estimates of approximate solutions for the boundary value problem is also given. The boundary value problem possesses many applications in mechanics and physics etc., for instance from (5.114) and (5.115), Chapter VI, [1], we see that Problem A of (0.1) possesses the important application to the shell and elasticity.

Dislocation Distribution Functions of the Edges of Mode I Crack under Several Boundary Conditions

Dislocation distribution functions of mode I dynamic crack subjected to two loads were studied by the methods of the theory of complex variable functions. By this way, the problems researched can be translated into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation. The analytical solutions attained relate to the crack propagation velocity and time, but the solutions have nothing to the other parameters. In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were gained, and variation rules of dislocation distribution functions were depicted.

A homogeneous Hilbert problem with discontinuous coefficients and two-side curling at infinity point of order 1/2 ≤ ρ < 1

We study a homogeneous Riemann—Hilbert boundary-value problem in the upper half of the complex plane with a countable set of coefficient discontinuities and two-side curling at infinity. We obtain a general solution in the case when the problem index has a power singularity of order ρ, 1/2 ≤ ρ < 1, and study the solvability conditions.

Factorizations, Riemann-Hilbert problems and the corona theorem

The solvability of the Riemann–Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener–Hopf-type factorization with bounded outer factors, but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting, when the factorization multiples belong to the algebra generated by the functions eλ(x ): =e iλx , λ ∈ R. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2 × 2

Riemann-Hilbert Problems for Multiply Connected Domains and Circular Slit Maps

A conformal mapping of a multiply connected circular domain onto a complex plane with circular slits is obtained. No restriction on the location of the boundary circles is assumed. The mapping is derived in terms of the uniformly convergent Poincare series by solution to a Riemann-Hilbert boundary value problem.

Dislocation Distribution Function of the Mode III Dynamic Crack Subjected to Moving Unit Step Load from a Point

Dislocation distribution functions of the edges of mode III propagation crack subjected to Moving unit step load from a point was studied by the methods of the theory of complex variable functions.By the methods, the problems researched can be facilely transformed into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements, dynamic stress intensity factor and dislocation distribution function were obtained by the methods of the theory of self-similar functions.In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were attained.

Dislocation Distribution Model of Mode I Dynamic Crack

Dislocation distribution functions of mode I dynamic crack subjected to two loads were studied by the methods of the theory of complex variable functions. By this way, the problems researched can be translated into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation. The analytical solutions attained relate to the crack propagation velocity and time, but the solutions have nothing to the other parameters. In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were gained, and variation rules of dislocation distribution functions were depicted.

On the Waterbag Continuum

The aim of this paper is to study the existence of a classical solution for the waterbag model with a continuum of waterbags, which can been viewed as an infinite dimensional system of first-order conservation laws. The waterbag model, which constitutes a special class of exact weak solution of the Vlasov equation, is well known in plasma physics, and its applications in gyrokinetic theory and laser–plasma interaction are very promising. The proof of the existence of a continuum of regular waterbags relies on a generalized definition of hyperbolicity for an integrodifferential hyperbolic system of equations, some results in singular integral operators theory and harmonic analysis, Riemann–Hilbert boundary value problems and energy estimates.

The problem of an interface crack with a rigid patch plate on part of its edge

A mixed boundary-value problem is solved for a piecewise-homogeneous elastic body with a rectilinear semi-infinite crack on the line where the materials are joined. A rigid patch plate (a reinforcing plate) of specified shape is attached to the upper edge of the crack on a finite interval adjacent to the crack tip. The edges of the crack are loaded with specified stresses. The body is stretched at infinity by a specified longitudinal stress. External forces with a given principal vector and moment act on the patch plate. The problem reduces to a Riemann-Hilbert boundary-value matrix problem with a piecewise-constant coefficient, the solution of which is explicitly constructed using a Gaussian hypergeometric function. The angle of rotation of the patch plate and the complex potentials describing the stress state of the body are found and the stress state of the body close to the ends of the patch plate, one of which is also simultaneously the crack tip, is investigated. Numerical examples are presented that illustrate the effect of the initial force parameters, the length of the patch plate and other parameters of the body on the angle of rotation of the patch plate and the stress state of the body.