Riemann-Hilbert Boundary Value Problem Research Articles

Factorization on a Riemann Surface in Scattering Theory

Motivated by a problem of scattering theory, the authors solve by quadratures a vector Riemann– Hilbert problem with the matrix coefficient of Chebotarev–Khrapkov type. The problem of matrix factorization reduces to a scalar Riemann–Hilbert boundary-value problem on a twosheeted Riemann surface of genus 3 that is topologically equivalent to a sphere with three handles. The conditions quenching an essential singularity of the solution at infinity lead to the classical Jacobi inversion problem. It is shown that this problem is equivalent to an algebraic equation of degree that coincides with the genus of the Riemann surface. A closedform solution of this nonlinear problem is found for genus 3. A normal matrix of factorization and the canonical matrix are constructed in explicit form. It is proved that the vector Riemann– Hilbert problem possesses zero partial indices and is, therefore, stable. The proposed technique is illustrated by a problem of scattering of sound waves by a perforated sandwich panel.

Atomic diffusion from a material surface into a grain boundary

A model problem of stress–induced atomic diffusion from the surface of a material into a semi–infinite grain boundary orthogonal to the surface is analysed. The governing equation is shown to be a singular integro–differential equation. An algorithm for an exact solution to this problem is proposed. It involves reduction to a functional–difference equation in a strip and then to a Riemann–Hilbert boundary–value problem for an open contour that admits solution by quadratures. Analytical continuation of the solution of the difference equation in the strip is constructed. The continuation is a meromorphic function in the exterior of the strip. Its poles form a countable set and are not periodic. In addition to a closed–form solution of the integro–differential equation, a procedure for finding a rapidly convergent series representation and an asymptotic expansion for large arguments are proposed. Asymptotic analysis and numerical results for the grain–boundary traction, the atomic flux in the boundary and the displacement rate are reported and discussed.

On the analytic inversion of functions, solution of transcendental equations and infinite self-mappings

The solution of seemingly simple transcendental equations is in effect equivalent to the general problem of analytical inversion of functions. Within a powerful and systematic method, based on the solution of an associated Riemann–Hilbert boundary value problem, beautiful explicit results for various inverse functions of physical importance have been found which inevitably take on the guise of integral representations of these functions. In an attempt to reduce one such solution to a standard-function expression which would then be easy to evaluate, we recognize an infinite ladder self-mapping solution. This new perspective, born out of complicated complex analysis, is straightforwardly and uniquely related to the systematic generation of fast converging expansions within the corresponding regions of single-valuedness of the inverse function.

A problem of the bending of a plate for a doubly-connected domain bounded by polygons

The problem of the bending of an isotropic elastic plate, bounded by two convex polygons is considered. It is assumed that the internal boundary of the plate is simply supported and normal bending moments act on each section of the external contour in such a way that the angle of rotation of the middle surface of the plate is a piecewise-constant function. With respect to the complex potentials, which express the bendings of the middle surface (Goursat's formula), the problem is reduced to a Riemann-Hilbert boundary-value problem for a circular ring, the solution of which is constructed in analytic form. Estimates are given of the behaviour of these potentials in the neighbourhood of the corner points.

The sufficient conditions under which the S-matrix orthogonal part is real and even

The sufficient conditions under which the S-matrix orthogonal part U(k) is real and even are proved. These conditions are required for generalization of the projection matrices method in the N×N Riemann–Hilbert boundary value problem, N=2,3,…,N<+∞. The dependence of the analytical properties of the orthogonal matrix U(k) on the S-matrix diagonal part is thoroughly considered. The examples in which the sufficient conditions are not realized and Re U(k)≠0, U(−k)=−U(k) are demonstrated.

A M/M/ queue in a semi-Markovian environment

We consider an M/M/1 queue in a semi-Markovian environment. The environment is modeled by a two-state semi-Markov process with arbitrary sojourn time distributions F 0 ( x ) and F 1 ( x ). When in state i = 0, 1, customers are generated according to a Poisson process with intensity λ i and customers are served according to an exponential distribution with rate μ i . Using the theory of Riemann-Hilbert boundary value problems we compute the z -transform of the queue-length distribution when either F 0 ( x ) or F 1 ( x ) has a rational Laplace-Stieltjes transform and the other may be a general --- possibly heavy-tailed --- distribution. The arrival process can be used to model bursty traffic and/or traffic exhibiting long-range dependence, a situation which is commonly encountered in networking. The closed-form results lend themselves for numerical evaluation of performance measures, in particular the mean queue-length.

The riemann — hilbert problem for elliptic systems of second order equations in infinite domains

There are many problems in mechanics and physics, where the mathematical models are boundary value problems for nonlinear elliptic systems of first and second order equations in unbounded domains. In this paper, we discuss the Riemann—Hilbert boundary value problem for nonlinear elliptic systems of second order equations in multiply connected domains including the point at infinity.

Exact solution of integro-differential equations of diffusion along a grain boundary

We analyse model problems of stress-induced atomic diffusion from a point source or from the surface of a material into an infinite or semi-infinite grain boundary, respectively. The problems are formulated in terms of partial differential equations which involve singular integral operators. The self-similarity of these equations leads to singular integro-differential equations which are solved in closed form by reduction to an exceptional case of the Riemann–Hilbert boundary-value problem of the theory of analytic functions on an open contour. We also give a series representation and a full asymptotic expansion of the solution in the case of large arguments. Numerical results are reported.

The generalized riemann-hilbert boundary value problem for nonlinear and nonhomogeneous complex polyharmonic equation in the sobolev space W n,p (D)

Given is a complex nonlinear and nonhomogeneous polyharmonic equation of order n in a simply-connected bounded domain D in the complex plane. It is proved, initially, the existence of a general solution in W 2 n,p( D). This is effected by first reducing the given equation into a system of integro- differential equations. Later the solvability of a generalized Riemann–Hilbert problem for the equation is established. The 2n boundary conditions imposed on the solution are transformed into 2n equivalent Riemann-Hilbert problems for 2n holomorphic functions. The solution is unique.

The generalized riemann—hilbert boundary value problem for polyharmonic functions in the sobolev spaces W 2n,P (D)

The article discusses the construction of a polyharmonic function Ф of order n defined on a simply-connected and bounded domain D, which satisfies 2n prescribed generalized Riemann–Hilbert boundary conditions on the boundary ∂D. The 2n boundary conditions are transformed into 2n classical Riemann–Hilbert boundary value problems for holomorphic functions G j (z)Q j (z)j= 1,…,n, defined in terms of the holormorphic functions ψ p (z),ϕ p (z),p = 0,1,…,n–1, which define the n-harmonic function Ф. The latter problems are solved, using the standard methods from the literature, and their solutions are used to determine ϕ p , ψ p ) and hence Ф.

The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$

Given is a nonlinear non-homogeneous polyanalytic differential equation of order n in a simply-connected domain D in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in W_{n,p}(D) by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into n classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature.

Boundary value problems for quaternionic monogenic functions on non-smooth sureaces

In this paper, analogous of the Compound Riemann-Hilbert boundary value problems are investigate for quaternionic monogenic functions. The solution (explicitly) of the problem is established over continuous surface, with little smoothness, which bounds a bounded domain of R3. In particular, smoothness property for high-dimensional Cauchy type integral are computed. We also use Zygmund type estimates to adapt existing one-variable complex results to ilustrate the Holder-boundedness of the singular integral operator on 2-dimensional Ahlfors regular surfaces. At the end, uniqueness of solution for the Riemann boundary value problem have already built taking as a base the general Operator Theory.

Linearization and Special Asymptotic Behaviour of the Second Painlevé Equation

Linearization of the initial value problem associated with the special second Painlevé equation is discussed using a different isomonodromic spectral problem than the one used in [1]. Further properties of the monodromy data [2, 3] are detected and these properties are used to reduce the problem to a linear singular integral equation via a Riemann–Hilbert boundary value problem on an imaginary line. A special asymptotic increasing solution of the Painlevé equation is constructed forx→−∞ from the above integral equation. Moreover, failure to extract asymptotics forx→+∞ is also mentioned.

The riemann-hilbert boundary value problem for nonlinear elliptic complex equations of nth order

This paper deals with the Riemann-Hilbert boundary value problem for nonlinear elliptic complex equations of n order in simply and multiply connected domains. Firstly we give the formulation of the Riemann-Hilbert problem and its well posedness and then we give the representation of solutions for the modified boundary value problem and prove its solvability, and finally derive solvability conditions of the original Kiemaiin-Hilbert problem. In the following, we divide into two cases of complex equations of 2nth order and 2n + 1th order to discuss.

A method for solving Riemann–Hilbert boundary value problems in nonreciprocal wave propagation

A new method for solving periodic boundary value problems in nonreciprocal wave propagation is described. A characteristic function χ0 and its complementary function χ̃0 are introduced after the particular problem has been reduced to a Riemann–Hilbert boundary value problem based on complex variable theory. The Taylor coefficients νn and ν̃n of χ0 and χ̃0,respectively, compose a system of linear equations to deduce a characteristic equation. A numerical example is presented for magnetostatic surface waves propagating in a ferrite film on which metallic stripes are deposited periodically.

On nonlinear riemann-hilbert boundary value problems for second order elliptic systems in the plane

In this paper the well-Known Newton's method approach, whose origin is [18] and is based on a-priori estimates, is used to solve a nonlinear Riemann-Hilbert boundary value problem for second order complex elliptic partial differential equations in the plane . An a-priori estimate is established for solutions to the related linear equations.

A Canonical Diffraction Problem With Two Media

A Wiener–Hopf equation in L2 being equivalent [5] to a boundary value problem (of the first kind) for a wave-scattering Sommerfeld half-plane Σ=ℝ+×{0} which faces two different media Ω-: x2<0, Ω+: x2>0, as a special configuration in [3], is solved by canonical Weiner–Hopf factorization of its L2-regular scalar symbol γo=γo- γo+. The factors are calculated by solving a Riemann–Hilbert boundary value problem on the semi-infinite branch cuts of tj(ξ):=(ξ2−k2j)1/2, kj∈ℂ++ for j=1,2: taken parallel to the imaginary axis. The procedure following this idea is known as the Wiener–Hopf–Hilbert(–Hurd) method [2] and requires the evaluation of elliptic-type integrals. Formula (3.7) seems not to be contained in tables of integrals.

On riemann–hilbert boundary problem in several complex variables

The Riemann-Hilbert boundary value problem for a holomorphic function in the polydomain and in the unit ball of is considered. In contrast to well known results in the case of a single complex variable for polydomain the cokernel of the problem turns to have always infinite dimension, while its kernel has the finite dimension. One of the features of the problem for the unit ball is that its boundary coefficient doesn't produce an index.

Application of the coupling problem to the study of the stressed state of a multiply connected isotropic half-plane

We consider a problem of the theory of elasticity for an isotropic half-plane with holes. By continuing analytically across an unstressed portion of the boundary and solving the resulting Riemann-Hilbert boundary-value problem we find a general representation of the complex potential. The unknown functions occurring in the potential and the coefficients of a polynomial are determined from the boundary conditions on the edges of holes and the equilibrium conditions under punches. As special cases we consider prescribed stresses and displacements on portions of the boundary and the action of punches with and without friction. Bibliography: 2 titles.

Some boundary value problems for a beltrami equation

Many properties of solutions to Beltrami equations were established by I.N. Vekua [11], B. Bojarski [3], A. Dzhuraev [5], R.P. Gilbert [6], H. Begehr [1] and others. Based on Cauchy-type integrals for example Riemann-Hilbert as well as Riemann boundary value problems were studied. Because the kernel of the Cauchy-type integral in general is not known, only existence and uniqueness results were given. In this paper some special cases for the Beltrami equation are considered. A corresponding Cauchy formula and theorem are given where the Cauchy type integral can be represented explicitly as in the analytic case (q= 0). Riemann-Hilbert boundary value problems are solved explicitly for the half-plane and for an ellipse when q is constant. Besides the Riemann-Schwarz reflection principle is established. Using a Wiener-Hopf integral equation and convolution-type dual integral equations the solutions of some nonlocal problems are given in explicit form. For ellipticity of (1) the given function q must satisfy |q|≠1. Without loss of generality is assumed because if |q|q0>1 one can consider equation (1) for .