Hilbert Boundary Value Problem with an Unknown Function on Arbitrary Infinite Straight Line

ON HILBERT BOUNDARY VALUE PROBLEMS WITH RADICAL

In this paper, a class of elliptic systems of four 1st order differential equations of the orthogonal type in R3 is considered. For such systems we study the issue of regularizability of the Riemann – Hilbert boundary value problem in an arbitrary limited simply-connected region with a smooth boundary in R3. Using the coefficients of the elliptic system and the matrix of the boundary operator, a special vector field is constructed, and its not entering the tangent plane in any point of the boundary provides the Lopatinski condition of the regularizability of the boundary value problem. The obtained condition permits to prove that the set of regularizable Riemann – Hilbert boundary value problems for the considered class of systems has two components of homotopic connectedness, and the index of an arbitrary regularizable problem equals to minus one.

The Hilbert boundary value problem corresponding to a convolution equation on a finite interval, with kernel belonging to a class singled out earlier by the author, is reduced to a system of integral equations. The solvability of this system in appropriate weighted spaces is studied and an algorithm for constructing a canonical matrix of solutions of the Hilbert problem from certain solutions of the system. Estimates of partial indices are given. Bibliography: 15 titles.

In this work, we consider an adaptive two-user random access scheme with multipacket reception (MPR) ca- pabilities. A user adapts its transmission characteristics based on the status of the other user. The users have external random traffic stored in their queues. We derive the stable throughput region of the system and the convexity conditions of this region. We derive analytically the queueing delay by formulating and solving a Riemann-Hilbert boundary value problem. Finally, we evaluate numerically the presented theoretical results.

Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12]. In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.

Fracture mechanical assessment of interface cracks with contact zones in piezoelectric bimaterials under thermoelectromechanical loadings II. Electrically impermeable interface cracks

We review here some recent results concerning the Cauchy problem for various incompressible models of Fluid Mechanics. Let us first recall the basic models: in order to avoid technicalities with boundary conditions, we shall consider the simple case of periodic boundary conditions. In other words, all unknown functions will be periodic on R 2 or on R 3 with a period T i in each variable x i (i= 1,2 or i = 1, 2, 3). Next, if u = u (x,t) denote the vectorfield corresponding to the velocity field, the incompressibility condition reads $$div u = 0.$$ The Euler Equation corresponds to the additional equation $$\frac{{\partial u}}{{\partial t}} + (u \cdot \nabla )u + \nabla p = 0$$ Where p is thr pressure field, another unknown (scalar) function that can be thought as a Lagrange multiplier corresponding to the constraint(1).

An interface crack between two semi-infinite piezoelectric/piezomagnetic media under out-of-plane mechanical load and in-plane electric and magnetic fields parallel to the crack faces is examined. A portion of the faces is electrically conducting and kept at a uniform magnetic potential, while the remaining portion is electrically and magnetically permeable. The coupled fields are represented by functions analytic in the plane outside the crack. With these representations, the mixed crack-face conditions lead to a combined Dirichlet–Riemann and Hilbert boundary-value problem, which is solved in closed form for arbitrary conductive versus permeable segment lengths. The solution yields explicit expressions for stresses, electric and magnetic fields, and the crack-face sliding (displacement jump). The singular behavior at both crack tips and at the transition between conducting and permeable zones is characterized, and intensity factors are defined accordingly. Parametric results illustrate how applied electric and magnetic fields modulate the fracture driving force; in particular, suitable magnetic loading can markedly reduce the mechanical stress intensity at the permeable tip. The formulas supply benchmark data for verification and enable design guidelines for tailoring electrode coverage and field application to mitigate interface fracture. The approach provides an analytic framework for mixed electromagnetic conditions in magnetoelectroelastic interface fracture.

In this paper, we deal with surfaces of revolution and their intersections. We start with the surfaces of revolution RS that have their axis along the x3–axis and find intersections with a line, a plane, and then intersection of two such RS. Furthermore, we apply formulas for the intersection with a line to determine the visibility of RS. Later we develop formulas for the intersection of two surfaces of revolution that have their axis along different arbitrary straight lines, and, as a special case, the intersections of two spheres and intersections of general surface of revolution with a sphere and a surface given by an equation. We apply our own software to the graphical representation of all the results we present.

Beam based alignment procedure for an undulator with superimposed F0D0 lattice

Analytical conformity is established between the wave functions corresponding to different potentials and angular momenta. An approach suggested permits construction of a new series of potentials and pertinent solutions to the Schrödinger equation in a closed analytical form for variable values of energy and angular momentum along arbitrary straight lines in the ((l + 1/2)2, E) plane.

In a general integral geometry problem, there are given the integrals of an unknown function over certain manifolds. The traditional statement of the problem consists in determining the integrand. We consider the case of an underdetermined problem when the unknown functions depend on a greater number of variables than the given integrals. These situations appear in a few applied problems when a rather complicated mathematical model is used and no a priori information is available. For overcoming the lack of appropriate data, we pose the problem of finding part of the information unknown; namely, we search only for the discontinuity surfaces of the integrand. The corresponding uniqueness theorem is proved. The present paper finalizes our studies into the case of integration over one-dimensional manifolds. In the previous articles we considered similar problems in the case of integration over straight lines. In this paper the same result is proved for the integration of unknown functions over unknown curves.

In the preceding chapter, we developed rather completely the regularity properties of weak and strong solutions of certain first order variational problems and second order differential equations involving only one unknown function. Complete results concerning the solutions of the corresponding problems involving several unknown functions were obtained only in the case where v = 2; these results were obtained by the writer before the war and were described in Chapter 1. In the preceding chapter some first differentiability results were obtained for certain systems of equations but these results did not imply the continuity of the first derivatives. In 1952, the writer presented a paper (Morrey [10]) at the Arden House Conference on Partial Differential Equations, in which it was proved that any vector solution of class C1 of a regular variational problem of class C n μ , n ≥ 2, was also of class C n μ . These results still leave a gap in the theory for systems, which can only be filled by an extension of the De Giorgi-Nash results, developed in § 5–3, to systems or some entirely new device. But, in proving the C n gm differentiability results for systems of equations, the writer was forced to use the very important formulas of F. John ([1], [3]) for fundamental solutions and to use methods which are appropriate for the discussion of elliptic systems of higher order. Hence the inclusion of this chapter in this book.KeywordsWeak SolutionElliptic SystemGeneral BoundaryPrincipal PartRegularity TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions at infinity is rigorously studied. Several new results are obtained: (i) The analyticity of the Jost eigenfunctions is investigated, and precise conditions on the potential that guarantee such analyticity are provided. (ii) The analyticity of the scattering coefficients is established. (iii) The behavior of the eigenfunctions and scattering coefficients at the branch points is discussed. (iv) New symmetries are derived for the analytic eigenfunctions (which differ from those in the scalar case). (v) These symmetries are used to obtain a rigorous characterization of the discrete spectrum and to rigorously derive the symmetries of the associated norming constants. (vi) The asymptotic behavior of the Jost eigenfunctions is derived systematically. (vii) A general formulation of the inverse scattering problem as a Riemann--Hilbert problem is presented. (viii) Precise results guaranteeing the existence and uniqueness of solutions of the Riemann--Hilbert problem are provided. (ix) Explicit relations among all reflection coefficients are given, and all entries of the scattering matrix are determined in the case of reflectionless solutions. (x) A compact, closed-form expression is presented for general soliton solutions, including any combination of dark-dark and dark-bright solitons. (xi) A consistent framework is formulated for obtaining solutions corresponding to double zeros of the analytic scattering coefficients, leading to double poles in the Riemann--Hilbert problem, and such solutions are constructed explicitly.

This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [ − 1 , + 1 ] . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.