L. P. Castro

On a Class of Wedge Diffraction Problems posted by Erhard Meister

A class of canonical wedge diffraction problems for Helmholtz equations was formulated by E. Meister in 1986 and subsequently treated by an operator theoretical approach in various publications of his research group including two of the authors. Certain subclasses of those problems, recognized of being unsolved, are subject of the present paper. Some of them are now solved explicitly by refined operator theoretical and analytical methods, others are reduced to systems of equations which contain so-called convolution type operators with symmetry. By a new factorization approach those are proved to be Fredholm in certain (fractional) Sobolev spaces, sometimes with necessary compatibility conditions. Several of the associated operators are therefore explicitly inverted and a number of new problems can be recognized reflecting the challenges of the present state-of-the-art.

Relations between convolution type operators on intervals and on the half-line

This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces,Hs,p, which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is that it provides explicit operator matrix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of properties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders:S − 1/p ∉ ℤ. For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented.

Aveiro Discretization Method in Mathematics: A New Discretization Principle

We found a very general discretization method for solving wide classes of mathematical problems by applying the theory of reproducing kernels. An illustration of the generality of the method is here performed by considering several distinct classes of problems to which the method is applied. In fact, one of the advantages of the present method—in comparison to other well-known and well-established methods—is its global nature and no need of special or very particular data conditions. Numerical experiments have been made, and consequent results are here exhibited. Due to the powerful results which arise from the application of the present method, we consider that this method has everything to become one of the next-generation methods of solving general analytical problems by using computers. In particular, we would like to point out that we will be able to solve very global linear partial differential equations satisfying very general boundary conditions or initial values (and in a somehow independent way of the boundary and domain). Furthermore, we will be able to give an ultimate sampling theory and an ultimate realization of the consequent general reproducing kernel Hilbert spaces. The general theory is here presented in a constructive way, and contains some related historical and concrete examples.

Effective conductivity of a composite material with non-ideal contact conditions

The effective conductivity of 2D doubly periodic composite materials with circular disjoint inclusions under non-ideal contact conditions on the boundary between material components is found. The obtained explicit formula for the effective conductivity contains all parameters of the considered model, such as the conductivities of matrix and inclusions, resistance coefficients, radii and centres of the inclusions and also the values of special Eisenstein functions. The method of functional equations is used to analyse the conjugation problem for analytic functions which is equivalently derived from the initial problem. Existence and uniqueness for the solution of the problem is obtained by using a reduction to a certain mixed boundary value problem for analytic functions in special functional spaces.

Reduction of singular integral operators with flip and their Fredholm property

This paper deals with singular integral operators with a reverting orientation Carleman shift defined on the classic Lebesgue space and having essentially bounded functions as coefficients. We will use similarity relations to show that the mentioned operators are equivalent tomatrix Toeplitz plus Hankel operators acting on the corresponding Hardy space. The main purpose is to extract Fredholm characteristics of the initial operators (in the form of necessary and sufficient conditions). Namely, Fredholm criteria are obtained for some of the operators under study when they have coefficients in the classes of continuous, piecewise continuous, and semi-almost-periodic functions. In addition, Fredholm index formulas are also provided in some of these cases.

How to Catch Smoothing Properties and Analyticity of Functions by Computers

We would like to propose a new method in view to catch smoothing properties and analyticity of functions by computers. Of course, in the strict sense, such goal is impossible. However, we would like to propose some practical method that may be applied for many concrete cases for some good functions (but not for bad functions, in a sense). Therefore, this may be viewed as a procedure proposal which includes numerical experiments for the just mentioned challenge and within a new method.

Source inversion of heat conduction from a finite number of observation data

This article deals with a backward heat conduction type problem. Namely, the Gaussian convolution is here analysed in a new way so that inverse source formulae to the heat conduction problem are obtained from a finite number of observation data at time and space points. In view of obtaining this main goal, different reproducing kernel Hilbert spaces, iteration schemes and Tikhonov regularization procedures are used and combined in an unified way.

Discrete linear differential equations

Abstract We propose new constructions of the approximate solutions of discrete linear differential equations and their inverse source problems. This is mainly based on a reproducing kernel Hilbert spaces approach where different types of spaces are naturally considered as a consequence of the method. Here, the major influence will be given by the Paley–Wiener and Sobolev spaces.

DIRICHLET-NEUMANN-IMPEDANCE BOUNDARY VALUE PROBLEMS ARISING IN RECTANGULAR WEDGE DIFFRACTION PROBLEMS

Boundary value problems originated by the diffraction of an electromagnetic (or acoustic) wave by a rectangular wedge with faces of possible different kinds are analyzed in a Sobolev space framework. The boundary value problems satisfy the Helmholtz equation in the interior (Lipschitz) wedge domain, and are also subject to different combinations of boundary conditions on the faces of the wedge. Namely, the following types of boundary conditions will be under study: Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, Impedance-Dirichlet, and Impedance-Neumann. Potential theory (combined with an appropriate use of extension operators) leads to the reduction of the boundary value problems to integral equations of Fredholm type. Thus, the consideration of single and double layer potentials together with certain reflection operators originate pseudo-differential operators which allow the proof of existence and uniqueness results for the boundary value problems initially posed. Furthermore, explicit solutions are given for all the problems under consideration, and regularity results are obtained for these solutions.

Singular Integral Equations on Piecewise Smooth Curves in Spaces of Smooth Functions

We prove the boundedness of the Cauchy singular integral operator in modified weighted Sobolev \( \mathbb{K}\mathbb{W}_{p}^{m}(\Gamma ,\rho ) \), Holder-Zygmund \( \mathbb{K}\mathbb{Z}_{\mu }^{0}(\Gamma ,\rho ) \) Bessel potential \( \mathbb{K}\mathbb{H}_{p}^{s}(\Gamma ,\rho ) \) and Besov \( \mathbb{K}\mathbb{B}_{{p,q}}^{s}(\Gamma ,\rho ) \) spaces under the assumption that the smoothness parameters m,μ,s are large. The underlying contour Γ is piecewise smooth with angular points and even with cusps. We obtain Fredholm criteria and an index formula for singular integral equations with piecewise smooth coefficients and complex conjugation in these spaces provided the underlying contour has angular points but no cusps. The Fredholm property and the index turn out to be independent of the integer parts of the smoothness parameters m,µ,s. The results are applied to an oblique derivative problem (the Poincare problem) in plane domains with angular points and peaks on the boundary.