Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the half-lines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well.     

Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility

Finite interval convolution operators with periodic kernel-functions are studied from the point of view of Fredholm properties and invertibility. These operators are associated with Wiener-Hopf operators with matrix-valued symbols defined on a space of functions whose domain is a contour consisting of two parallel straight-lines. For the Fredholm study a Wiener-Hopf operator is considered on a space of functions defined on a contour composed of two closed curves having a common multiple point. Invertibility of the finite interval operator is studied for a subclass of symbols related to the problem of wave diffraction by a strip grating.The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

On convolution equations with semi-almost periodic symbols on a finite interval

This note concerns a class of Wiener-Hopf operators on a finite interval, acting between Sobolev multi-index spaces. Necessary and sufficient conditions for such an operator to be Fredholm are given, as well as a formula for the index. The argument is based on a reduction procedure of convolution operators on a finite interval to operators of the same type on the half-line.supported by the Netherlands organization for scientific research (NWO)supported in part by NSF Grant 9101143     

Solvability of some boundary-value problems for polyharmonic equation with Hadamard-Marchaud boundary operator

We investigate solvability conditions for certain non-classical boundary-value problems for the polyharmonic equation. As the boundary operators we consider fractional differential operators in the sense of Hadamard-Marchaud. The considered problems generalize well-known Dirichlet and von Neumann boundary-value problems for boundary operators of fractional type.     

A note on first kind convolution equations on a finite interval

This note deals with a class of convolution operators of the first kind on a finite interval. Necessary and sufficient conditions for such an operator to be Fredholm are given. The argument is based on a process of reduction of convolution-type operators on a finite interval to operators of the same type on the half line.Research supported by the Netherlands organization for scientific research (NWO).     

Boundary Value Problems for the Helmholtz Equation in an Octant

We consider a class of boundary value problems for the three-dimensional Helmholtz equation that appears in diffraction theory. On the three faces of the octant, which are quadrants, we admit first order boundary conditions with constant coefficients, linear combinations of Dirichlet, Neumann, impedance and/or oblique derivative type. A new variety of surface potentials yields 3 × 3 boundary pseudodifferential operators on the quarterplane that are equivalent to the operators associated to the boundary value problems in a Sobolev space setting. These operators are analyzed and inverted in particular cases, which gives us the analytical solution of a number of well-posed problems. Dedicated to Vladimir G. Maz’ya on the occasion of his 70th birthday     

Resolvent kernels of Green's function kernels and other finite-rank modifications of Fredholm and Volterra kernels

Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green's functions for Sturm-Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.This research was sponsored by the United States Army under Contract No. DAA29-75-C-0024.     

Quasi-banded operators,convolutions with almost periodic or quasi-continuous data,and their approximations

We study the stability and Fredholm property of the finite sections of quasi-banded operators acting on L^p$L^p$ spaces over the real line. This family is significantly larger than the set of band-dominated operators, but still permits to derive criteria for the stability and results on the splitting property, as well as an index formula in the form as it is known for the classical cases. In particular, this class covers convolution type operators with semi-almost periodic and quasi-continuous symbols, and operators of multiplication by slowly oscillating, almost periodic or even more general coefficients.     

A Duduchava-Saginashvili's type theory for Wiener-Hopf plus Hankel operators

This paper presents a Duduchava-Saginashvili's type theory for Wiener-Hopf plus Hankel operators with semi-almost periodic Fourier symbols and acting between Lp Lebesgue spaces. This means the obtainment of one-sided invertibility and Fredholm property for these operators upon certain mean values of the representatives at infinity of their Fourier symbols. Additionally, a formula for the Fredholm index is provided by introducing a corresponding winding number of some new elements.     

Boundary and initial-value methods for solving Fredholm equations with semidegenerate kernels

We develop a new approach to the theory and numerical solution of a class of linear and nonlinear Fredholm equations. These equations, which have semidegenerate kernels, are shown to be equivalent to two-point boundary-value problems for a system of ordinary differential equations. Applications of numerical methods for this class of problems allows us to develop a new class of numerical algorithms for the original integral equation. The scope of the paper is primarily theoretical; developing the necessary Fredholm theory and giving comparisons with related methods. For convolution equations, the theory is related to that of boundary-value problems in an appropriate Hilbert space. We believe that the results here have independent interest. In the last section, our methods are extended to certain classes of integrodifferential equations.     

Non-quasielliptic boundary-value problems in a cylinder with regularly degenerate model problem on the cross-section

The author introduces a class of boundary-value problems in cylindrical domains. These problems are neither elliptic, nor parabolic, nor quasielliptic, but their main properties are inherited from problems of the said three types, namely, in a proper scale of weighted functional spaces, the operators of these problems are of Fredholm type and their solutions admit asymptotic expansions at infinity with power-exponential terms. The proof is based on precise estimates of solutions of some regularly degenerate model problems on the cross-section of the cylinder in stepwise norms depending on a small parameter.     

A fredholm study for convolution operators with piecewise continuous symbols on a union of a finite and a semi-infinite intervel 1

We study convolution type operators with kernels that have Fourier transforms in the class of piecewise continuous matrix functions. These convolution operators are assumed to act between Sobolev spaces defined on a union of a finite and a semi-infinite intervel. The main result is a criterion for the Fredholm property of these operators. An application to a problem related to diffraction theory is illustrated.     

Potential theory for problems of diffraction on a layer between two parallel planes

We investigate the boundary-value problems that appear when studying the diffraction of acoustic waves on obstacles in a layer between two parallel planes. By using potential theory, these boundary-value problems are reduced to the Fredholm integral equations given on the boundary of the obstacles. The theorems on existence and uniqueness are proved for the Fredholm equations obtained and, hence, for the boundary-value problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 647–662, May, 1993.     

The Fredholm Index of Locally Compact Band-dominated Operators on L^p \,(\user2{\mathbb{R}})

We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator A on and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of A. The results are applied to operators of convolution type with almost periodic symbol.     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

An Invertibility Criterion in a C*-Algebra Acting on the Hardy Space with Applications to Composition Operators

In this paper, we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators, we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application, we prove that for “quasi-parabolic” composition operators the spectra and the essential spectra are equal.     

On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics

Quasilinear parabolic functional differential equations containing multiple transformations of spatial variables are considered with the Neumann boundary-value conditions. Sufficient conditions of the Andronov-Hopf bifurcation of periodic solutions are obtained along with expansions of the solutions in powers of a small parameter. Spectral properties of the linearized elliptic operator of this problem are investigated. Necessary and sufficient conditions of normality are obtained for such operators. Examples illustrating their properties are given. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 21, Proceedings of the Seminar on Differential and Functional Differential Equations Supervised by A. Skubachevskii (Peoples’ Friendship University of Russia), 2007.     

On the algebraic classification of Fredholm integral operators

Using the well-known and specific connections between Fredholm integral equations, two-point boundary-value problems, and linear dynamics-quadratic cost control processes, we present a complete, independent set of algebraic invariants suitable for classifying a wide range of Fredholm integral operators with respect to a certain group of transformations. The group, termed theRiccati group, is naturally suggested by the control theoretic setting, but seems nonintuitive from a purely integral-equations point of view. Computational considerations resulting from this classification are discussed.     

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann problems with operators of a fractional order.     

Analysis of united boundary‐domain integro‐differential and integral equations for a mixed BVP with variable coefficient

The mixed (Dirichlet–Neumann) boundary‐value problem for the ‘Laplace’ linear differential equation with variable coefficient is reduced to boundary‐domain integro‐differential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs/BDIEs contain integral operators defined on the domain under consideration as well as potential‐type operators defined on open sub‐manifolds of the boundary and acting on the trace and/or co‐normal derivative of the unknown solution or on an auxiliary function. Some of the considered BDIDEs are to be supplemented by the original boundary conditions, thus constituting boundary‐domain integro‐differential problems (BDIDPs). Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP, as well as invertibility of the associated operators are investigated in appropriate Sobolev spaces. Copyright © 2005 John Wiley & Sons, Ltd.