Linear functional equations of the first, second, and third kind in L 2

Under consideration are the functional equations of the first, second, and third kind with operators in wide classes of linear continuous operators in L ₂ containing all integral operators. We propose methods for reducing these equations by linear invertible changes either to linear integral equations of the first kind with nuclear operators or to equivalent linear integral equations of the second kind with quasidegenerate Carleman kernels. Some various approximate methods of solution are applicable to the so-obtained integral equations.     

Measure-compact operators, almost compact operators, and linear functional equations in L p

Under study are the measure-compact operators and almost compact operators in L _p . We construct an example of a measure-compact operator that is not almost compact. Introducing two classes of closed linear operators in L _p , we prove that the resolvents of these operators are almost compact or measure-compact. We present methods for the reduction of linear functional equations of the second kind in L _p with almost compact or measure-compact operators to equivalent linear integral equations in L _p with quasidegenerate Carleman kernels.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

On Compactness of Regular Integral Operators in the Space <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

In this paper we obtain a sufficient condition for quite continuity of Fredholm type integral operators in the space L₁(a, b). Uniform approximations by operators with degenerate kernels of horizontally striped structures are constructed. A quantitative error estimate is obtained. We point out the possibility of application of the obtained results to second kind integral equations, including convolution equations on a finite interval, equations with polar kernels, one-dimensional equations with potential type kernels, and some transport equations in non-homogeneous layers.     

核属于H函数类的多维积分方程近似解直接方法的优化

本文我们确定了核属于 H函数类的多维第二类 Fredholm积分方程类在自适直接方法意义下的最优近似解的精确阶估计 ,并给出了最优算法 .     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

Potential-based numerical solution of Dirichlet problems for the Helmholtz equation

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.     

Functions orthogonal to polynomials and their application in axially symmetric problems in physics

We investigate properties of sets of functions comprising countably many elements A_n such that every function A_n is orthogonal to all polynomials of degrees less than n. We propose an effective method for solving Fredholm integral equations of the first kind whose kernels are generating functions for these sets of functions. We study integral equations used to solve some axially symmetric problems in physics. We prove that their kernels are generating functions that produce functions in the studied families and find these functions explicitly. This allows determining the elements of the matrices of systems of linear equations related to the integral equations for considering the physical problems.     

Direct methods for the approximate solution of operator equations with nonzero kernel

We justify direct methods for the approximate solution of linear operator equations with nonzero kernels and apply these methods to the justification of projective methods for the approximate solution of standard singular integral equations with Cauchy kernels and positive index on the unit disk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1202–1213, September, 1998.     

On a Certain Class of Integral Equations Associated with Hankel Transforms

This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber–Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber–Orr integral transforms of functions belonging to L ¹ and L ². These operators together with the proven Paley–Wiener’s theorem for the Weber–Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke–Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations.      

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

On the similarity of linear operators in L p to integral operators of the first and second kind

We construct an example of a compact operator of the third kind in L _p (p ≠ 2) not similar to any integral operator of the first or second kind. This example shows that not every linear equation of the third kind in L _p (p ≠ 2) can be reduced by an invertible continuous linear change to an equivalent integral equation of the first or second kind. The example also proves the impossibility of a characterization of integral and Carleman integral operators in L _p (p ≠ 2) in terms of the spectrum and its components.     

Constructive methods for certain Bergman operators

This paper is concerned with the class of linear partial differential equations of second order such that there exist Bergman operators with polynomial kernels (cf, [12]). In an earlier paper [ll] the authors have shown that these equations also admit differential operators as introduced by K. W. Bauer [I]. In the present paper, relations between different types of representations of solutions are investigated. These representations are of interest in developing a function theory of solutions; cf., for instance, K. W. Bauer [I] and S. Ruscheweyh [19]. They are also essential to global extensions of local results obtained by means of Bergman operators of the first kind. The inversion problem for those operators is solved, and it is shown that all solutions of equations of that class which are holomorphic in a domain of C² can be represented by operators with polynomial kernels. Furthermore, a construction principle for deriving the equations investigated by K. W. Bauer [2] is obtained; this yields corresponding representations of solutions by differential and integral operators in a systematic fashion     

Convergence and Rate of Approximation in BVΦ for a Class of Integral Operators

We obtain estimates and convergence results with respect to ?-variation in spaces BV_Φ for a class of linear integral operators whose kernels satisfy a general homogeneity condition. Rates of approximation are also obtained. As applications, we apply our general theory to the case of Mellin convolution operators, to that one of moment operators and finally to a class of operators of fractional order.     

On Volterra partial integral equations in the space of continuous functions of three variables

It is well known that any Volterra integral equation of the second kind with compact operator is uniquely solvable. Partial integral operators are not compact, even in the general case of continuous kernels. Unique solvability conditions for Volterra partial integral equations of the second kind in the space of continuous functions of three variables are considered. Conditions for a Volterra partial integral equation to be equivalent to a three-dimensional Volterra integral equation with compact operator are obtained. Continuum analogues of matrix equations for some problems of scattering theory are reduced to the Volterra partial integral equations under examination. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Rational interpolation and approximate solution of integral equations

In the space of continuous periodic functions, we construct interpolation rational operators, use them to obtain quadrature formulas with positive coefficients which are exact on rational trigonometric functions of order 2n, and suggest an algorithm for an approximate solution of integral equations of the second kind. We estimate the accuracy of the approximate solution via the best trigonometric rational approximations to the kernel and the right-hand side of the integral equation.     

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

A fast Fourier-Galerkin method for solving integral equations of second kind with weakly singular kernels

We propose in this paper a convenient way to compress the dense matrix representation of a compact integral operator with a weakly singular kernel under the Fourier basis. This compression leads to a sparse matrix with only ${\mathcal{O}}(n\log n)$ number of nonzero entries, where 2n+1 denotes the order of the matrix. Based on this compression strategy, we develop a fast Fourier-Galerkin method for solving second kind integral equations with weakly singular kernels. We prove that the approximate order of the truncated equation remains optimal and that the spectral condition number of the coefficient matrix of the truncated linear system is uniformly bounded. Furthermore, we develop a fast algorithm for solving the corresponding truncated linear system, which preserves the optimal order of the approximate solution with only ${\mathcal{O}}(n\log^{2}n)$ number of multiplications required. Numerical examples complete the paper.     

Approximation von operatorgleichungen mit reprokernen

In this paper we develop a method for the approximation of a broad class of operator equations by reproducing kernels. The relevant operators are defined on Hilbert spaces. Necessary and sufficient conditions for the convergence of the approximation are discussed in detail. The results can be applied-for example-to Fredholm integral operators of the first and second kind and to ordinary and partial differential operators of elliptic type. In this context we refer to [9] for methods to construct reproducing kernels.     

A note on integral representations of linear operators

In this paper we describe families of those bounded linear operators that are simultaneously unitarily equivalent to integral operators with smooth Carleman kernels. The singleton case of the main result implies that every integral operator is unitarily equivalent to an integral operator having smooth Carleman kernel.