Integral equations of the third kind with unbounded operators

We consider linear functional equations of the third kind in L ₂ with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.     

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 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.     

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.     

Solvability of a class of integro-differential equations of first order with variable coefficients

In the present paper, we consider a class of linear integro-differential equations of first order with a stochastic kernel and with variable coefficients on the semiaxis. These equations have important applications in physical kinetics. By combining special factorization methods with methods involving integral Fredholm equations of the second kind, we can construct solutions of such equations in the Sobolev space W ₁ ¹ (?⁺). In certain singular cases, we can also describe the structure of the obtained solutions.     

Fractional Multistep Methods for Weakly Singular Volterra Integral Equations of the First Kind with Perturbed Data

ABSTRACT

Fractional multistep methods were introduced by C. Lubich for the quadrature of Abel integral operators and the solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides. In the current paper, we consider the regularizing properties of these methods to solve the mentioned integral equations of the first kind for perturbed right-hand sides. Finally, numerical results are presented.     

Stability analysis of methods employing reducible rules for volterra integral equations of the first kind

The concept of (A ₀,S)-stability for Volterra integral equations of the second kind will be extended to that of the first kind equations. We will show that stability polynomials for methods employing reducible quadrature rules, as applied to the first kind equations, can be trivially obtained from the results for the second kind equations.     

Abstract potential operator and spectral methods for a class of degenerate evolution problems

The spectral theory of operators in Banach spaces is employed to treat a class of degenerate evolution equations. A basic role is played by the assumption that the Banach space under consideration may be expressed as a direct sum of two suitable subspaces. Two methods for solving the problem are studied. The first method is based on the expansion of the resolvent of a closed operator into Laurent series in a neighbourhood of 0. The second one makes use of the theory of abstract potential operators. In particular, an extension of the Hille-Yosida theorem on infinitesimal generators of (C₀) semigroups of linear operators is obtained. Some examples relative to operators appearing in many applications to partial differential equations are given.     

On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

General linear methods for Volterra integral equations

We investigate the class of general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.     

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.      

Approximation numbers of bounded operators

The theory of ideals of linear operators is well developed and has a lot of applications in theory and practise. The purpose of this paper is to give a first idea of a similar theory for bounded (nonlinear) operators. In view of applications we will not give an abstract (perhaps general nonsense) theory, but an example of a class λ_p of bounded operators with a structure similar to an $\text{L}$-module($\text{L}$ represents the class of all linear operators between Banach spaces), and applications to projection methods for solving equations with λ_p-type operators.     

Natural Volterra Runge-Kutta methods

A very general class of Runge-Kutta methods for Volterra integral equations of the second kind is analyzed. Order and stage order conditions are derived for methods of order p and stage order q = p up to the order four. We also investigate stability properties of these methods with respect to the basic and the convolution test equations. The systematic search for A- and V ₀-stable methods is described and examples of highly stable methods are presented up to the order p = 4 and stage order q = 4.     

Spline Collocation for Cordial Volterra Integral Equations

We study the convergence and convergence speed of two versions of spline collocation methods on the uniform grids for linear Volterra integral equations of the second kind with noncompact operators.     

Spline Collocation-Interpolation Method for Linear and Nonlinear Cordial Volterra Integral Equations

We study the convergence and convergence speed of the discontinuous spline collocation and collocation-interpolation methods on uniform grids for linear and nonlinear Volterra integral equations of the second kind with noncompact operators.     

On the integral equation method for the plane mixed boundary value problem of the Laplacian

Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ₂ and given Neumann data on the remaining part Γ₂ of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ₂ and the also desired values of the normal derivatie on Γ₁. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ₁ ∩ Γ₂ even for C^∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ₂ providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ₁ and Γ₂ augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).     

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     

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.     

Additive Schwarz algorithms for the p version of the Galerkin boundary element method

Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators. Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000