Numerical solution of Fredholm integral equations of first kind by two-dimensional trigonometric wavelets in holder space <Emphasis Type="Italic">C</Emphasis><Superscript>α</Superscript>([<Emphasis Type="Italic">a,b</Emphasis>])

In this article, we employ trigonometric wavelet bases to numerical solution of Fredholm integral equations of first kind in Holder space. Employment of Galerkin method for trigonometric wavelets in Fredholm integral equations of first kind has resulted in occurrence of two-dimensional trigonometric wavelets. Here, we present the convergence of two-dimensional trigonometric wavelets in numerical solution in Holder space C ^α([a, b]).     

Computation of the eigenvalues of Fredholm–Stieltjes integral equations

The Rayleigh–Ritz and the inverse iteration methods are used in order to compute the eigenvalues of Fredholm–Stieltjes integral equations, i.e. Fredholm equations with respect to suitable Stieltjes-type measures. Some applications to the so-called ‘charged’ (in German ‘belastete’) integral equation, and particularly the problem of computing the eigenvalues of a string charged by a finite number of cursors are given.     

A numerical method based on hybrid orthonormal Bernstein and improved block-pulse functions for solving Volterra–Fredholm integral equations

This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process.     

A main theorem of spectral relationships for Volterra–Fredholm integral equation of the first kind and its applications

This paper is concerned to derive the main theorem of spectral relationships of Volterra–Fredholm integral equation (V‐FIE) of the first kind in the space L₂[?1,1]×C[0,T], ?1?x?1, 0?t?T<1. The Fredholm integral (FI) term is considered in position and its kernel takes a logarithmic form multiplying by a continuous function. While Volterra integral (VI) term in time with a positive continuous kernel. Many important special and new cases can be established from the main theorem. Moreover, we use it to solve V‐FIE of the second kind in the same space. The numerical results are computed and the error is calculated using Maple 12. Copyright © 2010 John Wiley & Sons, Ltd.     

On an approach to the definition of a stochastic integral with respect to a Gaussian measure

We define a stochastic Riemann integral with respect to a Gaussian measure. The class of integrable functions is introduced in which there exists a solution of a stochastic Fredholm integral equation. It is shown by examples how to pass from the integral defined here to the Itô and Stratonovich integrals.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 100–108, 1986.     

On the solution of mixed problems for a fiber with a vertical circular cylindrical cavity

A method of solving paired integral equations that appear in considerations of mixed problems of elasticity and thermoelasticity theory is given, with the help of generalized integral Weber transforms. The paired equations are reduced to an integral Fredholm equation of the second kind on the semiaxis, which have a discontinuous kernel, or to Fredholm equations of the second kind on a finite interval and infinite systems of linear algebraic equations, which are normal in the sense of Poincare-Koch. As an example, contact problems for an inhomogeneous fiber with a cavity are considered. If the fiber is bonded with the elastic half-space, then a second appproach is realized, which is based on a reduction to an equation with a self-adjoint operator, for which some method of sequential iteractions and the Bubnov-Galerkin method are justified.Translated from Dinamicheskie Sistemy, No. 7, pp. 95–102, 1988.     

Pseudodifferential Operators on Periodic Graphs

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group \mathbb Zⁿ{{\mathbb {Z}}^n} . The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OPS ⁰ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.     

On the Fredholm radius of integral operators of potential theory

The Fredholm radius of the integral operators of the theory of n-dimensional harmonic potentials on surfaces with edges, acting in certain weighted Hölder type spaces, is found. Results for integral operators are derived from theorems on operators in certain auxiliary contact problems.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 109–133, 1990.     

Torsional oscillations of a circular stamp on an elastic layer with a cylindrical cavity

The problem of torsional oscillations of a stamp that is linked with an elastic stratum which contains a cylindrical cavity is considered. The problem is formulated in the form of conjugate integral equations that are related to the integral Weber transforms. The conjugate equations are reduced to an equivalent Fredholm equation of the second kind.Translated from Dinamicheskie Sistemy, No. 9, pp. 54–59, 1990.     

On linear Fredholm integro-differential equations of first kind

We study the question whether linear one-dimensional integro-differential equations with constant limits of integration (equations of Fredholm type) containing no free differential expression (equations of first kind) can be reduced to integral equations of first kind and to Fredholm integro-differential equations of second kind.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 20–27.     

A Fredholm Determinant Formula for Section Determinants of Bounded Operators

We show that the section determinant of e^A can be expressed, under certain conditions, by the Fredholm determinant of an integral operator. The kernel function of this integral operator is computed explicitly in terms of the operator A. As a simple consequence we derive a Weierstrass type product expansion for the section determinant.     

On stability of index of Fredholm complexes on the C* -algebra

This article deals with the index of Fredholm complexes of Λ-operators of the Hilbert Λ-modulus on C*-algebra. For this class of operators necessary and sufficient conditions in order to be a Fredholm, are obtained. Based on these results, a notion of Fredholm complex and its index is introduced. For this index, a stability theorem related to various perturbations is proved. In the second part of the article, a completation of a semigroup Fredholm complexes is analysed. It is proved that the group K _G (X, Λ) is the completation of G ? Λ-fibration of the above group on the compact space X.     

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.      

The basic functionsb andh for Fredholm integral equations with displacement kernels

In many branches of applied mathematics, including lateral inhibition in neural systems, radiation dosimetry, and optimal filtering of noisy signals, important roles are played by Fredholm integral equations with displacement kernels. Frequently, certain functionals on the solutions of the integral equations are as important as the solutions themselves. In this paper, it is shown that many such functionals may be expressedalgebraically in terms of two basic functions,b andh, and that these functions themselves are solutions of a certain Cauchy system and a system of singular integral equations.     

Imbedding methods for the integral equations for the optimal control of time-lag systems,I: Parameter dependence

The problem of the optimal control of systems with time lags can be reduced to a Fredholm integral equation. In this paper, it is shown that the Fredholm resolvent satisfies an initial-value problem in which a parameter serves as the independent variable. This reduction may be of computational utility.This work was partially supported by Sigma Delta Epsilon—Graduate Women in Science and NIH Grant No. GM-23732.     

On some geometric conditions of Fredholmity of one-dimensional singular integral operators

Here we consider one-dimensional singular integral operators with arbitrary bounded measurable coefficients. We study Fredholm properties of these operators inL _p spaces using geometric information on the cluster values of their coefficients.     

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Dynamical problems on longitudinal shear of a half-space with nonhomogeneities

Problems on diffraction of shear waves at cavities and rigid inclusions in a half-space with a fixed boundary and in one with a boundary which is free from forces are considered. The problem is reduced to a singular integral equation in the case of rigid inclusion in a half-space and to a Fredholm equation of the second kind in the case of a cavity.Translated from Dinamicheskie Sistemy, No. 9, pp. 47–54, 1990.     

On homotopy properties and indices of families of singular and bisingular operators with piecewise-continuous coefficients

We present the stable homotopy classification of families of Fredholm and invertible singular integral operators with piecewise-continuous coeffcients in weighted L _p -spaces with Khvedelidze weights. We obtain a new topological formula for the calculation of the indices of families. These results allowed the authors to investigate homotopy properties of bisingular operators.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol.9, Suzdal Conference-3, 2003.