System Of Fredholm Integral Equations Research Articles

A Nyström method for Fredholm integral equations with right-hand sides having isolated singularities

Fredholm integral equations on the interval [−1,1] with right-hand sides having isolated singularities are considered. The original equation is reduced to an equivalent system of Fredholm integral equations with smooth input functions. The Nystrom method is applied to the system after a polynomial regularization. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.

Approximate solution of a system of linear integral equations by the Taylor expansion method

A simple and efficient approximate technique is developed to obtain the solution to a system of linear integral equations. This technique is based on the Taylor expansion. The method has been successfully applied to determine approximate solutions of a system of Fredholm integral equations and Volterra integral equations of not only the second kind but also the first kind. The mth order approximation of the solution is exact up to a polynomial of degree equal to or less than m. Several illustrative examples are presented to show the effectiveness and accuracy of this method.

Modified homotopy perturbation method for solving system of linear Fredholm integral equations

In this paper, we present a modification to homotopy perturbation method for solving system of linear Fredholm integral equations (LFIEs). Some examples are given, revealing its effectiveness and convenience. Also the results obtained by modified homoperturbation method (MHPM) is superior to that obtained by Adomian decomposition method (ADM).

Boundary value problem for the Laplace equation outside cuts on the plane with different conditions of the third kind on opposite sides of the cuts

We consider a boundary value problem for the Laplace equation outside cuts on a plane. Boundary conditions of the third kind, which are in general different on different sides of each cut, are posed on the cuts. We show that the classical solution of the problem exists and is unique. We obtain an integral representation for the solution of the problem in the form of potentials whose densities are found from a uniquely solvable system of Fredholm integral equations of the second kind.

Boundary integral equations for viscous incompressible flows in porous media or past porous bodies

AbstractThe purpose of this paper is to present the existence and uniqueness result for a boundary value problem which describes the flow of a viscous incompressible fluid past an arbitrary porous particle with a Lyapunov boundary, which is embedded in a second porous medium, by using the Brinkman model and the potential theory. We use a boundary integral method that reduces the problem to a system of second kind Fredholm integral equations that has a unique solution in some Hölder spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary integral equations for two‐dimensional low Reynolds number flow past a porous body

AbstractIn this paper we use the method of matched asymptotic expansions in order to study the two‐dimensional steady flow of a viscous incompressible fluid at low Reynolds number past a porous body of arbitrary shape. One assumes that the flow inside the porous body is described by the Brinkman model, i.e. by the continuity and Brinkman equations, and that the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. By considering some indirect boundary integral representations, the inner problems are reduced to uniquely solvable systems of Fredholm integral equations of the second kind in some Sobolev or Hölder spaces, while the outer problems are solved by using the singularity method. It is shown that the force exerted by the exterior flow on the porous body admits an asymptotic expansion with respect to low Reynolds number, whose terms depend on the solutions of the abovementioned system of boundary integral equations. In addition, the case of small permeability of the porous body is also treated. Copyright © 2008 John Wiley & Sons, Ltd.

Effect of Residual Stresses on Fracture of Semi-Infinite Composites with Cracks

In the present study, an axisymmetric problem for a near-surface circular crack in a semi-infinite composite, which is subjected to initial (residual) stresses parallel to the crack plane, is considered. Normal and radial shear loads are applied. An analytical approach within the framework of the three-dimensional linearized mechanics of solids is used. The problems are reduced to systems of Fredholm integral equations of the second kind, where the solutions are identified with harmonic potential functions. The stress intensity factors at the crack tips for two types of composites are analyzed. These are a laminated composite made of aluminum/boron/silicate glass with epoxy-maleic resin and a carbon/plastics composite with stochastic reinforcement by short ellipsoidal carbon fibers. The influence of the residual stresses and of the geometrical parameters of the system are illustrated.

Mutual influence of two parallel coaxial cracks in a composite material with initial stresses

By using the approaches of three-dimensional linearized mechanics of deformed bodies, we study the axisymmetric problem of two parallel coaxial circular cracks in an infinite composite material with initial stresses acting in the planes of the cracks. The resolving system of Fredholm integral equations of the second kind is deduced and the representations of the stress intensity factors in the vicinity of the crack tips are obtained. The dependences of the stress intensity factors on the initial stresses and the distance between the cracks are determined. For two types of composite materials, namely, for a layered composite with isotropic layers and a composite with stochastic reinforcement by short ellipsoidal fibers, the stress intensity factors are computed and their dependences on the initial stresses, physicomechanical characteristics of the composites, and geometric parameters of the problem are investigated.

Approximate Solution of the System of Nonlinear Fredholm Integral Equations of the Second Kind Using Spline Function

This study deals with introducing spline function to find the approximate solution of the system of nonlinear Fredholm integral equations of the second kind. The benefit of spline functions was demonstrated by presenting several examples.

Nystrom method for systems of integral equations on the real semiaxis

In this paper, the authors introduce a Nystrom method for solving systems of Fredholm integral equations on the real semiaxis. They prove that the method is stable and convergent. Moreover, they show some numerical tests that confirm the error estimates. Finally, they discuss a specific application to an inverse scattering problem for the Schrodinger equation.

Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case

We consider the system of Fredholm integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{T}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$ and also the system of Volterra integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{t}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$ where T>0 is fixed and the nonlinearities h i (t,u 1,u 2,…,u n ) can be singular at t=0 and u j =0 where j∈{1,2,…,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ i u i (t)≥0 for t∈[0,1] and 1≤i≤n, where θ i ∈{1,−1} is fixed. We also include examples to illustrate the usefulness of the results obtained.

Boundary value problem for a coupled system of nonlinear fractional differential equations

In this work we discuss a boundary value problem for a coupled differential system of fractional order. The differential operator is taken in the Riemann–Liouville sense and the nonlinear term depends on the fractional derivative of an unknown function. By means of Schauder fixed-point theorem, an existence result for the solution is obtained. Our analysis relies on the reduction of the problem considered to the equivalent system of Fredholm integral equations.

Regularization methods for the analysis of EXAFS spectra of chemical complexes

The analysis of EXAFS data aimed at extracting information on the local atomic structure of materials is discussed. To identify the structure of interest, one has to solve a system of Fredholm integral equations of the first kind. The situation is considered in which three functions, that characterize the structure under study, are to be determined from one equation. In practice, in this case a nonuniqueness arises in the reconstruction of the unknown functions within a prescribed accuracy, so that additional information has to be used. For a broad range of materials, the functions that are reconstructed possess a rather specific form; namely, these functions appear to be a sum of several narrow peaks that can be approximated by δ functions. This qualitative information about the functions is non-classical. The proposed new analysis procedure is based on the classical theory of ill-posed problems and on using the available non-classical information. The efficiency of the algorithms proposed is illustrated by applying them to model and experimental data of a test compound consisting of uranium, silicon and oxygen, whose structure was previously identified by other authors.

Acoustic diffraction by a finite number of small soft bodies

The scattering problem of an incident plane sound wave by a finite number of small soft-sound arbitrarily shaped bodies is considered. A system of first-class Fredholm integral equations is posed which, for the small bodies, yields a system of potential equation sequences for power solution expansion. Reciprocity relations were exploited to construct the first and second order asymptotic models which give approximate solutions valid for a large number of small bodies. The leading two terms of the asymptotics for the farfield amplitude and the scattering cross section are obtained. The case of a several elliptic discs is considered in detail.

A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations

In this paper, an application of He’s homotopy perturbation (HPM) method is applied to solve the system of Fredholm and Volterra type integral equations, the results revealing that the HPM is very effective and simple.

Solution of a System of Linear Fredholm Integral Equations of the Second Kind by Iteration Methods

In this paper, we consider the linear system of Fredholm integral equation of the second kind. Three methods are used to solve this system, successive approximation method, Aitken's method depending on successive approximation method and a new procedure which is Aitken's method depending on Adomian decomposition method. A comparison between approximate and exact results for two numerical examples depending on the least-square error, are given to show the accuracy of the results obtained by using these methods.

Numerical treatment of second kind Fredholm integral equations systems on bounded intervals

In this paper the authors propose numerical methods to approximate the solutions of systems of second kind Fredholm integral equations. They prove that such methods are stable and convergent. Error estimates in weighted L p norm, 1 ⩽ p ⩽ + ∞ , are given and some numerical tests are shown.

On a circular shear crack in a semiinfinite composite with initial stresses

By the methods of the linearized three-dimensional mechanics of deformed bodies, we consider an axisymmetric problem of a circular subsurface crack of radial shear in a semiinfinite composite material with initial stresses. We deduce the resolving system of Fredholm integral equations of the second kind and the representations of the stress intensity factors in the vicinity of the crack tip depending on the initial stresses. For two types of composite materials (layered composites with isotropic layers and composites stochastically reinforced by short ellipsoidal fibers), we compute the stress intensity factors and study their dependence on the initial stresses, physicomechanical characteristics of the composites, and the geometric parameters of the problem.

Easy computational approach to solution of system of linear Fredholm integral equations

This paper is about the construction of a simple method to approximate the solution of system of linear Fredholm integral equations of the second kind based on Adomian’s decomposition method. Easy computations rather than successive integrations are used with simple algorithm. Some solved problems are given to show the efficiency of the method. The convergence of the method is also considered.

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.