System Of Fredholm Integral Equations Research Articles

A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method

Homotopy perturbation method is applied to the numerical solution for solving system of Fredholm integral equations. Comparison of the result obtained by the present method with that obtained by Taylor-series expansion method [K. Maleknejad, N. Aghazade, M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput. 175 (2006) 1229–1234] reveals that the present method is very effective and convenient.

Convergence of approximate solution of system of Fredholm integral equations

In this paper numerical solution of system of linear Fredholm integral equations by means of the Sinc-collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The exponential convergence rate of the method O ( e − k N ) is proved. The method is applied to a few test examples with continuous kernels to illustrate the accuracy and the implementation of the method.

Transient response of a piezoelectric layer with a penny-shaped crack under electromechanical impacts

In this paper, the dynamic response of a piezoelectric layer with a penny-shaped crack is investigated. The piezoelectric layer is subjected to an axisymmetrical action of both mechanical and electrical impacts. Two kinds of crack surface conditions, i.e., electrically impermeable and electrically permeable, are adopted. Based upon integral transform technique, the crack boundary value problem is reduced to a system of Fredholm integral equations in the Laplace transform domain. By making use of numerical Laplace inversion the time-dependent dynamic stress and electric displacement intensity factors are obtained, and the dynamic energy release rate is further derived. Numerical results are plotted to show the effects of both the piezoelectric layer thickness and the electrical impact loadings on the dynamic fracture behaviors of the crack tips.

Oscillations of an inhomogeneous elastic layer

A method is proposed to analyze wave fields in an elastic layer with elastic properties varying arbitrarily with depth. The method is based on reducing the boundary-value problem to a system of Fredholm integral equations of the second kind, which is then analyzed numerically. Some features of the structure of dispersion sets are analyzed and, in particular, their asymptotes are constructed.

Coupled Integral Equations for Diffraction by Profiled, Anisotropic, Periodic Structures

Recently Popov and Neviere have reformulated harmonic Maxwell's equations in Fourier space for periodic structures as a first-order system of differential equations. To ensure fast convergence of the Fourier series of the electromagnetic field, the authors utilized Li's theorems of Fourier factorization in their derivation. In this paper, Popov's and Neviere's equations are slightly modified to facilitate the derivation of an equivalent system of Fredholm integral equations (IEs) of the second kind. This system of coupled IEs involves only transversal components of the electromagnetic field and corresponds to the volume-type electric field integral equation (EFIE) and magnetic field integral equation (MFIE) for the particular case of periodic structures. Therefore, the proposed, fast-converging IEs are herein after referred to as the TEFIE and TMFIE. The numerical solution of the TEFIE and TMFIE can be obtained with conventional direct or iterative solution methods.

THE ADOMIAN DECOMPOSITION METHOD APPLIED TO THE FUZZY SYSTEM OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND

In this work, the Adomian decomposition(AD) method is applied to the Fuzzy system of linear Fredholm integral equations of the second kind(FFIE). First the crisp Fredholm integral equation is solved by AD method and then the crisp solution is fuzzified by extension principle. The proposed algorithm is illustrated by solving a numerical example.

Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method

In this paper, we present a Taylor-series expansion method for a second kind Fredholm integral equations system with smooth or weakly singular kernels. This method reduce the system of integral equations to a linear system of ordinary differential equation. After constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods. Finally, we show the efficiency of the method using some numerical examples.

On the boundary integral equation method for a mixed boundary value problem of the biharmonic equation§

This article is concerned with weak solution of a mixed boundary value problem for the biharmonic equation in the plane. Using Green's formula, the problem is converted into a system of Fredholm integral equations for the unknown data on different parts of the boundary. Existence and uniqueness of the solutions of the system of boundary integral equations are established in appropriate Sobolev spaces.

Electroelastic fracture mechanics analysis of central active piezoelectric transformer

The theory of linear piezoelectricity is applied to solve the plane strain electroelastic problem of a central active piezoelectric transformer in presence of a crack located normal to the interfaces. The ends of the crack are situated at equal distances away from the interfaces. The problem of an infinite central active piezoelectric transformer is formulated by means of integral transforms and reduced to the solution of a system of Fredholm integral equations of the second kind. Numerical calculations are carried out, and the fracture mechanics parameters such as stress intensity factor, energy release rate and energy density factor based on both permeable and impermeable crack models are shown graphically and compared.

Numerical solution of integral equations system of the second kind by Block–Pulse functions

This paper endeavors to formulate Block–Pulse functions to propose solutions for the Fredholm integral equations system. To begin with we describe the characteristic of Block–Pulse functions and will go on to indicate that through this method a system of Fredholm integral equations can be reduced to an algebraic equation. Numerical examples presented to illustrate the accuracy of the method.

Three positive solutions of a system of Fredholm integral equations

In this paper, existence criteria for at least three positive solutions to the following system of Fredholm integral equations: u i ( t ) = ∫ 0 1 g i ( t , s ) f i ( s , u 1 ( s ) , u 2 ( s ) , ... , u n ( s ) ) d s , t ∈ [ 0 , 1 ] , 1 ≤ i ≤ n , are established by using the Leggett-Williams fixed-point theorem. The generality of the results obtained is illustrated through applications to some well-known boundary value problems.

Eigenvalues of a system of Fredholm integral equations

We consider the following system of Fredholm integral equations: u i=λ ∫ 0 1 g i(t,s)P i(s,u 1(s),u 2(s),…,u n(s))ds, tϵ[0,1], 1≤i≤n, where λ > 0. Our aim is to determine those values of λ such that the above system has a constant-sign solution. In addition, explicit intervals for λ will be presented. The generality of the results obtained is illustrated through applications to several well-known boundary value problems. We also extend the above problem to that on the half-line [0, ∞) u i=λ ∫ 0 ∞ g i(t,s)P i(s,u 1(s),u 2(s),…,u n(s))ds, tϵ[0,∞], 1≤i≤n, Finally, both the systems above are extended to the general case when λ is replaced by λ i.

Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method

In this paper rationalized Haar functions are developed to approximate the solutions of the linear Fredholm integral equations system. Properties of rationalized Haar functions are first presented, the operational matrix of the product of rationalized Haar functions vector is utilized to reduce the computation of Fredholm integral equations system to some algebraic equations. Finally, numerical result are given which support the theoretical results. †E-mail: mirzaee@mail.iust.ac.ir

Coherent backscattering of light by a layer of discrete random medium

We consider the problem of backscattering of light by a layer of discrete random medium illuminated by an obliquely incident plane electromagnetic wave. The multiply scattered reflected radiation is assumed to consist of incoherent and coherent parts, the coherent part being caused by the interference of multiply scattered waves. Formulas describing the characteristics of the reflected radiation are derived assuming that the scattering particles are spherical. The formula for the incoherent contribution reproduces the standard vector radiative transfer equation. The interference contribution is expressed in terms of a system of Fredholm integral equations with kernels containing Bessel functions. The special case of the backscattering direction is considered in detail. It is shown that the angular width of the backscattering interference peak depends on the polar angle of the incident wave and on the azimuth angle of the reflection direction.

The decomposition method applied to systems of Fredholm integral equations of the second kind

In this paper, the Adomian decomposition method is applied to solve systems of linear and nonlinear Fredholm integral equations of the second kind. Convergence of the method is proved and some examples are presented to illustrate the method.

Sufficient conditions for the reduction of a BVP for PDE with non-local and global boundary conditions to Fredholm integral equations (on a rectangular domain)

With the expanding domain of application of mathematical methods at present time there often arise problems connected with the study of partial differential equations that do not belong to any of the classical type. The elucidation of the correct formulation of these problems and the study of the specific properties of the solutions of similar equations are closely related to the study of questions of a general nature. In this respect, a time dependent PDE on a bounded rectangular region D⊂ R 2 with a more general non-local and global boundary conditions is considered. Its equivalent boundary value problem through an application of the method of contour integral of Rasulov [Methods of Contour Integration, North-Holland, 1967] is obtained. First, by using the theory of fundamental solutions, the essential (or necessary) conditions are obtained. Using these conditions one is able to give sufficient conditions to reduce the problem to system of Fredholm integral equations of second kind with at most weak singularities.

On a fieldless method for the computation of induction-generated heat in 3D non-ferromagnetic metal bodies

The paper deals with the mathematical and computer modeling of the induction heating of non-ferromagnetic metal bodies in harmonic electromagnetic fields. One of the main advantages of the presented method is the elimination of the surrounding air from the electromagnetic model, which strongly reduces the necessity of meshing and simplifies the computation. The task is formulated as a non-stationary quasi-coupled problem, with respect to the temperature dependencies of all important material parameters. Distribution of the eddy currents and Joule losses in the metal body is solved by a system of second-kind Fredholm integral equations. Existence and uniqueness of solution for the continuous as well as discrete problem is shown. Convergence results for the numerical scheme are presented. The theoretical analysis is supplemented with two illustrative examples.

ANALYSIS OF THE BENDING OF A UNIFORMLY LOADED RECTANGULAR PLATE WITH VARIABLE WIDTH CORNER SUPPORTS

The bending of a rectangular elastic plate under uniform distributed load and simply supported at the corners by equal-leg angles is studied analytically. The width of the supporting legs can be varied symmetrically about the plate axes giving mixed boundary conditions with supported and free edges. The solution is set up by using the Le'vy–Na'dai approach and the mixed boundary equation at the edges are written in the form of dual series equations. By choosing the proper finite integral transform, the dual series equations can be written as a system of inhomogeneous Fredholm integral equations. The inhomogeneous Fredholm integral equations are reduced to a set of inhomogeneous algebraic simultaneous equations by using Simpson's rule and solved by Picard iteration. The deflection along the middle line of the rectangular plate with various lengths of simple support legs has been calculated.

Numerical solution of laplace's equation on non-simply connected regions on R 2

We consider the interior Dirichlet problem for Laplace's equation on a non-simply connected two-dimensional regions with smooth boundaries.The solution is sought as the real part of a holomorphic function on the region, given as Cauchy-type integral.The approximate double layer density function is found by solving a system of Fredholm integral equations of second kind.Because of the non-uniqueness of the solution of the system we solve it using a technique based on the solution of the “Modified Dirichlet problem”.The Nystrom's method coupled with the trapezoidal rule is used as numerical integration scheme.The linear system derived from the integral equation is solved using the conjugate gradient applied to the normal equation.Theoretical and computational details of the method are presented.

M-integral analysis for microcrack-damaged anisotropic composite materials

This paper presents an M-integral analysis for the microcracked anisotropic composite materials. By using an elementary solution derived for a single finite crack subjected to a concentrated force on crack faces, the problem of strong interacting, arbitrarily oriented and located microcracks in an anisotropic composite materials is reduced to a system of Fredholm integral equations. The crack-tip fracture parameters, such as the stress intensity factors, are evaluated from a numerical solution of the system of integral equations. Its dependence on the coordinate system, calculation, and physical interpretation of the M-integral are discussed in the interaction problem. Finally, a numerical example of the damage evaluation by the M-integral analysis is given.