Homogeneous Fredholm Integral Equation Research Articles

Green functions for four-point boundary value problems with applications to heterogeneous beams

The main objective of this study is to define the Green functions for four-point boundary value problems. It is a further aim to clarify what properties the Green functions have and to present a method for calculating the elements of these Green functions. The examples are related to two heterogeneous beams with four supports: the (first) [second] beam is (fixed)[pinned] at the endpoints while the intermediate supports are two rollers. Determination of the eigenfrequencies leads to four-point eigenvalue problems associated with homogeneous boundary conditions. Utilizing the Green functions that belong to these eigenvalue problems we can transform those into eigenvalue problems governed by homogeneous Fredholm integral equations. Then a numerical solution is computed by reducing the homogeneous Fredholm integral equations into algebraic eigenvalue problems.

On Data Dependency and Solutions of Nonlinear Fredholm Integral Equations with the Three-Step Iteration Method

In this study, the solution of the second type of homogeneous nonlinear Fredholm integral equations is investigated using a three-step iteration algorithm. In other words, it has been shown that the sequences obtained from this algorithm converge to the solution of the mentioned equations. Also, data dependency is obtained for the second type of homogeneous nonlinear Fredholm integral equations and this result is supported by an example.

Solving a singular integral equation for the one-dimensional Coulomb problem

A new integral equation that describes the behavior of the momentum space wave function for the one-dimensional Coulomb potential is proposed. The obtained result turned out to be a homogeneous Fredholm integral equation of the second kind and a singular integral equation, because its kernel has a singularity at some point in the momentum space. A nontriviality of the method of solving this singular integral equation lies in the application of the integral representation for its integral kernel. The technique applied in this paper made it possible to show that the wave function in the momentum representation is simultaneously a solution of the homogeneous Fredholm integral equation of the second kind and of the linear Volterra integral equation of the second kind. Since a linear Volterra integral equation of the second kind was easily transformed into a second order linear inhomogeneous differential equation with constant coefficients, the eigenfunctions and eigenvalues in the one-dimensional Coulomb problem were found without any difficulties. Such a circumstance may indicate the validity of the new integral equation and the proposed method of its solving.

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

It is our main objective to find the critical load for three beams with cross sectional heterogeneity. Each beam has three supports, of which the intermediate one is a spring support. Determination of the critical load for these beams leads to three point boundary value problems (BVPs) associated with homogeneous boundary conditions—the mentioned BVPs constitute three eigenvalue problems. They are solved by using a novel solution strategy based on the Green functions that belong to these BVPs: the eigenvalue problems established for the critical load are transformed into eigenvalue problems governed by homogeneous Fredholm integral equations with kernels that can be given in closed forms provided that the Green function of each BVP is known. Then the eigenvalue problems governed by the Fredholm integral equations can be manipulated into algebraic eigenvalue problems solved numerically by using effective algorithms. It is an advantage of the way we attack these problems that the formalism established and the results obtained remain valid for homogeneous beams as well. The numerical results for the critical forces can be applied to solve some stability problems in the engineering practice.

Green functions for three-point boundary value problems governed by differential equation systems with applications to Timoshenko beams

The present paper is devoted to the issue of the Green function matrices that belongs to some three-point boundary- and eigenvalue problems. A detailed definition is given for the Green function matrices provided that the considered boundary value problems are governed by a class of ordinary differential equation systems associated with homogeneous boundary and continuity conditions. The definition is a constructive one, i.e., it provides the means needed for calculating the Green function matrices. The fundamental properties of the Green function matrices—existence, symmetry properties, etc.—are also clarified. Making use of these Green functions, a class of three-point eigenvalue problems can be reduced to eigenvalue problems governed by homogeneous Fredholm integral equation systems. The applicability of the novel findings is demonstrated through a Timoshenko beam with three supports.

Vibration of an axially loaded heterogeneous fixed–fixed beam with an intermediate roller support

The present paper investigates how an axial load can change the natural frequencies of heterogeneous fixed–fixed beams with an intermediate roller support. The problem is treated as a three-point boundary value problem (eigenvalue problem) that is paired with homogeneous boundary conditions. The Green functions are determined for the unloaded and pre-loaded beams as well—in the later case, both for compression and tension. With these, the eigenvalue problems can be transformed into eigenvalue problems governed by a homogeneous Fredholm integral equations. It is then replaced by an algebraic eigenvalue problem, that is solved numerically with an effective solution algorithm which is based on the boundary element method.

BUCKLING OF BEAMS WITH A BOUNDARY ELEMENT TECHNIQUE

The present work is devoted to the buckling study of non-homogeneous fixed- fixed beams with intermediate spring support. The stability issue of these beams leads to three-point boundary value problems. If the Green functions of these boundary value problems are known, the differential equations of the stability problems that contain the critical load sought can be turned into eigenvalue problems given by homogeneous Fredholm integral equations. The kernel function of these equations can be calculated from the associated Green functions. The eigenvalue issues can be reduced to algebraic eigenvalue problems, which are subsequently solvable numerically with the use of an efficient algorithm from the boundary element method. Within this article, the critical load findings of these beams are compared to those obtained using commercial finite element software, and the results are in excellent correlation.

Stability of heterogeneous beams with three supports through Green functions

The present paper is devoted to the issue how the critical load of some heterogeneous beams with three supports can be determined by using Green functions. The stability problems of these beams are equivalent to three-point boundary value problems, paired with homogeneous boundary conditions. If the Green functions of these boundary value problems are known, the eigenvalue problems that provide the critical load can be transformed into eigenvalue problems governed by homogeneous Fredholm integral equations. The later eigenvalue problems can be reduced to algebraic eigenvalue problems which then can be solved numerically with effective algorithms.

Solutions for the vibration and stability problems of heterogenous beams with three supports using Green functions

The goal of this study is to calculate the eigenvalues that provide the eigenfre- quencies and the critical loads for two heterogeneous beams with three supports: the (first) [second] beam is (fixed)[pinned] at the left end, the intermediate support is a roller while the right end of the beams can move vertically but the rotation is prevented there. The beams are referred to as FrsRp and PrsRp beams. Determination of the (eigenfrequencies) [critical loads] leads to three point eigenvalue problems associated with homogeneous boundary con- ditions. With the Green functions that belong to these eigenvalue problems we can transform them into eigenvalue problems governed by homogeneous Fredholm integral equations. The eigenvalue problems can then be reduced to algebraic eigenvalue problems that are solvable numerically by utilizing effective solution algorithms.

A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L2-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C0-conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

Vibration of an axially loaded heterogeneous pinned-pinned beam with an intermediate roller support

The present paper is devoted to the issue of what effect the axial load (compressive or tensile) has on the eigenfrequencies of a heterogeneous pinned-pinned beam with an intermediate roller support (called a PrsP beam). This problem is a three point boundary value problem (eigenvalue problem) associated with homogeneous boundary conditions. If the Green functions of the three point boundary value problem (BVP) are known the eigenvalue problem that provide the eigenfrequencies for the beam loaded axially can be transformed into an eigenvalue problem governed by a homogeneous Fredholm integral equation. The later eigenvalue problems can be reduced to an algebraic eigenvalue problem which then can be solved numerically by using an effective solution algorithm which is based on the boundary element method.

Biorthogonal Systems of Analytic Functions Generated by a Regular Triangle

We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as ‘half’ of the fundamental set. Using the method of contracting mappings in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series.

On the eigenfrequencies of preloaded rotationally restrained extensible circular beams by Green’s functions

The article is devoted to the vibrations of heterogeneous curved beams with centerline extensibility accounted for. The end supports are rotationally restrained pins, and the effect of a central, either compressive or tensile, force (preload) is incorporated into the model. The coupled differential equations that govern the problem are derived from the principle of virtual work. These are replaced by a system of homogeneous Fredholm integral equations. The kernel of the integral equation system is the Green’s function matrix, which is given in closed form. The eigenvalue problem determined by the homogeneous Fredholm integral equation system is solved numerically, and the results obtained are presented graphically. If the spring stiffness tends to (zero) [infinity], the beam behaves as if it were (pinned–pinned) [fixed–fixed]. When the external force tends to zero, the model returns the eigenfrequencies of the free vibrations.

Source-function synthesis in the self-consistent problem of radiation

A new approach to a rigorous theory of aperture antennas is proposed, which is based on solution of the self-consistent problem of radiation. It is ascertained that this formulation of the problem leads to a system of homogeneous Fredholm integral equations of the second kind—i.e., a problem for eigenfunctions and eigenvalues that provides a basis for solving the problem of field generation in the open space. The passage to Fredholm integral equations of the first kind allows one to solve the task of synthesis of the source function for a given radiation field.

A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen–Loeve expansion of Gaussian random fields. We develop, validate, and discuss an algorithm based on the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor-product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals, allowing the implementation of a fast and efficient algorithm that provides considerable reduction in CPU time, when compared with classical Galerkin methods. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We illustrate the numerical solution of a diffusion problem with random input data with the present method. In this problem, accuracy was retained after dropping the coefficients below a threshold value that was numerically determined. A similar method with scaling functions rather than wavelet functions does not need a discrete wavelet transform and leads to eigenvalue systems with better conditioning but lower sparsity.

Necessary and Sufficient Conditions for the Solvability of Linear Boundary-Value Problems for the Fredholm Integrodifferential Equations

We propose a method for the investigation and solution of linear boundary-value problems for the Fredholm integrodifferential equations based on the partition of the interval and introduction of additional parameters. Every partition of the interval is associated with a homogeneous Fredholm integral equation of the second kind. The definition of regular partitions is presented. It is shown that the set of regular partitions is nonempty. A criterion for the solvability of the analyzed problem is established and an algorithm for finding its solutions is constructed. Integrodifferential equations are often encountered in applications as mathematical models of various processes in natural sciences. In the monograph [1], their role in the study of hereditary processes is indicated and a survey of the papers devoted to the boundary-value problems for the Volterra and Fredholm integrodifferential equations is presented. The solvability of various problems for the Fredholm integrodifferential equations and approximate methods for finding their solutions are studied by many authors [2–11]. Linear boundary-value problems for Fredholm integrodifferential equations with degenerate kernels are considered in the monograph [12]. Necessary and sufficient conditions for the solvability of boundary-value problems are established and the algorithms for finding the solutions of these problems are constructed. In the present paper, we consider a linear two-point boundary-value problem

Spectral element approximation of Fredholm integral eigenvalue problems

We consider the numerical approximation of homogeneous Fredholm integral equations of second kind, with emphasis on computing truncated Karhunen–Loève expansions. We employ the spectral element method with Gauss–Lobatto–Legendre (GLL) collocation points. Similar to the piecewise-constant finite elements, this approach is simple to implement and does not lead to generalized discrete eigenvalue problems. Numerical experiments confirm the expected convergence rates for some classical kernels and illustrate how this approach can improve the finite element solution of partial differential equations with random input data.

Fredholm's equations for subwavelength focusing

Subwavelength focusing (SF) is a very useful tool that can be carried out with the use of left hand materials for optics that involve the range of the microwaves. Many recent works have described a successful alternative procedure using time reversal methods. The advantage is that we do not need devices which require the complicated manufacture of left-hand materials; nevertheless, the theoretical mathematical bases are far from complete because before now we lacked an adequate easy-to-apply frame. In this work we give, for a broad class of discrete systems, a solid support for the theory of electromagnetic SF that can be applied to communications and nanotechnology. The very central procedure is the development of vector-matrix formalism (VMF) based on exploiting both the inhomogeneous and homogeneous Fredholm's integral equations in cases where the last two kinds of integral equations are applied to some selected discrete systems. To this end, we first establish a generalized Newmann series for the Fourier transform of the Green's function in the inhomogeneous Fredholm's equation of the problem. Then we go from an integral operator equation to a vector-matrix algebraic one. In this way we explore the inhomogeneous case and later on also the very interesting one about the homogeneous equation. Thus, on the one hand we can relate in a simple manner the arriving electromagnetic signals with those at their sources and we can use them to perform a SF. On the other hand, we analyze the homogeneous version of the equations, finding resonant solutions that have analogous properties to their counterparts in quantum mechanical scattering, that can be used in a proposed very powerful way in communications. Also we recover quantum mechanical operator relations that are identical for classical electromagnetics. Finally, we prove two theorems that formalize the relation between the theory of Fredholm's integral equations and the VMF we present here.

Spectral decomposition of random fields defined over the generalized Cantor set

Solutions of a set of homogeneous Fredholm integral equations that appear in field analyses on approximations of fractal sets are presented. If the support is a stochastic fractal, then the field will inherit this property. A simple representation of such functions in terms of a basis is desirable. A solution to this problem is provided by the Karhunen–Loeve decomposition of stochastic functions with known covariance. In the problem at hand, the covariance is defined by the fractal nature of the support. Here we provide a rule by which the spectral decomposition of the covariance is evaluated for any iteration/scale of a generalized Cantor set. This problem is identical to the solution of a Fredholm equation of second kind.

Eigenvalue method for computing size effect of cohesive cracks with residual stress, with application to kink-bands in composites

The previously developed eigenvalue method for computing the size effect of cohesive crack model is extended to the cohesive crack model with a finite residual stress. In this model, the structure size for which a specified relative length of kink-band corresponds to the maximum load is obtained as an eigenvalue of a homogeneous Fredholm integral equation. This new method is direct and much more efficient than the classical finite element approach in which the entire load-deflection history must be computed to obtain the maximum load. A secondary purpose of the paper is to apply the new method to the effect of structure size on the compressive strength of unidirectional fiber–polymer composites failing by propagation of kink-band with fiber microbuckling. The kink-band is simulated by a cohesive crack model with a linear compressive softening law and a finite residual stress. The simulation shows that the specimens tested have a negative–positive geometry, i.e., the energy release rate of the kink-band for a unit load first decreases but at a certain length of propagation begins to increase. Finally the effect of shape of the softening law of cohesive crack on the size effect curve is studied by using the new eigenvalue method. It is shown that, for a negative–positive geometry, the size effect on the peak load depends on the entire softening curve if the specimens is not too small.