Solution Of Fredholm Integral Equations Research Articles

Simulation of multi-dimensional random fields by Karhunen–Loève expansion

The Karhunen–Loève (K–L) expansion method is a powerful tool for simulating stationary and nonstationary, Gaussian and non-Gaussian stochastic processes with explicitly known covariance functions. Since the K–L expansion requires the solution of Fredholm integral equation of the second kind, it is generally not feasible to simulate multi-dimensional random fields. This is because even the numerical solution of the Fredholm integral multi-dimensional eigenvalue problem is difficult to obtain. In order to address this problem, this paper develops a consistent generalization of K–L expansion for multi-dimensional random field simulation. The new method decomposes an n-dimensional random field into a total of n stochastic processes, each can be represented by using the traditional K–L expansion. Thus, the developed method is embedded into the well-established framework of the K–L expansion for simulating stochastic process, and obviating the need for solving the multi-dimensional integral eigenvalue problems. Four examples, including random fields with different kinds of covariance functions, are used to demonstrate the application of the proposed method.

Interface Crack at Orthotropic Media

This paper deals with the diffraction of SH-waves by a Griffith crack located at the interface of two bonded dissimilar orthotropic half spaces. The mixed boundary value problem has been reduced to the solution of Fredholm integral equation of second kind by applying Fourier and Abel transforms. Stress intensity factor at the tip of the crack has been calculated by solving integral equation using perturbation method for low frequency and plotted against dimensionless frequency.

Graph theoretical methods for efficient stochastic finite element analysis of structures

Stochastic finite element method (StFEM) is a robust tool for uncertainty quantification of engineering systems having random properties. Nevertheless, the matrices involved in this method are very large compared to their deterministic counterparts. Thus, the computational aspects of StFEM are of great importance to be optimized. In this paper, an efficient StFEM is developed for analysis of structures. For this purpose, a method based on graph concepts is presented and extended to StFEM and recently developed stochastic spectral finite element method (StSFEM) procedures. Here, mathematical remedies are incorporated to enhance the analysis performance. Firstly, a graph theoretical method is presented for swift numerical solution of Fredholm integral equation arising from Karhunen–Loève expansion, which greatly reduces the existing computational cost, and can even be applied to the domain without symmetry. Secondly, a preconditioner is applied to decompose the matrices to Kronecker products of sub-matrices, and then graph product rules are utilized to solve the governing linear equation of cyclically symmetric models without inversing the final matrix, while only a small matrix is inversed instead. The proposed method provides significant improvement in the stochastic structural analysis. Illustrative examples demonstrate the efficiency and accuracy of the present method as a swift analysis.

Normal impact response of a saturated porous cylinder with a penny-shaped crack

The transient response of a saturated porous cylinder containing a penny-shaped crack and subjected to a suddenly applied normal loading is investigated. The general field equations of Biot’s theory are applied in the formulation in conjunction with the Laplace and integral transforms to reduce the mixed boundary-value problem to the solution of Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used, and the time-dependent magnitudes of the local stress field near the crack border measured in terms of the dynamic stress-intensity factor are computed for selected material property values. These results are presented and compared to the results of dry medium to find out the influence of pore fluid and its extent on the dynamic stress-intensity factor.

Multiscale approximation of the solution of weakly singular second kind Fredholm integral equation in Legendre multiwavelet basis

Numerical solution of Fredholm integral equation of second kind with weakly singular kernel is obtained in this paper by employing Legendre multi-wavelet basis. The low- and high-pass filters for two-scale relations involving Legendre multiwavelets having four or five vanishing moments of their wavelets have been derived and are used in the evaluation of integrals for the multiscale representation of the integral operator. Explicit expressions for the elements of the matrix associated with the multiscale representation are given. An estimate for the Hölder exponent of the solution of the integral equation at any point in its domain is obtained. A number of examples is provided to illustrate the efficiency of the method developed here.

ОБ ОСОБЕННОСТЯХ ИДЕНТИФИКАЦИИ НЕОДНОРОДНЫХ ПОРОУПРУГИХ ХАРАКТЕРИСТИК ПОЛОГО ЦИЛИНДРА

The coefficient inverse problem of finding poroelastic characteristics of ingomogeneous hollow cylinder is considered. Direct problem poroelasticity for the cylinder is solved by a method of adjustment. As additional information for solving the inverse problem appears measured displacement or pressure at the outer edge of the cylinder. To solve the inverse problem, based on the linearization method derived operator relations, establishing a relationship between a desired and measured characteristics. Poroelastictic characteristics restored in two phases. The first phase was an initial approximation. The second stage amendments to the recoverable characteristics are determined from the solution of Fredholm integral equations of the 1st kind. For solutions of Fredholm integral equations of the 1st kind used method of A.N. Tikhonov. Concrete examples of the reconstruction of the characteristics of inhomogeneous poroelastic cylinder are given. Keywords: poroelasticity, inverse problem, hollow cylinder, integral equations.

A modified approach to numerical solution of Fredholm integral equations of the second kind

A modified approach to obtain approximate numerical solutions of Fredholm integral equations of the second kind is presented. The error bound is explained by the aid of several illustrative examples. In each example, the approximate solution is compared with the exact solution, wherever possible, and an excellent agreement is observed. In addition, the error bound in each example is compared with the one obtained by the Nyström method. It is found that the error bound of the present method is smaller than the ones obtained by the Nyström method. Further, the present method is successfully applied to derive the solution of an integral equation arising in a special Dirichlet problem.

A novel stochastic-spectral finite element method for analysis of elastodynamic problems in the time domain

This article proposes a stochastically-tuned spectral finite element method (SFEM) which is applied to elastodynamic problems. Stochastic finite element method is an efficient numerical method incorporating randomness for uncertainty quantification of engineering systems. On the other hand, SFEM is an excellent remedy for solving dynamic problems with fine accuracy, which employs Lobatto polynomials leading to reduction of domain discretization and making diagonal mass matrices. The presented method simultaneously collects the advantages of the both methods in order to solve stochastically linear elastodynamic problems with suitable computational efficiency and accuracy. Furthermore, spectral finite element is also proposed for numerical solution of Fredholm integral equation associated with Karhunen–Loeve expansion followed by the presented hybrid method which enhances the efficiency of the methodology. Various types of numerical examples are prepared so as to demonstrate advantages of the proposed stochastic SFEM.

Coupled Fixed Point Theorems with New Implicit Relations and an Application

We introduce two new classes of implicit relations S and S′ where S′ is a proper subset of S, and these classes are more general than the class of implicit relations defined by Altun and Simsek (2010). We prove the existence of coupled fixed points for the maps satisfying an implicit relation in S. These coupled fixed points need not be unique. In order to establish the uniqueness of coupled fixed points we use an implicit relation S′, where S′⊂S. Our results extend the fixed point theorems on ordered metric spaces of Altun and Simsek (2010) to coupled fixed point theorems and generalize the results of Gnana Bhaskar and Lakshimantham (2006). As an application of our results, we discuss the existence and uniqueness of solution of Fredholm integral equation.

On the solution of Fredholm integral equations based on spline quasi-interpolating projectors

We use spline quasi-interpolating projectors on a bounded interval for the numerical solution of linear Fredholm integral equations of the second kind by Galerkin, Kantorovich, Sloan and Kulkarni schemes. We get theoretical results related to the convergence order of the methods, in case of quadratic and cubic spline projectors, and we describe computational aspects for the construction of the approximate solutions. Finally, we give several numerical examples, that confirm the theoretical results and show that higher orders of convergence can be obtained by Kulkarni’s scheme.

Emerging problems in approximation theory for the numerical solution of the nonlinear Schrödinger equation

We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schr?dinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solution of Fredholm integral equations and the identification of coefficients and parameters of monomial-exponential sums. The aim of the paper is twofold: propose a method for solving the above mentioned problems under particular hypothesis; arise interest in the issues illustrated to achieve an effective method for solving the problem under more general assumptions.

The estimation of solution of Fredholm integral equation in three variables

The aim of the present paper is to establish an existence result (Theorem 2.1) as well as an estimation result (Theorem 2.2) for a functional equation in three variables.

The Approximate Solution of Fredholm Integral Equations with Oscillatory Trigonometric Kernels

A method for approximating the solution of weakly singular Fredholm integral equation of the second kind with highly oscillatory trigonometric kernel is presented. The unknown function is approximated by expansion of Chebychev polynomial and the coefficients are determinated by classical collocation method. Due to the highly oscillatory kernels of integral equation, the discretised collocation equation will give rise to the computation of oscillatory integrals. These integrals are calculated by using recursion formula derived from the fundamental recurrence relation of Chebyshev polynomial. The effectiveness and accuracy of the proposed method are tested by numerical examples.

Superconvergent Nyström and degenerate kernel methods for Hammerstein integral equations

In a recent paper, we introduced new methods called superconvergent Nyström and degenerate kernel methods for approximating the solution of Fredholm integral equations of the second kind with a smooth kernel. In this paper, these methods are applied to numerically solve the Hammerstein equations. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤r−1, we prove that, as for Fredholm integral equations, the proposed methods exhibit convergence orders 3r and 4r for the iterated version. Several numerical examples are given to demonstrate the effectiveness of the current methods.

Solution of Fredholm integral equation of the first kind involving some dual integral equations

A solution of some dual integral equations with Bessel function as kernel for an infinite thick plate, is reduced a Fredholm integral equation of the first kind which is solved by iteration method.

Intervallic Coiflets for Numerical Calculation of Dynamic Stress Intensity Factor

There are lots of practical problems which are related to the solution of Fredholm integral equations of the second kind. The present work proposes intervallic Coiflets for solving the equations. Illustrative problem involving dynamic stress and electric fields of a cracked piezoelectric excited by anti-plane shear wave is addressed. Permeable boundary condition has been used to obtain a pair of dual integral equations of the symmetric and antisymmetric parts which can be reduced to the solutions of two Fredholm integral equations of the second kind. The dynamic stress intensity factor is expressed in terms of the right-end values of two unknown functions in Fredholm integral equations. The two unknown functions are solved by intervallic Coiflets which have less the endpoints error. And intervallic Coiflets have low calculation cost and high accuracy due to the wavelet expansion coefficients are exactly obtained without calculating the wavelet integrations. The calculation results agree well with the existing method, which show the high accuracy of the estimation and demonstrate validity and applicability of the method.

Using orthonormal wavelet basis in Petrov‐Galerkin method for solving Fredholm integral equations of the second kind

PurposeThe purpose of this paper is to explain the choice of Alpert multi‐wavelet as basis functions to discrete Fredholm integral equation of the second kind by using Petrov‐Galerkin method.Design/methodology/approachIn this process, two kinds of matrices are obtained from inner product between basis of test space and trial space; some of them are diagonal with positive elements and some others are invertible. These matrices depend on type of selection of test and trial space basis.FindingsIn this process, solution of Fredholm integral equation of the second kind is found by solving the generated system of linear equations.Originality/valueIn previous work, convergence of Petrov‐Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree. In point of computation, because of appearance of diagonal and invertible matrices, a small dimension system with a more accurate solution is obtained. The numerical examples illustrate these facts.

Numerical solution of Fredholm integral equations of first kind by two-dimensional trigonometric wavelets in holder space C α([a, b

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 Cα([a, b]).

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem II. High dimensional problems

In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral mean value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral mean value theorem. However the mean value theorem is valid for multiple integrals, we apply one dimensional integral mean value theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.

Numerical solution of Fredholm integral equations of the second kind by using fuzzy transforms

In this paper, we introduce an approach by using inverse fuzzy transforms based on the fuzzy partition with combination in collocation technique for the numerical solution of Fredholm integral equations of the second kind. The main advantage of this approach is to reduce the problem to the linear system of equations. We present the convergence theorem for this method. Finally, we give two examples to illustrate the efficiency of the proposed method.