Solution Of Fredholm Integral Equations Research Articles

Numerical solution of Fredholm integral equations of the first kind with singular logarithmic kernel and singular unknown function via monic Chebyshev polynomials

Numerical solution of Fredholm integral equations of the first kind with singular logarithmic kernel and singular unknown function via monic Chebyshev polynomials

Graphon Filters: Graph Signal Processing in the Limit

Graph signal processing is an emerging field which aims to model processes that exist on the nodes of a network and are explained through diffusion over this structure. Graph signal processing works have heretofore assumed knowledge of the graph shift operator. Our approach is to investigate the question of graph filtering on a graph about which we only know a model. To do this we leverage the theory of graphons proposed by L. Lovasz and B. Szegedy. We make three key contributions to the emerging field of graph signal processing. We show first that filters defined over the scaled adjacency matrix of a random graph drawn from a graphon converge to filters defined over the Fredholm integral operator with the graphon as its kernel. Second, leveraging classical findings from the theory of the numerical solution of Fredholm integral equations, we define the Fourier-Galerkin shift operator. Lastly, using the Fourier-Galerkin shift operator, we derive a graph filter design algorithm which only depends on the graphon, and thus depends only on the probabilistic structure of the graph instead of the particular graph itself. The derived graphon filtering algorithm is verified through simulations on a variety of random graph models.

Asymptotic expansion of iterated Galerkin solution of Fredholm integral equations of the second kind with Green's kernel

We consider a Fredholm integral equation of the second kind with kernel of the type of Green’s function. Iterated Galerkin method is applied to such an integral equation. For r≥1, a space of piecewise polynomials of degree ≤r−1 with respect to a uniform partition is chosen to be the approximating space. We obtain an asymptotic expansion for the iterated Galerkin solution at the partition points. Richardson extrapolation is used to increase the order of convergence. A numerical example is considered to illustrate our theoretical results.

A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression

In this paper, we develop a machine learning method with the Least Squares Support Vector Regression (LS-SVR) for the numerical solution of Fredholm integral equations. Two different approaches are proposed for training the network by using the shifted Legendre kernel, the collocation and Galerkin LS-SVR approaches. As with the standard LS-SVR for known dataset regression, the formulation of the method gives rise to an optimization problem. An equivalent system of algebraic equations is then derived and in linear cases discussed in terms of the sparsity of the matrices and computational efficiency. Finally, the method is carried out on some numerical examples, including nonlinear and multidimensional cases to show the accuracy and efficiency of the method.

The stability study of numerical solution of Fredholm integral equations of the first kind with emphasis on its application in boundary elements method

In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically for Laplace and Helmholtz equations by obtaining Fourier transform of singular fundamental solutions applied in BEM. Analytical result presented in this paper is an extension of stability idea of radial basis functions (RBFs) used to interpolate scattered data described by Wendland in [51]. Similar to the interpolation, it is proved here mathematically that integral operators which have radial kernels with positive Fourier transform are strictly positive definite. Thanks to the stability idea presented in [51], a positive lower bound for eigenvalues of these integral operators is found here, explicitly.

METHODS FOR SOLVING FREDHOLM INTEGRAL EQUATIONS

The article deals with integral equations that are widely used in various sections of physics (theory of waves on the surface of liquids, quantum mechanics, problems of spectroscopy, crystallography, acoustics, analysis and diagnostics of plasma, etc.), Geophysics (problems of gravimetry, kinematic problems of seismics), mechanics (vibrations of structures), etc. When the physics introduced aftereffect, it is not enough ordinary differential equations or partial differential equations, otherwise the initial data would determine the future state. To take into account the continuous sequence of previous States, we need to use integral and integro-differential equations, where the sign of the integral appears functions of parameters that characterize the system, which depend on time for some period preceding the moment under consideration. In this article we have considered the solution of Fredholm integral equations of the second kind by the method of successive approximations and the method of iterated nuclei.

The reverse interpolation and its application in the numerical solutions of Fredholm integral equations of the second kind

In this paper, we introduce a new interpolation method that is easy to use and its interpolating function can be explicitly expressed. This interpolation method can be used for a wide spectral type of functions, since it has the ability to change the form of the interpolating function. This interpolation inspired by the Shepard type interpolation. We examine the ability of this interpolation to piecewise functions and rational functions and compare it with Lagrange interpolation and linear interpolation by using Bernstein polynomials basis. We applied this interpolation for an integral equation, and we presented a linear system of equations to approximate the solution of a Fredholm integral equation by collocation method. In the collocation method, the Reverse Interpolation expresses the approximate solution in the form of a linear combination of some basic functions. Several examples are presented to illustrate the effectiveness of the proposed method and the numerical results confirm the desired accuracy.

Моноиды натуральных чисел в теоретико-числовом методе в приближенном анализе

В работе для каждого моноида M натуральных чисел определён новый класс периодических функций $$M_s^\alpha$$, который является подклассом известного класса Коробова периодических функций $$E_s^\alpha$$. Относительно нормы $$\|f(\vec{x})\|_{E_s^\alpha}$$ класс $$M_s^\alpha$$ является несепарабельным банаховым подпространством класса $$E_s^\alpha$$.Установлено, что класс $$M_s^\alpha$$ замкнут относительно действия интегрального оператора Фредгольма и на этом классе разрешимо интегральное уравнение Фредгольма второго рода. В работе получены оценки нормы образа интегрального оператора, которые содержат норму ядра и s-ю степень дзета-функции моноида M. Получены оценки на параметр $$\lambda$$, при которых интегральный оператор $$A_{\lambda,f}$$ является сжатием. Доказана теорема о представлении единственного решения интегрального уравнения Фредгольма второго рода в виде ряда Неймана.В работе рассмотрены вопросы решения дифференциального уравнения с частными производными с дифференциальным оператором $$Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)$$ в пространстве $$M^\alpha_{s}$$, который зависит от арифметических свойств спектра этого оператора.В работе обнаружен парадоксальный факт, что для моноида $$M_{q,1}$$ чисел сравнимых с 1 по модулю q квадратурная формула с параллелепипедальной сеткой для допустимого набора коэффициентов по модулю q точна на классе $$M_{q,1,s}^\alpha$$. Более того, это утверждение остается верным и для класса $$M_{q,a,s}^\alpha$$ с 1 < a < q, когда q - простое число. Так как функции из класса $$M_{q,a,s}^\alpha$$ с 1 < a < q не имеют нулевого коэффициента Фурье $$C(\vec{0})$$, то при простом q сумма значений функции по узлам соответствующей параллелепипедальной сетки будет нулевой.

A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle

In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate.

Nystrom method for solution of fredholm integral equations of the second kind under interval data

Nystrom method for solution of fredholm integral equations of the second kind under interval data

A Version of the Projection-Iterative Method for the Solution of Fredholm Integral Equations of the Second Kind

We discuss the problem of application of Alpert’s multiwavelets to the solution of Fredholm integral equations by the projection-iterative method.

The Piecewise Polynomial Collocation Method for The Solution of Fredholm Equation of Second Kind By Using AGE Iteration

In this study, we study a piecewise approximate solution of Fredholm integral equation of the second kind by using the collocation discretization scheme. The discretization scheme has been used to build up a piecewise approximation equation which is the approximation equation form a system of linear equation. In the effort to get the numerical solution, the Gauss-Seidel iterative method and Alternating Group Explicit (AGE) have been treated as a linear solver. In addition to that, the formulation and the implementation of AGE method are included. Based on the numerical finding, the result showed that Alternating Group Explicit (AGE) iterative method is better than the Gauss-Seidel method.

English

English

A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations

Abstract In this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

О специальном варианте метода подобластей решения интегральных уравнения Фредгольма второго рода

Предложен и теоретически обоснован специальный вариант метода подобластей на базе полиномов Канторовича приближенного решения интегральных уравнений Фредгольма второго рода в пространстве гладких функций. A special variant of the subdomain method based on Kantorovich polynomials is proposed and theoretically substantiated for the approximate solution of Fredholm integral equations of the second kind in the space of smooth functions.

On the Existence of Approximate Solution of Fredholm Integral Equation of the First Kind by Band-Limited Scaling Function

In this paper, we study the existence and uniqueness of the solution of Fredholm integral equation of the first kind with convolution type kernel. These results are based on band-limited scaling function which is generated by a class of band-limited wavelets. Since these band-limited functions are infinitely differentiable and possess rapid decay property, methods based on these functions would be highly accurate. Finally, convergence analysis has been discussed to validate the approximate solution.

Analysis of Nonclassical Fracture Problems for Prestressed Bodies with Interacting Cracks

We consider two types of nonclassical fracture mechanisms, namely, the fracture of cracked bodies with initial (residual) stresses acting along the crack planes and the fracture of solids under compression along parallel cracks. To investigate nonaxisymmetric and axisymmetric problems for infinite solids containing two parallel coaxial cracks or a periodic set of coaxial parallel cracks, we use a combined analytic-numerical method within the framework of the three-dimensional linearized mechanics of solids. The analysis involves the representation of stresses and displacements in the linearized theory via harmonic potential functions. With the use of the integral Fourier–Hankel transformations, the problems are reduced to the solution of Fredholm integral equations of the second kind. This approach allows us to investigate problems in a unified general form for compressible and noncompressible homogeneous isotropic or transversely isotropic elastic bodies with an arbitrary structure of the elastic potential, and the material specification of the model is carried out only in the stage of numerical analysis of the resolving equations obtained in the general form. The effects of initial stresses on the stress intensity factors are analyzed for highly elastic materials and layered composites (modeled as transversely isotropic elastic bodies). The “resonance-like” effects are revealed when compressive initial stresses reach the values corresponding to the local loss of stability of the material in the vicinity of cracks, which, according to the indicated combined method, allows one to determine the critical (limiting) load parameters under the conditions of compression of the body along the cracks. The conclusions concerning the dependences of the stress intensity factors and critical (limiting) parameters of compression on the geometric parameters of the problems are formulated as well as concerning the dependences on physical and mechanical characteristics of the materials.

High order iterative methods for matrix inversion and regularized solution of fredholm integral equation of first kind with noisy data

The motivation of the present work is to propose high order iterative methods with a recurrence formula for approximate matrix inversion and provide regularized solution of Fredholm integral equation of first kind with noisy data by an algorithm using the proposed methods. From the given family of methods of orders p = 7,11,15,19 are applied to solve problems of Fredholm integral equation of first kind. From the literature, iterative methods of same orders are used to solve the considered problems and numerical comparisons are shown through tables and figures.

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.