The numerical solution of Fredholm integral equations using product type quadrature formulas

Product type quadrature formulas are applied to obtain approximate solutions of Fredholm integral equations. A convergence theorem, and several numerical examples which demonstrate the efficacy of the technique, are presented.     

An adaptive numerical method for solving linear Fredholm integral equations of the first kind

A general procedure is presented for numerically solving linear Fredholm integral equations of the first kind. The approximate solution is expressed as a continuous piecewise linear (spline) function. The method involves collocation followed by the solution of an appropriately scaled stabilized linear algebraic system. The procedure may be used iteratively to improve the accuracy of the approximate solution. Several numerical examples are given.Supported in part by the Office of Naval Research under Contract No. NR 044-457.Supported in part by the National Science Foundation under Grant No. GJ-31827.     

On the numerical solution of Fredholm integral equations on unbounded intervals

The authors propose some numerical methods to solve Fredholm integral equations of the second kind on unbounded intervals. The proposed procedure includes projection methods and their discretized versions. Special attention is turned to the conditioning of the linear system corresponding to the finite-dimensional equation.     

Filter matrix based on interpolation wavelets for solving Fredholm integral equations

The interpolation wavelet is used to solve the Fredholm integral equation of the second kind in this study. Hence, by the extension of interpolation wavelets that [−1, 1] is divided to 2^N+1 (N ⩾  − 1) subinterval, we have polynomials with a degree less than M + 1 in each new interval. Therefore, by considering the two-scale relation the filter coefficients and filter matrix are used as the proof of theorems. The important point is interpolation wavelets lead to more sparse matrix when we try to solve integral equation by an approximate kernel decomposed to a lower and upper resolution. Using n-time, where (n ⩾ 2), two-scale relation in this method errors of approximate solution as O((2^−(N+1))ⁿ⁺¹). Also, the filter coefficient simplifies the proof of some theorems and the order of convergence is estimated by numerical errors.     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.     

Numerical solution for fuzzy Fredholm integral equations with upper-bound on error by splines interpolation

In this paper, a numerical procedure is proposed for the fuzzy linear Fredholm integral equations of the second kind by using splines interpolation. Besides, the convergence conditions and an upper-bound on error are derived. Finally, the advantages of the proposed method have been shown through simulation examples and comparison with Lagrange method.     

The numerical method of successive interpolations for Fredholm functional integral equations

A new numerical method for Fredholm functional integral equations is proposed. The method combines the fixed point technique with numerical integration and cubic spline interpolation. The convergence and the numerical stability of the method are proved and tested on some numerical examples.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

A method for accelerating the iterative solution of a class of Fredholm integral equations

The present paper extends the synthetic method of transport theory to a large class of integral equations. Convergence and divergence properties of the algorithm are studied analytically, and numerical examples are presented which demonstrate the expected theoretical behavior. It is shown that, in some instances, the computational advantage over the familiar Neumann approach is substantial.This authors acknowledge with pleasure conversations with Paul Nelson. Thanks are due also to Janet E. Wing, whose computer program was used in making the calculations reported in Section 8.This work was performed in part under the auspices of USERDA at the Los Alamos Scientific Laboratory of the University of California, Los Alamos, New Mexico.     

A natural interpolation formula for the numerical solution of singular integral equations with hilbert kernel

The direct quadrature method for the numerical solution of singular integral equations with Hilbert kernel is investigated and a very accurate natural interpolation formula for the approximation of the unknown function is proposed. It is further proved that this formula coincides with Nyström's natural interpolation formula for the Fredholm integral equation of the second kind equivalent to the original integral equation if the same quadrature rule is used in both cases.     

Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.     

An analysis of the numerical solution of Fredholm integral equations of the first kind

Summary This paper analyzes the numerical solution of Fredholm integral equations of the first kindTx=y by means of finite rank and other approximation methods replacingTx=y byT _N x=y _N ,N=1,2, .... The operatorsT andT _N can be viewed as operators from eitherL ₂[a, b] toL ₂[c,d] or as operators fromL [a, b] toL [c, d]. A complete analysis of the fully discretized problem as compared with the continuous problemTx=y is also given. The filtered least squares minimum norm solutions (LSMN) to the discrete problem and toT _N x=y are compared with the LSMN solution ofTx=y. Rates of convergence are included in all cases and are in terms of the mesh spacing of the quadrature for the fully discretized problem.     

A new numerical approach for the solution of nonlinear Fredholm integral equations system of second kind by using Bernstein collocation method

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B‐spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

A Numerical solution to the linear and nonlinear Fredholm integral equations using Legendre wavelet functions

Two methods for solving the Fredholm integral equation of the second kind in linear case, i.e. f (x) – λ ∫_a^b K (x,y)f (y)dy = g (x), and nonlinear case, i.e., f (x) = g (x) + λ ∫_a^b K (x,y)F (f (y))dy, are proposed. In order to solve the linear equation, the kernel K (x,y) as well as the functions f and g are initially approximated through Legendre wavelet functions. This leads to a system of linear equations its solution culminates in a solution to the Fredholm integral equation. In nonlinear case only K (x,y) is approximated by Legendre wavelet base functions. This leads to a separable kernel and makes it possible to employ a number of earlier methods in solving nonlinear Fredholm integral equation with separable kernels. Another feature of the proposed method is that it finds the solution as a function instead of specific solution points, what is done by the majority of the existing methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on the application of interpolatory quadrature rules to the numerical solution of singular integral equations

A Cauchy type singular integral equation of the first or the second kind can be numerically solved either directly or after its reduction (by the usual regularization procedure) to an equivalent Fredholm integral equation of the second kind. The equivalence of these two methods (that is, the equivalence both of the systems of linear algebraic equations to which the singular integral equation is reduced and of the natural interpolation formulae) is proved in this paper for a class of Cauchy type singular integral equations of the first kind and of the second kind (but with constant coefficients) for general interpolatory quadrature rules under sufficiently mild assumptions. The present results constitute an extension of a series of previous results concerning only Gaussian quadrature rules, based on the corresponding orthogonal polynomials and their properties.     

The numerical solution of Fredholm integral equations of the second kind with singular kernels

A numerical method is given for integral equations with singular kernels. The method modifies the ideas of product integration contained in [3], and it is analyzed using the general schema of [1]. The emphasis is on equations which were not amenable to the method in [3]; in addition, the method tries to keep computer running time to a minimum, while maintaining an adequate order of convergence. The method is illustrated extensively with an integral equation reformulation of boundary value problems for u–P(r ²)u=0; see [9].This research was supported in part by NSF grant GP-8554.