Regularized collocation method for Fredholm integral equations of the first kind

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An “unregularized” use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.     

Approximate solutions of Fredholm integral equations of the second kind

This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail.     

Fredholm equations of the first kind

A method is described for solving the Fredholm integral equation of the 1st kind by passage to the moment L-problem. The problem is reduced to a linear programming problem. A bound of the method for a particular normed space is derived. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 34–38.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

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 approximate solution of one-dimensional singular integral equations of first kind

We prove an estimate for the error in approximate solution of one-dimensional singular integral equations. The estimate is obtained by an approximation of the kernel. For a specific problem we give a comparison of numerical results. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Convergence analysis of a regularized approximation for solving Fredholm integral equations of the first kind

In this paper, we suggest a convergence analysis for solving Fredholm integral equations of the first kind using Tikhonov regularization under supremum norm. We also provide an a priori parameter choice strategy for choosing the regularization parameter and obtain an error estimate.     

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.     

An approximate solution of a Fredholm integral equation of the first kind by the residual method

A Tikhonov finite-dimensional approximation is applied to a Fredholm integral equation of the first kind. This allows using a variational regularization method with a regularization parameter from the residual principle and reducing the problem to a system of linear algebraic equations. The accuracy of the approximate solution is estimated with allowance for the error of the finitedimensional approximation of the problem. The use of this approach is illustrated by solving an inverse boundary value problem for the heat conduction equation.     

An algorithm based on the regularization and integral mean value methods for the Fredholm integral equations of the first kind

In this paper, an algorithm based on the regularization and integral mean value methods, to handle the ill-posed multi-dimensional Fredholm equations, is introduced. The application of this algorithm is based on the transforming the first kind equation to a second kind equation by the regularization method. Then, by converting the first kind to a second kind, the integral mean value method is employed to handle the resulting Fredholm integral equations of the second kind. The efficiency of the approach will be shown by applying the procedure on some examples.     

Solution of non-linear Fredholm integral equations of the first kind using modified homotopy perturbation method

In this paper, we present a modification to homotopy perturbation method for solving some non-linear Fredholm integral equations of the first kind. Solved problems reveal that the proposed method is very effective and simple and in some cases it gives the exact solution rather than the approximated one.     

On linear Fredholm integro-differential equations of first kind

We study the question whether linear one-dimensional integro-differential equations with constant limits of integration (equations of Fredholm type) containing no free differential expression (equations of first kind) can be reduced to integral equations of first kind and to Fredholm integro-differential equations of second kind.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 20–27.     

Polynomially based multi-projection methods for Fredholm integral equations of the second kind

In this paper, we propose a multi-projection and iterated multi-projection methods for Fredholm integral equations of the second kind with a smooth kernel using polynomial bases. We obtain super-convergence rates for the approximate solutions, more precisely, we prove that in M-Galerkin and M-collocation methods not only iterative solution approximates the exact solution u in the supremum norm with the order of convergence n^-4k, but also the derivatives of approximate the corresponding derivatives of u in the supremum norm with the same order of convergence, n being the degree of polynomial approximation and k being the smoothness of the kernel.     

Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind

In this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far.     

第二类Fredholm积分方程的泰勒展开解法

本进一步发展了用Taylor公式求解第二类Fredholm积分方程的方法，并给出了近似解的误差精度分析．     

Note on the integral mean value method for Fredholm integral equations of the second kind

We study the paper of Avazzadeh et al. [Z. Avazzadeh, M. Heydari, G.B., Loghmani, Numerical solution of Fedholm integral equations of the second kind by using integral mean value theorem, Appl. Math. Model. 35 (2011) 2374–2383] with the integral mean value method for Fredholm integral equations of the second kind. The objective of the note is threefold. First, we point out a basic error in the paper. Second, we find that the given numerical examples are only related to the special cases of Fredholm integral equations of the second kind with the degenerate kernels, which can be solved simply. Third, due to the basic error, our observations reveal that generally the suggested method should not be considered for a Fredholm integral equation of the second kind.     

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 is proved. The method is applied to a few test examples with continuous kernels to illustrate the accuracy and the implementation of the method.