Regularized Quadrature Methods for Fredholm Integral Equations of the First Kind (original) (raw)
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A Computational Approach to the Fredholm Integral Equation of the Second Kind
Proceedings of the World Congress on …, 2008
The Fredholm integral equation of the second kind is of widespread use in many realms of engineering and applied mathematics. Among the variety of numerical solutions to this equation, the qudrature method and its modification are remarkable. The latter aims at reducing the computational complexity of the quadrature method. In this paper, we present Mathematica programs that utilize the modified quadrature method to solve the equation.
2009
This is a personal perspective on the development of numerical methods for solving Fredholm integral equations of the second kind, discussing work being done principally during the 1950s and 1960s. The principal types of numerical methods being studied were projection methods (Galerkin, collocation) and Nystrom methods. During the 1950s and 1960s, functional analysis became the framework for the analysis of numerical methods for solving integral equations, and this inuenced the questions being asked. This paper looks at the history of the analyses being done at that time.
The Use of Lavrentiev Regularization Method in Fredholm Integral Equations of the First Kind
2019
The Fredholm integral equations of the first kind are often considered as ill-posed problems. The conventional way of solving them is to first convert them into the Fredholm integral equations of the second kind by means of a regularization method. This is followed by applying some standard techniques that are available for solving Fredholm integral equations of the second kind. This combination of two methods usually has some significant drawbacks in the sense that it may not produce a solution or produces only one solution after tedious calculations. The aim of this study is to remove these impediments once and for all for separable kernels and provide a closed-form expression for obtaining one or infinitely many solutions using the Lavrentiev regularization method. MSC: 47A52 • 45B05
Numerische Mathematik, 2020
This paper is concerned with the numerical approximation of Fredholm integral equations of the second kind. A Nyström method based on the anti-Gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces. The Nyström interpolants corresponding to the Gauss and the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation, under suitable assumptions which are easily verified for a particular weight function. Hence, an error estimate is available, and the accuracy of the solution can be improved by approximating it by an averaged Nyström interpolant. The effectiveness of the proposed approach is illustrated through different numerical tests.
On the Regularization of Fredholm Integral Equations of the First Kind
SIAM Journal on Mathematical Analysis, 1998
In this paper the problem of recovering a regularized solution of the Fredholm integral equations of the first kind with Hermitian and square-integrable kernels, and with data corrupted by additive noise, is considered. Instead of using a variational regularization of Tikhonov type, based on a priori global bounds, we propose a method of truncation of eigenfunction expansions that can be proved to converge asymptotically, in the sense of the L 2 -norm, in the limit of noise vanishing.
Numerical methods for Fredholm integral equations with singular right-hand sides
Advances in Computational Mathematics, 2010
Fredholm integral equations with the right-hand side having singularities at the endpoints are considered. The singularities are moved into the kernel that is subsequently regularized by a suitable one-to-one map. The Nyström method is applied to the regularized equation. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.
Approximate solutions of Fredholm integral equations of the second kind
2009
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.
Applied Mathematical Modelling, 2013
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.