Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules
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Numerische Mathematik (2020) 146:699–728 https://doi.org/10.1007/s00211-020-01163-7
Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules Patricia Díaz de Alba1,2 · Luisa Fermo1 · Giuseppe Rodriguez1 Received: 28 February 2020 / Revised: 25 October 2020 / Accepted: 26 October 2020 / Published online: 18 November 2020 © The Author(s) 2020
Abstract 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. Mathematics Subject Classification 65R20 · 65D30 · 42C05
1 Introduction Let us consider the following Fredholm integral equation of the second kind f (y) −
B
1
−1
k(x, y) f (x)w(x) d x = g(y),
y ∈ [−1, 1],
(1)
Patricia Díaz de Alba [email protected] Luisa Fermo [email protected] Giuseppe Rodriguez [email protected]
1
Department of Mathematics and Computer Science, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy
2
Gran Sasso Science Institute, viale Francesco Crispi 7, 67100 L’Aquila, Italy
123
700
P. Díaz de Alba et al.
where f is the unknown function, k and g are two given functions, and w(x) = (1 − x)α (1 + x)β
(2)
is the Jacobi weight with parameters α, β > −1. Several numerical methods have been described for the numerical approximation of the solution of Eq. (1) (collocation methods, projection methods, Galerkin methods, etc.) and have been extensively investigated in terms of stability and convergence in suitable function spaces, also according to the smoothness properties of the kernel k and the right-hand side g; see [2,6,7,9,17,25,29–32]. Most of these methods are based on the approximation of the integral appearing in (1) by means of the well-known Gauss quadrature formula, introduced by C. F. Gauss at the beginning of the nineteenth century [10] and considered one of the most significant discoveries in the field of numerical integration and in all of numerical analysis. As it is well known, it is an interpolatory formula having maximal algebraic degree of exactness, it is stable and convergent, and it provides one of the most important applications of orthogonal polynomials. Gauss’s discovery inspired other contemporaries, such as Jacobi and Christoffel, who developed Gauss’s method into new directions, and Heun, who generalized Gauss’s idea to ordinary differential equations opening the way to the discovery of Runge-Kutta methods. S
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