Abstract
In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given.
Highlights
Let the following be a Fredholm Integral Equation (FIE) of the second kind:f (y) = g(y) + μ f (x)k(x, y)ρ(x) dx, y ∈ (−1, 1), (1)−1 where ρ is a Jacobi weight, g and k are known functions defined in (−1, 1) and (−1, 1)2, respectively, μ is a non zero real parameter and f is the unknown function we want to look for
In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1, 1]
We introduce a Nyström method based on the extended product quadrature rule (8), calling it the Extended Nyström Method (ENM)
Summary
Let the following be a Fredholm Integral Equation (FIE) of the second kind:. −1 where ρ is a Jacobi weight, g and k are known functions defined in (−1, 1) and (−1, 1), respectively, μ is a non zero real parameter and f is the unknown function we want to look for. A high number of papers on the numerical methods for FIEs is disposable in the literature, and in the last two decades a deep attention was devoted, in the case under consideration, to the so-called “global approximation methods” They are essentially based on polynomial approximation and use zeros of orthogonal polynomials (see for instance [3,4] and the references therein). We propose here a Mixed Nyström scheme based on product quadrature rules of the “extended” type, i.e., based on the zeros of the polynomial pm+1(w)pm(w), where {pn(w)}n denotes the orthonormal sequence with respect to a suitable fixed Jacobi weight w [8].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
