Applied Numerical Mathematics 59 (2009) 2970–2989 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains ✩ Luisa Fermo Department of Mathematics and Computer Science, University of Basilicata, v.le dell’Ateneo Lucano 10, 85100 Potenza, Italy article info abstract Article history: Received 27 October 2008 Received in revised form 10 July 2009 Accepted 13 July 2009 Available online 15 July 2009 MSC: 65R20 45E05 Keywords: Fredholm integral equations Nyström method The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.  2009 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction This paper deals with the numerical treatment of the following class of Fredholm integral equations of the third kind on unbounded domains, p( y) f ( y) − μ b  a k(x, y) f (x) w(x) dx = g ( y), y ∈ (a, b), (1.1) where (a, b) = (0, ∞) or (a, b) = (−∞, +∞), f is the unknown, k, g and p are given functions, μ ∈ R, w(x) =|x| α e −|x| β , x ∈ (a, b), α > −1, β> 1 2 if (a, b) = (0, ∞) and β> 1 if (a, b) = (−∞, +∞). Moreover we will assume that the right-hand side g is a smooth function, the coefficient p has only one zero at the origin of the type y ι ,0 < ι < 1 and that the kernel k behaves like y λ log ℓ y,0 <λ< 1, ℓ  0 at the origin. The case when w ≡ 1 and the integral is defined on finite interval was extensively investigated (see, for instance, [24,22]) and there exists a wide literature about numerical methods, e.g. collocation and spline methods (see, for instance, [9,8,25, 26] and the bibliography therein) in order to approximate the solution of these equations in different spaces of functions. In this paper we examine the case when the given functions are defined on unbounded domains, have singularities at the origin and increase exponentially for x →∞. Hence in virtue of the smoothness of the given functions, we consider the above equation in a suitable weighted space equipped with the uniform norm. In order to approximate its solution (if it exists) we propose a strategy mainly consisting in three steps. ✩ Work supported by the research project PRIN 2006 “Numerical methods for structured algebra and applications” of the Italian Ministry for the University and Research. E-mail address: luisa.fermo@unibas.it. 0168-9274/$30.00  2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2009.07.002