### Abstract:

We investigate the numerical solution of two-dimensional Fredholm integral equations defined on the set S = [a, b] × [c, d], −∞ ≤ a < b ≤ ∞, −∞ ≤ c < d ≤ ∞,
f(x,y)−μ \int{k(x,y,s,t)f(s,t)w(s,t)}dsdt=g(x,y), (x,y)∈S, (1)
where w(x, y) := w1(x)w2(y) and w1, w2 are suitable weight functions defined on [a, b], [c, d] respectively, μ is a real number. k and g are given functions defined on ([a, b] × [c, d])^2 and [a, b]×[c, d] respectively, which are sufficiently smooth on the open sets but can have (algebraic) singularities on the finite boundaries and an exponential growth at ±∞ at most w.r.t. each variable. f is the unknown function.
Therefore S is intended to be an unbounded domain, for instance a quarter of the plane, a strip etc.
We introduce some Nystro ̈m methods based on cubature formulas obtained as tensor products of two Gaussian quadrature formulas w.r.t. the weights w1, w2. Due to the “unbounded- ness” of the domain we need to “truncate” the quadrature rules. The convergence, stability and well conditioning of the methods are proved in suitable weighted spaces of continuous functions. Some numerical examples illustrate the efficiency of the methods.