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A Nyström method for a class of Fredholm integral equations on the real semiaxis. (English) Zbl 1365.65285

Summary: A class of Fredholm integral equations of the second kind, with respect to the exponential weight function \(w(x)=\exp (-(x^{-\alpha }+x^\beta ))\), \(\alpha >0\), \(\beta >1\), on \((0,+\infty )\), is considered. The kernel \(k(x,y)\) and the function \(g(x)\) in such kind of equations, \[ f(x)-\mu \int _0^{+\infty }k(x,y)f(y)w(y)\mathrm {d}y =g(x), x\in (0,+\infty ), \] can grow exponentially with respect to their arguments, when they approach to \(0^+\) and/or \(+\infty \). We propose a simple and suitable Nyström-type method for solving these equations. The study of the stability and the convergence of this numerical method in based on our results on weighted polynomial approximation and “truncated” Gaussian rules, recently published in [G. Mastroianni et al., Acta Math. Hung. 142, No. 1, 167–198 (2014; Zbl 1299.41010); IMA J. Numer. Anal. 34, No. 4, 1654–1685 (2014; Zbl 1304.65116)] respectively. Moreover, we prove a priori error estimates and give some numerical examples. A comparison with other Nyström methods is also included.

MSC:

65R20 Numerical methods for integral equations
41A10 Approximation by polynomials
45B05 Fredholm integral equations
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