Numerical methods for hypersingular integrals on the real line. (English) Zbl 1372.65067

De Bonis, Maria Carmela; Occorsio, Donatella

Dolomites Res. Notes Approx. 10, Spec. Iss., 97-117 (2017).

Summary: In the present paper the authors propose two numerical methods to approximate Hadamard transforms of the type
\[ \mathbf{H}_p(fw_{\beta},t) =~~=\!\!\!\!\!\!\!\int_{\mathbb{R}} \frac{f(x)}{(x-t)^{p+1}} w_{\beta}(x)dx, \]
where \(p\) is a nonnegative integer and \(w_{\beta}(x) = e^{-|x|^{\beta}}\), \(\beta > 1\), is a Freud weight. One of the procedures employed here is based on a simple tool like the “truncated” Gaussian rule conveniently modified to remove numerical cancellation and overflow phenomena. The second approach is a process of simultaneous approximation of the functions \(\{\mathbf{H}_k (fw_{\beta}, t)\}^p_{k=0}\). This strategy can be useful in the numerical treatment of hypersingular integral equations. The methods are shown to be numerically stable and convergent and some error estimates in suitable Zygmund-type spaces are proved. Numerical tests confirming the theoretical estimates are given. Comparisons of our methods among them and with other ones available in literature are shown.

MSC:

65D30	Numerical integration
41A55	Approximate quadratures