A numerical method for finite-part integrals

In the present paper we introduce and study an extended product quadrature rule to approximate Hadamard finite part integrals of the type Hp( f U, t) = Z = +1 0 f (x) (x t)p+1 U(x)d x, t > 0, p 2 N, U(x) = ex x , 0. Hypersingular integrals arise in many contexts, such as singular and hypersingular boundary integral equations, which are tools for modeling many phenomena in different branches of the applied sciences. Here we derive an extended product rule and by a mixed combination with the one weight product rule introduced in [9], we propose a compound scheme of quadrature rules which allows a significant reduction in the number of evaluations of the density function f . Conditions assuring the stability and the convergence of the the mixed scheme in weighted uniform form are deduced. Some numerical experiments are also given, in order to highlight the efficiency of the mixed approach.