A fourth order product integration rule by using the generalized Euler-Maclaurin summation formula

The present paper deals with a variant of the generalized Euler-Maclaurin summation formula for the product integration rule. The main idea lies on introducing a mesh associated with the integral of the square root of the weight function. We construct the set of nodes and implement it for approximating the considered integrals. It is shown theoretically that the proposed quadrature rule is of fourth order. The results of the provided numerical examples confirm the theoretical prediction. Moreover, applications of the suggested scheme for approximating some real test problems such as weakly singular integrals, including a particular case of the Fermi-Dirac integral, are investigated.