An extended sigmoidal transformation technique for evaluating weakly singular integrals without splitting the integration interval

In this paper, we present an efficient transformation technique for accurate numerical evaluation of weakly singular integrals with interior singularities. The present transformation technique does not require any division of the integration interval. It is composed of two parts of a sigmoidal transformation whose tails coincide with a singular point smoothly up to the order of the sigmoidal transformation employed. Therefore, in using the standard Gauss quadrature rule, the present transformation has a feature in which the integration points are clustered into the interior singular point very closely, as in the case of endpoint singular integrals. The results of some numerical examples show the superiority of the present method over the existing methods.