THE TRAPEZOIDAL RULE WITH A NONLINEAR COORDINATE TRANSFORMATION FOR WEAKLY SINGULAR INTEGRALS

Abstract

It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation <TEX>$\Omega$</TEX><TEX>$_{m}$</TEX>(b;<TEX>$\chi$</TEX>), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation <TEX>$\Omega$</TEX><TEX>$_{m}$</TEX>(b;<TEX>$\chi$</TEX>), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation <TEX>$\Omega$</TEX><TEX>$_{m}$</TEX>(b;<TEX>$\chi$</TEX>).TEX>).