Gauss-Radau and Gauss-Lobatto quadratures with double end points
In this paper Gauss-Radau and Gauss-Lobatto quadrature rules are developed having end points of multiplicity 2. These formulae seem to be useful for special integrands as well as when applied to boundary value problems. First, the Gauss-Radau formula on [-1,1] is derived with a double node at \(-1,\) \[ \int^{1}_{-1}f(t)w(t)dt=\kappa^ R_ 0f(-1)+\kappa^ R_ 1f'(-1)+\sum^{n}_{\nu =1}\lambda^ R_{\nu}f(\tau^ R_{\nu})+E^ R_ n(f); \] this is done both for general weight functions and all four of Chebyshev type \((w(t)=(1-t^ 2)^{-1/2}\), \((1-t^ 2)^{1/2}\), \((1-t)^{-1/2}(1+t)^{1/2}\), \((1- t)^{1/2}(1+t)^{-1/2})\), yielding explicit expressions for \(\kappa^ R_ i\) and \(\lambda^ R_ i.\) In a second part, the analogon is carried out for Gauss-Lobatto rules on \([-1,1]\) with double nodes at -1 and 1, \[ \int^{1}_{- 1}f(t)w(t)dt=\kappa^ L_ 0f(-1)+\kappa^ L_ 1f'(- 1)+\sum^{n}_{\nu =1}\lambda^ L_{\nu}f(\tau^ L_{\nu})+\mu^ L_ 0f(1)-\mu^ L_ 1f'(-1)+E^ L_ n(f). \] Here, the symmetry of the problem is widely exploited. Finally, numerical examples are given demonstrating advantages of the new formulae in special cases.
scientific article; zbMATH DE number 4201462
scientific article