A global approximation method of Nystr\"om type is explored for the numerical solution of a class of nonlinear integral equations of the second kind. The cases of smooth and weakly singular kernels are both considered. In the first occurrence, the method uses a Gauss-Legendre rule whereas in the second one resorts to a product rule based on Legendre nodes. Stability and convergence are proved in functional spaces equipped with the uniform norm and several numerical tests are given to show the good performance of the proposed method. An application to the interior Neumann problem for the Laplace equation with nonlinear boundary conditions is also considered.

翻译：本文研究了一种Nyström型全局逼近方法，用于数值求解一类第二类非线性积分方程。同时考虑了核函数光滑和弱奇异两种情况。对于前者，该方法采用高斯-勒让德求积法则；对于后者，则采用基于勒让德节点的乘积法则。在配备一致范数的函数空间中证明了方法的稳定性和收敛性，并通过多个数值实验展示了所提方法的良好性能。文中还探讨了该方法在具有非线性边界条件的拉普拉斯方程内部诺伊曼问题中的应用。