In this paper, we consider Galerkin and multi-Galerkin methods and their iterated versions for solving the nonlinear Hammerstein-type integral equation on the half-line with sufficiently smooth kernels, using Laguerre polynomials as basis functions. We obtain optimal convergence results in iterated-Galerkin method in both infinity and weighted $L^2$-norms. We also obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods, respectively, in weighted $L^2$-norm. Numerical results are presented to validate the theoretical results.

中文翻译：

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使用拉盖尔多项式对半线上的非线性Hammerstein积分方程进行Galerkin和多Galerkin方法的收敛性分析

在本文中，我们考虑使用 Laguerre 多项式作为基函数来求解具有足够光滑核的半线上的非线性 Hammerstein 型积分方程的 Galerkin 和多 Galerkin 方法及其迭代版本。我们在无穷大和加权的迭代Galerkin方法中获得了最优收敛结果${\mathrm{大号}}^2$-规范。我们还分别获得了多Galerkin和迭代多Galerkin方法的超收敛结果，在加权${\mathrm{大号}}^2$-规范。给出了数值结果来验证理论结果。