In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{r, d\}}),\) whereas iterated discrete Galerkin/iterated discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{2r, d\}}),\) where \(n^{-1}\) is the maximum norm of the graded mesh and r denotes the order of the piecewise polynomial employed and \(d-1\) is the degree of precision of quadrature formula. We also show that iterated discrete multi-Galerkin/iterated discrete multi-collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{4r, d\}})\). Hence by choosing sufficiently accurate numerical quadrature rule, we show that the convergence rates in discrete projection and discrete multi-projection methods are preserved. Numerical examples are given to uphold the theoretical results.

在本文中，我们研究了使用分段多项式基函数在具有光滑核的半线上解决Hammerstein积分方程的离散投影方法。我们证明了离散Galerkin /离散搭配方法收敛到阶为\（{\ mathcal {O}}（n ^ {-min \ {r，d \}}），\）的精确解，而迭代离散Galerkin /迭代离散搭配方法以阶\（{\ mathcal {O}}（n ^ {-min \ {2r，d \}}），\）收敛到精确解，其中\（n ^ {-1} \）为最大值渐变网格的范数，r表示采用的分段多项式的阶数和\（d-1 \）是正交公式的精确度。我们还表明，迭代离散多重Galerkin /迭代离散多重配置方法以\（{\ mathcal {O}}（n ^ {-min \ {4r，d \}}））\）收敛到精确解。因此，通过选择足够精确的数值正交规则，我们证明了离散投影和离散多投影方法的收敛速度得以保留。通过数值算例证明了理论结果的正确性。