The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical methods. To address this, we propose a novel class of mapped Hermite functions, which are constructed by applying a mapping to Hermite polynomials.We establish fundamental approximation theory for the orthogonal functions. We propose MHFs-spectral collocation method and MHFs-smoothing transformation method to solve the two-point weakly singular FHIEs, respectively. Error analysis and numerical results demonstrate that our methods, based on the new orthogonal functions, are particularly effective for handling problems with weak singularities at two endpoints, yielding exponential convergence rate. We position this work as the first to directly study the mapped spectral method for multi-point singularity problems, to the best of our knowledge.

翻译：具有弱奇异核的Fredholm-Hammerstein积分方程在端点或边界处表现出多点奇异性。采用数值方法时，稠密离散化矩阵会导致较高的计算复杂度。为解决此问题，我们提出了一类新颖的映射埃尔米特函数，其通过对埃尔米特多项式应用映射构造而成。我们建立了该正交函数的基本逼近理论。我们分别提出了MHF-谱配置法和MHF-光滑变换法来求解两点弱奇异FHIEs。误差分析和数值结果表明，基于新正交函数的我们的方法对于处理两端点具有弱奇异性的问题特别有效，并能获得指数收敛速度。据我们所知，本研究是首个直接针对多点奇异性问题探讨映射谱方法的工作。