Meshless methods based on infinitely smooth radial kernels have the potential to provide spectrally accurate function approximations with enormous geometric flexibility in any number of dimensions. The highest accuracy can often be obtained when the shape parameter in the basis function is small. But as the shape parameter goes to zero, the standard RBF interpolant matrix becomes severely ill-conditioned. The ill-conditioning can be reduced using alternate bases. One of these alternative bases is the Hilbert–Schmidt SVD basis. The Hilbert–Schmidt SVD method offers a stable mechanism for converting a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space. In this paper, we apply the Gaussian Hilbert–Schmidt SVD basis functions method for solving the linear Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert–Schmidt SVD basis functions constructed on a set of disordered data. This approach reduces the solution of the problem under study to the solution of a linear system of algebraic equations. Also, the convergence of the proposed approach is analyzed. Lastly, several numerical experiments are presented to test the stability and accuracy of the proposed method.

基于无限平滑径向核的无网格方法有潜力提供光谱精确的函数近似值，在任何数量的维度上都具有巨大的几何灵活性。当基函数中的形状参数小时，通常可以获得最高的精度。但是，随着形状参数变为零，标准RBF插值矩阵变得病重。使用备用底座可以减少不适。这些替代基础之一是希尔伯特-施密特SVD基础。希尔伯特-施密特SVD方法提供了一种稳定的机制，可以将一组分散中心的近平坦内核转换为条件完全相同的良好基数。在本文中，我们将高斯Hilbert-Schmidt SVD基函数方法用于求解第二类线性Fredholm积分方程。该方法基于基于无序数据集的高斯Hilbert-Schmidt SVD基函数的离散搭配方法来估计解决方案。这种方法将正在研究的问题的求解简化为代数方程的线性系统的求解。此外，分析了该方法的收敛性。最后，通过数值实验验证了该方法的稳定性和准确性。