标题： 混合径向核用于求解弱奇异弗雷德霍姆积分方程：在无网格方法中平衡精度和稳定性 显示英文标题

标题： Hybrid Radial Kernels for Solving Weakly Singular Fredholm Integral Equations: Balancing Accuracy and Stability in Meshless Methods

摘要： 在过去几十年中，基于核的近似方法在解决科学和工程领域的不同问题方面取得了惊人的成功。 然而，当使用无限平滑核进行直接或标准计算方法时，理论上可以达到的精度与数值稳定性之间会出现冲突。 换句话说，当形状参数趋于零时，使用无限平滑核的标准基的运算矩阵会变得严重病态。 可以通过应用混合核来管理这种冲突。 混合核扩展了近似空间，并提供了高度的灵活性，以在精度和稳定性之间取得最佳平衡。 在当前的研究中，提供了一种创新方法，使用混合径向核（HRKs）在无网格方案中求解第二类弱奇异Fredholm积分方程（WSFIEs）。 该方法在离散配点技术中使用基于分散节点构建的混合核作为基函数。 这种方法将所研究的问题转化为线性代数方程组。 此外，利用粒子群优化（PSO）算法计算混合核的最佳参数，该算法基于最小化最大绝对误差（MAE）。 我们还研究了所建议方案的误差估计。 最后，通过进行各种数值实验来评估混合技术的准确性和有效性。 数值结果表明，与纯核相比，从混合核获得的估计在求解WSFIEs时显著更准确。 此外，结果显示混合基在各种形状参数值下保持稳定。 显示英文摘要

摘要： Over the past few decades, kernel-based approximation methods had achieved astonishing success in solving different problems in the field of science and engineering. However, when employing the direct or standard method of performing computations using infinitely smooth kernels, a conflict arises between the accuracy that can be theoretically attained and the numerical stability. In other words, when the shape parameter tends to zero, the operational matrix for the standard bases with infinitely smooth kernels become severely ill-conditioned. This conflict can be managed applying hybrid kernels. The hybrid kernels extend the approximation space and provide high flexibility to strike the best possible balance between accuracy and stability. In the current study, an innovative approach using hybrid radial kernels (HRKs) is provided to solve weakly singular Fredholm integral equations (WSFIEs) of the second kind in a meshless scheme. The approach employs hybrid kernels built on dispersed nodes as a basis within the discrete collocation technique. This method transforms the problem being studied into a linear system of algebraic equations. Also, the particle swarm optimization (PSO) algorithm is utilized to calculate the optimal parameters for the hybrid kernels, which is based on minimizing the maximum absolute error (MAE). We also study the error estimate of the suggested scheme. Lastly, we assess the accuracy and validity of the hybrid technique by carrying out various numerical experiments. The numerical findings show that the estimates obtained from hybrid kernels are significantly more accurate in solving WSFIEs compared to pure kernels. Additionally, it was revealed that the hybrid bases remain stable across various values of the shape parameters.