In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

中文翻译：

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使用高级重心拉格朗日插值公式求解第二类弱奇异 Fredholm 积分方程的计算方法

在这项研究中，我们应用了先进的重心拉格朗日插值公式来寻找第二类弱奇异 Fredholm 积分方程的插值解。核对两个变量进行两次插值，然后转化为五个矩阵的乘积；其中两个是单项式基矩阵。为了隔离内核的奇异性，我们基于在集成域上分布的两组不同节点的良好选择，开发了两种技术。每组特定于一个内核参数，因此内核值永远不会变为零或虚数。所呈现的两种技术的显着优点是能够在不应用搭配方法的情况下访问与插值解等效的代数线性系统。而且，研究了插值解的均值收敛和最大误差范数估计。发现所说明的四个示例的插值解强烈地一致地收敛到精确解。