Motivated by a recent method for approximate solution of Fredholm equations of the first kind, we develop a corresponding method for a class of Fredholm equations of the \emph{second kind}. In particular, we consider the class of equations for which the solution is a probability measure. The approach centres around specifying a functional whose gradient flow admits a minimizer corresponding to a regularized version of the solution of the underlying equation and using a mean-field particle system to approximately simulate that flow. Theoretical support for the method is presented, along with some illustrative numerical results.

翻译：受近期求解第一类Fredholm方程近似方法的启发，本文针对一类\emph{第二类}Fredholm方程发展了相应方法。特别地，我们研究解为概率测度的方程类型。该方法的核心在于构造一个泛函，其梯度流的最小化子对应于原方程的正则化解，并通过平均场粒子系统对该梯度流进行近似模拟。本文给出了该方法的理论支撑，并提供若干示例性数值结果。