论文标题
爱 - lieb积分方程：应用，理论，近似和计算
Love--Lieb integral equations: applications, theory, approximations, and computation
论文作者
论文摘要
本文主要与欺骗性的简单积分方程\ [ u(x) - \frac{1}π\int_{-1}^{1} \frac{α\, u(y)}{α^2+(x-y)^2} \, \rd y = 1, \quad -1 \leq x \leq 1, \] where $α$ is a real non-zero parameter and $u$ is the unknown 功能。该方程将其归类为具有连续内核的第二种弗雷德姆积分方程。因此，它属于有一个良好发展理论的一类方程。该理论表明，完全有一个连续的真实解决方案$ u $。尽管该解决方案在封闭形式中尚不清楚，但可以使用多种方法在数值上进行计算。所有这些都是好奇心，因为这不是因为整数方程在经典和量子物理学的几种情况下出现。我们回顾了有关这些应用的文献，调查可用的主要分析和数值工具，并调查用于构建近似解决方案的方法。当右侧的常数被给定函数替换时，我们还考虑相同的积分方程。
This paper is concerned mainly with the deceptively simple integral equation \[ u(x) - \frac{1}π\int_{-1}^{1} \frac{α\, u(y)}{α^2+(x-y)^2} \, \rd y = 1, \quad -1 \leq x \leq 1, \] where $α$ is a real non-zero parameter and $u$ is the unknown function. This equation is classified as a Fredholm integral equation of the second kind with a continuous kernel. As such, it falls into a class of equations for which there is a well developed theory. The theory shows that there is exactly one continuous real solution $u$. Although this solution is not known in closed form, it can be computed numerically, using a variety of methods. All this would be a curiosity were it not for the fact that the integral equation arises in several contexts in classical and quantum physics. We review the literature on these applications, survey the main analytical and numerical tools available, and investigate methods for constructing approximate solutions. We also consider the same integral equation when the constant on the right-hand side is replaced by a given function.
