第二类Fredholm方程的一种有理离散配置方法
A Rational Discrete Collocation Method for Second Kind Fredholm Equations
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中文总结 AI 辅助
提出用于求解第二类Fredholm积分方程的有理离散配置方法，基于再生核希尔伯特空间的有理插值方案，具有无实极点等特性，所得数值方法有稳定性等，实验证实其在处理挑战性核时比Nyström型方法更有效。
中文摘要 AI 辅助
在这项工作中，我们提出了一种新颖的离散配置方法，用于在配备一致范数的连续函数空间中数值求解第二类Fredholm积分方程。该方法基于最近在再生核希尔伯特空间的一般框架内开发的有理插值方案。这种有理逼近没有实极点，在任意雅可比节点处对目标函数进行插值，并具有一致有界的勒贝格常数。此外，它对所有连续函数以至少等于最佳一致多项式逼近的速率一致收敛。这些有趣的性质被所得的数值方法继承，该方法在对积分核的最小假设下建立了稳定性、收敛性和良好的条件数。一系列数值实验证实了理论结果，并表明在存在特别具有挑战性的核的情况下，所提出的方法为Nyström型方法提供了一种稳健且有效的替代方法。
英文摘要
In this work we present a novel discrete collocation method for the numerical solution of Fredholm integral equations of the second kind in the space of continuous functions equipped with the uniform norm. The method is based on a rational interpolation scheme recently developed within the general framework of reproducing kernel Hilbert spaces. This rational approximation has no real poles, interpolates the target function at arbitrary Jacobi nodes and exhibits uniformly bounded Lebesgue constants. Moreover, it converges uniformly for all continuous functions at a rate at least equal to that of the best uniform polynomial approximation. These interesting properties are inherited by the resulting numerical method, for which stability, convergence and good conditioning are established under minimal assumptions on the integral kernel. A series of numerical experiments confirm the theoretical findings and indicate that, in the presence of particularly challenging kernels, the proposed approach provides a robust and effective alternative to Nyström-type methods.
