论频谱上三维传输问题的弗雷德霍尔姆第一类边界积分方程的可解性

Pub Date : 2024-06-05 DOI:10.1134/s0012266124020058

A. A. Kashirin, S. I. Smagin

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引用次数: 0

摘要

摘要本文研究了两个弱奇异的弗雷德霍尔姆边界积分方程，每个方程都可以将三维亥姆霍兹传输问题简化为第一类问题。研究了这些方程在频谱上的性质，在频谱上这些方程的问题是不明确的。对于第一个方程，研究表明，如果它的解存在于频谱上，就可以找到传输问题的解。在这种情况下，第二个方程总是有无穷多个解，其中只有一个能给出传输问题的解。讨论了寻找积分方程近似解的插值法和有关的传输问题。

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On the Solvability of Fredholm Boundary Integral Equations of the First Kind for the Three-Dimensional Transmission Problem on the Spectrum

Abstract

The paper considers two weakly singular Fredholm boundary integral equations of the first kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on the spectra, where they are ill posed. For the first equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the transmission problem. The second equation in this case always has infinitely many solutions, with only one of them giving a solution of the transmission problem. The interpolation method for finding approximate solutions of the integral equations and the transmission problem in question is discussed.

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