arXiv 0812.16602008-12math.AP

一种求解流体载荷平板的新方法

A Novel Method of Solution for the Fluid Loaded Plate

Anthony C. L Ashton, A. S. Fokas

备注：: 17 pages, 3 figures. Minor adjustments made to the introduction
DOI：: 10.1098/rspa.2009.0170

摘要：

本文研究了在直线或半直线上描述的流体载荷板的控制方程的Cauchy问题。我们表明，在这两种情况下，可以解决的问题，采用统一的方法，边值问题介绍了作者在20世纪90年代末。Crighton et.使用拉普拉斯和傅里叶变换的组合。新方法避免了板的振幅在$L^1_{dt}(R^+)$中的先验假设的技术困难，并且还产生了更简单的解表示，这直接意味着问题是适定的。对于半直线上的问题，类似的分析产生了一个解的表示，但这个公式涉及两个未知函数。半线问题的主要困难在于这两个函数的特征化。通过采用所谓的整体关系，我们表明，这两个功能可以通过一个复值的卷积型积分方程的解决方案。该方程可以使用拉普拉斯变换以封闭形式求解。通过规定的初始数据$η_0$是在$H^3(R^+)$，我们的解决方案连续依赖于初始数据，因此，问题是适定的。

英文摘要：

We study the Cauchy problem associated with the equations governing a fluid loaded plate formulated on either the line or the half-line. We show that in both cases the problem can be solved by employing the unified approach to boundary value problems introduced by on of the authors in the late 1990s. The problem on the full line was analysed by Crighton et. al. using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in $L^1_{dt}(R^+)$ and furthermore yields a simpler solution representation which immediately implies the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, but this formula involves two unknown functions. The main difficulty with the half-line problem is the characterisation of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex valued integral equation of the convolution type. This equation can be solved in closed form using the Laplace transform. By prescribing the initial data $η_0$ to be in $H^3(R^+)$, we show that the solution depends continuously on the initial data, and hence, the problem is well-posed.

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