The basic elliptic equations in an equilateral triangle
The basic elliptic equations in an equilateral triangle
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等边三角形的基本椭圆方程
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
A. Fokas
中科院分区：
文献类型：
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作者：
G. Dassios;A. Fokas
在他深入和多产的调查热扩散，拉梅是导致调查的本征值和本征函数的拉普拉斯运营商在一个等边三角形。特别是，他得出明确的结果狄利克雷和诺依曼案件使用巧妙的变化的变量。相关的本征函数是一个复杂的无穷级数。在这里，我们首先表明，边界值问题与简单的边界条件，如狄利克雷和诺依曼问题，可以解决一个基本的方式。特别是，未知的诺依曼和狄利克雷边界值可以表示为一个傅立叶级数的狄利克雷和诺依曼问题，分别。我们的分析是基于所谓的全球关系，这是一个代数方程耦合的狄利克雷和诺依曼谱值的周边的三角形。正如Lamé正确地指出的那样，无穷级数不足以表示更复杂问题的解，例如混合边值问题。在本文中，我们表明，进一步利用全球的关系，这样的问题可以解决广义傅立叶积分。
In his deep and prolific investigations of heat diffusion, Lamé was led to the investigation of the eigenvalues and eigenfunctions of the Laplace operator in an equilateral triangle. In particular, he derived explicit results for the Dirichlet and Neumann cases using an ingenious change of variables. The relevant eigenfunctions are a complicated infinite series in terms of his variables. Here we first show that boundary-value problems with simple boundary conditions, such as the Dirichlet and the Neumann problems, can be solved in an elementary manner. In particular, the unknown Neumann and Dirichlet boundary values can be expressed in terms of a Fourier series for the Dirichlet and the Neumann problems, respectively. Our analysis is based on the so-called global relation, which is an algebraic equation coupling the Dirichlet and the Neumann spectral values on the perimeter of the triangle. As Lamé correctly pointed out, infinite series are inadequate for expressing the solution of more complicated problems such as mixed boundary-value problems. In this paper we show, further utilizing the global relation, that such problems can be solved in terms of generalized Fourier integrals.