Fourier–Mellin Transforms for Circular Domains
Fourier–Mellin Transforms for Circular Domains
复制标题
圆域的傅立叶-梅林变换
DOI：
10.1007/s40315-015-0139-6
复制
发表时间：
2015
影响因子：
2.1
通讯作者：
D. Crowdy
中科院分区：
文献类型：
--
作者：
D. Crowdy
本文导出了单连通圆域上解析函数的广义Fourier-Mellin变换。圆形域被认为是那些边界是圆弧的有限并集的域，包括直线边。这些结果是由Fokas和Kapaev（IMA J Appl Math 68：355-408，2003）导出的凸多边形（仅具有直线边缘）的广义傅立叶变换的圆形域的扩展。首先，一个新的，初级推导后者的结果多边形的基础上柯西积分公式和谱表示的柯西核。一旦建立了柯西核的适应谱表示，这种重新推导以自然的方式扩展到圆形域的情况。可以类似地处理具有圆弧和直线边缘的组合的边界的域。新导出的变换是经典的傅里叶变换和梅林变换一般圆形域的推广。通过实例说明了它们如何用于求解此类区域上的拉普拉斯方程边值问题。谱矩阵和基本轮廓的概念，这自然产生的配方，也介绍了。
Generalized Fourier–Mellin transforms for analytic functions defined in simply connected circular domains are derived. Circular domains are taken to be those with boundaries that are a finite union of circular arcs, including straight line edges. The results are an extension to circular domains of the generalized Fourier transforms for convex polygons (having only straight line edges) derived by Fokas and Kapaev (IMA J Appl Math 68:355–408, 2003). First, a new, elementary derivation of the latter result for polygons is given based on Cauchy’s integral formula and a spectral representation of the Cauchy kernel. This rederivation extends in a natural way to the case of circular domains once an adapted spectral representation of the Cauchy kernel is established. Domains with boundaries that are a combination of circular arc and straight line edges can be treated similarly. The newly derived transforms are generalizations of the classical Fourier and Mellin transforms to general circular domains. It is shown by example how they can be used to solve boundary value problems for Laplace’s equation in such domains. The notions of spectral matrix and fundamental contour, which arise naturally in the formulation, are also introduced.
一种新的变换方法二：极坐标下的全局关系与边值问题
DOI：
10.1098/rspa.2009.0513
发表时间：
2010
期刊：
Mathematical, Physical and Engineering Sciences
影响因子：
--
作者：
Spence E
通讯作者：
Spence E
影响因子：
2.1
作者：
Smitheman, S. A.;Spence, E. A.;Fokas, A. S.
通讯作者：
Fokas, A. S.