Separable Laplace equation, magic Toeplitz matrix, and generalized Ohm's law
Separable Laplace equation, magic Toeplitz matrix, and generalized Ohm's law
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DOI：
10.1016/j.amc.2006.02.030
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发表时间：
2006-10-15
影响因子：
4
通讯作者：
Demidenko, Eugene
中科院分区：
文献类型：
--
作者：
Demidenko, Eugene
我们说非齐次介质上的拉普拉斯方程是可分的，如果该介质可以表示为参数函数的乘积。结果表明，如果介质的电导率是角函数和半径函数的乘积，则二维圆盘上的电导率方程可以简化为一对常微分方程。讨论了几个重要的情况，如圆形和非圆形斑点状介质的导电性。我们推导出Neumann-to-Dirichlet映射的一个封闭形式，它产生了一组新的对称矩阵——我们称之为神奇的Toeplitz矩阵（一个行和列之和相同的对称Toeplitz矩阵）。导出了圆盘上的广义欧姆定律，该定律通过电阻矩阵将电流密度矢量与电压矢量联系起来。我们演示了一个可分离的拉普拉斯PDE如何应用于电阻抗断层扫描和乳腺癌检测。(c) 2006爱思唯尔公司版权所有。
We say that a Laplace equation on an inhomogeneous medium is separable if the medium can be represented as the product of the argument functions. It is shown that the conductivity equation on a 2-D disk is reduced to a couple of ordinary differential equations if the conductivity of the medium is the product of the angle and radius functions. Several important cases are discussed such as circular and a noncircular blob-like medium conductivity. We derive a closed form of the Neumann-to-Dirichlet map that gives rise to a new family of symmetric matrices-we call it the magic Toeplitz matrix (a symmetric Toeplitz matrix with the rows and columns summing to the same value). The generalized Ohm's law on the disk is derived, which relates the vector of the current density to the vector of voltages through the resistance matrix. We demonstrate how a separable Laplace PDE can be applied to electrical impedance tomography and breast cancer detection. (c) 2006 Elsevier Inc. All rights reserved.