Separable Laplace equation, magic Toeplitz matrix, and generalized Ohm’s law

Abstract

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.