Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions
type: Item
instance of: scholarly article
title: Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions
zbMATH Open document ID: 0596.30014
DOI: 10.1016/0377-0427(86)90141-X
author: R. J. Williams
author: Lloyd N. Trefethen
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Consider Laplace's equation on a polygonal domain with homogeneous oblique boundary conditions cos \(\theta\) \({}_ k u_ n(z)+\sin \theta_ k u_{\sigma}(z)=0\) on the edges \(\Gamma_ k\) of the polygon. If at the corners neither regularity nor boundedness is required, the solution space has infinite dimension. First, it is shown that every solution is the real part of an analytic function which maps each \(\Gamma_ k\) onto a straight line segment oriented at the angle \(\theta_ k\). Then the exact forms of the general solution are given, first, under the assumption of at most polynomial growth near the vertices and, second, under the assumption of boundedness of continuity. In both cases f is a composition of an ordinary Schwarz-Christoffel map for the polygon and a similar function obtained by solving a generalized Schwarz-Christoffel parameter problem. Two applications are treated in detail: The Hall effect on a polygonal conductor and a Brownian motion problem with oblique reflection, which comes up in queueing theory.
Mathematics Subject Classification ID: 30C30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 65N99
zbMATH DE Number: 3959863
zbMATH Keywords: Laplace's equation
zbMATH Keywords: Schwarz-Christoffel map
zbMATH Keywords: Hall effect
zbMATH Keywords: Brownian motion
reviewed by: Martin H. Gutknecht
describes a project that uses: SCPACK
MaRDI profile type: Publication
cites work: Q5535091
cites work: Q3042109
cites work: Classical free-streamline flow over a polygonal obstacle
cites work: A Generalization of the Schwarz-Christoffel Transformation
cites work: Q4023365
cites work: The diffusion approximation for tandem queues in heavy traffic
cites work: The stationary distribution of reflected Brownian motion in a planar region
cites work: Reflected Brownian motion on an orthant
cites work: A tandem storage system and its diffusion limit
cites work: Q3870044
cites work: Boundary-value problems for partial differential equations in non-smooth domains
cites work: Q3314210
cites work: Q3844606
cites work: Q5815016
cites work: Approximate behavior of tandem queues
cites work: Numerical Computation of the Schwarz–Christoffel Transformation
cites work: Brownian motion in a wedge with oblique reflection
cites work: Q5332327
cites work: Solution of the Field Problem of the Germanium Gyrator
cites work: Recurrence classification and invariant measure for reflected Brownian motion in a wedge
description: scientific article; zbMATH DE number 3959863