Paper Info
Title
Conformal mapping and bipolar coordinate for eccentric Laplace problems
Abstract
Boundary value problems on the eccentric annulus are quite complex and cannot directly be solved analytically using cartesian or polar coordinates. Many mathematical techniques have been used to solve such a problem by using conformal mapping and bipolar coordinate. In the literature, Carrier and Pearson [Partial differential equation-theory and technique. New York, Academic Press, 1976, pp 68-71], Muskhelishvili [Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen; 1953, pp 175- 179], Ling [Torsion of an eccentric circular tube, Technical Report, No. 1, Chinese Bureau of Aeronautical Research, 19401, Timoshenko and Goordier [Theory of Elasticity. New York, McGraw-Hill; 1972, pp 196-202], Shen [Null-held approach for Laplace problems with circular boundaries using degenerate kernels, Master thesis, National Taiwan Ocean University, Keelung, Taiwan, 2005], Lebedev et al. [Worked Problems in Applied Mathematics. New York, Dover; 1965] have solved this kind of problems using similar techniques. By using transformation in a transformed plane in the complex variable theory, we can obtain the analytical solution easily. We focus on the connection between conformal mapping and curvilinear coordinates, and figure out the relation to take integration by way of mapping in the complex plane. All the transformations and curvilinear coordinates can be unified using the viewpoint of conformal mapping. Their relationship among available methods can be constructed by translation, stretching, rotation and inversion. Finally, an example of eccentric domain is solved by using various mappings and curvilinear coordinates and their relations are linked. Not only geometry transformation is addressed but also the solution of the Laplace equation is obtained. C 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 17: 314-322, 2009; Published online in Wiley InterScience (www.interscience.wiley.com) DOI 10.1002/cae.20208
Year
DOI
Venue
2009
10.1002/cae.20208
COMPUTER APPLICATIONS IN ENGINEERING EDUCATION
Keywords
Field
DocType
conformal mapping,bipolar coordinate,complex plane,transformation,eccentric circle,Laplace equation
Log-polar coordinates,Mathematical analysis,Laplace's equation,Complex plane,Conformal map,Polar coordinate system,Curvilinear coordinates,Orthogonal coordinates,Mathematics,Cartesian coordinate system
Journal
Volume
Issue
ISSN
17.0
3.0
1061-3773
Citations 
PageRank 
References 
2
1.12
0
Authors
3
Name
Order
Citations
PageRank
Jeng-Tzong Chen1218.46
Ming-Hong Tsai262.31
Chein-Shan Liu37624.29