An analytic series method for Laplacian problems with mixed boundary conditions

Read, W.W. (2007) An analytic series method for Laplacian problems with mixed boundary conditions. Journal of Computational and Applied Mathematics, 209 (1). pp. 22-32.

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Mixed boundary value problems are characterised by a combination of Dirichlet and Neumann conditions along at least one boundary. Historically, only a very small subset of these problems could be solved using analytic series methods (“analytic” is taken here to mean a series whose terms are analytic in the complex plane). In the past, series solutions were obtained by using an appropriate choice of axes, or a co-ordinate transformation to suitable axes where the boundaries are parallel to the abscissa and the boundary conditions are separated into pure Dirichlet or Neumann form. In this paper, I will consider the more general problem where the mixed boundary conditions cannot be resolved by a co-ordinate transformation. That is, a Dirichlet condition applies on part of the boundary and a Neumann condition applies along the remaining section. I will present a general method for obtaining analytic series solutions for the classic problem where the boundary is parallel to the abscissa. In addition, I will extend this technique to the general mixed boundary value problem, defined on an arbitrary boundary, where the boundary is not parallel to the abscissa. I will demonstrate the efficacy of the method on a well known seepage problem.

Item ID: 2495
Item Type: Article (Research - C1)
ISSN: 0377-0427
Keywords: mixed boundary value problem; series solution; analytic solution; seepage; pseudo-spectral method
Date Deposited: 22 Jul 2009 01:10
FoR Codes: 04 EARTH SCIENCES > 0406 Physical Geography and Environmental Geoscience > 040603 Hydrogeology @ 40%
01 MATHEMATICAL SCIENCES > 0103 Numerical and Computational Mathematics > 010301 Numerical Analysis @ 40%
05 ENVIRONMENTAL SCIENCES > 0503 Soil Sciences > 050399 Soil Sciences not elsewhere classified @ 20%
SEO Codes: 97 EXPANDING KNOWLEDGE > 970101 Expanding Knowledge in the Mathematical Sciences @ 51%
96 ENVIRONMENT > 9609 Land and Water Management > 960999 Land and Water Management of Environments not elsewhere classified @ 49%
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