Laplace's equation in the exterior of a convex polygon. The equilateral triangle
Authors:
A. Charalambopoulos, G. Dassios and A. S. Fokas
Journal:
Quart. Appl. Math. 68 (2010), 645660
MSC (2000):
Primary 35C15, 35J05, 35J25
Published electronically:
September 21, 2010
MathSciNet review:
2761508
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Additional Information
Abstract: A general method for studying boundary value problems for linear and for integrable nonlinear partial differential equations in two dimensions was introduced in Fokas, 1997. For linear equations in a convex polygon (Fokas and Kapaev (2000) and (2003), and Fokas (2001)), this method: (a) expresses the solution in the form of an integral (generalized inverse Fourier transform) in the complex plane involving a certain function (generalized direct Fourier transform) that is defined as an integral along the boundary of the polygon, and (b) characterizes a generalized DirichlettoNeumann map by analyzing the socalled global relation. For simple polygons and simple boundary conditions, this characterization is explicit. Here, we extend the above method to the case of elliptic partial differential equations in the exterior of a convex polygon and we illustrate the main ideas by studying the Laplace equation in the exterior of an equilateral triangle. Regarding (a), we show that whereas is identical with that of the interior problem, the contour of integration in the complex plane appearing in the formula for depends on . Regarding (b), we show that the global relation is now replaced by a set of appropriate relations which, in addition to the boundary values, also involve certain additional unknown functions. In spite of this significant complication we show that, for certain simple boundary conditions, the exterior problem for the Laplace equation can be mapped to the solution of a Dirichlet problem formulated in the interior of a convex polygon formed by three sides.
 1.
G.
Dassios and A.
S. Fokas, The basic elliptic equations in an equilateral
triangle, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
461 (2005), no. 2061, 2721–2748. MR 2165508
(2006f:35048), http://dx.doi.org/10.1098/rspa.2005.1466
 2.
A.
S. Fokas and A.
A. Kapaev, A RiemannHilbert approach to the Laplace equation,
J. Math. Anal. Appl. 251 (2000), no. 2,
770–804. MR 1794770
(2001k:35241), http://dx.doi.org/10.1006/jmaa.2000.7052
 3.
A.
S. Fokas, A unified transform method for solving linear and certain
nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453
(1997), no. 1962, 1411–1443. MR 1469927
(98e:35007), http://dx.doi.org/10.1098/rspa.1997.0077
 4.
A.
S. Fokas and A.
A. Kapaev, On a transform method for the Laplace equation in a
polygon, IMA J. Appl. Math. 68 (2003), no. 4,
355–408. MR 1988152
(2004c:37176), http://dx.doi.org/10.1093/imamat/68.4.355
 5.
A.
S. Fokas, Twodimensional linear partial differential equations in
a convex polygon, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.
457 (2001), no. 2006, 371–393. MR 1848093
(2002j:35084), http://dx.doi.org/10.1098/rspa.2000.0671
 6.
A.
S. Fokas and M.
Zyskin, The fundamental differential form and boundaryvalue
problems, Quart. J. Mech. Appl. Math. 55 (2002),
no. 3, 457–479. MR 1919978
(2003f:35267), http://dx.doi.org/10.1093/qjmam/55.3.457
 7.
S.
R. Fulton, A.
S. Fokas, and C.
A. Xenophontos, An analytical method for linear elliptic PDEs and
its numerical implementation, J. Comput. Appl. Math.
167 (2004), no. 2, 465–483. MR 2064703
(2005d:65216), http://dx.doi.org/10.1016/j.cam.2003.10.012
 8.
A.
G. Sifalakis, A.
S. Fokas, S.
R. Fulton, and Y.
G. Saridakis, The generalized DirichletNeumann map for linear
elliptic PDEs and its numerical implementation, J. Comput. Appl. Math.
219 (2008), no. 1, 9–34. MR 2437692
(2009g:35042), http://dx.doi.org/10.1016/j.cam.2007.07.012
 9.
S.A. Smitheman, E.A. Spence and A.S. Fokas, ``A spectral collocation method for the Laplace and Modified Helmholtz equations in a convex polygon'', IMA J. Num. Anal. doi:10.1093/imanum/dm079
 1.
 G. Dassios and A.S. Fokas, ``The Basic Elliptic Equations in an Equilateral Triangle'', Proceedings of the Royal Society of London A, 461, pp. 27212748, 2005. MR 2165508 (2006f:35048)
 2.
 A.S. Fokas and A.A. Kapaev, ``A RiemannHilbert Approach to the Laplace Equation'', Journal of Mathematical Analysis and Applications, 251, pp. 770804, 2000. MR 1794770 (2001k:35241)
 3.
 A.S. Fokas, ``A Unified Transform Method for Solving Linear and Nonlinear PDE's'', Proceeding of the Royal Society of London A, 453, pp. 14111443, 1997. MR 1469927 (98e:35007)
 4.
 A.S. Fokas and A.A. Kapaev, ``On a Transform Method for the Laplace Equation in polygon'', IMA Journal in Applied Mathematics, 68, pp. 355408, 2003. MR 1988152 (2004c:37176)
 5.
 A.S. Fokas, ``TwoDimensional Linear Partial Differential Equations in a Convex Polygon'', Proceeding of the Royal Society of London A, 457, pp. 371393, 2001. MR 1848093 (2002j:35084)
 6.
 A.S. Fokas and M. Zyskin, ``The Fundamental Differential Form and BoundaryValue Problems'', Quarterly Journal of Mechanics and Applied Mathematics, 55, pp. 457479, 2002. MR 1919978 (2003f:35267)
 7.
 S.R. Fulton, A.S. Fokas and C.A. Xenophontos, ``An Analytical Method for Linear Elliptic PDEs and its Numerical Implementation'', Journal of Computational and Applied Mathematics, 167, pp. 465483, 2004. MR 2064703 (2005d:65216)
 8.
 A.G. Sifalakis, A.S. Fokas, S.R. Fulton and Y.G. Saridakis, ``The Generalized DirichletNeumann Map for Linear Elliptic PDEs and its Numerical Implementation'', J. Comp. Appl. Math, 219, 934, 2008. MR 2437692
 9.
 S.A. Smitheman, E.A. Spence and A.S. Fokas, ``A spectral collocation method for the Laplace and Modified Helmholtz equations in a convex polygon'', IMA J. Num. Anal. doi:10.1093/imanum/dm079
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Additional Information
A. Charalambopoulos
Affiliation:
Department of Material Science and Engineering, University of Ioannina, Greece
G. Dassios
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
DOI:
http://dx.doi.org/10.1090/S0033569X201001168X
PII:
S 0033569X(2010)01168X
Keywords:
Laplace equation,
equilateral triangle,
exterior domain
Received by editor(s):
January 29, 2009
Received by editor(s) in revised form:
February 18, 2009
Published electronically:
September 21, 2010
Additional Notes:
The second author is on leave from the University of Patras and ICEHT/FORTH Greece. His current address is the Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, GR 265 04 Patras, Greece
The present work was performed under the Marie Curie Chair of Excellence Project BRAIN, granted to the second and the third author by the European Commission under code number EXC 023928
Article copyright:
© Copyright 2010 Brown University
