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A. Fokas (2002)
A new transform method for evolution partial differential equationsIma Journal of Applied Mathematics, 67
A. Fokas, A. Kapaev (2000)
A Riemann–Hilbert Approach to the Laplace EquationJournal of Mathematical Analysis and Applications, 251
B. McCartin (2003)
Eigenstructure of the Equilateral Triangle, Part I: The Dirichlet ProblemSIAM Rev., 45
J. Bona, A. Fokas (2008)
Initial-boundary-value problems for linear and integrable nonlinear dispersive partial differential equationsNonlinearity, 21
Y. Antipov, A. Fokas (2005)
The modified Helmholtz equation in a semi-stripMathematical Proceedings of the Cambridge Philosophical Society, 138
A. Fokas, B. Pelloni (2008)
A transform method for evolution PDEs on a finite interval
A. Sifalakis, A. Fokas, S. Fulton, Y. Saridakis (2008)
The generalized Dirichlet-Neumann map for linear elliptic PDEs and its numerical implementationJournal of Computational and Applied Mathematics, 219
A. Fokas, A. Fokas (1997)
A unified transform method for solving linear and certain nonlinear PDEsProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453
A. Fokas (2008)
A Unified Approach To Boundary Value Problems
S. Smitheman, E. Spence, A. Fokas (2010)
A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygonIma Journal of Numerical Analysis, 30
M. Ablowitz, A. Fokas, Z. Musslimani (2006)
On a new non-local formulation of water wavesJournal of Fluid Mechanics, 562
A. Fokas (2000)
On the integrability of linear and nonlinear partial differential equationsJournal of Mathematical Physics, 41
M. Pinsky (1985)
Completeness of the Eigenfunctions of the Equilateral TriangleSiam Journal on Mathematical Analysis, 16
R. Terras, R. Swanson (1980)
Image methods for constructing Green’s functions and eigenfunctions for domains with plane boundariesJournal of Mathematical Physics, 21
David Smith (2011)
Well-posed two-point initial-boundary value problems with arbitrary boundary conditionsMathematical Proceedings of the Cambridge Philosophical Society, 152
B. McCartin (2004)
Eigenstructure of the equilateral triangle. Part III. The Robin problemInt. J. Math. Math. Sci., 2004
A. Its, E. Its, J. Kaplunov (2011)
Riemann–Hilbert Approach to the Elastodynamic Equation: Part ILetters in Mathematical Physics, 96
A. Fokas, N. Flyer, S. Smitheman, E. Spence (2009)
A semi-analytical numerical method for solving evolution and elliptic partial differential equationsJournal of Computational and Applied Mathematics, 227
G Dassios, AS Fokas (2005)
The basic elliptic equations in an equilateral triangleProc. R. Soc. Lond. A, 461
A. Fokas, A. Nachbin (2012)
Water waves over a variable bottom: a non-local formulation and conformal mappingsJournal of Fluid Mechanics, 695
Euan Spence, A. Fokas (2010)
A new transform method II: the global relation and boundary-value problems in polar coordinatesProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466
B. Pelloni (2005)
The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equationsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461
G. Dujardin (2009)
Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervalsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465
SR Fulton, AS Fokas, CA Xenophontos (2004)
An analytical method for linear elliptic PDEs and its numerical implementationJ. Comput. Appl. Math., 167
M. Práger (1998)
Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangleApplications of Mathematics, 43
D. Crowdy (2008)
An assembly of steadily translating bubbles in a Hele–Shaw channelNonlinearity, 22
B. Deconinck, T. Trogdon, V. Vasan (2014)
The Method of Fokas for Solving Linear Partial Differential EquationsSIAM Rev., 56
ACL Ashton, AS Fokas (2011)
A non-local formulation for rational water wavesJ. Fluid Mech., 689
K. Kalimeris, A. Fokas (2010)
The Heat Equation in the Interior of an Equilateral TriangleStudies in Applied Mathematics, 124
G. Dassios, A. Fokas (2004)
The basic elliptic equations in an equilateral triangleProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461
M. Hochbruck, Heinrich-Heine (2007)
Highly Oscillatory Problems
B. Pelloni (2004)
Well-posed boundary value problems for linear evolution equations on a finite intervalMathematical Proceedings of the Cambridge Philosophical Society, 136
B Pelloni (2005)
The spectral representation of two-point boundary-value problems for third order linear evolution partial differential equationsProc. R. Soc. Lond. A, 461
A. Fokas (2001)
Two–dimensional linear partial differential equations in a convex polygonProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 457
B. McCartin (2003)
Eigenstructure of the equilateral triangle
A. Fokas, A. Kapaev (2003)
On a transform method for the Laplace equation in a polygonIma Journal of Applied Mathematics, 68
G. Dassios (2007)
What non-linear methods offered to linear problems? The Fokas transform methodInternational Journal of Non-linear Mechanics, 42
Euan Spence, A. Fokas (2010)
A new transform method I: domain-dependent fundamental solutions and integral representationsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466
D. ben-Avraham, A. Fokas (2001)
Solution of the modified Helmholtz equation in a triangular domain and an application to diffusion-limited coalescence.Physical review. E, Statistical, nonlinear, and soft matter physics, 64 1 Pt 2
D. Crowdy, A. Fokas (2004)
Explicit integral solutions for the plane elastostatic semi-stripProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460
M. Pinsky (1980)
The Eigenvalues of an Equilateral TriangleSiam Journal on Mathematical Analysis, 11
S. Fulton, A. Fokas, C. Xenophontos (2004)
An analytical method for linear elliptic PDEs and its numerical implementationFuel and Energy Abstracts
(2009)
Novel analytical and numerical methods for elliptic boundary value problems
B. McCartin (2002)
Eigenstructure of the equilateral triangle, Part II: The Neumann problemMathematical Problems in Engineering, 8
(1833)
Mémoire sur la propagation de la chaleur dans les polyédres
B. Fornberg, N. Flyer (2011)
A numerical implementation of Fokas boundary integral approach: Laplace's equation on a polygonal domainProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467
AS Fokas (2008)
A Unified Approach to Boundary Value Problems. CBMS-NSF Regional Conference Series in Applied Mathematics
The eigenvalues of the Laplace operator for the Dirichlet, Neumann and Robin problems in the interior of an equilateral triangle were first obtained by Lamé. Here, we present a simple, unified approach for rederiving the above results and also obtain the eigenvalues for the oblique Robin and for certain Poincaré problems. The explicit formula for the Poincaré eigenvalues yields, via appropriate limits, the relevant formulae for the oblique Robin, Robin, Neumann and Dirichlet eigenvalues. The method introduced here is based on the analysis of the so-called global relation, which as shown recently in the literature, provides an effective tool for the study of boundary value problems.
Computational Methods and Function Theory – Springer Journals
Published: Dec 12, 2013
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