Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons

Colbrook, Matthew J.; Fokas, A. S.

doi:10.1016/j.apnum.2017.12.001

Cited by 9 publications

(4 citation statements)

References 27 publications

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“…By evaluating the approximate global relation, (2.14) and its Schwartz conjugate (which is given by taking the complex conjugate then replacing λ withλ) if the solution is real, at suitably chosen Fourier collocation points, λ i ∈ C, we can construct sufficiently many equations for the unknown constants {a j , b j }. Discussion of how optimal collocation points are chosen for bounded domains can be found in [10,19].…”

Section: (A) Dirichlet To Neumann Mapmentioning

confidence: 99%

The unified transform for mixed boundary condition problems in unbounded domains

2019

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This paper implements the unified transform to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener–Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. By approximating the known data and the unknown boundary values by suitable functions and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The method is illustrated for the modified Helmholtz and Helmholtz equations. In each case, comparisons between the traditional Wiener–Hopf approach, other spectral or boundary methods and the unified transform approach are discussed.

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Section: (A) Dirichlet To Neumann Mapmentioning

confidence: 99%

The unified transform for mixed boundary condition problems in unbounded domains

2019

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“…Such a situation is analogous to the coordinate-separability problems in analytically finding Laplace eigenvalues on polygonal domains, for which rectangular, elliptical and equilateral triangular domains are the most prominent examples where simple analytical approaches suffice. Beyond these examples, almost all polygonal domains (including the hexagon) require numerical study [ 76 ].…”

Section: Structured Domainsmentioning

confidence: 99%

Modern perspectives on near-equilibrium analysis of Turing systems

Krause

Gaffney

Maini

et al. 2021

Phil. Trans. R. Soc. A.

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In the nearly seven decades since the publication of Alan Turing’s work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction–diffusion theory. Some of these developments were nascent in Turing’s paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction–diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of ‘trivial’ base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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“…Such a situation is analogous to the coordinate-separability problems in analytically finding Laplace eigenvalues on polygonal domains, for which rectangular, elliptical, and equilateral triangular domains are the most prominent examples where simple analytical approaches suffice. Beyond these examples, almost all polygonal domains (including the hexagon) require numerical study [33].…”

Section: (C) Multi-domain Modelsmentioning

confidence: 99%

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

Krause,

Gaffney,

Maini

et al. 2021

Preprint

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In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations, and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasise important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality.

show abstract

Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons

Cited by 9 publications

References 27 publications

The unified transform for mixed boundary condition problems in unbounded domains

The unified transform for mixed boundary condition problems in unbounded domains

Modern perspectives on near-equilibrium analysis of Turing systems

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

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