Well-posed boundary value problems for linear evolution equations on a finite interval

BEATRICE PELLONI

doi:10.1017/S0305004103007205

Abstract

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We identify the class of smooth boundary conditions that yield an initial-boundary value problem admitting a unique smooth solution for the case of a dispersive linear evolution PDE of arbitrary order, in one spatial dimension, defined on a finite interval.

This result is obtained by an application of a spectral transform method, introduced by Fokas, which allows us to reduce the problem to the study of the singularities of the set of functions arising as the unique solution of a certain linear system.

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Crossref Citations

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Fokas, A. S. and Pelloni, B. 2005. Boundary Value Problems for Boussinesq Type Systems. Mathematical Physics, Analysis and Geometry, Vol. 8, Issue. 1, p. 59.

Pelloni, Beatrice 2005. The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 461, Issue. 2061, p. 2965.

Pelloni, Beatrice 2006. Linear and nonlinear generalized Fourier transforms. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 364, Issue. 1849, p. 3231.

Pelloni, Beatrice 2008. Boundary value problems for third-order linear PDEs in time-dependent domains. Inverse Problems, Vol. 24, Issue. 1, p. 015004.

Dujardin, Guillaume Michel 2009. Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 465, Issue. 2111, p. 3341.

Fokas, A S and Pelloni, B 2012. The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation. Nonlinearity, Vol. 25, Issue. 4, p. 1011.

SMITH, DAVID A. 2012. Well-posed two-point initial-boundary value problems with arbitrary boundary conditions. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 152, Issue. 3, p. 473.

Trogdon, Thomas and Deconinck, Bernard 2012. The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions. Applicable Analysis, Vol. 91, Issue. 3, p. 529.

Fokas, A S and Lenells, J 2012. The unified method: I. Nonlinearizable problems on the half-line. Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue. 19, p. 195201.

Pelloni, B. and Smith, D. A. 2013. Spectral theory of some non-selfadjoint linear differential operators. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 469, Issue. 2154, p. 20130019.

Fokas, A. S. Lenells, J. and Pelloni, B. 2013. Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip. Journal of Nonlinear Science, Vol. 23, Issue. 2, p. 241.

Fokas, A. S. and Glasser, M. L. 2013. The Laplace equation in the exterior of the Hankel contour and novel identities for hypergeometric functions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 469, Issue. 2157, p. 20130081.

Fokas, A. S. and Kalimeris, K. 2014. Eigenvalues for the Laplace Operator in the Interior of an Equilateral Triangle. Computational Methods and Function Theory, Vol. 14, Issue. 1, p. 1.

Sheils, N E and Smith, D A 2015. Heat equation on a network using the Fokas method. Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue. 33, p. 335001.

KESICI, EMINE PELLONI, BEATRICE PRYER, TRISTAN and SMITH, DAVID 2018. A numerical implementation of the unified Fokas transform for evolution problems on a finite interval. European Journal of Applied Mathematics, Vol. 29, Issue. 3, p. 543.

FOKAS, ATHANASSIOS S. and WANG, ZIPENG 2018. Generalised Dirichlet to Neumann maps for linear dispersive equations on half-line. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 164, Issue. 2, p. 297.

Miller, P.D. and Smith, D.A. 2018. The diffusion equation with nonlocal data. Journal of Mathematical Analysis and Applications, Vol. 466, Issue. 2, p. 1119.

Pelloni, Beatrice and Smith, David Andrew 2018. Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations. Studies in Applied Mathematics, Vol. 141, Issue. 1, p. 46.

Himonas, A. Alexandrou Mantzavinos, Dionyssios and Yan, Fangchi 2019. Initial-boundary value problems for a reaction-diffusion equation. Journal of Mathematical Physics, Vol. 60, Issue. 8,

Fokas, A. S. and van der Weele, M. C. 2021. The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions*. Studies in Applied Mathematics, Vol. 147, Issue. 4, p. 1339.

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Well-posed boundary value problems for linear evolution equations on a finite interval

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