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Generalised Dirichlet to Neumann maps for linear dispersive equations on half-line

Published online by Cambridge University Press:  27 February 2017

ATHANASSIOS S. FOKAS  and
ZIPENG WANG
ATHANASSIOS S. FOKAS
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA. United Kingdom. e-mail: [email protected]
ZIPENG WANG
Affiliation:
Cambridge Centre for Analysis, University of Cambridge, Cambridge, CB3 0WAUnited Kingdom. e-mail: [email protected]
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Abstract

A large class of initial-boundary problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit representations are presented for the generalised Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values when an essential set of initial and boundary data is given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 297 - 324
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Fokas, A. S. A unified transform method for solving linear and certain nonlinear PDEs. Proc. Royal Soc. A, 453 (1997), 1411–1443.Google Scholar
[2] Fokas, A. S. On the integrability of linear and nonlinear. PDEs. J. Math. Physics 41 (2000), 4188–4237.CrossRefGoogle Scholar
[3] Fokas, A. S. A unified approach to boundary value problems. CBMS-NSF 78, Regional Conference Series in Applied Mathematics (2008).CrossRefGoogle Scholar
[4] Fokas, A. S. and Spence, E. A.. Synthesis as opposed to separation of variables. SIAM Review 54 (2012).Google Scholar
[5] Deconinck, B., Trogdon, T. and Vasan, V.. The method of Fokas for solving linear partial differential equations. SIAM Review 56 (1) (2014), 159–186.Google Scholar
[6] Fokas, A. S. A new transform method for evolution PDEs. IMA J. Appl. Math. 67 (2002), 1–32.Google Scholar
[7] Mantzavinos, D. and Fokas, A. S.. The unified method for the heat equation: I non-separable boundary conditions and non-local constraints in one dimension. European J. Appl. Math. 24 (6) (2013), 857–886.Google Scholar
[8] Fokas, A. S. and Pelloni, B.. A transform method for linear evolution PDEs on a finite interval. IMA J. Appl. Math. 70 (4) (2005), 564–587.CrossRefGoogle Scholar
[9] Fokas, A. S. and Pelloni, B.. Two-point boundary value problems for linear evolution equations. Math. Proc. Camb. Phil. Soc. 131 (2001), 521–543.Google Scholar
[10] Pelloni, B. Well-posed boundary value problems for linear evolution equations on a finite interval . Math. Proc. Camb. Phil. Soc. 136 (2004), 361–382.CrossRefGoogle Scholar
[11] Pelloni, B. The spectral representation of two-point boundary value problems for third order linear evolution partial differential equations. Proc. Royal Soc. London, Series A 461 (2005), 2965–2984.Google Scholar
[12] Trogdon, T. and Deconinck, B.. The solution of linear constant coefficient evolution PDEs with periodic boundary conditions. Applicable Analysis 91 (3) (2012), 529–544.CrossRefGoogle Scholar
[13] Fokas, A. S. and Schultz, P. F.. The long time asymptotics of moving boundary problems using an ehrenpreis type representation and its riemann hilbert nonlinerisation. Comm. Pure Appl. Math. LVI (2001), 1–40.Google Scholar
[14] Bona, J. and Fokas, A. S.. Initial boundary value problems for linear and nonlinear dispersive PDE's . Nonlinearity 21 (2008), 195–230.Google Scholar
[15] Dujardin, G. M. Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite interval. Proc. Royal Soc. A 456 (2009), 3341–3360.Google Scholar
[16] Flyer, N. and Fokas, A. S.. A hybrid analytical numberical method for solving evolution partial differential equations: I the half-line. Proc. Royal Soc. A 464 (2008), 1823–1849.Google Scholar
[17] Vetra-Carvalho, S. The Computation of Spectral Representations for Evolution PDE: Theory and Application (AbeBooks, 2010).Google Scholar
[18] Fokas, A. S. and Sung, L. Y.. Initial-boundary value problems for linear dispersive evolution equations on the half line. http://www.math.sc.edu/~imip/99papers/9911.ps.Google Scholar
[19] Fokas, A. S., Himonas, A. and Mantzavinos, D.. The nonlinear Schrödinger equation on the half-line. Preprint.Google Scholar
[20] Fokas, A. S. and Kalimeris, K.. The heat equation in the interior of an equilateral triangle. Stud. Appl. Math. 124 (2010), 283–305.Google Scholar
[21] Mantzavinos, D. and Fokas, A. S.. The unified transform for the heat equation. II non-separable boundary conditions in two dimensions. Preprint.Google Scholar
[22] Fokas, A. S. From Green to Lax via Fourier. Recent Advances in Nonlinear Partial Differential Equations and Applications (AMS, Providence, Rhode Island, 2007), 87–118.Google Scholar
[23] Pelloni, B. and Smith, D. A.. Spectral theory of some non-selfadjoint linear differential operators. Proc. Royal Soc. London, Series A, 469 (2013), 20130019.Google Scholar
[24] Fokas, A. S. and DeLillo, S.. The unified transform for linear, linearizable and integrable nonlinear partial differential equations, 89 (2014), (038004).Google Scholar
[25] Spence, E. A. Boundary Value Problems for Linear Elliptic PDEs. PhD Thesis. University of Cambridge (2008).Google Scholar
[26] Dimakos, M. Boundary value problems for evolution and elliptic PDEs, PhD Thesis. University of Cambridge. (2008).Google Scholar
[27] Smith, D. A. Well-posed two-point initial boundary value problems with arbitrary boundary conditions. Math. Proc. Camb. Phil. Soc. 152 (2012), 473–496.Google Scholar
[28] Asvestas, M., Sifalakis, A. G., Papadopoulou, E. P. and Saridakis, Y. G.. Fokas method for a multidomain linear reaction-diffusion equation with discontinuous diffusivity, J. Physics: Conference Series 490 (1) (2014), 012143.Google Scholar