Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
EnglishFrançais
Mathematical Proceedings of the Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society

Article contents

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: T.Fokas@damtp.cam.ac.uk
ZIPENG WANG
Affiliation:
Cambridge Centre for Analysis, University of Cambridge, Cambridge, CB3 0WA United Kingdom. e-mail: zipeng.wang@cantab.net
Corresponding
E-mail address:
T.Fokas@damtp.cam.ac.uk
zipeng.wang@cantab.net
Get access
Rights & Permissions[Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

[1] Fokas, A. S. A unified transform method for solving linear and certain nonlinear PDEs. Proc. Royal Soc. A, 453 (1997), 1411–1443.CrossRefGoogle 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).Google Scholar
[4] Fokas, A. S. and Spence, E. A.. Synthesis as opposed to separation of variables. SIAM Review 54 (2012).CrossRefGoogle 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.CrossRefGoogle Scholar
[6] Fokas, A. S. A new transform method for evolution PDEs. IMA J. Appl. Math. 67 (2002), 1–32.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 2
Total number of PDF views: 71 *
View data table for this chart

* Views captured on Cambridge Core between 27th February 2017 - 22nd January 2021. This data will be updated every 24 hours.

Hostname: page-component-76cb886bbf-frjnl Total loading time: 0.212 Render date: 2021-01-22T04:29:41.926Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *