Time-periodic linear boundary value problems on a finite interval
Authors:
A. S. Fokas, B. Pelloni and D. A. Smith
Journal:
Quart. Appl. Math. 80 (2022), 481-506
MSC (2020):
Primary 35B10, 35B40, 35G16; Secondary 35P20
DOI:
https://doi.org/10.1090/qam/1615
Published electronically:
March 24, 2022
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Additional Information
Abstract: We study the large time behaviour of the solution of a linear dispersive PDEs posed on a finite interval, when the prescribed boundary conditions are time periodic. We use the approach pioneered by A. S. Fokas and J. Lenells in The unified method: II. NLS on the half-line with $t$-periodic boundary conditions, J. Phys. A 45 (2012) for nonlinear integrable PDEs and then applied to linear problems on the half-line in A. S. Fokas and M. C. van der Weele, The unified transform for evolution equations on the half-line with time-periodic boundary conditions, Stud. Appl. Math. 147 (2021) to characterise necessary conditions for the solution of such a problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the second-order cases and establishing new ones for the third order one.
References
- Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, The method of Fokas for solving linear partial differential equations, SIAM Rev. 56 (2014), no. 1, 159–186. MR 3246302, DOI 10.1137/110821871
- Guillaume Michel Dujardin, Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2111, 3341–3360. With supplementary data available online. MR 2545299, DOI 10.1098/rspa.2009.0194
- A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494
- A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411–1443. MR 1469927, DOI 10.1098/rspa.1997.0077
- A. S. Fokas, A unified approach to boundary value problems, CBMS-SIAM, 2008.
- A. S. Fokas and M. C. van der Weele, The unified transform for evolution equations on the half-line with time-periodic boundary conditions, Stud. Appl. Math. 147 (2021), no. 4, 1339–1368, DOI 10.1111/sapm.12452.
- J. Lenells and A. S. Fokas, The unified method: II. NLS on the half-line with $t$-periodic boundary conditions, J. Phys. A 45 (2012), no. 19, 195202, 36. MR 2924498, DOI 10.1088/1751-8113/45/19/195202
- A. S. Fokas and B. Pelloni, A transform method for linear evolution PDEs on a finite interval, IMA J. Appl. Math. 70 (2005), no. 4, 564–587. MR 2156459, DOI 10.1093/imamat/hxh047
- James W. Hopkins, Some convergent developments associated with irregular boundary conditions, Trans. Amer. Math. Soc. 20 (1919), no. 3, 245–259. MR 1501125, DOI 10.1090/S0002-9947-1919-1501125-5
- D. Jackson, Expansion problems with irregular boundary conditions, Proc. Amer. Acad. Arts Sci. 51 (1915), no. 7, 383–417.
- Rudolph E. Langer, The asymptotic location of the roots of a certain transcendental equation, Trans. Amer. Math. Soc. 31 (1929), no. 4, 837–844. MR 1501516, DOI 10.1090/S0002-9947-1929-1501516-9
- R. E. Langer, On the zeros of exponential sums and integrals, Bull. Amer. Math. Soc. 37 (1931), no. 4, 213–239. MR 1562129, DOI 10.1090/S0002-9904-1931-05133-8
- John Locker, Spectral theory of non-self-adjoint two-point differential operators, Mathematical Surveys and Monographs, vol. 73, American Mathematical Society, Providence, RI, 2000. MR 1721499, DOI 10.1090/surv/073
- Beatrice Pelloni, The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2061, 2965–2984. MR 2165521, DOI 10.1098/rspa.2005.1474
- David A. Smith, Well-posed two-point initial-boundary value problems with arbitrary boundary conditions, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 3, 473–496. MR 2911141, DOI 10.1017/S030500411100082X
- D. A. Smith, Well-posedness and conditioning of 3rd and higher order two-point initial-boundary value problems, arXiv:1212.5466 [math.AP], 2012.
References
- Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, The method of Fokas for solving linear partial differential equations, SIAM Rev. 56 (2014), no. 1, 159–186. MR 3246302, DOI 10.1137/110821871
- Guillaume Michel Dujardin, Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2111, 3341–3360. With supplementary data available online. MR 2545299, DOI 10.1098/rspa.2009.0194
- A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494
- A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411–1443. MR 1469927, DOI 10.1098/rspa.1997.0077
- A. S. Fokas, A unified approach to boundary value problems, CBMS-SIAM, 2008.
- A. S. Fokas and M. C. van der Weele, The unified transform for evolution equations on the half-line with time-periodic boundary conditions, Stud. Appl. Math. 147 (2021), no. 4, 1339–1368, DOI 10.1111/sapm.12452.
- J. Lenells and A. S. Fokas, The unified method: II. NLS on the half-line with $t$-periodic boundary conditions, J. Phys. A 45 (2012), no. 19, 195202, 36. MR 2924498, DOI 10.1088/1751-8113/45/19/195202
- A. S. Fokas and B. Pelloni, A transform method for linear evolution PDEs on a finite interval, IMA J. Appl. Math. 70 (2005), no. 4, 564–587. MR 2156459, DOI 10.1093/imamat/hxh047
- James W. Hopkins, Some convergent developments associated with irregular boundary conditions, Trans. Amer. Math. Soc. 20 (1919), no. 3, 245–259. MR 1501125, DOI 10.2307/1988866
- D. Jackson, Expansion problems with irregular boundary conditions, Proc. Amer. Acad. Arts Sci. 51 (1915), no. 7, 383–417.
- Rudolph E. Langer, The asymptotic location of the roots of a certain transcendental equation, Trans. Amer. Math. Soc. 31 (1929), no. 4, 837–844. MR 1501516, DOI 10.2307/1989566
- R. E. Langer, On the zeros of exponential sums and integrals, Bull. Amer. Math. Soc. 37 (1931), no. 4, 213–239. MR 1562129, DOI 10.1090/S0002-9904-1931-05133-8
- John Locker, Spectral theory of non-self-adjoint two-point differential operators, Mathematical Surveys and Monographs, vol. 73, American Mathematical Society, Providence, RI, 2000. MR 1721499, DOI 10.1090/surv/073
- Beatrice Pelloni, The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2061, 2965–2984. MR 2165521, DOI 10.1098/rspa.2005.1474
- David A. Smith, Well-posed two-point initial-boundary value problems with arbitrary boundary conditions, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 3, 473–496. MR 2911141, DOI 10.1017/S030500411100082X
- D. A. Smith, Well-posedness and conditioning of 3rd and higher order two-point initial-boundary value problems, arXiv:1212.5466 [math.AP], 2012.
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Additional Information
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
MR Author ID:
67825
Email:
t.fokas@cam.ac.uk
B. Pelloni
Affiliation:
Heriot-Watt University & Maxwell Institute for the Mathematical Sciences, Edinburgh, United Kingdom
MR Author ID:
637645
ORCID:
0000-0003-2961-7613
Email:
b.pelloni@hw.ac.uk
D. A. Smith
Affiliation:
Yale-NUS College & Department of Mathematics, National University of Singapore, Singapore
MR Author ID:
975701
ORCID:
0000-0002-3525-3142
Email:
dave.smith@yale-nus.edu.sg
Received by editor(s):
September 16, 2021
Received by editor(s) in revised form:
January 5, 2022
Published electronically:
March 24, 2022
Article copyright:
© Copyright 2022
Brown University
