Fokas’s Unified Transform Method for linear systems
Authors:
Bernard Deconinck, Qi Guo, Eli Shlizerman and Vishal Vasan
Journal:
Quart. Appl. Math. 76 (2018), 463-488
DOI:
https://doi.org/10.1090/qam/1484
Published electronically:
September 28, 2017
MathSciNet review:
3805037
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, as they arise symmetrically in the global relations. This allows the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. Wave equations are treated separately in an appendix.
References
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- A. C. L. Ashton and A. S. Fokas, Elliptic equations with low regularity boundary data via the unified method, Complex Var. Elliptic Equ. 60 (2015), no. 5, 596–619. MR 3326268, DOI https://doi.org/10.1080/17476933.2014.964227
- B. Deconinck, B. Pelloni, and N. Sheils. Non-steady-state heat conduction in composite walls. Proc. Royal Soc. London A, 470(2165):20130605, 2014.
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- A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems, Math. Phys. Anal. Geom. 8 (2005), no. 1, 59–96. MR 2136653, DOI https://doi.org/10.1007/s11040-004-1650-6
- F. Mandl and G. Shaw. Quantum field theory. John Wiley & Sons, New York, NY, 2010.
- D. Mantzavinos, M. G. Papadomanolaki, Y. G. Saridakis, and A. G. Sifalakis, Fokas transform method for a brain tumor invasion model with heterogeneous diffusion in $1+1$ dimensions, Appl. Numer. Math. 104 (2016), 47–61. MR 3478186, DOI https://doi.org/10.1016/j.apnum.2014.09.006
- J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50(10):2061–2070, 1962.
- B. Pelloni and D. A. Pinotsis, The Klein-Gordon equation in a domain with time-dependent boundary, Stud. Appl. Math. 121 (2008), no. 3, 291–312. MR 2458539, DOI https://doi.org/10.1111/j.1467-9590.2008.00416.x
- Beatrice Pelloni and Dimitrios A. Pinotsis, The Klein-Gordon equation on a half line: a Riemann-Hilbert approach, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 3, 334–344. MR 2452444, DOI https://doi.org/10.2991/jnmp.2008.15.s3.32
- B. Pelloni and D. A. Pinotsis, Moving boundary value problems for the wave equation, J. Comput. Appl. Math. 234 (2010), no. 6, 1685–1691. MR 2644159, DOI https://doi.org/10.1016/j.cam.2009.08.016
- P. A. Treharne and A. S. Fokas, Boundary value problems for systems of linear evolution equations, IMA J. Appl. Math. 69 (2004), no. 6, 539–555. MR 2101843, DOI https://doi.org/10.1093/imamat/69.6.539
- Vishal Vasan and Bernard Deconinck, Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. 33 (2013), no. 7, 3171–3188. MR 3007743, DOI https://doi.org/10.3934/dcds.2013.33.3171
References
- Unified transform method. http://unifiedmethod.azurewebsites.net/. Accessed: 2015-10-24.
- Mark J. Ablowitz and Athanassios S. Fokas, Complex variables: introduction and applications, 2nd ed., Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2003. MR 1989049
- A. C. L. Ashton and A. S. Fokas, Elliptic equations with low regularity boundary data via the unified method, Complex Var. Elliptic Equ. 60 (2015), no. 5, 596–619. MR 3326268, DOI https://doi.org/10.1080/17476933.2014.964227
- B. Deconinck, B. Pelloni, and N. Sheils. Non-steady-state heat conduction in composite walls. Proc. Royal Soc. London A, 470(2165):20130605, 2014.
- 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 https://doi.org/10.1137/110821871
- 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 https://doi.org/10.1098/rspa.1997.0077
- A. S. Fokas, A new transform method for evolution partial differential equations, IMA J. Appl. Math. 67 (2002), no. 6, 559–590. MR 1942267, DOI https://doi.org/10.1093/imamat/67.6.559
- Athanassios S. Fokas, A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2451953
- A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems, Math. Phys. Anal. Geom. 8 (2005), no. 1, 59–96. MR 2136653, DOI https://doi.org/10.1007/s11040-004-1650-6
- F. Mandl and G. Shaw. Quantum field theory. John Wiley & Sons, New York, NY, 2010.
- D. Mantzavinos, M. G. Papadomanolaki, Y. G. Saridakis, and A. G. Sifalakis, Fokas transform method for a brain tumor invasion model with heterogeneous diffusion in $1+1$ dimensions, Appl. Numer. Math. 104 (2016), 47–61. MR 3478186, DOI https://doi.org/10.1016/j.apnum.2014.09.006
- J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50(10):2061–2070, 1962.
- B. Pelloni and D. A. Pinotsis, The Klein-Gordon equation in a domain with time-dependent boundary, Stud. Appl. Math. 121 (2008), no. 3, 291–312. MR 2458539, DOI https://doi.org/10.1111/j.1467-9590.2008.00416.x
- Beatrice Pelloni and Dimitrios A. Pinotsis, The Klein-Gordon equation on a half line: a Riemann-Hilbert approach, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 3, 334–344. MR 2452444, DOI https://doi.org/10.2991/jnmp.2008.15.s3.32
- B. Pelloni and D. A. Pinotsis, Moving boundary value problems for the wave equation, J. Comput. Appl. Math. 234 (2010), no. 6, 1685–1691. MR 2644159, DOI https://doi.org/10.1016/j.cam.2009.08.016
- P. A. Treharne and A. S. Fokas, Boundary value problems for systems of linear evolution equations, IMA J. Appl. Math. 69 (2004), no. 6, 539–555. MR 2101843, DOI https://doi.org/10.1093/imamat/69.6.539
- Vishal Vasan and Bernard Deconinck, Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. 33 (2013), no. 7, 3171–3188. MR 3007743, DOI https://doi.org/10.3934/dcds.2013.33.3171
Additional Information
Bernard Deconinck
Affiliation:
Department of Applied Mathematics, University of Washington, Campus Box 353925, Seattle, WA, 98195
Email:
deconinc@uw.edu
Qi Guo
Affiliation:
Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, CA 90095-1555
Email:
qiguo@math.ucla.edu
Eli Shlizerman
Affiliation:
Department of Applied Mathematics & Department of Electrical Engineering, University of Washington, Campus Box 353925, Seattle, WA, 98195
MR Author ID:
758251
Email:
shlizee@uw.edu
Vishal Vasan
Affiliation:
International Centre for Theoretical Sciences, Bengaluru, India
MR Author ID:
992275
Email:
vishal.vasan@icts.res.in
Received by editor(s):
April 28, 2017
Received by editor(s) in revised form:
August 10, 2017
Published electronically:
September 28, 2017
Article copyright:
© Copyright 2017
Brown University
