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A numerical implementation of the unified Fokas transform for evolution problems on a finite interval

Published online by Cambridge University Press:  23 November 2017

EMINE KESICI ,
BEATRICE PELLONI ,
TRISTAN PRYER  and
DAVID SMITH
EMINE KESICI
Affiliation:
Department of Mathematics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey email: emine.dlger@gmail.com
BEATRICE PELLONI
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK email: b.pelloni@hw.ac.uk
TRISTAN PRYER
Affiliation:
Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK email: t.pryer@reading.ac.uk
DAVID SMITH
Affiliation:
Division of Science, Yale-NUS College, 28 College Ave West (RC3), 01-501, Singapore 138533 email: dave.smith@yale-nus.edu.sg
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Abstract

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We present the numerical solution of two-point boundary value problems for a third-order linear PDE, representing a linear evolution in one space dimension. To our knowledge, the numerical evaluation of the solution so far could only be obtained by a time-stepping scheme, that must also take into account the issue, generically non-trivial, of the imposition of the boundary conditions. Instead of computing the evolution numerically, we evaluate the novel solution representation formula obtained by the unified transform, also known as Fokas transform. This representation involves complex line integrals, but in order to evaluate these integrals numerically, it is necessary to deform the integration contours using appropriate deformation mappings. We formulate a strategy to implement effectively this deformation, which allows us to obtain accurate numerical results.

Keywords

Initial-boundary value problemsAiry equationnon-periodic problemsunified transformFokas transform
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 3 , June 2018 , pp. 543 - 567
Copyright
Copyright © Cambridge University Press 2017 

References

[1] Fokas, A. S. & Pelloni, B. (2001) Two-point boundary value problems for linear evolution equations. Math. Proc. Camb. Philos. Soc. 131, 521–543.CrossRefGoogle Scholar
[2] Fokas, A. S. & Pelloni, B. (2005) A transform method for linear evolution PDEs on a finite interval. IMA J. Appl. Math. 70, 564–587.CrossRefGoogle Scholar
[3] Pelloni, B. (2004) Well-posed boundary value problems for linear evolution equations on a finite interval. Math. Proc. Camb. Philos. Soc. 136, 361–382.CrossRefGoogle Scholar
[4] Pelloni, B. (2005) 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, 2965–2984.Google Scholar
[5] Smith, D. A. (2011) Spectral Theory of Ordinary and Partial Linear Differential Operators on Finite Intervals, Phd, University of Reading.Google Scholar
[6] Fokas, A. S. (2008) A Unified Approach to Boundary Value Problems, CBMS-SIAM.CrossRefGoogle Scholar
[7] Fokas, A. S. & Pelloni, B. (2015) Unified Transform for Boundary Value Problems: Applications and Advances, Vol. 141, SIAM.Google Scholar
[8] Ablowitz, M. J. & Fokas, A. S. (1997) Complex Variables, Cambridge Texts in Applied Mathematics, Cambridge University Press.Google Scholar
[9] Its, A. (2003) The Riemann–Hilbert problem and integrable systems. Not. AMS 50 (11), 1389–1400.Google Scholar
[10] Pelloni, B. & Smith, D. A. (2016) Evolution PDEs and augmented eigenfunctions. Half line. J. Spectral Theory 6, 185–213.CrossRefGoogle Scholar
[11] Fokas, A. S. & Smith, D. A. (2016) Evolution PDEs and augmented eigenfunctions. Finite interval. Adv. Differ. Equ. 21 (7/8), 735–766.Google Scholar
[12] Fokas, A. S. & Sung, L. Y. (1999) Initial-boundary value problems for linear dispersive evolution equations on the half-line, (unpublished).Google Scholar
[13] Olver, P. J. (2010) Dispersive quantization. Amer. Math. Mon. 117 (7), 599–610.CrossRefGoogle Scholar
[14] Pelloni, B. (2005) The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 461 (2061), 2965–2984.Google Scholar
[15] Flyer, N. & Fokas, A. S. (2008) A hybrid analytical-numerical method for solving evolution partial differential equations. I. The half-line. Proc. R. Soc. A 464, 1823–1849.CrossRefGoogle Scholar
[16] Langer, R. E. (1931) The zeros of exponential sums and integrals. Bull. Amer. Math. Soc. 37, 213–239.CrossRefGoogle Scholar
[17] Smith, D. A. (2012) Well-posed two-point initial-boundary value problems with arbitrary boundary conditions. Math. Proc. Camb. Philos. Soc. 152, 473–496.CrossRefGoogle Scholar
[18] Deconinck, B. & Trogdon, T. (2012) The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions. Appl. Anal. 91 (3), 529–544.Google Scholar
[19] Biondini, G. & Trogdon, T. (2017) Gibbs phenomenon for dispersive pdes on the line. SIAM J. Appl. Math. 77 (3), 813–837.CrossRefGoogle Scholar
[20] Birkhoff, G. D. (1908) Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math. Soc. 9, 373–395.CrossRefGoogle Scholar
[21] Trefethen, L. N., Wiedeman, J. A. C. & Schmelzer, T. (2006) Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670.CrossRefGoogle Scholar
[22] Trefethen, L. N. & Weideman, J. A. C. (2014) The exponentially convergent trapezoidal rule. SIAM Rev. 56 (3), 385–458.CrossRefGoogle Scholar
[23] Papatheodorou, T. S. & Kandili, A. N. (2009) Novel numerical techniques based on fokas transforms, for the solution of initial boundary value problems. J. Comput. Appl. Math. 227, 75–82.CrossRefGoogle Scholar