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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 

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