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The unified method for the heat equation: I. non-separable boundary conditions and non-local constraints in one dimension

Published online by Cambridge University Press:  18 July 2013

DIONYSSIOS MANTZAVINOS  and
ATHANASSIOS S. FOKAS
DIONYSSIOS MANTZAVINOS
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK emails: mantzavinos.1@nd.edu, T.Fokas@damtp.cam.ac.uk
ATHANASSIOS S. FOKAS
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK emails: mantzavinos.1@nd.edu, T.Fokas@damtp.cam.ac.uk
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Abstract

We use the heat equation as an illustrative example to show that the unified method introduced by one of the authors can be employed for constructing analytical solutions for linear evolution partial differential equations in one spatial dimension involving non-separable boundary conditions as well as non-local constraints. Furthermore, we show that for the particular case in which the boundary conditions become separable, the unified method provides an easier way for constructing the relevant classical spectral representations avoiding the classical spectral analysis approach. We note that the unified method always yields integral expressions which, in contrast to the series or integral expressions obtained by the standard transform methods, are uniformly convergent at the boundary. Thus, even for the cases that the standard transform methods can be implemented, the unified method provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of typical boundary value problems.

Keywords

initial-boundary value problemsheat equationunified methodnon-separable boundary conditionsnon-local constraints

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Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 6 , December 2013 , pp. 857 - 886
Copyright
Copyright © Cambridge University Press 2013 

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