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The unified transform for the heat equation: II. Non-separable boundary conditions in two dimensions

Published online by Cambridge University Press:  04 June 2015

D. MANTZAVINOS  and
A. S. FOKAS
D. MANTZAVINOS
Affiliation:
Department of Mathematics, The State University of New York at Buffalo, Buffalo, NY, USA email: dionyssi@buffalo.edu
A. S. FOKAS
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK email: tf227@cam.ac.uk
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Abstract

We use the two-dimensional heat equation as an illustrative example to show that the unified transform is capable of constructing analytical solutions for linear evolution partial differential equations (PDEs) in two spatial dimensions involving non-separable boundary conditions. Such non-separable boundary value problems apparently cannot be solved by the usual transforms. We note that the unified transform always yields integral expressions which, in contrast to the expressions obtained by the usual transforms, have the advantage that are uniformly convergent at the boundary. Thus, even for the cases of separable boundary value problems where the usual transforms can be implemented, the unified transform provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of a typical boundary value problem, by extending the approach of Flyer and Fokas (2008Proc. R. Soc.464, 1823–1849). This work is the two-dimensional continuation of the heat equation with oblique Robin boundary conditions which was analysed in Mantzavinos and Fokas (2013Eur. J. Appl. Math.24(6), 857–886.

Keywords

initial-boundary value problemstwo-dimensional heat equationunified transform methodnon-separable boundary conditionsoblique Neumann boundary conditions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 6 , December 2015 , pp. 887 - 916
Copyright
Copyright © Cambridge University Press 2015 

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References

[1]Fokas, A. S. (1997) A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. A. 453, 1411–1443.CrossRefGoogle Scholar
[2]Fokas, A. S. (2000) On the integrability of linear and nonlinear PDEs, J. Math. Phys. 41, 4188–4237.CrossRefGoogle Scholar
[3]Fokas, A. S. & Spence, E. A. (2012) Synthesis as opposed to separation of variables. SIAM Rev. 54, no. 2, 291–324.CrossRefGoogle Scholar
[4]Fokas, A. S. & Kalimeris, K. (2010) The heat equation in the interior of an equilateral triangle, stud. Appl. Math. 124, 283–305.Google Scholar
[5]Mantzavinos, D. & Fokas, A. S. (2013) The unified method for the heat equation: I. non-separable boundary conditions and non-local constraints in one dimension. Eur. J. Appl. Math. 24 (6), 857–886.CrossRefGoogle Scholar
[6]Antipov, Y. & Fokas, A. S. (2004) A transform method for the modified helmholtz equation on the semi-strip. Math. Proc. Cam. Phil. Soc. 137, 339–365.Google Scholar
[7]Flyer, N. & Fokas, A. S. (2008) A hybrid analytical-numerical method for solving evolution partial differential equations: I. The half-line. Proc. R. Soc. 464, 1823–1849.CrossRefGoogle Scholar