The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions

Abstract

We implement the new transform method for solving boundary-value problems developed by Fokas for periodic boundary conditions. The approach presented here is neither a replacement for classical methods nor is it necessarily an improvement. However, in addition to establishing that periodic problems can indeed be solved by the new transform method (which enhances further its scope and applicability), our implementation also has the advantage that it yields a new simpler approach to computing the limit from the periodic Cauchy problem to the Cauchy problem on the line.

Keywords:

• linear partial differential equations
• constant coefficients
• periodic problem

AMS Subject Classifications::

• 35C15
• 35G16
• 30E20

Acknowledgements

We thank A.S. Fokas for the suggestions which lead to this work and V. Vasan for useful comments. The authors gratefully acknowledge support from the National Science Foundation under grants NSF-DMS-VIGRE-0354131 (TT), NSF-DMS-0604546 (BD) and NSF-DMS-1008001 (TT,BD). Any opinions, findings and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the funding sources.