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On the diffusion equation and diffusion wavelets on flat cylinders and the n ‐torus
Author(s) -
Bernstein Swanhild,
Ebert Svend,
Kraußhar Rolf Sören
Publication year - 2010
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1369
Subject(s) - mathematics , torus , mathematical analysis , diffusion equation , lipschitz continuity , operator (biology) , dirac operator , diffusion , geometry , physics , quantum mechanics , biochemistry , chemistry , economy , repressor , transcription factor , economics , gene , service (business)
In this paper we study the solutions to the diffusion equation on some conformally flat cylinders and on the n ‐torus. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the parabolic Dirac operator. We study their fundamental properties, give representation formulas of all these solutions and develop some integral representation formulas. In particular we set up a Green type formula for the solutions to the homogeneous diffusion equation on cylinders and tori. Then we also treat the inhomogeneous diffusion equation diffusion with prescribed boundary conditions in Lipschitz domains on these manifolds. As main application, we construct well localized diffusion wavelets on this class of cylinders and tori by means of multiperiodic eigensolutions to the parabolic Dirac operator. We round off with presenting some concrete numerical simulations for the three dimensional case. Copyright © 2010 John Wiley & Sons, Ltd.