ANZIAM J.
45 (2004), 333348

Conservation laws for secondorder parabolic partial differential equations


D. Pidgeon
Institute of Fundamental Sciences,
Mathematics
Massey University
New Zealand


M. VliegHulstman
Institute of Fundamental Sciences,
Mathematics
Massey University
New Zealand
M.Vlieg@massey.ac.nz




Abstract

Conservation laws for partial differential
equations can be characterised by an operator,
the characteristic and a condition involving the
adjoint of the Fréchet derivatives of this
operator and the operator defining the partial
differential equation. This approach was
developed by Anco and Bluman and we exploit it to
derive conditions for secondorder parabolic
partial differential equations to admit
conservation laws. We show that such partial
differential equations admit conservation laws
only if the time derivative appears in one of two
ways. The adjoint condition, however, is a
biconditional, and we use this to prove necessary
and sufficient conditions for a certain class of
partial differential equations to admit a
conservation law.

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