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Conservation laws for second-order parabolic partial differential equations

Published online by Cambridge University Press:  17 February 2009

B. Van Brunt ,
D. Pidgeon ,
M. Vlieg-Hulstman  and
W. D. Halford
B. Van Brunt
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, New Zealand; e-mail: B.vanBrunt@massey.ac.nz, W.D.Halford@massey.ac.nz and M.Vlieg@massey.ac.nz.
D. Pidgeon
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, New Zealand; e-mail: B.vanBrunt@massey.ac.nz, W.D.Halford@massey.ac.nz and M.Vlieg@massey.ac.nz.
M. Vlieg-Hulstman
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, New Zealand; e-mail: B.vanBrunt@massey.ac.nz, W.D.Halford@massey.ac.nz and M.Vlieg@massey.ac.nz.
W. D. Halford
Affiliation:
Institute of Fundamental Sciences, Mathematics, Massey University, New Zealand; e-mail: B.vanBrunt@massey.ac.nz, W.D.Halford@massey.ac.nz and M.Vlieg@massey.ac.nz.
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Abstract

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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 second-order 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.

Type
Research Article
Information
The ANZIAM Journal , Volume 45 , Issue 3 , January 2004 , pp. 333 - 348
Copyright
Copyright © Australian Mathematical Society 2004

References

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