Description: Talk by Prof. Fazal Mahomed (Wits) given to the School of Physics on 6/10/09

Title: Invariant Characterization of Scalar Linear (1+1) Parabolic Equations

Abstract: We obtain a complete invariant characterization of scalar
linear (1+1) parabolic equations under equivalence transformations
for all the four canonical forms. Firstly semi-invariants under
change of independent and dependent variables and the construction
of the relevant transformations that relate the two parabolic
equations are given. Then necessary and sufficient conditions for a
(1+1) parabolic equation, in terms of the coefficients of the
equation, to be reducible via local equivalence transformations to
the one-dimensional classical heat equation and the second Lie canonical
equation are presented. These invariant conditions provide practical criteria for reduction to the respective canonical equations. Also the construction of the transformation formulas that do the reductions are provided. We further show how one can transform a (1+1) parabolic equation to the third and fourth Lie canonical forms thus providing invariant
criteria for a parabolic equation to have two and one nontrivial
symmetries as well. Ample examples are given to illustrate the
various results.

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