Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations

DOI

10.2991/jnmp.2008.15.s1.10How to use a DOI?

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 changes 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 Lie canonical equation ut = uxx + Au/x2, A a nonzero constant, 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.

Copyright

© 2008, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Download article (PDF)

TY  - JOUR
AU  - Fazal M. Mahomed
PY  - 2008
DA  - 2008/08/01
TI  - Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 112
EP  - 123
VL  - 15
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2008.15.s1.10
DO  - 10.2991/jnmp.2008.15.s1.10
ID  - Mahomed2008
ER  -