Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations
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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.
- © 2008, the Authors. Published by Atlantis Press.
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Cite this article
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 -