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SIGMA 3 (2007), 019, 14 pages math-ph/0702033
https://doi.org/10.3842/SIGMA.2007.019
Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations
Valentyn Tychynin a, Olga Petrova b and Olesya Tertyshnyk b
a) Prydniprovs'ka State Academy of Civil Engineering
and Architecture, 24a Chernyshevsky Str., Dnipropetrovsk, 49005 Ukraine
b) Dnipropetrovsk National University, 13 Naukovyi Per., Dnipropetrovsk, 49050 Ukraine
Received January 06, 2006, in final form January 17, 2007; Published online February 06, 2007
Abstract
We have constructed new formulae for generation of
solutions for the nonlinear heat equation and for the Burgers equation
that are based on linearizing nonlocal transformations and on nonlocal
symmetries of linear equations. Found nonlocal symmetries and
formulae of nonlocal nonlinear superposition of solutions of
these equations were used then for construction of chains of exact solutions.
Linearization by means of the Legendre transformations of a
second-order PDE with three independent variables allowed to
obtain nonlocal superposition formulae for solutions of this
equation, and to generate new solutions from group invariant
solutions of a linear equation.
Key words:
Lie classical symmetry; nonlocal symmetries; formulae for generation of solutions; nonlinear superposition principle.
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