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Symmetry 2015, 7(3), 1410-1435; https://doi.org/10.3390/sym7031410

Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems

1
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
2
Institute of Mathematics, National Academy of Science of Ukraine, 3 Tereshchenkivs’ka Street, Kyiv 01601, Ukraine
*
Author to whom correspondence should be addressed.
Academic Editor: Sergei Odintsov
Received: 30 June 2015 / Revised: 3 August 2015 / Accepted: 11 August 2015 / Published: 17 August 2015
(This article belongs to the Special Issue Lie Theory and Its Applications)
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Abstract

A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case) is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, followed as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1 + 2)-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proven that there is a special exponent, k ≠ -2, for the power diffusivity uk when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case k ≠ -2. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity uk and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena) to (1 + 1)-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on the geometry of the domain are presented. View Full-Text
Keywords: Lie symmetry; Q-conditional symmetry; nonlinear boundary-value problem; nonlinear diffusion; exact solution Lie symmetry; Q-conditional symmetry; nonlinear boundary-value problem; nonlinear diffusion; exact solution
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Cherniha, R.; King, J.R. Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems. Symmetry 2015, 7, 1410-1435.

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