Lie and conditional symmetries of a class of nonlinear (1+2)-dimensional boundary value problems

TL;DR

This paper introduces a new definition of conditional invariance for boundary value problems, enabling symmetry classification and reduction of nonlinear (1+2)-dimensional diffusion problems, with applications to problems in semi-infinite domains.

Contribution

It proposes a generalized definition of conditional invariance for boundary value problems, unifies previous approaches, and applies it to classify symmetries of nonlinear diffusion equations in semi-infinite domains.

Findings

Identified a special exponent k=-2 where additional symmetries occur.

Reduced complex (1+2)-dimensional problems to simpler (1+1)-dimensional forms.

Demonstrated the influence of domain geometry on Lie invariance.

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, follow 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 proved that there is a special exponent, $k=-2$, for the power diffusivity $u^k$ when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case $k\not=-2$. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity $u^k$ 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 geometry of the domain are presented.