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

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 u k 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 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 the geometry of the domain are presented.

A (1+2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions

“…Moreover, BVP (6) involves conditions at infinity, so one cannot apply the definition [13,14] in order to examine Lie invariance of this problem. Here we adapt for such purpose the definition proposed in [15].…”

“…Let us consider the following change of variables, which was used in [15] for the similar purposes, in order to examine items (d)-(f)…”

“…In order to solve (15) by using the classical technique, we should specify an initial profile. Let us set for simplicity V (0, y) = V 0 = const.…”

Lie and Conditional Symmetry of Nonlinear Boundary Value Problems: Definitions, Algorithms and Applications

“…In our recent papers [1,2], a new definition of Lie and conditional invariance of BVPs with a wide range of boundary conditions (including those at infinity and moving surfaces) was formulated and an algorithm for finding such symmetries for the given class of BVPs was determined. The definition and algorithm were applied to some classes of nonlinear (including multidimensional) BVPs arising in physical and biological applications in order to show their efficiency (see [1][2][3] and the references cited therein).…”

“…The definition and algorithm were applied to some classes of nonlinear (including multidimensional) BVPs arising in physical and biological applications in order to show their efficiency (see [1][2][3] and the references cited therein). As a result, Lie and conditional symmetries for several BVPs were completely described, reductions to BVPs of lower dimensionality were constructed and examples of exact solutions with physical/biological meaning were found.…”