Lie symmetries and exact solutions of nonlinear equations of heat conductivity with convection term

Abstract

The Lie symmetries of nonlinear diffusion equations with convection term are completely described. The Lie ansatzes and exact solutions of a certain nonlinear generalization of the Murray equation are constructed. An example of the family of non-Lie solutions of the Murray equation is given.

New Q-conditional symmetries and exact solutions of some reaction–diffusion–convection equations arising in mathematical biology

“…It follows from [23] (see Theorem 1) that Eq. (2) admits a non-trivial Lie algebra only in the cases listed below.…”

“…In papers [12,23], a complete description of Lie symmetries was obtained for the RDC equation (1). It follows from [23] (see Theorem 1) that Eq.…”

“…Consider case (I). Substituting (23) into (21), we immediately establish that the function C(u) can be at maximum a cubic polynomial with respect to the variable u, i.e. :…”

Lie symmetries and form-preserving transformations of reaction–diffusion–convection equations

“…It should be stressed that this paper is the remarkable work because the first time the problem of Lie symmetry classification was completely solved for a class of non-linear partial differential equations. Several papers were devoted to extension of Ovsiannikov's result on other equations of the form (1) [8][9][10][11][12][13][14]. The papers [8] and [13,14] contain the complete description of Lie symmetries for (1) with B = 0 and B = 0, respectively, so that the results obtained here will be compared with those from [8] and [13,14].…”

On Exact Solutions of Nonlinear Diffusion Equations

“…(1) was given in [3], and an analysis of the conditional symmetry of Eq. (1) was carried out in [4].…”