Group Classification and Generalized Conditional Symmetry Reduction of the Nonlinear Diffusion–Convection Equation with a Nonlinear Source

Abstract

The generalized conditional symmetry method, which can be considered a generalization of the conditional symmetry method, is used to study the nonlinear diffusion᎐convection equations with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the convective term and the source term, which permit the generalized conditional symmetry reductions. A number of examples are considered and some exact solutions are constructed via the compatibility of the generalized conditional symmetry and the considered equation.

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

“…We have also checked that Solutions (84) and (85) with λ = 0 produce the solutions obtained in [30] (see Formulae (3.11a), (3.11b), (3.11c) therein). In the paper [42], exact solutions of Equation (56) were constructed using the generalized conditional symmetries. The first, second and third solutions listed in Table 4 [42] are nothing else but particular cases of the exact solution (92) of the nonlinear Equation (56) with λ 2 = 0, while the fourth and fifth solutions from Table 4 [42] are also obtainable by the Q-conditional symmetries (see Solutions (82) and (83)).…”

“…It should be pointed out that several sets of non-Lie solutions of the RDC Equation (56) were independently constructed in [40,42] (see also Section 5.2 in [7]) using other methods. It was established much later [38] (comparison with the solutions obtained in [42] is presented above) that these solutions are also obtainable via the relevant Q-conditional symmetries.…”

Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

“…The approach and its several extensions are illustrated in the books [35,36] and the papers [32,33,37,38]. One of the multiple applications of the Lie symmetry method is the similarity reduction of PDEs to ones with fewer variables.…”

Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

“…This method was developed by Fokas, Zhdanov and Qu et al, and has been applied to study the functional separation of variables for various nonlinear equation [47,48]. In order to implement the method effectively, let us review some basic notations of the functionally separation solutions and generalized conditional symmetry [46][47][48]. (3) is said to be functionally separable if there exist functions q(u), ϕ(t), and ψ(x) such that…”