Abstract
This review is devoted to search for Lie and Q-conditional (nonclassical) symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is derived via two different algorithms in order to show that the result depends essentially on the type of equivalence transformations used for the classification. Moreover, a complete description of Q-conditional symmetries for PDEs from the class in question is also presented. It is shown that all the well-known results for reaction-diffusion equations with exponential nonlinearities follow as particular cases from the results derived for this class of reaction-diffusion-convection equations. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found by means of different techniques. Finally, an application of the exact solutions for solving boundary-value problems arising in population dynamics is presented.
Highlights
It is well known that nonlinear evolution PDEs play a crucial role in mathematical modeling of a wide range of processes in natural, social and life sciences
The first appearance of a PDE with the exponential nonlinearity in the diffusion coefficient probably was in the Ovsiannikov paper [12] devoted to the Lie symmetry classification (LSC) of nonlinear diffusion equations
This section is devoted to the construction of exact solutions of the RDC equations with exponential nonlinearities using Q-conditional symmetries derived in the previous section
Summary
It is well known that nonlinear evolution PDEs play a crucial role in mathematical modeling of a wide range of processes in natural, social and life sciences. In contrast to the evolution PDEs with power-law nonlinearities, there are not many papers devoted to the examination of reaction-diffusion-convection (RDC) equations with exponential nonlinearities. The first appearance of a PDE with the exponential nonlinearity in the diffusion coefficient probably was in the Ovsiannikov paper [12] devoted to the Lie symmetry classification (LSC) of nonlinear diffusion equations This equation has the form: ut = (eu u x ) x (5). There are some recent papers devoted to the study of the evolution PDEs with the exponential nonlinearity by different mathematical methods (see, e.g., [13,14,15] and the references cited therein). We unite all the known results about Lie and Q-conditional symmetry, exact solutions and their applications for the following class of RDC equations: ut = (enu u x ) x + λemu u x + C (u),. We present some conclusions and highlight new results obtained in this review
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