A Diffusion Equation with Exponential Nonlinearity Recant Developments

The purpose of this paper is to analyze in detail a special nonlinear partial differential equation (nPDE) of the second order which is important in physical, chemical and technical applications. The present nPDE describes nonlinear diffusion and is of interest in several parts of physics, chemistry and engineering problems alike. Since nature is not linear intrinsically the nonlinear case is therefore the general. We determine the classical Lie point symmetries including algebraic properties whereas similarity solutions are given as well as nonlinear transformations could derived. In addition, we discuss the nonclassical case which seems to be not solvable. Moreover we show how one can deduce approximate symmetries modeling the nonlinear part and we deduce new generalized symmetries of lower symmetry. The analysis allows one to deduce wider classes of solutions either of practical and theoretical usage in different domains of science and engineering.