Classification of the Extended Symmetries of the Inhomogeneous Nonlinear Diffusion Equation

Abstract

Using Lie group methods, we analyse nonlinear diffusion equations in an inhomogeneous medium f(x)u_t=(g(x)D(u)u_x)_x with arbitrary diffusion coefficient D(u), and arbitrary thermal coefficients f(x) and g(x), which have a wide spectrum of applications in many areas of science. The Lie-group-based similarity method leads to a classification of the diffusion and thermal coefficients according to its symmetry properties. With the help of the adjoint representation, the optimal system of similarity reductions is calculated. Exact similarity solutions of the second-order ordinary differential equatiors resulting from the reductions are demonstrated by examples.