Lie symmetry classification analysis for nonlinear coupled diffusion

The group properties of the (1 + 1)-dimensional matrix diffusion equation of the form with respect to point symmetries are given. It is shown that in particular cases the group properties of this equation are similar to those of the corresponding scalar equation. Namely, the Lie algebra is extended for power and exponential functions, with additional extensions when powers of and -2 exist. However, a specific form of is shown to exist for cases admitting both infinite and finite groups. The result obtained is used to construct new group-invariant solutions for impulsive boundary inputs. Using two similarity variables, a reduction is applied to the coupled diffusion of temperature and volumetric moisture content in porous media under periodic boundary conditions. A perturbation method is employed to obtain an explicit solution for the case when can be expressed exponentially in terms of the similarity variables.