Linearity of the steady state of a nonlinear anisotropic diffusion process

The nonlinear equation describing the steady state of a nonlinear anisotropic diffusion process can be transformed to a special type of linear partial differential equation using the Kirchhoff transformation together with the hodograph transformation. The method of separation of variables is employed to obtain exact solutions. A solution with a domain bounded by two ellipses is studied in detail for the materials which have a linear conductivity function in one direction and a constant conductivity function in the other direction. Comparison of the adopted approach with the similarity reduction approach is also presented, though the obtained results cannot be easily found via the latter approach. A special case is found in which the transformed linear equation can be reduced further to the Laplace equation.