Separable Solutions to Non-Linear Anisotropic Diffusion Equation in Elliptic Coordinates

The steady non-linear diffusion equation is often encountered in natural sciences and engineering. Whilst isotropic and axially symmetric solutions are elementary, the complexity of many systems requires a more realistic description. The studies seeking to break the symmetry via a generalisation to elliptic domains are rare, and despite a series of preluding simplifications, eventuate in computational methods and results instead of analytical methods. We will prove that two- and three-dimensional exact solutions of the non-linear diffusion equation exist in elliptic coordinates subject to an arbitrary piecewise constant azimuthal anisotropy. The general purpose is to delineate the conditions under which the anisotropic solutions persist in curvilinear orthogonal coordinate systems. This research is a part of long-term study which has brought up the question of existence of the solution in other coordinate systems after being done in polar and spherical coordinates.