Metal surface evolution was described by a nonlinear fourth-order partial differential equation for curvature-driven flow. The standard boundary conditions for grain-boundary grooving, at a grain–grain–fluid triple intersection, involved a prescribed slope at the groove axis. The well-known similarity reduction was no longer valid when the dihedral angle and surface diffusivity depended upon time due to variation of the surface temperature. A non-linear fourth-order model was adapted which could be deduced from symmetry analysis to be integrable; equivalent to the fourth-order linear diffusion equation. The connection between classical symmetries and separation of variables permitted the development of a correction to the self-similar approximation as a power series in a time-like variable.

Temperature-Dependent Surface Diffusion Near a Grain Boundary. P.Broadbridge, J.M.Goard: Journal of Engineering Mathematics, 2011 66[1-3], 87-102