Degree Name

Doctor of Philosophy


Department of Mathematics


Members of a hierarchy of integrable nonlinear evolution equations are applied problem in solidification and various problems in surface redistribution of crystalline materials.

The members of the hierarchy are related to the well known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of concentration. We find that any linear combination of members of the hierarchy linearizable and the evolution equations formed represent a rotation of the diffusion potential modelled by the linear equation.

The base member of the hierarchy is employed to obtain an analytic solution describing the temperature distribution and position of any number of phase boundaries as a material cools on an effectively semi-infinite base material. Each is initially homogeneous and at a uniform temperature. The solution method may incorporate any materials with temperature dependent thermal properties undergoing any number of phase changes. As an example, we incorporate transitions through five phases of iron with nonlinear heat conduction, as the iron cools copper base.

We adapt various other members to derive a new integrable nonlinear equation which models the surface evolution of an arbitrarily oriented theoretical anisotropic material by the concomitant action of evaporation-condensation and surface diffusion. This theoretical anisotropic material behaves like a liquid crystal and the same constitutive relations as those of the constant coefficient linear equation used in the small slope approximation for isotropic materials.

The integrable nonlinear equation may be used to advantage as test cases for numerical schemes. Its form has many attributes of the nonlinear governing equation for an isotropic material.

Closed form solutions are constructed, where the theoretical anisotropic material may be arbitrarily oriented, for the evolution of a ramped surface by concomitant evaporation-condensation and surface diffusion, and the development of a single symmetric grain boundary groove by surface diffusion.

The solution for the ramped surface showed, unlike the solution when the small slope approximation is assumed, that a vertical tangent may develop for the case where the initial inclination is large.

In the development of a single symmetric grain boundary groove by surface diffusion, the theoretical anisotropic material has a well defined solution even the limiting case of a groove which has a root that is vertical and it is found case that the groove growth rate is finite. As an isotropic material is a special of a liquid crystal, then it is proposed that the groove growth rate for an isotropic material is finite also.

Lastly, by a novel application of the new integrable nonlinear equation, we derive a solution for the development of a single symmetric grain boundary groove by surface diffusion for an isotropic material. A solution is achieved by partitioning the surface into subintervals delimited by lines of constant slope. Within subinterval, the advance of the surface is described by the new integrable evolution equation. The model is capable of incorporating the actual nonlinearity arbitrarily closely. The surface profile is determined for various values of the central groove slope including the limiting case of a groove which has a root that is vertical.