Modeling the Diffusion of Heat Energy within Composites of Homogeneous Materials using the Uncertainty Principle

The purpose of this project is to model the diffusion of heat energy in one space dimension, such as within a rod, in the case where the heat flow is through a medium consisting of two or more homogeneous materials. The challenge of creating such a mathematical model is that the diffusivity will be represented using a piecewise constant function, because the diffusivity changes based on the material. The resulting model cannot be solved using analytical methods, and is impractical to solve using existing numerical methods, thus necessitating a novel approach. The approach presented in this thesis is to represent the solution as a linear combination of wave functions that change frequencies at the boundaries of different materials. It will be demonstrated that by using the Uncertainty Principle to construct a basis of such functions, in conjunction with a numerical method that is ideally suited to work with them, a mathematical model for heat diffusion through different materials can be solved much more efficiently than with other well-established methods from the literature