Contemporary materials science abounds with novel processing methods. Devices such as lasers, microwave sources, and electron beam guns, provide unprecedented control over the deposition of energy within a material. The modern materials scientist has the ability to deposit energy volumetrically, to precisely control the location of energy deposition within a material, and to deposit energy in extremely short intervals of time. While making possible numerous thermal processing methods, these devices also push the limits of our understanding of the response of materials to energy deposition. In order to optimize and control these processing methods, it becomes necessary to further our understanding of this response.

Here, we investigate several problems, motivated by the study of thermal processing methods, whose analyses further our understanding of these new parameter regimes. First, we consider two classes of problems arising in microwave processing of ceramics. These problems are characterized by volumetric energy deposition and a weak coupling between thermal diffusion and electromagnetic wave propagation. Next, we investigate a sequence of problems motivated by and arising in the study of an electron beam joining process. These problems are characterized by rapid volumetric energy deposition and a strong coupling between thermal diffusion and elastic wave propagation.