Work on this project continues mathematical research on models of physical systems in which emphasis is placed on effects which are non-local in time, in space or both. Applications are expected in the analysis of thermal and elastic properties of materials with memory, electromagnetic and elastic scattering and population models. The mathematical formulations involve differential equations, Volterra and Fredholm integral equations and combinations of these. Three areas will receive attention. The first concerns models which involve operators influenced by memory effects and are represented by integral operators with singular kernels. Efforts will be made to understand relations between problems and solutions as the order of the singularity changes. Both qualitative behavior as well as numerical approximation of solutions will be sought. Another line of investigation will consider interface problems in which parabolic-hyperbolic equations describe the same phenomenon on either side of a region's boundary. In trying to understand these problems a new concept of absorbing boundaries has evolved which shows great theoretical as well as numerical promise - but at this time there is no firm understanding as to why the boundaries work so well. Finally, in work related to population diffusion, studies will continue on reaction-diffusion processes in which directed diffusion rather than random diffusion is the controlling hypothesis. Models of this type have been used to analyze single species populations diffusing to avoid crowding, extensions to age-dependent situations, models for epidemics and for predator- prey situations. Much of the theoretical work has lagged behind the numerical because basic existence questions remain unanswered. The search will continue.
