Topic description: Partial differential equations and systems have been successfully used to describe the spatiotemporal evolution of complex physical or biological systems. For example, nonlinear reaction-reaction equations have been applied to morphogenesis pattern formations, epidemic spreading, and oscillatory chemical reactions. Numerical computation is an important part of these scientific investigations because a fully rigorous analysis of the models is usually not possible. Development of more realistic models and advanced computational tools are of current research interests in many important branches of science and engineering. For example, integro-difference equations, cross-diffusion systems, higher order partial differential equations, interacting particle systems, and nonlinear matrix models have been proposed as alternates of traditional diffusion models. Also of interest is the computation of bifurcation diagrams of steady state solutions of these models.
Research opportunities: The numerical approach to these models leads to new research problems, and many of them are accessible to undergraduate students with a background in numerical analysis. A natural student project would involve applying a newly-developed technique called validated continuation to various models, studying when the validation procedure fails, and using this information to adapt and improve the numerical procedures. A project in this direction would require the learning and implementation of widely-used continuation techniques as well as analytical techniques and interval arithmetic computations for bounding error.
Suggested prerequisites: Math 302, Math 426, experience with Maple and/or Matlab and C++
Contact: Sarah Day