NICOLE BUCZKOWSKI

I studied nonlocal models at UNL under the supervision of Petronela and Mikil during my PhD work at UNL. Classical derivatives have been crucial in classical mechanics since their formulation by Newton and Leibniz in the late 1600s, but rely on our ability to take limits which require information at a point. Thus if there is a discontinuity in the function or in the domain (e.g. a plate being fractured), we can not take the derivative of a function anymore (we an sometimes leverage weak derivatives, but in some cases these may still not be enough.) Instead of considering information at a point, we focus on gathering information around a point instead. This makes it so nonlocal models are not only capable of capturing one-scale of interactions like classical models but multiple scales through their kernels. There are many applications, including a personal favorite of mine peridynamics (from the Greek peri meaning around and dynamics, meaning force or power), introduced by Stewart Silling. Peridynamics reformulates continuum mechanics, but allows for discontinuities which is extremely beneficial in fracture mechanics.

Below is a list of publications I have helped author

• Two nonlocal biharmonic operators
• Sensitivity analysis for solutions to heterogeneous nonlocal systems
• Nonlocal physics informed neural networks

"PDEs stands for Partial Differential Equations. A partial differential equation is an equation that has partial derivatives in it. Pretty cool!" Leilani Pai, who has definetly taken PDEs