Dr. Du earned his Ph.D. in Mathematics (1988) from Carnegie Mellon University. Before Columbia, he was most recently the Verne M. Willaman Professor of Mathematics and Professor of Materials Science and Engineering at Penn State University. Recognition for his work includes Frame Faculty Teaching Award (1992), Feng Kang prize in scientific computing (2005), SIAM Outstanding Paper prize (2016), ACM Gordon Bell prize finalist (2016), and his selection as a 2013 SIAM Fellow for contributions to applied and computational mathematics with applications in materials science, computational geometry, and biology.
We present a mathematical framework of nonlocal models of mechanics and diffusion processes characterized by a horizon parameter which measures the range of nonlocal interactions and/or memory effects. We study various properties of the nonlocal models and also explore their various limits. In particular, we show their close connections to classical local PDE models in the limit when the horizon parameter shrinks to zero and to global fractional PDEs in the limit when the horizon parameter tends to infinity. We also discuss the coupling of models characterized by different scales.