PDE & Dynamical Systems

Partial differential equations (PDEs) are one of the most fundamental tools for describing continuum phenomena in the sciences and engineering. Early work on PDEs, in the 1700s, was motivated by problems in fluid mechanics, wave motion, and electromagnetism. Since that time, the range of applications of PDEs has expanded rapidly. For example, PDEs are used in mathematical models of weather and climate, in medical imaging technologies, in the design of new composite materials, in models of elementary particle interaction and of the formation of galaxies, in models of cancerous tumor growth or of blood flow in the heart, in simulating semiconductor devices, in models of bacterial colonies, in models of financial markets and asset price bubbles, in describing the flocking behavior of birds and fish. PDEs also have played an important part in the development of other branches of mathematics, including harmonic analysis, differential geometry, probability, optimization and control theory. The phenomena described by PDEs are as complex as the world around us; the mathematical techniques needed to study PDEs are very diverse.

Many of the faculty in the Duke mathematics department conduct research on partial differential equations, using a variety of techniques. Some of this work focusses on the geometric structure of PDEs, on the analytical properties of their solutions, or on the way in which solutions depend on parameters. Some faculty are developing computer algorithms for approximating solutions to complicated PDEs. Other faculty are developing new mathematical models based on PDEs in the context of applications in fluid mechanics, photonics, materials science, and biology.

Faculty

J. Thomas Beale

Professor Emeritus of Mathematics

Keywords in this area
boundary integral methods, motion of fluid interfaces, equations of incompressible flow, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow

Robert Bryant

Philip Griffiths Professor of Mathematics

Keywords in this area
differential geometry, holonomy, exterior differential systems, integrability, symplectic geometry

John Everett Dolbow

Professor of Civil and Environmental Engineering

Keywords in this area
evolving interfaces, finite element methods, phase field methods

Gregory Joseph Herschlag

Visiting Assistant Professor of Mathematics

Keywords in this area
biological modeling, physical modeling, fluids flow across dynamical channels, kinetic equations, surface catalysis, molecular dynamics, stochastic boundary conditions

Anita T. Layton

Robert R. & Katherine B. Penn Professor of Mathematics

Keywords in this area
mathematical biology, mathematical physiology, mathematical modeling, kidney physiology, renal hemodynamics, diabetes, multiscale modeling, fluid-structure interactions, computational fluid dynamics, numerical partial differential equations, feedback control, systems biology

Jian-Guo Liu

Professor of Physics

Jianfeng Lu

Associate Professor of Mathematics

Keywords in this area
multiscale modeling and simulations, numerical analysis, calculus of variations, kinetic equations, Schroedinger equations

Jonathan Christopher Mattingly

Professor of Mathematics

Keywords in this area
Stochastic and Random PDEs, Stochastic Dynamical Systems, Malliavin Calculus, Fluid Mechanics, Approximating invariant measures

James H. Nolen

Associate Professor of Mathematics

Keywords in this area
reactive diffusion equations & applications, homogenization of partial differential equations, random media, asymptotic analysis

Arlie O. Petters

Benjamin Powell Professor of Mathematics in Trinity College of Arts and Sciences

Keywords in this area
gravity, light, mathematical finance

Marc Daniel Ryser

Visiting Assistant Professor in the Department of Mathematics

Keywords in this area
bone biology, pattern formation

Mark A. Stern

Professor of Mathematics

Keywords in this area
geometric analysis, elliptic partial differential equations

Thomas Peter Witelski

Professor in the Department of Mathematics

Keywords in this area
fluid dynamics, nonlinear partial differential equations, dynamical systems, perturbation methods