Applied Analysis

The department is a leader in the analysis of systems of nonlinear differential equations and dynamical systems that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.

Applied analysis is an important area of research in the Department of Mathematics.

Analysis of Partial Differential Equations Arising in Fluid Dynamics, Hyperbolic Conservation Laws, Etc.

Wang's research area is analysis of partial differential equations arising in fluid dynamics, hyperbolic conservation laws, elastodynamics, geometry, kinetic theory, biology, etc. He works on nonlinear PDE problems including transonic flows, free boundary problems, vortex sheets, turbulence, isometric embeddings,  boundary layer, regularity, and so on.

Free Boundary Problems in Mathematical Finance

Chadam's recent research efforts have been focused on the study of free boundary problems that arise in mathematical finance. With his colleague, Xinfu Chen, their students, and foreign collaborators, he has studied early exercise boundaries for American style options analytically and numerically. In addition to giving precise estimates for the location of these boundaries, the work provided the first rigorous proof of the existence and uniqueness of the solution to the mathematical problem for the American put in the nonlinear integral equation formulation as well as convexity of its early exercise boundary. These methods have been carried over in a unified manner to treat a wide range of similar problems, including inverse first crossing problems in credit default and optimal strategies for prepayment of mortgages. Present research in this area is directed toward the study of default contagion in the context of higher dimensional value-of-firm models and multiple boundaries in callable convertible bonds.

Chadam and his students also are interested in a variety of other problems, such as pricing and hedging equity-linked securities and calibrating jump-diffusion and stochastic volatility models to electricity prices and using them to price futures contracts and swing options.

Lattice Models of Phase Transitions in Crystalline Solids

Vainchtein's research program seeks to advance the understanding of the dynamics of phase transitions in crystalline solids from the perspective of mesoscopic and microscopic frameworks. It focuses on a series of prototypical lattice models of increasing complexity with the ultimate goal of developing a quasicontinuum theory that captures the essential features of phase nucleation, interface kinetics, and the associated energy dissipation. The mathematical problems can often be reduced to quasilinear advance-delay differential-difference equations with bi-stable nonlinearity.

Nonlinear Analysis, Partial Differential Equations and Calculus of Variations.

Lewicka's research areas are nonlinear analysis, partial differential equations and calculus of variations. She has obtained results on the well-posedness and stability of systems of conservation laws and reaction-diffusion equations. Currently, her research focuses on the mathematical theory of elasticity with connections to Riemannian geometry and with an eye on the applications in morphogenesis of growth.

Nonlinear Differential Equations

Chen studies a wide variety of problems on such topics such as non-linear partial differential equations of parabolic and elliptic type, ordinary differential equations and dynamical systems, free boundary problems and interfacial dynamics, singular perturbation and asymptotic expansions, and mathematical finance.

Phase Field Equations; Renormalization and Scaling in Differential Equations

Prof. Caginalp and collaborators developed many aspects of the phase field equations that describe interfaces using a smooth transition. A large number of free boundary problems including the classical Stefan model, the surface tension and kinetics model and Cahn-Hilliard have been shown to be distinguished limits of the phase field equations. The method has also been used to derive the limiting equations for alloy solidification. Recently, Chen, Eck and Caginalp have proposed a new phase field model that they proved differs from the sharp interface problem at only the second order (in interface thickness) leading to computations that are highly accurate.

Prof. Caginalp's initial paper on the phase field equations is the second most cited paper in the Archives RMA during the 20 year period 1984-2004.

The renormalization and scaling research focuses on techniques to calculate the exponents associated with large time and space behavior for the heat equation with nonlinear source terms, for example. Results have also been obtained for large time behavior of a solidification interface.

Prof. Caginalp also works in the area of Mathematical Finance and Economics (see publications under that heading).

Transient and Multiple Timescale Dynamics

Rubin works to develop and apply dynamical systems techniques to study transient dynamics and to study complicated dynamics including canards, mixed-mode oscillations and bursting in multiple timescale and hybrid systems.