Applied Analysis
Group members: Professors Caginalp, Chadam, Chen, Lewicka, Rubin, Troy, Vainchtein, and Wang
The department is a leader in the analysis of systems of nonlinear differential equations 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. Research in these areas is aided by a variety of computational facilities, including Sun Microsystem work stations and the massively parallel machines at the Pittsburgh Supercomputing Center.
Applied analysis is an important area of research in the Department of Mathematics. List below are a number of the current projects.
Phase Field Equations; Renormalization and Scaling in Differential Equations (Caginalp).
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).
Free Boundary Problems in Mathematical Finance (Chadam, Chen)
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.
Nonlinear Differential Equations (Chen)
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.
Pattern Formation in Coupled Cell Networks (Rubin)
The general topic of Rubin's research is spatio-temporal pattern formation in coupled cell networks. The overall goal of this research is to understand how the intrinsic dynamics of network elements interact with the architecture and properties of coupling to drive network activity.
To this end, Rubin primarily uses and develops techniques in geometric singular perturbation theory, bifurcation theory, Evans function theory for stability analysis, as well as map-based reduction methods. Much of his work is motivated by biological applications, particularly those arising from networks of neurons. Some of Rubin's recent work, for example, has considered transitions in activity patterns in respiratory pacemaker networks, analysis of tremor and deep brain stimulation in Parkinson's disease, spike-timing dependent synaptic plasticity, and reduced models of the inflammatory response.
Pattern Formation in Wilson-Cowan Networks (Troy)
The general topic of Troy's research is spatio-temporal pattern formation in coupled Wilson-Cowan networks. The overall goal of this research is to understand how different types of waves form. These include spirals and periodic traveling waves.
Lattice Models of Phase Transitions in Crystalline Solids (Vainchtein)
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.
Conservation Laws (Wang)
Wang's research area is nonlinear partial differential equations and applied mathematics. His current research focuses on multi-dimensional problems arising in conservation laws, compressible fluid flows, magnetohydrodynamics, geometry, and related applications in charge transport, plasmas, and combustion.