Differential Equations and Applied Mathematics
Main Focus of Research Interests
- Qualitative and Quantitative Theory of Ordinary, Functional and Partial Differential Equations, and systems of coupled differential equations arising in mathematical physics, engineering and applied sciences. These encompass both linear and nonlinear models of heat conduction, wave phenomena, material science, mathematical finance, electron density theory, quantum chemistry, suspension flows, nonlinear mechanics and elasticity, fluid dynamics, thermo-elasticity, viscoelasticity, etc;
- Control Theory as directed to the above models, to include optimal control theory; differential game theory; stabilization and controllability theories, in particular long time behavior and some aspects of dynamical systems; inverse problems.
The group holds weekly research seminars with presentations from faculty and graduate students. The aim is to introduce graduate students to active research areas and guide them to a dissertation topic. Past seminars can be found on Events Calendar page.
In order to carry research in these areas, foundational courses need to be mastered as well as core areas. The particular track is tailored to fit the student's interests under the advisor's guidance. PhD students are encouraged to submit their work to research-oriented journals.
- Math 6391 - Partial Differential Equation I
- Math 7350 - Real Variables I
- Math 7351 - Real Variables II
- Math 7/8355 - Functional Analysis I
- Math 7361 - Complex Analysis
- Math 7375 - Methods Math Physics I
- Math 7376 - Mthds Math Physics II
- Math 7/8393 - Differential Equations/Applications
- Math 7/8395 - Theory of Differential Equations
- Math 7/8501 - Nonlinear Wave Phenomena
- Math 7/8504 - Partial Differential Equations
For more information on course descriptions see the graduate catalog.
PhD students are encouraged to attend various conferences in their research areas. This is in addition to the conferences and seminars frequently hosted by the Department.
PhD students of the advisors in the group have secured academic tenured positions in top rated research universities such as UC Berkeley, Northwestern University, Univ of Vanderbilt, North Carolina State, University of Nebraska, Clemson University, Texas Tech; postdocs positions at MIT, Univ. of Minnesota, UCLA, USC, Oregon State, Tapper School of Business at Carnegie Mellon, Mittag Leffler, NSF, AFOSR, and Army Fellowships; as well as high tech jobs in the private sector and at federal institutions such as: Martin Lockeed, NSA, Northrop Grumman, financial institutions (WellFargo, Suntrust); and also postdoc and regular positions abroad at INRIA (Paris), Companie General Geophysique (Paris), L'Ecole Central (Paris). The highlight is Daniel Tataru, presently Professor at UC Berkeley, recipient of AMS Bocher prize in 2002, invited speaker at ICM in Beijing and most recently elevated to membership in the American Academy of Art and Sciences.
Placement of former PhD students advised by the faculty in this group include academic jobs at research universities as well as at high-tech industrial, financial and research companies both in US and abroad.
Summer funding may be available for some advanced PhD students.
Several faculty members serve with no less than 30 cumulative appointments on Editorial Boards of main stream journals in the area of Applied Analysis, Applied Mathematics, Differential Equations and Control Theory, to include also the position as editor-in chief.
Below are research interests and two representative or recent publications from each member.Hongqiu Chen, Associate Professor
Research Interests: Partial differential equations and fluid mechanics.
- Conservative discontinuous Galerkin methods for the generalized Korteweg-de-Vries equation (with J.L. Bona, O.A. Karakashian and Y. Xing), Mathematics of Computation 82 (2013) pp. 1401-1432.
- Periodic traveling-wave solutions of nonlinear dispersive evolution equations (with J. L. Bona), Discrete and Continuous Dynamical Systems 33 (2013) pp. 4841-4873.
Research Interests: Mathematical physics, continuum mechanics, financial mathematics, quantum mechanical density functional theory, applied analysis and partial differential equations.
- A Cahn-Hillard model in a domain with non-permeable walls, Physica D: Nonlinear Phenomena 240 (2011), 754-766 (with A. Miranville and G. Schimperna)
- Chaotic solutions for the Black-Scholes equation, Proceedings of the American Mathematical Society, 140 (2012), 2043-2052 (with H. Emamirad and J.A. Goldstein)
Research Interests: Linear and nonlinear PDE, operator theory, stochastic processes, mathematical physics, classical analysis, applied analysis, financial mathematics.
- Abstract wave equations and associated Dirac-type operators (with F.Gesztesy, H. Holden and G. Teschl), Ann. Mat. Pura Appl. (4) 191 (2012),631-676 .
- Asymptotic parabolicity for strongly damped wave equations (with G.Fragnelli, G. R. Goldstein and S. Romanelli), in Spectral Analysis, Differential Equations and Mathematical Physics, A Festschrift in Honor of Fritz Gesztesy's 60th Birthday, Proceedings of Symposia in Pure Mathematics, Vol. 87, Amer. Math. Soc. (2013), 119-131.
Research Interests: Ordinary and functional differential equations, STEM Education.
- Parallel theories of integral equations (with T.A. Burton), Nonlinear Studies, 17 (2010), 177-197
- The effectiveness of blended instruction in core postsecondary mathematics courses, (with A. Bargagliotti-Lead, F. Botelho, J. Gleason, and A. Windsor), The International Journal for Technology in Mathematics Education, 19, no. 3 (2012), 83-94
Research Interests: Applied analysis, partial differential equations, dynamical systems, fluid mechanics.
- Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models (with J.D. Evans), J. Non-Newtonian Fluid Mech. 154 (2008), 39-46.
- "Non-failure of filaments and global existence for the equations of fiber spinning" (with M. Renardy), IMA J. Appl. Math. 76 (2011), 834-846.
Research Interests: Applied Mathematics. Evolutionary linear and nonlinear partial differential equations. Mathematical control theory and optimization theory. Long time behavior and theory of attractors. Boundary stabilization, boundary controllability problems for linear and nonlinear parabolic and hyperbolic PDE's, as well as for coupled systems of PDE's equations arising in structural acoustic, thermoelastic systems, fluid structure and flow structure interactions with an interface.
- Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping (with I. Chueshov ), Memoires of AMS, 2008, vol 195., 183 pp
- On Well-Posedness and Small Data Global Existence for an Interface Damped Free Boundary Fluid-Structure Model (with M. Ignatova, I. Kukavica, and A. Tuffaha), to appear in Nonlinearity, 2014.
Research Interests: Control Theory, Partial Differential Equations.
- Backward uniqueness of the C_0-semigroup associated with a parabolic-hyperbolic coupled Stokes-Lame PDE system (with G.Avalos), Trans.Amer.Math.Soc. Vol 88(9) (2009), 1357-1396.
- Boundary control and boundary inverse theory for non-homogeneous second order hyperbolic equations: a common Carleman estimates approach (with S.Liu), HCDTE Lectures Notes. Part I. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations, AIMS on Applied Mathematics Vol 6, (2013), pp 227-339, American Institute of Mathematical Sciences.