| September 3 | Organizational Meeting
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| September 10 |
Weizhang Huang
University of Kansas Convergence of finite element solution of elliptic differential equations using a posteriori equidistributing meshes Abstract Equidistributing meshes are known as optimal meshes in the numerical solution of differential equations, and they are typically defined using the exact solution or some information of the exact solution. In this talk I will present a framework of defining equidistributing meshes in an a posteriori manner, namely, based on computed solutions. Convergence analysis of finite element solution using such a posteriori equidistributing meshes is also given. |
| September 17 |
Anna Ghazaryan
University of Kansas Traveling waves in the models for combustion of solid and high density liquid fuels. I Abstract We consider combustion wavefronts arising in the model for high Lewis number combustion processes such as combustion of high density liquid fuels. An efficient method for the proof of the existence and uniqueness of the combustion front is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with the infinite Lewis number which corresponds to the burning of solid fuels. The question of stability of the fronts is more complicated. I will discuss issues and recent results on the stability analysis that arise at both the linear and nonlinear level. |
| September 24 |
Anna Ghazaryan
University of Kansas Traveling waves in the models for combustion of solid and high density liquid fuels. II Abstract We consider combustion wavefronts arising in the model for high Lewis number combustion processes such as combustion of high density liquid fuels. An efficient method for the proof of the existence and uniqueness of the combustion front is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with the infinite Lewis number which corresponds to the burning of solid fuels. The question of stability of the fronts is more complicated. I will discuss issues and recent results on the stability analysis that arise at both the linear and nonlinear level. |
| October 1 |
Doug Wright
Drexel University Shooting and exit manifolds for pulse interactions in one dimensional reaction-diffusion equations Abstract Many systems possess asymptotically stable traveling wave solutions which are, in some sense, spatially localized. Experiment, numerical simulation and intuition lead us to believe that there ought to be solutions of the original equation which are very nearly the linear superposition of multiple pulses. Constructing such solutions is a first step towards understanding collisions between pulses. We construct a two-dimensional manifold of such solutions (the two parameters should be though of the location of each of the pulses) and show that this manifold is dynamically stable. |
| October 8 |
Vahagn Manukian
University of Kansas Traveling waves for a thin liquid film with surfactant on an inclined plane Abstract We show the existence of traveling wave solutions for a lubrication model of surfactant-driven flow of a thin liquid film down an inclined plane in various parameter regimes via geometric singular perturbation theory. This is a joint work with Stephen Schecter. |
| October 22 |
Professor Cheng-Hsiung Hsu
National Central University, Taiwan On solutions of the compressible Euler equation Abstract In this talk, I will survey some our previous works on the existence of solutions of the compressible Euler equations, and report our recent progress in the study of viscous standing asymptotic states of the compressible nozzle flows. |
| October 29 |
Professor Ting-Hui Yang
Tamkang University, Taiwan Traveling Wave Solutions of Delayed Lattice Differential System in Lotka-Volterra Type Abstract We consider the existence of traveling plane wave solutions of a class of delayed lattice differential system in Lotka-Volterra type. Employing the techniques of cross iteration method coupled with the explicit construction of upper and lower solutions in the theory of weak quasi-monotone dynamical systems, we obtain a critical speed, and show the existence of traveling plane wave solutions connecting two different equilibria when the wave speeds are less than the critical speed. |
| November 5 |
Aslihan Demirkaya
University of Kansas L2 bounds for Radially Symmetric Solutions of 3D-Kuramoto-Sivashinsky Equation Abstract We consider radially symmetric solutions for 3D-Kuramoto-Sivashinsky equation in an annulus {r &isin &real: 0 < r0&le r &le R0}. By using Lyapunov function approach, we show that supt &rarr &infin||u ||L2 [r0,R0]&le C(R0-r0)3/2. Similar results hold for any dimension n and the exponent 3/2 remains the same. |
| November 12 |
Dimitri Breda
University of Udine, Italy On computing the spectrum of mixed functional differential equations Abstract Information on the asymptotic behaviour of a traveling wave, arising for instance in lattice differential equations or in neural transmission models, can be recovered by looking into the spectrum of a suitable operator associated to the relevant advanced-retarded functional differential equation. In the retarded case the asymptotic information is contained in the spectrum of the infinitesimal generator of the semigroup of solution operators. Opposite, in the mixed case such a standard approach is not possible and rather, a direct sum decomposition of the state space allows for considering two separated (backward and forward) evolutions. Nevertheless, all information is contained in the spectrum of the associated exponential dichotomy. After recalling the necessary theoretical background, we present an exponentially accurate numerical approach to reduce the computation of the above infinite-dimensional spectra to a standard matrix eigenvalue problem. Local asymptotic stability properties of wide classes of linear and autonomous retarded and mixed functional differential equations can thus be quantitatively analyzed. |
| November 19 |
Weishi Liu
University of Kansas A tent structure for stability-gain turning points Abstract |
| December 3 |
Mohamed Badawy
University of Kansas Lyapunov exponents and the Sacker-Sell spectrum for infinite dimensional dynamical systems Abstract Lyapunov, more than a century ago, introduced what proved to be a very important tool in the study of the stability of solutions of linear nonautonomous dynamical systems. Lyapunov exponents give us an idea about the asymptotic exponential rate of growth (or decay) of the solutions of the dynamical system. Several numerical methods have already been developed, along with error analysis results, to approximate these exponents for finite dimensional systems. In 1978, Sacker & Sell introduced another spectrum to study linear dynamical systems. It is based on the concept of "Exponential dichotomy". Exponential dichotomy plays a fundamental role in many studies of dynamical systems and has been a widely used assumption in several numerical works. Although finite dimensional systems are relevant in many applications, in several other cases it is most natural to deal with infinite dimensional systems over a Banach or a Hilbert space. In this presentation, we will talk briefly about some properties of the above spectra in the finite dimensional case, and then explain how these spectra can be generalized to study linear infinite dimensional systems. |
| December 10 |
Charles Lamb
University of Kansas Fredholm alternative for implicitly defined functional differential equations of mixed type Abstract Implicitly defined functional differential equations of mixed type can, for instance, arise from applying Kirchoff's laws to a chain of cells analyzing current flow. We will discuss the Fredholm alternative for this type of problem. We will begin with a discussion about how the implicitly defined case may arise, and related background material. In addition, we will finish by applying the Fredholm alternative to an implicitly defined equation. |
(Link to CAM seminar talks in Spring, 2008)
(Link to CAM seminar talks in Fall, 2007)
(Link to CAM seminar talks in Spring, 2007)
(Link to CAM seminar talks in Fall, 2006)
(Link to CAM seminar talks in other years.)
(Link to Numerical Analysis Group Webpage)