Analysis
Department of Mathematics and Statistics
Texas Tech University
There has been a growing interest in hybrid dynamical systems in recent years. Such systems often undergo vector field switching and/or state jumps. By introducing the notions of persistent limit set and persistent mode, we extend the classical LaSalle's invariance principle to hybrid systems exhibiting both impulses and switching. A weak invariance principle is established for such systems, under a weak dwell-time condition on the impulsive and switching signals. This weak invariance principle is then applied to derive asymptotic stability criteria for impulsive switched systems. As an application, we investigate a switched SEIR epidemic model with pulse treatment and establish sufficient conditions for the global asymptotic stability of the disease-free solution under weak dwell-time signals.
This seminar is co-sponsored with the BioMath group.
Zoom link
Meeting ID: 968 5126 2664
Passcode: TTU
We study the anisotropic Forchheimer-typed flows for compressible fluids in porous media. The first half of the paper is devoted to understanding the nonlinear structure of the anisotropic momentum equations. Unlike the isotropic flows, the important monotonicity properties are not automatically satisfied in this case. Therefore, various sufficient conditions for them are derived and applied to the experimental data. In the second half of the paper, we prove the existence and uniqueness of the steady state flows subject to a nonhomogeneous Dirichlet boundary condition. It is also established that these steady states, in appropriate functional spaces, have local Hölder continuous dependence on the forcing function and the boundary data. This is a joint work with Thinh Kieu (University of North Georgia, Gainesville Campus.)
We study the anisotropic Forchheimer-typed flows for compressible fluids in porous media. Unlike the isotropic flows, the important monotonicity properties are not automatically satisfied in this case. Therefore, various sufficient conditions for them are derived and applied to the experimental data. Next, we prove the existence and uniqueness of the steady state flows subject to a nonhomogeneous Dirichlet boundary condition. It is also established that these steady states, in appropriate functional spaces, have local Hölder continuous dependence on the forcing function and the boundary data. This is a joint work with Thinh Kieu (University of North Georgia, Gainesville Campus.)
This talk is about a modified quasi-reversibility method for computing the exponentially unstable solution of a terminal-boundary value parabolic problem with noisy data. As a PDE-based approach, this variant relies on adding a suitable perturbing operator to the original PDE and consequently, on gaining the corresponding fine stabilized operator. The designated approximate problem is a forward-like one. This new approximation is analyzed in a variational framework, where the finite element method can be applied. With respect to each noise level, the Faedo-Galerkin method is benefited to study the weak solvability of the approximate problem. Relying on the energy-like analysis coupled with a suitable Carleman weight, convergence rates in L2–H1 of the proposed method are obtained when the true solution is sufficiently smooth.
Several problems on the logarithmic capacity of configurations consisting of n ≥ 3 disks in R 2 and Newtonian capacity of n balls in R 3 will be discussed. The most general and difficult problem here is to identify configurations which minimize the logarithmic capacity or Newtonian capacity under certain restrictions on configurations of disks or balls. Also we will discuss maximization problems for these capacities. In particular, I will prove that the linear string maximizes the logarithmic capacity among all strings consisting of n disks and that the circular necklace maximizes the logarithmic capacity over the set of all necklaces consisting of n disks, each of radius one. Similar problems for the Newtonian capacity of constellations of balls in R 3 will be also discussed.
The Muskat problem (also known as the Hele-Shaw problem with gravity) models the evolution of the interface between two different fluids in porous media. We introduce a mathematical model for this problem using viscosity solutions theory for integro-differential equations, and discuss a new result on the global well-posedness of the corresponding Hamilton-Jacobi-Bellmann equation with bounded, uniformly continuous initial data, in all dimensions. This is a joint work with Russell Schwab and Olga Turanova.
This Analysis seminar may be attended at 4:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 966 4439 4032
Passcode: 946975
Various changing phenomena around us can be described as dynamical systems. A dynamical system could be deterministic or random, continuous or discrete time. Each of the four combinations are different mathematical objects and have their own theoretical study. They may however be compared based on the collection of sample-paths they generate. This talk will present some old and new connections between these four kinds of dynamics, and the space of sample paths.
This talk focuses on the construction, analysis, and derivation of a new class of exponential methods — specifically, two-derivative exponential Runge--Kutta (TDEXPRK) methods — designed for stiff PDEs. Interestingly, while TDEXPRK methods are constructed based on a fixed linearization of the vector field similar to exponential Runge--Kutta (EXPRK) methods, they require a significantly smaller number of order conditions than EXPRK methods. This enables the derivation of high-order and efficient schemes with only a few stages. Numerical examples for both one- and two-dimensional problems are provided to validate the accuracy and efficiency of TDEXPRK schemes compared to state-of-the-art exponential methods such as EXPRK and exponential Rosenbrock methods. This work is a collaboration with my PhD student, Hoang V. Nguyen.
A well known result by George Pólya asserts that among all the simply connected domains with fixed area A containing the point z0, the disk of radius p A/π centered at z0 maximizes the conformal radius. In this talk, I will define the geometric mean R2(Ω, E) of the conformal radius R(Ω, a) over the compact subsets E ⊂ Ω with positive area and the geometric mean R1(Ω, γ) of the conformal radius R(Ω, a) over the rectifiable curves γ ⊂ Ω with positive length. Two problems will be introduced and the first problem is about obtaining an upper bound for R2(Ω, E) over all Ω of area A and all compact sets E ⊂ Ω of area 0 < A0 < A and the second problem is about obtaining an upper and lower bound for R1(Ω, γ). Similar problems will be considered for triangles and quadrilaterals. The extremal configurations of these cases will also be discussed. In addition to that, the hyperbolic conformal center of a compact set contained in the unit disk will be introduced and some properties of hyperbolic conformal center will be addressed.
A new heuristic mathematical model was proposed for accurate forecasting of prices of stock options. This new technique uses the Black-Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black-Scholes equation was solved in the positive direction of the time variable. This ill-posed initial boundary value problem was solved by the so-called Quasi-Reversibility Method (QRM). In our current work, we used the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate. This talk emphasizes convergence analysis.