Browsing by Author "Gao, Guangyue"
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- Some Controllability and Stabilization Problems of Surface Waves on Water with Surface tensionGao, Guangyue (Virginia Tech, 2015-12-23)The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation wt + wxxx - wxxxxx - n∑j=1 ajwjwx = 0, w(x, 0) = w0(x) posed on a periodic domain x ∈ [0, 2π] with boundary conditions wix(0, t) = wix(2π, t), i = 0, 2, 3, 4 and an L2-stabilizing feedback control law wx(2π, t) = αwx(0, t) + (1 - α)wxxx(0; t) where n is a fixed positive integer, aj, j = 1, 2, ... n, α are real constants, and |α| < 1. It is shown that for w0(x) ∈ H1α(0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T]; H1α(0, 2π)) with a conserved volume of w, [w] = ∫2π0 w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w0(x)]/(2π) as t → + ∞. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case.
- A Stochastic Model for The Transmission Dynamics of Toxoplasma GondiiGao, Guangyue (Virginia Tech, 2016-05-05)Toxoplasma gondii (T. gondii) is an intracellular protozoan parasite. The parasite can infect all warm-blooded vertebrates. Up to 30% of the world's human population carry a Toxoplasma infection. However, the transmission dynamics of T. gondii has not been well understood, although a lot of mathematical models have been built. In this thesis, we adopt a complex life cycle model developed by Turner et al. and extend their work to include diffusion of hosts. Most of researches focus on the deterministic models. However, some scientists have reported that deterministic models sometimes are inaccurate or even inapplicable to describe reaction-diffusion systems, such as gene expression. In this case stochastic models might have qualitatively different properties than its deterministic limit. Consequently, the transmission pathways of T. gondii and potential control mechanisms are investigated by both deterministic and stochastic model by us. A stochastic algorithm due to Gillespie, based on the chemical master equation, is introduced. A compartment-based model and a Smoluchowski equation model are described to simulate the diffusion of hosts. The parameter analyses are conducted based on the reproduction number. The analyses based on the deterministic model are verified by stochastic simulation near the thresholds of the parameters.