Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 318-330). Available also in a digital version from Dissertation Abstracts.

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University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

This thesis explores higher order numerical methods for solving fractional differential equations.

No description available.

Zlomkový kalkulus je matematická disciplína zabývající se vlastnostmi derivací a integrálů neceločíselných řádů (nazývaných zlomkové derivace a integrály, zkráceně diferintegrály) a metodami řešení diferenciálních rovnic obsahujících zlomkové derivace neznámé funkce (tzv. zlomkovými diferenciálními rovnicemi). V této práci představujeme standardní přístupy k definicím zlomkového kalkulu a důkazy některých základních vlastností diferintegrálů. Dále uvádíme krátký přehled metod řešení některých lineárních zlomkových diferenciálních rovnic a vymezujeme hranice jejich použitelnosti. Na závěr si všímáme některých fyzikálních aplikací zlomkového kalkulu.

The goal of this thesis is to present a comprehensive study of the parabolic and hyperbolic Anderson models with constant initial condition, driven by a Gaussian noise which is fractional in space with index H > 1/2 or H < 1/2, and is either white in time, or fractional in time with index H_0 > 1/2. As a preliminary step, we study the linear stochastic heat and wave equations with the same type of noise. In the case H_0 > 1/2 and H < 1/2, we present a new result, regarding the solution of the parabolic Anderson model with general initial condition given by a measure.

Vehicle platooning is an important innovation in the automotive industry that aims at improving safety, mileage, effciency, and the time needed to travel. This research focuses on the various aspects of vehicle platooning, one of the important aspects being analysis of different control strategies that lead to a stable and robust platoon. Safety of passengers being a very important consideration, the control design should be such that the controller remains robust under uncertain environments. As a part of the Department of Energy (DOE) project, this research also tries to show a demonstration of vehicle platooning using robots. In an automated highway scenario, a vehicle platoon can be thought of as a string of vehicles, following one another as a platoon. Being equipped by wireless communication capabilities, these vehicles communicate with one another to maintain their formation as a platoon, hence are "cognitive." Autonomous capable vehicles in tightly spaced, computer-controlled platoons will lead to savings in energy due to reduced aerodynamic forces, as well as increased passenger comfort since there will be no sudden accelerations or decelerations. Impacts in the occurrence of collisions, if any, will be very low. The greatest benefit obtained is, however, an increase in highway capacity, along with reduction in traffic congestion, pollution, and energy consumption. Another aspect of this project is the automated electric transportation (AET). This aims at providing energy directly to vehicles from electric highways, thus reducing their energy consumption and CO2 emission. By eliminating the use of overhead wires, infrastructure can be upgraded by electrifying highways and providing energy on demand and in real time to moving vehicles via a wireless energy transfer phenomenon known as "wireless inductive coupling." The work done in this research will help to gain an insight into vehicle platooning and the control system related to maintaining the vehicles in this formation.

We consider a stochastic functional differential equation with infinite memory driven by a fractional Brownian motion with Hurst parameter $H>1/2$. We prove an existence and uniqueness result of the solution to the stochastic differential equation. We investigate the dependence of the solution on the initial condition and the existence of finite moments of the solution. Furthermore we generalize these results to wider classes of stochastic differential equations. The stochastic integral with respect to fractional Brownian motion is defined as a pathwise Riemann-Stieltjes integral.