Mathematical Sciences
http://hdl.handle.net/11141/40
2019-04-08T17:21:57ZOn the Qualitative Theory of the Nonlinear Parabolic p-Laplacian Type Reaction-Diffusion Equations
http://hdl.handle.net/11141/2758
On the Qualitative Theory of the Nonlinear Parabolic p-Laplacian Type Reaction-Diffusion Equations
Jeli, Roqia Abdullah
This dissertation presents full classification of the evolution of the interfaces and asymptotics of the local solution near the interfaces and at infinity for the nonlinear second
order parabolic p-Laplacian type reaction-diffusion equation of non-Newtonian elastic
filtration
ut −
(
|ux|
p−2
ux
)
x
+buβ = 0, p > 1, β > 0. (1)
Nonlinear partial differential equation (1) is a key model example expressing competition between nonlinear diffusion with gradient dependent diffusivity in either slow
(p > 2) or fast (1 < p < 2) regime and nonlinear state dependent reaction (b > 0) or
absorption (b < 0) forces. If interface is finite, it may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface,
expressed in terms of the parameters p, β,sign b, and asymptotics of the initial function
near its support. In the fast diffusion regime strong domination of the diffusion causes
infinite speed of propagation and interfaces are absent. In all cases with finite interfaces
we prove the explicit formula for the interface and the local solution with accuracy up
to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are generaliii
ization of the methods developed in U.G. Abdulla & J. King, SIAM J. Math. Anal., 32,
2(2000), 235-260; U.G. Abdulla, Nonlinear Analysis, 50, 4(2002), 541-560 and based
on rescaling laws for the nonlinear PDE and blow-up techniques for the identification of
the asymptotics of the solution near the interfaces, construction of barriers using special
comparison theorems in irregular domains with characteristic boundary curves.
Thesis (Ph.D.) - Florida Institute of Technology, 2018
2018-11-01T00:00:00ZFluctuation Analysis in Stochastic System with Parallel Queues
http://hdl.handle.net/11141/2756
Fluctuation Analysis in Stochastic System with Parallel Queues
Merie, Ahmed I.
In this dissertation, We analyze a complex queueing system with a single
server operating in three different modes and, dependent on circumstances,
servicing two different queues simultaneously. There are different
switching policies that specify when the server takes one or two queues.
Main techniques are based on fluctuation analysis. We study an enhanced
hysteretic control system, with primary and secondary queues and random
batch service. When the primary queue down-crosses r the server operates
on two parallel lines servicing them asynchronously until the primary line
of remaining units is processed or the number of serviced secondary units is
at least S which ever comes first. The server waits thereafter is the total
quantity of primary units is less than R (In chapter II we assumed N = 1).
The server capacity of primary units is limited by R with two options:
r ≤ R ≤ N and R > N. Using fluctuation analysis we obtain closed-form
distributions of available units during key periods of time and the steady
state distribution of the primary units. We illustrate analytical tractability by
numerous analytical and computational examples.
Thesis (Ph.D.) - Florida Institute of Technology, 2018
2018-07-01T00:00:00ZSome new nonlinear second-order boundary value problems on an arbitrary domain
http://hdl.handle.net/11141/2739
Some new nonlinear second-order boundary value problems on an arbitrary domain
Alsaedi, Ahmed; Alsulami, Mona; Agarwal, Ravi P.; Ahmad, Bashir
In this paper, we develop the existence theory for nonlinear second-order ordinary differential equations equipped with new kinds of nonlocal non-separated type integral multi-point boundary conditions on an arbitrary domain. Existence results are proved with the aid of fixed point theorems due to Schaefer, Krasnoselskii, and Leray–Schauder, while the uniqueness of solutions for the given problem is established by means of contraction mapping principle. Examples are constructed for the illustration of the obtained results. Ulam-stability is also discussed for the given problem. A variant of the problem involving different boundary data is also discussed. Finally, we introduce an associated boundary value problem involving integro-differential equations and discuss the uniqueness of its solutions.
2018-12-01T00:00:00ZTime Sensitive Functionals of Marked Random Measures in Real Time
http://hdl.handle.net/11141/2693
Time Sensitive Functionals of Marked Random Measures in Real Time
Frisbee, Kizza M. Nandyose
In this dissertation, we study marked random measures that model stochastic
networks (under attacks), status of queueing systems during vacation modes, responses
to cancer treatments (such as chemotherapy and radiation), hostile actions
in economics and warfare. We extend the recently developed time sensitivity technique
for investigating the processes’ behavior about a fixed threshold to a novel
time sensitive technique in three important directions: (1) real-time monotone
stochastic processes; (2) two-dimensional signed random measures; and (3) antagonistic
stochastic games with two active players and one passive player. The need
for the time sensitive feature in our study (i.e., an analytical association with real-time
parameter ) allows stochastic control implementation in sharp contrast with
time insensitive analysis very often occurring in the literature. To reach our objectives,
we proceed with the classical approach of fluctuation analysis of a particle
running through a random grid of a convex set that the particle is trying to escape
using stand-alone techniques of stochastic expansion and Laplace transform. We
investigate the status of the processes upon as well as the statuses at each time in
a given observation time interval. For the monotone process, we target the first passage time, pre-first passage time, the status of the associated continuous time
parameter process between these two epochs, and the status of the process upon
these two epochs. We obtain analytically tractable formulas and demonstrate them
on special cases of marked Poisson processes. Inspired by the monotone result, we
embellish it to a two-dimensional signed random measure with position dependent
marking. The real-valued component of the associated marked point process is
non-monotone presenting an analytical challenge. We manage to investigate various
characteristics of that component, including the nth drop or a sharp surge that
find applications to finance (like option trading) and risk theory. Finally, we apply
the technique to a class of antagonistic stochastic games of three players A,B, and
T, of whom the first two are active and the third is a passive player. The active
players exchange hostile attacks of random magnitudes with each other and also
with player T exerted at random times. At some point (ruin time), one of the two
active players is ruined, when the cumulative damages become unsustainable. We
obtain the a closed form of the joint distribution functional representing the status
of all players upon and also at each time prior to . We illustrate the game on a
number of practical models, including stock option trading and queueing systems
with vacations and (N,T)-policy.
Thesis (Ph.D.) - Florida Institute of Technology, 2018
2018-10-01T00:00:00Z