Ph.D. Theses (Mathematics)http://hdl.handle.net/1957/157392015-07-05T04:59:09Z2015-07-05T04:59:09ZThe Variable Speed Wave Equation and Perfectly Matched LayersKim, Dojinhttp://hdl.handle.net/1957/563082015-07-02T15:54:59Z2015-06-04T00:00:00ZThe Variable Speed Wave Equation and Perfectly Matched Layers
Kim, Dojin
A perfectly matched layer (PML) is widely used to model many different types of wave propagation in different media. It has been found that a PML is often very effective and also easy to set, but still many questions remain.
We introduce a new formulation from regularizing the classical Un-Split PML of the acoustic wave equation and show the well-posedness and numerical efficiency. A PML is designed to absorb incident waves traveling perpendicular to the PML, but there is no effective absorption of waves traveling with large incident angles. We suggest one method to deal with this problem and show well-posedness of the system, and some numerical experiments. For the 1-d wave equation with a constant speed equipped a PML, stability and the exponential decay rate of energy has been proved, but the question for variable sound speed equation remained open. We show that the energy decays exponentially in the 1-d PML wave equation with variable sound speed.
Most PML wave equations appear as a first-order hyperbolic system with as a zero-order perturbation. We introduce a general formulation and show well-posedness and stability of the system. Furthermore we develop a discontinuous Galerkin method and analyze both the semi-discrete and fully discretized system and provide a priori error estimations.
Graduation date: 2016
2015-06-04T00:00:00Z3D cone beam reconstruction formulas for the transverse-ray transform with source points on a curveWongsason, Patchareehttp://hdl.handle.net/1957/517542014-08-27T20:49:23Z2014-07-23T00:00:00Z3D cone beam reconstruction formulas for the transverse-ray transform with source points on a curve
Wongsason, Patcharee
3D vector tomography has been explored and results have been achieved in the last few decades. Among these was a reconstruction formula for the solenoidal part of a vector field from its Doppler transform with sources on a curve. The Doppler transform of a vector field is the line integral of the component parallel to the line. In this work, we shall study the transverse ray transform of a vector field, which instead integrates over lines the component of the vector field perpendicular to the line. We provide a reconstruction procedure for the transverse ray transform of a vector field with sources on a curve fulfilling Tuy’s condition of order 3. We shall recover both the potential and solenoidal parts. We present two steps for the reconstruction. The first one is to reconstruct the solenoidal part and the techniques we use are inspired by work of Katsevich and Schuster. A procedure for recovering the potential part will be the second step. The main ingredient is the difference between the measured data and the reprojection of the solenoidal part. We also provide a variation of the Radon inversion formula for the vector part of a quaternionic-valued function (or vector field) and an inversion formula in cone-beam setting with sources on the sphere.
Graduation date: 2015
2014-07-23T00:00:00ZMathematical treatment and simulation of methane hydrates and adsorption modelsMedina, Francis Patriciahttp://hdl.handle.net/1957/504242014-07-15T18:00:13Z2014-05-13T00:00:00ZMathematical treatment and simulation of methane hydrates and adsorption models
Medina, Francis Patricia
In this dissertation we develop mathematical treatment for two important applications: (i) evolution of methane in coalbeds with the associated phenomena of adsorption, and (ii) formation of methane hydrates in seabed. We use simplified models for (i) and (ii) since we are more interested in qualitative properties of the solutions rather than direct applications to engineering.
For methane hydrates we focus on a scalar problem with diffusion only, and we discuss it as a nonlinear parabolic problem in a single variable with monotone operators. We show how the problem can be cast in the framework of a free boundary problem. The particular nonlinearity that we deal with comes from a constraint on one of the variables. For the simplified model of methane hydrates, we establish well-posedness of the problem in an abstract weak setting. We also perform simulations with a novel approach based on semismooth Newton methods. We demonstrate convergence rates of the numerical approximation which are similar to those for Stefan free boundary value problem.
On the other hand, for adsorption problems, we focus on their structure as systems of conservation laws, with equilibrium and non-equilibrium type nonlinearities, where the latter are associated with microscale diffusion. We also work with an unusual type isotherm called Ideal Adsorbate Solution, which is defined implicitly. For the IAS adsorption system, we show sufficient conditions that render the system hyperbolic. We also construct numerical approximations for equilibrium and nonequilibrium models.
Graduation date: 2015
2014-05-13T00:00:00ZCanonical states in quantum statistical mechanicsKvarda, Robert Edwardhttp://hdl.handle.net/1957/475232014-04-21T18:12:14Z1965-08-26T00:00:00ZCanonical states in quantum statistical mechanics
Kvarda, Robert Edward
This report presents a characterization of the quantum mechanical
analog of the Gibbs canonical density. The approach is
based on a method developed by D.S. Carter for the case of classical
statistical mechanics, which considers composite mechanical systems
composed of mechanically and statistically independent components.
After a brief introductory chapter, Chapter II outlines how
the case of classical mechanics may be described in terms of the
usual measure theoretic treatment of probability. The necessary
statistical background of quantum mechanics is then discussed in
Chapter III, relying on the classic treatment of J. von Neumann and
the more recent work of G. W. Mackey. The basic idea of probability
measure in quantum mechanics differs from that in classical measure
theory, for the measure is defined on a non-Boolean lattice consisting
of all closed linear subspaces of a Hilbert space. Because of this
difference, the classical theory of product measures does not apply. Chapter IV presents a detailed treatment of probability measures for
composite quantum systems.
The analog of the Gibbs canonical density is characterized in
Chapter V, by considering a large collection Q of noninteracting
quantum systems, each of which is in an equilibrium statistical state.
The set Q, the Hamiltonian operator for each system, and the
equilibrium states are assumed to have certain properties which are
given as axioms.
The axioms require each Hamiltonian operator to have a pure
point spectrum. It is assumed, without loss of generality, that the
lowest characteristic value of each Hamiltonian is zero. The set Q
is assumed to be closed under the formation of pairwise mechanically
independent composite systems. This implies that the set D of all
Hamiltonian spectra is closed under addition. It is further assumed
that D is closed under positive differences. The final requirement
on the set Q is that it contain certain "harmonic oscillators".
More precisely, for each positive λεD, Q must contain a system
whose Hamiltonian has the spectrum {nλ : n=0,1,2,[superscript ...]}. The usual
assumption is made that each density operator is a function of the
system Hamiltonian. Finally, it is assumed that for each composite
system in Q, with two mechanically independent components, the
component systems are statistically independent.
It is shown that these assumptions imply that each member of Q is in a canonical state at a temperature which is the same for all
systems. The possibility of zero absolute temperature is included.
Graduation date: 1966
1965-08-26T00:00:00Z