published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

published_or_final_version / Civil Engineering / Master / Master of Philosophy

published_or_final_version / Industrial and Manufacturing Systems Engineering / Master / Master of Philosophy

A new mathematical model for the fundamental mode of a propagating Gaussian beam is presented. The model is two-fold, consisting of a mathematical expression and a corresponding geometrical representation which interprets the expression in the light of geometrical optics. The mathematical description arises from the (0,0) order of a new family of exact, closed-form solutions to the scalar Helmholtz equation. The family consists of nonseparable functions in the oblate spheroidal coordinate system and can easily be transformed to a different set of solutions in the prolate spheroidal coordinate system, where the (0,0) order is a spherical wave. This transformation consists of two substitutions in the coordinate system parameters and represents a more general method of obtaining a Gaussian beam from a spherical wave than assuming a complex point source on axis. Further, each higher-order member of the family of solutions possesses an amplitude consisting of a finite number of higher-order terms with a zero-order term that is Gaussian. The geometrical interpretation employs the skew-line generator of a hyperboloid of one sheet as a ray-like element on a contour of constant amplitude in the Gaussian beam. The geometrical characteristics of the skew line and the consequences of treating it as a ray are explored in depth. The skew line is ultimately used to build a nonorthogonal coordinate system which allows straight-line propagation of a Gaussian beam in three-dimensional space. Highlights of the research into other methods used to model a propagating Gaussian beam--such as complex rays, complex point sources and complex argument functions--are reviewed and compared with this work.

A model of premixed lean Hydrogen-Oxygen flame is studied by singular perturbation techniques based on high activation energy. The model is built from four reaction steps consisting of two chain branching steps, a chain propagating step, and a recombination step. The analysis, in this case, gives rise to a layer phenomenon different from what is currently seen in combustion literature. First, there is a basic layer similar to those obtained for the one step reaction model. Then embedded in the first layer is a thinner layer giving rise to an interesting system of nonlinear boundary value problems. This system of nonlinear problems does not meet standard existence criterium and also involves an unknown parameter. Hence existence results are called for. Existence is proved for both the boundary value problem and the unknown parameter, and numerical solutions are obtained in support of the existence results. A numerical estimate of the unknown parameter is obtained. A generalization of the model for different reaction parameter ranges is made. Two new thin layers emerge. The structure of one of the new thin layers turns out to be exactly the same as that just described, hence the existence results do carry over. The boundary value problem resulting from the second of the new thin layers turned out to be quite simple and a solution could be written down explicitly.

The topic of this dissertation, modeling and identification of nonlinear oscillation, represents an area of mathematical systems theory that has received little attention in the past. Primarily, the types of oscillation of interest are those found in biological systems where theoretical foundations for mathematical models are insufficient. These oscillations are also observed in other systems including electrical, mechanical, and chemical. The contributions of this dissertation are a generalized class of autonomous differential equations that are found to exhibit stable limit cycles, and an investigation of a method of system identification that can be used to estimate the model parameters. Here the observed signal is modeled as the response of a nonlinear system that can be described by differential equations. Modeling the signal in this way shifts the emphasis from signal characteristics, such as spectral content, to system characteristics, such as parameter values and system structure. This shift in emphasis may provide a better method for monitoring complex systems that exhibit periodic behavior such as patients under anesthesia. A class of autonomous differential equations, called the generalized oscillator models, are presented as one nᵗʰ-order differential equations with nonlinear coefficients. The coefficients are chosen to change sign depending on the magnitude of the phase variables. The coefficients are negative near the origin and positive away from the origin. Motivated by the generalized Routh-Hurwitz criterion, this coefficient sign changing produces the desired oscillation. Properties of the generalized oscillator model are investigated using the describing function method of analysis and numerical simulation. Several descriptive examples are presented. Based on the generalized oscillator model as a set of candidate models, the system identification problem is formed as a mathematical programming problem. The method of quasilinearization is investigated as method of solving the identification problem. Two examples are presented that demonstrate the method. It is shown that in general, the method of quasilinearization as a solution to the system identification problem will not converge regardless of the initial starting point. This result indicates that although the quasilinearization method is useful for solving two-point boundary value problems, it is not useful (in its present form) for solving the system identification problem.

The problem of utilizing blending techniques to control coal quality at the production-consumption phase is considered. Three blending models were developed to provide coal of high thermal content and low pollutants. With the aid of operational mine planning, coal is blended at the coal producing mines such that the best quality of coal is mined during a planning period, while meeting the management production objectives. The first model developed uses 0-1 programming formulation to select potential working areas of a mine on the basis of predicted grade values obtainable from geostatistics. A second model developed combines economically coals produced by different suppliers to meet the specification of a power plant. The second model uses a linear programming formulation to develop coal purchasing strategy. Finally, a multiobjective programming technique is used to determine the tonnages of coal which must be cleaned from various sources (e.g. stockpiles) in order to result in clean coal of high thermal content and low sulfur content. The two objectives used are minimization of total sulfur and maximization of total Btu. Both the operational mine planning and coal purchasing models were tested on actual mine data. The study demonstrated the capability of controlling coal quality by blending technique with the aid of the three models. This can be translated into dollar savings to both the coal producer and the coal consumer.

We address the problem of the consistency between discrete and continuous models for density-dependent size-structured populations. Some earlier works have discussed the consistency of density independent age and size-structured models. Although the issue of consistency between these models has raised interest in recent years, it has not been discussed in depth, perhaps because of the non-linear nature of the equations involved. We construct a numerical scheme of the continuous model and show that the transition matrix of this scheme has the form of the standard discrete model. The construction is based on the theory of Upwind Numerical Schemes for non-Linear Hyperbolic Conservation Laws with one important difference, that we do have a non-linear source at the boundary; interestingly, this case has not been explored in depth from the purely mathematical point of view. We prove the consistency, non-linear stability and hence convergence of the numerical scheme which guarantee that both models yield results that are completely consistent with each other. Several examples are worked out: a simple linear age-structured problem, a density-independent size-structured problem and a non-linear size-structured problem. These examples confirm the convergence just proven theoretically. An ample revision of relevant biological and computational literature is also presented and used to establish realistic restrictions on the objects under consideration and to prepare significant examples to illustrate our points.

In this research, we are concerned with obtaining characteristics of a scattering object from transient input-output data. The input is a transient pulse with broad bandwidth. The output is the field scattered by the object. Specifically, we consider two classical structures: First a single lossless dielectric slab backed by a perfect conductor; second, a double-layer lossless dielectric slab backed by a perfect conductor. We begin with two generic solution forms: First, the ray-optic form, which emphasizes local object features; second, the singularity expansion method (SEM) form, which emphasizes object resonances. Using these two forms, we generate a variety of solutions for each structure. For the single slab, we obtain five solution forms for the transient response as follows: The ray-optic, the SEM, two hybrids, and one closed. We find that the input signal plays an important role in the results. We believe the specific hybrid solutions for the slab are new. For the double slab, we find four solution forms as follows: Two ray-optic and two quasi-hybrid. The quasi-hybrid solutions involve a ray-optic expansion in one slab and SEM in the other. We believe the quasi-hybrid forms are new and lead to some interesting comparisons with work by other researchers. As a result of critical study of the various solution types, we reach some conclusions concerning determination of parameters that classify an object (the parametric inverse problem). We find that a given SEM pole set does not always correspond to a unique object. In addition, we show that it is often not possible to relate SEM poles to object size or constitution. Because of these facts, it is necessary to add knowledge of the specific form factor of the object to permit classification. We include some conclusions concerning object identification and point out some areas for future research.

We analyze a dissipative map of the plane. The map was initially defined by Ikeda as a model for bistable behavior in an optical ring cavity. Our analysis is based upon an examination of attracting sets and basins of attraction. The primary tools utilized in the analysis are stable and unstable manifolds of fixed and periodic saddle points. These manifolds determine boundaries of basins of attraction, and the extent and evolution of attracting sets. We perform extensive numerical iterations of the map with a central focus on sudden changes in the topological nature of attractors and basins. Our analysis concentrates on the destruction of the lower branch attractor as a prominent example of attractor/basin interaction. This involves an examination of a possible link between two fixed points L and M, namely the heteroclinic connection Wᵘ(L) ∩ Wˢ(M) ≠ 0. We use two different methods to approach this question. Although the Ikeda map is used as the working model throughout, both of the techniques apply to a more general class of dissipative maps satisfying certain hypotheses. The first of these techniques analyzes Wˢ(M) when Wᵘ(M) ∩ Wˢ(M) ≠ 0, with the result that Wˢ(M) is found to invade some minimum limiting region for Wᵘ(M) ∩ Wˢ(M) ≠ 0 arbitrarily close to tangency. The second approach is more topological in nature. We define a mesh of subregions to bridge the spatial gap between the points L and M, and concentrate on the occurrence of Wᵘ(L) ∩ Wˢ(M) ≠ 0 (destruction of the attractor). The first main result is a necessary condition for the heteroclinic connection in terms of the behavior of the map on these subregions. The second result is a sequence of sufficient conditions for this link. There remains a gap between these two conditions, and in the final sections we present numerical investigations indicating that the concept of intersection links between subregions is useful to resolve cases near the boundary of the destruction region.