The chemical oxygen-iodine laser (COIL) has several important military and industrial applications. The concern of this work is do develop a partial differential equation model describing optical behavior in the COIL. Optical behavior of the COIL has traditionally been investigated via a ray tracing method. Photons are represented as discrete particles, and their behavior is described by the geometry of the system. We develop an optical model wherein photons have a wave description. In order to construct the mathematical model, we utilize the theory of paraxial wave optics and Gaussian beams. Doing so allows us to incorporate physical effects such as diffusion/diffraction and refraction into the model. After describing the optical model, we present numerical methods for obtaining approximate solutions to the model in the cases of one and two transverse directions. Results are presented illustrating the efficacy of the numerical methods. / Ph. D.

A mathematical study has been undertaken to model various kinds of granular flows including the perfect plasticity flow and the viscous elasto-plasticity flow. The work is mainly based on the double-shearing theory originated by Spencer and developed by many others. The focus of the project is on the formulation of the theory, the construction of mathematical models and the development of robust simulation techniques.Based on a general formulation of the double-shearing theory, the perfect plasticity flow is shown to be governed by a set of highly nonlinear first order hyperbolic partial differential equations with two distinct characteristics. A sophisticated numerical algorithm is then developed based on the method of characteristics to determine the stress discontinuity and the velocity and stress fields. With the method developed, a numerical study is then undertaken to model the flow of granular materials in a hopper in the presence of stress discontinuity and to investigate the influence of various parameters on the distribution of hopper wall pressures.Utilising the double shearing theory, a set of stress-strain constitutive equations in explicit form has been derived, which makes it possible to formulate the double-shearing theory within the framework of the finite element method. Thus, consequently, a sophisticated finite element technique has been developed to solve the general boundary value problem governing the viscous elasto-plasticity flows obeying the double-shearing theory. Numerical implementation of the frictional boundary condition is also presented. The model is then illustrated with a numerical example demonstrating the influence of wall friction on the distribution of pressures on silo walls throughout the dynamic process of material discharge from silos.

Nine populations of sixty-eight individuals representing the distribution of Petrophytum caespitosum (Nutt.) Rydb. (Rosaceae) in the Intermountain Region were investigated for taxonomically significant variation using numerical techniques. On the basis of 19 morphological characters 3 subspecific taxa are recognized. Two taxa are existing subspecies: P. caespitosum (Nutt.) Rydb. var. caespitosum and P. caespitosum (Nutt.) Rydb. var. elatius (S. Wats.) Tides. One new subspecies is described: P. caespitosum (Nutt.) Rydb. var. latifolium. The ecology of the species is discussed. The floral and vegetative anatomy are described. Chromosome counts of 3 populations have given n = 9. A list of herbarium material is given. The numerical analyses were run on an IBM S/360 computer using standardized data to find the coefficients of correlation between individuals of the same population and the coefficients of correlation between pairs of populations. The results of the numerical analyses agree with more traditional methods.

Boundary value problems required for modelling plane wave propagation in electrically anisotropic and inhomogeneous media relevant to the surface impedance methods in electromagnetic geophysics are formally posed and treated. For a homogeneous TM-type wave propagating in a half space with both vertical and horizontal inhomogeneities where the TM-type wave is aligned with one of the elements of the conductivity tensor, it is shown using exact solutions that the shearing term in the homogeneous Helmholtz equation for inclined anisotropic media: [Equation 1], unequivocally vanishes and solutions need only be sought to the homogeneous Helmholtz equation for biaxial media: [Equation 2]. This implies that those problems posed with an inclined uniaxial conductivity tensor can be identically stated with a fundamental biaxial conductivity tensor, provided that the conductivity values are the reciprocal of the diagonal terms from the Euler rotated resistivity tensor: [Equation 3], [Equation 4], [Equation 5]. The applications of this consequence for numerical methods of solving arbitrary two-dimensional problems for a homogeneous TM-type wave is that they need only to approximate the homogeneous Helmholtz equation and neglect the corresponding shearing term. The self-consistent impedance method, a two-dimensional finite-difference approximation based on a network analogy, is demonstrated to accurately solve for problems with inclined uniaxial anisotropy using the fundamental biaxial anisotropy equivalence. The problem of a homogeneous plane wave at skew incidence upon an inclined anisotropic half space is then formally treated. In the half space, both TM- and TE-type waves are coupled and the linearly polarised incident TM- and TE-type waves reflect TE- and TM-type components. Equations for all elements of the impedance tensor are derived for both TM- and TE-type incidence. This offers potential as a method of predicting the direction of anisotropic strike from tensor impedance measurements in sedimentary environments.

<p> Contemporary approaches to data analysis suffer from being both time-consuming and subjective; however, the application of numerical techniques to the automated (non-interactive) analysis of gamma ray spectra often leads to considerably improved performance. The foundations and limitations of such techniques lie in the applicability of certain mathematical operations such as deconvolution, and the careful study of stochastic models. The use of digital filters as a method of enhancing detector response has been applied to a triple-coincidence counting arrangement, after modelling undesired physical effects. An objective background estimation method has been described based on the statistical nature of nuclear measurements. Finally, the application of such techniques is demonstrated with a package of FORTRAN programs designed to be used in a variety of situations with minimal modifications. </p> / Thesis / Master of Science (MSc)

This dissertation considers an American put option written on a single underlying which does not pay dividends, for which no closed form solution exists. As a conse- quence, numerical techniques have been developed to estimate the value of the Amer- ican put option. These include analytical approximations, tree or lattice methods, ¯nite di®erence methods, Monte Carlo simulation and integral representations. We ¯rst present the mathematical descriptions underlying these numerical techniques. We then provide an examination of a selection of algorithms from each technique, including implementation details, possible enhancements and a description of the convergence behaviour. Finally, we compare the estimates and the execution times of each of the algorithms considered.

The principle of coordinate invariance states that all physical laws must be formulated in a mathematical form that is independent of the geometrical properties of any particular coordinate system. Embracing this principle is the key to understand how to systematically incorporate curved material interfaces into a numerical solution of Maxwell’s equations. This dissertation describes how to generate a coordinate invariant representation of Maxwell’s equations in differential form, and it demonstrates why employing such representation is crucial to the development of robust finite-difference discretisations with consistent global error properties. As part of this process, two original contributions are presented that address the issue of constructing finite-difference approximations at the locations of material interfaces. The first contribution is a domain-decomposition procedure to enforce the tangential field continuity conditions with a second-order local truncation error that can be applied in 2-D or 3-D. The second contribution is a similar domain-decomposition procedure that enforces the tangential field continuity conditions with a local truncation of order 2L—where L is an integer greater or equal to one—but that can only be applied in 1-D. To conclude, the dissertation also describes the interesting connection that exists between the use of a coordinate invariant representation of Maxwell’s equations to design artificial materials and the use of the same representation to model curved material interfaces in a finite-difference discretisation.

The principle of coordinate invariance states that all physical laws must be formulated in a mathematical form that is independent of the geometrical properties of any particular coordinate system. Embracing this principle is the key to understand how to systematically incorporate curved material interfaces into a numerical solution of Maxwell’s equations. This dissertation describes how to generate a coordinate invariant representation of Maxwell’s equations in differential form, and it demonstrates why employing such representation is crucial to the development of robust finite-difference discretisations with consistent global error properties. As part of this process, two original contributions are presented that address the issue of constructing finite-difference approximations at the locations of material interfaces. The first contribution is a domain-decomposition procedure to enforce the tangential field continuity conditions with a second-order local truncation error that can be applied in 2-D or 3-D. The second contribution is a similar domain-decomposition procedure that enforces the tangential field continuity conditions with a local truncation of order 2L—where L is an integer greater or equal to one—but that can only be applied in 1-D. To conclude, the dissertation also describes the interesting connection that exists between the use of a coordinate invariant representation of Maxwell’s equations to design artificial materials and the use of the same representation to model curved material interfaces in a finite-difference discretisation.

PhD (Applied Mathematics) / Department of Mathematics and Applied Mathematics / The work presented in this thesis is the application of the recently introduced numerical techniques, namely the spectral quasi-linearization method (SQLM) and the bivariate spectral quasi-linearization method (BSQLM), in solving problems arising in fluid flow. Firstly, we use the SQLM to solve the highly non-linear one dimensional Bratu problem. The results obtained are compared with exact solution and previously published results using the B-spline method, Picard’s Green’s Embedded Method and the iterative finite difference method. The results obtained show that the SQLM is highly accurate and computationally efficient. Secondly, we use the bivariate spectral quasi-linearization method to solve the two dimensional Bratu problem. Since the exact solution of the two-dimensional Bratu problem is unknown, the results obtained are compared with those previously published results using the finite difference method and the weighted residual method. Thirdly, we use the BSQLM to study numerically the boundary layer flow of a third grade non-Newtonian fluid past a vertical porous plate. We use the Jeffrey fluid as a typical fluid which shows non-Newtonian characteristics. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations which are then solved using BSQLM. The influence of some thermo-physical parameters namely, the ratio relaxation to retardation times parameter, Prandtl number, Schmidt number and the Deborah number is investigated. Also investigated is the influence of the ratio of relaxation to retardation times, Schmidt number and the Prandtl number on the skin friction, heat transfer rate and the mass transfer rate. The results obtained show that increasing the Schmidt number decelerates the fluid flow, reduces the skin friction, heat and mass transfer rates and strongly depresses the fluid concentration whilst the temperature is increased. The fluid velocity, the skin friction, heat and mass transfer rates are increased with increasing values of the relaxation to retardation parameter whilst the fluid temperature and concentration are reduced. Using the the solution based errors, it was shown that the BSQLM converges to the solution only after 5 iterations. The residual error infinity norms showed that BSQLM is very accurate by giving an error of order of 10−4 within 5 iterations. Lastly we propose a model of the non-Newtonian fluid flow past a vertical porous plate in the presence of thermal radiation and chemical reaction. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations. The BSQLM is used to solve the system of equations. We investigate the influence of the ratio of relaxation to retardation parameter, Schmidt number, Prandtl number, thermal radiation parameter, chemical reaction iv parameter, Nusselt number, Sherwood number, local skin fiction coefficient on the fluid concentration, fluid temperature as well as the fluid velocity. From the study, it is noted that the fluid flow velocity, the local skin friction coefficient, heat and mass transfer rate are increased with increasing ratio of relaxation to retardation times parameter whilst the fluid concentration is depressed. Increasing the Prandtl number causes a reduction in the velocity and temperature of the fluid whilst the concentration is increased. Also, the local skin friction coefficient and the mass transfer rates are depressed with an increase in the Prandtl number. An increase in the chemical reaction parameter decreases the fluid velocity, temperature and the concentration. Increasing the thermal radiation parameter has an effect of decelerating the fluid flow whilst the temperature and the concentration are slightly enhanced. The infinity norms were used to show that the method converges fast. The method converges to the solution within 5 iterations. The accuracy of the solution is checked using residual errors of the functions f, and . The errors show that the BSQLM is accurate, giving errors of less than 10−4, 10−7 and 10−8 for f, and , respectively, within 5 iterations. / NRF

Hierarchical Markovian Models (HMMs) are composed of multiple low level models (LLMs) and high level model (HLM) that defines the interaction among LLMs. The essence of the HMM approach is to model the system at hand in the form of interacting components so that its (larger) underlying continous-time Markov chain (CTMC) is not generated but implicitly represented as a sum of Kronecker products of (smaller) component matrices. The Kronecker structure of an HMM induces nested block partitionings in its underlying CTMC. These partitionings may be used in block versions of classical iterative methods based on splittings, such as block SOR (BSOR), to solve the underlying CTMC for its stationary vector. Therein the problem becomes that of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of a particular partitioning. This paper shows that in each HLM state there may be diagonal blocks with identical off-diagonal parts and diagonals differing from each other by a multiple of the identity matrix. Such diagonal blocks are named candidate blocks. The paper explains how candidate blocks can be detected and how the can mutually benefit from a single real Schur factorization. It gives sufficient conditions for the existence of diagonal blocks with real eigenvalues and shows how these conditions can be checked using component matrices. It describes how the sparse real Schur factors of candidate blocks satisfying these conditions can be constructed from component matrices and their real Schur factors. It also demonstrates how fill in- of LU factorized (non-candidate) diagonal blocks can be reduced by using the column approximate minimum degree algorithm (COLAMD). Then it presents a three-level BSOR solver in which the diagonal blocks at the first level are solved using block Gauss-Seidel (BGS) at the second and the methods of real Schur and LU factorizations at the third level. Finally, on a set of numerical experiments it shows how these ideas can be used to reduce the storage required by the factors of the diagonal blocks at the third level and to improve the solution time compared to an all LU factorization implementation of the three-level BSOR solver.