There are many different methods available for the solution of the heat equation and the choice of which to use is dependent upon the nature of the problem and the specific regions of the domain where the temperature is required. In the case of melting or freezing problems it is usual for the position of the boundary, at which change of physical state (phase change) occurs, to be of greater interest than the temperature at particular points. Again there are several solution methods enabling the tracking of the moving interface between the physical states of the material. For this work we begin with the isotherm migration method, which first appeared in the 1970s but is less frequently cited now. We first solve problems in one dimension with no phase change using the isotherm migration method, which is in itself new work, since all references we have found allude to it as a tool for the solution of phase change problems. We test the method using a variety of examples to explore the difficulties and challenges it produces, and we find it to be robust and tolerant of errors. We then combine it with the Laplace transform method, a well-established technique for solving ordinary and partial differential equations, in which the number of independent variables is reduced by one. The solution is then transformed back into the time domain using a suitable numerical process. The Laplace transform isotherm migration method is a new process, not mentioned previously to our knowledge, and it produces results which are comparable with the isotherm migration method. The new process is applied to one-dimensional phase change problems,where we find that due to the mathematics at the phase change boundary, we are required to make a modification to the usual manner of operating the Laplace transform. This is novel as far as we are aware. Our method is applied to a variety of problems and produces satisfactory results. We then move on to a two-dimensional setting where we find the situation to be much more complex and challenging, as it requires interpolation and curve-fitting processes. Finally we examine the possiblity of speeding up the calculation time using the Laplace transform isotherm migration method by setting problems in a parallel environment and using an MPI platform. This has not been previously attempted and we are able to show a measure of success in our objective.

Time-domain NMR, in one and higher dimensionalities, makes routine use of inversion algorithms to generate results called \T2-distributions' or joint distributions in two (or higher) dimensions of other NMR parameters, T1, diffusivity D, pore size a, etc. These are frequently referred to as \Inverse Laplace Transforms' although the standard inversion of the Laplace Transform long-established in many textbooks of mathematical physics does not perform (and cannot perform) the calculation of such distributions. The operations performed in the estimation of a \T2-distribution' are the estimation of solutions to a Fredholm Integral Equation (of the First Kind), a different and more general object whose discretization results in a standard problem in linear algebra, albeit suffering from well-known problems of ill-conditioning and computational limits for large problem sizes. The Fredholm Integral Equation is not restricted to exponential kernels; the same solution algorithms can be used with kernels of completely different form. On the other hand, (true) Inverse Laplace Transforms, treated analytically, can be of real utility in solving the diffusion problems highly relevant in the subject of NMR in porous media.

Please read the abstract in the 00front section of this document. / Thesis (PhD)--University of Pretoria, 2011. / Mathematics and Applied Mathematics / unrestricted

Knowledge of the indoor propagation of sound has many important applications including acoustic prediction in homes, office buildings, stores, and schools, and the design of concert halls, auditoriums, classrooms, and factories. At low frequencies, interior acoustics are analyzed with the wave equation, but significant computational expense imposes an upper frequency limit. Thus, energy methods are often sought for high frequency analysis. However, conventional energy methods are significantly limited by vast simplifications or computational costs. Therefore, new improvements are still being sought. The basis of this dissertation is a recently developed mathematical model for interior acoustics known as the acoustic diffusion model. The model extends statistical methods in high frequency acoustics to predict the spatial distribution of acoustic energy in the volume over time as a diffusion process. Previously, solutions to the acoustic diffusion model have been limited to one dimensional (1-D) analytical solutions and to the use of the finite element method (FEM). This dissertation focuses on a new, efficient method for solving the acoustic diffusion model based on a boundary element method (BEM) solution using the Laplace transform. First, a Laplace domain solution to the diffusion model is obtained using the BEM. Then, a numerical inverse Laplace transform is used to efficiently compute the time domain response. The diffusion boundary element-Laplace transform solution (BE-LTS) is validated through comparisons with Sabine theory, ray tracing, and a diffusion FEM solution. All methods demonstrate excellent agreement for three increasingly complex acoustic volumes and the computational efficiency of the BE-LTS is exposed. Structural coupling is then incorporated in the diffusion BE-LTS using two methods. First, a simple transmission coefficient separating two acoustic volumes is implemented. Second, a structural power flow model represents the coupling partition separating acoustic volumes. The validation of these methods is successfully performed in an example through comparisons with statistical theory, a diffusion FEM solution, ray tracing, and experimental data. Finally, the diffusion model and the BE-LTS are shown to possess capabilities beyond that of room acoustics. The acoustic transmission through a heat exchanger, acoustic foam, and mufflers is successfully modeled using the diffusion BE-LTS and compared to experimental data. / Ph. D.

Copper (Cu) is an essential trace element in ruminants such as the cow. Its deficiency leads to a number of debilitating symptoms and can eventually cause death. Secondary Cu deficiency is caused by the presence of chemical antagonists such as the thiomolybdates (TMs) (MoOxS4-x2-; x = 0 3). TMs form in the rumen and then form tightly bound insoluble complexes with Cu. These complexes are then excreted and the Cu is unavailable to the animal. The TMs and their effects on ruminant Cu are the focus of this thesis.<p>This study includes three main bodies of work. First, a method for quantifying the composition of TM mixtures in solution was developed. Second, the rate constants of TM interconversions were determined. Third, reactions between Cu and the TMs were studied. This work is part of an ongoing study in the Reid group to model Cu-ligand speciation in the rumen. <p>Thiomolybdates form by successive replacement of molybdate (MoO42-) O ligands with S. This makes synthesis of the uncontaminated compounds difficult. Mixtures of TMs are best analyzed by UV-visible spectroscopy despite the considerable spectral overlap. The combination of these two issues makes the characterization of the spectra difficult. In this study, a method was developed to establish the spectra of the pure TMs despite cross-contamination. Multivariate curve fitting methods were used to establish the composition of synthesized samples simultaneously with the molar absorptivities of the pure TMs. This was done using a Beers Law model. The composition was determined from a fit using assumed molar absorptivities. The absorptivities were then refined using the composition results. These processes were successively reiterated until both were optimized. The optimized absorptivities could then be used to determine the composition of any TM solution mixture. <p>The kinetics of TM formation under biologically relevant conditions were followed by UV-visible spectroscopy. The spectra were then analyzed, using the method developed above, to determine all TM concentrations. Curve fitting methods were used to simultaneously determine the rate constants of all processes that occurred. Integrated rate equations used in the fitting process were developed using the Laplace Transform method. Rate constants were determined at varying pH, ionic strengths and temperatures. From this information it was shown that TM formation and hydrolysis occurs via an associative mechanism. It was also determined that H2S and not HS- was the sulfide nucleophile in TM formation reactions.<p>This study was then extended to include reactions involving Cu and the TMs. First, the UV-visible spectra of the reaction products of Cu with each TM were characterized. This allowed the kinetics of reactions between Cu and the TMs to be followed. This reaction has been found to proceed via an intermediate. Molar absorptivities for this intermediate were optimized along with the appropriate rate constants. This was done using a combination of mathematical simulations and the curve fitting methods used for the TM kinetics. The resulting rate constants can be compared with those obtained previously in the group using a Cu ion selective electrode. <p>These rate constants were used in conjunction with the TM formation rate constants to perform simulations. The results of these simulations provide a picture of what is expected to occur in the rumen. Reactions were also performed in which TM formation occurred in the presence of Cu. This was then extended to mimic bovine feeding habits. Here, portions of the reactions mixture was removed at various time intervals and replaced with fresh reagents. These experiments were used to gain a qualitative picture of Cu speciation in the presence of the TMs over time. The work presented in this thesis provides a crucial step toward understanding the problem of bovine copper deficiency.

Copper (Cu) is an essential trace element in ruminants such as the cow. Its deficiency leads to a number of debilitating symptoms and can eventually cause death. Secondary Cu deficiency is caused by the presence of chemical antagonists such as the thiomolybdates (TMs) (MoOxS4-x2-; x = 0 3). TMs form in the rumen and then form tightly bound insoluble complexes with Cu. These complexes are then excreted and the Cu is unavailable to the animal. The TMs and their effects on ruminant Cu are the focus of this thesis.<p>This study includes three main bodies of work. First, a method for quantifying the composition of TM mixtures in solution was developed. Second, the rate constants of TM interconversions were determined. Third, reactions between Cu and the TMs were studied. This work is part of an ongoing study in the Reid group to model Cu-ligand speciation in the rumen. <p>Thiomolybdates form by successive replacement of molybdate (MoO42-) O ligands with S. This makes synthesis of the uncontaminated compounds difficult. Mixtures of TMs are best analyzed by UV-visible spectroscopy despite the considerable spectral overlap. The combination of these two issues makes the characterization of the spectra difficult. In this study, a method was developed to establish the spectra of the pure TMs despite cross-contamination. Multivariate curve fitting methods were used to establish the composition of synthesized samples simultaneously with the molar absorptivities of the pure TMs. This was done using a Beers Law model. The composition was determined from a fit using assumed molar absorptivities. The absorptivities were then refined using the composition results. These processes were successively reiterated until both were optimized. The optimized absorptivities could then be used to determine the composition of any TM solution mixture. <p>The kinetics of TM formation under biologically relevant conditions were followed by UV-visible spectroscopy. The spectra were then analyzed, using the method developed above, to determine all TM concentrations. Curve fitting methods were used to simultaneously determine the rate constants of all processes that occurred. Integrated rate equations used in the fitting process were developed using the Laplace Transform method. Rate constants were determined at varying pH, ionic strengths and temperatures. From this information it was shown that TM formation and hydrolysis occurs via an associative mechanism. It was also determined that H2S and not HS- was the sulfide nucleophile in TM formation reactions.<p>This study was then extended to include reactions involving Cu and the TMs. First, the UV-visible spectra of the reaction products of Cu with each TM were characterized. This allowed the kinetics of reactions between Cu and the TMs to be followed. This reaction has been found to proceed via an intermediate. Molar absorptivities for this intermediate were optimized along with the appropriate rate constants. This was done using a combination of mathematical simulations and the curve fitting methods used for the TM kinetics. The resulting rate constants can be compared with those obtained previously in the group using a Cu ion selective electrode. <p>These rate constants were used in conjunction with the TM formation rate constants to perform simulations. The results of these simulations provide a picture of what is expected to occur in the rumen. Reactions were also performed in which TM formation occurred in the presence of Cu. This was then extended to mimic bovine feeding habits. Here, portions of the reactions mixture was removed at various time intervals and replaced with fresh reagents. These experiments were used to gain a qualitative picture of Cu speciation in the presence of the TMs over time. The work presented in this thesis provides a crucial step toward understanding the problem of bovine copper deficiency.

Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical finance. For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE techniques. To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the design of parallel programs and the parameter values used. To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference solutions needed to determine the accuracy of the LTFD solutions. Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas.

O presente trabalho teve como objetivo a implementação de técnicas de Ressonância Magnética Nuclear (RMN) para o estudo de meios porosos em geral, com potenciais aplicações no estudo de rochas, ossos, polímeros e materiais cerâmicos porosos. Devido à heterogeneidade e à complexidade desses meios, torna-se importante correlacionar variados parâmetros físicos que contenham informações tanto sobre a dinâmica molecular quanto sobre a estrutura físico-química dos poros e dos fluidos que os permeiam. Utilizando técnicas de RMN, é possível investigar esses parâmetros através da medida dos coeficientes de difusão e dos tempos de relaxação longitudinal e transversal dos fluidos presentes nos poros, que apresentam diferentes estados dinâmicos, variando desde um líquido isotrópico até um líquido com alta viscosidade ou organizado devido à sua interação com a superfície dos poros ou com partículas presentes nos fluidos. Para isso, foram empregadas técnicas de RMN baseadas especialmente em medidas de tempos de relaxação transversal que, quando adequadamente combinadas, permitem a construção de mapas bidimensionais que correlacionam propriedades da dinâmica molecular de fluidos saturantes nessas amostras em instantes diferentes do experimento, provendo informações sobre as dimensões e escalas temporais envolvidas na dinâmica dos fluidos saturantes dos meios porosos. Esses mapas foram obtidos a partir de dados experimentais obtidos com rochas petrolíferas saturadas com água via algoritmos de inversão da transformada de Laplace bidimensional. Este trabalho foi realizado utilizando a infraestrutura existente nos laboratórios de RMN do IFSC, sendo grande parte do hardware e software desenvolvidos durante o projeto de doutorado. Foi desenvolvido um programa para processamento da transformada inversa de Laplace de dados unidimensionais, sendo este utilizado para o estudo de propriedades de adesão de pastilhas de gesso quando submetidas a diferentes pressões. Além disso, foram desenvolvidos modelos computacionais para simular essas propriedades relacionadas aos experimentos utilizados para o estudo dos meios porosos. Parte deste trabalho contou com a realização de dos estudos na Victoria University of Wellington em colaboração com o professor Paul T. Callaghan através do Programa de Doutorado e Estágio no Exterior (PDEE) da CAPES. / This study aimed to implement Nuclear Magnetic Resonance (NMR) for the study of porous media in general, with potential applications in the study of rocks, bones, porous polymers and ceramics. Due to the heterogeneity and complexity of these media, it becomes important to correlate various physical parameters that contain information about both the molecular dynamics on the structure and physical chemistry of pore fluid and the permeate. Using NMR techniques, it is possible to investigate these parameters by measuring diffusion coefficients, transverse and longitudinal relaxation times of fluids within the pores, which have different dynamical states, ranging from an isotropic liquid to a liquid with high viscosity or organized due to its interaction with the surface of the pores or particles suspended in fluid. For this, we employed NMR techniques based especially on measurements of transverse relaxation times, which when properly combined, allow the construction of two-dimensional maps that correlate properties of the molecular dynamics of fluids saturating these samples at different times of the experiment, providing information on dimensions and time scales involved in the dynamics of fluids saturating the porous media. These maps were obtained from experimental data on water-saturated oil rocks via inversion of two-dimensional Laplace transform. This work was performed using the existing infrastructure in the IFSC NMR laboratories and much of the hardware and software was developed during the doctoral project. We developed a program for processing the one-dimensional inverse Laplace transform data, which was used for studying the adhesion properties of gypsum pellets when subjected to different pressures. Additionally, we developed computational models to simulate physical-chemical properties related to the experiments used for the studies of fluids within porous media. Part of this work included studies performed in Victoria University of Wellington in collaboration with Professor Paul T. Callaghan through the Programa de Doutorado e Estágio no Exterior (PDEE) from CAPES.

O contínuo Tempo Randon Walk desenvolvido por Scher e Montroll é generalizado a fim de incluir as transições de taxas de espaço e o tempo. Tomando o limite contínuo, no passeio de equações aleatórias, uma equação geral de uma carga espacial transportada é obtida. A equivalência entre hopping e transporte através de estados expandidos com armadilhas é mostrado para realizar no espaço uma carga caso. Uma análise dos transportes um processo com dois processos simultâneos condução, por um armadilhagem e alargado a outros estados e por hopping através de armadilhas, é realizada. O Método Substituição Funcional (EFM) é introduzida a fim de obter resultados semi-markofian de Markofian soluções. Soluções específicas são obtidas em campo caso o alto e aproximadas queridos no espaço cobrar caso. Uma aproximação analítica de solução para o potencial superficial de decadência de uma sólida com armadilhas cobrado pela Corona é dada pela utilização do FSM. / The continuous Time Randon Walk developed by Scher and Montroll is generalized in order t o include space and time transitions rates. Taking the continuum limit in the random walk equations a general equation for space charge transport is obtained. The equivalence between hopping and transport via extended states with traps is shown to hold in the space charge case. An analysis of a transport process with two simultaneous conduction process, one by trapping and extended states and the other by hopping via traps, is carried out. The Functional Substitution Method (FSM) is introduced in order to get semi-markofian results from Markofian solutions. Specific exact solutions are obtained in the high field case and approximated ones in the space charge case. An approximated analytical solution for the superficial potential decay of a solid with traps charged by Corona is given by use of FSM.

In this study, piezometric heads in a composite aquifer composed of an alluvial deposit having a width adjacent to a semi-infinite fractured rock are determined. One dimensional transient flow induced by a constant discharge pumping rate from a stream intersecting alluvial part of the aquifer is considered. Parts of the aquifer are homogeneous andisotropic. Equations of flow, initial and boundary conditions are converted to dimensionless forms for graphical presentation and the interpretation of results independent of discharge and head inputs specific to the problem. Equations are solved first in Laplace domain and Laplace domain solutions are inverted numerically to real time domain by utilizing Stehfest algorithm.For this purpose, a set of subroutines in VBA Excel are developed. This procedure is verified by application of code to flow in semi-infinite homogeneous aquifer under constant discharge for which analytical solution is available in literature. VBA codes are also developed for two special cases of finite aquifer with impervious and with recharge boundary on the right hand side. Results of composite aquifer solutions with extreme tranmissivity values are compared with these two cases for verification of methodology and sensivity of results.