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1 
Aspects of the Laplace transform isotherm migration methodRadford, Linda Elisabeth January 2008 (has links)
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 wellestablished 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 onedimensional 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 twodimensional setting where we find the situation to be much more complex and challenging, as it requires interpolation and curvefitting 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.

2 
What are, and what are not, Inverse Laplace TransformsFordham, Edmund J., Venkataramanan, Lalitha, Mitchell, Jonathan, Valori, Andrea 11 September 2018 (has links)
Timedomain NMR, in one and higher dimensionalities, makes routine use of inversion algorithms to generate results called \T2distributions' 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 longestablished 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 \T2distribution' 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 wellknown problems of illconditioning 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.

3 
Contribution to qualitative and constructive treatment of the heat equation with domain singularitiesChin, P.W.M. (Pius Wiysanyuy Molo) 13 February 2012 (has links)
Please read the abstract in the 00front section of this document. / Thesis (PhD)University of Pretoria, 2011. / Mathematics and Applied Mathematics / unrestricted

4 
An Energy Diffusion Model for Interior Acoustics with Structural Coupling Using the Laplace Transform Boundary Element SolutionCorcoran, Joseph Michael 13 June 2013 (has links)
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 (1D) 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 elementLaplace transform solution (BELTS) 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 BELTS is exposed.
Structural coupling is then incorporated in the diffusion BELTS 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 BELTS 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 BELTS and compared to experimental data. / Ph. D.

5 
Kinetics of Thiomolybdate and CopperThiomolybdate Interconversion ProcessesClark, Rhett Jason 29 September 2008
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) (MoOxS4x2; 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 Culigand 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 UVvisible 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 crosscontamination. 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 UVvisible 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 UVvisible 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.

6 
Kinetics of Thiomolybdate and CopperThiomolybdate Interconversion ProcessesClark, Rhett Jason 29 September 2008 (has links)
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) (MoOxS4x2; 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 Culigand 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 UVvisible 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 crosscontamination. 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 UVvisible 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 UVvisible 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.

7 
EFFICIENT LARGE SCALE TRANSIENT HEAT CONDUCTION ANALYSIS USING A PARALLELIZED BOUNDARY ELEMENT METHODErhart, Kevin 13 September 2006 (has links)
A parallel domain decomposition Laplace transform Boundary Element Method, BEM, algorithm for the solution of largescale transient heat conduction problems will be developed. This is accomplished by building on previous work by the author and including several new additions (most noteworthy is the extension to 3D) aimed at extending the scope and improving the efficiency of this technique for largescale problems. A Laplace transform method is utilized to avoid time marching and a Proper Orthogonal Decomposition, POD, interpolation scheme is used to improve the efficiency of the numerical Laplace inversion process. A detailed analysis of the Stehfest Transform (numerical Laplace inversion) is performed to help optimize the procedure for heat transfer problems. A domain decomposition process is described in detail and is used to significantly reduce the size of any single problem for the BEM, which greatly reduces the storage and computational burden of the BEM. The procedure is readily implemented in parallel and renders the BEM applicable to largescale transient conduction problems on even modest computational platforms. A major benefit of the Laplace space approach described herein, is that it readily allows adaptation and integration of traditional BEM codes, as the resulting governing equations are time independent. This work includes the adaptation of two such traditional BEM codes for steadystate heat conduction, in both two and three dimensions. Verification and validation example problems are presented which show the accuracy and efficiency of the techniques. Additionally, comparisons to commercial Finite Volume Method results are shown to further prove the effectiveness. / M.S.M.E. / Department of Mechanical, Materials and Aerospace Engineering; / Engineering and Computer Science / Mechanical Engineering

8 
Semianalytical Solution for Multiphase Fluid Flow Applied to CO2 Sequestration in Geologic Porous MediaMohamed, Ahmed 16 December 2013 (has links)
The increasing concentration of CO_(2) has been linked to global warming and changes in climate. Geologic sequestration of CO_(2) in deep saline aquifers is a proposed greenhouse gas mitigation technology with potential to significantly reduce atmospheric emissions of CO_(2). Feasibility assessments of proposed sequestration sites require realistic and computationally efficient models to simulate the subsurface pressure response and monitor the injection process, and quantify the risks of leakage if there is any. This study investigates the possibility of obtaining closed form expressions for spatial distribution of CO_(2) injected in brine aquifers and gas reservoirs.
Four new semianalytical solutions for CO_(2) injection in brine aquifers and gas reservoirs are derived in this dissertation. Both infinite and closed domains are considered in the study. The first solution is an analysis of CO_(2) injection into an initially brinefilled infinite aquifer, exploiting self–similarity and matched asymptotic expansion. The second is an expanding to the first solution to account for CO_(2) injection into closed domains. The third and fourth solutions are analyzing the CO_(2) injection in infinite and closed gas reservoirs. The third and fourth solutions are derived using Laplace transform. The brine aquifer solutions accounted for both Darcyian and nonDarcyian flow, while, the gas reservoir solutions considered the gas compressibility variations with pressure changes.
Existing analytical solutions assume injection under constant rate at the wellbore. This assumption is problematic because injection under constant rate is hard to maintain, especially for gases. The modeled injection processes in all aforementioned solutions are carried out under constant pressure injection at the wellbore (i.e. Dirichlet boundary condition). One major difﬁculty in developing an analytical or semianalytical solution involving injection of CO_(2) under constant pressure is that the flux of CO_(2) at the wellbore is not known. The way to get around this obstacle is to solve for the pressure wave first as a function of flux, and then solve for the flux numerically, which is subsequently plugged back into the pressure formula to get a closed form solution of the pressure. While there is no simple equation for wellbore flux, our numerical solutions show that the evolution of flux is very close to a logarithmic decay with time. This is true for a large range of the reservoir and CO_(2) properties.
The solution is not a formation specific, and thus is more general in nature than formationspecific empirical relationships. Additionally, the solution then can be used as the basis for designing and interpreting pressure tests to monitor the progress of CO_(2) injection process. Finally, the infinite domain solution is suitable to aquifers/reservoirs with large spatial extent and low permeability, while the closed domain solution is applicable to small aquifers/reservoirs with high permeability.

9 
Solution Of One Dimensional Transient Flow In Composite Aquifers Using Stehfest AlgorithmBakar, Urun 01 September 2010 (has links) (PDF)
In this study, piezometric heads in a composite aquifer composed of an alluvial deposit having a width adjacent to a semiinfinite 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 semiinfinite 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.

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A transformada de Laplace e algumas aplicaçõesLustosa, José Ivelton Siqueira Lustosa 26 May 2017 (has links)
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Previous issue date: 20170526 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior  CAPES / In this work, we study the Laplace Transform and explore its application in solving
some linear ordinary di erential equations, which model various phenomena in
the areas of Physics, Engineering, Industrial Automation and Mathematics itself.
Such knowledge is of great importance in higher education courses covering such
areas. We present the de nition, properties and main results involving the Laplace
Transform and address several problems in the areas mentioned above. / Neste trabalho, estudamos a Transformada de Laplace e exploramos sua aplica
ção na resolução de algumas equações diferenciais ordinárias lineares, as quais
modelam vários fenômenos nas áreas de Física, Engenharia, Automação Industrial
e na própria Matemática. Tais conhecimentos são de suma importância em cursos
superiores que abrangem tais áreas. Apresentamos a de nição, propriedades
e principais resultados envolvendo a Transformada de Laplace e abordamos vários
problemas nas áreas citadas anteriormente.

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