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Wilkerson, Dorian. ""Mathermatical Analysis of a Truly Nonlinear Oscillator Differential Equation"". DigitalCommons@Robert W. Woodruff Library, Atlanta University Center, 2009. http://digitalcommons.auctr.edu/dissertations/101.
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Stahl, Levi Russell. "OBJECT ORIENTED DEVELOPMENT OF A MATHEMATICAL EQUATION EDITOR". MSSTATE, 2005. http://sun.library.msstate.edu/ETD-db/theses/available/etd-07062005-173340/.
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Computers since their inception have been used to solve engineering problems. Toward support of next-generation, customizable, generalized software, a mathematical equation editor has been designed, developed, and tested using object oriented (OO) programming techniques. The motivating purpose of this equation editor is to allow a user to graphically define mathematical equations to be solved in a computational partial differential equation-based problem solving environment. The OO scripting language Python was used in conjunction with the OO GUI toolkit Qt to create the editor. Analysis of the underlying abstraction of a general equation yielded the key concept of an information-holding bounding box. Such boxes hierarchically contain every character and symbol in an equation. Specific rules were formulated to spatially arrange a set of boxes into a properly formatted equation. Robust insertion logic of alphanumeric characters, mathematical symbols, and common function names was implemented for intuitive point-and-click equation building.
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Sakamoto, Shota. "Mathematical analysis of global solutions to the Boltzmann equation". 京都大学 (Kyoto University), 2017. http://hdl.handle.net/2433/225680.
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Tzou, Leo. "Linear and nonlinear analysis and applications to mathematical physics /". Thesis, Connect to this title online; UW restricted, 2007. http://hdl.handle.net/1773/5761.
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Flegg, Jennifer Anne. "Mathematical modelling of chronic wound healing". Thesis, Queensland University of Technology, 2009. https://eprints.qut.edu.au/40164/1/Jennifer_Flegg_Thesis.pdf.
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Chronicwounds fail to proceed through an orderly process to produce anatomic and functional integrity and are a significant socioeconomic problem. There is much debate about the best way to treat these wounds. In this thesis we review earlier mathematical models of angiogenesis and wound healing. Many of these models assume a chemotactic response of endothelial cells, the primary cell type involved in angiogenesis. Modelling this chemotactic response leads to a system of advection-dominated partial differential equations and we review numerical methods to solve these equations and argue that the finite volume method with flux limiting is best-suited to these problems. One treatment of chronic wounds that is shrouded with controversy is hyperbaric oxygen therapy (HBOT). There is currently no conclusive data showing that HBOT can assist chronic wound healing, but there has been some clinical success. In this thesis we use several mathematical models of wound healing to investigate the use of hyperbaric oxygen therapy to assist the healing process - a novel threespecies model and a more complex six-species model. The second model accounts formore of the biological phenomena but does not lend itself tomathematical analysis. Bothmodels are then used tomake predictions about the efficacy of hyperbaric oxygen therapy and the optimal treatment protocol. Based on our modelling, we are able to make several predictions including that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds, treatment should continue until healing is complete and finding the right protocol for an individual patient is crucial if HBOT is to be effective. Analysis of the models allows us to derive constraints for the range of HBOT protocols that will stimulate healing, which enables us to predict which patients are more likely to have a positive response to HBOT and thus has the potential to assist in improving both the success rate and thus the cost-effectiveness of this therapy.
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Karlsson, Olle. "The Black-Scholes Equation and Formula". Thesis, Uppsala universitet, Analys och tillämpad matematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-200441.
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Pierantozzi, Mariano. "Mathematical modeling for Thermodynamics: Thermophysical Properties and Equation of State". Doctoral thesis, Università Politecnica delle Marche, 2015. http://hdl.handle.net/11566/242931.
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Nelle moderne società multiculturali e multidisciplinari, sempre di più si devono adottare delle prospettive più ampie possibili.
In questa tesi, si è tentato di adottare un metodo multidisciplinare che coinvolgesse non solo la matematica e la fisica, ma anche la chimica, la statistica, e più in generale l’ingegneria.
Gli aspetti toccati sono quelli delle proprietà termofisiche della materia e delle equazioni di stato dei gas (EOS). Le proprietà termofisiche analizzate sono: tensione superficiale, conduttività termica, viscosità, dei liquidi e dei gas ed il secondo coefficiente del viriale. Dopo la raccolta dei dati sperimentali, essi sono stati analizzati con varie tecniche statistiche che trasformassero i dati grezzi in dati più attendibili. Dopo lo studio delle equazioni della letteratura si è proceduto con uno studio di sensibilità dei dati per vedere quali proprietà fisiche avessero maggiore impatto sulle proprietà studiate. Infine si è cercata un’equazione che potesse rappresentare nel migliore modo possibile i dati sperimentali.
Si sono sempre preferite equazioni scalate ad equazioni puramente empiriche, in modo da avere non solo l’aderenza ai dati sperimentali, ma anche il rispetto dell’aspetto chimico-fisico. Dall’analisi dei residui, confrontandoci con le migliori equazioni in letteratura, i nostri risultati sono sempre stati migliori, tanto che hanno avuto dignità di pubblicazione nelle maggiori riviste del settore.
Discorso a parte per le EOS.
Analizzando la letteratura, ciò che subito è saltato all’occhio è che cercare la migliore equazione possibile è impossibile! Oppure come dice Martin parafrasando una frase della favola Biancaneve: “Specchio specchio delle mie brame, qual è la più bella del reame?”
Abbiamo scelto la modifica dell’equazione Carnahan-Starling-De Santis. Tramite tecniche di minimizzazione multi obiettivo si sono migliorate le performance di tal equazione proprio intorno al punto critico.
Questi sono gli aspetti principali toccati in questo lavoro di tesi, che di là dai risultati, pur buoni ottenuti, mi ha aperto il mondo della ricerca.Abstract In the modern multicultural and multidisciplinary society, always adopting more and more wider prospective than before. In this thesis, we try to adopt a multidisciplinary method, which involves Mathematics, Physics, but also Chemistry, Statistics, and in general the scientific engineering. The aspects explained are thermo physical properties, and Equations of State (EOS) of gases. Regarding thermo physical properties have been analysed Surface Tension, Thermal Conductivity, Viscosity, and the second virial coefficient. On this arguments, the work had been subdivided between the gathering of experimental data, the analysing of data with statistical techniques transforming them to more reliable data than row. The second step was to collect the equations of literature. Then we went ahead studying the sensibility of data to find out which physical properties could have bigger impact to property examined. At the end, we looked for an equation that could represent experimental data in a better way. We always preferred the scaled equations that respect chemical and physical aspects, to the empirical ones. Comparing our results with better equations in literature, our results are always better, in fact all of the have been published in the best international journals on this subject. A separate discussion is that of EOS. Analyzing the previous literature, the first thing that came to our minds was that to find the best possible equation is impossible. Or as Martin wrote copying words of the famous fables Snow White: “Mirror mirror on the wall, who is the fairest of them all?”. We choose to modify The Carnahan-Starling-De Santis (CSD) equation of state, a parametrich equation with good results in the calculation of Vapor Liquid Equilibrium. Due to multi objective minimization techniques the performance of CSD has been improved. These are the principals aspect brought to light in this research, which apart from the results, with good results has opened to me the world of research.
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Beech, Robert. "Extensions of the nonlinear Schrödinger equation using Mathematica". Thesis, View thesis, 2009. http://handle.uws.edu.au:8081/1959.7/46572.
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The aim of this thesis is to investigate the theory of the extensions of the Nonlinear Schrödinger Equation (NLSE), concentrating on the following main points: Developing further analytical techniques and properties under relativistic conditions. This thesis demonstrates numerical techniques that can be used to form numerical codes that can be applied to the very recent need for source and industrial application of laser-driven ion sources for ion implantation. The analytical and numerical evaluations of the nonlinear mechanisms are measured utilising various techniques that include computer packages such as Mathematica(R) [Wolfram 2003]1, Maple™ 9 [2003] and C++© [Strousop 2003]. This project expands the author’s present undergraduate honours research work on the theory of Schrödinger equations. The breaking of light waves: in the course of this research the breaking of light waves was the first new phenomenon to be encountered. The highest authority on this subject [Zakharov and Shabat 1972], Prof. Zakharov, advised me [Zakharov 2004] that this topic was at that time not researched in any detail. It was envisaged that entering more fully into this area of research using Mathematica version 5 [Wolfram 2003], which had been recently released (June 2003) and which is uniquely adapted for such research, would be the most profitable direction to go. The intention was to research the behaviour of radiation from the soliton in respect of the higher order (dispersion) term in the NLSE. This research was expected to reveal its properties and consequences and possibly new ways in which this radiation can be predicted, controlled, eliminated or otherwise profitably manipulated. These results are considered vital to the uses of solitons, particularly in optical fibre telecommunications. Numerical artefacts: At this juncture the direction of the research changed in a way that had not been anticipated. The compilation and execution of Mathematica codes, now advanced to the use of new techniques and iterative methods such as the Split-Step Method, had been anticipated to clearly show the existence of secondary and possibly tertiary radiation attending the soliton. It had also been anticipated that this would confirm the theory that this radiation attended only solitons resulting from the cubic, and odd numbered, higher-order NLSE. The first assumption simply did not materialise and the second was not at all up to expectations. At best, the results coming from this line of investigation could only show that solitons derived from the even numbered, or quadratic higher-order NLSEs were in some ways fundamentally different from the odd numbered or cubic ones. These setbacks all resulted from a phenomenon, hitherto unanticipated as a problem to this program of research, namely ‘numerical artefact’ in Mathematica [Beech and Osman 2005: 1369; See Appendix I Paper 3]. This reduced Paper 3 [ibid] ‘Effects of higher order dispersion terms in the nonlinear Schrödinger Equation’ from a serious contribution in this field to a scathing criticism of the use of iterative methods in computerised mathematics.
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Ahmad, Ferhana. "A stochastic partial differential equation approach to mortgage backed securities". Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:ee33aa2d-b9fa-4cc4-a399-5f681966bc77.
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The market for mortgage backed securities (MBS) was active and fast growing from the issuance of the first MBS in 1981. This enabled financial firms to transform risky individual mortgages into liquid and tradable market instruments. The subprime mortgage crisis of 2007 shows the need for a better understanding and development of mathematical models for these securities. The aim of this thesis is to develop a model for MBS that is flexible enough to capture both regular and subprime MBS. The thesis considers two models, one for a single mortgage in an intensity based framework and the second for mortgage backed securities using a stochastic partial differential equation approach. In the model for a single mortgage, we capture the prepayment and default incentives of the borrower using intensity processes. Using the minimum of the two intensity processes, we develop a nonlinear equation for the mortgage rate and solve it numerically and present some case studies. In modelling of an MBS in a structural framework using stochastic PDEs (SPDEs), we consider a large number of individuals in a mortgage pool and assume that the wealth of each individual follows a stochastic process, driven by two Brownian mo- tions, one capturing the idiosyncratic noise of each individual and the second a common market factor. By defining the empirical measure of a large pool of these individuals we study the evolution of the limit empirical measure and derive an SPDE for the evolution of the density of the limit empirical measure. We numerically solve the SPDE to demonstrate its flexibility in different market environments. The calibration of the model to financial data is the focus of the final part of thesis. We discuss the different parameters and demonstrate how many can be fitted to observed data. Finally, for the key model parameters, we present a strategy to estimate them given observations of the loss function and use this to determine implied model parameters of ABX.HE.
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Sum, Kwok-wing Anthony, i 岑國榮. "Partial differential equation based methods in medical image processing". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B38958624.
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Tsandzana, Afonso Fernando. "Homogenization of some new mathematical models in lubrication theory". Doctoral thesis, Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-59629.
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We consider mathematical modeling of thin film flow between two rough surfaces which are in relative motion. For example such flows take place in different kinds of bearings and gears when a lubricant is used to reduce friction and wear between the surfaces. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation, which describes the motion of viscous fluids. In thin domains several approximations are possible which lead to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict vorous important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field. In hydrodynamic lubrication the effect of surface roughness is not negligible, because in practical situations the amplitude of the surface roughness are of the same order as the film thickness. Moreover, a perfectly smooth surface does not exist in reality due to imperfections in the manufacturing process. Therefore, any realistic lubrication model should account for the effects of surface roughness. This implies that the mathematical modeling leads to partial differential equations with coefficients that will oscillate rapidly in space and time. A direct numerical computation is therefore very difficult, since an extremely dense mesh is needed to resolve the oscillations due to the surface roughness. A natural approach is to do some type of averaging. In this PhD thesis we use and develop modern homogenization theory to be able to handle the questions above. Especially, we use, develop and apply the method based on the multiple scale expansions and two-scale convergence. The thesis is based on five papers (A-E), with an appendix to paper A, and an extensive introduction, which puts these publications in a larger context. In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness
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Krupp, Armin Ulrich. "Mathematical modelling of membrane filtration". Thesis, University of Oxford, 2017. http://ora.ox.ac.uk/objects/uuid:ae6dd9e4-a862-4476-a8d9-35156848297f.
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In this thesis, we consider four different problems in membrane filtration, using a different mathematical approach in each instance. We account for the fluid-driven deformation of a filtercake using nonlinear poroelasticity in Chapter 2. By considering feeds with very high and very low particle concentrations, we introduce a quasi-static caking model that provides a suitable approximation to the full model for the physically realistic concentration regimes. We illustrate the agreements and differences between our model and the existing conventional cake-filtration law. In Chapter 3, we introduce a stochastic model for membrane filtration based on the quantised nature of the particles and show how it can be applied for feeds with different particle types and membranes with an interconnected pore structure. This allows us to understand the relation between the effects of clogging on the level of an individual pore and on the macroscopic level of the entire membrane. We conclude by explaining the transition between the discrete and continuous model based on the Fokker--Planck equation. In Chapter 4, we consider the inverse problem of determining the underlying filtration law from the spreading speed of a particle-laden gravity current. We first couple the theory of gravity currents with the stochastic model developed in Chapter~3 to determine a filtration law from a given set of experiments. We then generalise this idea for the porous medium equation, where we show that the position of the front follows a power law for the conventional filtration laws, which allows us to infer the clogging law in certain instances. We conclude the thesis by showing in Chapter 5 how we can combine experimental measurements for the clogging of a depth filter and simple fluid dynamics to accurately predict the pressure distribution in a multi-capsule depth filter during a filtration run.
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Tsang, Cheng-hou Alan, i 曾正豪. "Dynamics of waves and patterns of the complex Ginburg Landau and soliton management models: localized gain andeffects of inhomogeneity". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2011. http://hub.hku.hk/bib/B46975433.
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Dimitry, Johan. "A mathematical study of convertible bonds". Thesis, KTH, Farkost och flyg, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-151312.
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A convertible bond (CB) is a financial derivative, a so called hybrid security. It is an issued contract from a company or a government, which is paid for up-front. The contract yields a known amount at the specified maturity date, unless the holder chooses to convert it into an amount of the underlying asset. This kind of financial products can have complex features affecting the contract price and the optimal exercising situation. The partial differential equation (PDE) approach used for pricing financial derivatives makes it possible to describe convertible bonds with a physical model, a reversed diffusion described by a parabolic PDE. One can sometimes find both analytical and numerical solutions for this type of PDEs and interpret the solutions from a financial point of view, as they suggest predictable behaviour of the contract price.
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Bäck, Viktor. "Localization of Multiscale Screened Poisson Equation". Thesis, Uppsala universitet, Algebra och geometri, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-180928.
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Steinle, Peter John. "Finite difference methods for the advection equation /". Title page, table of contents and abstract only, 1993. http://web4.library.adelaide.edu.au/theses/09PH/09phs8224.pdf.
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Hwang, Heungsun 1969. "Structural equation modeling by extended redundancy analysis". Thesis, McGill University, 2000. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=36954.
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A new approach to structural equation modeling based on so-called extended redundancy analysis (ERA) is proposed. In ERA, latent variables are obtained as exact linear combinations of observed variables, and model parameters are estimated by consistently minimizing a single criterion. As a result, the method can avoid limitations of covariance structure analysis (e.g., stringent distributional assumptions, improper solutions, and factor score indeterminacy) in addition to those of partial least squares (e.g., the lack of a global optimization procedure). The method is simple yet versatile enough to fit more complex models; e.g., those with higher-order latent variables and direct effects of observed variables. It can also fit a model to more than one sample simultaneously. Other relevant topics are also discussed, including data transformations, missing data, metric matrices, robust estimation, and efficient estimation. Examples are given to illustrate the proposed method.
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Olivares, Nicole Michelle. "Accuracy of Wave Speeds Computed from the DPG and HDG Methods for Electromagnetic and Acoustic Waves". PDXScholar, 2016. http://pdxscholar.library.pdx.edu/open_access_etds/2920.
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We study two finite element methods for solving time-harmonic electromagnetic and acoustic problems: the discontinuous Petrov-Galerkin (DPG) method and the hybrid discontinuous Galerkin (HDG) method.
The DPG method for the Helmholtz equation is studied using a test space normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitrary positive scaling parameter. We find that, as the parameter approaches zero, better results are obtained, under some circumstances. A dispersion analysis on the multiple interacting stencils that form the DPG method shows that the discrete wavenumbers of the method are complex, explaining the numerically observed artificial dissipation in the computed wave approximations. Since the DPG method is a nonstandard least-squares Galerkin method, its performance is compared with a standard least-squares method having a similar stencil.
We study the HDG method for complex wavenumber cases and show how the HDG stabilization parameter must be chosen in relation to the wavenumber. We show that the commonly chosen HDG stabilization parameter values can give rise to singular systems for some complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. For real wavenumbers, results from a dispersion analysis for the Helmholtz case are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method shows that its wavenumber errors are an order smaller than those of the HDG method.
We conclude by presenting some contributions to the development of software tools for using the DPG method and their application to a terahertz photonic structure. We attempt to simulate field enhancements recently observed in a novel arrangement of annular nanogaps.
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Pankratova, Iryna. "Convection-diffusion equation in unbounded cylinder and related homogenization problems". Licentiate thesis, Luleå : Luleå tekniska universitet, 2009. http://pure.ltu.se/ws/fbspretrieve/2579688.
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Sagheer, Muhammad. "Mathematical analysis and numerical solutions of an integral equation arising from population dynamics". Thesis, University of Sussex, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.420495.
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Caunce, James Frederick Physical Environmental & Mathematical Sciences Australian Defence Force Academy UNSW. "Mathematical modelling of wool scouring". Awarded by:University of New South Wales - Australian Defence Force Academy. School of Physical, Environmental and Mathematical Sciences, 2007. http://handle.unsw.edu.au/1959.4/38650.
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Wool scouring is the first stage of wool processing, where unwanted contaminants are removed from freshly shorn wool. In most scouring machines wool is fed as a continuous mat through a series of water-filled scour and rinse bowls which are periodically drained. The purpose of this project is to mathematically model the scour bowl with the aim of improving efficiency. In this thesis four novel models of contaminant concentration within a scour bowl are developed. These are used to investigate the relationships between the operating parameters of the machine and the concentration of contamination within the scour bowl. The models use the advection-diffusion equation to simulate the settling and mixing of contamination. In the first model considered here, the scour bowl is simulated numerically using finite difference methods. Previous models of the scouring process only considered the average steady-state concentration of contamination within the entire scour bowl. This is the first wool scouring model to look at the bowl in two dimensions and to give time dependent results, hence allowing the effect of different drainage patterns to be studied. The second model looks at the important region at the top of the bowl - where the wool and water mix. The governing equations are solved analytically by averaging the concentration vertically assuming the wool layer is thin. Asymptotic analysis on this model reveals some of the fundamental behaviour of the system. The third model considers the same region by solving the governing equations through separation of variables. A fourth, fully two-dimensional, time dependent model was developed and solved using a finite element method. A model of the swelling of grease on the wool fibres is also considered since some grease can only be removed from the fibre once swollen. The swelling is modelled as a Stefan problem, a nonlinear diffusion equation with two moving boundaries, in cylindrical coordinates. Both approximate, analytical and a numerical solutions are found.
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Zhang, Quanju. "Ordinary differential equation methods for some optimization problems". HKBU Institutional Repository, 2006. http://repository.hkbu.edu.hk/etd_ra/710.
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PARLETTE, EDWARD BRUCE. "GENERALIZED FUNCTION SOLUTIONS TO THE FOKKER-PLANCK EQUATION". Diss., The University of Arizona, 1985. http://hdl.handle.net/10150/187933.
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In problems involving highly forward-peaked scattering, the Boltzmann transport equation can be simplified using the Fokker-Planck model. The purpose of this project was to develop an analytical solution to the resulting Fokker-Planck equation. This analytical solution can then be used to benchmark numerical transport codes. A numerical solution to the Fokker-Planck equation was also developed. The analytical solution found is a generalized function. It satisfies the purpose of the project with two limitations. The first limitation is that the solution can only be evaluated for certain sources. The second limitation is that the solution can only be evaluated for small times. The moments of the Fokker-Planck equation can be evaluated for any time. The numerical solution developed works for all sources and all times. The analytical solution, then, provides an accurate and precise benchmark under certain conditions. The numerical solution provides a less accurate benchmark under all conditions.
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Borggaard, Jeffrey T. "The sensitivity equation method for optimal design". Diss., Virginia Tech, 1994. http://hdl.handle.net/10919/38563.
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In this work, we introduce the Sensitivity Equation Method (SEM) as a method
for approximately solving infinite dimensional optimal design problems. The SEM
couples a trust-region/quasi-Newton optimization algorithm with gradient information
provided by apprOXimately solving the sensitivity equation for (design) sensitivities.
The sensitivity equation is (in the problems considered here) a partial differential
equation (POE) which describes the influence of a design parameter on the state of
the system. It is shown that obtaining design sensitivities from the sensitivity equation
has advantages over finite difference and semi-analytical methods in that there is no need to remesh or compute mesh sensitivities (even if the domain is parameter
dependent), the sensitivity equation is a linear POE for the sensitivities and can be
approximated in an efficient manner using the same approximation scheme used to
approximate the states.
The applicability of the SEM to shape optimization problems, where the state is
described by the Euler equations, is studied in detail. In particular, we prove convergence of the method for a one dimensional test problem. These results are used to speculate on the applicability of the method for more complex problems. Finally. we solve a two dimensional forebody simulator design problem (for use in wind tunnel experiments) using the SEM, which is shown to be a very efficient method for this problem.Ph. D.
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Miller, Susan J. "The mathematics of longing: Exploring the interface between science and theatre by translating mathematical theorems into a play script". Thesis, Queensland University of Technology, 2020. https://eprints.qut.edu.au/201671/1/Susan_Miller_Thesis.pdf.
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This PhD investigated how a playwright could translate what is elemental and elegant about a series of disparate mathematical equations into a play script such that the translations were expressed as an overall unified narrative that makes an overarching comment about science and humanity. By combining cultural translation and ekphrasis, a new methodology model was created for writing my PhD play entitled The Mathematics of Longing. Furthermore, the play script was presented in a non-traditional publishing format that visually invokes the subtext of the play, and potentially influences future production mis en scenes of the play text.
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El-Hachem, Maud. "Mathematical models of biological invasion". Thesis, Queensland University of Technology, 2022. https://eprints.qut.edu.au/232864/1/Maud_El-Hachem_Thesis.pdf.
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This thesis studies mathematical models of a population of cells invading the surrounding environment or another living population. A classical single-species model is reformulated using a moving boundary to track the position of the moving front of the invading population. The moving boundary is also used to separate two populations. Other models studied are coupled partial differential equations to describe the interaction of a population with another one. Different types of interaction are represented: the degradation of healthy skin by cancer and the growth of bone tissue on substrate.
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Thomson, Stuart. "Mathematical modelling of elastoplasticity at high stress". Thesis, University of Oxford, 2017. http://ora.ox.ac.uk/objects/uuid:a7d565c6-abeb-4932-8c1e-aebc38da7584.
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This thesis is concerned with the mathematical modelling of elastic-plastic deformation in regimes of stress far exceeding the yield stress. Such scenarios are typically encountered in violent impact testing, where millimetre-thick samples of metal are subjected to pressures on the order of the bulk modulus of the material. We begin with an overview of violent impact testing, with particular attention paid to a specific class of experiments known as isentropic compression experiments (ICEs), which will provide motivation for the mathematical modelling and analysis in subsequent chapters. In chapter 2, by appealing to sound notions from rational mechanics and thermodynamics, we construct a mathematical model which aims to encapsulate the essential phenomena involved in violent elastic-plastic deformation. This is followed in chapter 3 with a numerical analysis of the mathematical model in uniaxial strain, which is the geometry relevant ICEs. In chapters 4 and 5, we corroborate the observations made in chapter 3 via a systematic mathematical analysis. In particular, our focus will be on the elastic and plastic waves that can propagate through finite metal samples during isentropic compression. Finally, in chapter 6, we explore the applicability of our model to other geometries, specifically the radially axisymmetric expansion of a circular cavity embedded in an infinite elastic-plastic medium. We conclude with a summary of our findings and suggest some avenues for future investigation.
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Chen, Yongpin, i 陈涌频. "Surface integral equation method for analyzing electromagnetic scattering in layered medium". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2011. http://hub.hku.hk/bib/B4775283X.
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Surface integral equation (SIE) method with the kernel of layered medium Green's
function (LMGF) is investigated in details from several fundamental aspects. A
novel implementation of discrete complex image method (DCIM) is developed to
accelerate the evaluation of Sommerfeld integrals and especially improve the far
field accuracy of the conventional one. To achieve a broadband simulation of thin
layered structure such as microstrip antennas, the mixed-form thin-stratified
medium fast-multipole algorithm (MF-TSM-FMA) is developed by applying
contour deformation and combining the multipole expansion and plane wave
expansion into a single multilevel tree. The low frequency breakdown of the
integral operator is further studied and remedied by using the loop-tree
decomposition and the augmented electric field integral equation (A-EFIE), both
in the context of layered medium integration kernel. All these methods are based
on the EFIE for the perfect electric conductor (PEC) and hence can be applied in
antenna and circuit applications. To model general dielectric or magnetic objects,
the layered medium Green's function based on pilot vector potential approach is
generalized for both electric and magnetic current sources. The matrix
representation is further derived and the corresponding general SIE is setup.
Finally, this SIE is accelerated with the DCIM and applied in quantum optics,
such as the calculation of spontaneous emission enhancement of a quantum
emitter embedded in a layered structure and in the presence of nano scatterers.published_or_final_version
Electrical and Electronic Engineering
Doctoral
Doctor of Philosophy
29
Shi, Bin 1966. "Identification of the material constitutive equation for simulation of the metal cutting process". Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=115709.
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This study presents a novel methodology to characterize material plastic behavior within a practical range of stresses, strains, strain rates, and temperatures encountered in the metal cutting process. The methodology is based on integrating a newly developed analytical model with quasi-static tests and orthogonal cutting experiments that incorporate a laser heating system. Friction and heat transfer models are developed to describe the tribological and thermal interactions at the tool-chip interface. These models are implemented in a FEM package in order to improve the accuracy of the simulation of the machining process.The new analytical model, which is developed to predict the distributions of the stress, the strain, the strain rate, and the temperature in the primary shear zone, is based on conceptual considerations, as well as characterization of the plastic deformation process through comprehensive FEM simulations.
Orthogonal cutting experiments at room temperature and preheated conditions were carefully designed. While the cutting tests at room temperature provided the constitutive data encountered in the primary shear zone, the preheated cutting tests were designed to capture the material behavior at the high level of temperature and strain encountered in the secondary shear zone. In these preheated cutting tests, a laser beam was employed. Quasi-static tests were also utilized to identify some of the coefficients in the constitutive equations, in order to improve the convergence to a unique solution for the constitutive law.
Evaluation criteria were developed to assess the performance of constitutive equations. Based on the developed methodology and the evaluation criteria, a new constitutive equation for Inconel 718 has been proposed. This constitutive equation was further validated by Split Hopkinson Pressure Bar (SHPB) tests and cutting tests in conjunction with FEM simulations. The SHPB test data show an excellent agreement with the proposed material model. The cutting tests and the FEM simulation results also proved the validity of the proposed material constitutive law.
30
Cuifeng, Wei. "Improved Finite Analytic Methods for Solving Advection-dominated Transport Equation in Highly Variable Velocity Field". PDXScholar, 1995. https://pdxscholar.library.pdx.edu/open_access_etds/4922.
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Solute transport studies frequently rely on numerical solutions of the classical advection-diffusion equation. Unfortunately, solutions obtained with traditional finite difference and finite element techniques typically exhibit excessive numerical diffusion or spurious oscillation when advection dominates, especially when velocity field is highly variable. One recently developed technique, the finite analytic method, offers an attractive alternative. Finite analytic methods utilize local analytic solutions in discrete elements to obtain the algebraic representations of the governing partial differential equations, thus eliminating the truncation error in the finite difference and the use of approximating functions in the finite element method. The finite analytic solutions have been shown to be stable and numerically robust for advection-dominated transport in heterogeneous velocity fields. However, the existing finite analytic methods for solute transport in multiple dimensions have the following disadvantages. First, the method is computationally inefficient when applied to heterogeneous media due to the complexity of the formulation. Second, the evaluation of finite analytic coefficients is when the Peclet number is large. Third, the method introduces significant numerical diffusion due to inadequate temporal approximation when applied to transient problems. This thesis develops improved finite analytic methods for two-dimensional steady as well as unsteady solute transports in steady velocity fields. For steady transport, the new method exploits the advantages of the existing finite analytic and finite difference methods. The analytically difficult diffusion terms are approximated by finite difference and numerically difficult advection and reaction terms are treated analytically in a local element in deriving the numerical schemes. The new finite analytic method is extended to unsteady transport through application of Laplace transformation. Laplace transformation converts the transient equation to a steady-state expression that can be solved with the steady version of the improved finite analytic method. Numerical inversion of the transformed variables is used to recover solute concentration in the physical space-time domain. The effectiveness and accuracy of the new finite analytic method is demonstrated through stringent test examples of two dimensional steady-state transport in highly variable velocity fields. The results clearly demonstrated that the improved finite analytic methods are efficient, robust and accurate.
31
Liashenko, A. A., i T. M. Onishenko. "The actual use of mathematical analisis in the reserch of the equation of body movements". Thesis, Sumy State University, 2018. http://essuir.sumdu.edu.ua/handle/123456789/67687.
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The methods of mathematical analysis allow us to investigate the functional dependencies between any quantities in any processes and phenomena (both natural and social), thus making them predictable for any values of the initial parameters and for any number of them.
32
Alzaix, Benjamin. "Mathematical and numerical analysis of the Herberthson integral equation dedicated to electromagnetic plane wave scattering". Thesis, Bordeaux, 2017. http://www.theses.fr/2017BORD0578/document.
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Cette thèse porte sur la diffraction d’une onde plane électromagnétique par une surface lisse parfaitement conductrice (PEC). Elle présente l’analyse des propriétés d’une nouvelle formulation des trois principales équations intégrales de frontières de la théorie de la diffraction électromagnétique (EFIE, MFIE et CFIE). L’idée est d’adapter les équations intégrales conventionnelles à la diffraction d’une onde plane en supposant que la fonction de phase de l’onde plane incidente détermine la fonction de phase de la distribution de courant induit sur la surface.L’idée d’utiliser la phase dans la diffraction d’ondes planes a déjà été étudiée pour les hautes fréquences, notamment dans les thèses de Zhou (1995) et Darrigrand (2002) qui adaptèrent les espaces d’approximation des éléments finis. Dans cette thèse, cependant, nous suivons une formulation plus récente, donnée par Herberthson (2008), où la fonction de phase est incorporée dans la distribution du noyau des opérateurs intégraux.En présentant les versions modifiées de l’EFIE et de la MFIE (dénommées HEFIE et HMFIE)dans des espaces fonctionnels appropriés, nous prouvons ici l’existence d’une solution unique à cette formulation spécifique et présentons une mise en oeuvre pratique originale qui tire parti de l’expérience acquise sur l’EFIE/MFIE. Par la suite, nous explorons une propriété importante offerte par ces nouvelles formulations: la possibilité de réduire le nombre de degrés de liberté requis pour obtenir une solution précise du problèmeThis thesis is about the scattering of an electromagnetic plane wave incidenton a perfectly conducting smooth surface. It presents the analysis of the properties of a newformulation of the three principal boundary integral equations of electromagnetic scattering theory(EFIE, MFIE and CFIE). The basic idea is to adapt the conventional integral equations toplane-wave scattering by supposing that the phase function of an incident plane wave determinesthe phase function of the induced boundary current distribution.This idea of using the phase in plane wave scattering has previously been studied in highfrequencyscattering, in particular in the theses by Zhou (1995) and Darrigrand (2002) whoadapt the finite element approximation spaces. In this thesis, though, we follow a more recentformulation, given by Herberthson (2008), where the phase function is incorporated in the kerneldistribution of the integral operators.Presenting the modified version of the EFIE and the MFIE (denoted HEFIE and HMFIE) inappropriate function spaces, we prove the existence of a unique solution to this specific formulationand developp an original practical implementation which takes advantage of the gainedexperience on the EFIE/MFIE. Then, we explore another important property provided by thenew formulations: the possibility to reduce the number of degrees of freedom required to get anaccurate solution of the problem
33
Khalmanova, Dinara Khabilovna. "A mathematical model of the productivity index of a well". Diss., Texas A&M University, 2004. http://hdl.handle.net/1969.1/301.
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Motivated by the reservoir engineering concept of the productivity index of a producing oil well in an isolated reservoir, we analyze a time dependent functional, diffusive capacity, on the solutions to initial boundary value problems for a parabolic equation. Sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions. The dependence of the constant diffusive capacity on the type of the boundary condition (Dirichlet, Neumann or third-type boundary condition) is investigated using a known variational principle and confirmed numerically for various geometrical settings. An important comparison between two principal constant values of a diffusive capacity is made, leading to the establishment of criteria when the so-called pseudo-steady-state and boundary-dominated productivity indices of a well significantly differ from each other. The third type boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well. The questions of stabilization and uniqueness of the time independent values of the diffusive capacity are addressed. The derived formulas are used in numerical study of evaluating the productivity index of a well in a general three-dimensional reservoir for a variety of well configurations.
34
Rosado, Linares Jesús. "Analysis of some diffusive and kinetic models in mathematical biology and physics". Doctoral thesis, Universitat Autònoma de Barcelona, 2010. http://hdl.handle.net/10803/3113.
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Johansen, Jonathan Frederick. "Mathematical modelling of primary alkaline batteries". Thesis, Queensland University of Technology, 2007. https://eprints.qut.edu.au/16412/1/Jonathan_Johansen_Thesis.pdf.
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Three mathematical models, two of primary alkaline battery cathode discharge, and one of primary alkaline battery discharge, are developed, presented, solved and investigated in this thesis. The primary aim of this work is to improve our understanding of the complex, interrelated and nonlinear processes that occur within primary alkaline batteries during discharge.
We use perturbation techniques and Laplace transforms to analyse and simplify an existing model of primary alkaline battery cathode under galvanostatic discharge. The process highlights key phenomena, and removes those phenomena that have very little effect on discharge from the model. We find that electrolyte variation within Electrolytic Manganese Dioxide (EMD) particles is negligible, but proton diffusion within EMD crystals is important. The simplification process results in a significant reduction in the number of model equations, and greatly decreases the computational overhead of the numerical simulation software. In addition, the model results based on this simplified framework compare well with available experimental data.
The second model of the primary alkaline battery cathode discharge simulates step potential electrochemical spectroscopy discharges, and is used to improve our understanding of the multi-reaction nature of the reduction of EMD. We find that a single-reaction framework is able to simulate multi-reaction behaviour through the use of a nonlinear ion-ion interaction term.
The third model simulates the full primary alkaline battery system, and accounts for the precipitation of zinc oxide within the separator (and other regions), and subsequent internal short circuit through this phase. It was found that an internal short circuit is created at the beginning of discharge, and this self-discharge may be exacerbated by discharging the cell intermittently. We find that using a thicker separator paper is a very effective way of minimising self-discharge behaviour.
The equations describing the three models are solved numerically in MATLABR, using three pieces of numerical simulation software. They provide a flexible and powerful set of primary alkaline battery discharge prediction tools, that leverage the simplified model framework, allowing them to be easily run on a desktop PC.
36
Johansen, Jonathan Frederick. "Mathematical modelling of primary alkaline batteries". Queensland University of Technology, 2007. http://eprints.qut.edu.au/16412/.
Pełny tekst źródłaStreszczenie:
Three mathematical models, two of primary alkaline battery cathode discharge, and one of primary alkaline battery discharge, are developed, presented, solved and investigated in this thesis. The primary aim of this work is to improve our understanding of the complex, interrelated and nonlinear processes that occur within primary alkaline batteries during discharge. We use perturbation techniques and Laplace transforms to analyse and simplify an existing model of primary alkaline battery cathode under galvanostatic discharge. The process highlights key phenomena, and removes those phenomena that have very little effect on discharge from the model. We find that electrolyte variation within Electrolytic Manganese Dioxide (EMD) particles is negligible, but proton diffusion within EMD crystals is important. The simplification process results in a significant reduction in the number of model equations, and greatly decreases the computational overhead of the numerical simulation software. In addition, the model results based on this simplified framework compare well with available experimental data. The second model of the primary alkaline battery cathode discharge simulates step potential electrochemical spectroscopy discharges, and is used to improve our understanding of the multi-reaction nature of the reduction of EMD. We find that a single-reaction framework is able to simulate multi-reaction behaviour through the use of a nonlinear ion-ion interaction term. The third model simulates the full primary alkaline battery system, and accounts for the precipitation of zinc oxide within the separator (and other regions), and subsequent internal short circuit through this phase. It was found that an internal short circuit is created at the beginning of discharge, and this self-discharge may be exacerbated by discharging the cell intermittently. We find that using a thicker separator paper is a very effective way of minimising self-discharge behaviour. The equations describing the three models are solved numerically in MATLABR, using three pieces of numerical simulation software. They provide a flexible and powerful set of primary alkaline battery discharge prediction tools, that leverage the simplified model framework, allowing them to be easily run on a desktop PC.
37
Dyhr, Benjamin Nicholas. "The Chordal Loewner Equation Driven by Brownian Motion with Linear Drift". Diss., The University of Arizona, 2009. http://hdl.handle.net/10150/195702.
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Schramm-Loewner evolution (SLE(kappa)) is an important contemporary tool for identifying critical scaling limits of two-dimensional statistical systems. The SLE(kappa) one-parameter family of processes can be viewed as a special case of a more general, two-parameter family of processes we denote SLE(kappa, mu). The SLE(kappa, mu) process is defined for kappa>0 and real numbers mu; it represents the solution of the chordal Loewner equations under special conditions on the driving function parameter which require that it is a Brownian motion with drift mu and variance kappa. We derive properties of this process by use of methods applied to SLE(kappa) and application of Girsanov's Theorem. In contrast to SLE(kappa), we identify stationary asymptotic behavior of SLE(kappa, mu). For kappa in (0,4] and mu > 0, we present a pathwise construction of a process with stationary temporal increments and stationary imaginary component and relate it to the limiting behavior of the SLE(kappa, mu) generating curve. Our main result is a spatial invariance property of this process achieved by defining a top-crossing probability for points in the upper half plane with respect to the generating curve.
38
Tsang, Suk-chong, i 曾淑莊. "A numerical study of coupled nonlinear Schrödinger equations arising in hydrodynamics and optics". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2003. http://hub.hku.hk/bib/B26652651.
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Shek, Cheuk-man Edmond, i 石焯文. "The continuous and discrete extended Korteweg-de Vries equations and their applications in hydrodynamics and lattice dynamics". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2006. http://hub.hku.hk/bib/B36925585.
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Persson, Leif. "Quasi-radial solutions of the p-harmonic equation in the plane and their stream functions". Licentiate thesis, Luleå tekniska universitet, 1988. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-25699.
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Liu, Rongsheng. "Global existence in L1 for the square-well kinetic equation". Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/40106.
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An attractive square-well is incorporated into the Enskog equation, in order to model the kinetic theory of a moderately dense gas with intermolecular potential. The existence of solutions to the Cauchy problem in L¹. global in time and for arbitrary initial data. is proved.
A simple derivation of the square-well kinetic equation is given. Lewis's method is used~ which starts from the Liouville equation of statistical mechanics. Then various symmetries of the collisional integrals are established. An H-theorem for entropy, mass, and momentum conservation is obtained, as well as an energy estimate, and key gain-loss estimates.
Approximate equations for the square-well kinetic equation are constructed that preserve symmetries of the collisional integral. Existence of nonnegative solutions of the approximate equations and weak compactness are obtained. The velocity averaging lemma of Golse is then a principal tool in demonstrating the convergence of the approximate solutions to a solution of the renormalized square well kinetic equation. The existence of weak solution of the initial value problem for the square well kinetic equation is thus proved.Ph. D.
42
Church, Kevin. "Applications of Impulsive Differential Equations to the Control of Malaria Outbreaks and Introduction to Impulse Extension Equations: a General Framework to Study the Validity of Ordinary Differential Equation Models with Discontinuities in State". Thesis, Université d'Ottawa / University of Ottawa, 2014. http://hdl.handle.net/10393/31874.
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Impulsive differential equations are often used in mathematical modelling to simplify
complicated hybrid models. We propose an inverse framework inspired by impulsive
differential equations, called impulse extension equations, which can be used as a tool to determine when these impulsive models are accurate. The linear theory is the
primary focus, for which theorems analoguous to ordinary and impulsive differential
equations are derived. Results explicitly connecting the stability of impulsive differential equations to related impulse extension equations are proven in what we call time scale consistency theorems. Opportunities for future research in this direction are discussed.
Following the work of Smith? and Hove-Musekwa on malaria vector control by
impulsive insecticide spraying, we propose a novel autonomous vector control scheme based on human disease incidence. Existence and stability of periodic orbits is established. We compare the implementation cost of the incidence-based control to a fixed-time spraying schedule. Hybrid control strategies are discussed.
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Liu, Xing. "Rigorous exponential asymptotics for a nonlinear third order difference equation". Connect to this title online, 2004. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1101927781.
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Thesis (Ph. D.)--Ohio State University, 2004.Title from first page of PDF file. Document formatted into pages; contains viii, 140 p.; also includes graphics. Includes bibliographical references (p. 139-140).
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Yang, Yang. "Two-dimensional dynamic analysis of functionally graded structures by using meshfree boundary-domain integral equation method". Thesis, University of Macau, 2015. http://umaclib3.umac.mo/record=b3335354.
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Mat, Isa Zaiton. "Mathematical modelling of fumigant transport in stored grain". Thesis, Queensland University of Technology, 2014. https://eprints.qut.edu.au/75420/1/Zaiton_Mat%20Isa_Thesis.pdf.
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Computational fluid dynamics, analytical solutions, and mathematical modelling approaches are used to gain insights into the distribution of fumigant gas within farm-scale, grain storage silos. Both fan-forced and tablet fumigation are considered in this work, which develops new models for use by researchers, primary producers and silo manufacturers to assist in the eradication grain storage pests.
46
Agaba, Peter. "Optimal Control Theory and Estimation of Parameters in a Differential Equation Model for Patients with Lupus". TopSCHOLAR®, 2019. https://digitalcommons.wku.edu/theses/3118.
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System Lupus Erythematosus (SLE) is a chronic inflammatory autoimmune disorder that affects many parts of the body including skin, joints, kidneys, brains and other organs. Lupus Nephritis (LN) is a disease caused by SLE. Given the complexity of LN, we establish an optimal treatment strategy based on a previously developed mathematical model.For our thesis work, the model variables are: Immune Complexes (I), Pro-inflammatory mediators (P), Damaged tissue (D), and Anti-inflammatory mediators (A). The analysis in this research project focuses on analyzing therapeutic strategies to control damage using both parameter estimation techniques (integration of data to quantify any uncertainties associated with parameters) and optimal control with the goal of minimizing time spent on therapy for treating damaged tissue by LN.
47
Akram, Sina. "Evaluation of the Effects of Vegetated Buffer Strips on Fate of Sediment with Mathematical Models". Thesis, Griffith University, 2014. http://hdl.handle.net/10072/366674.
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Several mathematical models have been developed in recent years to simulate hydrologic and sediment transport processed through vegetated buffer strips. Some of these models are empirically-based equations with limited applicability to conditions other than those under which they have been developed and the input data collected and used. There is a considerable amount of uncertainty in the application of existing process-based models as they consist of empirical sub-models in their structure. The inconsistency between the assumed section of erosion and deposition in and around the grass strip beds in the existing models is also a major limitation. Three independent predictive models are developed and presented in this thesis to fill in the gaps and to improve our predictive capabilities for hydrology and sediment transport across grass strips.
The first model developed in this study is a process-based one capable of simulating surface water profile in the upstream section and within grass strips. The data from the experiments carried out earlier in Griffith University’s GUTSR facilities were used to develop a relationship between the hydraulic roughness and flow characteristics and to develop the model itself. The gradually varied flow equation was used to apply the new relationship and to calculate the flow characteristics in the upstream section and within the grass strip bed.Thesis (PhD Doctorate)
Doctor of Philosophy (PhD)
Griffith School of Environment
Science, Environment, Engineering and Technology
Full Text
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Saleemi, Asima Parveen. "Finite Difference Methods for the Black-Scholes Equation". Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48660.
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Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.
49
Prieto, Moreno Kernel Enrique. "Novel mathematical techniques for structural inversion and image reconstruction in medical imaging governed by a transport equation". Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/novel-mathematical-techniques-for-structural-inversion-and-image-reconstruction-in-medical-imaging-governed-by-a-transport-equation(b45f5566-daa7-4d47-a982-cf479e360c6f).html.
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Since the inverse problem in Diffusive Optical Tomography (DOT) is nonlinear and severely ill-posed, only low resolution reconstructions are feasible when noise is added to the data nowadays. The purpose of this thesis is to improve image reconstruction in DOT of the main optical properties of tissues with some novel mathematical methods. We have used the Landweber (L) method, the Landweber-Kaczmarz (LK) method and its improved Loping-Landweber-Kaczmarz (L-LK) method combined with sparsity or with total variation regularizations for single and simultaneous image reconstructions of the absorption and scattering coefficients. The sparsity method assumes the existence of a sparse solution which has a simple description and is superposed onto a known background. The sparsity method is solved using a smooth gradient and a soft thresholding operator. Moreover, we have proposed an improved sparsity method. For the total variation reconstruction imaging, we have used the split Bregman method and the lagged diffusivity method. For the total variation method, we also have implemented a memory-efficient method to minimise the storage of large Hessian matrices. In addition, an individual and simultaneous contrast value reconstructions are presented using the level set (LS) method. Besides, the shape derivative of DOT based on the RTE is derived using shape sensitivity analysis, and some reconstructions for the absorption coefficient are presented using this shape derivative via the LS method.\\Whereas most of the approaches for solving the nonlinear problem of DOT make use of the diffusion approximation (DA) to the radiative transfer equation (RTE) to model the propagation of the light in tissue, the accuracy of the DA is not satisfactory in situations where the medium is not scattering dominant, in particular close to the light sources and to the boundary, as well as inside low-scattering or non-scattering regions. Therefore, we have solved the inverse problem in DOT by the more accurate time-dependant RTE in two dimensions.
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Ting, Lycretia Englang. "Sturm-Liouville theory". CSUSB ScholarWorks, 1996. https://scholarworks.lib.csusb.edu/etd-project/1206.
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