Dissertations / Theses on the topic 'Free boundary model'
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Tambyah, Tamara. "A novel free boundary mathematical model of epithelial tissues with mechanobiological coupling." Thesis, Queensland University of Technology, 2020. https://eprints.qut.edu.au/205987/1/Tamara_Tambyah_Thesis.pdf.
Full textTeles, da Silva A. F. "Application of boundary integral methods to the study of steep free surface waves." Thesis, University of Bristol, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.279782.
Full textHeitzman, Michael Thomas Chicone Carmen Charles. "A free boundary gas dynamic model as a two-body field theory problem." Diss., Columbia, Mo. : University of Missouri--Columbia, 2009. http://hdl.handle.net/10355/7017.
Full textGlasner, Karl. "Multilayered Equilibria in a Density Functional Model of Copolymer-solvent Mixtures." SIAM PUBLICATIONS, 2017. http://hdl.handle.net/10150/624087.
Full textSidahmed, Abdelmgid Osman Mohammed. "Mesh free methods for differential models in financial mathematics." Thesis, University of the Western Cape, 2011. http://etd.uwc.ac.za/index.php?module=etd&action=viewtitle&id=gen8Srv25Nme4_3917_1319185202.
Full textWanntorp, Henrik. "Optimal Stopping and Model Robustness in Mathematical Finance." Doctoral thesis, Uppsala : Department of Mathematics, Uppsala University, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-9516.
Full textFormentin, Marco. "Two problems concerning interacting systems: 1. On the purity of the free boundary condition Potts measure on Galton-Watson trees 2. Uniform propagation of chaos in some spin-flip models." Doctoral thesis, Università degli studi di Padova, 2009. http://hdl.handle.net/11577/3426494.
Full textSommario: Un approccio rigoroso a questioni di Fisica Statistica spesso produce interessanti problemi matematici. Questa tesi di dottorato è composta da due parti. La prima non interseca la seconda, ma entrambe stanno sul confine tra Teoria della Probabilità e Meccanica Statistica. • La prima parte tratta il problema della ricostruzione per catene di Markov su alberi di tipo Galton-Watson. Miglioriamoi risultati precedentemente ottenuti da Mossel e Peres, sia per catene simmetriche che fortemente asimmetriche. Dimostriamo una condizione sufficiente della forma Q(d)c(M) < 1 per la non ricostruzione di catene diMarkov a q-stati sull’albero. Qui c(M) è una costante che dipende dalla matrice di transizione M e Q(d) è la media del numero di figli per vertice nell’albero di Galton-Watson. Questo risultato è equivalente alla purezza della misura libera di Gibbs. Quando consideriamo il caso del modello di Potts assumiamo anche questo punto di vista. Il teorema è valido anche per catene non reversibili. Nel caso del modello di Ising il nostro risultato produce la correta soglia di ricostruzione, nel caso di catene (fortemente) asimmetriche dove si sa che il bound di Kesten-Stigum non è esatto il metodo usato dà risultati numerici migliori. • Nella seconda parte diamo delle stime uniformi nel tempo per la propagazione del caos in alcuni modelli di spin con interazione a campo medio che presentano transizione di fase. Il primo è il modello dinamico di Curie-Weiss, che può essere considerato come il più semplice esempio di sistema con interazione a campo medio. Il secondo è un modello recentemente impiegato per spiegare i meccanismi del rischio di credito; esso descrive l’evoluzione temporale di indicatori finaziari per un gruppo di aziende interagenti quotate sul mercato. Anche se abbiamo trattato modelli specifici, crediamo che il metodo funzioni piuttosto in generale e che sia applicabile anche ad altre classi di modelli. Una limitazione sostanziale dei nostri risultati è che valgono solo nel caso sottocritico, che corrisponde, nel linguaggio della Meccanica Statistica, al regime di alta temperatura.
Bu, Tianren. "Option pricing under exponential jump diffusion processes." Thesis, University of Manchester, 2018. https://www.research.manchester.ac.uk/portal/en/theses/option-pricing-under-exponential-jump-diffusion-processes(0dab0630-b8f8-4ee8-8bf0-8cd0b9b9afc0).html.
Full textJiang, An. "American Spread Option Pricing with Stochastic Interest Rate." BYU ScholarsArchive, 2016. https://scholarsarchive.byu.edu/etd/5987.
Full textCoelho, Afonso Valente Ricardo de Seabra. "American options and the Black-Scholes Model." Master's thesis, Instituto Superior de Economia e Gestão, 2020. http://hdl.handle.net/10400.5/20735.
Full textOs problemas de apreçamento de opções têm sido um dos principais assuntos de em Matemática Financeira, desde a criação desse conceito nos anos 70. Mais especificamente, as opções americanas são de grande interesse nesta área do conhecimento porque são matematicamente muito mais complexas do que as opções europeias padrão e o modelo de Black-Scholes não fornece, na maioria dos casos, uma fórmula explícita para a determinação do preço deste tipo de opções. Nesta dissertação, mostramos como o estudo de opções americanas conduz à análise de problemas de fronteira livre devido à possibilidade de exercício antecipado, onde nosso principal objetivo é encontrar o preço de exercício ótimo. Também apresentamos a reformulação do problema em termos de um problema de complementaridade linear e de desigualdade variacional parabólica. Além disso, também abordamos a caracterização probabilística das opções americanas com base no conceito de tempos de paragem ótima. Essas formulações, aqui tratadas em termos analíticos ou probabilísticos, podem ser muito úteis na aplicação de métodos numéricos ao problema de precificação de opções do estilo americano, uma vez que, na maioria dos casos, é quase impossível encontrar soluções explícitas. Além disso, utilizamos o Método da Árvore Binomial, que é um método numérico muito simples do ponto de vista matemático, para ilustrar alguns aspectos da teoria estudada ao longo desta tese e para comparar as opções americanas com as opções europeias e bermudas, por meio de alguns exemplos numéricos.
Option pricing problems have been one of the main focuses in the field of Mathematical Finance since the creation of this concept in the 1970s. More specifically, American options are of great interest in this area of knowledge because they are much more complex mathematically than the standard European options and the Black-Scholes model cannot give an explicit formula to value this style options in most cases. In this dissertation, we show how pricing American options leads to free boundary problems because of the possibility of early exercise, where our main goal is to find the optimal exercise price. We also present how to reformulate the problem into a linear complementarity problem and a parabolic variational inequality. Moreover, we also address the probabilistic characterization of American options based on the concept of stopping times. These formulations, here viewed from the analytical and probabilistic point of view, can be very useful for applying numerical methods to the problem of pricing American style options since, in most cases, it is almost impossible to find explicit solutions. Furthermore, we use the Binomial Tree Method, which is a very simple numerical method from the mathematical point of view, to illustrate some aspects of the theory studied throughout this thesis and to compare American options with European and Bermudan Options, by means of a few numerical examples.
info:eu-repo/semantics/publishedVersion
Cummings, Linda Jane. "Free boundary models in viscous flow." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.339364.
Full textXin, W. (Weidong). "Continuum electrostatics of biomolecular systems." Doctoral thesis, University of Oulu, 2008. http://urn.fi/urn:isbn:9789514287602.
Full textKiss, Andrew Elek, and Andrew Kiss@anu edu au. "Dynamics of laboratory models of the wind-driven ocean circulation." The Australian National University. Research School of Earth Sciences, 2001. http://thesis.anu.edu.au./public/adt-ANU20011018.115707.
Full textHipp, Hans Christoph 1959. "Numerical investigation of mode interaction in free shear layers." Thesis, The University of Arizona, 1988. http://hdl.handle.net/10150/276871.
Full textLopez, Gerald Gabriel. "The impact of interconnect process variations and size effects for gigascale integration." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/31781.
Full textCommittee Chair: Jeffrey A. Davis; Committee Co-Chair: James D. Meindl; Committee Member: Azad J. Naeemi; Committee Member: Dennis W. Hess; Committee Member: George F. Riley; Committee Member: Linda S. Milor. Part of the SMARTech Electronic Thesis and Dissertation Collection.
Ashour, Osama Naim. "Receptivity to free stream acoustic disturbances due to a roughness element on a flat plate." Thesis, This resource online, 1993. http://scholar.lib.vt.edu/theses/available/etd-09052009-040628/.
Full textJoubert, Dominique. "Numerical methods for pricing American put options under stochastic volatility / Dominique Joubert." Thesis, North-West University, 2013. http://hdl.handle.net/10394/10202.
Full textMSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013
Muthedath, Premkumar. "Numerical study of nonlinear free-surface flows." Thesis, This resource online, 1992. http://scholar.lib.vt.edu/theses/available/etd-07212009-040300/.
Full textJuszczuk, Agnieszka Beata, and Evgeniya Tkacheva. "Revision Moment for the Retail Decision-Making System." Thesis, Högskolan i Halmstad, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-6191.
Full textAlamichel, Claire. "Study of the influence of the cell nucleus on cell motility." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM023.
Full textCell motility is a phenomenon involved in many biological processes such as cancer propagation, immune response, wound healing and embryonic development. This phenomenon is ensured by a cell's ability to deform from a symmetrical, non-polarised configuration to an asymmetrical, polarised configuration, and to maintain this asymmetrical configuration. In this thesis, we focus on the role of the nucleus in this phenomenon. To this end, a free boundary model in dimension 2 is introduced. The cell is modelled by an incompressible fluid with a rigid structure modelling the nucleus. Polarity markers are present in the fluid and these actively transduce the forces applied by and on the cytoskeleton. The proposed model also models the effect of undercooling and the external environment on cell motility. The different components of the model are studied separately in the thesis.We are studying the influence of the nucleus on cell motility. We consider various models, including a deformable model, a rigid model and two toy models. We prove that these models admit stationary states. Using linear stability analysis, we demonstrate that there exists a threshold above which the radially symmetric stationary state is unstable. For each of these models, a finite element numerical scheme is developed. The numerical results obtained highlight the link between the position of the nucleus in the cell and the polarisation of the cell. They are in qualitative agreement with biological observations. Trajectory analysis is also carried out. An analogous model in dimension 1, which is a non-local and non-linear Fokker-Planck equation, is introduced. We show that it is well posed. The existence of stationary states and travelling waves is also studied. A second model in dimension 1 based on a second modelling is studied.The effect of undercooling on cell motility is studied. By proving the existence of travelling waves and stationary states, we illustrate that undercooling has a stabilising effect. The existence of travelling waves is proved using a bifurcation theorem. We also demonstrate, using linear stability analysis, that there exists a threshold above which the stationary state is unstable.The influence of the presence of attractive external signals on cell motility is also investigated. We prove the existence of a stationary state and the existence of a range of parameters for which it is stable and another for which it is unstable. Numerically, we illustrate that there is competition between the forces induced by the polarity markers and those induced by the external signal.This thesis also includes a work carried out during the CEMRACS 2022 summer school. A model of one of the mechanisms taking place at the membrane—endocytosis—is presented. After studying the model, a finite volume scheme is presented. This provides results that are consistent with biological results
Bouselmi, Aych. "Options américaines et processus de Lévy." Phd thesis, Université Paris-Est, 2013. http://tel.archives-ouvertes.fr/tel-00944239.
Full textDallaston, Michael C. "Mathematical models of bubble evolution in a Hele-Shaw Cell." Thesis, Queensland University of Technology, 2013. https://eprints.qut.edu.au/63701/1/Michael_Dallaston_Thesis.pdf.
Full textAltaie, Huda. "Nouvelle technique de grilles imbriquées pour les équations de Saint-Venant 2D." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4220/document.
Full textMost flows in the rivers, seas, and ocean are shallow water flow in which the horizontal length andvelocity scales are much larger than the vertical ones. The mathematical formulation of these flows, so called shallow water equations (SWEs). These equations are a system of hyperbolic partial differentialequations and they are effective for many physical phenomena in the oceans, coastal regions, riversand canals. This thesis focuses on the design of a new two-way interaction technique for multiple nested grids 2DSWEs using the numerical methods. The first part of this thesis includes, proposing several ways to develop the derivation of shallow water model. The complete derivation of this system from Navier-Stokes equations is explained. Studying the development and evaluation of numerical methods by suggesting new spatial and temporal discretization techniques in a standard C-grid using an explicit finite difference method in space and leapfrog with Robert-Asselin filter in time which are effective for modeling in oceanic and atmospheric flows. Several numerical examples for this model using Gaussian level initial condition are implemented in order to validate the efficiency of the proposed method. In the second part of our work, we are interested to propose a new two-way interaction technique for multiple nested grids to solve ocean models using four choices of higher restriction operators (update schemes) for the free surface elevation and velocities with high accuracy results. Our work focused on the numerical resolution of SWEs by nested grids. At each level of resolution, we used explicit finite differences methods on Arakawa C-grid. In order to be able to refine the calculations in troubled regions and move them into quiet areas, we have considered several levels of resolution using nested grids. This makes it possible to considerably increase the performance ratio of the method, provided that the interactions (spatial and temporal) between the grids are effectively controlled. In the third part of this thesis, several numerical examples are tested to show and verify twoway interaction technique for multiple nested grids of shallow water models can works efficiently over different periods of time with nesting 3:1 and 5:1 at multiple levels. Some examples for multiple nested grids of the tsunami model with nesting 5:1 using moving boundary conditions are tested in the fourth part of this work
Kimmerle, Sven-Joachim. "Macroscopic diffusion models for precipitation in crystalline gallium arsenide." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2009. http://dx.doi.org/10.18452/16060.
Full textBased on a thermodynamically consistent model for precipitation in gallium arsenide crystals including surface tension and bulk stresses by Dreyer and Duderstadt, we propose two different mathematical models to describe the size evolution of liquid droplets in a crystalline solid. The first model treats the diffusion-controlled regime of interface motion, while the second model is concerned with the interface-controlled regime of interface motion. Our models take care of conservation of mass and substance. These models generalise the well-known Mullins-Sekerka model for Ostwald ripening. We concentrate on arsenic-rich liquid spherical droplets in a gallium arsenide crystal. Droplets can shrink or grow with time but the centres of droplets remain fixed. The liquid is assumed to be homogeneous in space. Due to different scales for typical distances between droplets and typical radii of liquid droplets we can derive formally so-called mean field models. For a model in the diffusion-controlled regime we prove this limit by homogenisation techniques under plausible assumptions. These mean field models generalise the Lifshitz-Slyozov-Wagner model, which can be derived from the Mullins-Sekerka model rigorously, see Niethammer et al., and is well-understood. Mean field models capture the main properties of our system and are well adapted for numerics and further analysis. We determine possible equilibria and discuss their stability. Numerical evidence suggests in which case which one of the two regimes might be appropriate to the experimental situation.
David, Noemi. "Incompressible limit and well-posedness of PDE models of tissue growth." Electronic Thesis or Diss., Sorbonne université, 2022. https://accesdistant.sorbonne-universite.fr/login?url=https://theses-intra.sorbonne-universite.fr/2022SORUS235.pdf.
Full textBoth compressible and incompressible porous medium models have been used in the literature to describe the mechanical aspects of living tissues, and in particular of tumor growth. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems of Hele-Shaw type where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. Then, a coupled system of equations describes the cell density and the nutrient concentration. For this reason, the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, also recently applied to related problems, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state
Marth, Wieland. "Hydrodynamic Diffuse Interface Models for Cell Morphology and Motility." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-204651.
Full textDiese Dissertation beschäftigt sich mit mathematischen Modellen zur Beschreibung von Gleichgewichts- und dynamischen Zuständen von verallgemeinerten biologischen Zellen. Die Zellen werden dabei als thermodynamisches System aufgefasst, bei dem Strömungseffekte innerhalb und außerhalb der Zelle zusammen mit einem Helfrich-Modell für Zellmembranen kombiniert werden. Schließlich werden durch einen Energie-Variations-Ansatz die Evolutionsgleichungen für die Zelle hergeleitet. Es ergeben sie dabei Mehrphasen-Systeme, die Strömungseffekte mit einem freien Randwertproblem, das zusätzlich physikalischen Einflüssen wie Biegung und Oberflächenspannung unterliegt, vereinen. Um solche Probleme effizient zu lösen, wird in dieser Arbeit die Diffuse-Interface-Methode verwendet. Ein Vorteil dieser Methode ist, dass es sehr einfach möglich ist, Modelle, die verschiedenste Prozesse beschreiben, miteinander zu vereinen. Dies erlaubt es, komplexe biologische Phänomene, wie zum Beispiel Zellmotilität oder auch die kollektive Bewegung von Zellen, zu beschreiben. In den Modellen für Zellmotilität wird ein biologisches Netzwerk-Modell für GTPasen oder auch ein Active-Polar-Gel-Modell, das die Aktinfilamente im Inneren der Zellen als Flüssigkristall auffasst, mit dem Multi-Phasen-Modell kombiniert. Beide Modelle erlauben es, komplexe Vorgänge bei der selbst hervorgerufenen Bewegung von Zellen, wie das Vorantreiben der Zellmembran durch Aktinpolymerisierung oder auch die Kontraktionsbewegung des Zellkörpers durch kontraktile Spannungen innerhalb des Zytoskelets der Zelle, zu verstehen. Weiterhin ist die kollektive Bewegung von vielen Zellen von großem Interesse, da sich hier viele nichtlineare Phänomene zeigen. Um das Diffuse-Interface-Modell für eine Zelle auf die Beschreibung mehrerer Zellen zu übertragen, werden mehrere Phasenfelder eingeführt, die die Zellen jeweils kennzeichnen. Schließlich werden die Zellen durch ein lokales Abstoßungspotential gekoppelt. Das Modell wird angewendet, um White blood cell margination, das die Annäherung von Leukozyten an die Blutgefäßwand bezeichnet, zu verstehen. Dieser Prozess wird dabei bestimmt durch den komplexen Zusammenhang zwischen Kollisionen, den jeweiligen mechanischen Eigenschaften der Zellen, sowie deren Auftriebskraft innerhalb der Adern. Die Simulationen zeigen, dass diese Annäherung sich in bestimmten Gebieten des kardiovaskulären Systems stark vermindert, in denen die Blutströmung das Stokes-Regime verlässt. Schließlich wird das Active-Polar-Gel-Modell mit dem Modell für die kollektive Bewegung vom Zellen kombiniert. Dies macht es möglich, die kollektive Bewegung der Zellen und den Einfluss von Hydrodynamik auf diese Bewegung zu untersuchen. Es zeigt sich dabei, dass der Zustand der kollektiven gerichteten Bewegung sich spontan aus der Neuausrichtung der jeweiligen Zellen durch inelastische Kollisionen ergibt. Obwohl die Hydrodynamik einen großen Einfluss auf solche Systeme hat, deuten die Simulationen nicht daraufhin, dass Hydrodynamik die kollektive Bewegung vollständig unterdrückt. Weiterhin wird in dieser Arbeit gezeigt, wie die stark gekoppelten Systeme numerisch gelöst werden können mit Hilfe der Finiten-Elemente-Methode und wie die Effizienz der Methode gesteigert werden kann durch die Anwendung von Operator-Splitting-Techniken und Problemparallelisierung mittels OPENMP
Kome, Melvin Njumbe. "Well testing in gas hydrate reservoirs." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2015. http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-160567.
Full text"A Two Strain Spatiotemporal Mathematical Model of Cancer with Free Boundary Condition." Doctoral diss., 2014. http://hdl.handle.net/2286/R.I.25882.
Full textDissertation/Thesis
Doctoral Dissertation Applied Mathematics 2014
Vinayan, Vimal. "A Boundary Element Method for the strongly nonlinear analysis of ventilating water-entry and wave-body interaction problems." 2009. http://hdl.handle.net/2152/14765.
Full texttext
"Experimental investigation of turbulent thermal convection with slip-free boundary conditions." 2010. http://library.cuhk.edu.hk/record=b5894454.
Full text"September 2010."
Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.
Includes bibliographical references (leaves 52-57).
Abstracts in English and Chinese.
Zhao, Xiaozheng = Hua yi bian jie tiao jian xia tuan liu re dui liu de shi yan yan jiu / Zhao Xiaozheng.
Abstract --- p.i
摘要 --- p.ii
Acknowledgement --- p.iv
Contains --- p.iv
List of Figures --- p.vii
List of Tables --- p.xi
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Turbulence --- p.1
Chapter 1.2 --- Turbulent Rayleigh-Benard Convection --- p.2
Chapter 1.2.1 --- Physical Picture --- p.2
Chapter 1.2.2 --- Governing Equations and Characteristic Parameters --- p.5
Chapter 1.2.3 --- Nu Scaling --- p.7
Chapter 1.2.4 --- Boundary Layer --- p.8
Chapter 1.3 --- Motivations of the Present Work --- p.10
Chapter 2 --- Experimental Setup --- p.13
Chapter 2.1 --- The Convection Cell --- p.13
Chapter 2.2 --- Temperature Probe and Translation Stage --- p.15
Chapter 2.3 --- Calibration of the Thermistors --- p.17
Chapter 2.4 --- Data Acquisition Units --- p.18
Chapter 2.5 --- The Working Fluids --- p.19
Chapter 2.6 --- Heat Leakage Prevention --- p.21
Chapter 3 --- Heat Transfer and Thermal Boundary Layer Measurement --- p.23
Chapter 3.1 --- The Setup and Experimental Procedure --- p.23
Chapter 3.2 --- The Mean Temperature and Temperature Fluctuation Profiles across the Interfaces --- p.24
Chapter 3.2.1 --- Profiles across the Water-FC77 Interface --- p.24
Chapter 3.2.2 --- Profiles across the FC77-Mercury Interface --- p.27
Chapter 3.3 --- Nu Results --- p.29
Chapter 3.3.1 --- Results Obtained with Assumption of Pure Conduction --- p.30
Chapter 3.3.2 --- Results from Mean Temperature Profile --- p.32
Chapter 3.3.3 --- Comparison of the Two Methods --- p.33
Chapter 3.4 --- Boundary Layer Thickness --- p.37
Chapter 3.5 --- Summary --- p.39
Chapter 4 --- Influence of Flow in the Water (Mercury) Layer on the FC77 Layer --- p.41
Chapter 4.1 --- Experimental Setup --- p.41
Chapter 4.2 --- Main Results --- p.42
Chapter 4.3 --- Probability Density Function and Temperature Oscillation --- p.44
Chapter 4.4 --- Summary --- p.50
Chapter 5 --- Conclusions and Perspective --- p.51
Chapter 5.1 --- Conclusions --- p.51
Chapter 5.2 --- Perspective for Future Work --- p.52
"Analytic approximations to the free boundary and multi-dimensional problems in financial derivatives pricing." 2014. http://repository.lib.cuhk.edu.hk/en/item/cuhk-1291256.
Full textFor each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function.
For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark prices obtained by numerical integration or Monte Carlo simulation. By exploiting an explicit relationship between the option price and the underlying probability distribution, we further derive an approximate distribution function for the general basket-spread variable. It can be used to approximate the transition probability distribution of any linear combination of correlated GBMs. Finally, an implicit perturbation is applied to reduce the pricing errors by factors of up to 100. When compared against the existing methods, the basket-spread option formula coupled with the implicit perturbation turns out to be one of the most robust and accurate approximation methods.
本論文為金融衍生產品定價的兩類問題作出了研究。第一類是自由邊界問題,它可以制定一個受制於自由邊界條件的偏微分方程式(PDE),雖然當中自由邊界條件的函數形式是已知的,但自由邊界的位置是未知的,只能通過為實際解施加連續性條件作隱式確定。這裡為兩個具體問題進行了研究,分別是固定利率按揭合約(fixed-rate mortgages)定價和方差恆彈性模型的美式期權(CEV American options)定價。第二類是多維問題,它涉及到多個相關隨機變量及他們引申出的多維PDE。這裡為一個典型例子進行了研究,稱為籃子差異期權(basket-spread options),其基礎資產價格由相關的幾何布朗運動驅動。我們為這三個問題提出了解析近似解。
對於上述的自由邊界問題,我們提出了一項參數移動邊界來近似模仿未知的自由邊界,使原來的自由邊界問題轉化為移動邊界問題,從而提出一種解析近似解。控制移動邊界的參數是通過滿足近似解的一階導數連續性條件來定。得到了解析近似解令當中的衍生產品定價和避險參數能有效快速地計算出,相比於有限差分法(finite-difference method),精度保持了但計算時間顯著降低。再透過應用一個多階段方案,將移動邊界重鑄成一項分段光滑的連續函數,能有系統地將近似解的結果逼近有限差分法的結果。
對於上述的多維問題,我們從Kirk(1995)的二維差異期權(spread option)近似解定價公式推廣到多維的籃子差異期權。由於最終的定價公式是封閉形式,所有避險參數也從而得到封閉式近似解。從一些模擬例子顯示出,近似解的定價和避險參數,與通過數值積分法(numerical integration)或蒙地卡羅模擬法(Monte Carlo simulation)獲得的基準值比較,只有小於百分之一的誤差。此外,透過利用一種期權價格和相關基礎變量的概率分佈關係,我們進一步推論出一項籃子差異變量的近似解分佈函數,這可應用到任何多維幾何布朗運動的線性組合變量分佈。最後,我們提出一種隱式攝動方法,把定價誤差減少高達一百倍,跟現有的近似解定價方法相比,這是其中一種最健全和準確的籃子差異期權定價方法。
Lau, Chun Sing = 自由邊界和多維的金融衍生產品定價問題 : 解析近似解 / 劉振聲.
Thesis Ph.D. Chinese University of Hong Kong 2014.
Includes bibliographical references (leaves 174-186).
Abstracts also in Chinese.
Title from PDF title page (viewed on 12, September, 2016).
Lau, Chun Sing = Zi you bian jie he duo wei de jin rong yan sheng chan pin ding jia wen ti : jie xi jin si jie / Liu Zhensheng.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
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