First-passage time of Duffing oscillator under combined harmonic and white-noise excitations

The first-passage time of Duffing oscillator under combined harmonic and white-noise excitations is studied. The equation of motion of the system is first reduced to a set of averaged Ito stochastic differential equations by using the stochastic averaging method. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. Numerical results for two resonant cases with several sets of parameter values are obtained and the analytical results are verified by using those from digital simulation.

A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).

THEORETICAL TREATMENT OF A3B5 CHLORIDE VAPOR-PHASE EPITAXY - GROWTH, DOPING, OPTIMIZATION

A theoretical description of chloride vapour-phase epitaxy (CVPE) has been proposed which contains two-dimensional (2D) gas-dynamic equations for transport of reactive components and kinetic equations for surface growth processes connected by nonlinear adiabatic boundary conditions. No one of these stages is supposed to be the limiting one. Calculated variations of growth rate and impurity concentrations along the growing layer fit experimental data well.

Coking kinetics on the catalyst during alkylation of fcc off-gas with benzene to ethylbenzene

The production of ethylbenzene from the alkylation of dilute ethylene in fee off-gases with benzene has been commercialized in China over a newly developed catalyst composed of ZSM-5/ZSM-11 co-crystallized zeolite. The duration of an operation cycle of the commercial catalyst could be as long as 180 days. The conversion of ethylene could attain higher than 95%, while the amount of coke deposited on the catalyst was only about 10 wt.%. Thermogravimetry (TG) was used to study the coking behavior of the catalyst during the alkylation of fee off-gas with benzene to ethylbenzene. Based on effects of reaction time, reaction temperature, reactants and products on coking during the alkylation process, it is found that the coking rate during the alkylation procedure follows the order: ethylbenzene > ethylene > propylene > benzene for single component, and benzene-ethylene > benzene-propylene for bi-components under the same reaction condition. Furthermore, the coking kinetic equations for benzene-ethylene, benzene-propylene and ethylbenzene were established. (C) 2003 Elsevier B.V. All rights reserved.

Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions

Gough, John, (2004) 'Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions', Communications in Mathematical Physics 254 pp.498-512 RAE2008

A population model of folate-mediated one-carbon metabolism.

BACKGROUND: Previous mathematical models for hepatic and tissue one-carbon metabolism have been combined and extended to include a blood plasma compartment. We use this model to study how the concentrations of metabolites that can be measured in the plasma are related to their respective intracellular concentrations. METHODS: The model consists of a set of ordinary differential equations, one for each metabolite in each compartment, and kinetic equations for metabolism and for transport between compartments. The model was validated by comparison to a variety of experimental data such as the methionine load test and variation in folate intake. We further extended this model by introducing random and systematic variation in enzyme activity. OUTCOMES AND CONCLUSIONS: A database of 10,000 virtual individuals was generated, each with a quantitatively different one-carbon metabolism. Our population has distributions of folate and homocysteine in the plasma and tissues that are similar to those found in the NHANES data. The model reproduces many other sets of clinical data. We show that tissue and plasma folate is highly correlated, but liver and plasma folate much less so. Oxidative stress increases the plasma S-adenosylmethionine/S-adenosylhomocysteine (SAM/SAH) ratio. We show that many relationships among variables are nonlinear and in many cases we provide explanations. Sampling of subpopulations produces dramatically different apparent associations among variables. The model can be used to simulate populations with polymorphisms in genes for folate metabolism and variations in dietary input.

Utilization of the three-dimensional volcano surface to understand the chemistry of multiphase systems in heterogeneous catalysis

CO hydrogenation is used as a model system to understand why multiphase catalysts are chemically important in heterogeneous catalysis. By including both adsorption and subsequent surface reactions, kinetic equations are derived with two fundamental properties, the chemisorption energies of C and O (Delta H-C and Delta H-O, respectively). By plotting the activity against Delta H-C and Delta H-O, a 3-D volcano surface is obtained. Because of the constraint between Delta H-C and Delta H-O on monophase systems, a maximum can be achieved. However, if multiphase systems are used, such a constraint can be released and the global maximum may be achieved.

PHOTOMINERALISATION OF 4-CHLOROPHENOL SENSITIZED BY TITANIUM-DIOXIDE - A STUDY OF THE INTERMEDIATES

The photomineralisation of 4-chlorophenol (4-CP) sensitised by Degussa P25 TiO2 in O2-saturated solution represents a possible standard test system in semiconductor-sensitised photomineralisation studies. As part of a detailed examination of this photosystem, the results of the temporal variations in the concentrations of 4-CP, CO2, Cl- and the major organic intermediates, namely, 4-chlorocatechol (4-CC), hydroquinone (HQ), benzoquinone and 4-chlororesorcinol, are reported. The observed variations in [4-CP], [4-CC], [HQ] and [CO2] fit those predicted by a kinetic model which utilises kinetic equations with a Langmuir-Hinshelwood form and assumes that there are three major possible routes in which the photogenerated hydroxyl radicals can react with 4-CP, ie. 4-CP --> 4-CC, 4-CP --> HQ and 4-CP --> (unstable intermediate) --> CO2 and that these routes have the following probabilities of occurring: 48%, 10% and 42%.

Sinkkirikasteen liuotusprosessin kineettinen mallinnus

Työssä on tehty kineettinen simulointimalli sinkkirikasteen liuotusprosessista. Prosessi on pieni osa Kokkolan sinkinvalmistusprosessia, jonka muita osia ovat: pasutus, neutraaliliuotus, konversio, liuospuhdistus ja elektrolyysi. Rikasteen liuotukseen tulee konversioprosessin liuos ja liuotuksesta lähtevä neste menee takaisin neutraaliliuotukseen. Saostunut jarosiitti läjitetään. Kokkolan liuotusprosessi koostuu liettoreaktorista ja kahdesta neljän liuotusreaktorin sarjasta. Liuotukseen syötetään paluuhappoa liettoreaktoriin ja liuotuspiirien ensimmäisiin liuotusreaktoreihin. Happea syötetään kaikkiin liuotusreaktoreihin. Prosessin mallintamiseen käytettiin Aspen Plus-simulointiohjelmaa, johon pystyttiin syöttämään kineettisiä yhtälöitä. Reaktionopeusyhtälöitä käytettiin raudan hapetuksen, sulfidien liuotuksen ja jarosiitiin saostumisen mallintamiseen, eli kaikkiin liuotusreaktoreissa tapahtuviin reaktioihin. Kineettiset yhtälöt etsittiin kirjallisuudesta. Liettoreaktori puolestaan mallinnettiin syöttämällä ohjelmaan reaktioyhtälöt ja antamalla niille etenemisasteet. Jarosiitin liukenemisesta työssä on tehty laboratoriokokeita, koska aiheesta ei kirjallisuudesta löytynyt kineettistä tietoa. Liuotuskokeissa käytetyn kiintoaineen kuitenkin todettiin sisältävän liikaa götiittiä, että tuloksista olisi voitu laskea kinetiikkaa jarosiitin liukenemiselle. Simulointimallilla laskettiin yksi tapaus vertailukohdaksi, johon malliin tehtyjä muutoksia verrattiin. Mallilla tutkittiin konversiosta tulevan jarosiitin määrän vaikutusta, reaktorikoon merkitystä ja rikasteen liuotuksen sekä jarosiitin saostuksen reaktionopeuksien muutoksen vaikutuksia. Käytetyillä kineettisillä yhtälöillä reaktioiden todettiin tarvitsevan vain ¾ käytetystä reaktiotilavuudesta, rikasteen liuotusnopeuden kohtalaisen pienellä hidastamisen todettiin vähentävän sinkin saantoa ja jarosiitin saostuksen reaktionopeuden kasvulla todettiin myös olevan negatiivinen vaikutus sinkin saantoon. Simulointimallissa käytettyjen reaktionopeusyhtälöiden varmentaminen kokeilla todettiin tarpeelliseksi, sillä jo kohtalaisen pienillä muutoksilla havaittiin olevan merkitystä prosessin toimivuuteen. Lisäksi todettiin jarosiitin liukenemisen huomioimisen olevan tarpeen.

Sampling Interval and estimated Betas : Implications for the Presence of Transitory Components in Stock Prices

We provide a theoretical framework to explain the empirical finding that the estimated betas are sensitive to the sampling interval even when using continuously compounded returns. We suppose that stock prices have both permanent and transitory components. The permanent component is a standard geometric Brownian motion while the transitory component is a stationary Ornstein-Uhlenbeck process. The discrete time representation of the beta depends on the sampling interval and two components labelled \"permanent and transitory betas\". We show that if no transitory component is present in stock prices, then no sampling interval effect occurs. However, the presence of a transitory component implies that the beta is an increasing (decreasing) function of the sampling interval for more (less) risky assets. In our framework, assets are labelled risky if their \"permanent beta\" is greater than their \"transitory beta\" and vice versa for less risky assets. Simulations show that our theoretical results provide good approximations for the means and standard deviations of estimated betas in small samples. Our results can be perceived as indirect evidence for the presence of a transitory component in stock prices, as proposed by Fama and French (1988) and Poterba and Summers (1988).

New simulation schemes for the Heston model

Les titres financiers sont souvent modélisés par des équations différentielles stochastiques (ÉDS). Ces équations peuvent décrire le comportement de l'actif, et aussi parfois certains paramètres du modèle. Par exemple, le modèle de Heston (1993), qui s'inscrit dans la catégorie des modèles à volatilité stochastique, décrit le comportement de l'actif et de la variance de ce dernier. Le modèle de Heston est très intéressant puisqu'il admet des formules semi-analytiques pour certains produits dérivés, ainsi qu'un certain réalisme. Cependant, la plupart des algorithmes de simulation pour ce modèle font face à quelques problèmes lorsque la condition de Feller (1951) n'est pas respectée. Dans ce mémoire, nous introduisons trois nouveaux algorithmes de simulation pour le modèle de Heston. Ces nouveaux algorithmes visent à accélérer le célèbre algorithme de Broadie et Kaya (2006); pour ce faire, nous utiliserons, entre autres, des méthodes de Monte Carlo par chaînes de Markov (MCMC) et des approximations. Dans le premier algorithme, nous modifions la seconde étape de la méthode de Broadie et Kaya afin de l'accélérer. Alors, au lieu d'utiliser la méthode de Newton du second ordre et l'approche d'inversion, nous utilisons l'algorithme de Metropolis-Hastings (voir Hastings (1970)). Le second algorithme est une amélioration du premier. Au lieu d'utiliser la vraie densité de la variance intégrée, nous utilisons l'approximation de Smith (2007). Cette amélioration diminue la dimension de l'équation caractéristique et accélère l'algorithme. Notre dernier algorithme n'est pas basé sur une méthode MCMC. Cependant, nous essayons toujours d'accélérer la seconde étape de la méthode de Broadie et Kaya (2006). Afin de réussir ceci, nous utilisons une variable aléatoire gamma dont les moments sont appariés à la vraie variable aléatoire de la variance intégrée par rapport au temps. Selon Stewart et al. (2007), il est possible d'approximer une convolution de variables aléatoires gamma (qui ressemble beaucoup à la représentation donnée par Glasserman et Kim (2008) si le pas de temps est petit) par une simple variable aléatoire gamma.

Problèmes de premier passage et de commande optimale pour des chaînes de Markov à temps discret.

Nous considérons des processus de diffusion, définis par des équations différentielles stochastiques, et puis nous nous intéressons à des problèmes de premier passage pour les chaînes de Markov en temps discret correspon- dant à ces processus de diffusion. Comme il est connu dans la littérature, ces chaînes convergent en loi vers la solution des équations différentielles stochas- tiques considérées. Notre contribution consiste à trouver des formules expli- cites pour la probabilité de premier passage et la durée de la partie pour ces chaînes de Markov à temps discret. Nous montrons aussi que les résultats ob- tenus convergent selon la métrique euclidienne (i.e topologie euclidienne) vers les quantités correspondantes pour les processus de diffusion. En dernier lieu, nous étudions un problème de commande optimale pour des chaînes de Markov en temps discret. L’objectif est de trouver la valeur qui mi- nimise l’espérance mathématique d’une certaine fonction de coût. Contraire- ment au cas continu, il n’existe pas de formule explicite pour cette valeur op- timale dans le cas discret. Ainsi, nous avons étudié dans cette thèse quelques cas particuliers pour lesquels nous avons trouvé cette valeur optimale.

Studies on the dynamics and rheology of dilute suspensions of Brownian particles in simple shear flow under the action of an external force

We present a novel approach to computing the orientation moments and rheological properties of a dilute suspension of spheroids in a simple shear flow at arbitrary Peclct number based on a generalised Langevin equation method. This method differs from the diffusion equation method which is commonly used to model similar systems in that the actual equations of motion for the orientations of the individual particles are used in the computations, instead of a solution of the diffusion equation of the system. It also differs from the method of 'Brownian dynamics simulations' in that the equations used for the simulations are deterministic differential equations even in the presence of noise, and not stochastic differential equations as in Brownian dynamics simulations. One advantage of the present approach over the Fokker-Planck equation formalism is that it employs a common strategy that can be applied across a wide range of shear and diffusion parameters. Also, since deterministic differential equations are easier to simulate than stochastic differential equations, the Langevin equation method presented in this work is more efficient and less computationally intensive than Brownian dynamics simulations.We derive the Langevin equations governing the orientations of the particles in the suspension and evolve a procedure for obtaining the equation of motion for any orientation moment. A computational technique is described for simulating the orientation moments dynamically from a set of time-averaged Langevin equations, which can be used to obtain the moments when the governing equations are harder to solve analytically. The results obtained using this method are in good agreement with those available in the literature.The above computational method is also used to investigate the effect of rotational Brownian motion on the rheology of the suspension under the action of an external force field. The force field is assumed to be either constant or periodic. In the case of con- I stant external fields earlier results in the literature are reproduced, while for the case of periodic forcing certain parametric regimes corresponding to weak Brownian diffusion are identified where the rheological parameters evolve chaotically and settle onto a low dimensional attractor. The response of the system to variations in the magnitude and orientation of the force field and strength of diffusion is also analyzed through numerical experiments. It is also demonstrated that the aperiodic behaviour exhibited by the system could not have been picked up by the diffusion equation approach as presently used in the literature.The main contributions of this work include the preparation of the basic framework for applying the Langevin method to standard flow problems, quantification of rotary Brownian effects by using the new method, the paired-moment scheme for computing the moments and its use in solving an otherwise intractable problem especially in the limit of small Brownian motion where the problem becomes singular, and a demonstration of how systems governed by a Fokker-Planck equation can be explored for possible chaotic behaviour.

Microstructural evolution of primary crystallization in diffusion-controlled grain growth

A model has been developed for evaluating grain size distributions in primary crystallizations where the grain growth is diffusion controlled. The body of the model is grounded in a recently presented mean-field integration of the nucleation and growth kinetic equations, modified conveniently in order to take into account a radius-dependent growth rate, as occurs in diffusion-controlled growth. The classical diffusion theory is considered, and a modification of this is proposed to take into account interference of the diffusion profiles between neighbor grains. The potentiality of the mean-field model to give detailed information on the grain size distribution and transformed volume fraction for transformations driven by nucleation and either interface- or diffusion-controlled growth processes is demonstrated. The model is evaluated for the primary crystallization of an amorphous alloy, giving an excellent agreement with experimental data. Grain size distributions are computed, and their properties are discussed.