In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model.

Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Bismuth ferrite (BFO) is a multiferroic material with cross-correlation between magnetic and electric orders. With no applied external fields the spin structure of BFO is anitferromagnetic and cycloidal. This ordering prevents the detection of the weak ferromagnetism known to exist in the material. The application of magnetic and electric fields of suitable strength and direction is capable of compelling the Fe3+ spins to align in a homogeneous, antiferromagnetic fashion. This report details how numerical methods were used to simulate the spin alignment of a BFO system under different fields. The results were compiled into electric field-magnetic field phase diagrams of BFO to show the divide between cycloidal and homogeneous systems. / Graduate / 0607 / 0611 / marca@uvic.ca

A robust computational framework is presented for the risk-neutral valuation of capital guarantees written on discretely-reallocated portfolios following the Constant Proportion Portfolio Insurance (CPPI) strategy. Aiming to address the (arguably more realistic) cases where analytical results are unavailable, this framework accommodates risky-asset jumps, volatility surfaces, borrowing restrictions, nonuniform reallocation schedules and autonomous CPPI floor trajectories. The two-asset state space representation developed herein facilitates visualising the CPPI strategy, which in turn provides insight into grid design and interpolation. It is demonstrated that given a deterministic process for the risk-free rate, the pricing problem can be cast as solving cascading systems of 1D partial integro-differential equations (PIDEs). This formulation’s stability and monotonicity are studied. In addition to making more sense financially, the limited borrowing variant of the CPPI strategy is found to be better suited than the classical (unlimited borrowing) counterpart for bounded-domain calculations. Consequently, it is demonstrated how the unlimited borrowing problem can be approximated by imposing an artificial borrowing limit. For implementation validation, analytical solutions to special cases are derived. Numerical tests are presented to demonstrate the versatility of this framework.

There is often freedom in choosing the location of actuators on systems governed by partial differential equations. The actuator locations should be selected in order to optimize the performance criterion of interest. The main focus of this thesis is to consider H-∞-performance with state-feedback. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output of a full-information plant. Optimal H-∞-disturbance attenuation as a function of actuator location is used as the cost function. It is shown that the corresponding actuator location problem is well-posed. In practice, approximations are used to determine the optimal actuator location. Conditions for the convergence of optimal performance and the corresponding actuator location to the exact performance and location are provided. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied. Systems of large model order arise in a number of situations; including approximation of partial differential equation models and power systems. The system descriptions are sparse when given in descriptor form but not when converted to standard first-order form. Numerical calculation of H-∞-attenuation involves iteratively solving large H-∞-algebraic Riccati equations (H-∞-AREs) given in the descriptor form. An iterative algorithm that preserves the sparsity of the system description to calculate the solutions of large H-∞-AREs is proposed. It is shown that the performance of our proposed algorithm is similar to a Schur method in many cases. However, on several examples, our algorithm is both faster and more accurate than other methods. The calculation of H-∞-optimal actuator locations is an additional layer of optimization over the calculation of optimal attenuation. An optimization algorithm to calculate H-∞-optimal actuator locations using a derivative-free method is proposed. The results are illustrated using several examples motivated by partial differential equation models that arise in control of vibration and diffusion.

No description available.

In this research, a new technique for modeling periodic structures in the finite-difference time-domain (FDTD) method is developed, and the technique is applied to the study of structural color in insects. Various recent supplements to the FDTD method, such as a nearly-perfect plane-wave injector and convolutional perfectly matched layer boundary condition, are used. A method for implementing the FDTD method on a parallel, distributed-memory computer cluster is given. To model a periodic structure, a single periodic cell is terminated by periodic boundary conditions (PBCs). A new technique for incorporating PBCs into the FDTD method is presented. The simplest version of the technique is limited to two-dimensional, singly-periodic geometries. The accuracy is demonstrated by comparing to independent results calculated with a frequency-domain, mode-matching method. The periodic FDTD method is then extended to the more general case of three-dimensional, doubly-periodic problems. This extension requires additional steps and imposes new limitations. The computational cost and limitations of the method are presented. Certain species of butterflies exhibit structural color, which is caused by quasi-periodic structures on the scales covering the wings. Numerical experiments are performed to develop a technique for modeling quasi-periodic structures using the periodic FDTD method. The observed structural color of butterflies is then calculated from the electromagnetic data using colorimetric theory. Three types of butterflies are considered. The first type are from the Morpho genus. These are typically a brilliant, almost metallic, blue color. The second type is the Troides magellanus, which exhibits an interplay of structural and pigmentary color, but the structural color is only visible near grazing incidence. The final type is the Ancyluris meliboeus, which exhibits iridescence on the ventral side. For all cases, the effects of changing the dimensions of various structural elements are considered. Finally, some earlier work on the design of TEM horn antennas is presented. The TEM horn is a simple and popular antenna, but only limited design information is available in the literature. A parametric study was performed, and the results are given. A complete derivation of the characteristic impedance of the basic antenna is also presented.

Κύριο αντικείμενο μελέτης της παρούσας μεταπτυχιακής διπλωματικής εργασίας είναι οι περιστρεφόμενοι αστέρες νετρονίων. Λόγω του ότι οι κλασικές διαταρακτικές μέθοδοι που εφαρμόζονται για την εύρεση της ακτίνας ενός περιστρεφόμενου πολυτροπικού μοντέλου περιορίζονται, από την επιφάνεια του αστέρα, αναπτύξαμε μία μέθοδο για τον υπολογισμό ποσοτήτων πέραν αυτού του ορίου. Αυτή η γενικευμένη μέθοδος χρησιμοποιεί τις μετανευτώνειες παραμέτρους ως όρους διαταραχής. Υλοποιώντας έναν κώδικα σε γλώσσα προγραμματισμού Fortran, υπολογίσαμε εκτεταμένους πίνακες ποσοτήτων και σταθερών. Μέσω της γενικευμένης αυτής μεθόδου επιτυγχάνεται η εύρεση της ακριβούς τιμής της ακτίνας ενός τέτοιου μοντέλου καθώς και ο καθορισμός της κρίσιμης παραμέτρου περιστροφής, η οποία αποτελεί μία μετανευτώνεια παράμετρο. Ο υπολογισμός της κρίσιμης παραμέτρου διαταραχής επιτυγχάνεται με ευκολία, κυρίως εκ του λόγου ότι η μέθοδος έχει υπολογίσει εκτεταμένους πίνακες συναρτησιακών τιμών. Οι υπολογιζόμενες κρίσιμες παράμετροι διαταραχής είναι μεγαλύτερες των αντιστοίχων τιμών της βιβλιογραφίας (κυρίως σε σύγκριση με τους Fahlman-Anand [55]), και φαίνεται να συμφωνούν καλύτερα με τις τιμές που υπολογίζονται από τις λεγόμενες επαναληπτικές μεθόδους. Τα αποτελέσματα επαληθεύουν με μεγάλη ακρίβεια τιμές συναρτήσεων και παραμέτρων σε σύγκριση με την κλασική βιβλιογραφία. Η παρούσα εργασία χωρίζεται σε πέντε μέρη, τα οποία αναπτύσσονται στα κεφάλαια 1, 2, 3, 4 και 5. Στο πρώτο κεφάλαιο, περιγράφεται ο αστέρας νετρονίων ως αστροφυσικό αντικείμενο. Δίνεται βάρος τόσο στη δομή του όσο και στα φυσικά χαρακτηριστικά του. Η ύπαρξη των αστέρων νετρονίων είναι απόλυτα συνδεδεμένη με τους πάλσαρς. Αυτοί αποτελούν ένα «ζωντανό» παράδειγμα περιστρεφομένων αστέρων νετρονίων; έτσι, γίνεται αναφορά στις φυσικές ιδιότητες και στις διεργασίες που πραγματοποιούνται σε αυτούς. Στο δεύτερο κεφάλαιο, αναφερόμαστε στις καταστατικές εξισώσεις που διέπουν το εσωτερικό των αστέρων νετρονίων, και στην έννοια του πολυτρόπου. Αφενός μεν, διότι δεν μπορεί να πραγματοποιηθεί μία μελέτη για αυτούς τους αστέρες χωρίς να υιοθετηθεί κάποια καταστατική εξίσωση, αφετέρου δε διότι μία από τις πλέον ενδεικτικές για την περιγραφή τους (και την οποία εμείς υιοθετούμε) είναι αυτή του πολυτρόπου. Επιπλέον, αναλύουμε τις εξισώσεις που διέπουν το αδιατάρακτο πολυτροπικό μοντέλο, όπως και αυτές που περιγράφουν το αντίστοιχο διαταραγμένο, σύμφωνα με τη θεωρία που ανέπτυξε ο Chandrasekhar. Στο τρίτο κεφάλαιο, χρησιμοποιούμε τη Γενική Θεωρία της Σχετικότητας στη μελέτη του πολυτροπικού μοντέλου, εστιάζοντας κυρίως στον τρόπο με τον οποίο τροποποιείται η κλασική θεώρηση, στο πώς μετασχηματίζονται η βασικές ποσότητες του μοντέλου, και στο πώς προκύπτουν οι σχέσεις της μετανευτώνειας προσέγγισης. Εξάγουμε τις μετανευτώνειες εξισώσεις της υδροδυναμικής και αναπτύσσουμε το διαταρακτικό μοντέλο επίλυσης, από το οποίο προκύπτουν οι εξισώσεις που επιλύουμε αριθμητικά. Στο τέταρτο κεφάλαιο, κάνουμε αναφορά στις διάφορες αριθμητικές μεθόδους που έχουν αναπτυχθεί για την μελέτη του σχετικιστικά περιστρεφόμενου πολυτροπικού μοντέλου. Στο πέμπτο κεφάλαιο, παρουσιάζουμε πίνακες αποτελεσμάτων και ενδιαφέρουσες γραφικές παραστάσεις. Δίνουμε επίσης ορισμένες αλγοριθμικές λεπτομέρειες για το πρόγραμμά μας. Συγκεκριμένα, γενικεύουμε τη μέθοδο των μετανευτωνείων προσεγγίσεων και αναλύουμε τα πλεονεκτήματα της. Ακολούθως, παραθέτουμε μία περιγραφή της αριθμητικής διαπραγμάτευσης της μεθόδου και την πορεία υλοποίησής της. Τέλος, παρατίθενται οι πίνακες των αποτελεσμάτων και τα τελικά συμπεράσματα. / In the present Thesis, we study rotating neutron stars. Due to the fact that the classical perturbation methods implemented to compute the radius of a polytropic rotating model are restricted by the star's surface, we develop a method for continuing integrations beyond this limit. This general approach utilises the postnewtonian parameters in terms of disturbance. By the application of a code written in Fortran, we calculate extensive tables of quantities and constants. Furthermore, we compute the radius as well as the critical rotation parameter, which plays the role of a postnewtonian term. This Thesis is organized in five chapters. In the first chapter, the neutron star is presented as an astrophysical object. Its structure and physical characteristics are of a great importance. Moreover, the existence of neutron stars is linked to pulsars, which are "living" examples of rotating neutron stars. Therefore, the physical characteristics of these objects are discussed in this chapter. The second chapter refers to the equations that describe the structure of the neutron stars and to the concept of polytropes. First, due to the difficulty in implementing a study for these stars without the adoption of any equation of state as well as due to the most indicative one for their description which is that of the polytrope. Second, the equations that refer to the undistorted and those that describe the corresponding distorted configurations are analysed in this chapter, in accordance with Chandrasekhar's perturbation theory. In the third chapter, the General Theory of Relativity is used to the study of the polytropic model, focusing on how the classical theory is corrected, on how the basic model's quantities are transformed and on how the equations of the postnewtonian approach are derived. The equations to be solved result from the latter ones. Furthermore, a a discussion on the various numerical methods that have been developed for studying the relativistic rotating polytopric model is given in the fourth chapter. In the fifth chapter of this Thesis, a number of tables illustrating results as well as some interesting diagrams are included. Certain algorithmic details for our program are given. We also discuss the generalisation of the postnewtonian approach and its advantages.

Thesis (MEng)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The management of physical assets has become a popular eld of study over recent years and is being acknowledged in multiple disciplines world wide. In this project, research on Physical Asset Management (PAM), maintenance and decision making are presented. PAM is a complex subject and requires the participation of multiple disciplines in order to successfully manage physical assets. Moreover, the management of maintenance makes a big contribution in achieving successful PAM. Decision making is a core element to manage maintenance e ciently, both on strategic and operational level. Various methods and techniques can be used to aid the decision making process such as, using past experience, xed decision making techniques and techniques involving numerical calculations, to mention only a few. However, using numerical calculations to make decisions are not very popular. This is due to various reasons, for example the inherent complexity of the mathematics and the time required to execute such calculations are disliked. People tend to avoid complex numerical calculations and rather rely on past experience and discussion of circulating opinions to make decisions. This is not ideal and can lead to inconsistent and inaccurate decisions. In this project, the importance of numerical decision making is researched, especially in maintenance related decisions. The focus is placed on the simpli cation of numerical decision making techniques with the aim to make it easy and quick to use to support operational PAM decisions. Di erent decisions regarding PAM, especially decisions with regards to managing maintenance in order to achieve PAM, are discussed by means of a literature study. This is done to clarify the applicability of using numerical decision making techniques to support this type of decisions. A few di erent available numerical techniques are highlighted that can be used to support the decision making process. The decisions together with numerical decision making techniques are evaluated in order to combine the most appropriate techniques in a simpli ed manner. The purpose of this is that it can be used by anyone with the necessary knowledge of a speci c system or operation. As a result a simpli ed numerical decision making toolbox is developed that can support maintenance related decision. This toolbox is applied to a real life situation by means of a case study, made possible by Anglo American Platinum Limited (Amplats). An evaluation and validation of the toolbox is done through the case study to conclude wether it has value in practice or not. / AFRIKAANSE OPSOMMING: Die bestuur van siese bates het die afgelope paar jaar 'n gewilde studieveld geword en word erken in verskeie dissiplines reg oor die w^ereld. In hierdie projek word navorsing gedoen oor Fisiese Bate Bestuur (FBB), instandhouding en besluitneming. FBB is 'n komplekse onderwerp en vereis die deelname van verskeie dissiplines om sukses te behaal. Die bestuur van instandhouding maak 'n groot bydrae tot suksesvolle FBB. 'n Kern element van doeltre ende instandhouding is besluitneming, beide op strategiese en operasionele vlak. Verskillende metodes en tegnieke kan gebruik word om die besluitnemingsproses te ondersteun soos byvoorbeeld om gebruik te maak van ondervinding en vorige gebeurtenisse, vaste besluitnemingstegnieke, tegnieke wat numeriese berekeninge gebruik en nog meer. Die gebruik van numeriese metodes om besluite te neem is nie baie gewild nie. Dit is as gevolg van verskeie redes soos byvoorbeeld die inherente kompleksiteit en ingewikkeldheid van die wiskunde en ook die tyd wat benodig word om sulke berekeninge uit te voer. Mense is geneig om ingewikkelde numeriese berekeninge te vermy en eerder staat te maak op vorige ervaring en die bespreking van menings om besluite te neem. Dit is nie ideaal nie en kan lei tot onkonsekwente besluite, of selfs verkeerde besluite. In hierdie projek is die belangrikheid van numeriese besluitneming nagevors, veral in die onderhoudsverwante besluite. Die fokus word geplaas op die vereenvoudiging van die numeriese besluitnemings tegnieke. Die doel is om dit op so 'n manier te vereenvoudig dat dit maklik en vinnig is om te gebruik vir operasionele FBB besluite. Verskillende besluite oor FBB, veral besluite met betrekking tot instandhouding om suksesvolle FBB te bereik, word bespreek deur middel van 'n literatuurstudie. Die literatuurstudie ondersoek die toepaslikheid van die gebruik van numeriese besluitnemingstegnieke vir hierdie soort besluite. 'n Paar verskillende beskikbare numeriese tegnieke wat gebruik kan word om die besluitnemingsproses te ondersteun word uitgelig. Die besluite, saam met numeriese besluitnemingtegnieke, word ge evalueer om die mees gepaste tegnieke te kombineer in 'n vereenvoudigde manier. Uiteindelik moet dit deur enige iemand met die nodige kennis van 'n spesi eke stelsel of proses gebruik kan word. As resultaat is 'n vereenvoudigde numeriese besluitnemingstegniekkombinasie ontwikkel wat besluite verwant aan instandhouding kan ondersteun. Hierdie tegniek-kombinasie word toegepas in 'n werklike situasie deur middel van 'n gevallestudie, wat moontlik gemaak is deur Anglo American Platinum Limited. 'n Evaluering en validering van die tegniek-kombinasie word gedoen in die gevallestudie om te bepaal of dit wel waarde het in die praktyk of nie.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho o processo não linear de transmissão de calor condução-radiação é abordado num contexto bidimensional plano e simulado com o uso de um esquema linear em diferenças finitas. O problema original é tratado como o limite de uma sequencia de problemas lineares, do tipo condução-convecção. Este limite, cuja existência é comprovada, é facilmente obtido a partir de procedimentos básicos, accessíveis a qualquer estudante de engenharia, permitindo assim o emprego de hipóteses mais realistas, já que não se tem o limitante matemático para a abordagem numérica de uma equação diferencial parcial elíptica. Neste trabalho foi resolvido o problema de condução de calor em regime permanente em uma placa com condições de contorno convectivas e radioativas utilizando-se o software MatLab, vale ressaltar, que a mesma metodologia é aplicável para geometrias mais complexas. / In this work the nonlinear conduction-radiation heat transfer process is considered under a plane two dimensional assumption and simulated by means of a finite difference linear scheme. The original problem is regarded as the limit (which always exists) of a sequence of linear problems like the conduction-convection ones. Such a limit is reached in an easy way by means of standard procedures, available for any undergraduate engineering student, allowing the employment of more realistic hypotheses, since the mathematical complexities are not a constraint for simulating the elliptic partial differential equation. This work solved the problem of heat conduction in steady state conditions on a plate with convective and radioactive contour using MatLab software, it is noteworthy that the same methodology is applicable to more complex geometries.