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This thesis discusses the use of perturbation theory in the context of financial mathematics, in particular on the use of matched asymptotic expansions in option pricing. Our methods are applied to the ordinary Black-Scholes model for illustration. In this simple example of the Black-Scholes model an exact solution is available, so it is in fact not neccessary to apply the method of asymptotic expansions on this model. However, in case we do apply the method, two artificial layers have to be constructed. Making smart choices for the local variables leads to a transformation of the equations into a heat equation, which can easily be solved. Finally, the results are compared to a Taylor expansion of the exact solution to see that this method is very accurate. After this first instructive model, the method of matched asymptotic expansions is applied to two more advanced models based on papers by Sam Howison and Patrick Hagan et al.. Here, different choices for the scalings are made. The former discusses a fast mean-reverting stochastic volatility model that turns out to have many open ends. In Howison's paper quite a lot of assumptions and simplifications are made. Unfortunately, often the motivation for them is not explicitly given in the paper, and in some cases we even think these assumptions and simplifications are incorrect. The latter examines a new three-parameter stochastic volatility model that successfully prices back the volatility smile as observed in the market nowadays, and that is commonly used. The derivation of this model is the main focus of this thesis. The resulting expression for the implied volatility under the SABR model is obtained by considering the forward and backward Kol-mogorov equations per order in epsilon, making some smart choices for local variables and functions in order to transform them into an equation that looks like a heat equation, which is easier to solve. Recommendations for further investigation on these models would be to consider several different choices for the scalings and see which one works best.

In this paper the stability properties of the vibrations of a singular degree of freedom oscillator with a periodically and multi-stepwise time-varying mass are studied. The free vibrations and the vibrations due to two types of forcing are investigated.

In this report an attempt is made to analyse how a damped Timoshenko beam is affected by an external force.

In this thesis the free and forced vibrations of a single degree of freedom oscillator with a periodically time-varying mass have been studied. Linear and weakly non-linear oscillator equations have been considered. The forced vibrations of the oscillator are partly due to small masses which are T-periodically hitting and leaving the oscillator with T-periodic velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator and the shape of the oscillator will periodically vary in time. The effect of a damping term (in the linear oscillator equation) on the solutions also has been considered. For the free vibrations the minimal damping rates have been computed for which the oscillator is always stable. Also cases with external, harmonic forcing have been investigated in detail for the linear oscillator equation, and interesting resonance conditions have been found. As simple model to describe the rain-wind induced oscillations of a cable, an initial value problem for an oscillator equation with a Rayleigh type of non-linearity has been studied.By applying a straight-forward perturbation method the problem has been solved approximately on a time-interval of length T. In all cases studied in this thesis initial value problems for oscillator equations have been formulated. The constructed solutions on a time-interval of length T or the approximations of the solutions on the same time-interval have been used to construct maps. By using these maps (i.e. by using a system of difference equations) the stability properties of the solutions have been determined. The instability regions in the parameter space have been computed partly analytically and partly numerically. Some phase-space figures for the weakly non-linear problem have been computed numerically to show a number of interesting bifurcations, and to show the rich dynamics of the problem.

In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for nonlinear difference equations are constructed by using the recently developed perturbation method based on invariance vectors. The asymptotic approximations of the solutions of the oscillator equations can be derived from these asymptotic approximations of the first integrals. It is shown that all invariance factors have to satisfy a functional equation. One of the main difficulties in finding first integrals for a system of first order difference equations is solving the aforementioned functional equation. In this thesis, we consider a functional equation which is related to a system of two first order, linear ordinary difference equations. Linear transformations and an adapted version of the method of separation of variables is used to construct the general solution of this functional equation. A perturbation method based on invariance factors and multiple scales is also presented for weakly nonlinear, regularly perturbed systems of ordinary difference equations. Asymptotic approximations of first integrals are constructed on long iteration-scales, that is, on iteration-scales of order ϵ-1, where ϵ is a small parameter. To show how this perturbation method works, the method is applied to a Van der Pol equation, and a Rayleigh equation. We also apply an improved version of the multiple scales perturbation method to a general system of weakly nonlinear, regularly perturbed ordinary difference equations including linear, quadratic, and qubic terms. Such systems arise as a result of the discretization of a system of nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations. As an example, in the study of the forced vibrations of a (damped) linear sdofo with a time-varying mass a system of two nonlinear ordinary difference equations is obtained to describe the stability properties of the oscillator. In such oscillators the forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. In our procedure, asymptotic approximations of the solutions of the difference equations are constructed which are valid on long iteration scales. In this thesis it is shown that the presented perturbation method based on invariance vectors can be applied to weakly nonlinear oscillator equations, which are "close" to integrable equations (that is, are integrable in the unperturbed case).

To solve problems in geotechnical engineering often numerical methods such as the Finite Element Method (FEM) are used. This method can be applied for example for the calculation of the strength of dikes, the determination of the stability of (rail)road embankments, the prediction of deformations due to landfills, or the analysis of subsurface constructions such as foundations, excavation pits and tunnels. Executing these numerical calculations frequently unreliable results are observed, which are the consequence of non-converging or unstable solutions. Indeed, often the source of the unexpected behaviour remains unknown. The present research aims to explain one of the possible causes, i.e. the influence of the applied material model on the behaviour of the numerical solution. In soil mechanics the elasto-plastic Mohr-Coulomb material model (including hardening and softening) is very commonly used. In this research the equations of static equilibrium, on which the FEM formulation is based, are analysed and solved completely analytically. For this purpose the method of separation of variables is used, in an adapted and extended version, which allows for the solution of a larger class of problems than generally assumed. Using this method the complete analytical solution is derived for linear elasticity as well as for Mohr-Coulomb elasto-plasticity. The necessary and sufficient conditions for uniqueness and stability of the solution are determined. These conditions allow for the determination and clarification of the parameter limits for the applicability of those material models. Using the results of this research the limits of applicability of the two considered material models can be determined for numerical applications as e.g. the Finite Element Method.

In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for the nonlinear differential equations are constructed by using the recently developed perturbation method based on integrating vectors. The existence and the stability of time-periodic solutions can be determined from these asymptotic approximations of the first integrals. Also asymptotic approximations of the solutions of the oscillator equations can be derived from these asymptotic approximations of the first integrals. Not only autonomous oscillator equations but also nonautonomous equations can be treated. In this thesis it is shown that the presented perturbation method based on integrating vectors can be applied to weakly and strongly nonlinear oscillator equations, which are "close" to integrable equations (that is, are integrable in the unperturbed case). In combination with a phase-space analysis and a Poincareturn-map technique the presented perturbation method gives a good insight in the global behavior of the solutions of the oscillator equations. All nonlinear oscillator equations which are studied in this thesis are simple model-equations describing the galloping oscillations of iced overhead power transmission lines in a windfield.

In this thesis the dynamics of inclined stretched strings has been studied. These elastic strings are suspended between a fixed support and a vibrating support. Longitudinal vibrations and transversal (in- and out-of-plane) vibrations of the strings have both been considered. In this thesis it has been assumed that the excitation-forces applied at the vibrating support can act in the longitudinal direction and/or in the (in-plane) transversal direction. It also has been assumed that the bending stiffness of the string is negligible and that the tension in the string is sufficiently large such that the sag of the string due to gravity is small. For certain excitation-frequencies it turns out that complicated internal resonances occur, and for these frequencies the small excitation-forces can lead to large amplitude-responses of the string. The parameters, that is, the properties of the string that give rise to these resonances have been determined analytically and numerically by using the averaging method, the Galerkin truncation method, and the linearisation method.

Elastic structures are susceptible to wind- and earthquake-induced vibrations. These vibrations can damage a structure or cause human discomfort. To suppress structural vibrations, various types of damping mechanisms, active or passive, can be applied. In this thesis the model of a weakly damped, standing Euler-Bernoulli beam in a (turbulent) wind-field and the model of a standing Timoshenko beam will be used as a simple model of a tall building. These models will be used to study the stabilizing effect of dampers which are installed at the top of the beam (the so-called boundary dampers), the self-weight effect of a beam on its stability, and the possibly destabilizing effect due to galloping (a dynamic wind response). In this thesis two passive control methods will be applied to the Euler-Bernoulli beam. Moreover, the string-equation will be used to study the dynamics of a cable with an end-mass and subjected to boundary damping. The model of a tensioned beam will be used to examine the damping behavior of a tensioned cable with small bending stiffness and an attached tuned mass damper. The vibrations of these beam and string models can be described by (stochastic) initial-boundary value problems. The problem will be stochastic if a beam in a turbulent wind-field is studied. It is assumed that the damping effect, the self-weight effect, and the wind-force in these problems are small but not negligible. The multiple-timescales perturbation method, the method of separation of variables, and a combination of the Galerkin truncation method and a numerical scheme, will be used to construct (explicit) approximations of the beam-like problems. The Laplace transform method will be applied tothe string-like problem. In this way a so-called characteristic equation has been obtained. This equation have been solved by using (adapted) classical perturbation methods. For both control methods, the uniform stability of an Euler-Bernoulli beam subjected to boundary damping has been established and it has been concluded that these strategies can be used effectively to damp the wind-induced vibrations of a standing Euler-Bernoulli beam. Furthermore, it has been found that the self-weight effect on the frequencies and damping rates of an Euler-Bernoulli and Timoshenko beam is small. For the string problem approximations of the damping rates have been constructed. These have been used to conclude that a string with an end-mass can be damped uniformly by applying boundary damping. Lastly, for the tensioned beam with attached damper it has been shown that small bending stiffness only slightly influences the damping rates of the cable.

In this thesis a mathematical analysis has been given for model which describes the transversal vibrations of belt systems. The belt speed is assumed to be time-varying and to be small compared to the wave speed. Not only linear string-like or beam-like models but also nonlinear models have been studied. In all cases initial-boundary value problems are formulated, and are investigated by using multiple time-scales perturbation methods. Formal approximations of the solutions are constructed and it is shown whether or not mode interactions between vibration modes occur for specific values of the belt parameters. For some linear models instabilities in the solution occur, which disappear when nonlinear terms are included in the model. It is also shown for what parameter values in the nonlinear models a simplification in the formulation of the problem (based on Kirchhoff's assumption) can (or can not) be used.

In axially moving structures like conveyor belt systems, magnetic tapes, and so on, vibrations occur due to the presence of different kinds of imperfections in the systems. For these structures internal resonances can lead to severe vibrations. Resonance free conveyor belt systems can be constructed if the frequencies of the system are known. In this thesis two models for an axially moving continuum have been studied: a string-like model (mathematically modelled by a wave equation) and a beam-string-like model (mathematically modelled by the Euler-Bernoulli beam equation with an additional tension term). The corresponding initial-boundary value problems have been solved approximately or exactly by using the multiple timescales perturbation method or the Laplace transform method. Exact solutions and formal approximations of the solutions for some string-like problems have been found in the form of Fourier series. It has been shown that the truncation method for the string-like problem can not be applied in order to obtain an accurate solution or approximation on a long time-scale. A new model approach describing the transient ``from string to beam'' behaviour, based on the calculations of the natural frequencies of the system has been proposed. The influence of the bending stiffness on the stability properties of the solution of the problem has been studied. An important implication of the results as presented in this thesis is that for these types of axially moving continua problems the use of only string-like models is not appropriate. To describe the dynamics of these types of problems correctly one has to include (small) bending stiffness in the model. It has also been shown that the introduction of a damping term does not solve the truncation problem for the string-like equation, at least if the damping is assumed to be small. A possible energy transfer between transversal and longitudinal vibration-directions has been studied in case of a sudden stoppage of the conveyor belt system. It has been shown that this energy transfer can occur depending on the existing internal resonances in the system which are determined by the belt system parameters.

On the design and implementation of integral-conservative numerical integration schemes for few-body problems in astrodynamics. Focuses on exact and approximate energy and angular-momentum integrals in the Jacobi 3-body problem, and related Jacobi-type integrals in the circular restricted 3-body problem and a 4-body model for ballistic lunar capture. Includes a self-contained discussion of necessary astrodynamics and mathematics background, as well as a discussion of the application of these techniques to ballistic lunar capture trajectories for small satellites.

On the determination of approximations of first integrals using the method of integrating vectors for ODE systems, as applied to few-body gravitational problems. Considers the Jacobi 3-body problem, the circular restricted 3-body problem and a 4-body model for ballistic lunar capture. Also discusses the application of these techniques to numerical solutions of the ODE systems using methods designed to preserve exact and approximate first integrals, such as those developed using the method of integrating vectors.