Dans cette thèse, nous nous intéressons à deux approximations hyperboliques des équations de Navier-Stokes incompressibles en dimensions 2 et 3 d'espace. Dans un premier temps, on considère une perturbation hyperbolique de l'équation de la chaleur, introduite par Cattaneo en 1949, pour remédier au paradoxe de la propagation instantanée de cette équation. En 2004, Brenier, Natalini et Puel remarquent que la même perturbation, qui consiste à rajouter ε∂tt à l'équation, intervient en relaxant les équations d'Euler. En dimension 2, les auteurs montrent que, pour des sonnées régulières et sous certaines hypothèses de petitesse, la solution globale de la perturbation converge vers l'unique solution globale de (NS). En 2007, Paicu et Raugel améliorent les résultats de [BNP] en étendant la théorie à la dimension 3 et en prenant des données beaucoup moins régulières. Nous avons obtenu des résultats de convergence, avec données de régularité quasi-critique, qui complètent et prolongent ceux de [BNP] et [PR]. La seconde approximation que l'on considère est un nouveau modèle hyperbolique à vitesse de propagation finie. Ce modèle est obtenu en pénalisant la contrainte d'incompressibilité dans la perturbation de Cattaneo. Nous démontrons que les résultats d'existence globale et de convergence du précédent modèle sont encore vérifiés pour celui-ci. / In this work, we are interested in two hyperbolic approximations of the 2D and 3D Navier-Stokes equations. The first model we consider comes from Cattaneo's hyperbolic perturbation of the heat equation to obtain a finite speed of propagation equation. Brenier, Natalini and Puel studied the same perturbation as a relaxed version of the 2D Euler equations and proved that the solution to this relaxation converges towards the solution to (NS) with smooth data, provided some smallness assumptions. Later, Paicu and Raugel improved their results, extending the theory to the 3D setting and requiring significantly less regular data. Following [BNP] and [PR], we prove global existence and convergence results with quasi-critical regularity assumptions on the initial data. In the second part, we introduce a new hyperbolic model with finite speed of propagation, obtained by penalizing the incompressibility constraint in Cattaneo's perturbation. We prove that the same global existence and convergence results hold for this model as well as for the first one.

A method is developed for solving the two dimensional Helmholtz equation in a ductwith varying cross-section region bounded by a curved top and flat bottom, having oneregion inside. To compute the propagation of sound waves in a curved duct with a curvedinternal interface is difficult problem. One method is to transform the wave equation intoa solvable form and making the curved interface plane. To this end a local orthogonaltransformation is developed for the varying cross-sectional duct having one medium inside.This transformation is first used to make the curved top of the waveguide flat andto transform the Helmholtz equation into an initial value problem. Later on the local orthogonaltransformation is developed for a waveguide having two media inside with flattop, a flat bottom and a curved interface. This local orthogonal transformation is used toflatten the interface and also to transform the Helmholtz equation into a simple, solvableordinary differential equation. In this paper we present operator formulation for the partwith flat bottom and curved top including a curved interface. In the ordinary differentialequation with operators in coefficients, obtained after the transformation, all the operationsrelated to the transverse variable are treated as operators while the derivative withrespect to the range variable is kept.

In the modern world, because of increasing oil prices and the need to control greenhouse gas emission, a new interest in the production of electric cars is coming about. One of the products is a charging point for electric cars, at which electric cars can be recharged by a plug in cable. Usually people are required to pay for the electricity after recharging the electric cars. Today, the payment is handled by using SMS or through the parking system. There is now an opportunity, in cooperation with AES (the company with which we are working), to equip the pole with GPRS, and this requires development and maintenance of the antenna. The project will include data analysis of the problem, measurements and calculations. In this work, we are computing energy flow of the wave due to the location of the antenna inside the box. We need to do four steps. First, we take a set of points (determined by the computational mesh) that have the same distance from the antenna in the domain. Second, we calculate the angles between the ground and the points in the set. Third, we do an angle-energy plot, to analyse which angle can give the maximum energy. And last, we need to compare the maximum energy value of different position of the antenna. We are going to solve the problem in Matlab, based on the Maxwell equation and the Helmholtz equation, which is not time-dependent.

Lienard Equations serve as the elegant models for oscillating circuits. Motivated by this fact, this thesis addresses the stability property of a class of delayed Lienard equations. It shows the existence of the Hopf bifurcation around the steady state. It has both practical and theoretical importance in determining the criticality of the Hopf bifurcation. For such purpose, center manifold analysis on the bifurcation line is required. This thesis uses operator differential equation formulation to reduce the infinite dimensional delayed Lienard equation onto a two-dimensional manifold on the critical bifurcation line. Based on the reduced two-dimensional system, the so called Poincare-Lyapunov constant is analytically determined, which determines the criticality of the Hopf bifurcation. Numerics based on a Matlab bifurcation toolbox (DDE-Biftool) and Matlab solver (DDE-23) are given to compare with the theoretical calculation. Two examples are given to illustrate the method.

A ferromagnetic particle with single domain, at some kinds of applied field (at some angle or strangth), the particle's free energy would be two state model. The rate of barrier crossing could be solve by Fokker-Planck equation .And use master equation to find out the Total rate between two potential well. In this thysis, we use the upper method to simulate particle's magnetic moment under time varying magnetic field at fixed angle or fixed magnetic applied rotate the particle. In numerical method, we use the back Euler method to prevent the divergence of the calculation.

published_or_final_version / Physics / Master / Master of Philosophy

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

Internal solitary waves (ISWs) are commonly observed in the ocean, and they play important roles in many ways, such as transport of mass and various nutrients through propagation. The fluids considered in this thesis are assumed to be incompressible, inviscid, non-diffusive and to be weakly affected by the Earth's rotation. Comparisons of the evolution of an initial solitary wave predicted by a fully nonlinear model, IGW, and two weakly-nonlinear wave equations, the Ostrovsky equation and a new alternative Ostrovsky equation, are done. Resolution tests have been run for each of the models to confirm that the current choices of the spatial and time steps are appropriate. Then we have run three numerical simulations with varying initial wave amplitudes. The rigid-lid approximation has been used for all of the models. Stratification, flat bottom and water depth stay the same for all three simulations. In the simulation analysis, we use the results from the IGW as the standard. Both of the two weakly nonlinear models give fairly good predictions regarding the leading wave amplitudes, shapes of the wave train and the propagation speeds. However, the weakly nonlinear models over-predict the propagation speed of the leading solitary wave and that the alternative Ostrovsky equation gives the worst prediction. The difference between the two weakly nonlinear models decreases as the initial wave amplitude decreases.

A family of parameterized Thue equations is defined as F_{t,s,...}(X, Y ) = m, m ∈ Z where F_{t,s,...}(X,Y) is a form in X and Y with degree greater than or equal to 3 and integer coefficients that are parameterized by t, s, . . . ∈ Z. A variety of these families have been studied by different authors. In this thesis, we study the following families of Thue inequalities |sx3 −tx2y−(t+3s)xy2 −sy3|≤2t+3s, |sx4 −tx3y−6sx2y2 +txy3 +sy4|≤6t+7s, |sx6 − 2tx5y − (5t + 15s)x4y2 − 20sx3y3 + 5tx2y4 +(2t + 6s)xy5 + sy6| ≤ 120t + 323s, where s and t are integers. The forms in question are “simple”, in the sense that the roots of the underlying polynomials can be permuted transitively by automorphisms. With this nice property and the hypergeometric functions, we construct sequences of good approximations to the roots of the underlying polynomials. We can then prove that under certain conditions on s and t there are upper bounds for the number of integer solutions to the above Thue inequalities.

The Vlasov equation describes the evolution of the distribution function of particles in phase space (x,v), where the particles interact with long-range forces, but where shortrange "collisional" forces are neglected. A space plasma consists of low-mass electrically charged particles, and therefore the most important long-range forces acting in the plasma are the Lorentz forces created by electromagnetic fields. What makes the numerical solution of the Vlasov equation a challenging task is that the fully three-dimensional problem leads to a partial differential equation in the six-dimensional phase space, plus time, making it hard even to store a discretised solution in a computer’s memory. Solutions to the Vlasov equation have also a tendency of becoming oscillatory in velocity space, due to free streaming terms (ballistic particles), in which steep gradients are created and problems of calculating the v (velocity) derivative of the function accurately increase with time. In the present thesis, the numerical treatment is limited to one- and two-dimensional systems, leading to solutions in two- and four-dimensional phase space, respectively, plus time. The numerical method developed is based on the technique of Fourier transforming the Vlasov equation in velocity space and then solving the resulting equation, in which the small-scale information in velocity space is removed through outgoing wave boundary conditions in the Fourier transformed velocity space. The Maxwell equations are rewritten in a form which conserves the divergences of the electric and magnetic fields, by means of the Lorentz potentials. The resulting equations are solved numerically by high order methods, reducing the need for numerical over-sampling of the problem. The algorithm has been implemented in Fortran 90, and the code for solving the one-dimensional Vlasov equation has been parallelised by the method of domain decomposition, and has been implemented using the Message Passing Interface (MPI) method. The code has been used to investigate linear and non-linear interaction between electromagnetic fields, plasma waves, and particles.