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The subject of this Thesis is to examine nontrivial solutions of boundary value problems for second order ODEs with unilateral jumping nonlinearities. Considered problems can be interpreted as models of a simple beam supported by three types of elastic obstacles. Couples of eigenvalues and eigenfunctions are found and discussed with respect to parameter which represents the strength of the obstacle.

Neste trabalho apresentamos duas caracterizações para o K-funcional do tipo Peetre sobre os espaços compactos 2-homogêneos. Provamos a equivalência no sentido assintótico entre o módulo de suavidade de ordem fracionária e o K-funcional do tipo Peetre, e a equivalência deste último com o raio de aproximação de um operator multiplicativo definido para este propósito. Como consequência obtivemos a desigualdade de Marchaud, neste contexto. Estes resultados generalizam os equivalentes, e bem conhecidos, sobre o contexto esférico. As caracterizações foram aplicadas para mostrar que uma condição abstrata de Hölder, ou de diferenciabilidade de ordem finita, sobre núcleos que geram operadores integrais positivos, implica a obtenção de uma taxa de decrescimento polinomial para suas sequências de autovalores. / We prove two characterization for the Peetre type K-functional on M, a compact two-point homogeneous space. One in terms the rate of approximation of a family of multipliers operator defined to this purpose, and another in terms of the fractional moduli of smoothness. As a direct consequence of those we obtained the Marchaud inequality on this framework. These extend the well known results on the spherical setting. The characterizations are employed to show that an abstract Hölder condition or finite order of differentiability condition imposed on kernels generating certain operators implies a sharp decay rates for their eigenvalues sequences.

In this thesis three methods LMGS, TQR and GR are applied to solve an inverseeigenvalue problem. We list the numerical results and compare the accuracy of the computed Jacobi matrix $T$ and the associated orthogonal matrix $Q$, wherethe columns of $Q^T$ are the eigenvectors of $T$. In the application of this inverse eigenvalue problem, the Fourier coefficients of $h(x)=e^x$ relative to the orthonormal polynomials associatedwith $T$ are evaluated, and these values are used to compute the least squarescoefficients of $h$ relative to the Chebyshev polynomials. We list thesenumerical results and compare them as our conclusion.

Dans cette thèse, nous menons à l'aide de la méthode des équations intégrales des analyses de sensitivité de solutions ou de spectres de l'équation de conductivité par rapport aux variations géométriques ou de paramètres de l'équation. En particulier, nous considérons le problème de conductivité dans des milieux à forts contrastes, le problème de perturbation du bord d'une inclusion de conductivité, le problème de valeurs propres du Laplacien dans des domaines perturbés et le problème d'ouverture de gap dans le spectre des cristaux photoniques.

<p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p>

Rational Krylov methods for model order reduction are studied. A dual rational Arnoldi method for model order reduction and a rational Krylov method for model order reduction and eigenvalue computation have been implemented. It is shown how to deflate redundant or unwanted vectors and how to obtain moment matching. Both methods are designed for generalised state space systems---the former for multiple-input-multiple-output (MIMO) systems from finite element discretisations and the latter for single-input-single-output (SISO) systems---and applied to relevant test problems. The dual rational Arnoldi method is designed for generating real reduced order systems using complex shift points and stabilising a system that happens to be unstable. For the rational Krylov method, a forward error in the recursion and an estimate of the error in the approximation of the transfer function are studie. A stability analysis of a heat exchanger model is made. The model is a nonlinear partial differential-algebraic equation (PDAE). Its well-posedness and how to prescribe boundary data is investigated through analysis of a linearised PDAE and numerical experiments on a nonlinear DAE. Four methods for generating reduced order models are applied to the nonlinear DAE and compared: a Krylov based moment matching method, balanced truncation, Galerkin projection onto a proper orthogonal decomposition (POD) basis, and a lumping method. / QC 20101013

We consider the problem of maximizing the second smallest eigenvalue of the weighted Laplacian of a (simple) graph over all nonnegative edge weightings with bounded total weight. We generalize this problem by introducing node significances and edge lengths. We give a formulation of this generalized problem as a semidefinite program. The dual program can be equivalently written as embedding problem. This is fifinding an embedding of the n nodes of the graph in n-space so that their barycenter is at the origin, the distance between adjacent nodes is bounded by the respective edge length, and the embedded nodes are spread as much as possible. (The sum of the squared norms is maximized.) We proof the following necessary condition for optimal embeddings. For any separator of the graph at least one of the components fulfills the following property: Each straight-line segment between the origin and an embedded node of the component intersects the convex hull of the embedded nodes of the separator. There exists always an optimal embedding of the graph whose dimension is bounded by the tree-width of the graph plus one. We defifine the rotational dimension of a graph. This is the minimal dimension k such that for all choices of the node significances and edge lengths an optimal embedding of the graph can be found in k-space. The rotational dimension of a graph is a minor monotone graph parameter. We characterize the graphs with rotational dimension up to two.

In this study, various stability control systems are developed to remove the lateral instability of a conventional articulated steer vehicle (ASV) during the oscillatory yaw motion or “snaking mode”. First, to identify the nature of the instability, some analyses are performed using several simplified models. These investigations are mainly focused on analyzing the effects of forward speed and of two main subsystems of the vehicle, the steering system and tires, on the stability. The basic insights into the stability behavior of the vehicle obtained from the stability analyses of the simplified models are verified by conducting some simulations with a virtual prototype of the vehicle in ADAMS. To determine the most critical operating condition with regard to the lateral stability and to identify the effects of vehicle parameters on the stability, various studies are performed by introducing some modifications to the simplified models. Based on these studies, the disturbed straight-line on-highway motion with constant forward speed is recognized as the most critical driving condition. Also, the examinations show that when the vehicle is traveling with differentials locked, the vehicle is less prone to the instability. The examinations show that when the vehicle is carrying a rear-mounted load having interaction with ground, the instability may happen if the vehicle moves on a relatively good off-road surface. Again, the results gained from the analyses related to the effects of the vehicle parameters and operating conditions on the stability are verified using simulations in ADAMS by making some changes in the virtual prototype for any case. To stabilize the vehicle during its most critical driving condition, some studies are directed to indicate the shortcomings of passive methods. Alternative solutions, including design of different types of stability control systems, are proposed to generate a stabilizing yaw moment. The proposed solutions include an active steering system with a classical controller, an active torque vectoring device with a robust full state feedback controller, and a differential braking system with a robust variable structure controller. The robust controllers are designed by using simplified models, which are also used to evaluate the ability to deal with the uncertainties of the vehicle parameters and its variable operating conditions. These controllers are also incorporated into the virtual prototype, and their capabilities to stabilize the vehicle in different operating conditions and while traveling on different surfaces during the snaking mode are shown.

In this study, various stability control systems are developed to remove the lateral instability of a conventional articulated steer vehicle (ASV) during the oscillatory yaw motion or “snaking mode”. First, to identify the nature of the instability, some analyses are performed using several simplified models. These investigations are mainly focused on analyzing the effects of forward speed and of two main subsystems of the vehicle, the steering system and tires, on the stability. The basic insights into the stability behavior of the vehicle obtained from the stability analyses of the simplified models are verified by conducting some simulations with a virtual prototype of the vehicle in ADAMS. To determine the most critical operating condition with regard to the lateral stability and to identify the effects of vehicle parameters on the stability, various studies are performed by introducing some modifications to the simplified models. Based on these studies, the disturbed straight-line on-highway motion with constant forward speed is recognized as the most critical driving condition. Also, the examinations show that when the vehicle is traveling with differentials locked, the vehicle is less prone to the instability. The examinations show that when the vehicle is carrying a rear-mounted load having interaction with ground, the instability may happen if the vehicle moves on a relatively good off-road surface. Again, the results gained from the analyses related to the effects of the vehicle parameters and operating conditions on the stability are verified using simulations in ADAMS by making some changes in the virtual prototype for any case. To stabilize the vehicle during its most critical driving condition, some studies are directed to indicate the shortcomings of passive methods. Alternative solutions, including design of different types of stability control systems, are proposed to generate a stabilizing yaw moment. The proposed solutions include an active steering system with a classical controller, an active torque vectoring device with a robust full state feedback controller, and a differential braking system with a robust variable structure controller. The robust controllers are designed by using simplified models, which are also used to evaluate the ability to deal with the uncertainties of the vehicle parameters and its variable operating conditions. These controllers are also incorporated into the virtual prototype, and their capabilities to stabilize the vehicle in different operating conditions and while traveling on different surfaces during the snaking mode are shown.