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Concentration phenomena for singularly perturbed problems on two dimensional domains. / CUHK electronic theses & dissertations collectionJanuary 2007 (has links)
Firstly, we establish the existence of a solution u epsilon concentrating along a curve Gammaepsilon near the nondegenerate Gamma, exponentially small in epsilon at any positive distance from the curve, provided epsilon is small and away from certain critical numbers. The concentrating curve Gammaepsilon will collapse to Gamma as epsilon → 0. / In this thesis, we consider the following problem 32Duu+up= 0 and u>0 in W , 6u6n= 0 on 6W, where O is a bounded domain in R2 with smooth boundary, epsilon is a small positive parameter, nu denotes the outward normal of O and p > 1. Let Gamma be a straight line intersecting orthogonally with ∂O at exactly two points. We use the infinite dimensional LyapunovSchmidt reduction method, introduced by M. del Pino, M. Kowalczyk and J. Wei in [14], to deal with the noninvertibility caused by the critical eigenvalues of the linearized operator in the perturbed problems and then construct interior concentration layers near Gamma, which interact with the boundary. Moreover, the method of successive improvements of the approximation helps us decompose the interaction between the boundary and the interior layers. / Secondly, for any given integer N with N ≥ 2 and for small epsilon away from certain critical numbers, we construct another solution uepsilon exhibiting N concentration layers at mutual distances O(epsilon∣ ln epsilon∣), whose concentration set will approach the nondegenerate and nonminimal Gamma as epsilon → 0, provided that the exponent p ≥ 2. Asymptotic location of these layers is governed by a Toda type system. / Yang, Jun. / "July 2007." / Adviser: Juncheng Wei. / Source: Dissertation Abstracts International, Volume: 6901, Section: B, page: 0357. / Thesis (Ph.D.)Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 129136). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

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Perturbations singulières pour des EDP linéaires et non linéaires en presence de discontinuitésHamouda, Makram 21 December 2001 (has links) (PDF)
Ma thèse porte sur l'étude des couches limites et de perturbations singulières (\textit{i.e.} des problèmes caractérisés par la présence d'un petit paramètre qui tend vers zéro) dans des conditions plus délicates que d'habitude, à savoir lorsque la solution limite n'est pas régulière. Je considère ainsi deux classes de problèmes réguliers associes à un laplacien et à un bilaplacien, et un problème non linéaire dérivé du problème de Plateau (surfaces minimas), pour lequels la fonction limite possède une singularité (discontinuité simple pour les premiers problèmes, dérivée normale infinie sur certaines parties de la frontière pour le second).\\ La première partie de cette thèse est consacrée à l'étude de deux modèles linéaires singuliers associés à des perturbations singulières pour des EDPs ayant une fonction source singulière. Ce type d'équations fait l'objet de plusieurs applications, par exemple les problèmes de flambement en élasticité, les tourbillons singuliers en mécanique des fluides, le problème de la charge critique pour une poutre ou une plaque élastoplastique, le problème du contrôle automatique de la trajectoire d'un mobile et le problème du bord arrière pour l'écoulement autour d'une aile. De manière classique, la présence d'un petit paramètre dans des équations aux dérivées partielles entraîne, dans certains cas, l'apparition d'une couche limite classique près du bord du domaine pour la solution dite régularisée. Cependant, si on considère en plus une fonction source discontinue (voire une distribution), on constate que de nouvelles couches limites apparaissent à l'intérieur du domaine; l'étude de cellesci constitue le principal but de cette première partie. Dans la deuxième partie, on s'intéresse à l'étude du problème des surfaces minimales sur une couronne. Pour certaines classes de données au bord, ce problème n'admet pas de solution et sa solution faible dite ``généralisée'' admet une dérivée infinie. On introduit alors une méthode de régularisation elliptique qui entraîne une couche limite près du bord. Le résultat fondamental de cette partie consiste à donner explicitement une approximation pour cette solution régularisée.

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Robust computational methods for twoparameter singular perturbation problemsElago, David January 2010 (has links)
<p>This thesis is concerned with singularly perturbed twoparameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.</p>

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Robust computational methods for twoparameter singular perturbation problemsElago, David January 2010 (has links)
<p>This thesis is concerned with singularly perturbed twoparameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.</p>

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Generalized TitchmarshWeyl functions and super singular perturbationsNeuner, Christoph January 2015 (has links)
In this thesis we study certain singular SturmLiouville differential expressions from an operator theoretic point of view.In particular we are interested in expressions that involve strongly singular potentials as introduced by Gesztesy and Zinchenko.On the ODE side, analyzing these expressions involves the socalled $m$functions, often generalized Nevanlinna functions, who encapsulate spectral information of the underlying problem.The aim of the two papers in this thesis is to further understanding on the operator theory side.In the first paper, we use a model for super singular perturbations to describe a family of induced selfadjoint realizations of a perturbed Schr\"o\din\ger operator, i.e., with a potential of the form $c/x^2 + q$ where $q$ is a perturbation.Following the unperturbed example of Kurasov and Luger, we find that the socalled $Q$function appearing in this approach is in good agreement with the above named $m$function.Furthermore, we show that the operator model can be chosen such that $Q \equiv m$.In the second paper, we present a negative result in this area, namely that the supersingular perturbations model cannot be used for all strongly singular potentials.For a potential with a stronger singularity at the origin, namely $1/x^4$, we discuss the asymptotic behaviour of the Weyl solution at zero.It turns out that this function cannot be regularized appropriately and the operator model breaks down.

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Modeling a proton exchange membrane fuel cell stackDeLashmutt, Timothy E. January 2008 (has links)
Thesis (M.S.)Ohio University, November, 2008. / Title from PDF t.p. Includes bibliographical references.

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Higher order numerical methods for singular perturbation problems. /Munyakazi, Justin Bazimaziki. January 2009 (has links) (PDF)
Thesis (M.Sc. (Dept. of Mathematics, Faculty of Natural Sciences))University of the Western Cape, 2009. / Bibliography: leaves 180195.

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Robust computational methods for twoparameter singular perturbation problemsElago, David January 2010 (has links)
Magister Scientiae  MSc / This thesis is concerned with singularly perturbed twoparameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results. / South Africa

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Singularperturbation analysis of climbcruisedash optimizationShankar, Uday J. 15 November 2013 (has links)
The method of singularperturbation analysis is applied to the determination of rangefueltime optimal aircraft trajectories.
The problem is shown to break down into three subproblems which are studied separately. In particular, the inner layer containing the altitude pathangle dynamics is analyzed in detail. The outer solutions are discussed in an earlier work.
As a step forward in solving the ensuing nonlinear twopoint boundaryvalue problem, linearization of the equations is suggested. Conditions for the stability of the linearized boundarylayer equations are discussed. Also, the question of parameter selection to fit the solution to the split boundary conditions is resolved. Generation of feedback laws for the angleofattack from the linear analysis is discussed.
Finally, the techniques discussed are applied to a numerical example of a missile. The linearized feedback solution is compared to the exact solution obtained using a multiple shooting method. / Master of Science

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Fast halfloop maneuvers for the F/A18 fighter aircraft using a singular pertubation feedback control lawGarrett, Frederick Earl 12 April 2010 (has links)
The primary purpose of this study is to develop a nonlinear feedback control law for the F / AI8 fighter aircraft that performs a fast halfloop maneuver. This feedback law is developed using a singular perturbation approach. A secondary purpose of this study is to establish a baseline for time optimal halfloop maneuvers. The singular perturbation approach makes it possible to develop a state feedback control law which rotates the velocity vector through one hundred and eighty degrees at a maximum equilibrium pitch rate with a nearly constant angle of attack. The response of the aircraft to the control law is compared to simulations of halfloop maneuvers generated at NASA Langley Research Center. / Master of Science

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