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Ubostad, Nikolai Høiland. "The Infinity Laplace Equation." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2013. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-20686.
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In this thesis, we prove that the Infinity-Laplace equation has a unique solution in the viscosity sense. We prove existence by approximating the equation by the p-Laplace equation, and uniqueness will be shown by use of the Theorem on Sums. We will also show that the viscosity solutions of the Infinity-Laplace equation enjoys comparison with cones, and vice versa.
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Fejne, Frida. "The p-Laplace equation – general properties and boundary behaviour." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-359721.
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Mansour, Gihane. "Méthode de décomposition de Domaine pour les équations de Laplace et de Helmholtz : Equation de Laplace non linéaire." Paris 13, 2009. http://www.theses.fr/2009PA132013.
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L'objectif de ce travail est double : D'une part, la résolution à l'aide de la méthode de décomposition de domaine, de l'équation de Poisson et de l'équation de Helmholtz, avec donnée de Dirichlet homogène au bord. D'autre part, l'étude de l'équation de Laplace, avec donnée non linéaire g au bord en se basant sur la méthode du Min-Max. Dans la première partie, nous introduisons les outils indispensables sur lesquels nous nous sommes appuyés pour aborder les équations à résoudre et nous présentons deux méthodes indirectes de résolution de l'équation de Poisson: l'algorithme de Dirichlet-Neumann pénalisé barycentriquement et l'algorithme de Dirichlet-Neumann symétrisé, donné par le problème couplé. Le premier schéma a été proposé et démontré convergent par A. Quarteroni et A. Valli. Nous élaborons dans ce mémoire une nouvelle démonstration de convergence de l'algorithme. Le second schéma est nouveau : la condition de Dirichlet-Neumann est symétrisé. Nous montrons la convergence de cet algorithme vers le problème global. Les études théoriques ont montré que les deux méthodes discrétisées convergent et des estimations d'erreur portant sur l'ordre de la convergence ont été établies. Les résultats déjà trouvés ont été validés par les essais numériques, en utilisant le logiciel Comsol pour le maillage, avec le solveur de Matlab. Notons que l'algorithme symétrisé converge plus rapidement que celui pénalisé. Nous étudions ensuite le problème de Helmholtz avec données mixtes sur le bord actif, qui fournit le cadre du travail nécessaire pour examiner l'algorithme introduit par M. Balabane. Nous analysons les résultats théoriques obtenus et nous testons l'algorithme numériquement. Les essais décèlent une saturation de cette méthode pour le maillage considéré. De plus, cette méthode converge très lentement dans un voisinage de la fréquence résonnante. Une dégradation de la convergence est relevée quand la géométrie du domaine est complexe. Dans la deuxième partie, nous exposons une généralisation de l'étude faite par K. Medville et A. Vogelius, pour la résolution de l'équation de Laplace avec donnée non linéaire au bord. Dans le cas où la fonction est sous-linéaire, nous montrons que le problème admet au moins une solution. L'unicité est obtenue en imposant une condition de monotonie sur la fonction sous-linéaire. Dans le cas sur-linéaire, le nombre de solutions du problème dépend du signe du coefficient multipliant la fonctionThis work is divided into two parts : First, a domain decomposition method for the resolution of the Poisson equation and the Helmholtz equation in a bounded domain,with Dirich let boundary condition. Second, The study of the Laplace equation, with non linear boundary condition g. Using the Min-Max method. First, we elaborate some essential tools to introduce our equations, then we present two indirect methods for solving the Poisson equation : there laxed barycentric Dirichlet-Neumann algorithm and the symmetric Dirichlet-Neumann algorithm. The first algorithm was introduced and studied by A. Quarteroni, A. Valli. We present in this work a new proof of its convergence. The second scheme presented is new : we give asymmetric version of the Dirichlet-Neumann condition. We prove that this algorithm is convergent. The theoretical results show that both of the discretization methods are convergent and estimation son the error of convergence are given. We test the two methods numerically, using Comsol with Matlab solver. We notice that the symmetric method converges faster than the barycentric one
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Rockstroh, Parousia. "Boundary value problems for the Laplace equation on convex domains with analytic boundary." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/273939.
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In this thesis we study boundary value problems for the Laplace equation on do mains with smooth boundary. Central to our analysis is a relation, known as the global relation, that couples the boundary data for a given BVP. Previously, the global re lation has primarily been applied to elliptic PDEs defined on polygonal domains. In this thesis we extend the use of the global relation to domains with smooth boundary. This is done by introducing a new transform, denoted by F_p, that is an analogue of the Fourier transform on smooth convex curves. We show that the F_p-transform is a bounded and invertible integral operator. Following this, we show that the F_p-transform naturally arises in the global relation for the Laplace equation on domains with smooth boundary. Using properties of the F_p-transform, we show that the global relation defines a continuously invertible map between the Dirichlet and Neumann data for a given BVP for the Laplace equation. Following this, we construct a numerical method that uses the global relation to find the Neumann data, given the Dirichlet data, for a given BVP for the Laplace equation on a domain with smooth boundary.
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Masur, Gökce Tuba. "An Adaptive Surface Finite Element Method for the Laplace-Beltrami Equation." Thesis, KTH, Numerisk analys, NA, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-202764.
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In this thesis, we present an adaptive surface finite element method for the Laplace-Beltrami equation. The equation is known as the manifold equivalent of the Laplace equation. A surface finite element method is formulated for this partial differential equation which is implemented in FEniCS, an open source software project for automated solutions of differential equations. We formulate a goal-oriented adaptive mesh refinement method based on a posteriori error estimates which are established with the dual-weighted residual method. Some computational examples are provided and implementation issues are discussed.I den här rapporten presenterar vi en adaptiv finite elementmetod för Laplace-Beltrami ekvationen. Ekvationen är känd som Laplace ekvation på ytor. En finita elementmetod för ytor formuleras för denna partiella differentialekvation vilken implementeras i FEniCS, en open source mjukvara för automatiserad lösning av differentialekvationer. Vi formulerar en mål-orienterad adaptiv nätförfinings-metod baserad på a posteriori feluppskattningar etablerade med hjälp av metoden för dual-viktad residual. Beräkningsexempel presenteras och implementeringen diskuteras
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Ricciotti, Diego. "Regularity of solutions of the p-Laplace equation in the Heisenberg group." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2013. http://amslaurea.unibo.it/5708/.
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Correia, Joaquim, Costa Fernando da, Sackmone Sirisack, and Khankham Vongsavang. "Burgers' Equation and Some Applications." Master's thesis, Edited by Thepsavanh Kitignavong, Faculty of Natural Sciences, National University of Laos, 2017. http://hdl.handle.net/10174/26615.
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In this thesis, I present Burgers' equation and some of its applications. I consider the inviscid and the viscid Burgers' equations and present different analytical methods for their study: the Method of Characteristics for the inviscid case, and the Cole-Hopf Transformation for theviscid one.
Two applications of Burgers' equations are given: one in simple models of Traffic Flow (which have been introduced independently by Lighthill-Whitham and Richards) and another in Coagulation theory (in which we use Laplace Transform to obtain Burgers' equations from the original coagulation integro-differential equation). In both applications we consider only analytical methods.
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Consiglio, Armando. "Time-fractional diffusion equation and its applications in physics." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/13704/.
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In physics, process involving the phenomena of diffusion and wave propagation have great relevance; these physical process are governed, from a mathematical point of view, by differential equations of order 1 and 2 in time. By introducing a fractional derivatives of order $\alpha$ in time, with $0 < \alpha < 1$ or
$1 <= \alpha <= 2$, we lead to process in mathematical physics which we may refer to as fractional phenomena; this is not merely a phenomenological procedure providing an additional fit parameter.
The aim of this thesis is to provide a description of such phenomena adopting a mathematical approach to the fractional calculus.
The use of Fourier-Laplace transform in the analysis of the problem leads to certain special functions, scilicet transcendental functions of the Wright type, nowadays known as M-Wright functions.
We will distinguish slow-diffusion processes ($0 < \alpha < 1$) from intermediate processes ($1 <=\alpha <= 2$), and we point out the attention to the applications of fractional calculus in certain problems of physical interest, such as the Neuronal Cable Theory.
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Chin, P. W. M. (Pius Wiysanyuy Molo). "Contribution to qualitative and constructive treatment of the heat equation with domain singularities." Thesis, University of Pretoria, 2011. http://hdl.handle.net/2263/28554.
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Pichon, Eric. "Novel Methods for Multidimensional Image Segmentation." Diss., Georgia Institute of Technology, 2005. http://hdl.handle.net/1853/7504.
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Artificial vision is the problem of creating systems capable of processing visual information. A fundamental sub-problem of artificial vision is image segmentation, the problem of detecting a structure from a digital image. Examples of segmentation problems include the detection of a road from an aerial photograph or the determination of the boundaries of the brain's ventricles from medical imagery. The extraction of structures allows for subsequent higher-level cognitive tasks. One of them is shape comparison. For example, if the brain ventricles of a patient are segmented, can their shapes be used for diagnosis? That is to say, do the shapes of the extracted ventricles resemble more those of healthy patients or those of patients suffering from schizophrenia?
This thesis deals with the problem of image segmentation and shape comparison in the mathematical framework of partial differential equations. The contribution of this thesis is threefold:
1. A technique for the segmentation of regions is proposed. A cost functional is defined for regions based on a non-parametric functional of the distribution of image intensities inside the region. This cost is constructed to favor regions that are homogeneous. Regions that are optimal with respect to that cost can be determined with limited user interaction.
2. The use of direction information is introduced for the segmentation of open curves and closed surfaces. A cost functional is defined for structures (curves or surfaces) by integrating a local, direction-dependent pattern detector along the structure. Optimal structures, corresponding to the best match with the pattern detector, can be determined using efficient algorithms.
3. A technique for shape comparison based on the Laplace equation is proposed. Given two surfaces, one-to-one correspondences are determined that allow for the characterization of local and global similarity measures. The local differences among shapes (resulting for example from a segmentation step) can be visualized for qualitative evaluation by a human expert. It can also be used for classifying shapes into, for example, normal and pathological classes.
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Tang, Yanfei. "Stratification in Drying Particle Suspensions." Diss., Virginia Tech, 2019. http://hdl.handle.net/10919/87435.
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This thesis is on molecular dynamics studies of drying suspensions of bidisperse nanoparticle mixtures. I first use an explicit solvent model to investigate how the structure of the dry film depends on the evaporation rate of the solvent and the initial volume fractions of the nanoparticles. My simulation results show that the particle mixtures stratify according to their sizes when the suspensions are quickly dried, consistent with the prediction of recent theories. I further show that stratification can be controlled using thermophoresis induced by a thermal gradient imposed on the drying suspension. To model larger systems on longer time scales, I explore implicit solvent models of drying particle suspensions in which the solvent is treated as a uniform viscous background and the liquid-vapor interface is replaced by a potential barrier that confines all the solutes in the solution. Drying is then modeled as a process in which the location of the confining potential is moved. In order to clarify the physical foundation of this moving interface method, I analyze the meniscus on the outside of a circular cylinder and apply the results to understand the capillary force experienced by a spherical particle at a liquid-vapor interface. My analyses show that the capillary force is approximately linear with the displacement of the particle from its equilibrium location at the interface. An analytical expression is derived for the corresponding spring constant that depends on the surface tension and lateral span of the interface and the particle radius. I further show that with a careful mapping, both explicit and implicit solvent models yield similar stratification behavior for drying suspensions of bidisperse particles. Finally, I apply the moving interface method based on an implicit solvent to study the drying of various soft matter solutions, including a solution film of a mixture of polymers and nanoparticles, a suspension droplet of bidisperse nanoparticles, a solution droplet of a polymer blend, and a solution droplet of diblock copolymers.PHD
Drying is a ubiquitous phenomenon. In this thesis, I use molecular dynamics methods to simulate the drying of a suspension of a bidisperse mixture of nanoparticles that have two different radii. First, I use a model in which the solvent is included explicitly as point particles and the nanoparticles are modeled as spheres with finite radii. Their trajectories are generated by numerically solving the Newtonian equations of motion for all the particles in the system. My simulations show that the bidisperse nanoparticle mixtures stratify according to their sizes after drying. For example, a “small-on-top” stratified film can be produced in which the smaller nanoparticles are distributed on top of the larger particles in the drying film. I further use a similar model to demonstrate that stratification can be controlled by imposing a thermal gradient on the drying suspension. I then map an explicit solvent system to an implicit one in which the solvent is treated as a uniform viscous background and only the nanoparticles are kept. The physical foundation of this mapping is clarified. I compare simulations using the explicit and implicit solvent models and show that similar stratification behavior emerge in both models. Therefore, the implicit solvent model can be applied to study much larger systems on longer time scales. Finally, I apply the implicit solvent model to study the drying of various soft matter solutions, including a solution film of a mixture of polymers and nanoparticles, a droplet of a bidisperse nanoparticle suspension, a solution droplet of a polymer blend, and a droplet of a diblock copolymer solution.
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Smith, John Matthew Smith. "The Schrodinger Equation of a Particle in a Time Dependent Electric Field: Case Studies." The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1532045633954294.
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Hein, Hoernig Ricardo Oliver. "Green's functions and integral equations for the Laplace and Helmholtz operators in impedance half-spaces." Phd thesis, Ecole Polytechnique X, 2010. http://pastel.archives-ouvertes.fr/pastel-00006172.
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Dans cette thèse on calcule la fonction de Green des équations de Laplace et Helmholtz en deux et trois dimensions dans un demi-espace avec une condition à la limite d'impédance. Pour les calculs on utilise une transformée de Fourier partielle, le principe d'absorption limite, et quelques fonctions spéciales de la physique mathématique. La fonction de Green est après utilisée pour résoudre numériquement un problème de propagation des ondes dans un demi-espace qui est perturbé de manière compacte, avec impédance, en employant des techniques des équations intégrales et la méthode d'éléments de frontière. La connaissance de son champ lointain permet d'énoncer convenablement la condition de radiation dont on a besoin. Des expressions pour le champ proche et lointain de la solution sont données, dont l'existence et l'unicité sont discutées brièvement. Pour chaque cas un problème benchmark est résolu numériquement. On expose étendument le fond physique et mathématique et on inclut aussi la théorie des problèmes de propagation des ondes dans l'espace plein qui est perturbé de manière compacte, avec impédance. Les techniques mathématiques développées ici sont appliquées ensuite au calcul de résonances dans un port maritime. De la même façon, ils sont appliqués au calcul de la fonction de Green pour l'équation de Laplace dans un demi-plan bidimensionnel avec une condition à la limite de dérivée oblique.
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Heinen, Ismael Rodrigo. "Soluções analíticas da equação de difusão de nêutrons geral por técnicas de transformadas integrais." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2009. http://hdl.handle.net/10183/18298.
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No presente trabalho são apresentadas soluções analíticas das equações de difusão de nêutrons bidimensionais com dois grupos de energia, a saber, nêutrons rápidos e térmicos em uma placa com propriedades homogêneas. Alem disso, são resolvidos detalhadamente os problemas onde a placa homogênea é substituída por duas e quatro regiões, tornando-os não-homogêneos. A partir da aplicação da transformada de Laplace e da Técnica da Transformada Integral Generalizada (GITT), respectivamente, é resolvida em uma forma analítica o problema de autovalor resultante para o fluxo de nêutrons. No problema heterogêneo são usados filtros para homogenizar as condições de contorno não-homogêneas. Esta é a condição para a aplicação da GITT. Os três problemas mencionados acima são resolvidos aplicando primeiramente a GITT, o qual reduz a dimensão da equação de difusão, seguida da aplicação da transformada de Laplace, o qual reduz a ordem da equação. Deste procedimento, resulta um sistema de equações algébricas dependente das constantes de integração. 0 sistema é resolvido usando a técnica da eliminação de Gauss. Os fluxos transformados pela GITT são recuperados invertendo-se analiticamente a transformada de Laplace usando a expansão de Heaviside, os quais ainda dependem das constantes de integração. A partir da aplicação das condições de contorno e de interface (para os problemas não-homogêneos) obtém-se um sistema de equações algébricas homogêneas, de onde é determinado o fator de multiplicação efetivo Keff pelo método da bissecção. As constantes de integração são determinadas fazendo use da potencia prescrita da placa. Assim, os fluxos de nêutrons transformados pela GITT ficam determinados e os fluxos de nêutrons rápidos e térmicos são recuperados através da formula da inversa da GITT, usando a expansão do potencial. Resultados são comparados com a solução do método de diferenças finitas.In the present work we present analytical solutions of the bi-dimensional neutron diffusion equation with two energy groups, i.e. fast and thermal neutrons in a sheet with homogeneous properties. Further we solve the detailed problem where the homogeneous sheet is substituted by two and four regions, rendering the problem a non-homogeneous one. Upon application of the Laplace transform and Generalized Integral Transform Tecnique (GITT), respectively, we solve in an analytical fashion the resulting eigenvalue problem for the neutron flux. In the heterogeneous problem, we use filter functions in order to homogenize the non-homogeneous boundary conditions. This is a condition for the application of GITT. We solve the three problems mentioned above applying first GITT, which reduces the dimension of the diffusion equation followed by the Laplace transform, which reduces the order of the equation. This procedure yields a non-homogeneous algebraic system depending on integration constants. The system is solved using the elimination technique by Gauss. The transformed fluxes by GITT are recovered upon inverting analytically the Laplace transform using Heaviside's expansion which depend still on the integration constants. Upon application of the boundary and interface conditions (for the non-homogeneous problem) one obtains a system of homogeneous algebraic equations, where we determine the effective multiplication factor keff by the bisection method. The integration constants are determined making use of the predefined power of the sheet. Thus the neutron fluxes transformed by GITT are determined and the fast and thermal neutron flux are recovered by the inverse formula of GITT, using the potential expansion. Results are compared to the solution by the finite difference method.
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Santos, Fabiana Alves dos. "Espectro de variedades completas e não-compactas." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25815.
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SANTOS, Fabiana Alves dos. Espectro de variedades completas e não-compactas. 2017. 39 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-13T14:03:43Z No. of bitstreams: 1 2017_tese_fasantos.pdf: 609112 bytes, checksum: 0bbcd05e8e335e0ecb00510e212c4e79 (MD5)
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On this work we study the espectrum of Laplace-Beltrami operator on the warped Riemannian manifold Mn = R_r Sn1, whose warping function is smooth, positive, periodic, with period a and satis_es r0 = min r(t) < p n 1a=_. We show that spectrum there no eingevalue, is formed by a union of closed intervals, and, from the peridicity of r, using the classical Hill's Equations Theory, we conclude the existence of gaps.
Neste trabalho caracterizamos o espectro do operador de Laplace-Beltrami na variedade warped Mn = R_r Sn1 cuja função warping _e suave, positiva, periódica, de período a, e satisfaz r0 = min r(t) < p n 1a=_. Mostramos que tal espectro não possui autovalores, é escrito como a união de intervalos e, da periodicidade de r, utilizamos a clássica teoria a cerca dos operados de Hill, e concluímos e existência de gaps no espectro de M.
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Poignard, Clair. "Méthodes asymptotiques pour le calcul des champs électromagnétiques dans des milieux à couches minces.Application aux cellules biologiques." Phd thesis, Université Claude Bernard - Lyon I, 2006. http://tel.archives-ouvertes.fr/tel-00124110.
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Dans cette thèse, nous présentons des méthodes asymptotiques mathématiquement justifiées permettant de connaître les champs
électromagnétiques dans des milieux à couches minces hétérogènes.
La motivation de ce travail est le calcul du champ électrique dans des
cellules biologiques composées d'un cytoplasme conducteur entouré
d'une fine membrane très isolante.
Nous remplaçons la membrane, lorsque son épaisseur est infiniment
petite, par des conditions de transmission ou des conditions aux
limites appropriées et nous estimons l'erreur commise par ces
approximations.
Pour les basses fréquences, nous considérons l'équation quasistatique
donnant le potentiel dont dérive le champ. A l'aide d'un
calcul en géométrie circulaire nous obtenons les expressions explicites
du potentiel et nous en déduisons les asymptotiques du champ
électrique, en fonction de l'épaisseur de la couche mince, avec des
estimations de l'erreur. Nous estimons ensuite la différence entre le
champ réel et le champ statique. Puis nous généralisons notre
développement asymptotique à une géométrie quelconque.
La deuxième partie de cette thèse traite des moyennes fréquences :
nous donnons le développement asymptotique de la solution de
l'équation de Helmholtz lorsque l'épaisseur de la membrane tend vers
0. Tous ces précédents résultats sont illustrés par des calculs par
éléments finis.
Enfin, pour les hautes fréquences, nous construisons une condition
d'impédance pseudodifférentielle permettant de concentrer l'effet de
la couche sur son bord intérieur. Nous concluons cette thèse par un
problème de diffraction à haute fréquence d'une onde incidente par
un disque de petite taille. A l'aide d'une analyse pseudodifférentielle,
nous bornons la norme de la trace du champ diffracté à distance fixe
de l'inhomogénéité en fonction de la taille de l'objet et de l'onde
incidente.
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Gemmrich, Simon. "A first-kind boundary integral study to solve the Laplace-Beltrami equation on a subsurface of the unit sphere and a multigrid algorithm for the acoustic single layer equation." Thesis, McGill University, 2009. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=66653.
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This dissertation is a study of two independent problems from the common research area of boundary element methods used to solve elliptic boundary value problems. Both topics focus mainly on a single layer approach. For the first project we consider the Laplace-Beltrami Dirichlet problem on a subsurface of the unit sphere in R^3. We derive and analyze a boundary element method for a first-kind integral equation to solve this boundary value problem. The method can be used to study the motion of point vortices on a sphere with impenetrable walls; we compare our approach with previous methods in this field. We derive rigorous error estimates for approximations of the solution to the integral equation in appropriate Sobolev spaces which yield global error estimates for the solution of the boundary value problem. Moreover, we support the theoretical results with numerical evidence gathered from test cases. The second project is concerned with a multigrid preconditioning strategy for the acoustic single layer equation in two dimensions. As proposed in [6], we reformulate the boundary element method in a weaker base inner product and then use a V-cycle multigrid scheme with a Richardson type smoother. We provide a full convergence analysis for the proposed multigrid algorithm based on an analogous result for the single layer equation corresponding to the Laplace operator. Numerical experiments underline the performance of the algorithm. Moreover, we conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the corresponding eigenfunctions.Cette dissertation est une étude de deux problèmes indépendants provenant du domaine commun de la recherche des méthodes d'éléments finis de frontière pour résoudre des problèmes aux limites elliptiques. Les deux sujets portent principalement sur l'approche de l'opérateur simple couche.Pour le premier projet nous considérons le problème de Laplace-Beltrami Dirichlet sur une sous-surface de la sphère de rayon 1. Nous dérivons et analysons une méthode à équations intégrales du premier type pour résoudre ce problème aux limites. Cette méthode peut être utilisée pour étudier le mouvement de tourbillons ponctuels sur une sphère avec des murs impénétrables; nous comparons notre approche avec d'autres méthodes connues pour ce problème. Nous dérivons rigoureusement des estimations d'erreur pour les équations intégrales dans les espaces de Sobolev appropriés, ce qui donne des estimations globales de l'erreur pour la solution du problème aux limites. De plus, nous appuyons ces résultats théoriques grâce à des simulations numériques obtenues à partir de tests.Le deuxième projet porte sur une stratégie de préconditionnement multigrille pour l'équation de simple couche acoustique en deux dimensions. Tel que proposé dans [6], nous transférons la formulation en termes d'éléments finis de frontière à un produit scalaire plus faible et utilisons par la suite une méthode multigrille V-cycle avec un lisseur de type Richardson. Nous fournissons une analyse complète de convergence pour l'algorithme proposé basée sur un résultat analogue pour l'équation de simple couche correspondant à l'opérateur de Laplace. Des simulations numériques soulignent la performance de l'algorithme. De plus, nous faisons une étude numérique de l'effet du produit scalaire faible sur le comportement oscillatoire des fonctions propres correspondantes.
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Erhart, Kevin. "EFFICIENT LARGE SCALE TRANSIENT HEAT CONDUCTION ANALYSIS USING A PARALLELIZED BOUNDARY ELEMENT METHOD." Master's thesis, University of Central Florida, 2006. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2973.
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A parallel domain decomposition Laplace transform Boundary Element Method, BEM, algorithm for the solution of large-scale transient heat conduction problems will be developed. This is accomplished by building on previous work by the author and including several new additions (most note-worthy is the extension to 3-D) aimed at extending the scope and improving the efficiency of this technique for large-scale problems. A Laplace transform method is utilized to avoid time marching and a Proper Orthogonal Decomposition, POD, interpolation scheme is used to improve the efficiency of the numerical Laplace inversion process. A detailed analysis of the Stehfest Transform (numerical Laplace inversion) is performed to help optimize the procedure for heat transfer problems. A domain decomposition process is described in detail and is used to significantly reduce the size of any single problem for the BEM, which greatly reduces the storage and computational burden of the BEM. The procedure is readily implemented in parallel and renders the BEM applicable to large-scale transient conduction problems on even modest computational platforms. A major benefit of the Laplace space approach described herein, is that it readily allows adaptation and integration of traditional BEM codes, as the resulting governing equations are time independent. This work includes the adaptation of two such traditional BEM codes for steady-state heat conduction, in both two and three dimensions. Verification and validation example problems are presented which show the accuracy and efficiency of the techniques. Additionally, comparisons to commercial Finite Volume Method results are shown to further prove the effectiveness.M.S.M.E.
Department of Mechanical, Materials and Aerospace Engineering;
Engineering and Computer Science
Mechanical Engineering
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Chandra, Santanu. "A NUMERICAL STUDY FOR LIQUID BRIDGE BASED MICROGRIPPING AND CONTACT ANGLE MANIPULATION BY ELECTROWETTING METHOD." University of Akron / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=akron1197299987.
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Ugail, Hassan. "Method of trimming PDE surfaces." Elsevier, 2006. http://hdl.handle.net/10454/2648.
Full textAbstract:
A method for trimming surfaces generated as solutions to Partial Differential Equations
(PDEs) is presented. The work we present here utilises the 2D parameter
space on which the trim curves are defined whose projection on the parametrically
represented PDE surface is then trimmed out. To do this we define the trim curves
to be a set of boundary conditions which enable us to solve a low order elliptic
PDE on the parameter space. The chosen elliptic PDE is solved analytically, even
in the case of a very general complex trim, allowing the design process to be carried
out interactively in real time. To demonstrate the capability for this technique we
discuss a series of examples where trimmed PDE surfaces may be applicable.
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Childers, Kristen Snyder. "Generalizations of a Laplacian-Type Equation in the Heisenberg Group and a Class of Grushin-Type Spaces." Scholar Commons, 2011. http://scholarcommons.usf.edu/etd/3042.
Full textAbstract:
In [2], Beals, Gaveau and Greiner find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This fundamental solution is based on the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces [4] and the Heisenberg group [6]. In this thesis, we look to generalize the work in [2] for a p-Laplace-type equation. After discovering that the "natural" generalization fails, we find two generalizations whose solutions are based on the fundamental solution to the p-Laplace equation.
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Norotte, Cyrille. "De la biologie du développement à l'ingénierie tissulaire : impression de vaisseaux sanguins." Paris 6, 2009. http://www.theses.fr/2009PA066717.
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Les maladies cardiovasculaires représentent actuellement une des causes principales de mortalité et nécessitent souvent le recours à la chirurgie vasculaire reconstructrice. Nous exploitons ici des processus développementaux comparables aux phénomènes associés aux liquides (tels que la fusion de tissus, leur enveloppement mutuel ou les phénome��nes de « sorting » cellulaire) dans le but de fabriquer des vaisseaux sanguins de petit calibre, comportant des couches cellulaires distinctes. En particulier, nous montrons que les tensions de surface associées aux trois types cellulaires vasculaires majeurs (cellules endothéliales, cellules musculaires lisses, et fibroblastes), et calculées en utilisant la solution exacte de l’équation de Laplace, guident leur ségrégation en couches distinctes in vitro. Une nouvelle technologie de prototypage rapide, appelée « bioprinting », permet de guider l’auto-assemblage des différents types cellulaires vasculaires en structures tissulaires tubulaires de géométrie sur mesure, de tubes vasculaires simples à des arbres vasculaires complexes, potentiellement utilisables pour la médecine régénératrice.
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Li, Boning. "Extending the scaled boundary finite-element method to wave diffraction problems." University of Western Australia. School of Civil and Resource Engineering, 2007. http://theses.library.uwa.edu.au/adt-WU2007.0173.
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[Truncated abstract] The study reported in this thesis extends the scaled boundary finite-element method to firstorder and second-order wave diffraction problems. The scaled boundary finite-element method is a newly developed semi-analytical technique to solve systems of partial differential equations. It works by employing a special local coordinate system, called scaled boundary coordinate system, to define the computational field, and then weakening the partial differential equation in the circumferential direction with the standard finite elements whilst keeping the equation strong in the radial direction, finally analytically solving the resulting system of equations, termed the scaled boundary finite-element equation. This unique feature of the scaled boundary finite-element method enables it to combine many of advantages of the finite-element method and the boundaryelement method with the features of its own. ... In this thesis, both first-order and second-order solutions of wave diffraction problems are presented in the context of scaled boundary finite-element analysis. In the first-order wave diffraction analysis, the boundary-value problems governed by the Laplace equation or by the Helmholtz equation are considered. The solution methods for bounded domains and unbounded domains are described in detail. The solution process is implemented and validated by practical numerical examples. The numerical examples examined include well benchmarked problems such as wave reflection and transmission by a single horizontal structure and by two structures with a small gap, wave radiation induced by oscillating bodies in heave, sway and roll motions, wave diffraction by vertical structures with circular, elliptical, rectangular cross sections and harbour oscillation problems. The numerical results are compared with the available analytical solutions, numerical solutions with other conventional numerical methods and experimental results to demonstrate the accuracy and efficiency of the scaled boundary finite-element method. The computed results show that the scaled boundary finite-element method is able to accurately model the singularity of velocity field near sharp corners and to satisfy the radiation condition with ease. It is worth nothing that the scaled boundary finite-element method is completely free of irregular frequency problem that the Green's function methods often suffer from. For the second-order wave diffraction problem, this thesis develops solution schemes for both monochromatic wave and bichromatic wave cases, based on the analytical expression of first-order solution in the radial direction. It is found that the scaled boundary finiteelement method can produce accurate results of second-order wave loads, due to its high accuracy in calculating the first-order velocity field.
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Paditz, Ludwig. "Using ClassPad-technology in the education of students of electricalengineering (Fourier- and Laplace-Transformation)." Proceedings of the tenth International Conference Models in Developing Mathematics Education. - Dresden : Hochschule für Technik und Wirtschaft, 2009. - S. 469 - 474, 2012. https://slub.qucosa.de/id/qucosa%3A1799.
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By the help of several examples the interactive work with the ClassPad330 is considered. The student can solve difficult exercises of practical applications step by step using the symbolic calculation and the graphic possibilities of the calculator. Sometimes several fields
of mathematics are combined to solve a problem. Let us consider the ClassPad330 (with the actual operating system OS 03.03) and discuss on some new exercises in analysis, e.g. solving a linear differential equation by the help of the Laplace transformation and using the inverse Laplace transformation or considering the Fourier transformation in discrete time (the Fast Fourier Transformation FFT and the inverse FFT). We use the FFT- and IFFT-function to study periodic signals, if we only have a sequence generated by sampling the time signal. We know several ways to get a solution. The techniques for studying practical applications fall into the following three categories: analytic, graphic and numeric. We can use the
Classpad software in the handheld or in the PC (ClassPad emulator version of the handheld).
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Makhmudov, O. I., and I. E. Niyozov. "The Cauchy problem for the Lame system in infinite domains in R up(m)." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2996/.
Full textAbstract:
We consider the problem of analytic continuation of the solution of the multidimensional Lame system in infinite domains through known values of the solution and the corresponding strain tensor on a part of the boundary, i.e,the Cauchy problem.
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Paditz, Ludwig. "Using ClassPad-technology in the education of students of electrical engineering (Fourier- and Laplace-Transformation)." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-80814.
Full textAbstract:
By the help of several examples the interactive work with the ClassPad330 is considered. The student can solve difficult exercises of practical applications step by step using the symbolic calculation and the graphic possibilities of the calculator. Sometimes several fields
of mathematics are combined to solve a problem. Let us consider the ClassPad330 (with the actual operating system OS 03.03) and discuss on some new exercises in analysis, e.g. solving a linear differential equation by the help of the Laplace transformation and using the inverse Laplace transformation or considering the Fourier transformation in discrete time (the Fast Fourier Transformation FFT and the inverse FFT). We use the FFT- and IFFT-function to study periodic signals, if we only have a sequence generated by sampling the time signal. We know several ways to get a solution. The techniques for studying practical applications fall into the following three categories: analytic, graphic and numeric. We can use the
Classpad software in the handheld or in the PC (ClassPad emulator version of the handheld).
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Alvarellos, Jose. "Fundamental Studies of Capillary Forces in Porous Media." Diss., Georgia Institute of Technology, 2004. http://hdl.handle.net/1853/5314.
Full textAbstract:
The contact angle defined by Young's equation depends on the ratio between solid and liquid surface energies. Young's contact angle is constant for a given system, and cannot explain the stability of fluid droplets in capillary tubes. Within this framework, large variations in contact angle and explained aassuming surface roughness, heterogeneity or contamination. This research explores the static and dynamic behavior of fluid droplets within capillary tubes and the variations in contact angle among interacting menisci. Various cases are considered including wetting and non-wetting gluids, droplets in inclined capillary tubes or subjected to a pressure difference, within one-dimensional and three-dimensional capillary systems, and under static or dynamic conditions (either harmonic fluid pressure or tube oscillation). The research approach is based on complementary analytical modeling (total energy formulation) and experimental techniques (microscopic observations). The evolution of meniscus curvatures and droplet displacements are studied in all cases. Analytical and experimental results show that droplets can be stable within capillary tubes even under the influence of an external force, the resulting contact angles are not constant, and bariations from Young's contact angle aare extensively justified as menisci interaction. Menisci introduce stiffness, therefore two immiscible Newtonian fluids behave as a Maxwellian fluid, and droplets can exhibit resonance or relaxation spectral features.
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I, Makhmudov O., and Niyozov I. E. "Regularization of the Cauchy Problem for the System of Elasticity Theory in R up (m)." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2998/.
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Souza, José Luiz de. "A h-adaptabilidade no Método dos Elementos de Contorno (MEC): algumas considerações sobre singularidades, hipersingularidades e hierarquia." Universidade de São Paulo, 1999. http://www.teses.usp.br/teses/disponiveis/18/18134/tde-13122017-152526/.
Full textAbstract:
O principal objetivo deste trabalho é estudar as singularidades e hipersingularidades existentes nas formulações: singular - clássica - e hipersingular no Método dos Elementos de Contorno (MEC). Também é proposto um esquema residual h-adaptativo para a solução numérica do problema físico governado pela equação de Laplace. Usa-se malha poligonal, juntamente, com funções de interpolação - distribuição - de forma, dos tipos: constantes e lineares. Para controlar o erro a posteriori, é considerado o valor do resíduo, fora dos pontos de colocação. Também é testada uma técnica de quadratura numérica chamada adaptativa, específica para subelementos, no sentido de verificar se a precisão no cálculo das integrais com singularidades é melhorada. O uso de funções hierárquicas é discutido na forma de um algoritmo para atualização da matriz principal do sistema linear.The main purpose of this work is to study the existing singularities and hypersingularities in the Boundary Element Method (BEM) with singular - classical - and hypersingular formulations. Also, an h-adaptive residual scheme for the numerical solution of the physical problem, driven by Laplace equation, is proposed. Boundary polygonal mesh, with constant and linear interpolation - distribution - shape functions together are used. To control the a posteriori error, is considered the residue value outside the collocation points. Also, a sub-element specific adaptive numerical quadrature technique, in an effort to verify if the precision when dealing with integrals possessing singularities is increased, is tested. The use of hierarchical functions is discussed, as an algorithm to update the linear system main matrix.
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Mat, Isa Zaiton. "Mathematical modelling of fumigant transport in stored grain." Thesis, Queensland University of Technology, 2014. https://eprints.qut.edu.au/75420/1/Zaiton_Mat%20Isa_Thesis.pdf.
Full textAbstract:
Computational fluid dynamics, analytical solutions, and mathematical modelling approaches are used to gain insights into the distribution of fumigant gas within farm-scale, grain storage silos. Both fan-forced and tablet fumigation are considered in this work, which develops new models for use by researchers, primary producers and silo manufacturers to assist in the eradication grain storage pests.
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Musolino, Paolo. "Singular perturbation and homogenization problems in a periodically perforated domain. A functional analytic approach." Doctoral thesis, Università degli studi di Padova, 2012. http://hdl.handle.net/11577/3422452.
Full textAbstract:
This Dissertation is devoted to the singular perturbation and homogenization analysis
of boundary value problems in the periodically perforated Euclidean space. We investigate the behaviour of the solutions of boundary value problems for the Laplace, the Poisson, and the Helmholtz equations, as parameters related to diameter of the holes or the size of the periodicity cells tend to 0.
The Dissertation is organized as follows.
In Chapter 1, we present two known constructions of a periodic analogue of the fundamental solution of the Laplace equation and we introduce the periodic layer and volume potentials for the Laplace equation and some basic results of periodic potential theory. Chapter 2 is devoted to singular perturbation and homogenization problems for the Laplace and the Poisson equations with Dirichlet and Neumann boundary conditions. In Chapter 3 we consider the case of (linear and nonlinear) Robin boundary value problems for the Laplace equation, while in Chapter 4 we analyze (linear and nonlinear) transmission problems. In Chapter 5 we apply the results of Chapter 4 in order to prove the real analyticity of the effective conductivity of a periodic dilute composite. Chapter 6 is dedicated to the construction of a periodic analogue of the fundamental solution of the Helmholtz equation and of the corresponding periodic layer potentials. In Chapter 7 we collect some results of spectral theory for the Laplace operator in periodically perforated domains. In Chapter 8 we investigate singular perturbation and homogenization problems for the Helmholtz equation with Neumann boundary conditions. In Chapter 9 we consider singular perturbation and homogenization problems with Dirichlet boundary conditions for the Helmholtz equation, while in Chapter 10 we study (linear and nonlinear) Robin boundary value problems. Chapter 11 is devoted to the study of periodic layer potentials for general second order differential operators with constant coefficients. At the end of the Dissertation we have enclosed some Appendices with some results that we have exploited.Questa Tesi è dedicata all'analisi di problemi di perturbazione singolare e omogeneizzazione nello spazio Euclideo periodicamente perforato. Studiamo il comportamento delle soluzioni di problemi al contorno per le equazioni di Laplace, di Poisson e di Helmholtz al tendere a 0 di parametri legati al diametro dei buchi o alla dimensione delle celle di periodicità. La Tesi è organizzata come segue. Nel Capitolo 1, presentiamo due costruzioni note di un analogo periodico della soluzione fondamentale dell'equazione di Laplace, e introduciamo potenziali di strato e di volume periodici per l'equazione di Laplace e alcuni risultati basilari di teoria del potenziale periodica. Il Capitolo 2 è dedicato a problemi di perturbazione singolare e omogeneizzazione per le equazioni di Laplace e Poisson con condizioni al bordo di Dirichlet e Neumann. Nel Capitolo 3 consideriamo il caso di problemi al contorno di Robin (lineari e nonlineari) per l'equazione di Laplace, mentre nel Capitolo 4 analizziamo problemi di trasmissione (lineari e nonlineari). Nel Capitolo 5 applichiamo i risultati del Capitolo 4 al fine di provare l'analiticità della conduttività effettiva di un composto periodico. Il Capitolo 6 è dedicato alla costruzione di un analogo periodico della soluzione fondamentale dell'equazione di Helmholtz e dei corrispondenti potenziali di strato. Nel Capitolo 7 raccogliamo alcuni risultati di teoria spettrale per l'operatore di Laplace in domini periodicamente perforati. Nel Capitolo 8 studiamo problemi di perturbazione singolare e di omogeneizzazione per l'equazione di Helmholtz con condizioni al contorno di Neumann. Nel Capitolo 9 consideriamo problemi di perturbazione singolare e di omogeneizzazione con condizioni al contorno di Dirichlet per l'equazione di Helmholtz, mentre nel Capitolo 10 studiamo problemi al contorno di Robin (lineari e nonlineari). Il Capitolo 11 è dedicato allo studio di potenziali di strato periodici per operatori differenziali generali del secondo ordine a coefficienti costanti. Alla fine della Tesi abbiamo incluso delle Appendici con alcuni risultati utilizzati.
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Ladeia, Cibele Aparecida. "A equação de transferência radiativa condutiva em geometria cilíndrica para o problema do escape do lançamento de foguetes." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/152738.
Full textAbstract:
Nesta contribuição apresentamos uma solução para a equação de transferência radiativa condutiva em geometria cilíndrica. Esta solução é aplicada para simular a radiação e campo de temperatura juntamente com o transporte de energia radiativa e condutiva proveniente do escape liberado em lançamentos de foguetes. Para este fim, discutimos uma abordagem semianalítica reduzindo a equação original, que é contínua nas variáveis angulares, numa equação semelhante ao problema SN da transferência radiativa condutiva. A solução é construída usando um método de composição por transformada de Laplace e o método da decomposição de Adomian. O esquema recursivo ´e apresentado para o sistema de equações de ordenadas duplamente discretas juntamente com as dependências dos parâmetros e suas influências sobre a convergência heurística da solução. A solução obtida, em seguida, permite construir o campo próximo relevante para caracterizar o termo fonte para problemas de dispersão ao ajustar os parâmetros do modelo, tais como, emissividade, refletividade, albedo e outros, em comparação com a observação, que são relevantes para os processos de dispersão de campo distante e podem ser manipulados de forma independente do presente problema. Além do método de solução, também relatamos sobre algumas soluções e simulações numéricas.In this contribution we present a solution for the radiative conductive transfer equation in cylinder geometry. This solution is applied to simulate the radiation and temperature field together with conductive and radiative energy transport originated from the exhaust released in rocket launches. To this end we discuss a semi-analytical approach reducing the original equation, which is continuous in the angular variables, into an equation similar to the SN radiative conductive transfer problem. The solution is constructed using a composite method by Laplace transform and Adomian decomposition method. The recursive scheme is presented for the doubly discrete ordinate equations system together with parameter dependencies and their influence on heuristic convergence of the solution. The obtained solution allows then to construct the relevant near field to characterize the source term for dispersion problems when adjusting the model parameters such as emissivity, reflectivity, albedo and others in comparison to the observation, that are relevant for far field dispersion processes and may be handled independently from the present problem. In addition to the solution method we also report some solutions and numerical simulations.
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Abramov, Vilen. "Stopping Times Related to Trading Strategies." Kent State University / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=kent1209080577.
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Feng, Xue. "Modélisation numérique par éléments finis d'un problème aéroacoustique en régime transitoire : application à l'équation de Galbrun." Phd thesis, Université de Technologie de Compiègne, 2013. http://tel.archives-ouvertes.fr/tel-00935560.
Full textAbstract:
Les travaux de cette thèse concernent la modélisation et la simulation numérique de la propagation d'ondes acoustiques en présence d'un écoulement. Le modèle retenu pour ces études est l'équation de Galbrun. Les travaux faits sur l'équation de Galbrun ont essentiellement porté sur le régime harmonique. En revanche, la plupart des études mathématiques et numériques du problème de l'aéroacoustique est en régime transitoire. C'est pourquoi, il est intéressant pour nous d'étudier l'équation de Galbrun en régime transitoire. Pour résoudre cette équation en régime transitoire, notre approche a reposé sur la transformée de Laplace, qui nous permet de faire l'échange entre le domaine harmonique et le domaine réel. Un autre sujet abordé dans cette thèse est celui du traitement des conditions aux limites non réfléchissantes en écoulement uniforme et non-uniforme. Nous proposons la méthode PML pour l'équation de Galbrun. Inspirée par la méthode de Hu, nous proposons un nouveau modèle PML associé à l'équation de Galbrun, qui a toujours conduit à une solution exponentiellement décroissante dans la couche, même en présence d'ondes inverses. Les simulations acoustiques montrent étonnamment d'erreur de convergence pour les deux modèles classiques et nouveaux. Nous validons notre modèle PML à travers plusieurs exemples numériques dans l'écoulement uniforme et non-uniforme. Le dernier objectif est de proposer des modèles de sources aéroacoustiques associées à l'équation de Galbrun. Après une présentation en détail des modèles existants, on adapte une méthode hybride (EIF) à l'équation de Galbrun. Pour assurer la validité de l'approche globale, certains tests classiques sont choisis parmi la littérature et les résultats sont comparés avec les approches existantes et les solutions analytiques.
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Thompson, Jeremy R. (Jeremy Ray). "Physical Motivation and Methods of Solution of Classical Partial Differential Equations." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc277898/.
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We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.
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Avelin, Benny. "Boundary Behavior of p-Laplace Type Equations." Doctoral thesis, Uppsala universitet, Analys och tillämpad matematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-198008.
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This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations. Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary. Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough. Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough. In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling. Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case. In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ. Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.
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Pester, M., and S. Rjasanow. "A parallel version of the preconditioned conjugate gradient method for boundary element equations." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800455.
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The parallel version of precondition techniques is developed for
matrices arising from the Galerkin boundary element method for
two-dimensional domains with Dirichlet boundary conditions.
Results were obtained for implementations on a transputer network
as well as on an nCUBE-2 parallel computer showing that iterative
solution methods are very well suited for a MIMD computer. A
comparison of numerical results for iterative and direct solution
methods is presented and underlines the superiority of iterative
methods for large systems.
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Putot, Sylvie. "Calcul des capacités parasites dans les interconnexions des circuits intégrés par une méthode de domaines fictifs." Phd thesis, Université Joseph Fourier (Grenoble ; 1971-2015), 2001. http://www.theses.fr/2001GRE10015.
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Cette these presente une methode performante pour le calcul des capacites parasites dues aux interconnexions des circuits integres. Il s'agit de calculer la charge des conducteurs, comme la derivee normale a la surface de ces conducteurs, du potentiel solution de l'equation de laplace sur des couches horizontales, la valeur du potentiel etant fixee constante sur chaque conducteur. La difficulte de la resolution numerique provient de la complexite des structures : sur une portion de circuit d'une surface d'un centimetre carre et d'une hauteur de quelques microns, il peut y avoir plus d'un kilometre d'interconnexions, c'est-a-dire de fils conducteurs enchevetres. Une methode de domaines fictifs avec multiplicateurs de lagrange surfaciques est utilisee. Elle donne une formulation mixte du probleme, couplant le potentiel sur un domaine parallelepipedique contenant le circuit, et la charge a la surface des conducteurs. Nous en proposons une approximation, qui tient compte du saut du gradient du potentiel a travers la surface des conducteurs dans la discretisation du potentiel, tout en menant a un systeme que l'on peut resoudre par une methode rapide. Cette approximation garantit une bonne convergence du calcul de la charge vers la valeur reelle, sans condition de compatibilite contraignante entre les maillages de volume et de surface. Une implementation efficace en dimension 3, avec laquelle nous avons effectue des tests numeriques sur des structures reelles, permet de montrer l'interet de la methode, en temps de calcul et en place memoire.
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Kisela, Tomáš. "Basics of Qualitative Theory of Linear Fractional Difference Equations." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2012. http://www.nusl.cz/ntk/nusl-234025.
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Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.
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Andreevska, Irena. "Mathematical modeling and analysis of options with jump-diffusion volatility." [Tampa, Fla.] : University of South Florida, 2008. http://purl.fcla.edu/usf/dc/et/SFE0002343.
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Šafařík, Jan. "Slabě zpožděné systémy lineárních diskrétních rovnic v R^3." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2018. http://www.nusl.cz/ntk/nusl-378908.
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Dizertační práce se zabývá konstrukcí obecného řešení slabě zpožděných systémů lineárních diskrétních rovnic v ${\mathbb R}^3$ tvaru \begin{equation*} x(k+1)=Ax(k)+Bx(k-m), \end{equation*} kde $m>0$ je kladné celé číslo, $x\colon \bZ_{-m}^{\infty}\to\bR^3$, $\bZ_{-m}^{\infty} := \{-m, -m+1, \dots, \infty\}$, $k\in\bZ_0^{\infty}$, $A=(a_{ij})$ a $B=(b_{ij})$ jsou konstantní $3\times 3$ matice. Charakteristické rovnice těchto systémů jsou identické s charakteristickými rovnicemi systému, který neobsahuje zpožděné členy. Jsou získána kriteria garantující, že daný systém je slabě zpožděný a následně jsou tato kritéria specifikována pro všechny možné případy Jordanova tvaru matice $A$. Systém je vyřešen pomocí metody, která ho transformuje na systém vyšší dimenze, ale bez zpoždění \begin{equation*} y(k+1)=\mathcal{A}y(k), \end{equation*} kde ${\mathrm{dim}}\ y = 3(m+1)$. Pomocí metod lineární algebry je možné najít Jordanovy formy matice $\mathcal{A}$ v závislosti na vlastních číslech matic $A$ and $B$. Tudíž lze nalézt obecné řešení nového systému a v důsledku toho pak odvodit obecné řešení počátečního systému.
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Hansson, Mattias. "Numerical experiments with FEMLAB® to support mathematical research." Thesis, Linköping University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-3724.
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Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(u) ≡ u2xuxx + 2uxuyuxy + u2yuyy = 0. For numerical reasons ∆q(u) = div (|▼u|q▼u) = 0, which (formally) approaches as ∆∞(u) = 0 as q → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(u) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.
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Ngounda, Edgard. "Numerical Laplace transformation methods for integrating linear parabolic partial differential equations." Thesis, Stellenbosch : University of Stellenbosch, 2009. http://hdl.handle.net/10019.1/2735.
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Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009.ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative method for the numerical solution of PDEs. Effective methods for the numerical inversion are based on the approximation of the Bromwich integral. In this thesis, a numerical study is undertaken to compare the efficiency of the Laplace inversion method with more conventional time integrator methods. Particularly, we consider the method-of-lines based on MATLAB’s ODE15s and the Crank-Nicolson method. Our studies include an introductory chapter on the Laplace inversion method. Then we proceed with spectral methods for the space discretization where we introduce the interpolation polynomial and the concept of a differentiation matrix to approximate derivatives of a function. Next, formulas of the numerical differentiation formulas (NDFs) implemented in ODE15s, as well as the well-known second order Crank-Nicolson method, are derived. In the Laplace method, to compute the Bromwich integral, we use the trapezoidal rule over a hyperbolic contour. Enhancement to the computational efficiency of these methods include the LU as well as the Hessenberg decompositions. In order to compare the three methods, we consider two criteria: The number of linear system solves per unit of accuracy and the CPU time per unit of accuracy. The numerical results demonstrate that the new method, i.e., the Laplace inversion method, is accurate to an exponential order of convergence compared to the linear convergence rate of the ODE15s and the Crank-Nicolson methods. This exponential convergence leads to high accuracy with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the Crank-Nicolson method as the results show. Finally, we apply with satisfactory results the inversion method to the axial dispersion model and the heat equation in two dimensions.
AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die benadering van die Bromwich integraal. In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer op MATLAB se ODE15s en die Crank-Nicolson metode. Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode. Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie, waar ons die interpolasie polinoom invoer sowel as die konsep van ’n differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word. Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste van al hierdie metodes word verbeter met die LU sowel as die Hessenberg ontbindings. Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode, akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson metode. Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.
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Green, Edward L. "Spectral theory of laplace-beltrami operators with periodic metrics." Diss., Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/29187.
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Lustosa, José Ivelton Siqueira Lustosa. "A transformada de Laplace e algumas aplicações." Universidade Federal da Paraíba, 2017. http://tede.biblioteca.ufpb.br:8080/handle/tede/9332.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this work, we study the Laplace Transform and explore its application in solving some linear ordinary di erential equations, which model various phenomena in the areas of Physics, Engineering, Industrial Automation and Mathematics itself. Such knowledge is of great importance in higher education courses covering such areas. We present the de nition, properties and main results involving the Laplace Transform and address several problems in the areas mentioned above.
Neste trabalho, estudamos a Transformada de Laplace e exploramos sua aplica ção na resolução de algumas equações diferenciais ordinárias lineares, as quais modelam vários fenômenos nas áreas de Física, Engenharia, Automação Industrial e na própria Matemática. Tais conhecimentos são de suma importância em cursos superiores que abrangem tais áreas. Apresentamos a de nição, propriedades e principais resultados envolvendo a Transformada de Laplace e abordamos vários problemas nas áreas citadas anteriormente.
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Yang, Qianqian. "Novel analytical and numerical methods for solving fractional dynamical systems." Thesis, Queensland University of Technology, 2010. https://eprints.qut.edu.au/35750/1/Qianqian_Yang_Thesis.pdf.
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During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.
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Fitzharris, Andrew. "Parallel solution of diffusion equations using Laplace transform methods with particular reference to Black-Scholes models of financial options." Thesis, University of Hertfordshire, 2014. http://hdl.handle.net/2299/14109.
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Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical finance. For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE techniques. To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the design of parallel programs and the parameter values used. To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference solutions needed to determine the accuracy of the LTFD solutions. Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas.
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Dutriaux, Antoine. "Analyse et modèles dynamiques non commutatifs sur l'espace de q-Minkowski." Phd thesis, Université de Valenciennes et du Hainaut-Cambresis, 2008. http://tel.archives-ouvertes.fr/tel-00289899.
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Cette thèse se place dans le cadre du vaste domaine s'intitulant géométrie non commutative, domaine dont l'étude est motivée par l'opinion courante des mathématiciens et physiciens selon laquelle les méthodes de la géométrie non commutative peuvent être utiles pour décrire certains processus dynamiques à l'échelle de Planck. Aussi l'objectif principal de cette thèse est de généraliser quelques modèles dynamiques définis sur l'espace de Minkowski sur son q-analogue. Des tentatives d'introduire des modèles dynamiques qui seraient covariants par rapport à l'action de groupes quantiques ont été entrepris juste après la création de la théorie sur les groupes quantiques par Drinfeld. Les modèles les plus intéressants sont ceux qui sont liés au q-analogue de l'espace de Minkowski. C'est P. Kulish qui définit cette algèbre comme étant un cas particulier d'une algèbre appelée modified Reflection Equation Algebra (mREA) elle-même liée à un opérateur appelé symétrie de Hecke. Nous définissons donc certains modèles dynamiques qui sont des déformations de modèles classiques, l'espace des phases de nos modèles déformés n'est autre alors que notre espace de q-Minkowski. Nous recherchons par la suite des intégrales de mouvement de ces dynamiques, ce qui nous amène à définir des analogues de l'énergie et du vecteur de Runge-Lenz. Nous généralisons pour terminer les équations aux dérivées partielles de la théorie des champs et en particulier l'opérateur de Maxwell.
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Ngounda, Edgard. "Efficient numerical methods based on integral transforms to solve option pricing problems." Thesis, University of the Western Cape, 2012. http://hdl.handle.net/11394/4223.
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Philosophiae Doctor - PhDIn this thesis, we design and implement a class of numerical methods (based on integral transforms) to solve PDEs for pricing a variety of financial derivatives. Our approach is based on spectral discretization of the spatial (asset) derivatives and the use of inverse Laplace transforms to solve the resulting problem in time. The conventional spectral methods are further modified by using piecewise high order rational interpolants on the Chebyshev mesh within each sub-domain with the boundary domain placed at the strike price where the discontinuity is located. The resulting system is then solved by applying Laplace transform method through deformation of a contour integral. Firstly, we use this approach to price plain vanilla options and then extend it to price options described by a jump-diffusion model, barrier options and the Heston’s volatility model. To approximate the integral part in the jump-diffusion model, we use the Gauss-Legendre quadrature method. Finally, we carry out extensive numerical simulations to value these options and associated Greeks (the measures of sensitivity). The results presented in this thesis demonstrate the spectral accuracy and efficiency of our approach, which can therefore be considered as an alternative approach to price these class of options.
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Fischer, Emily M. "Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace-Beltrami Equations on the Unit Sphere." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/hmc_theses/62.
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I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.