DIFFERENTIAL EQUATIONS IN PARTIAL DERIVATIVES

Author: GALÁN DEL SASTRE PEDRO.
Year: 2003.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: FACULTAD DE CIENCIAS MATEMÁTICAS.
Place of preparation: FACULTAD DE CIENCIAS MATEMÁTICAS.

Summary: This memory has developed a new model barotrópico movement of the ocean, whose main characteristics is focused on retaining the effect bathymetry exercises on the dynamics of the oceans. This model is clear from the Navier-Stokes equations for three-dimensional incompressible flows moving rotation. Taking variables as average speeds on each horizontal water column, it shows the new models EDP (now two-dimensional) models where the bathymetry is considered flat, it is only the well-known formulation vorticidad-función flow. Under certain assumptions physical, is also a study on the existence and uniqueness of solutions of the model, as well as on the existence of global atractores. Then there is a numerical scheme (using the method of characteristics, together with the method of elements fínitos) applied to several case studies. First, we studied an ocean idealized with constant depth. It validates the scheme semilgrangiano against one of the most frequently used in all kinds of meteorological and oceanographic models: the outline Leap-Frog. Secondly deals with the North Atlantic Ocean, where both efforts wind regarded as bathymetry, are derived from databases realistic. Some other examples are studied also to validate the model, which stresses the Mediterranean Sea. Finally, it describes in detail the method of Functions Ortogonales Empirical, and the methods p-utilizan the atractores numeric calculated using the schemes described above, in order to obtain a global basis in the space of solutions, which is capable of retaining the information described in all of them. Once built the base, we make a projection Galerkin, which generates a dynamic system size (dimension finite), ie an EDO, which is equivalent to other model line. Specifically, this study focuses on obtaining the bifurcation diagram of the model on two case studies examined using the horizontal and constant viscosity parameter bifurcation (as this, the more uncertainty introduced parameter).

Author: PELLICER SABADI MARTA.
Year: 2003.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: SALA D'ACTES DE LA FME.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Author: VERDERA RIBAS JUAN.
Year: 2003.
University: POMPEU FABRA [www.upf.edu].
Place of defense: DEPARTAMENTO DE TECNOLOGIA.
Place of preparation: DEPARTAMENTO DE TECNOLOGIA.

Summary: This thesis presents two interpolation algorithms applied to a number of problems in digital image processing. The first algorithm fills holes in images. It applies to problems such as the restoration of old photographs, zoom images, removing subtitles, advertising and unwanted objects. The expansion of 2D to 3D allows fill volumes in 3D images. With applications in the restoration of films, interpolation tomography, or reconstruction of holes in surfaces triangularizadas. The second algorithm generates a merger P & Xs imaging satellite. From an image at high resolution panchromatic and multispectral creates a low-resolution multispectral image at high resolution. Both algorithms are based on the theory of morphology Mathematics, essentially on the principle that sets the topographic map of an image as the instrument that gives full description of the geometry of images to gray levels. His calculation is performed by minimizing the energy of the proposed functional in solving equations of second order partial derivatives.

Author: NARBONA REINA GLADYS.
Year: 2004.
University: SEVILLA [www.us.es].
Place of defense: E.T.S. DE ARQUITECTURA.
Place of preparation: FACULTAD DE MATEMÁTICAS.

Summary: This memory is several aspects of the numerical modeling of physical systems in the field of fluid mechanics. Specifically, we have been dedicated to the study of those physical systems that can be modeled by problems convection, diffusion phenomenon where convection is dominante.Y in the mathematical aspect, the effect of convection dominant makes makes them so difficult to approximate eficaz.En this field we have tried in this paper some of the most effective methods of resolution both in the theoretical and practical aspect. The thesis is divided into two partes.En the first one is the theoretical analysis of distributional methods applied to the discretization of the term of convección.Hemos given a precise meaning to the definition of these methods and characterization of its main propiedades.Se analyzes also an extension of the Ways Distributivos exposed LDA N-esquema and ISPs to flow tridimensionales.A then conducted an interpretaciÂ'n of these schemes as methods Petrov-Galerkin, through the definition of an operator of interpolation distributed uniformly stable and with interpolation error estimated depending on the size of the class. This interpretation coupled with the good properties of PSI scheme (positive and second order), you can perform the adaptation of the standard analysis of convergence and error estimates in standard H1 to the problem of convection difusioón and rule H1 in speed and L2 in pressure for the Navier Stokes equations. In the second part we made the effective resolution of some problems with implementation in Architecture and ingeniería.En First, the numerical modeling of a flowmeter térmico.Se involve studying the evolution of the temperature reached for the water to flow around a plate heat and compare the results with experimental data. Problems with an interest in architecture are related to the interior design and exteriro of edificios.Hemos solved two problems: cooling a floor of a building (both natural ventilation as a mechanism artificial refrigeration) and the dissemination of contminantes in the atmosphere in a domain where three buildings involved and the continuing expulsion by one of a gas contaminante.Estos problems have been resolved through software Freefem + +, free distribución.También we have made the development of a graphical interface using the Matlab package, in order to facilitate the graphical display of the results obtained by calculating Freefem + +.

Author: TAJDINE ANAS.
Year: 2004.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: UNIVERSIDAD COMPLUTENSE DE MADRID FAC. DE CC. MATEMÁTICAS.
Place of preparation: UNIVERSIDAD COMPLUTENSE DE MADRID.

Summary: The thesis explores different conditions on the terms of responsiveness and flow of the fronter to get a reaction diffusion equation with those terms presents existence comprehensive solutions, dispatividad or explosion in finite time. Moreover explores the problem periodic linear parabolic and semilineal where the boundary condition is also regularly and discusses various conditions that guarantee the existence of solutions for the newspaper. The thesis makes important contributions to the subject and improves many existing results in the literature.

Author: WILLIE ROBERT.
Year: 2004.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: UNIVERSIDAD COMPLUTENSE DE MADRID. FAC. DE CC. MATEMÁTICAS.
Place of preparation: UNIVERSIDAD COMPLUTENSE DE MADRID.

Summary: The thesis deals with the asymptotic behavior of solutions of systems of reaction - diffusion equations with nonlinear boundary conditions when the information is very big throughout the domain. In particular, studies the interaction between the status of non-linear contour and disseminating high throughout the domain. It is also a chapter in the case of a wave equation with conditions Robin linear behavior is studied and when the speed of propagation goes to infinity. As important features of the work, it is the equations that contain the potential presence of unique unsigned defined inside domain as at the border and the fact that the spread is not constant. The equation is a limit system on linear ordinary differential equations, where the non-linearity collects information from both the non-linearity of the interior domain, the non-linearity of the border and geometry own domain. The problem with the theory is interesting, the results are new and interesting constitute a contribution to the issue of dissemination high.

Author: LÓEZ-MESAS COLOMINA FERNANDO.
Year: 2004.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: FACULTAD DE CIENCIAS MATEMÁTICAS.
Place of preparation: FACULTAD DE CIENCIAS MATEMÁTICAS.

Summary: The thesis is devoted primarily to establish and explore the principles of non-regular analysis in the context of varieties riemannianas (both finite and infinite dimension) and to obtain applications of these findings to certain equations Hamilton-Jacobi raised in this context. The report begins with a brief summary of the ideas and results basic varieties riemannianas which will be used over the job. Then, in the second chapter, you get an interesting Begin Variacional Diferenciable in varieties riemannianas, in line with the principle Variacional of Deville, Godefroy and Zizler. To do this, it introduces the concept of pre-variety "uniformly mesetable" and are important examples of this type of varieties. The third chapter is devoted to the theorems of Rolle. Given that, as was known, Rolle theorem fails in infinite dimensions, you get an approximation in varieties riemannianas extending this context the result similar Banach spaces. Here are obtained approximate versions of Rolle's theorem for subdiferencial proximal (Hilbert spaces) and the generalized gradient (in Banach spaces), demonstrating that in these cases the theorem "exact" also fails in infinite dimension. In Chapter four is establishing the foundations of subdiferenciabilidad of Fréchet of functions defined in varieties riemannianas and discusses their basic properties with respect to the amount and composition product, including a principle of minimizing disrupted for the amount and the difference of duties. It also obtained here two inequities in the average value for functions subdiferenciables, extending results of Deville and Godefroy in this context. Finally, in this chapter discusses the properties of diferenciabilidad of the functions defined in convex varieties riemannanas, and it is shown that any continuing role and convex is subdiferenciable at all points and distinguishable in a dense package. The fifth and final chapter is dedicated to the equations Hamilton-Jacobi in varieties riemannianas. It introduces the concept of "solution viscosity" in this context, and the study developed over the report is applied here to obtain theorems of existence and uniqueness of this type of solution in certain classes of equations, both stationary and evolution .

Author: USERO MAINER DAVID.
Year: 2004.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: FACULTAD DE CC. MATEMÁTICAS.
Place of preparation: FACULTAD DE MATEMÁTICAS.

Summary: The objective of this work is to study the solutions of nonlinear equations involving non-local terms or random type disorder. The non-local systems are characterized by the appearance of different scales in the problem, which in the cases studied, leading to approach equations integro-diferenciales nonlinear. The first part discusses localized wave equations with terms of dispersion integro-diferenciales with nuclei type Hilbert transform resulting fractional and potential Kac-Baker. The second part examines systems that present terms nonlinear integral type memory and disipativos. The third part modeled a series of problems arising with simple fractional temporary in the sense Caputo. Specifically explores solutions for a harmonic oscillator subjected to this behavior temporarily. The fourth section discusses excitation type solitón under the action of potential random and with no correlation. It also includes a section explaining the numerical methods used for the integration of differential equations and to study their characteristics.

Author: MOYA LÁZARO NANCY-ROSA.
Year: 2004.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: UNIVERSIDAD COMPLUTENSE DE MADRID. FAC. DE CC. MATEMÁTICAS.
Place of preparation: UNIVERSIDAD COMPLUTENSE DE MADRID.

Summary: The thesis deals with a systematic and comprehensive study of the properties of qualitative reaction diffusion equations spaces with weight. It will get relevant results on properties of semigrupos linear partners, and subsequently obtained general conditions for the existence overall heights uniforms, compactness asontótica of the solution. Testing and the existence of the attractor and estimating its size HAUSDORFF. The thesis makes many important contributions and improved results in the previous topics discussed.

Author: HARO ORTEGA GLÒRIA.
Year: 2004.
University: POMPEU FABRA [www.upf.edu].
Place of defense: DEPARTAMENTO DE TECNOLOGÍA.
Place of preparation: UNIVERSIDAD POMPEU FABRA.

Summary: Topics covered in this thesis are on the one hand, the numerical simulation of the equations shallow waters (shallow waters) and on the other, solving some problems of image processing. The first part of the thesis is devoted to the resolution of the numerical equations shallow waters. We propose a combined treatment using the technique of double decomposition flow Marquina (extended to the case not homogeneous) when the two adjacent states are not coming and decomposition in a single case. The combined verifies ownership C accurate. Moreover, we propose a special treatment at the fronts dry / wet and in situations generation dry zone. The second topic is the digital simulation of the American Night ( "Day for Night"). The proposed algorithm simulates a nocturnal image from an image daytime considering various aspects of visual perception night. To simulate the loss of visual acuity proposes an equation in partial derivatives that simulates the principle of spatial summation of fotoreceptores located in the retina. The restoration of holes ( "inpainting") surfaces is the subject of the third. This suggests several approaches based on the geometric curvature average. They are also used two methods of interpolation: resolution of the Laplace equation and the method AMLE (Absolutely Minimization Lipschitz Extension). Finally, we devoted a part to the restoration of satellite imagery. We propose a problem variational model complete image acquisition. It also contains a term based on the Change Total towards regularizing the solution. The method proposed restoration gets obtain a regular collection of samples from a sampling irregular, while eliminating noise, deconvolucinando image and making zoom.

Author: CAPELLA KORT ANTONIO.
Year: 2004.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: SALA D'ACTES DE L'FME.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Author: GUERRERO RODRÍGUEZ SERGIO.
Year: 2004.
University: SEVILLA [www.us.es].
Place of defense: FACULTAD DE MATEMÁTICAS.
Place of preparation: FACULTAD DE MATEMÁTICAS.

Summary: This memory is essentially divided into five parts. In the first, there is a detailed introduction on the contents of this new memory, whereas at the end of the same describes other work done in parallel with the thesis. In the second and third part includes two work on the cotnrolabilidad of the equation of heat semilineal with boundary conditions Fourier-type linear and nonlinear This is a result of overall controllability when control acts on the system so distributed. The third chapter corresponds to a job in which we tested the exact controllability local systems Navier-Stokes and Boussinesq with few controls. In the latter part of this memory, includes work for a result of controllability accurate system of Lamé anisotropic elasticity.

Author: Taboada Vazquez Raquel.
Year: 2005.
University: A CORUÑA [www.udc.es].
Place of defense: E.T.S. Caminos, Canales y Puertos.
Place of preparation: E.T.S. Caminos, Canales y Puertos.

Summary: Classically the shallow water equations are obtained from the ecuaciopnes Euler or Navier-Stokes simplificadoras using certain assumptions. These assumptions are not always adequately justified, which leads to a large variety of models, but it is clear which of them is the 'best'. In this thesis are obtained using the method of developments asintóticos, different models of shallow waters with and without viscosity, one-dimensional and two-dimensional shape of a rigorous and without making the usual assumptions a priori. To apply the method of developments asintóticos, we identified a small dimensionless parameter (related to the depth domain) and we make a change to a variable domain of the independent parameter. We assume then that the solution to the Euler equations or Navier-Stokes supports development in power series of small parameter and the first estimate terms of such development. We built an approximation of the solution from the development in terms of power series rid calculated and the change of variable, so we get a pattern of shallow water. In the model obtendio from Euler equations of the horizontal velocity depends on the depth if the vorticity is not zero, which suponte an interesting novelty respect to the models found in the literature. The model of the shallow waters without viscosity propose that generalizes the classical model and makes it possible to calculate the exact solutions of the Euler equations in linear z, whereas the classical model only makes solutions with constant z. If part of the Navier-Stokes equations, the model that we include a new term viscosity. We have our model compared with other analytical and numerical models that can be found in literature, obtaining the model we propose improvements (or in the worst case equals) the results of other models. Ultimately, the proposed models are an improvement over the models found in the literature in the sense that the viscosity model incorporates a unit of the depth that allows better approximate the Euler equations and viscous model incorporates a new term viscosity justified by the method of developments asintóticos and numerically.

Author: Cañizo Rincón José Alfredo.
Year: 2005.
University: GRANADA [www.ugr.es].
Place of defense: Facultad de Ciencias.
Place of preparation: Universidad de Granada.

Summary: This paper deals mainly with the problem of the existence of solutions for two equations different equations continuous coagulación-fragmentación and equation Wigner-Poisson-Fokker-Planck. In addition, we study some aspects of qualitative behavior of the equations coagulación-fragmentación. The thesis is organized as follows: in this introduction briefly presented the context of both equations and the main results obtained. In Chapters 2 -- 4 give some preliminary results that are necessary for the further development of the equations continuous coagulación-fragmentación

Author: VIDAL LÓPEZ ALEJANDRO.
Year: 2005.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: UNIVERSIDAD COMPLUTENSE DE MADRID. FAC. CC. DE MATEMÁTICAS.
Place of preparation: UNIVERSIDAD COMPLUTENSE DE MADRID.

Summary: This thesis addresses problems related to the asymptotic behavior of solutions of equations of evolution semilineales of parabolic type. The equations can be autonomous or non-autonomous domain-general (bounded or not) and boundary conditions to be covered are general, even allowing for non-linear boundary conditions. As a general feature of the problems considered in this study emphasize the following: 1 - The equations are raised in an area of Banach ordered and generate a semigrupo monotonous. 2 - The equations have a structure disipativa. It assumes that there are two elements of the ordered phase space, so that the whole dynamic asymptotically falls within the range defined by these two points. 3 - The semigrupos non-linear equations to proceed parabolic type, have properties regularly and therefore some asymptotic compactness. In a first apart from the thesis, we studied the case of autonomous equations. These conditions allow prove the existence of two solutions extremales, minimal and maximal, which are points of balance and that the whole dynamic asymptotic System (attractor) is contained in the range bounded by these points. This result is used to obtain dynamic of a simpler and more unified form of a series of results known existence of solutions to various problems of elliptic type, the results of authors like H. Amann, D.de Figueiredo, PL Liones, etc.. In some cases, these results are even improved. In a second part of the thesis discusses the case of non-self-governing equations, and discusses properties similar to this case. By relying equation of time, there was no overall points this concept of balance and must be replaced with complete solution (defined for all time). This will test the existence of solutions extremales aminimales and maximal so that the dynamics asymptotic is also bounded by these two points. In this context, the concept of attraction has to be changed and used the concept of pull-back attraction. One learns in this case conditions guaranteeing the existence and uniqueness of solutions overall positive not warped and benchmarked. In the latter part of the report discusses two issues of high localized dissemination covered the results of the first part of the thesis. In the first of the problems the area where the spread goes to infinity has contact with the border and Robien Dirichlet problem. The second area that has contact only with the border Robin. We analyze the problem and limit the behavior of the solutions extremales under this type of disturbance. It also analyzes the behavior of the attractor system.

Author: CASAS TORIBIO LUIS VICENTE.
Year: 2005.
University: NACIONAL DE EDUCACIÓN A DISTANCIA [www.uned.es].
Place of defense: E.T.S. INGENIEROS INDUSTRIALES.
Place of preparation: E.T.S. DE INGENIEROS INDUSTRIALES.

Summary: In this paper we find the great effectiveness of the method in Generalized Finite Differences in any kind of dominance and status Border (Dirichlet, Neumann and Robin) in solving differential equations in partial derivatives. In this PhD thesis discusses the influence of the method of essential parameters, such as the number of nodes of the star, the criterion for selecting the star, the weighting function and stability for the equations that are time dependent. The analysis includes solutions for different types of equations with different types of boundary conditions. It presents a comprehensive study of the resolution of partial differential equations arising in time-dependent, using the method and using explicit finite difference widespread. This study covers the stability, for one, two and three dimensions, parabolic equations and hyperbolic. Lastly is the study of a post of error estimator for the Generalized Finite Difference Method. It also presents a novel method for adaptive method Generalized Finite Differences in three variables, analyzing the different parameters of influence. The method includes the implementation of adaptive algorithm to the resolution of various differential equations in partial derivatives of 2Â fourth order in three variables.

Author: SÁNCHEZ MUÑOZ ISABEL.
Year: 2005.
University: SEVILLA [www.us.es].
Place of defense: E.T.S. DE INGENIERÍA INFORMÁTICA.
Place of preparation: FACULTAD DE MATEMÁTICAS.

Summary: This work is a contribution to the study of compressible flows taking place in turbulent regime, mainly in the aspect of the modeling and numerical simulation of such flows. First, it develops a new version of the PPM model for gas flows perfect weakly viscous, which comes in the family of two models of turbulence equations (for the kinetic energy and helicity middle of the turbulent disturbance). The main contribution of this model, other earlier, is to consider the turbulent disruption in the class of functions whose regular periodicity cell can be transformed under homotecia. This fact alone shows that the model is valid for turbulence obtained locally homogeneous. Moreover, it produces dependence of the explicit terms of closure regarding the kinetic energy and helicity middle of the turbulent disturbance. Secondly, it builds a new model of turbulence understandable as a model hybrid model k-epsilon classic model PPB obtained previously, which incorporates effects that each genetic models trafficking in a more appropriate way. Specifically, the impact of turbulent diffusion is modeled as in the model k-epsilon and transitional effects and shearing between structures of different size being developed in the flow are modeled by homogenization. In the two-dimensional version of this model, the terms of closure are determined on a single, continuous connection of the field half. The validation of the numerical model is resolving a layer of compressible supersonic mixing. The results obtained confirm that the model predicts an appropriate average quantities of the main flow and also provides qualitative information relevant to the transitional structure of the turbulence. Finally, an analysis of passage to the limit for a unique equation convección-difusión with boundary conditions of mixed type and Robin Newman. This study is motivated for the non-standard form on which are imposed boundary conditions in the numerical scheme developed by the hybrid model and assure that it is a worthwhile procedure as a result of convergence achieved.

Author: OLIVE FARRE MARIA DEL CARME.
Year: 2005.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: SALA D'ACTES FME. EDIFICI U, CAMPUS SUD.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Author: AGUARELES CARRERO MARIA.
Year: 2006.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: SALA D'ACTES FME.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Summary: Many physical systems have the property that its dynamics is driven by some kind of spatial diffusion that is in competition with a reaction, like for instance two chemical species that react at the same time that there is a diffusion of each of them into the other. This interplay between reaction and diffusion produce non-homogeneous patterns that can sometimes be very rich. The mathematical models that describe this kind of behaviours are usually nonlinear partial differential equations whose solutions represent these patterns. In this thesis we focus on an specific reaction-diffusion equation that is the so-called general complex Ginzburg-Landau equation that is used as a model for oscillatory systems in extended domains. In particular we are interested in the type of patterns in two dimensions that arise when the solutions have a non-vanishing Brouwer degree. These patterns have the property that they exhibit rotating waves in the shape of spirals, which means that the contour lines arrange in the shape of spirals that emerge from the points where the solution vanishes. When the solution vanishes only at one point all the time dependence appears as a frequency term so the solutions can be expressed as a function of the polar radius and n. .ƒÖt where n is the degree of the solution, . is the polar angle and ƒÖ is the frequency of the wave. Therefore, these solutions can be expressed in terms of a system of ordinary differential equations. These solutions do only exist with a given frequency, and as a consequence and due to the existence of a dispersion relation, the wavenumber far from the origin, the so-called asymptotic wavenumber, is also unique. When the solutions have more than one isolated zero, the condition on the degree of the function has the effect of producing several spirals that emerge from the different zeros of the solution. These spirals evolve in time keeping their structure but moving around on the plane. In this work we use asymptotic analysis techniques to derive laws of motion for the centres of the spirals and we show that the time evolution of these patterns is slow and, being 1/ƒÃ the relative separation of the spirals, the time scale for the their dynamics is given byƒÃ 2 / | logƒÃ |. These laws of motion are different depending on the relation between the parameters of the complex Ginzburg-Landau equation and the relative separation of the spirals. We show that the way these laws change as the spirals separate or approach is highly singular. We also show that the asymptotic wavenumber in the case of multiple spirals is as well unique and that it evolves in time at the same rate as the velocity of the centres.