Dissertations / Theses on the topic 'Symmetry of solution'

Symmetry properties of solutions to some nonlinear Schroedinger equations are investigated. In particular, here the Laplace operator is perturbed by singular potentials which do not belong to the Kato class. A result of symmetry breaking of solutions is obtained provided a preliminary theorem about biradial solutions is stated. Further, a problem involving the biharmonic operator and exponential nonlinearity in dimension 4 is studied, connecting degree counting formulas with direct methods of calculus of variations via Morse theory and deformation lemmas.

Cette thèse porte sur l’étude des solutions périodiques du problème des N-tourbillons à vorticité positive. Ce problème, formulé par Helmholtz il y a plus de 160 ans, possède une histoire très riche et reste un domaine de recherche très actif. Pour un nombre quelconque de tourbillons et sans contrainte sur les vorticités, ce système n’est pas intégrable au sens de Liouville : on ne peut trouver de solution périodique non triviale par des méthodes explicites. Dans cette thèse, à l’aide de méthodes variationnelles, nous prouvons l’existence d’une infinité de solutions périodiques non triviales pour un système de N tourbillons à vorticités positives. De plus, lorsque les vorticités sont des nombres rationnels positifs, nous montrons qu’il n’existe qu’un nombre fini de niveaux d’énergie sur lesquels un équilibre relatif pourrait exister. Enfin, pour un système de N-tourbillons identiques, nous montrons qu’il existe une infinité de chorégraphies simples
This thesis focuses on the study of the periodic solutions of the N-vortex problem of positive vorticity. This problem was formulated by Helmholtz more than 160 years ago and remains an active research field. For an undetermined number of vortices and general vorticities the system is not Liouville integrable and periodic solutions cannot be determined explicitly, except for relative equilibria. By using variational methods, we prove the existence of infinitely many non-trivial periodic solutions for arbitrary N and arbitrary positive vorticities. Moreover, when the vorticities are positive rational numbers, we show that there exists only finitely many energy levels on which there might exist a relative equilibrium. Finally, for the identical N-vortex problem, we show that there exists infinitely many simple choreographies

In addition to previous publications, the paper presents the analytical solution of a special boundary value problem which arises in the context of elasticity theory for an extended constitutive law and a non-conservative symmetric ansatz. Besides deriving the general analytical solution, a specific form for linear boundary conditions is given for user convenience.

The analytical solution of a given inhomogeneous boundary value problem of a linear, elliptic, inhomogeneous partial differential equation and a set of inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions is derived in the present paper. In the context of elasticity theory, the problem arises for a non-conservative symmetric ansatz and an extended constitutive law shown earlier. For convenient user application, the scalar function expressed in cylindrical coordinates is primarily obtained for the general case before being expatiated on a special case of linear boundary conditions.

Questa tesi è incentrata sullo studio di equazioni differenziali alle derivate parziali di tipo ellittico. La prima parte della tesi riguarda la regolarità delle soluzioni stabili per un'equazione nonlineare con il p-Laplaciano, in un dominio limitato dello spazio Euclideo. La tecnica è basata sull'uso di disuguaglianze di tipo Hardy-Sobolev su ipersuperfici, del quale viene approfondito lo studio. Nella seconda parte viene preso in esame un problema nonlocale di tipo Dirichlet-Neumann. Studiamo la simmetria unidimensionale di alcune sottoclassi di soluzioni stabili, ottenendo risultati in dimensione n=2, 3. Inoltre, studiamo il comportamento asintotico dell'operatore associato a questo problema nonlocale, usando tecniche di Γ-convergenza.
This thesis deals with the study of elliptic PDEs. The first part of the thesis is focused on the regularity of stable solutions to a nonlinear equation involving the p-Laplacian, in a bounded domain of the Euclidean space. The technique is based on Hardy-Sobolev inequalities in hypersurfaces involving the mean curvature, which are also investigated in the thesis. The second part concerns, instead, a nonlocal problem of Dirichlet-to-Neumann type. We study the one-dimensional symmetry of some subclasses of stable solutions, obtaining new results in dimensions n=2, 3. In addition, we carry out the study of the asymptotic behaviour of the operator associated with this nonlocal problem, using Γ-convergence techniques.

In Reaction-Diffusion systems, some parameters can influence the behavior of other parameters in that system. Thus reaction diffusion equations are often used to model the behavior of biological phenomena. The Fitzhugh Nagumo partial differential equation is a reaction diffusion equation that arises both in population genetics and in modeling the transmission of action potentials in the nervous system. In this paper we are interested in finding solutions to this equation. Using Lie groups in particular, we would like to find symmetries of the Fitzhugh Nagumo equation that reduce this non-linear PDE to an Ordinary Differential Equation. In order to accomplish this task, the non-classical method is utilized to find the infinitesimal generator and the invariant surface condition for the subgroup where the solutions for the desired PDE exist. Using the infinitesimal generator and the invariant surface condition, we reduce the PDE to a mildly nonlinear ordinary differential equation that could be explored numerically or perhaps solved in closed form.

We are concerned with iterative solvers for large and sparse skew-symmetric linear systems. First we discuss algorithms for computing incomplete factorizations as a source of preconditioners. This leads to a new Crout variant of Gaussian elimination for skew-symmetric matrices. Details on how to implement the algorithms efficiently are provided. A few numerical results are presented for these preconditioners. We also examine a specialized preconditioned minimum residual solver. An explicit derivation is given, detailing the effects of skew-symmetry on the algorithm.

Orientador: Lucas Catão de Freitas Ferreira
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Nesta tese, estudamos dois tipos de EDPs com potenciais singulares críticos, a saber, uma equação elíptica com operador poliharmônico e a equação do calor linear. Para a primeira, pesquisamos existência e propriedades qualitativas das soluções no espaço $\mathcal{H}_{k,\vec{\alpha}}$ que é uma soma de espaços $L^{\infty}$ com peso, o qual parece ser um espaço mínimo para o tipo de potencial singular considerado. Investigamos um conceito de simetria para soluções que estende o de simetria radial e satisfaz uma ideia de invariância em torno das singularidades. Para a segunda, uma estratégia baseada na transformada de Fourier é empregada para obter resultados de boa-colocação global e comportamento assintótico de soluções, sem hipóteses de pequenez e sem utilizar a desigualdade de Hardy. Em particular, obtemos boa-colocação de soluções para o caso do potencial monopolar $V(x)=\frac{\lambda}{\left\vert x\right\vert ^{2}}$ com $\left\vert \lambda\right\vert <\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. Este valor limiar é o mesmo obtido em resultados de boa-colocação global em $L^2$ que utilizam desigualdades de Hardy e estimativas de energia. Desde que não existe uma relação de inclusão entre $L^{2}$ e $PM^{k}$, nossos resultados indicam que $\lambda_{\ast}$ é intrínseco da EDP e independe de uma particular abordagem. Palavras-chave: Equações elípticas, equação do calor, potencial singular, existência, simetria, autossimilaridade, comportamento assintótico
Abstract: In this thesis, we study two types of PDEs with critical singular potentials, namely, an elliptic equation with polyharmonic operator and the linear heat equation. For the first, we obtain existence and qualitative properties of solutions in $\mathcal{H}_{k,\vec{\alpha}}$-spaces which are a sum of weighted $L^{\infty}$-spaces, and seem to be a minimal framework for the potential profile of interest. We investigate a concept of symmetry for solutions which extends radial symmetry and carries out an idea of invariance around singularities. For the second, a strategy based on the Fourier transform is employed to obtain results of global well-posedness and asymptotic behavior of solutions, without smallness hypotheses and without using Hardy inequality. In particular, well-posedness of solutions is obtained for the case of the monopolar potential $V(x)=\frac{\lambda}{\left\vert x\right\vert ^{2}}$ with $\left\vert \lambda\right\vert <\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. This threshold value is the same one obtained for the global well-posedness of $L^{2}$-solutions by means of Hardy inequalities and energy estimates. Since there is no inclusion relation between $L^{2}$ and $PM^{k}$, our results indicate that $\lambda_{\ast}$ is intrinsic of the PDE and independent of a particular approach. Keywords: Elliptic equation, heat equation, singular potential, existence, symmetry, self-similarity, asymptotic behavior
Doutorado
Matematica
Doutora em Matemática

Thesis (MSc (Chemistry and Polymer Science))--University of Stellenbosch, 2006.
This study concerns the investigation of the isomerization of different metallocene catalysts in solution, and the effects thereof on the microstructure of polypropylenes prepared with these catalysts. Two C2 symmetric ansa metallocenes, ethylene-bis(indenyl) zirconium dichloride (EI) and dimethylsilyl-bis(2-methyl benzoindenyl) zirconium dichloride (MBI) were exposed, in solution, to both sunlight and UV radiation. The rac-meso isomerization of these catalysts were followed by 1H NMR spectroscopy. The reaching of a photostationary state is described, as well as the effect of isomerization of these catalysts in solution on the polymerization of propylene. Results show that metallocene structure has an effect on the isomerization rate and photostationary state. Results also show that the wavelength of light plays a role in the isomerization process. Effects on stereochemistry and molecular weight of the formed polymer as well as the catalyst activity is described and discussed. In addition the effect of activating the catalysts with MAO before exposure to light is discussed.

Desenvolve-se uma técnica para a extração de auto-pares relacionados com a solução de problemas de Elementos Finitos. O algoritmo consiste no uso dos métodos da Iteração Inversa e Gradiente Conjugado para a obtenção do vetor solução associado ao menor auto-valor. As soluções do auto-sistema são calculadas sequencialmente pela modificação da matriz dos coeficientes das equações de equilíbrio do problema através do uso de uma técnica de Deflação. O uso extensivo desta técnica introduz auto-valores múltiplos na matriz dos coeficientes, tornando necessário proceder-se a uma combinação dos dois métodos. É efetuado também um estudo para encontrar vetores iniciais apropriados a serem utilizados pelos métodos. O algoritmo foi implementado e alguns resultados de resolução de exemplos são apresentados, para ilustrar o seu desempenho.
A vector iterative technique is developed for the extraction of eigenpairs related to the solution of finite element problems. The algorithm consists of using inverse iteration and conjugate gradient methods so as to obtain the solution vector associated to the smallest eigenvalue. Eigensolutions are sequentially calculated by replacing the coefficient matrix in the problem equilibrium equation using a deflation technique. The extensive usage of this technique, introduces multiple eigenvalue in the coefficient matrix, requiring a procedure to combine both methods. Also, a study is performed to find the appropriate starting vector to be used with methods. The algorithm has been implemented and the results of some example solutions are given that yield insight into its predictive capabilities.

We consider the four-derivative modification to the Einstein-Hilbert action of general relativity, without a cosmological constant. Higher derivative terms are interesting because they make the theory renormalisable (but non-unitary) and because they appear generically in quantum gravity theories. We consider the classical, static, spherically symmetric solutions, and try to enumerate all solution families. We find three families in expansions around the origin: one corresponding to the vacuum, another which contains the Schwarzschild family, and another which does not appear in generic theories with other number of derivatives but seems to be the correct description of solutions coupled to positive matter in the four-derivative theory. We find three special families in expansions around a non-zero radius, corresponding to normal horizons, wormholes and exotic horizons. We study many examples of matter-coupled solutions to the theory linearised around flat space, which corroborate our arguments. We are assisted by use of a "no-hair" theorem that certain conditions imply that $R=0$, which is applicable in many cases including asymptotically flat space-times with horizons. The Schwarzschild black hole still exists in the theory, but a second branch of black hole solutions is found that can have both positive and negative mass, and that coincide with the Schwarzschild black holes at a single mass. The space of asymptotically flat solutions is probed numerically by shooting inwards from a weak-field solution at large radius, and the behaviour at small radius is classified into the families of series solutions (most of which make an appearance). The results are inconclusive but show several interesting features for further study.

Differential equations, in particular the nonlinear ones, are commonly used in formulating most of the fundamental laws of nature as well as many technological problems, among others. This makes the need for methods in finding closed form solutions to such equations all-important. In this thesis we study Lie symmetry methods for some nonlinear ordinary differential equations (ODE). The study focuses on identifying and using the underlying symmetries of the given first order nonlinear ordinary differential equation. An extension of the method to higher order ODE is also discussed. Several illustrative examples are presented.
Differentialekvationer, framförallt icke-linjära, används ofta vid formulering av fundamentala naturlagar liksom många tekniska problem. Därmed finns det ett stort behov av metoder där det går att hitta lösningar i sluten form till sådana ekvationer. I det här arbetet studerar vi Lie symmetrimetoder för några icke-linjära ordinära differentialekvationer (ODE). Studien fokuserar på att identifiera och använda de underliggande symmetrierna av den givna första ordningens icke-linjära ordinära differentialekvationen. En utvidgning av metoden till högre ordningens ODE diskuteras också. Ett flertal illustrativa exempel presenteras.

Since Schwarzschild found the first solution of the Einstein’s equations, more than 800 solutions were found. Solutions of Einstein’s equations are classified according to their Lie algebras of isometries and their isotropy subalgebras. Solutions were taken from the USU electronic library of solutions of Einstein’s field equations and the classification used Maple code developed at USU. This classification adds to the data contained in the library of solutions and provides additional tools for addressing the equivalence problem for solutions to the Einstein field equations. In this thesis, homogeneous spacetimes, hypersurface-homogeneous spacetimes, Robinson-Trautman solutions, and some famous black hole solutions have been classified.

We investigate the construction of analytic approximations to the solutions of the nonlinear anti-symmetric quadratic oscillator. The procedure used is based on the method of generalized harmonic balance as formulated by Mickens. We prove that all solutions to the differential equation are bounded and periodic. This result is based upon the use of a correct definition of the absolute value of a variable and the use of this definition to calculate its derivative.

The dynamics of soliton pulses for use in nonlinear optical devices is mathematically modelled by Maxwell-Bloch systems of equations for the interaction of light with a uniform distribution of quantum-mechanical atoms. We study the Reduced Maxwell-Bloch (RMB) equations occurring when an ensemble of rotationally symmetric 3-level atoms is assumed. The model applies for on and off-resonance conditions and is completely integrable using Inverse Scattering theory, since it arises as the compatibility condition of a 3 x 3 AKNS-system. Furthermore this integrability remains valid for all timescales of the optical field because only the “one-way wave approximation” is required during the derivation. Solutions are constructed in two ways: 1. Darboux-Bäcklund transforms are applied, generating single soliton pulses of ultrashort (< 1ps) duration, and families of elliptically polarised 2-solitons not possible in lower dimensional problems. 2. A general Inverse Scattering scheme is developed and tested. The Direct Scattering Problem is dealt with first to obtain a complete set of scattering data. Subsequently the Inverse Problem is solved both formally and then in explicit closed form for the special case that the reflection coefficients vanish for real values of the spectral parameter. In this case the main result is a determined system of n linear algebraic equations which yield the n-soliton of our RMB-system. It is confirmed that the 1-solitons found by means of Darboux transform are precisely the same as those given by the full mechanism of Inverse Scattering. Finally we calculate the invariants of the motion for the RMB-equations, and derive an evolution equation giving the variation with propagation distance of the invariant functionals when the original RMB-system is modified by an arbitrary perturbing term. As an application dissipative effects on 1-solitons are considered.

Security issues are a major concern, especially with real-time applications that are very time- sensitive because of file size. Cryptography is one of the main techniques used to provide the necessary data security. As is well known, the concept of the cryptography is to convert the plaintext into an unreadable version on the sender side by using an encryption key. At the receiver side, the ciphertext is returned into its original situation. Of particular relevance is key space viz. the number of possible keys that can be used to generate the key from the keys container. This is very important against brute force attacks. Many algorithms have been published; however, they feature limitations pertaining to security level, encryption key size, performance speed or incompatibility with the real-time applications. In this thesis, innovative 4D tesseract-based solutions (T-key) are suggested, implemented, tested and shown to overcome the drawbacks of most existing algorithms. They are based on a symmetric block cipher technique and have a large key-space (e.g. 384-bit). In-this thesis, three key size are suggested as examples, which are tested and evaluated: T-{O,l}128, T-{O,1}256 and T-{O,1}512. Moreover, four lightweight coding rounds are applied to create these keys from the 4D tesseract key containers. The execution speed of the T-key was compared with the AES cipher, and the results indicated that the first two encryption key sizes suggested here are faster than the first two encryption keys of AES. Furthermore, the security evaluations carried out on the suggested algorithm showed full resistance to statistical attacks and a comprehensive pass of correlation coefficient tests.

Characterizing the long-time behavior of solutions to the Einstein field equations remains an active area of research today. In certain types of coordinates the Einstein equations form a coupled system of quasilinear wave equations. The investigation of the nature and properties of solutions to these equations lies in the field of geometric analysis. We make several contributions to the study of solution dynamics near singularities. While singularities are known to occur quite generally in solutions to the Einstein equations, the singular behavior of solutions is not well-understood. A valuable tool in this program has been to prove the existence of families of solutions which are so-called asymptotically velocity term dominated (AVTD). It turns out that a method, known as the Fuchsian method, is well-suited to proving the existence of families of such solutions. We formulate and prove a Fuchsian-type theorem for a class of quasilinear hyperbolic partial differential equations and show that the Einstein equations can be formulated as such a Fuchsian system in certain gauges, notably wave gauges. This formulation of Einstein equations provides a convenient general framework with which to study solutions within particular symmetry classes. The theorem mentioned above is applied to the class of solutions with two spatial symmetries -- both in the polarized and in the Gowdy cases -- in order to prove the existence of families of AVTD solutions. In the polarized case we find families of solutions in the smooth and Sobolev regularity classes in the areal gauge. In the Gowdy case we find a family of wave gauges, which contain the areal gauge, such that there exists a family of smooth AVTD solutions in each gauge.

Le travail présenté est dédié à des problèmes d'EDP non linéaires. L'idée principale est de construire des solutions régulières á certaines EDPs elliptiques et hypo-elliptiques et étudier leur propriétés qualitatives. Dans une première partie, on considère un problème sur-critique du type $$-Delta u = lambda e^u$$ avec $lambda > 0$ posé dans un domaine extérieur avec conditions de Dirichlet homogènes. Une réduction en dimension finie permet de prouver l'existence d'un nombre infini de solutions régulières quand $lambda$ est assez petit. Dans une deuxième partie, on étudie la concentration de solutions d'un problème non local $$(-Delta)^s u = u^{p pm epsilon}, u>0, epsilon > 0$$ dans un domaine borné, régulier sous conditions de Dirichlet homogènes. Ici, on prend $0 < s < 1$ et $p:=(N+2s)/(N-2s)$, l'exposant de Sobolev critique. Une réduction en dimension finie dans des espaces fonctionnels bien choisis est utilisée. La partie principale de la fonction réduite est donnée en termes des fonctions de Green et Robin sur le domaine. On prouve que l'existence de solutions dépend des points critiques de la fonction susmentionnée augmentée d'une condition de non-dégénérescence. Enfin, on considère un problème non local dans le groupe de Heisenberg $H$. On s'intéresse à des propriétés de rigidité des solutions stables de $(-Delta_H)^s v = f(v)$ sur $H$, $s in (0,1)$. Une inégalité de type Poincaré connectée à un problème dégénéré dans $R^4_+$ est prouvée. Au travers d'une procédure d'extension, cette inégalité est utilisée pour donner un critère sous lequel les lignes de niveaux de la solution de l'EDP sont des surfaces minimales dans $H$
This work is devoted to nonlinear PDEs. The aim is to find regular solutions to some elliptic and hypo-elliptic PDEs and study their qualitative properties. The first part deals with the supercritical problem $$ -Delta u = lambda e^u,$$ $lambda > 0$, in an exterior domain under zero Dirichlet condition. A finite-dimensional reduction scheme provides the existence of infinitely many regular solutions whenever $lambda$ is sufficiently small.The second part is focused on the existence of bubbling solutions for the non-local equation $$ (-Delta)^s u =u^p, ,u>0,$$in a bounded, smooth domain under zero Dirichlet condition; where $0 0$ small). To this end, a finite-dimensional reduction scheme in suitable functional spaces is used, where the main part of the reduced function is given in terms of the Green's and Robin's functions of the domain. The existence of solutions depends on the existence of critical points of such a main term together with a non-degeneracy condition.In the third part, a non-local entire problem in the Heisenberg group $H$ is studied. The main interests are rigidity properties for stable solutions of $$(-Delta_H)^s v = f(v) in H,$$ $s in (0,1)$. A Poincaré-type inequality in connection with a degenerate elliptic equation in $R^4_+$ is provided. Through an extension (or ``lifting") procedure, this inequality will be then used to give a criterion under which the level sets of the above solutions are minimal surfaces in $H$, i.e. they have vanishing mean $H$-curvature

Exponential potentials can be associated both with expanding and collapsing cos- mologies: whereas power-law inflation can be obtained for a nearly flat exponential potential, a stable contracting cosmology (which, by the way, can solve the hori- zon problem) can be obtained for a negative sufficiently steep exponential potential. Motivated by these results, we study the Einstein equation for a scalar field with an exponential potential in a static and spherically symmetric spacetime. For the same parameters which describe a stable contracting cosmology, we find a non- asymptotically flat black hole and we study its properties.

Includes abstract.
Includes bibliographical references.
In this thesis we use the 1+1+2 covariant approach to General Relativity to study exact solutions and perturbations of rotationally symmetric spacetimes in f(R) gravity, one of the most widely studied classes of fourth order gravity. We begin by introducing f(R) theories of gravity and present the general equations for these theories. We investigate the problem of matching different regions of spacetime, shedding light on the problem of constructing realistic inhomogeneous cosmologies in the context of f(R) gravity. We also study strong lensing in these fourth order theories of gravity derive the lens mass and magnification for the gravitational lens system. We provide an extensive review of both the 1+3 and 1+1+2 covariant approaches to f(R) theories of gravity and give the full system of evolution, propagation and constraint equations of LRS spacetimes. We then determine the conditions for the existence of spherically symmetric vacuum solutions of these fourth order field equations and prove a Jebsen-Birkhoff like theorem for f(R) theories of gravity and the necessary conditions required for the existence of Schwarzschild solution in these theories.

Exact solution of the Schrö
dinger equation with some potentials is obtained. The normal and supersymmetric cases are considered. Deformed ring-shaped potential is solved in the parabolic and spherical coordinates. By taking appropriate values for the parameter q, similar results are obtained for Hulthé
n and exponential type screened potentials. Similarly, Morse, Pö
schl-Teller and Hulthé
n potentials are solved for the supersymmetric case. Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is also studied. The Nikiforov-Uvarov and Hamiltonian Hierarchy methods are used in the calculations. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. Results are in good agreement with ones obtained before.

El tema principal de esta tesis es el desarrollo de técnicas de actualización de precondicionadores para resolver sistemas lineales de gran tamaño y dispersos Ax=b mediante el uso de métodos iterativos de Krylov. Se consideran dos tipos interesantes de problemas. En el primero se estudia la solución iterativa de sistemas lineales no singulares y antisimétricos, donde la matriz de coeficientes A tiene parte antisimétrica de rango bajo o puede aproximarse bien con una matriz antisimétrica de rango bajo. Sistemas como este surgen de la discretización de PDEs con ciertas condiciones de frontera de Neumann, la discretización de ecuaciones integrales y métodos de puntos interiores, por ejemplo, el problema de Bratu y la ecuación integral de Love. El segundo tipo de sistemas lineales considerados son problemas de mínimos cuadrados (LS) que se resuelven considerando la solución del sistema equivalente de ecuaciones normales. Concretamente, consideramos la solución de problemas LS modificados y de rango incompleto. Por problema LS modificado se entiende que el conjunto de ecuaciones lineales se actualiza con alguna información nueva, se agrega una nueva variable o, por el contrario, se elimina alguna información o variable del conjunto. En los problemas LS de rango deficiente, la matriz de coeficientes no tiene rango completo, lo que dificulta el cálculo de una factorización incompleta de las ecuaciones normales. Los problemas LS surgen en muchas aplicaciones a gran escala de la ciencia y la ingeniería como, por ejemplo, redes neuronales, programación lineal, sismología de exploración o procesamiento de imágenes. Los precondicionadores directos para métodos iterativos usados habitualmente son las factorizaciones incompletas LU, o de Cholesky cuando la matriz es simétrica definida positiva. La principal contribución de esta tesis es el desarrollo de técnicas de actualización de precondicionadores. Básicamente, el método consiste en el cálculo de una descomposición incompleta para un sistema lineal aumentado equivalente, que se utiliza como precondicionador para el problema original. El estudio teórico y los resultados numéricos presentados en esta tesis muestran el rendimiento de la técnica de precondicionamiento propuesta y su competitividad en comparación con otros métodos disponibles en la literatura para calcular precondicionadores para los problemas estudiados.
The main topic of this thesis is updating preconditioners for solving large sparse linear systems Ax=b by using Krylov iterative methods. Two interesting types of problems are considered. In the first one is studied the iterative solution of non-singular, non-symmetric linear systems where the coefficient matrix A has a skew-symmetric part of low-rank or can be well approximated with a skew-symmetric low-rank matrix. Systems like this arise from the discretization of PDEs with certain Neumann boundary conditions, the discretization of integral equations as well as path following methods, for example, the Bratu problem and the Love's integral equation. The second type of linear systems considered are least squares (LS) problems that are solved by considering the solution of the equivalent normal equations system. More precisely, we consider the solution of modified and rank deficient LS problems. By modified LS problem, it is understood that the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Rank deficient LS problems are characterized by a coefficient matrix that has not full rank, which makes difficult the computation of an incomplete factorization of the normal equations. LS problems arise in many large-scale applications of the science and engineering as for instance neural networks, linear programming, exploration seismology or image processing. Usually, incomplete LU or incomplete Cholesky factorization are used as preconditioners for iterative methods. The main contribution of this thesis is the development of a technique for updating preconditioners by bordering. It consists in the computation of an approximate decomposition for an equivalent augmented linear system, that is used as preconditioner for the original problem. The theoretical study and the results of the numerical experiments presented in this thesis show the performance of the preconditioner technique proposed and its competitiveness compared with other methods available in the literature for computing preconditioners for the problems studied.
El tema principal d'esta tesi és actualitzar precondicionadors per a resoldre sistemes lineals grans i buits Ax=b per mitjà de l'ús de mètodes iteratius de Krylov. Es consideren dos tipus interessants de problemes. En el primer s'estudia la solució iterativa de sistemes lineals no singulars i antisimètrics, on la matriu de coeficients A té una part antisimètrica de baix rang, o bé pot aproximar-se amb una matriu antisimètrica de baix rang. Sistemes com este sorgixen de la discretització de PDEs amb certes condicions de frontera de Neumann, la discretització d'equacions integrals i mètodes de punts interiors, per exemple, el problema de Bratu i l'equació integral de Love. El segon tipus de sistemes lineals considerats, són problemes de mínims quadrats (LS) que es resolen considerant la solució del sistema equivalent d'equacions normals. Concretament, considerem la solució de problemes de LS modificats i de rang incomplet. Per problema LS modificat, s'entén que el conjunt d'equacions lineals s'actualitza amb alguna informació nova, s'agrega una nova variable o, al contrari, s'elimina alguna informació o variable del conjunt. En els problemes LS de rang deficient, la matriu de coeficients no té rang complet, la qual cosa dificultata el calcul d'una factorització incompleta de les equacions normals. Els problemes LS sorgixen en moltes aplicacions a gran escala de la ciència i l'enginyeria com, per exemple, xarxes neuronals, programació lineal, sismologia d'exploració o processament d'imatges. Els precondicionadors directes per a mètodes iteratius utilitzats més a sovint són les factoritzacions incompletes tipus ILU, o la factorització incompleta de Cholesky quan la matriu és simètrica definida positiva. La principal contribució d'esta tesi és el desenvolupament de tècniques d'actualització de precondicionadors. Bàsicament, el mètode consistix en el càlcul d'una descomposició incompleta per a un sistema lineal augmentat equivalent, que s'utilitza com a precondicionador pel problema original. L'estudi teòric i els resultats numèrics presentats en esta tesi mostren el rendiment de la tècnica de precondicionament proposta i la seua competitivitat en comparació amb altres mètodes disponibles en la literatura per a calcular precondicionadors per als problemes considerats.
Guerrero Flores, DJ. (2018). On Updating Preconditioners for the Iterative Solution of Linear Systems [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/104923
TESIS

Coupled nonlinear Schrodinger equations (CNLS) govern many physical phenomena, such as nonlinear optics and Bose-Einstein condensates. For their wide applications, many studies have been carried out by physicists, mathematicians and engineers from different respects. In this dissertation, we focused on standing wave solutions, which are of particular interests for their relatively simple form and the important roles they play in studying other wave solutions. We studied the multiplicity of this type of solutions of CNLS via variational methods and bifurcation methods. Variational methods are useful tools for studying differential equations and systems of differential equations that possess the so-called variational structure. For such an equation or system, a weak solution can be found through finding the critical point of a corresponding energy functional. If this equation or system is also invariant under a certain symmetric group, multiple solutions are often expected. In this work, an integer-valued function that measures symmetries of CNLS was used to determine critical values. Besides variational methods, bifurcation methods may also be used to find solutions of a differential equation or system, if some trivial solution branch exists and the system is degenerate somewhere on this branch. If local bifurcations exist, then new solutions can be found in a neighborhood of each bifurcation point. If global bifurcation branches exist, then there is a continuous solution branch emanating from each bifurcation point. We consider two types of CNLS. First, for a fully symmetric system, we introduce a new index and use it to construct a sequence of critical energy levels. Using variational methods and the symmetric structure, we prove that there is at least one solution on each one of these critical energy levels. Second, we study the bifurcation phenomena of a two-equation asymmetric system. All these bifurcations take place with respect to a positive solution branch that is already known. The locations of the bifurcation points are determined through an equation of a coupling parameter. A few nonexistence results of positive solutions are also given

In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of a dissipative nonlinear modified Klein-Gordon equation in an open sphere around the origin, with constant internal and external damping coefficients and nonlinear term of the form G' (w) = w ^p, with p an odd number greater than 1. We prove that our scheme is consistent of quadratic order, and provide a necessary condition for it to be stable order n. Part of our study will be devoted to study the effects of internal and external damping.

The supersymmetric solutions of PT-/non-PT symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the SchrÄ
odinger equation with the deformed Morse, Hulth¶
en, PÄ
oschl-Teller, Hyperbolic Kratzer-like, Screened Coulomb, and Exponential-Cosine Screened Coulomb (ECSC) potentials. The Hamiltonian hi- erarchy method is used to get the real energy eigenvalues and corresponding wave functions.

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.

This thesis is a review of research done over the course of the past 4 years, divided into two unrelated parts. The rst is set in the context of Bagger-Lambert-Gustavsson models, based on 3-Lie algebras. In particular I will describe theories with metric 3-algebras of inde nite signature: these present elds with negative kinetic terms. The problem can be solved by gaugeing away the non-physical degrees of freedom, to obtain other well understood theories. I will show how this procedure can be easily applied for 3-algebra metrics of any inde nite signature. Part II of this thesis focuses on solutions of topologically massive gravity (TMG): particular attention is devoted to warped AdS3 black holes, which are discussed in great detail. I will present a novel analysis of the near horizon geometries of these solutions. I further propose an approach for searching for new solutions to 3-dimensional gravity based on conformal symmetry. This approach is able to yield most of the known axisymmetric stationary TMG backgrounds.

My thesis deals with the study of elliptic PDE. It is divided into two parts, the first one concerning a nonlinear equation involving the p-Laplacian, and the second one focused on a nonlocal problem. In the first part, we study the regularity of stable solutions to a nonlinear equation involving the p-Laplacian in a bounded domain. This is the nonlinear version of the widely studied semilinear equation involving the classical Laplacian. Stable solutions to semilinear equations have been very recently proved to be bounded, and therefore smooth, up to dimension n=9 by Cabré, Figalli, Ros-Oton, and Serra. This result is known to be optimal by counterexamples in higher dimensions. In the case of the p-Laplacian, the boundedness of stable solutions is conjectured to hold up to a critical dimension depending on p. Examples of unbounded stable solutions are known if the dimension exceeds the critical one. Moreover, in the radial case or under strong assumptions on the nonlinearity, stable solutions are proved to be bounded in the optimal dimension range. We prove the boundedness of stable solutions under a new condition on n and p, which is optimal in the radial case, and more restrictive in the general one. It improves the known results in the field, and it is the first example, concerning the p-Laplacian, of a technique providing both a result in the nonradial case and the optimal result in the radial case. In the first part, we also investigate Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, all containing a mean curvature term. Our motivation comes from several applications of these inequalities to the study of a priori estimates for stable solutions. Specifically, we give a simplified proof of the celebrated Michael-Simon and Allard inequality, we obtain two new forms of the Hardy inequality on hypersurfaces, and an improved Hardy inequality in the Poincaré sense. In the second part of this thesis, we deal with a Dirichlet to Neumann problem arising in a model for water waves. The system is described by a diffusion equation in a slab of fixed height, containing a weight that depends on a parameter a belonging to (-1,1). The top of the slab is endowed with a 0-Neumann condition, while on the bottom we have a Dirichlet datum and an equation involving a smooth nonlinearity. The system can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum, by defining a suitable Dirichlet to Neumann operator. First, we prove a Liouville theorem that establishes the one dimensional symmetry of stable solutions, provided that a control on the growth of the energy associated with the problem is satisfied. As a consequence, we obtain the 1D symmetry of stable solutions to our problem in dimension 2. For n=3, we establish sharp energy estimates for both the energy minimizers and the monotone solutions, deducing the 1D symmetry of these classes of solutions, by an application of our Liouville theorem. Concerning this problem, we also investigate the nature of the associated Dirichlet to Neumann operator. First, we deduce its expression as a Fourier operator, which was known only in the case a=0. This result highlights the mixed nature of the operator, which is nonlocal, but not purely fractional. To better understand the dual behaviour of the operator, we provide a G-convergence result for an energy functional associated with the operator. Specifically, as a G-limit of our energy functional we find a mere interaction energy when a is greater than 0, and the classical perimeter when a is smaller or equal than 0. We point out that the threshold a=0 that we obtain here, as well as the G-limit behaviour for nonpositive values of a, is common to other nonlocal problems treated in the literature. On the contrary, the limit functional that we obtain in the other case appears to be new and structurally different from other nonlocal energy functionals that have been investigated in the literature.
Mi tesis se encaja en el estudio de las EDPs elípticas. Está dividida en dos partes: la primera trata una ecuación no-lineal con el p-Laplaciano, la segunda de un problema no-local. En la primera parte, estudiamos la regularidad de las soluciones estables de una ecuación no lineal con el p-Laplaciano en un dominio acotado. Esta ecuacion es la versión no-lineal de la ámpliamente estudiada ecuacion semilineal con el Laplaciano. Cabré, Figalli, Ros-Oton, y Serra han demostrado recientemente que las soluciones estables de las ecuaciones semilineales son acotadas, y por tanto regulares, hasta la dimensión 9. Este resultado es optimal. En el caso del p-Laplaciano, la regularidad de las soluciones estables se conjetura de ser cierta hasta una dimension critica y, de hecho, se conocen ejemplos de soluciones no acotadas cuando la dimension llega al valor critico. Además, se ha demostrado que en el caso radial o assumiendo hipótesis fuertes sobre la no-linealidad las soluciones estables son acotadas hasta la dimension critica. En el primer capítulo, demostramos que las soluciones estables son acotadas, bajo una nueva condición en n y p, que es optimal en el caso radial, y más restrictiva en el caso general. Esta investigación mejora conocidos resultados del tema y es el primer ejemplo, para el p-Laplaciano, de un método que produce un resultado para el caso general y un resultado optimal en el caso radial. En la primera parte, nos ocupamos también de las desigualdades funcionales del tipo Hardy y Sobolev sobre hipersuperfícies del espacio Euclideo, todas conteniendo un término de curvatura media. Nuestra motivación proviene de varias apliaciones que tienen estas desigualdades en el estudio de estimaciones para las soluciones estables. En detalle, damos una demostración simple de la conocida desigualdad de Michael-Simon y Allard, obtenemos dos formas nuevas de la desigualdad de Hardy sobre hipersuperfícies, y otra desigualdad de Hardy-Poincaré. En la segunda parte, nos ocupamos de un problema de Dirichlet-Neumann que emerge de un modelo para las ondas en el agua. El sistema se describe con una ecuación de difusión en una tira de altura fija, que contiene un parámetro a en (-1,1). La parte superior de la tira es dotada de una condicion 0 de Neumann, mientras en la parte inferior tenemos un dato de Dirichlet y una ecuación con una nonlinearidad regular. Este problema puede ser reformulado como una ecuación no-local sobre la componente dotada del dato de Dirichlet, definiendo un operador de Dirichlet-Neumann apropiado. Primero, demostramos un teorema del tipo Liouville, que garantiza la simetría unidimensional de las soluciones monótonas, asumiendo un control sobre el crecimiento de la energía asociada. Como consecuencia, obtenemos la simetría 1D de las soluciones estables en dimension 2. Para n=3, obtenemos estimaciónes optimales de la energía para las soluciones que minimizan la energía y para las soluciones monótonas. Estas estimaciones nos conducen a la simetría 1D de estas clases de soluciones, aplicando nuestro teorema del tipo Liouville. Relativo a este problema, estudiamos también la naturaleza del operador de Dirichlet-Neumann. Primero, deducimos su expresión como operador de Fourier, que anteriormente solo se conocía para a=0. Este resultado evidencia la naturaleza del operador, que es no-local pero no puramente fraccionaria. Estudiamos en profundidad este comportamiento mixto del operador a través del estudio de la G-convergencia de un funcional energía asociado al operador. Demostramos la G-convergencia de nuestro funcional a un límite que corresponde a una energía de interacción pura cuando a en (0,1) y al perímetro clásico cuando a en (-1,0]. El límite a=0, así como el G-límite para el régimen a en (-1,0], es común a otros problemas no-locales tratados en la literatura. Al contrario, el funcional límite en el régimen puramente no-local es nuevo y diferente a otros funciona
Questa tesi si occupa di equazioni differenziali alle derivate parziali di tipo ellittico. È divisa in due parti: la prima riguarda un’equazione nonlineare per il p-Laplaciano, mentre la seconda è incentrata su un problema nonlocale, che può essere formulato per mezzo di un operatore di Dirichlet-Neumann collegato con il Laplaciano frazionario. Nella prima parte, studiamo la regolarità delle soluzioni stabili dell’equazione nonlineare per il p-Laplaciano dove W è un dominio limitato, p 2 (1,+¥) e f è una nonlinearità C1. Questa equazione è la versione nonlineare dell’equazione semilineare 􀀀������������Du = f (u) in un dominio limitato W Rn, che è stata ampiamente studiata in letteratura. Molto recentemente, Cabré, Figalli, Ros-Oton, e Serra [38] hanno dimostrato che le soluzioni stabili delle equazioni semilineari sono limitate, e quindi regolari, in dimensione n 9. Questo risultato è ottimale, dato che esempi di soluzioni illimitate e stabili sono noti in dimensione n 10. Inoltre, i risultati in [38] forniscono una risposta completa ad un annoso problema aperto, proposto da Brezis e Vázquez [25], sulla regolarità delle soluzioni estremali dell’equazione 􀀀������������Du = l f (u). Queste ultime sono infatti esempi non banali di soluzioni stabili di equazioni semilineari, che possono essere limitate o illimitate in dipendenza della dimensione n, del dominio W, e della nonlinearità f . In questa tesi studiamo la limitatezza delle soluzioni stabili di (0.4), che si congettura essere vera fino alla dimensione n < p + 4p/(p 􀀀������������ 1). Sono infatti noti esempi di soluzioni stabili e illimitate quando n p + 4p/(p 􀀀������������ 1), anche quando il dominio è la palla unitaria. Inoltre, nel caso radiale o assumendo ipotesi forti sulla nonlinearità, è stato dimostrato che le soluzioni stabili di (0.4) sono limitate quando n < p + 4p/(p 􀀀������������ 1). Nel Capitolo 1 della tesi dimostriamo una nuova stima L¥ a priori per le soluzioni stabili di (0.4), assumendo una nuova condizione su n e p, che è ottimale nel caso radiale e più restrittiva nel caso generale. Il nostro risultato migliora ciò che è noto in letteratura e ed è il primo esempio di tecnica che produce sia un risultato nel caso non radiale sia il risultato ottimale nel caso radiale. Per ottenere questo risultato estendiamo al caso del p-Laplaciano una tecnica sviluppata da Cabré [30] per il caso classico del problema, con p = 2. La strategia si basa su una disuguaglianza di Hardy sugli insiemi di livello della soluzione, combinata con una disuguaglianza di tipo geometrico per le soluzioni stabili di (0.4). Nella prima parte della tesi ci occupiamo anche di disuguaglianze funzionali di tipo Hardy e Sobolev, su ipersuperfici dello spazio euclideo. Nel fare ciò siamo motivati dalle varie applicazioni di questo tipo di risultati allo studio di stime a priori per le soluzioni stabili, sia nel caso semilineare che nel caso nonlineare ...

La tesi è dedicata allo studio di varie proprietà qualitative possedute dalle soluzioni di equazioni ellittiche poste nello spazio euclideo. L'attenzione principale del lavoro è rivolta a soluzioni intere di equazioni anisotrope/eterogenee che mostrano qualche genere di proprietà di simmetria e, in particolare, che posseggono simmetria unidimensionale. L'elaborato è diviso in due parti. La prima parte è riservata ad equazioni alle derivate parziali locali, mentre la seconda si concentra su di una classe meno usuale di equazioni non locali, determinate da operatori integrali.
The thesis is concerned with the study of several qualitative properties shared by the solutions of elliptic equations set in the Euclidean space. The main focus of the work is on entire solutions of anisotropic/heterogeneous equations that show some kind of symmetric properties and, in particular, that possess one-dimensional symmetry. The dissertation is divided into two parts. The first part deals with local partial differential equations, while the second one addresses a class of less familiar nonlocal equations driven by integral operators.

In everyday situations individuals make decisions. For example, a tourist usually chooses a crowded or recommended restaurant to have dinner. Perhaps it is an individual decision, but the observed pattern of decision-making is a collective phenomenon. Collective behaviour emerges from the local interactions that give rise to a complex pattern at the group level. In our example, the recommendations or simple copying the choices of others make a crowded restaurant even more crowded. The rules of interaction between individuals are important to study. Such studies should be complemented by biological experiments. Recent studies of collective phenomena in animal groups help us to understand these rules and develop mathematical models of collective behaviour. The most important communication mechanism is positive feedback between group members, which we observe in our example. In this thesis, we use a generic experimentally validated model of positive feedback to study collective decision-making. The first part of the thesis is based on the modelling of decision-making associated to the selection of feeding sites. This has been extensively studied for ants and slime moulds. The main contribution of our research is to demonstrate how such aspects as "irrationality", speed and quality of decisions can be modelled using differential equations. We study bifurcation phenomena and describe collective patterns above critical values of a bifurcation points in mathematical and biological terms. In the second part, we demonstrate how the primitive unicellular slime mould Physarum Polycephalum provides an easy test-bed for theoretical assumptions and model predictions about decision-making. We study its searching strategies and model decision-making associated to the selection of food options. We also consider the aggregation model to investigate the fractal structure of Physarum Polycephalum plasmodia.

Fel serie i tryckt bok /Wrong series in the printed book

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
This master thesis is concerned to nonlinear elliptic problem with mono-polar anisotropic potential u + u|u|p−1 + v (x)u + f(x) = 0 in Rn u(x) - 0, as |x| - 00 provided n > 3 and p > n n−2 . These results, between others things, deals with sub-critical, critical and super-critical nonlinearity. We obtain well-posedness of solutions, regularity in c2(Rn), symmetries and asymptotic behavior of solutions in singular spaces Hk. We employ Banach fixed technique and a theorem of regularity elliptic to get those results, this technique does not need of the Hardy type inequalities and variational methods.
Nesta dissertação estudamos o problema elíptico u + u|u|p−1 + v (x)u + f(x) = 0 em Rn u(x) - 0, quando |x| - 00 sujeito a restrições n > 3 e p > n n−2 , cobrindo os casos sub-críticos, críticos e super-críticos. Obtemos boa-colocação de soluções, regularidade, simetrias de soluções e comportamento assintótico em espaços singulares Hk. Empregamos um argumento de ponto fixo em Hk e Ek ao invés de usar desigualdades do tipo Hardy e métodos variacionais.

Cette étude se concentre sur la compréhension de la physique des instabilités dans différents écoulements de cisaillement, particulièrement la cavité entraînée et la cavité thermocapillaire, où l'écoulement d'un fluide incompressible est assuré soit par le mouvement d’une ou plusieurs parois, soit par des contraintes d’origine thermique.Un code spectral a été validé sur le cas très étudié de la cavité entrainée par une paroi mobile. Il est démontré dans ce cas que l'écoulement transit d'un régime stationnaire à un instationnaire au-delà d'une valeur critique du nombre de Reynolds. Ce travail est le premier à donner une interprétation physique de l'évolution non monotonique du nombre de Reynolds critique en fonction du facteur d'aspect. Lorsque le fluide est entraîné par deux parois mobiles, la cavité entraînée possède un plan de symétrie particulièrement sensible. Des solutions asymétriques peuvent être observés en plus de la solution symétrique au-dessus d'une certaine valeur du nombre de Reynolds. La transition oscillatoire entre la solution symétrique et les solutions asymétriques est expliquée physiquement par les forces en compétition. Dans le cas asymétrique, l'évolution de la topologie permet à l'écoulement de rester stationnaire avec l'augmentation du nombre de Reynolds. Lorsque l'équilibre est perdu une instabilité se manifeste par l'apparition d'un régime oscillatoire dans l'écoulement asymétrique.Dans une cavité thermocapillaire rectangulaire avec une surface libre, Smith et Davis prévoient deux types d'instabilités convectives thermiques: des rouleaux longitudinaux stationnaires et des ondes hydrothermales instationnaires. L'apparition de ses instabilités a été mis en évidence à plusieurs reprises expérimentalement et numériquement. Alors que les applications impliquent souvent plus d'une surface libre, il semble qu'il y ait peu de connaissances sur l'écoulement thermocapillaire entraînée avec deux surfaces libres. Un film liquide libre soumis à des contraintes thermocapillaires possède un plan de symétrie particulier comme dans le cas de la cavité entrainée par deux parois mobiles. Une étude de stabilité linéaire avec deux profils de vitesse pour le film liquide libre est présentée avec différents nombres de Prandtl. Au-delà d'un nombre de Marangoni critique, il est découvert que ces états de base sont sensibles à quatre types d'instabilités convectives thermiques qui peuvent conserver ou briser la symétrie du système. Les mécanismes qui permettent de prédire ces instabilités sont également découverts et interpréter en fonction de la valeur du nombre de Prandtl du fluide. La comparaison avec les travaux de Smith et Davis est faite. Une simulation numérique directe permet de valider les résultats obtenus avec l'étude de stabilité de linéaire
This study focuses on the understanding of the physics of different instabilities in driven cavities, specifically the lid-driven cavity and the thermocapillarity driven cavity where flow in an incompressible fluid is driven either due to one or many moving walls or due to surface stresses that appear from surface tension gradients caused by thermal gradients. A spectral code is benchmarked on the well-studied case of the lid-cavity driven by one moving wall. In this case, It is shown that the flow transit form a steady regime to unsteady regime beyond a critical value of the Reynolds number. This work is the first to give a physical interpretation of the non-monotonic evolution of the critical Reynolds number versus the size of the cavity. When the fluid is driven by two facing walls moving in the same direction, the cavity possesses a plane of symmetry particularly sensitive. Thus, asymmetrical solutions can be observed in addition to the symmetrical solution above a certain value of the Reynolds number. The oscillatory transition between the symmetric solution and asymmetric solutions is explained physically by the forces in competition. In the asymmetric case, the change of the topology allows the flow to remain steady with increasing the Reynolds number. When the equilibrium is lost, an instability manifests by the appearance of an oscillatory regime in the asymmetric flow. In a rectangular cavity thermocapillary with a free surface, Smith and Davis found two types of thermal convective instabilities: steady longitudinal rolls and unsteady hydrothermal waves. The appearance of its instability has been highlighted repeatedly experimentally and numerically. While applications often involve more than a free surface, it seems that there is little knowledge about the thermocapillary driven flow with two free surfaces. A free liquid film possesses a particular plane of symmetry as in the case of the two-sided lid-driven cavity. A linear stability analysis for the free liquid film with two velocity profiles is presented with various Prandtl numbers. Beyond a critical Marangoni number, it is observed that these basic states are sensitive to four types of thermal convective instabilities, which can keep or break the symmetry of the system. Mechanisms that predict these instabilities are discovered and interpreted according to the value of the Prandtl number of the fluid. Comparison with the work of Smith and Davis is made. A direct numerical simulation is done to validate the results obtained with the linear stability analysis

ABSTRACT In jet problems the conserved quantity plays a central role in the solution process. The conserved quantities for laminar jets have been established either from physical arguments or by integrating Prandtl's momentum boundary layer equation across the jet and using the boundary conditions and the continuity equation. This method of deriving conserved quantities is not entirely systematic and in problems such as the wall jet requires considerable mathematical and physical insight. A systematic way to derive the conserved quantities for jet °ows using conservation laws is presented in this dissertation. Two-dimensional, ra- dial and axisymmetric °ows are considered and conserved quantities for liquid, free and wall jets for each type of °ow are derived. The jet °ows are described by Prandtl's momentum boundary layer equation and the continuity equation. The stream function transforms Prandtl's momentum boundary layer equation and the continuity equation into a single third- order partial di®erential equation for the stream function. The multiplier approach is used to derive conserved vectors for the system as well as for the third-order partial di®erential equation for the stream function for each jet °ow. The liquid jet, the free jet and the wall jet satisfy the same partial di®erential equations but the boundary conditions for each jet are di®erent. The conserved vectors depend only on the partial di®erential equations. The derivation of the conserved quantity depends on the boundary conditions as well as on the di®erential equations. The boundary condi- tions therefore determine which conserved vector is associated with which jet. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived. This approach gives a uni¯ed treatment to the derivation of conserved quantities for jet °ows and may lead to a new classi¯cation of jets through conserved vectors. The conservation laws for second order scalar partial di®erential equations and systems of partial di®erential equations which occur in °uid mechanics are constructed using di®erent approaches. The direct method, Noether's theorem, the characteristic method, the variational derivative method (mul- tiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example. The conservation laws for the non-linear di®usion equa- tion for the spreading of an axisymmetric thin liquid drop, the system of two partial di®erential equations governing °ow in the laminar two-dimensional jet and the system of two partial di®erential equations governing °ow in the laminar radial jet are discussed via these approaches. The group invariant solutions for the system of equations governing °ow in two-dimensional and radial free jets are derived. It is shown that the group invariant solution and similarity solution are the same. The similarity solution to Prandtl's boundary layer equations for two- dimensional and radial °ows with vanishing or constant mainstream velocity gives rise to a third-order ordinary di®erential equation which depends on a parameter. For speci¯c values of the parameter the symmetry solutions for the third-order ordinary di®erential equation are constructed. The invariant solutions of the third-order ordinary di®erential equation are also derived.

碩士
國立成功大學
系統及船舶機電工程學系碩博士班
92
The coupling of solutions at discrete points is strong by using the differential quadrature element method (DQEM). Thus, convergence and accurate can be assured by using less discrete points and less arithmetic operations which can reduce the computer CPU time required. 　　Like FEM, in using DQEM to solve a problem the domain is separated into many elements. The DQ discretization is carried out on an element-basis. The discretized governing differential or partial differential equations defined on the elements, transition conditions on inter-element boundaries and boundary conditions are assembled to obtain an overall algebraic system. 　　In this work, the DQEM analysis model of shear-deformable axisymmetric circular plates on Pasternak elastic foundation is developed, and the related computer problems is implemented. Problems of static deformation are analyzed. They prove that the developed DQEM analysis model is excellent。

The essential characteristics of large systems is their high dimensionality due to which conventional control techniques fail to give reasonable solutions with reasonable computational efforts. A number of large systems encountered in practice are composed of subsystems with similar dynamics interconnected in a symmetrical fashion. The analysis and control of a large system with these particular features must take advantage of the existing structural properties to achieve computational simplifications of the overall problem. The focus of this thesis is the feedback design and analysis of large systems possessing the property of spatial symmetry. Specifically, the problems of controller design and analysis for infinite dimensional toeplitz systems and their finite dimensional analogs, circulant systems, are studied. These spatially symmetric systems are special classes of large systems. The first part of this thesis is focused on the development of formal controller design methodologies which take advantage of the properties of the circulant matrices. The key to this development is the use of the FFT algorithm to diagonalize circulant matrices. The resulting controller design methodologies are computationally attractive and easily applicable to large systems with circulant symmetry. More specifically, the H$\sb2$ and H$\sb{\infty}$ controller synthesis problems are studied in detail and are shown to decompose into lower order independent problems. The second part of this work concentrates on proving that certain finite order toeplitz systems are asymptotically equivalent in an appropriate sense to circulant systems. This result justifies the use of circulant control design techniques for certain toeplitz systems. Moreover, the closed loop effects of controlling a toeplitz system with a controller designed for its asymptotically equivalent circulant system are analyzed. The application of the developed theoretical results to a realistic example is the focus of the last part of the thesis. The adaptive optics system used in this example is modeled by a transfer function matrix with toeplitz symmetry. The computational efficiency of the controller design methodologies developed in this thesis is illustrated by designing a series of controllers for this system.