Dissertationen zum Thema „Integral equation theories“

The thesis focuses on the prediction of solvation thermodynamics using integral equation theories. Our main goal is to improve the approach using a rational correction. We achieve it by extending recently introduced pressure correction, and rationalizing it in the context of solvation entropy. The improved model (to which we refer as advanced pressure correction) is rather universal. It can accurately predict solvation free energies in water at both ambient and non-ambient temperatures, is capable of addressing ionic solutes and salt solutions, and can be extended to non-aqueous systems. The developed approach can be used to model processes in biological systems, as well as to extend related theoretical models further.

The principal theorem of this thesis is a theorem by Peano on the existence of a solution to a certain integral equation. The two primary notions underlying this theorem are uniform convergence and equi-continuity. Theorems related to these two topics are proved in Chapter II. In Chapter III we state and prove a classical existence and uniqueness theorem for an integral equation. In Chapter IV we consider the approximation on certain functions by means of elementary expressions involving "bent line" functions. The last chapter, Chapter V, is the proof of the theorem by Peano mentioned above. Also included in this chapter is an example in which the integral equation has more than one solution. The first chapter sets forth basic definitions and theorems with which the reader should be acquainted.

In this thesis, we consider the massive field theories in 1+1 dimensions known as affine Toda quantum field theories. These have the special property that they possess an infinite number of conserved quantities, a feature which greatly simplifies their study, and makes extracting exact information about them a tractable problem. We consider these theories both in the full space (the bulk) and in the half space bounded by an impenetrable boundary at x = 0. In particular, we consider their fundamental objects: the scattering matrices in the bulk, and the reflection factors at the boundary, both of which can be found in a closed form. In Chapter 1, we provide a general introduction to the topic before going on, in Chapter 2, to consider the simplest ATFT—the sine-Gordon model—with a boundary. We begin by studying the classical limit, finding quite a clear picture of the boundary structure we can expect in the quantum case, which is introduced in Chapter 3. We obtain the bound-state structure for all integrable boundary conditions, as well as the corresponding reflection factors. This structure turns out to be much richer than had hitherto been imagined. We then consider more general ATFTs in the bulk. The sine-Gordon model is based on a(^(1))(_1), but there is an ATFT for any semi-simple Lie algebra. This underlying structure is known to show up in their S-matrices, but the path back to the parameters in the Lagrangian is still unclear. We investigate this, our main result being the discovery of a "generalised bootstrap" equation which explicitly encodes the Lie algebra into the S-matrix. This leads to a number of new S-matrix identities, as well as a generalisation of the idea that the conserved charges of the theory form an eigenvector of the Cartan matrix. Finally our results are summarised in Chapter 5, and possible directions for further study are highlighted.

Determination des fonctions de green des structures microrubans bicouches, calculees a l'aide de la theorie des milieux stratifies, et des densites de courant de surfaces qui sont evaluees par la methode des moments. Ainsi, l'impedance d'entree de l'antenne et son diagramme de rayonnement s'obtiennent facilement. Mise en evidence et quantification du rayonnement parasite

Les enjeux liés au réchauffement climatique poussent à la recherche de solvants plus respectueux de l'environnement. Le CO₂ supercritique (scCO₂) est un candidat prometteur, en raison de sa non-toxicité et de la facilité de son recyclage. Il est déjà utilisé dans des processus industriels chimiques tels que la séparation et l'extraction. De plus, les propriétés de solvatation peuvent être ajustées par des variations de pression. Or, son pouvoir solvatant, surtout dans le cas de solutés polaires, est très faible, mais il peut être accru en phase supercritique. Afin de mieux comprendre la corrélation entre les variations de pression et le pouvoir solvatant du scCO₂, il est important de disposer d'un outil efficace pour prédire les propriétés de solvatation dans différentes conditions thermodynamiques et en présence de divers solutés. Pour cela, nous faisons appel à la théorie de la fonctionnelle de la densité moléculaire (MDFT) qui offre une alternative prometteuse aux simulations numériques, combinant une modélisation microscopique précise avec des calculs ultra-rapides (100 000 fois plus rapides que la dynamique moléculaire). Dans l'approximation du fluide homogène de référence, la fonctionnelle MDFT est divisée en quatre parties : la partie idéale, l'interaction externe soluté/solvant, l'interaction homogène solvant/solvant et le terme de bridge. L'interaction homogène solvant/solvant nécessite les fonctions de corrélation directe en phase condensée du solvant, qui peuvent être calculées à partir de simulations de dynamique moléculaire (MD) coûteuses, ou de théories d'équations intégrales moléculaires, approximatives mais rapides. Différentes approximations existent pour le terme de bridge, qui peut également être paramétré sur les propriétés thermodynamiques du solvant pur. Le développement de la MDFT en tant qu'outil puissant pour étudier la solvatation dans le scCO₂ nécessite la construction d'équations intégrales moléculaires précises pour le scCO₂. Dans un premier temps, nous avons étudié les fonctions de corrélation directe exactes du scCO₂ obtenues à partir de MD et celles des équations intégrales moléculaires les plus simples, l'approximation hypernetted chain (HNC). Ensuite, nous avons déterminé les propriétés de solvatation pour des solutés atomiques et moléculaires par MDFT, alimentée par les simulations MD pour une condition thermodynamique particulière. Parallèlement, nous avons effectué des simulations MD pour tester la validité de nos résultats. Enfin, nous avons étudié d'autres conditions thermodynamiques pour déterminer l'énergie libre de solvatation d'un soluté de CO₂ dans le scCO₂ (c'est-à-dire le potentiel chimique) par MDFT alimentée soit par MD soit par HNC. Nous avons réussi à déterminer les propriétés de solvatation en quelques minutes et de manière aussi précise que les simulations MD
Climate change issues drive the search for more environmentally friendly solvents. Supercritical CO₂ (scCO₂) is a promising candidate due to its non-toxicity and ease of recycling, despite its low solvation power for polar solutes. It is already used in industrial chemical processes such as separation and extraction. Moreover, solvation properties can be adjusted by pressure variations. To better understand the correlation between pressure variations and the solvation power of scCO₂, it is essential to have an efficient tool to predict solvation properties under different thermodynamic conditions and in the presence of various solutes. For this, we turn to molecular density functional theory (MDFT), which offers a promising alternative by combining precise microscopic modeling with ultra-fast calculations (100,000 times faster than molecular dynamics). In the homogeneous reference approach, the MDFT functional is divided into four parts: the ideal part, the external solute/solvent interaction, the homogenous solvent/solvent interaction and the bridge term. The homogenous solvent/solvent interaction requires the direct correlation functions of the bulk solvent, which can be calculated from either expensive MD simulations or fast but approximate molecular integral equation theories. Different approximations exist for the bridge term, which can also be parametrized from the thermodynamic properties of the pure solvent. In this work, we first investigated the exact direct correlation functions of scCO₂ obtained from MD and those from the simplest molecular integral equations, the hypernetted-chain (HNC) approximation. We also fit two standard bridge terms using the equation of state of CO₂ obtained from MD. Next, we determine the solvation properties for atomic and molecular solutes using MDFT, fed by MD simulations for a particular thermodynamic condition. Simultaneously, we conduct MD simulations to test the validity of our results. Finally, we explore other thermodynamic conditions to determine the free energy of solvation of a CO₂ solute in scCO₂ (i.e., the chemical potential) using MDFT, fed either by MD or HNC. We successfully determine solvation properties in a few minutes with accuracy comparable to MD simulations

La finalite de ce travail est la modelisation de l'interaction d'une onde electromagnetique plane en regime harmonique avec des composantes simples d'une cible complexe. Ceci dans le but de caracteriser soit la contribution de chaque element de la structure au champ diffracte, soit le couplage entre elements distincts. Une application particuliere concerne la surface equivalente radar, monostastique et bistatique, de ces structures. L'utilisation d'equations integrales de surface permet d'obtenir les courants electrique et magnetique induits par l'onde incidente. Ces equations integrales sont resolues numeriquement par la methode des moments, pour laquelle la surface des objets, de forme quelconque, est modelisee par des triangles plans. Plusieurs types d'objet sont etudies avec cette methode

After giving a brief background on the subject we introduce in section two the Phase-Integral Method of Fröman & Fröman in terms of the platform function of Yngve and Thidé. In section three we derive a different form of the radial Bohr-Sommerfeld condition in terms of the apsidal angle of the corresponding classical motion. Using the derived expression, we then show how easily one can calculate the exact energy eigenvalues of the hydrogen atom and the isotropic three-dimensional harmonic oscillator, we also derive an expression for higher order quantization condition. In section four we derive an expression for the angular frequencies of a restricted (0≤φ≤β) soap bubble and also give a proposition concerning the parameters l and m of the associated Legendre differential equation.
Vi använder Fröman & Frömans Fas-Integral Metod tillsammans med Yngve & Thidés plattformfunktion för att härleda kvantiseringsvilkoret för högre ordningar. I sektion tre skriver vi Bohr-Sommerfelds kvantiseringsvillkor på ett annorlunda sätt med hjälp av den så kallade apsidvinkeln (definierad i samma sektion) för motsvarande klassiska rörelse, vi visar också hur mycket detta underlättar beräkningar av energiegenvärden för väteatomen och den isotropa tredimensionella harmoniska oscillatorn. I sektion fyra tittar vi på en såpbubbla begränsad till området 0≤φ≤β för vilket vi härleder ett uttryck för dess (vinkel)egenfrekvenser. Här ger vi också en proposition angående parametrarna l och m tillhörande den associerade Legendreekvationen.

La méthode d'intégration produit a été proposée pour résoudre des équations linéaires de Fredholm de deuxième espèce singulières dont la solution exacte est régulière, au moins continue. Dans ce travail on adapte cette méthode à des équations dont la solution est juste intégrable. On étudie également son extension au cas non linéaire posé dans l'espace des fonctions intégrables. Ensuite, on propose une autre manière de mettre en oeuvre la méthode d'intégration produit : on commence par linéariser l'équation par une méthode de type Newton puis on discrétise les itérations de Newton par la méthode d'intégration produit
The product integration method has been proposed for solving singular linear Fredholm equations of the second kind whose exact solution is smooth, at least continuous. In this work, we adapt this method to the case where the solution is only integrable. We also study the nonlinear case in the space of integrable functions. Then, we propose a new version of the method in the nonlinear framework : we first linearize the eqaution by a Newton type method and then discretize the Newton iterations by the product integration method

This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.

Thesis (Ph. D.)--Ohio State University, 2003.
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Cette these decrit la synthese de reseaux lineaires de dipoles couples electromagnetiquement (dipoles c. E. M. ) avec prise en compte des couplages. Cette synthese est ensuite appliquee a la realisation d'un diagramme de rayonnement forme du type depression spherique

Dans cette thèse, tout d’abord, nous faisons l’Analyse Mathématique du modèle exact du chauffage radiatif d’un corps semi-transparent $\Omega$ par une source radiative noire qui l’entoure. Il s’agit donc d’étudier le couplage d’un système d’Equations de Transfert Radiatif avec condition au bord de réflectivité indépendantes avec une équation de conduction de la chaleur non linéaire avec condition limite non linéaire de type Robin. Nous prouvons l’existence et l’unicité de la solution et nous démontrons des bornes uniformes sur la solution et les intensités radiatives dans chaque bande de longueurs d’ondes pour laquelle le corps est semi-transparent, en fonction de bornes sur les données, Deuxièmement, nous considérons le problème du contrôle optimal de la température absolue à l’intérieur du corps semi-transparent $\Omega$ en agissant sur la température absolue de la source radiative noire qui l’entoure. À cet égard, nous introduisons la fonctionnelle coût appropriée et l’ensemble des contrôles admissibles $T_{S}$, pour lesquels nous prouvons l’existence de contrôles optimaux. En introduisant l’espace des états et l’équation d’état, une condition nécessaire de premier ordre pour qu’un contrôle $T_{S}$ : t ! $T_{S}$ (t) soit optimal, est alors dérivée sous la forme d’une inéquation variationnelle en utilisant le théorème des fonctions implicites et le problème adjoint. Ensuite, nous considérons le problème de l’existence et de l’unicité d’une solution faible des équations de la thermoviscoélasticité dans une formulation mixte de type Hellinger- Reissner, la nouveauté par rapport au travail de M.E. Rognes et R. Winther (M3AS, 2010) étant ici l’apparition de la viscosité dans certains coefficients de l’équation constitutive, viscosité qui dépend dans ce contexte de la température absolue T(x, t) et donc en particulier du temps t. Enfin, nous considérons dans ce cadre le problème du contrôle optimal de la déformation du corps semi-transparent $\Omega$, en agissant sur la température absolue de la source radiative noire qui l’entoure. Nous prouvons l’existence d’un contrôle optimal et nous calculons la dérivée Fréchet de la fonctionnelle coût réduite
This thesis begins with a rigorous mathematical analysis of the radiative heating of a semi-transparent body made of glass, by a black radiative source surrounding it. This requires the study of the coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. We prove existence and uniqueness of the solution, and give also uniform bounds on the solution i.e. on the absolute temperature distribution inside the body and on the radiative intensities. Now, we consider the temperature $T_{S}$ of the black radiative source S surrounding the semi-transparent body $\Omega$ as the control variable. We adjust the absolute temperature distribution (x, t) 7! T(x, t) inside the semi-transparent body near a desired temperature distribution Td(·, ·) during the time interval of radiative heating ]0, tf [ by acting on $T_{S}$. In this respect, we introduce the appropriate cost functional and the set of admissible controls $T_{S}$, for which we prove the existence of optimal controls. Introducing the State Space and the State Equation, a first order necessary condition for a control $T_{S}$ : t 7! $T_{S}$ (t) to be optimal is then derived in the form of a Variational Inequality by using the Implicit Function Theorem and the adjoint problem. We come now to the goal problem which is the deformation of the semi-transparent body $\Omega$ by heating it with a black radiative source surrounding it. We introduce a weak mixed formulation of this thermoviscoelasticity problem and study the existence and uniqueness of its solution, the novelty here with respect to the work of M.E. Rognes et R. Winther (M3AS, 2010) being the apparition of the viscosity in some of the coefficients of the constitutive equation, viscosity which depends on the absolute temperature T(x, t) and thus in particular on the time t. Finally, we state in this setting the related optimal control problem of the deformation of the semi-transparent body $\Omega$, by acting on the absolute temperature of the black radiative source surrounding it. We prove the existence of an optimal control and we compute the Fréchet derivative of the associated reduced cost functional

Le contexte d'étude est celui de la Compatibilité ÉlectroMagnétique (CEM). L'objectif de la CEM est, comme son nom l'indique, d'assurer la compatibilité entre une source de perturbation électromagnétique et un système électronique victime. Or, la prédiction de ces niveaux de perturbation ne peut pas s'effectuer à l'aide d'un simple calcul analytique, en raison de la géométrie qui est généralement complexe pour le système que l'on étudie, tel que le champ à l'intérieur d'un cockpit d'avion par exemple. En conséquence, nous sommes contraints d'employer des méthodes numériques, dans le but de prédire ce niveau de couplage entre les sources et les victimes. Parmi les nombreuses méthodes numériques existantes à ce jour, les méthodes Multi-Domaines (MD) sont très prisées. En effet, elles offrent la liberté aux utilisateurs de choisir la méthode numérique la plus adaptée, en fonction de la zone géométrique à calculer. Au sein de ces méthodes MD, la " Domain Decomposition Method " (DDM) présente l'avantage supplémentaire de découpler chacun de ces domaines. En conséquence, la DDM est particulièrement intéressante, vis-à-vis des méthodes concurrentes, en particulier sur l'aspect du coût numérique. Pour preuve, l'ONERA continue de développer cette méthode qui ne cesse de montrer son efficacité depuis plusieurs années, notamment pour le domaine des Surfaces Équivalentes Radar (SER) et des antennes. L'objectif de l'étude est de tirer profit des avantages de cette méthode pour des problématiques de CEM. Jusqu'à maintenant, de nombreuses applications de CEM, traitées par le code DDM, fournissaient des résultats fortement bruités. Même pour des problématiques électromagnétiques très simples, des problèmes subsistaient, sans explication convaincante. Ceci justifie cette étude. Le but de cette thèse est de pouvoir appliquer ce formalisme DDM à des problématiques de CEM. Dans cette optique, nous avons été amenés à redéfinir un certain nombre de conventions, qui interviennent au sein de la DDM. Par ailleurs, nous avons développé un modèle spécifique pour les ouvertures, qui sont des voies de couplage privilégiées par les ondes, à l'intérieur des cavités que représentent les blindages. Comme les ouvertures sont, en pratique, de petites dimensions devant la longueur d'onde, on s'est intéressé à un modèle quasi-statique. Nous proposons alors un modèle, qui a été implémenté, puis validé. Suite à ce modèle, nous avons développé une méthode originale, basée sur un calcul en deux étapes, permettant de ne plus discrétiser le support des ouvertures dans les calculs 3D.

The Lie theory of extended groups is a practical tool in the analysis of differential equations, particularly in the construction of solutions. A formalism of the Lie theory is given and contrasted with Noether's theorem which plays a prominent role in the analysis of differential equations derivable from a Lagrangian. The relationship between the Lie and Noether approach to differential equations is investigated. The standard separation of Lie point symmetries into Noetherian and nonNoetherian symmetries is shown to be irrelevant within the context of nonlocality. This also emphasises the role played by nonlocal symmetries in such an approach.
Thesis (M.Sc.)-University of Natal, Durban, 1997.

In this thesis, we consider a new generalization of the Fourier transform, depending on four complex parameters and all the powers of the Fourier transform. This new transform is studied in some Lebesgue spaces. In fact, taking into account the values of the parameters of the operator, we can have very different kernels and so, the corresponding operator is studied in different Lebesgue spaces, accordingly with its kernel. We begin with the characterization of each operator by its characteristic polynomial. This characterization serves as a basis for the study of the forthcoming properties. Following this, we present, for each case, the spectrum of the corresponding operator, necessary and sufficient conditions for which the operator is invertible, Parseval-type identities and conditions for which the operator is unitary and an involution of order n. After this, we contruct new convolutions associated with those operators and obtain the corresponding factorization identities and some norm inequalities. By using these new operators and convolutions, we construct new integral equations and study their solvability. In this sense, we have equations generated by the studied operators and also a class of equations of convolution-type depending on multi-dimensional Hermite functions. Furthermore, we study the solvability of classical integral equations, using the new operators and convolutions, namely a class of Wiener-Hopf plus Hankel equations, whose solution is written in terms of a Fourier-type series. For one case of this generalization of the Fourier transform, that only depends on the cosine and sine Fourier transforms, we obtain PaleyWiener and Wiener’s Tauberian results, using the associated convolution and a new translation induced by that convolution. Heisenberg uncertainty principles for the one-dimensional case and for the multi-dimensional case are obtained for a particular case of the introduced operator. At the end, as an application outside of mathematics, we obtain a new result in signal processing, more properly, in a filtering processing, by applying one of our new convolutions.
Nesta tese, consideramos uma nova generalização da transformação de Fourier, dependente de quatro parâmetros complexos e de todas as potências da transformação de Fourier. Esta nova transformação é estudada em alguns espaços de Lebesgue. De facto, tendo em conta os valores dos parâmetros, podemos ter núcleos muito diferentes e assim, o correspondente operador é estudado em diferentes espaços de Lebesgue, de acordo com o seu núcleo. Começamos com a caracterização de cada operador pelo seu polinómio característico. Esta caracterização serve de base para o estudo das propriedades seguintes. Seguindo isto, apresentamos, para cada caso, o espetro do correspondente operador, condições necessárias e suficientes para as quais o operador é invertível, identidades do tipo de Parseval e condições para as quais o operador é unitário e uma involução de ordem n. Depois disto, construímos novas convoluções associadas àqueles operadores e obtemos as correspondentes identidades de factorização e algumas desigualdades da norma. Usando estes novos operadores e convoluções, construímos novas equações integrais e estudamos a sua solvabilidade. Neste sentido, temos equações geradas pelos operadores estudados e também uma classe de equações do tipo de convolução dependendo de funções de Hermite multidimensionais. Além disso, estudamos a solvabilidade de equações integrais clássicas, usando os novos operadores e convoluções, nomeadamente uma classe de equações de Wiener-Hopf mais Hankel, cuja solução é escrita em termos de uma série do tipo de Fourier. Para um caso desta generalização da transformação de Fourier, que depende apenas das transformações de Fourier do cosseno e do seno, obtemos resultados de Paley-Wiener e resultados Tauberianos de Wiener, usando a convolução associada e uma nova translação induzida por essa convolução. Princípios de incerteza de Heisenberg para os casos unidimensional e multidimensional são obtidos para um caso particular do operador introduzido. No final, como uma aplicação fora da matemática, obtemos um novo resultado em processamento de sinal, mais propriamente, num processo de filtragem, por aplicação de uma das nossas novas convoluções.
Programa Doutoral em Matemática Aplicada