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Dissertations / Theses on the topic 'Nonlocal equations in time'
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Hariz, Belgacem Khader. "Higher-order Embedding Formalism, Noether’s Theorem on Time Scales and Eringen’s Nonlocal Elastica." Electronic Thesis or Diss., Pau, 2022. https://theses.hal.science/tel-03981833.
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En mathématiques, le calcul des variations est un ensemble de méthodes permettant la détermination de solutions à des problèmes d'optimisation des quantités traduites en termes de fonctionnelle. De nombreuses applications existent, notamment dans la recherche de courbes ou de surfaces minimales. Les systèmes dynamiques considérés sont de natures diverses (équations différentielles, intégrales ou stochastiques) et modélisent des problèmes d'origines multiples : aérospatiale, automobile, biologie, économie, médecine, etc. Le théorème de Noether présente un fort intérêt puisqu'il propose une loi de conservation explicite (traduisant souvent une quantité physique comme l'énergie totale ou le moment angulaire en mécanique classique) qui permet de réduire ou d'intégrer l'équation différentielle associée par quadrature. L'objectif de ma thèse contient de nombreux thèmes, dans le premier but nous allons : *) donner le théorème de Noether discret dans le cadre ”time scale” (Le formalisme lagrangien et hamiltonien). Le passage de la nature discrète à la nature continue de la structure la morphologie est d'un intérêt primordial en physique pour comprendre comment la microstructure peut influencer les propriétés macroscopiques du matériau à plus grande échelle. Ce passage peut être modélisé par un système discret appelé 'Hencky's chain' et l'équation du mouvement est donnée par des équations aux différences non linéaires et cette équation ne possède pas de Lagrangien. Le deuxième but nous allons : *) donner les structures lagrangienne, hamiltonienne via le facteur intégrant et trouver la solution analytique de l'équation non locale au sens d'Eringen (nonlocalité différentielle d'Eringen, 1983). Le troisième but nous allons : *) étudier l'existence des formulations variationnelles via le principe de Brezis Ekeland-Nayroles (Gery de Saxce) - application sur la formulation 4D développée par E. Rouhaud pour l'étude des déformations des matériaux *) développer des schémas numériques qui respectent certaines particularités. En particulier, un schéma permettent de mettre en œuvre la théorie 4D développée par E. Rouhaud. *) applications numériques et théoriques sur le problème des déformations des matériauxThe aim of this thesis is to deal with the connection between continuous and discrete versions of a given object. This connection can be studied in two different directions: one going from a continuous setting to a discrete analogue, and in a symmetric way, from a discrete setting to a continuous one. The first procedure is typically used in numerical analysis in order to construct numerical integrators and the second one is typical of continuous modeling for the study of micro-structured materials.In this manuscript, we focus our attention on three distinct problems. In the first part, we propose a general framework precising different ways to derive a discrete version of a differential equation called discrete embedding formalism.More precisely, we exhibit three main discrete associate: the differential, integral or variational structure in both classical and high-order approximations.The second part focuses on the preservation of symmetries for discrete versions of Lagrangian and Hamiltonian systems, i.e., the discrete analogue of Noether's theorem.Finally, the third part applies these results in mechanics, i.e., the problem studied by N. Challamel, Kocsis and Wang called Eringen's nonlocal elastica equation which can beobtained by the continualization method. Precisely, we construct a discrete version of Eringen's nonlocal elastica then we study the difference with Challamel's proposal
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Figueroa, Iglesias Susely. "Integro-differential models for evolutionary dynamics of populations in time-heterogeneous environments." Thesis, Toulouse 3, 2019. http://www.theses.fr/2019TOU30098.
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Cette thèse porte sur l'étude qualitative de plusieurs équations paraboliques de type Lotka-Volterra issues de la biologie évolutive et de l'écologie, équations qui prennent en compte un taux de croissance périodique en temps et un phénomène de compétition non locale. Dans une première partie nous étudions d'abord la dynamique des populations phénotypiquement structurées sous l'effet des mutations et de la sélection dans des environnements qui varient périodiquement en temps, puis nous étudions l'impact d'un changement climatique sur ces populations, en considérant que les conditions environnementales varient selon une tendance linéaire, mais de manière oscillatoire. Dans les deux problèmes nous commençons par étudier le comportement en temps long des solutions. Ensuite nous utilisons une approche basée sur les équations de Hamilton-Jacobi pour l'étude asymptotique de ces solutions en temps long lorsque l'effet des mutations est petit. Nous prouvons que lorsque l'effet des mutations disparaît, la densité phénotypique de la population se concentre sur un seul trait (qui varie linéairement avec le temps dans le deuxième modèle), tandis que la taille de la population oscille périodiquement. Pour le modèle de changement climatique nous fournissons également un développement asymptotique de la taille moyenne de la population et de la vitesse critique menant à l'extinction de la population, ce qui est lié à la dérivation d'un développement asymptotique de la valeur propre de Floquet en fonction du taux de diffusion. Dans la deuxième partie, nous étudions quelques exemples particuliers de taux de croissance en donnant des solutions explicites et semi-explicites au problème, et nous présentons quelques illustrations numériques pour le modèle périodique. De plus, étant motivés par une expérience biologique, nous comparons deux populations évoluant dans des environnements différents (constants ou périodiques). En outre, nous présentons une comparaison numérique entre les modèles stochastiques et déterministes pour le phénomène de transfert horizontal des gènes. Dans un contexte Hamilton-Jacobi, nous parvenons à reproduire numériquement le sauvetage évolutif d'une petite population que nous observons dans le modèle stochastiqueThis thesis focuses on the qualitative study of several parabolic equations of the Lotka-Volterra type from evolutionary biology and ecology taking into account a time-periodic growth rate and a non-local competition term. In the initial part we first study the dynamics of phenotypically structured populations under the effect of mutations and selection in environments that vary periodically in time and then the impact of a climate change on such population considering environmental conditions which vary according to a linear trend, but in an oscillatory manner. In both problems we first study the long-time behaviour of the solutions. Then we use an approach based on Hamilton-Jacobi equations to study these long-time solutions asymptotically when the effect of mutations is small. We prove that when the effect of mutations vanishes, the phenotypic density of the population is concentrated on a single trait (which varies linearly over time in the second model), while the population size oscillates periodically. For the climate change model we also provide an asymptotic expansion of the mean population size and of the critical speed leading to the extinction of the population, which is closely related to the derivation of an asymptotic expansion of the Floquet eigenvalue in terms of the diffusion rate. In the second part we study some particular examples of growth rates by providing explicit and semi-explicit solutions to the problem and present some numerical illustrations for the periodic model. In addition, being motivated by a biological experiment, we compare two populations evolved in different environments (constant or periodic). In addition, we present a numerical comparison between stochastic and deterministic models modelling the horizontal gene transfer phenomenon. In a Hamilton-Jacobi context, we are able to numerically reproduce the evolutionary rescue of a small population that we observe in the stochastic model
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Belin, Théo. "On the free boundary of a forward-backward parabolic equation." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM040.
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Dans cette thèse, nous nous intéressons à un problème parabolique de type avance-rétrograde ainsi que la frontière libre qui en découle. L'équation modélise un changement de phase dirigé par un problème de Stefan couplé avec un opérateur d'hystérésis non local en temps. Notre étude s'occupe de questions théoriques et numériques soulevées par ce type d'équations non locales en temps, notamment autour de la frontière libre.Premièrement nous établissons une équivalence entre des inégalités d'entropie associées au problème avance-rétrograde et une formulation faible de l'opérateur d'hystérésis. Cette découverte motive la construction d'un schéma numérique de type volumes finis, en dimension quelconque d'espace, dont nous démontrons la convergence vers une solution. La compacité de la suite de solutions approchées repose sur l'inégalité de Hilpert. Des expériences numériques en dimensions 1 et 2 étayent ces résultats et montrent le comportement la frontière libre.Ensuite nous établissons un cadre général de solutions de viscosité pour des équations de propagation de front qui sont non locales en espace et en temps. Elles peuvent notamment inclure un couplage avec une équation d'évolution interne. Un théorème de comparaison strict ainsi qu'un théorème d'existence issu de la méthode de Perron sont démontrés. Le problème de Stefan ainsi que quelques variations de ce problème rentrent dans ce cadre général.Enfin, motivés par l'étude des équations paraboliques en domaines variables en temps apparaissant dans les couplages des équations de front, nous démontrons de nouveaux résultats de régularité maximale dans les espaces de Lebesgues. Un intérêt particulier est porté sur l'estimation précise de la constante de régularité pour les opérateurs non-autonomes et relativement continus. Ces résultats sont à l'origine de nouvelles hypothèses de croissance garantissant l'existence de solutions fortes et globales à des problèmes quasi-linéaires abstraits sur un interval en temps bornéIn this thesis, we focus on a forward-backward parabolic problem and the free boundary arising from it. The equation models a phase change driven by a Stefan problem coupled with a time nonlocal hysteresis operator. Our study deals with some theoretical and numerical aspects raised by this type of time nonlocal equation, in particular regarding the free boundary.First, we establish an equivalence between entropy inequalities associated with the problem and a weak formulation of the hysteresis operator. This discovery motivates the construction of a finite-volume numerical scheme whose convergence to a solution is shown. The compactness of the sequence of approximate solutions is based on Hilpert's inequality. Numerical experiments in dimensions 1 and 2 support these results and illustrate the behaviour of the free boundary.Next we establish a general framework of viscosity solutions for front propagation problems which are nonlocl in space and time. They may include a coupling with a bulk evolution equation. A strict comparison theorem and an existence theorem derived from Perron's method are proved. The Stefan problem and some variations of it fall within this general framework.Finally, motivated by the study of parabolic equations in time-varying domains appearing in couplings of front propagation problems, we prove new results of maximal regularity in Lebesgue spaces. Of particular interest is the precise estimation of the regularity constant for nonautonomous and relatively continuous operators. These results lead to new growth conditions guaranteeing the existence of strong global solutions to abstract quasi-linear problems on a bounded time interval
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Freitas, Pedro S. C. de. "Some problems in nonlocal reaction-diffusion equations." Thesis, Heriot-Watt University, 1994. http://hdl.handle.net/10399/1401.
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Begg, Ronald Evan. "Cell-population growth modelling and nonlocal differential equations." Thesis, University of Canterbury. Mathematics and Statistics, 2007. http://hdl.handle.net/10092/1165.
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Aspects of the asymptotic behaviour of cell-growth models described by partial differential equations, and systems of partial differential equations, are considered. The models considered describe the evolution of the size-distribution or age-distribution of a population of cells undergoing growth and division. First, the relationship between the behaviour, with and without dispersion, of a single-compartment size-distribution model of cell-growth with fixed-size cell division (where cells can only divide at a single, critical size) is considered. In this model dispersion accounts for stochastic variation in the growth process of each individual cell. Existence, uniqueness and the asymptotic stability of the solution is shown for a size-distribution model of cell-growth with dispersion and fixed-size cell division. The conditions for the analysis to hold for a more general class of division behaviours are also discussed. A class of nonlocal ordinary differential equations is studied, which contains as a subset the nonlocal ordinary differential equations describing the steady size-distributions of a single-compartment model of cell-growth. Existence of solutions to these equations is found to be implied by the existence of 'upper' and 'lower' solutions, which also provide bounds for the solution. A multi-compartment, age-distribution model of cell-growth is studied, which describes the evolution of the age-distribution of cells in different phases of cell-growth. The stability of the model when periodic solutions exist is examined. Sufficient conditions are given for the existence of stable steady age-distributions, as well as for stable periodic solutions. Finally, a multi-compartment age-size distribution model of cell-growth is studied, which describes the evolution of the age-size distribution of cells in different phases of cell-growth. Sufficient conditions are given for the existence of steady age-size distributions. An outline of the analysis required to prove stability of the steady age-size distributions of the model is also given. The analysis is based on ideas introduced in the previous chapters.
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Wang, Ying. "Contributions to local and nonlocal elliptic differential equations." Tesis, Universidad de Chile, 2015. http://repositorio.uchile.cl/handle/2250/134657.
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Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaEsta tesis doctoral está dividida en cuatro partes. La primera parte está dedicada al estudio de la simetría radial y las propiedades de monotonicidad de soluciones positivas de ecuaciones elípticas fraccionarias en la bola unitaria o en todo el espacio, usando el método de planos móviles. En la segunda parte, se consideran propiedades de decaimiento y simetría de las soluciones positivas para ecuaciones integro-diferenciales en todo el espacio. Estudiamos el decaimiento, construyendo super y subsoluciones apropiadas y usamos el método de los planos móviles para probar las propiedades de simetría. La tercera parte es investigar la existencia y unicidad de soluciones débiles de la ecuación del calor fraccionaria, involucrando medidas de Radon. Más aún, analizamos el comportamiento asintótico de la solución débil cuando la medida de Radon es la masa de Dirac. En la cuarta parte, estudiamos la existencia de soluciones a problemas elípticos no lineales que provienen del modelamiento de dispositivos de sistemas micro-electromecánicos en el caso en que la membrana elástica entra en contacto con la placa inferior en la frontera. Mostramos, en este caso, como el decaimiento de la membrana afecta la existencia de soluciones y la tensión pull-in.
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Felsinger, Matthieu [Verfasser]. "Parabolic equations associated with symmetric nonlocal operators / Matthieu Felsinger." Bielefeld : Universitätsbibliothek Bielefeld, 2013. http://d-nb.info/1042557322/34.
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Wu, Lijiang. "Nonlocal Interaction Equations in Heterogeneous and Non-Convex Environments." Research Showcase @ CMU, 2015. http://repository.cmu.edu/dissertations/474.
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Hollender, Julian. "Lévy-Type Processes under Uncertainty and Related Nonlocal Equations." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-211795.
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The theoretical study of nonlinear expectations is the focus of attention for applications in a variety of different fields — often with the objective to model systems under incomplete information. Especially in mathematical finance, advances in the theory of sublinear expectations (also referred to as coherent risk measures) lay the theoretical foundation for modern approaches to evaluations under the presence of Knightian uncertainty. In this book, we introduce and study a large class of jump-type processes for sublinear expectations, which can be interpreted as Lévy-type processes under uncertainty in their characteristics. Moreover, we establish an existence and uniqueness theory for related nonlinear, nonlocal Hamilton-Jacobi-Bellman equations with non-dominated jump terms.
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Topp, Paredes Erwin. "Some results for nonlocal elliptic and parabolic nonlinear equations." Tesis, Universidad de Chile, 2014. http://www.repositorio.uchile.cl/handle/2250/129978.
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Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática\quad Esta tesis est\'a dedicada al estudio de propiedades cualitativas de ecuaciones el\'ipticas degeneradas donde la difusi\'on es puramente no local, y se lleva a cabo en el contexto de la teor\'ia de soluciones viscosas. La primera parte de la tesis trata el estudio de propiedades de compacidad de una familia de \textsl{operadores no locales de orden cero}, es decir, operadores el\'ipticos no locales definidos a trav\'es de una medida finita. Consideramos un familia uni-param\'etrica de operadores de orden cero de la forma \begin \mathcal_\epsilon(u, x) = \int_ [u(x + z) - u(x)]K_\epsilon(z)dz, \end donde, para cada $\epsilon \in (0,1)$, $K_\epsilon \in L^1(\mathbb^N)$ es una funci\'on radialmente sim\'etrica y positiva. Configuramos nuestro problema de manera que $\mathcal_\epsilon$ aproxime el Laplaciano fraccionario cuando $\epsilon \to 0^+$, lo que implica que la norma $L^1$ de $K_\epsilon$ es no acotada a medida que $\epsilon \to 0^+$. Como primer resultado de esta parte obtenemos un m\'odulo de continuidad en espacio-tiempo para la familia de soluciones acotadas de la ecuaci\'on del calor no local en el plano asociada a $\mathcal_\epsilon$ que es independiente de $\epsilon \in (0,1)$. El segundo resultado de esta parte considera un problema de Dirichlet en un dominio acotado $\Omega \subset \mathbb^N$ asociado a $\mathcal_\epsilon$, y concluimos la compacidad de la familia de soluciones acotadas $\_\epsilon$ para estos problemas de Dirichlet encontrando un m\'odulo de continuidad com\'un en $\bar$ para $\_\epsilon$, que es independiente de $\epsilon$. \medskip La segunda parte de la tesis est\'a relacionada con la existencia y unicidad, regularidad y comportamiento a grandes tiempos para ecuaciones no locales con t\'erminos de gradiente dominantes. Comenzamos con la existencia y unicidad de una ecuaci\'on de Hamilton-Jacobi de la forma \begin{equation*} \begin{array}{rll} \lambda u - \mathcal{I}(u) + H(x, Du) & = 0 \quad & \mbox{en} \ \Omega \\ u & = \varphi \quad & \mbox{en} \ \Omega^c, \end{array} \end{equation*} donde el Hamiltoniano $H$ tiene una \textsl{forma de Bellman}. Estructuramos el problema de manera que el operador no local $\mathcal{I}$ es de orden menor que $1$ y por lo tanto puede aparecer una p\'erdida de la condici\'on de borde. En la segunda secci\'on de esta parte, consideramos $H$ coercivo con un crecimiento en el gradiente m\'as fuerte que el orden de la difusi\'on del operador no local. El resultado principal en este caso es la continuidad H\"older para \textsl{subsoluciones} para este problema. Estabilidad de las estimaciones de regularidad cuando $\lambda \to 0$ permiten concluir el comportamiento asint\'otico erg\'odico cuando $t \to \infty$ para el problema parab\'olico asociado en el toro. En esta tarea, principios del m\'aximo fuertes son de importancia mayor en el an\'alisis asint\'otico. Finalmente, adaptamos los resultados obtenidos en las primeras dos secciones de esta parte de la tesis para obtener el comportamiento a grandes tiempos para el problema de Cauchy-Dirichlet asociado a $H$ en las formas Bellman y coercivo. En este caso, la influencia del dato exterior en la ecuaci\'on a trav\'es del t\'ermino no local hace que el problema parab\'olico aproxime al correspondiente problema estacionario cuando $t \to \infty$.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1260127.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1256047.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1263903.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1259927.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1256106.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1259887.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1264045.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1259907.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1263963.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1263925.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1260134.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1260141.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1264186.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1263985.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1265363.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1264023.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1266687.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1265263.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1265283.
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REMONATO, FILIPPO. "Analytical and numerical bifurcation methods for nonlocal wave equations." Doctoral thesis, Università degli studi di Pavia, 2018. http://hdl.handle.net/11571/1256126.
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Balagué, Guardia Daniel. "Qualitative properties of stationary states of some nonlocal interaction equations." Doctoral thesis, Universitat Autònoma de Barcelona, 2013. http://hdl.handle.net/10803/120508.
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En aquesta tesi estudiem l'estabilitat d'estats estacionaris d'alguns models d'interacció, de fragmentació i de comportament de col·lectius. Tots aquests models comparteixen la propietat de no localitat i l'existència d'un funcional de Lyapunov. En el cas de les equacions d'interacció i dels models de col·lectius que considerem, hi ha en comú el terme no local ∇𝑊��∗𝜌��, on 𝑊�� és potencial d'interacció, i 𝜌�� la densitat de partícules en espai. El cas de les equacions de fragmentació és una mica diferent: són equacions integro-diferencials amb un terme no local donat per l'operador de fragmentació, una integral d'un nucli contra la densitat de partícules.
Comencem amb una introducció a l'equació d'agregació amb potencial d'interacció repulsor-atractor i radial. Deduïm alguns resultats d'existència i convergència cap a estats estacionaris en forma de capes esfèriques. Busquem mínims de l'energia del funcional de Lyapunov per tal de trobar estats estacionaris per l'equació, i estudiem la in/estabilitat d'aquests estats estacionaris en particular. Propietats de confinament sobre les equacions d'agregació són estudiades en el Capítol 3. Demostrem que les solucions són de suport compacte i estan ficades en una bola grossa fixada per a tot temps. Continuem la recerca en les equacions d'agregació en el Capítol 4, on s'estudia la dimensió dels mínims locals pel funcional de l'energia d'interacció.
Un altre problema que estudiem és el comportament asimptòtic de les equacions de creixement-fragmentació. Al Capítol 5 donarem estimacions sobre els perfils asimptòtics i provarem una desigualtat de forat espectral. Aquests models no són un flux gradient respecte del funcional de l'energia. Tanmateix, es pot trobar un funcional de Lyapunov que utilitzarem per a demostrar convergència exponencialment ràpida de les solucions cap als perfils asimptòtics demostrant una desigualtat d'entropia - dissipació d'entropia. Aquesta tècnica ens dóna estabilitat dels estats estacionaris demostrant convergència cap a mínims locals i, a més, ens permet estimar la ràtio de la convergència cap a l'equilibri.
Acabem aquesta tesi amb els resultats del Capítol 6, on estudiem dos models de segon ordre de partícules pel comportament de col·lectius. Ens referim a aquests sistemes com a models basats en individus (IBMs), que és el llenguatge comú que s'utilitza per aquest tipus de models. Demostrem l'estabilitat de dues solucions particulars: la manada en forma d'anell i la rotació en forma d'anell. Relacionarem l'estabilitat d'aquestes solucions per aquests dos models de segon ordre amb l'estabilitat dels anells del model de primer ordre, la versió discreta de l'equació d'agregació del Capítol 2.In this dissertation, we study the stability of stationary states for some interaction equations and for fragmentation and swarming models. All these models share the common property of nonlocality and the existence of a Lyapunov functional. In the case of the interaction equations and the models for swarming that we consider, they have in common the nonlocal interaction term ∇𝑊�∗𝜌� where 𝑊� is the interaction potential, and 𝜌� the density of particles in space. The case of the fragmentation equations is a bit different: they are integro-differential equations, with the nonlocal term given by the fragmentation operator, an integral of a kernel against the density of particles. We start with an introduction to aggregation equations, with repulsive-attractive radial interaction potential. We derive some existence results and convergence to spherical shell stationary states. We look for local minimizers of the interaction Lyapunov functional in order to find stable stationary states of the equation. We study radial ins/stability of these particular stationary states. For these aggregation models we will make use of the gradient flow structure that they have. Confinement properties of solutions of aggregation equations under certain conditions on the interaction potential are studied in Chapter 3. We show that solutions remain compactly supported in a large fixed ball for all times. We continue our research in aggregation equations in Chapter 4, where we characterize the dimensionality of local minimizers of the interaction energy. Another problem that we study is the asymptotic behavior of growth-fragmentation models. In Chapter 5, we give estimates on asymptotic profiles and a spectral gap inequality for growth-fragmentation equations. These models are not a gradient flow of a particular energy functional. However, they have a Lyapunov functional that we use to prove exponentially fast convergence of solutions to the asymptotic profiles by showing an entropy - entropy dissipation inequality. This technique gives us stability of the stationary states proving convergence to the local minimizers and it allows for estimates on the rate of convergence to equilibrium. We finish this thesis with the results in Chapter 6, where we study two second order particle systems for swarming. We refer to these systems as individual based models (IBMs), which is the common language used in swarming. We prove the stability of two particular solutions: flock rings and mill rings. We relate the stability of these ring solutions of the second order models with the stability of the rings of a first order model, the discrete version of the aggregation equation of Chapter 2.
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Rang, Marcus [Verfasser]. "Regularity results for nonlocal fully nonlinear elliptic equations / Marcus Rang." Bielefeld : Universitätsbibliothek Bielefeld, 2013. http://d-nb.info/103805026X/34.
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Elghandouri, Mohammed. "Approximate Controllability for some Nonlocal Integrodifferential Equations in Banach Spaces." Electronic Thesis or Diss., Sorbonne université, 2024. http://www.theses.fr/2024SORUS189.
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La théorie du contrôle, domaine interdisciplinaire, étudie le comportement des systèmes dynamiques dans le but de réguler leur fonctionnement. La théorie du contrôle mathématique, un sous-domaine spécialisé, se concentre sur l'utilisation de méthodes mathématiques pour analyser ces comportements et concevoir des contrôleurs. Cela implique l'application d'équations différentielles, d'algèbre linéaire, d'optimisation et d'outils mathématiques variés pour modéliser, réguler et comprendre le comportement des systèmes dynamiques. Ces systèmes trouvent de vastes applications dans des domaines tels que la robotique, l'automatisation, l'aérospatiale, le génie électrique, les systèmes mécaniques, la biologie et les sciences sociales. Leur description nécessite souvent des modèles complexes tels que des équations aux dérivées partielles, des équations différentielles fonctionnelles et d'autres modèles de dimension infinie, rendant leur analyse essentielle, mais complexe pour la recherche. L'application de la théorie du contrôle pour analyser et réguler le comportement de ces systèmes a récemment attiré une attention significative. Cette thèse se concentre sur l'étude de la contrôlabilité approchée de certains systèmes dynamiques de dimension infinie décrits par des équations intégrodifférentielles. Structurée en trois chapitres, la thèse examine spécifiquement la contrôlabilité approchée des équations d'évolution intégrodifférentielles avec des conditions non locales. Le premier chapitre présente des outils fondamentaux, notamment la théorie des opérateurs résolvants, les applications multi-valuées, l'application de dualité, la théorie du contrôle mathématique et d'autres concepts essentiels à l'établissement des résultats. Le deuxième chapitre se concentre précisément sur la contrôlabilité approchée des équations d'évolution intégrodifférentielles semi-linéaires avec des conditions non locales de la forme w(0)=w0+g(w). En supposant que la partie linéaire soit nulle et approximativement contrôlable, la théorie des opérateurs résolvants est utilisée pour présenter les résultats principaux. Le troisième chapitre se concentre sur l'investigation de l'existence de solutions faibles et de la contrôlabilité approchée des systèmes d'évolution intégrodifférentielles avec des conditions non locales prenant des valeurs multiples w(0) appartient w0+g(w). Des conditions suffisantes pour l'existence et la contrôlabilité approchée sont établies à l'aide de la théorie des opérateurs résolvants. Un critère général de contrôlabilité de Kalman est introduit pour étudier la contrôlabilité approchée dans les cas linéaires. Des exemples illustratifs sont fournis tout au long de ces chapitres pour appuyer nos résultatsControl theory is an interdisciplinary field that addresses the behavior of dynamical systems with the primary goal of managing their output. A specialized subset of this is mathematical control theory, which focuses on utilizing mathematical methods to analyze system behavior and design controllers. This involves applying differential equations, linear algebra, optimization, and various mathematical tools to comprehend, model, and regulate system behavior. These systems have extensive applications across robotics, automation, aerospace, electrical engineering, mechanical systems, robotics, biological and social systems, among others. Described by complex models such as partial differential equations, functional differential equations, and other infinite-dimensional models, these systems pose intricate challenges, rendering the analysis of their behavior a pivotal and intricate area of research. In recent years, the application of control theory to analyze and regulate the behavior of these systems has attracted significant attention. This thesis aims to investigate the approximate controllability of certain infinite-dimensional dynamical systems described by integrodifferential equations. The thesis is structured into three chapters, each addressing the problem of achieving approximate controllability in integrodifferential evolution equations equipped with nonlocal conditions. The first chapter introduces fundamental tools critical to establishing our main findings, including the theory of resolvent operators, multi-valued maps, duality mapping, mathematical control theory, and other essential concepts. Chapter 2 specifically focuses on the approximate controllability of semilinear integrodifferential evolution equations with nonlocal conditions of the form w(0)=w0+g(w). Here, assuming the linear part is precisely null and approximately controllable, we employ resolvent operator theory to present our main results. Chapter 3 centers on investigating the existence of mild solutions and the approximate controllability of integrodifferential evolution systems with multi-valued nonlocal conditions (w(0) belongs w0+g(w)). By using resolvent operator theory, we establish sufficient conditions for both existence and controllability. Introducing a general Kalman controllability criterion, we examine approximate controllability in linear cases and subsequently demonstrate it in nonlinear cases. Throughout these chapters, we provide illustrative examples to support our main findings
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Garbin, F., F. Garbin, A. Levano, and R. Arciniega. "Bending Analysis of Nonlocal Functionally Graded Beams." Institute of Physics Publishing, 2020. http://hdl.handle.net/10757/651836.
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In this paper, we study the nonlocal linear bending behavior of functionally graded beams subjected to distributed loads. A finite element formulation for an improved first-order shear deformation theory for beams with five independent variables is proposed. The formulation takes into consideration 3D constitutive equations. Eringen's nonlocal differential model is used to rewrite the nonlocal stress resultants in terms of displacements. The finite element formulation is derived by means of the principle of virtual work. High-order nodal-spectral interpolation functions were utilized to approximate the field variables, which minimizes the locking problem. Numerical results and comparisons of the present formulation with those found in the literature for typical benchmark problems involving nonlocal beams are found to be satisfactory and show the validity of the developed finite element model.
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Endal, Jørgen. "Nonlinear fractional convection-diffusion equations, with nonlocal and nonlinear fractional diffusion." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2013. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-22955.
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We study nonlinear fractional convection-diffusion equations with nonlocal and nonlinear fractional diffusion. By the idea of Kru\v{z}kov (1970), entropy sub- and supersolutions are defined in order to prove well-posedness under the assumption that the solutions are elements in $L^{\infty}(\mathbb{R}^d\times (0,T))\cap C([0,T];L_\text{loc}^1(\mathbb{R}^d))$. Based on the work of Alibaud (2007) and Cifani and Jakobsen (2011), a local contraction is obtained for this type of equations for a certain class of L\'evy measures. In the end, this leads to an existence proof for initial data in $L^{\infty}(\mathbb{R}^d)$
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Serra, Montolí Joaquim. "Elliptic and parabolic PDEs : regularity for nonlocal diffusion equations and two isoperimetric problems." Doctoral thesis, Universitat Politècnica de Catalunya, 2014. http://hdl.handle.net/10803/279290.
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The thesis is divided into two parts.
The first part is mainly concerned with regularity issues for integro-differential (or nonlocal) elliptic and parabolic equations.
In the same way that densities of particles with Brownian motion solve second order elliptic or parabolic equations, densities of particles with Lévy diffusion satisfy these more general nonlocal equations. In this context, fully nonlinear nonlocal equations arise in Stochastic control problems or differential games.
The typical example of elliptic nonlocal operator is the fractional Laplacian, which is the only translation, rotation and scaling invariant nonlocal elliptic operator. There many classical regularity results for the fractional Laplacian ---whose ``inverse'' is the Riesz potential. For instance, the explicit Poisson kernel for a ball is an ``old'' result, as well as the linear solvability theory in L^p spaces.
However, very little was known on boundary regularity for these problems. A main topic of this thesis is the study of this boundary regularity, which is qualitatively very different from that for second order equations. We stablish a new boundary regularity theory for fully nonlinear (and linear) elliptic integro-differential equations. Our proofs require a combination of original techniques and appropriate versions of classical ones for second order equations (such as Krylov's method).
We also obtain new interior regularity results for fully nonlinear parabolic nonlocal equation with rough kernels. To do it, we develop a blow up and compactness method for viscosity solutions to fully nonlinear equations that allows us to prove regularity from Liouville type theorems.This method is a main contribution of the thesis.
The new boundary regularity results mentioned above are crucially used in the proof of another main result of the thesis: the Pohozaev identity for the fractional Laplacian. This identity is has a flavor of integration by parts formula for the fractional Laplacian, with the important novely there appears a local boundary term (this was unusual with nolocal equations).
In the second part of the thesis we give two instances of interaction between isoperimetry and Partial Differential Equations. In the first one we use the Alexandrov-Bakelman-Pucci method for elliptic PDE to obtain new sharp isoperimetric inequalities in cones with densities by generalizing a proof of the classical isoperimetric inequality due to Cabré. Our new results contain as particular cases the classical Wulff inequality and the isoperimetric inequality in cones of Lions and Pacella.
In the second instance we use the isoperimetric inequality and the classical Pohozaev identity to establish a radial symmetry result for second order reaction-diffusion equations. The novelty here is to include discontinuous nonlinearities. For this, we extend a two-dimensional argument of P.-L. Lions from 1981 to obtain now results in higher dimensionsLa tesi està dividida en dues parts. La primera part es centra principalment en questions de regularitat per equacions integro - iferencials (o no locals) el·líptiques i parbòliques. De la mateixa manera que les densitats de partícules amb un moviment Brownià resolen equacions el·líptiques o parbòliques de segon ordre, les densitats de partícules amb una difusió de tipus Lévy resolen aquestes equacions no locals més generals. En aquest context, les equacions completament no lineals sorgeixen de problemes de control estocàstic o "differential games''. L'exemple típic d'operador el·liptic no local és el laplacià fraccionari, el qual és l'únic d'aquests operadors que és invariant per translacions, rotacions, i reescalament. Hi ha molts resultats clàssics de regularitat per el laplacià fraccionari --- "l'invers'' del qual és el potencial de Riesz. Per exemple, el nucli de Poisson (explícit) per la bola és un resultat "vell'', així com la teoria de resolubilitat en espais L^p. No obstant això, se sabia ben poc sobre la regularitat a la vora per a aquests problemes. Un tema principal d'aquesta tesi és l'estudi d'aquesta regularitat a la vora, que és qualitativament molt diferent de la de les equacions de segon ordre . A la tesi s'estableix una nova teoria regularitat a la vora per completament no lineals ( i lineals ) equacions integro - diferencials el·líptiques . Les nostres demostracions requeixen una combinació de tècniques originals i versions apropiades de les clàssiques equacions de segon ordre ( com ara el mètode de Krylov ). També obtenim nous resultats de regularitat interior per equacions parabòliques no locals completament no lineals i amb "rough kernels''. A tal efecte, desenvolupem un mètode de blow-up i compacitat per a equacions completament no lineals que en permet provar regularitat a partir de teoremes de tipus Liouville. Aquest mètode és una contribució principal de la tesi. Els nous resultats de regularitat a la vora esmentats anteriorment són essencials en la prova d'un altre resultat principal de la tesi: la identitat Pohozaev per al Laplacià fraccionari. Aquesta identitat recorda a una fórmula d'integració per parts, però amb el Laplacià fraccionari. La novetat important és que apareix un terme de vora locals (això era inusual amb equacions no locals) . A la segona part de la tesi que donem dos exemples d'interacció entre isoperimetria i Equacions en Derivades Parcials. En el primer, s'utilitza el mètode d'Alexandrov- Bakelman-Pucci per a EDP el·líptiques a fi d'obtenir noves desigualtats isoperimètriques en cons convexos amb densitats, generalitzant una prova de la desigualtat isoperimètric clàssica de X. Cabré. Els nostres nous resultats contenen com a casos particularsla desigualtat clàssica de Wulff i la desigualtat isoperimètrica en cons de Lions-Pacella. En el segon exemple s'utilitza la desigualtat isoperimètrica i la identitat Pohozaev clàssica per establir un resultat de simetria radial per equacions de reacció-difusió de segon ordre. La novetat en aquest cas és que s'inclouen no-linealitats discontínues. Per a provar aquest resultat, estenem un argument en dues dimensions de P.-L. Lions de 1981 i podem obtenir ara resultass en dimensions superiors.
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Fu, Xiaoming. "Reaction-diffusion Equations with Nonlinear and Nonlocal Advection Applied to Cell Co-culture." Thesis, Bordeaux, 2019. http://www.theses.fr/2019BORD0216/document.
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Cette thèse est consacrée à l’étude d’une classe d’équations de réaction-diffusion avec advection non-locale. La motivation vient du mouvement cellulaire avec le phénomène de ségrégation observé dans des expérimentations de co-culture cellulaire. La première partie de la thèse développe principalement le cadre théorique de notre modèle, à savoir le caractère bien posé du problème et le comportement asymptotique des solutions dans les cas d'une ou plusieurs espèces.Dans le Chapitre 1, nous montrons qu'une équation scalaire avec un noyau non-local ayant la forme d'une fonction étagée, peut induire des bifurcations de Turing et de Turing-Hopf avec le nombre d’ondes dominant aussi grand que souhaité. Nous montrons que les propriétés de bifurcation de l'état stable homogène sont intimement liées aux coefficients de Fourier du noyau non-local.Dans le Chapitre 2, nous étudions un modèle d'advection non-local à deux espèces avec inhibition de contact lorsque la viscosité est égale à zéro. En employant la notion de solution intégrée le long des caractéristiques, nous pouvons rigoureusement démontrer le caractère bien posé du problème ainsi que la propriété de ségrégation d'un tel système. Par ailleurs, dans le cadre de la théorie des mesures de Young, nous étudions le comportement asymptotique des solutions. D'un point de vue numérique, nous constatons que sous l'effet de la ségrégation, le modèle d'advection non-locale admet un principe d'exclusion.Dans le dernier Chapitre de la thèse, nous nous intéressons à l'application de nos modèles aux expérimentations de co-culture cellulaire. Pour cela, nous choisissons un modèle hyperbolique de Keller-Segel sur un domaine borné. En utilisant les données expérimentales, nous simulons un processus de croissance cellulaire durant 6 jours dans une boîte de pétri circulaire et nous discutons de l’impact de la propriété de ségrégation et des distributions initiales sur les proportions de la population finaleThis thesis is devoted to the study for a class of reaction-diffusion equations with nonlocal advection. The motivation comes from the cell movement with segregation phenomenon observed in cell co-culture experiments. The first part of the thesis mainly develops the theoretical framework of our model, namely the well-posedness and asymptotic behavior of solutions in both single-species and multi-species cases.In Chapter 1, we show a single scalar equation with a step function kernel may display Turing and Turing-Hopf bifurcations with the dominant wavenumber as large as we want. We find the bifurcation properties of the homogeneous steady state is closed related to the Fourier coefficients of the nonlocal kernel.In Chapter 2, we study a two-species nonlocal advection model with contact inhibition when the viscosity equals zero. By employing the notion of the solution integrated along the characteristics, we rigorously prove the well-posedness and segregation property of such a hyperbolic nonlocal advection system. Besides, under the framework of Young measure theory, we investigate the asymptotic behavior of solutions. From a numerical perspective, we find that under the effect of segregation, the nonlocal advection model admits a competitive exclusion principle.In the last Chapter, we are interested in applying our models to a cell co-culturing experiment. To that aim, we choose a hyperbolic Keller-Segel model on a bounded domain. By utilizing the experimental data, we simulate a 6-day process of cell growth in a circular petri dish and discuss the impact of both the segregation property and initial distributions on the finial population proportions
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Nguyen, Thi Tuyen. "Comportement en temps long des solutions de quelques équations de Hamilton-Jacobi du premier et second ordre, locales et non-locales, dans des cas non-périodiques." Thesis, Rennes 1, 2016. http://www.theses.fr/2016REN1S089/document.
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La motivation principale de cette thèse est l'étude du comportement en temps grand des solutions non-bornées d'équations de Hamilton-Jacobi visqueuses dans RN en présence d'un terme d'Ornstein-Uhlenbeck. Nous considérons la même question dans le cas d'une équation de Hamilton-Jacobi du premier ordre. Dans le premier cas, qui constitue le cœur de la thèse, nous généralisons les résultats de Fujita, Ishii et Loreti (2006) dans plusieurs directions. La première est de considérer des opérateurs de diffusion plus généraux en remplaçant le Laplacien par une matrice de diffusion quelconque. Nous considérons ensuite des opérateurs non-locaux intégro-différentiels de type Laplacien fractionnaire. Le second type d'extension concerne le Hamiltonien qui peut dépendre de x et est seulement supposé sous-linéaire par rapport au gradientThe main aim of this thesis is to study large time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in RN in presence of an Ornstein-Uhlenbeck drift. We also consider the same issue for a first order Hamilton-Jacobi equation. In the first case, which is the core of the thesis, we generalize the results obtained by Fujita, Ishii and Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a non-local integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear
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Ma, Ding Henderson Johnny. "Uniqueness implies uniqueness and existence for nonlocal boundary value problems for fourth order differential equations." Waco, Tex. : Baylor University, 2005. http://hdl.handle.net/2104/3577.
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Gray, Michael Jeffery Henderson Johnny L. "Uniqueness implies uniqueness and existence for nonlocal boundary value problems for third order ordinary differential equations." Waco, Tex. : Baylor University, 2006. http://hdl.handle.net/2104/4185.
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Schley, David. "The effects of time delays and nonlocal nonlinearities in population models." Thesis, University of Surrey, 1999. http://epubs.surrey.ac.uk/844051/.
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Time delays and non-local nonlinearities are an important aspect of many biological and ecological systems. Not only are they highly relevant from an accurate modelling point of view, but their consequences can be extensive, including the introduction of new dynamics. In this thesis we consider simple models to derive generic results, concerned with novel aspects of delays and spatial effects. In addition we consider some specific models, in an attempt to make any existing non-local effects more relevant to the original problem. Since in many cases these are initial enquiries considering the possible dynamics of a system, we predominantly concern ourselves with the stability of, and solutions bifurcating from, (uniform) equilibria.
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Zhang, You-Kuan. "A quasilinear theory of time-dependent nonlocal dispersion in geologic media." Diss., The University of Arizona, 1990. http://hdl.handle.net/10150/185039.
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A theory is presented which accounts for a particular aspect of nonlinearity caused by the deviation of plume "particles" from their mean trajectory in three-dimensional, statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities. Quasilinear expressions for the time-dependent nonlocal dispersivity and spatial covariance tensors of ensemble mean concentration are derived, as a function of time, variance σᵧ² of log hydraulic conductivity, degree of anisotropy, and flow direction. One important difference between existing linear theories and the new quasilinear theory is that in the former transverse nonlocal dispersivities tend asymptotically to zero whereas in the latter they tend to nonzero Fickian asymptotes. Another important difference is that while all existing theories are nominally limited to situations where σᵧ² is less than 1, the quasilinear theory is expected to be less prone to error when this restriction is violated because it deals with the above nonlinearity without formally limiting σᵧ². The theory predicts a significant drop in dimensionless longitudinal dispersivity when σᵧ² is large as compared to the case where σᵧ² is small. As a consequence of this drop the real asymptotic longitudinal dispersivity, which varies in proportion to σᵧ² when σᵧ² is small, is predicted to vary as σᵧ when σᵧ² is large. The dimensionless transverse dispersivity also drops significantly at early dimensionless time when σᵧ² is large. At late time this dispersivity attains a maximum near σᵧ² = 1, varies asymptotically at a rate proportional to σᵧ² when σᵧ² is small, and appears inversely proportional to σᵧ when σᵧ² is large. The actual asymptotic transverse dispersivity varies in proportion to σᵧ⁴ when σᵧ² is small and appears proportional to σᵧ when σᵧ² is large. One of the most interesting findings is that when the mean seepage velocity vector μ is at an angle to the principal axes of statistical anisotropy, the orientation of longitudinal spread is generally offset from μ toward the direction of largest log hydraulic conductivity correlation scale. When local dispersion is active, a plume starts elongating parallel to μ. With time the long axis of the plume rotates toward the direction of largest correlation scale, then rotates back toward μ, and finally stabilizes asymptotically at a relatively small angle of deflection. Application of the theory to depth-averaged concentration data from the recent tracer experiment at Borden, Ontario, yields a consistent and improved fit without any need for parameter adjustment.
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Lorz, Alexander Stephan Richard. "Partial differential equations modelling biophysical phenomena." Thesis, University of Cambridge, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.609381.
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Pagliardini, Dayana. "Fractional minimal surfaces and Allen-Cahn equations." Doctoral thesis, Scuola Normale Superiore, 2018. http://hdl.handle.net/11384/85738.
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In recent years fractional operators have received considerable attention both in pure and applied mathematics. They appear in biological observations, finance, crystal dislocation, digital image reconstruction and minimal surfaces.
In this thesis we study nonlocal minimal surfaces which are boundaries of sets minimizing certain integral norms and can be interpreted as a non-infinitesimal version of classical minimal surfaces. In particular, we consider critical points, with or withouth constraints, of suitable functionals, or approximations through diffuse models as the
Allen-Cahn’s.
In the first part of the thesis we prove an existence and multiplicity result for critical points of the fractional analogue of the Allen-Cahn equation in bounded domains.
We bound the functional using a standard nonlocal tool: we split the domain in two regions and we analyze the three significative interactions. Then, the proof becomes an application of a classical Krasnoselskii’s genus result.
Then, we consider a fractional mesoscopic model of phase transition i.e. the fractional Allen-Cahn equation with the addition of a mesoscopic term changing the ‘pure phases’ ±1 in periodic functions. We investigate geometric properties of the interface of the associated minimal solutions. Then we construct minimal interfaces lying to a strip
of prescribed direction and universal width. We provide a geometric and variational technique adapted to deal with nonlocal interactions.
In the last part of the thesis, we study functionals involving the fractional perimeter.
In particular, first we study the localization of sets with constant nonlocal mean curvature and small prescribed volume in an open bounded domain, proving that these sets are ‘sufficiently close’ to critical points of a suitable potential. The proof is an application of the Lyupanov-Schmidt reduction to the fractional perimeter.
Finally, we consider the fractional perimeter in a half-space. We prove the existence of a minimal set with fixed volume and some of its properties as intersection with the hyperplane {xN = 0}, symmetry, to be a graph in the xN-direction and smoothness.
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Zhang, Chaoen. "Long time behaviour of kinetic equations." Thesis, Université Clermont Auvergne (2017-2020), 2019. http://www.theses.fr/2019CLFAC056.
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Cette thèse est consacrée au comportement à long terme de l'équation cinétique de Fokker-Planck et de l'équation de McKean-Vlasov. Le manuscrit est composé d'une introduction et de six chapitres. L'équation cinétique de Fokker-Planck est un exemple de base de la théorie de l'hypocoercivité de Villani qui affirme la décroissance exponentielle dans le temps en l'absence de coercivité. Dans son mémoire AMS, Villani a prouvé l'hypocoercivité de l'équation cinétique de Fokker-Planck en H^1(\mu), L^2(\mu) ou entropie. Cependant, une condition sur la bornitude de l'Hessien de l'hamiltonien a été imposée dans le cas entropique. Nous montrons au chapitre 2 comment nous pouvons affaiblir cette hypothèse par des multiplicateurs bien choisis à l'aide d'une inégalité de Sobolev logarithmique pondérée. Nous montrons que nos conditions sont satisfaites sous certaines conditions pratiques de fonction de Lyapunov.Dans le chapitre 4, nous appliquons les idées de Villani et certaines conditions de Lyapunov pour prouver l'hypocoercivité en H^1 pondéré dans le cas d'une interaction de champ moyen avec un taux de convergence exponentielle indépendant du nombre de particules. Pour cet objectif nous devons établir l'inégalité de Poincaré uniforme (sur le nombre de particules) et rendre une estimation connue de Villani qui était dimension-dépendante, dimension-indépendante.Au chapitre 6, nous étudions la contraction hypocoercive de la distance L^2-Wasserstein et nous retrouvons le taux optimal dans le cas du potentiel quadratique. La méthode est basée sur la dérivée en temps de la distance de Wasserstein. Au chapitre 7, le théorème d'hypoercivité de Villani dans l'espace H^1 pondéré est généralisé aux espaces H^k pondérés par une norm auxiliaire avec des termes mélangés bien choisis.L'équation de McKean-Vlasov est une équation diffusive non linéaire non locale. Il est bien connu qu'il a une structure de gradient-flot. Cependant, les résultats connus dépendent fortement des hypothèses de convexité. De telles hypothèses sont notamment assouplies dans les chapitres 3 et 5 où nous prouvons la convergence exponentielle vers l'équilibre respectivement en énergie libre et la distance L^1-Wasserstain, sous la condition de Dobrushin-Zegarlinski de l'absence de phase de transition. Notre approche est basée sur la théorie de la limite de champ moyen. Autrement dit, nous étudions le système d'un grand nombre de particules avec une interaction du type champ-moyen, puis passons à la limite par la propagation de chaosThis dissertation is devoted to the long time behaviour of the kinetic Fokker-Planck equation and of the McKean-Vlasov equation. The manuscript is composed of an introduction and six chapters.The kinetic Fokker-Planck equation is a basic example for Villani's hypocoercivity theory which asserts the exponential decay in large time in the absence of coercivity. In his memoir, Villani proved the hypocoercivity for the kinetic Fokker-Planck equation in either weighted H^1, weighted L^2 or entropy.However, a boundedness condition of the Hessian of the Hamiltonian was imposed in the entropic case. We show in Chapter 2 how we can get rid of this assumption by well-chosen multipliers with the help of a weighted logarithmic Sobolev inequality. Such a functional inequality can be obtained by some tractable Lyapunov condition.In Chapter 4, we apply Villani's ideas and some Lyapunov conditions to prove hypocoercivity in weighted H^1 in the case of mean-field interaction with a rate of exponential convergence independent of the number N of particles. For proving this we should prove the Poincaré inequality with a constant independent of N, and rends a dimension dependent boundeness estimate of Villani dimension-free by means of the stronger uniform log-Sobolev inequality and Lyapunov function method. In Chapter 6, we study the hypocoercive contraction in L^2-Wasserstein distance and we recover the optimal rate in the quadratic potential case. The method is based on the temporal derivative of the Wasserstein distance.In Chapter 7, Villani's hypoercivity theorem in weighted H^1 space is extended to weighted H^k spaces by choosing carefully some appropriate mixed terms in the definition of norm of H^k.The McKean-Vlasov equation is a nonlinear nonlocal diffusive equation. It is well-Known that it has a gradient flow structure. However, the known results strongly depend on convexity assumptions. Such assumptions are notably relaxed in Chapter 3 and Chapter 5 where we prove the exponential convergence to equilibrium respectively in free energy and the L^1-Wasserstain distance. Our approach is based on the mean field limit theory. That is, we study the associated system of a large numer of paricles with mean-field interaction and then pass to the limit by propagation of chaos
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Morian, Christina. "Partial differential equations on time scales /." free to MU campus, to others for purchase, 2000. http://wwwlib.umi.com/cr/mo/fullcit?p9974665.
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Cao, Xinlin. "Geometric structures of eigenfunctions with applications to inverse scattering theory, and nonlocal inverse problems." HKBU Institutional Repository, 2020. https://repository.hkbu.edu.hk/etd_oa/754.
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Inverse problems are problems where causes for desired or an observed effect are to be determined. They lie at the heart of scientific inquiry and technological development, including radar/sonar, medical imaging, geophysical exploration, invisibility cloaking and remote sensing, to name just a few. In this thesis, we focus on the theoretical study and applications of some intriguing inverse problems. Precisely speaking, we are concerned with two typical kinds of problems in the field of wave scattering and nonlocal inverse problem, respectively. The first topic is on the geometric structures of eigenfunctions and their applications in wave scattering theory, in which the conductive transmission eigenfunctions and Laplacian eigenfunctions are considered. For the study on the intrinsic geometric structures of the conductive transmission eigenfunctions, we first present the vanishing properties of the eigenfunctions at corners both in R2 and R3, based on microlocal analysis with the help of a particular type of planar complex geometrical optics (CGO) solution. This significantly extends the previous study on the interior transmission eigenfunctions. Then, as a practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter. For the study on the geometric structures of Laplacian eigenfunctions, we separately discuss the two-dimensional case and the three-dimensional case. In R2, we introduce a new notion of generalized singular lines of Laplacian eigenfunctions, and carefully investigate these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We provide an accurate and comprehensive quantitative characterization of the relationship. In R3, we study the analytic behaviors of Laplacian eigenfunctions at places where nodal or generalized singular planes intersect, which is much more complicated. These theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal (polyhedral) setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Our second topic is concerning the fractional partial differential operators and some related nonlocal inverse problems. We present some prelimilary knowledge on fractional Sobolev Spaces and fractional partial differential operators first. Then we focus on the simultaneous recovery results of two interesting nonlocal inverse problems. One is simultaneously recovering potentials and the embedded obstacles for anisotropic fractional Schrödinger operators based on the strong uniqueness property and Runge approximation property. The other one is the nonlocal inverse problem associated with a fractional Helmholtz equation that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. The main method utilized here is the low-frequency asymptotics combining with the variational argument. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.
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Ried, Tobias [Verfasser], and D. [Akademischer Betreuer] Hundertmark. "On some nonlinear and nonlocal effective equations in kinetic theory and nonlinear optics / Tobias Ried ; Betreuer: D. Hundertmark." Karlsruhe : KIT-Bibliothek, 2017. http://d-nb.info/1147485097/34.
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Andersson, Ulf. "Time-Domain Methods for the Maxwell Equations." Doctoral thesis, Stockholm : Tekniska högsk, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3094.
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Xie, Xiaoliang. "Large time-stepping methods for higher order time-dependent evolution equations." [Ames, Iowa : Iowa State University], 2008.
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