Dissertations / Theses on the topic 'Nonlocal equation'
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Alshomrani, Ali. "Analysis and approximation of a nonlocal equation." Thesis, Heriot-Watt University, 2014. http://hdl.handle.net/10399/2769.
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Khan, K. B. "The nonlocal-nonlinear-Schroedinger-equation model of superfluid '4He." Thesis, University of Exeter, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.267224.
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Jachimavičienė, Justina. "Solution of a pseudoparabolic equation with nonlocal integral conditions by the finite difference method." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2013~D_20130220_160906-66807.
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The thesis analyzes the third-order one-dimensional pseudoparabolic equations with two types of nonlocal conditions. The stability of difference schemes for this problem was studied using the analysis of the spectrum structure of a difference operator with nonlocal conditions. The analysis of the increased accuracy difference schemes for third-order one-dimensional and two-dimensional pseudoparabolic equations with integral conditions has been made. The thesis considers a two-dimensional pseudoparabolic equation with nonlocal integral conditions in one coordinate direction. This problem was solved by a locally one-dimensional method. The stability of a difference scheme has been investigated based on the spectrum structure. The doctoral disertation investigates three-layer difference schemes for one-dimensional pseudoparabolic equations with various, including nonlocal, conditions. Also, the conditions for the stability of three-layer explicit difference schemes have been explored.Disertacijoje išnagrinėta trečiosios eilės vienmatė pseudoparabolinė lygtis su dviejų tipų nelokaliosiomis sąlygomis. Šiems uždaviniams spręsti sudarytos skirtuminės schemos, kurių stabilumas tiriamas, taikant skirtuminių operatorių su nelokaliosiomis sąlygomis spektro struktūrą. Trečiosios eilės vienmatėms ir dvimatėms pseudoparabolinėms lygtims su integralinėmis sąlygomis sudarytos ir išnagrinėtos padidinto tikslumo skirtuminės schemos. Išnagrinėta dvimatė pseudoparabolinė lygtis su nelokaliosiomis integralinėmis sąlygomis viena koordinačių kryptimi. Tokiam uždaviniui spręsti pritaikytas ir išnagrinėtas lokaliai vienmatis metodas, ištirtos šio metodo stabilumo sąlygos. Taip pat išnagrinėtos: trisluoksnės skirtuminės schemos vienmatei pseudoparabolinei lygčiai su įvairiomis, taip pat ir nelokaliosiomis, sąlygomis; trisluoksnių išreikštinių skirtuminių schemų stabilumo sąlygos.
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BRASSEUR, JULIEN. "ANALYSIS OF SOME NONLOCAL MODELS IN POPULATION DYNAMICS." Doctoral thesis, Università degli Studi di Milano, 2018. http://hdl.handle.net/2434/597755.
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This thesis is mainly devoted to the mathematical analysis of some nonlocal models arising in population dynamics. In general, the study of these models meets with numerous difficulties owing to the lack of compactness and of regularizing effects. In this respect, their analysis requires new tools, both theoretical and qualitative. We present several results in this direction.
In the first part, we develop a functional analytic toolbox which allows one to handle some quantities arising in the study of these models. In the first place, we extend the characterization of Sobolev spaces due to Bourgain, Brezis and Mironescu to low regularity function spaces of Besov type. This results in a new theoretical framework that is more adapted to the study of some nonlocal equations of Fisher-KPP type. In the second place, we study the regularity of the restrictions of these functions to hyperplanes. We prove that, for a large class of Besov spaces, a surprising loss of regularity occurs. Moreover, we obtain an optimal characterization of the regularity of these restrictions in terms of spaces of so-called “generalized smoothness”.
In the second part, we study qualitative properties of solutions to some nonlocal
reaction-diffusion equations set in (possibly) heterogeneous domains. In collaboration with J. Coville, F. Hamel and E. Valdinoci, we consider the case of a perforated domain which consists of the Euclidean space to which a compact set, called an “obstacle”, is removed. When the latter is convex (or close to being convex), we prove that the solutions are necessarily constant. In a joint work with J. Coville, we study in greater detail the influence of the geometry of the obstacle on the classification of the solutions. Using tools of the type of those developed in the first part of this thesis, we construct a family of counterexamples when the obstacle is no longer convex. Lastly, in a work in collaboration with S. Dipierro, we study qualitative properties of solutions to nonlinear elliptic systems in variational form. We establish various monotonicity results in a fairly general setting that covers both local and fractional operators.
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Roman, Svetlana. "Green's functions for boundary-value problems with nonlocal boundary conditions." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20111227_092148-01085.
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In the dissertation, second-order and higher-order differential and discrete equations with additional conditions which are described by linearly independent linear functionals are investigated. The solutions to these problems, formulae and the existence conditions of Green's functions are presented, if the general solution of a homogeneous equation is known. The relation between two Green's functions of two nonhomogeneous problems for the same equation but with different additional conditions is obtained. These results are applied to problems with nonlocal boundary conditions.
In the introduction the topicality of the problem is defined, the goals and tasks of the research are formulated, the scientific novelty of the dissertation, the methodology of research, the practical value and the significance of the results are presented.
m-order differential problem and its Green's function are investigated in the first chapter. The relation between two Green's functions and the existence condition of Green's function are obtained.
In the second chapter, the main definitions and results of the first chapter are formulated for the second-order differential equation with additional conditions. In the examples the application of the received results is analyzed for problems with nonlocal boundary conditions in detail.
In the third chapter, the second-order difference equation with two additional conditions is considered. The expression of Green's function and its existence... [to full text]Disertacijoje tiriami antros ir aukštesnės eilės diferencialinis ir diskretusis uždaviniai su įvairiomis, tame tarpe ir nelokaliosiomis, sąlygomis, kurios yra aprašytos tiesiškai nepriklausomais tiesiniais funkcionalais. Pateikiamos šių uždavinių Gryno funkcijų išraiškos ir jų egzistavimo sąlygos, jei žinoma homogeninės lygties fundamentalioji sistema. Gautas dviejų Gryno funkcijų sąryšis uždaviniams su ta pačia lygtimi, bet su papildomomis sąlygomis. Rezultatai pritaikomi uždaviniams su nelokaliosiomis kraštinėmis sąlygomis. Įvadiniame skyriuje aprašyta tiriamoji problema ir temos objektas, išanalizuotas temos aktualumas, išdėstyti darbo tikslai, uždaviniai, naudojama tyrimų metodika, mokslinis darbo naujumas ir gautų rezultatų reikšmė, pateikti ginamieji teiginiai ir darbo rezultatų aprobavimas. m-tosios eilės diferencialinis uždavinys ir jo Gryno funkcija nagrinėjami pirmajame skyriuje. Surastas uždavinio sprendinys, išreikštas per Gryno funkciją. Pateikta Gryno funkcijos egzistavimo sąlyga. Antrajame skyriuje pateikti pirmojo skyriaus pagrindiniai apibrėžimai ir rezultatai antros eilės diferencialinei lygčiai. Pavyzdžiuose išsamiai išanalizuotas gautų rezultatų pritaikymas uždaviniams su nelokaliosiomis kraštinėmis sąlygomis. Trečiajame skyriuje nagrinėjama antros eilės diskrečioji lygtis su dviem sąlygomis. Surastos diskrečiosios Gryno funkcijos išraiška ir jos egzistavimo sąlyga. Taip pat pateiktas dviejų Gryno funkcijų sąryšis, kuris leidžia surasti diskrečiosios... [toliau žr. visą tekstą]
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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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Cozzi, M. "QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS." Doctoral thesis, Università degli Studi di Milano, 2016. http://hdl.handle.net/2434/345873.
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La tesi è dedicata allo studio di varie proprietà qualitative possedute dalle soluzioni di equazioni ellittiche poste nello spazio euclideo. L'attenzione principale del lavoro è rivolta a soluzioni intere di equazioni anisotrope/eterogenee che mostrano qualche genere di proprietà di simmetria e, in particolare, che posseggono simmetria unidimensionale.
L'elaborato è diviso in due parti. La prima parte è riservata ad equazioni alle derivate parziali locali, mentre la seconda si concentra su di una classe meno usuale di equazioni non locali, determinate da operatori integrali.The thesis is concerned with the study of several qualitative properties shared by the solutions of elliptic equations set in the Euclidean space. The main focus of the work is on entire solutions of anisotropic/heterogeneous equations that show some kind of symmetric properties and, in particular, that possess one-dimensional symmetry. The dissertation is divided into two parts. The first part deals with local partial differential equations, while the second one addresses a class of less familiar nonlocal equations driven by integral operators.
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DE, LUCA ALESSANDRA. "On some nonlocal issues: unique continuation from the boundary and capillarity problems for anisotropic kernels." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/378950.
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Lo scopo della presente tesi è quello di discutere i risultati ottenuti durante i miei studi di dottorato, principalmente rivolti a problemi non locali.
Per prima cosa ci occupiamo di principi di continuazione unica forte ed espansioni asintotiche locali in determinati punti del bordo per soluzioni di due diverse classi di equazioni ellittiche.
In particolare, partiamo con lo studio di una classe di equazioni ellittiche frazionarie
in un dominio limitato sotto una condizione al contorno di Dirichlet omogenea esterna. Per fare ciò, sfruttiamo la procedura di estensione di Caffarelli-Silvestre, grazie alla quale il problema non locale può essere riformulato in modo equivalente come problema locale in una dimensione in più, generando un problema con condizioni miste. Dopodichè, utilizziamo un'idea classica di Garofalo e Lin per ottenere una condizione di raddoppio tramite una formula di monotonia per la funzione di Almgren.
Per superare le difficoltà legate alla perdita di regolarità in corrispondenza della transizione tra le regioni di Dirichlet e di Neumann, introduciamo una nuova tecnica basata su un argomento di approssimazione, che ci permette di derivare una cosiddetta identità di tipo Pohozaev necessaria per stimare la derivata della funzione di Almgren . Otteniamo così un risultato di continuazione unica forte nel contesto locale, che viene a sua volta
combinato con l’analisi di blow-up per dedurre espansioni asintotiche locali e, di conseguenza, una continuazione unica forte anche nel contesto non locale.
Inoltre forniamo anche un risultato di continuazione unica forte dal bordo di una fessura per le soluzioni di una classe specifica di equazioni ellittiche del secondo ordine in un dominio limitato aperto con una frattura, su cui è assegnata una condizione al contorno di Dirichlet omogenea. Questo problema locale è correlato a un caso particolare dello studio descritto sopra, in virtù di una forte connessione tra questo tipo di problemi e i problemi al contorno con condizioni miste.
Nella presente dissertazione, trattiamo anche una teoria della capillarità non locale. In particolare, consideriamo nuclei di interazione più generali che sono possibilmente anisotropici e non necessariamente invarianti rispetto allo stesso riscalamento. In particolare, la perdita di invarianza è modellata utilizzando due diversi esponenti frazionari per tenere conto della possibilità che il contenitore e l'ambiente presentino caratteristiche diverse rispetto alle interazioni delle particelle. Determiniamo inoltre una legge di Young non locale per l'angolo di contatto tra la gocciolina e la superficie del contenitore e discutiamo la solvibilità e l’unicità della soluzione dell'equazione corrispondente in termini di nuclei di interazione e del relativo coefficiente di adesione.The aim of the present thesis is to discuss the results obtained during my PhD studies, mainly devoted to nonlocal issues. We first deal with strong unique continuation principles and local asymptotic expansions at certain boundary points for solutions of two different classes of elliptic equations. We start the investigation by a class of fractional elliptic equations in a bounded domain under some outer homogeneous Dirichlet boundary condition. To do this, we exploit the Caffarelli-Silvestre extension procedure, which allows us to get an equivalent formulation of the nonlocal problem as a local problem in one dimension more, consisting in a mixed Dirichlet-Neumann boundary value problem. Then, we use a classical idea by Garofalo and Lin to obtain a doubling-type condition via a monotonicity formula for a suitable Almgren-type frequency function. To overcome the difficulties related to the lack of regularity at the Dirichlet-Neumann junction, we introduce a new technique based on an approximation argument, which leads us to derive a so-called Pohozaev-type identity needed to estimate the derivative of the Almgren function. Thus we gain a strong unique continuation result in the local context, which is in turn combined with blow-up arguments to deduce local asymptotics and, consequently, a strong unique continuation result in the nonlocal setting as well. We also provide a strong unique continuation result from the edge of a crack for the solutions to a specific class of second order elliptic equations in an open bounded domain with a fracture, on which a homogeneous Dirichlet boundary condition is prescribed, in the presence of potentials satisfying either a negligibility condition with respect to the inverse-square weight or some suitable integrability properties. This local problem is related to a particular case of the setting described above, by virtue of a strong connection between this type of problems and the mixed Dirichlet-Neumann boundary value problems. We also treat a capillarity theory of nonlocal type. In our setting, we consider more general interaction kernels that are possibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance is modeled via two different fractional exponents in order to take into account the possibility that the container and the environment present different features with respect to particle interactions. We determine a nonlocal Young's law for the contact angle between the droplet and the surface of the container and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coefficient.
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Šimkevičiūtė, Jolanta. "Parabolinės lygties su nelokaliąja daugiataške sąlyga sprendimas baigtinių skirtumų metodu." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20110615_110310-18441.
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Darbe nagrinėjamas parabolinių lygčių su nelokaliąja daugiataške sąlyga ir tikrinių reikšmių uždaviniai antrosios eilės paprastajam diferencialiniam operatoriui. Uždavinio specifika yra ta, kad vietoje vienos arba abiejų klasikinių kraštinių sąlygų duota nelokalioji sąlyga. Tokio tipo kraštiniai uždaviniai diferiancialinėms lygtims paskutiniaisiais metais gana intensyviai nagrinėjami matematinėje literatūroje. Darbe naudojamas M. Sapagovo ir A. Štikono 2005 straipsnio metodika tikrinių reikšmių savybėms tirti.The parabolic equation with nonlocal multi-point condition and the eigenvalue problem for differential operation with nonlocal multi-point condition is investigated in the work. Nonlocal condition is given instead one or both classical boundary conditions. These problems are investigated in the mathematical literature in recent years. The method of analysis to eigenvalue of the article [5] by M. Sapagovas and A. Štikonas on 2005 are used in the work.
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Jakubėlienė, Kristina. "Solution of a two-dimensional parabolic equation with an integral condition by the finite-difference method." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2013~D_20130521_163755-33966.
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The aim of the work is to analyze the finite difference method for solving two-dimensional parabolic equation with an integral boundary condition. The alternating direction method for solving the problem of this kind is analyzed. This method is applied the alternating direction method for solving two-dimensional parabolic equation with two nonlocal integral condition is analyzed. Solution of the problem is found by resolving an additional linear system of equations of lower order . Structure of the spectrum for difference operator with nonlocal condition is analyzed. In order to analyze stability of two-dimensional parabolic equation with one integral condition the structure of spectrum is analyzed. Influence of nonlocal condition for structure of the spectrum is determined. The finite difference method for elliptic problem is constructed.Darbo tikslas - išnagrinėti dvimatės parabolinio tipo lygties su nelokaliąja integraline sąlyga sprendimą baigtinių skirtumų metodu. Išnagrinėtas kintamųjų krypčių metodo algoritmas tokiam uždaviniui spręsti. Išnagrinėtas dvimatės parabolinės lygties su keliomis nelokaliosiomis integralinėmis kraštinėmis sąlygomis sprendimas kintamųjų krypčių metodu. Uždavinio sprendinys randamas papildomai išsprendžiant neaukštos eilės algebrinę tiesinių lygčių sistemą, kuri sudaroma panaudojant nelokaliąsias integralines sąlygas. Išanalizuota skirtuminio operatoriaus su nelokaliosiomis sąlygomis spektro struktūra. Spektro struktūra išanalizuota tuo tikslu, kad galima būtų išnagrinėti dvimačio parabolinio uždavinio su viena nelokaliąja integraline sąlyga sprendžiamo kintamųjų krypčių ar lokaliai vienmačiu metodu, stabilumą. Nustatyta nelokaliosios sąlygos įtaka spektro struktūrai. Sudarytas elipsinio uždavinio su papildoma nelokaliąja sąlyga sprendimo algoritmas.
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Abatangelo, N. "Large Solutions for Fractional Laplacian Operators." Doctoral thesis, Università degli Studi di Milano, 2015. http://hdl.handle.net/2434/320258.
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The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the boundary of the prescribed domain. We first remark the existence of a large class of harmonic functions with a boundary blow-up and we characterize them in terms of a new notion of degenerate boundary trace. Via some integration by parts formula, we then provide a weak theory of Stampacchia's sort to extend the linear theory to a setting including these functions: we study the classical questions of existence, uniqueness, continuous dependence on the data, regularity and asymptotic behaviour at the boundary. Afterwards we develop the theory of semilinear problems, by adapting and generalizing some sub- and supersolution methods. This allows us to build the fractional counterpart of large solutions in the elliptic PDE theory of nonlinear equations, giving sufficient conditions for the existence. The thesis is concluded with the definition and the study of a notion of nonlocal directional curvatures.
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MIRAGLIO, PIETRO. "ESTIMATES AND RIGIDITY FOR STABLE SOLUTIONS TO SOME NONLINEAR ELLIPTIC PROBLEMS." Doctoral thesis, Università degli Studi di Milano, 2020. http://hdl.handle.net/2434/704717.
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Questa tesi è incentrata sullo studio di equazioni differenziali alle derivate parziali di tipo ellittico. La prima parte della tesi riguarda la regolarità delle soluzioni stabili per un'equazione nonlineare con il p-Laplaciano, in un dominio limitato dello spazio Euclideo. La tecnica è basata sull'uso di disuguaglianze di tipo Hardy-Sobolev su ipersuperfici, del quale viene approfondito lo studio. Nella seconda parte viene preso in esame un problema nonlocale di tipo Dirichlet-Neumann. Studiamo la simmetria unidimensionale di alcune sottoclassi di soluzioni stabili, ottenendo risultati in dimensione n=2, 3. Inoltre, studiamo il comportamento asintotico dell'operatore associato a questo problema nonlocale, usando tecniche di Γ-convergenza.This thesis deals with the study of elliptic PDEs. The first part of the thesis is focused on the regularity of stable solutions to a nonlinear equation involving the p-Laplacian, in a bounded domain of the Euclidean space. The technique is based on Hardy-Sobolev inequalities in hypersurfaces involving the mean curvature, which are also investigated in the thesis. The second part concerns, instead, a nonlocal problem of Dirichlet-to-Neumann type. We study the one-dimensional symmetry of some subclasses of stable solutions, obtaining new results in dimensions n=2, 3. In addition, we carry out the study of the asymptotic behaviour of the operator associated with this nonlocal problem, using Γ-convergence techniques.
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Roman, Svetlana. "Gryno funkcijos uždaviniams su nelokaliosiomis kraštinėmis sąlygomis." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20111227_092259-85107.
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Disertacijoje tiriami antros ir aukštesnės eilės diferencialinis ir diskretusis uždaviniai su įvairiomis, tame tarpe ir nelokaliosiomis, sąlygomis, kurios yra aprašytos tiesiškai nepriklausomais tiesiniais funkcionalais. Pateikiamos šių uždavinių Gryno funkcijų išraiškos ir jų egzistavimo sąlygos, jei žinoma homogeninės lygties fundamentalioji sistema. Gautas dviejų Gryno funkcijų sąryšis uždaviniams su ta pačia lygtimi, bet su papildomomis sąlygomis. Rezultatai pritaikomi uždaviniams su nelokaliosiomis kraštinėmis sąlygomis.
Įvadiniame skyriuje aprašyta tiriamoji problema ir temos objektas, išanalizuotas temos aktualumas, išdėstyti darbo tikslai, uždaviniai, naudojama tyrimų metodika, mokslinis darbo naujumas ir gautų rezultatų reikšmė, pateikti ginamieji teiginiai ir darbo rezultatų aprobavimas.
m-tosios eilės diferencialinis uždavinys ir jo Gryno funkcija nagrinėjami pirmajame skyriuje. Surastas uždavinio sprendinys, išreikštas per Gryno funkciją. Pateikta Gryno funkcijos egzistavimo sąlyga.
Antrajame skyriuje pateikti pirmojo skyriaus pagrindiniai apibrėžimai ir rezultatai antros eilės diferencialinei lygčiai. Pavyzdžiuose išsamiai išanalizuotas gautų rezultatų pritaikymas uždaviniams su nelokaliosiomis kraštinėmis sąlygomis.
Trečiajame skyriuje nagrinėjama antros eilės diskrečioji lygtis su dviem sąlygomis. Surastos diskrečiosios Gryno funkcijos išraiška ir jos egzistavimo sąlyga. Taip pat pateiktas dviejų Gryno funkcijų sąryšis, kuris leidžia surasti diskrečiosios... [toliau žr. visą tekstą]In the dissertation, second-order and higher-order differential and discrete equations with additional conditions which are described by linearly independent linear functionals are investigated. The solutions to these problems, formulae and the existence conditions of Green's functions are presented, if the general solution of a homogeneous equation is known. The relation between two Green's functions of two nonhomogeneous problems for the same equation but with different additional conditions is obtained. These results are applied to problems with nonlocal boundary conditions. In the introduction the topicality of the problem is defined, the goals and tasks of the research are formulated, the scientific novelty of the dissertation, the methodology of research, the practical value and the significance of the results are presented. m-order differential problem and its Green's function are investigated in the first chapter. The relation between two Green's functions and the existence condition of Green's function are obtained. In the second chapter, the main definitions and results of the first chapter are formulated for the second-order differential equation with additional conditions. In the examples the application of the received results is analyzed for problems with nonlocal boundary conditions in detail. In the third chapter, the second-order difference equation with two additional conditions is considered. The expression of Green's function and its existence... [to full text]
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Zdanytė, Vaida. "Trisluoksnės skirtuminės schemos parabolinei lygčiai su integraline sąlyga spręsti." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2014. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2014~D_20140611_153535-55131.
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Magistriniame darbe tiriama trisluoksnė skirtuminė schema parabolinei lygčiai su integraline sąlyga. Aprašomi metodai skaitiniai diferencialinių kraštinių uţdavinių su nelokaliosiomis sąlygomis. Atlikto magistrinio darbo rezultatas papildo iki šiol kitų mokslininkų gautus rezultatus tiriant trisluoksnę skirtuminę schemą. Magistro darbą sudaro: įvadas, uţdavinio formulavimas, 4 pagrindinės dalys, uždavinio sprendimas bei išvados. Įvadiniame skyriuje aptariamas temos aktualumas ir darbo tikslas, nurodomi naudojamo tyrimo metodai. Antrajame skyriuje suformuluojamas diferencialinis ir skirtuminis uždavinys su nelokaliąja integraline sąlyga. Trečiajame skyriuje užrašoma trisluoksnė schema kanoniniu pavidalu. Ketvirtajame skyriuje suvedame trisluoksnę schemą į dvisluoksnę. Penktajame skyriuje pateikiamas neišreikštinių skirtuminių lygčių algoritmas. Šeštajame nagrinėjama išreikštinė trisluoksnė schema bei jos algoritmas. Septintajame skyriuje tiriame matricos spektro struktūrą. Aštuntajame sprendžiamas konkretus uždavinys. Pateikiamos viso darbo bendrosios išvados.In this master thesis there was investigated difference scheme for parabolic equation with integral condition. Numerical methods for solution differential boundary value problem nonlocal conditions methods investigated. Results of this completed work supplements by other scientists until now received results of investigation of three- layer difference scheme. Master thesis consists of introduction, problem formulation, four main chapter, numerical experiment and conclusions. Introductory chapter discusses relevance of the topic and the goal of this work, specifies methods that were used for this investigation. The second chapter formulates the differential task with nonlocal integral condition. In the third chapter is written a three- layer scheme in canonical form. In the fourth chapter the three-layer scheme reduce to the two-layers scheme. The fifth chapter presens the algorithm of realization of impicit scheme. The sixth chapter presents explicit three-layer scheme. The seventh chapter studies the structure of the matrix spectrum. There are presented all the general conclusions of the work.
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Šiaulytė, Austėja. "Parabolinės lygties su nelokaliąja integraline Robino sąlyga išreikštinė skirtuminė schema." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2013~D_20130617_182830-33054.
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Magistro darbe yra tiriama parabolinės lygties su nelokaliąja integraline Robino sąlyga skirtuminė schema. Skirtuminės schemos stabilumui nagrinėti naudojama skirtuminio operatoriaus su nelokaliąja sąlyga spektro struktūros tyrimo metodika bei Maple programa, skirta kompiuteriniams eksperimentams atlikti. Atlikto magistro darbo rezultatai papildo iki šiol kitų mokslininkų gautus rezultatus tiriant parabolinių lygčių su nelokaliosiomis sąlygomis išreikštinių skirtuminių schemų tyrimus. Magistro darbą sudaro: įvadas, šešios pagrindinės dalys bei išvados. Įvadiniame skyriuje aptariamas temos aktualumas ir darbo tikslas, nurodomi naudojami tyrimo metodai. Antrajame ir trečiajame skyriuose suformuluojama parabolinės lygties su nelokaliąja integraline Robino išreikštinė skirtuminė schema bei jos pakankamoji stabilumo sąlyga. Ketvirtajame, penktajame ir šeštajame skyriuose randamas išreikštinės schemos stabilumas įvairiais atvejais bei pateikiama gautų rezultatų analizė. Septintajame skyriuje atliktas skaitinis eksperimentas. Pateikiamos viso darbo bendrosios išvados.In the master work, explicit difference scheme for parabolic equation with nonlocal integral Robin condition, is considered. Stability condition of difference scheme is used to examine spectrum structure of differential operator with nonlocal condition and software of Maple, which perform of sacred to the computer experiment. My the master work extends and suplements the results of other scientists in analysis for explicit difference scheme for parabolic equation with nonlocal conditions. The master work consists of the introduction, six chapters and the conclusions. In the introduction the topicality of the problem and object of work are defined, also methods of analysis is presented. In the second and third chapters, explicit difference scheme for parabolic equation with nonlocal integral Robin condition is formulated and also the sufficient stability condition of the difference sheme. In the fourth, fifth and the sixth chapters the stability explicit difference scheme is considered and analysis the results is presented. In the seventh chapter the numerical experiment is used. The conlusions are presented.
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Garšvaitė, Skaistė. "Dvimatės elipsinės lygties su nelokaliąja sąlyga sprendimas baigtinių skirtumų metodu." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2008. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2008~D_20080619_122640-56101.
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Šiame darbe nagrinėjame elipsinės lygties stačiakampėje srityje su nelokaliąja sąlyga sprendimą baigtinių skirtumų metodu. Sprendžiame dvimates skirtumines lygčių sistemas, jas gavome pakeitę diferencialinę lygtį skirtumine. Trumpai apžvelgtas maksimumo principas ir sprendinio radimas iteraciniais metodais bei tikrinių reikšmių radimas dvimačiu atveju. Įvertinta skirtuminės lygčių sistemos paklaida, kuri gaunama sprendžiant elipsinę lygtį skirtuminiu metodu. Darbo pabaigoje išspręstas konkretus uždavinys.In this work we consider two dimensional elliptic equation on the rectangle with non local condition by finite difference method. We solve two dimensional equations instead one intricate differential equation. A short review of maximum principle and solution finding with iteration method, and the proper account finding with two dimensional case. Estimated differential equationerror, this making calculate elliptic equation difference method. Finally we solve particilar example with different steps.
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Khamitova, Raisa. "Symmetries and conservation laws." Doctoral thesis, Växjö : Växjö University Press, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:vxu:diva-2587.
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Tapdigoglu, Ramiz. "Inverse problems for fractional order differential equations." Thesis, La Rochelle, 2019. http://www.theses.fr/2019LAROS004/document.
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Dans cette thèse, nous nous intéressons à résoudre certains problèmes inverses pour des équations différentielles aux dérivées fractionnaires. Un problème inverse est généralement mal posé. Un problème mal posé est un problème qui ne répond pas à l’un des trois critères de Hadamard pour être bien posé, c’est-à-dire, soit l’existence, l’unicité ou une dépendance continue aux données n'est plus vraie, à savoir, des petits changements dans les données de mesure entraînent des changements indéfiniment importants dans la solution. La plupart des difficultés à résoudre des problèmes mal posés sont causées par l’instabilité de la solution. D’autre part, les équations différentielles fractionnaires deviennent un outil important dans la modélisation de nombreux problèmes de la vie réelle et il y a eu donc un intérêt croissant pour l’étude des problèmes inverses avec des équations différentielles fractionnaires. Le calcul fractionnaire est une branche des mathématiques qui fait référence à l’extension du concept de dérivation classique à la dérivation d’ordre non entier. Calculer une dérivée fractionnaire à un certain moment exige tous les processus précédents avec des propriétés de mémoire. C’est l’avantage principal du calcul fractionnaire d’expliquer les processus associés aux systèmes physiques complexes qui ont une mémoire à long terme et / ou des interactions spatiales à longue distance. De plus, les équations différentielles fractionnaires peuvent nous aider à réduire les erreurs découlant de paramètres négligés dans la modélisation des phénomènes physiquesIn this thesis, we are interested in solving some inverse problems for fractional differential equations. An inverse problem is usually ill-posed. The concept of an ill-posed problem is not new. While there is no universal formal definition for inverse problems, Hadamard [1923] defined a problem as being ill-posed if it violates the criteria of a well-posed problem, that is, either existence, uniqueness or continuous dependence on data is no longer true, i.e., arbitrarily small changes in the measurement data lead to indefinitely large changes in the solution. Most difficulties in solving ill-posed problems are caused by solution instability. Inverse problems come into various types, for example, inverse initial problems where initial data are unknown and inverse source problems where the source term is unknown. These unknown terms are to be determined using extra boundary data. Fractional differential equations, on the other hand, become an important tool in modeling many real-life problems and hence there has been growing interest in studying inverse problems of time fractional differential equations. The Non-Integer Order Calculus, traditionally known as Fractional Calculus is the branch of mathematics that tries to interpolate the classical derivatives and integrals and generalizes them for any orders, not necessarily integer order. The advantages of fractional derivatives are that they have a greater degree of flexibility in the model and provide an excellent instrument for the description of the reality. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time, i.e., calculating timefractional derivative at some time requires all the previous processes with memory and hereditary properties
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Jakubėlienė, Kristina. "Dvimatės parabolinės lygties su integraline sąlyga sprendimas baigtinių skirtumų metodu." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2013~D_20130521_163742-43540.
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Darbo tikslas - išnagrinėti dvimatės parabolinio tipo lygties su nelokaliąja integraline sąlyga sprendimą baigtinių skirtumų metodu. Išnagrinėtas kintamųjų krypčių metodo algoritmas tokiam uždaviniui spręsti. Išnagrinėtas dvimatės parabolinės lygties su keliomis nelokaliosiomis integralinėmis kraštinėmis sąlygomis sprendimas kintamųjų krypčių metodu. Uždavinio sprendinys randamas papildomai išsprendžiant neaukštos eilės algebrinę tiesinių lygčių sistemą, kuri sudaroma panaudojant nelokaliąsias integralines sąlygas. Išanalizuota skirtuminio operatoriaus su nelokaliosiomis sąlygomis spektro struktūra. Spektro struktūra išanalizuota tuo tikslu, kad galima būtų išnagrinėti dvimačio parabolinio uždavinio su viena nelokaliąja integraline sąlyga sprendžiamo kintamųjų krypčių ar lokaliai vienmačiu metodu, stabilumą. Nustatyta nelokaliosios sąlygos įtaka spektro struktūrai. Sudarytas elipsinio uždavinio su papildoma nelokaliąja sąlyga sprendimo algoritmas.The aim of the work is to analyze the finite difference method for solving two-dimensional parabolic equation with an integral boundary condition. The alternating direction method for solving the problem of this kind is analyzed. This method is applied the alternating direction method for solving two-dimensional parabolic equation with two nonlocal integral condition is analyzed. Solution of the problem is found by resolving an additional linear system of equations of lower order . Structure of the spectrum for difference operator with nonlocal condition is analyzed. In order to analyze stability of two-dimensional parabolic equation with one integral condition the structure of spectrum is analyzed. Influence of nonlocal condition for structure of the spectrum is determined. The finite difference method for elliptic problem is constructed.
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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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Novickij, Jurij. "Hiperbolinės lygties su nelokaliosiomis kraštinėmis sąlygomis skirtuminio sprendinio stabilumas." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2014. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2012~D_20140704_175149-44355.
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Darbo tikslas — ištirti baigtiniu skirtumu metodo antrosios eiles hiperbolinio tipo diferencialinei lygciai su nelokaliosiomis integralinemis kraštinemis salygomis stabiluma. Siekiant numatyto tikslo buvo sprendžiami šie uždaviniai: • išnagrinetas antrosios eiles hiperbolines lygties trisluoksnes skirtumines schemos suvedimas i dvisluoksne skirtumine schema; • išanalizuotas skirtuminio operatoriaus perejimo matricos spektras; • gauta pakankamoji skirtumines schemos stabilumo salyga, nusakoma nelokaliuju salygu parametrais; • atlikti skaitiniai eksperimentai, patvirtinantys teorines išvadas. Nurodyta stabilumo salyga yra esmine, sprendžiant hiperbolinio tipo uždavinius su pakankamai didelemis T reikšmemis. Skirtuminio operatoriaus perejimo matricos spektro tyrimo metodika gali buti pritaikyta placios klases diferencialiniu lygciu su nelokaliosiomis salygomis stabilumui tirti.On the stability of an explicit difference scheme for hyperbolic equation with integral conditions. The aim of the work is stability analysis of solution of finite difference method for hyperbolic equations. Trying to achieve formulated aim these tasks were solved: • a method of transformation of three-layered finite difference scheme into two-layered one was investigated; • a spectrum of transition matrix subject to the properties of second order differential operator Lambda was studied; • stability conditions of hyperbolic type equations with nonlocal conditions subject to boundary parameters were obtained; • numerical experiments, confirming theoretical derivations were made. Derived results could be used to solve one-dimensional tasks with hyperbolic equations in different sciences, to analyse spectrum structure of mathematical models and construct new numerical methods for solving hyperbolic PDEs.
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Druet, Pierre-Etienne. "Analysis of a coupled system of partial differential equations modeling the interaction between melt flow, global heat transfer and applied magnetic fields in crystal growth." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2009. http://dx.doi.org/10.18452/15893.
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Hauptthema der Dissertation ist die Analysis eines nichtlinearen, gekoppelten Systems partieller Differentialgleichungen (PDG), das in der Modellierung der Kristallzüchtung aus der Schmelze mit Magnetfeldern vorkommt. Die zu beschreibenden Phenomäne sind einerseits der im elektromagnetisch geheizten Schmelzofen erfolgende Wärmetransport (Wärmeleitung, -konvektion und -strahlung), und andererseits die Bewegung der Halbleiterschmelze unter dem Einfluss der thermischen Konvektion und der angewendeten elektromagnetischen Kräfte. Das Modell besteht aus den Navier-Stokeschen Gleichungen für eine inkompressible Newtonsche Flüssigkeit, aus der Wärmeleitungsgleichung und aus der elektrotechnischen Näherung des Maxwellschen Systems. Wir erörtern die schwache Formulierung dieses PDG Systems, und wir stellen ein Anfang-Randwertproblem auf, das die Komplexität der Anwendung widerspiegelt. Die Hauptfrage unserer Untersuchung ist die Wohlgestelltheit dieses Problems, sowohl im stationären als auch im zeitabhängigen Fall. Wir zeigen die Existenz schwacher Lösungen in geometrischen Situationen, in welchen unstetige Materialeigenschaften und nichtglatte Trennfläche auftreten dürfen, und für allgemeine Daten. In der Lösung zum zeitabhängigen Problem tritt ein Defektmaß auf, das ausser der Flüssigkeit im Rand der elektrisch leitenden Materialien konzentriert bleibt. Da eine globale Abschätzung der im Strahlungshohlraum ausgestrahlten Wärme auch fehlt, rührt ein Teil dieses Defektmaßes von der nichtlokalen Strahlung her. Die Eindeutigkeit der schwachen Lösung erhalten wir nur unter verstärkten Annahmen: die Kleinheit der gegebenen elektrischen Leistung im stationären Fall, und die Regularität der Lösung im zeitabhängigen Fall. Regularitätseigenschaften wie die Beschränktheit der Temperatur werden, wenn auch nur in vereinfachten Situationen, hergeleitet: glatte Materialtrennfläche und Temperaturunabhängige Koeffiziente im Fall einer stationären Analysis, und entkoppeltes, zeitharmonisches Maxwell für das transiente Problem.The present PhD thesis is devoted to the analysis of a coupled system of nonlinear partial differential equations (PDE), that arises in the modeling of crystal growth from the melt in magnetic fields. The phenomena described by the model are mainly the heat-transfer processes (by conduction, convection and radiation) taking place in a high-temperatures furnace heated electromagnetically, and the motion of a semiconducting melted material subject to buoyancy and applied electromagnetic forces. The model consists of the Navier-Stokes equations for a newtonian incompressible liquid, coupled to the heat equation and the low-frequency approximation of Maxwell''s equations. We propose a mathematical setting for this PDE system, we derive its weak formulation, and we formulate an (initial) boundary value problem that in the mean reflects the complexity of the real-life application. The well-posedness of this (initial) boundary value problem is the mainmatter of the investigation. We prove the existence of weak solutions allowing for general geometrical situations (discontinuous coefficients, nonsmooth material interfaces) and data, the most important requirement being only that the injected electrical power remains finite. For the time-dependent problem, a defect measure appears in the solution, which apart from the fluid remains concentrated in the boundary of the electrical conductors. In the absence of a global estimate on the radiation emitted in the cavity, a part of the defect measure is due to the nonlocal radiation effects. The uniqueness of the weak solution is obtained only under reinforced assumptions: smallness of the input power in the stationary case, and regularity of the solution in the time-dependent case. Regularity properties, such as the boundedness of temperature are also derived, but only in simplified settings: smooth interfaces and temperature-independent coefficients in the case of a stationary analysis, and, additionally for the transient problem, decoupled time-harmonic Maxwell.
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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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