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Dissertationen zum Thema „Diffusion equations“

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Veröffentlicht am 4. Juni 2021
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1

Ta, Thi nguyet nga. "Sub-gradient diffusion equations." Thesis, Limoges, 2015. http://www.theses.fr/2015LIMO0137/document.

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Ce mémoire de thèse est consacrée à l'étude des problèmes d'évolution où la dynamique est régi par l'opérateur de diffusion de sous-gradient. Nous nous intéressons à deux types de problèmes d'évolution. Le premier problème est régi par un opérateur local de type Leray-Lions avec un domaine borné. Dans ce problème, l'opérateur est maximal monotone et ne satisfait pas la condition standard de contrôle de la croissance polynomiale. Des exemples typiques apparaît dans l'étude de fluide non-Neutonian et aussi dans la description de la dynamique du flux de sous-gradient. Pour étudier le problème nous traitons l'équation dans le contexte de l'EDP non linéaire avec le flux singulier. Nous utilisons la théorie de gradient tangentiel pour caractériser l'équation d'état qui donne la relation entre le flux et le gradient de la solution. Dans le problème stationnaire, nous avons l'existence de la solution, nous avons également l'équivalence entre le problème minimisation initial, le problème dual et l'EDP. Dans l'équation de l'évolution, nous proposons l'existence, l'unicité de la solution. Le deuxième problème est régi par un opérateur discret. Nous étudions l'équation d'évolution discrète qui décrivent le processus d'effondrement du tas de sable. Ceci est un exemple typique de phénomènes auto-organisés critiques exposées par une slope critique. Nous considérons l'équation d'évolution discrète où la dynamique est régie par sous-gradient de la fonction d'indicateur de la boule unité. Nous commençons par établir le modèle, nous prouvons existence et l'unicité de la solution. Ensuite, en utilisant arguments de dualité nous étudions le calcul numérique de la solution et nous présentons quelques simulations numériques<br>This thesis is devoted to the study of evolution problems where the dynamic is governed by sub-gradient diffusion operator. We are interest in two kind of evolution problems. The first problem is governed by local operator of Leray-Lions type with a bounded domain. In this problem, the operator is maximal monotone and does not satisfied the standard polynomial growth control condition. Typical examples appears in the study of non-Neutonian fluid and also in the description of sub-gradient flows dynamics. To study the problem we handle the equation in the context of nonlinear PDE with singular flux. We use the theory of tangential gradient to characterize the state equation that gives the connection between the flux and the gradient of the solution. In the stationary problem, we have the existence of solution, we also get the equivalence between the initial minimization problem, the dual problem and the PDE. In the evolution one, we provide the existence, uniqueness of solution and the contractions. The second problem is governed by a discrete operator. We study the discrete evolution equation which describe the process of collapsing sandpile. This is a typical example of Self-organized critical phenomena exhibited by a critical slop. We consider the discrete evolution equation where the dynamic is governed by sub-gradient of indicator function of the unit ball. We begin by establish the model, we prove existence and uniqueness of the solution. Then by using dual arguments we study the numerical computation of the solution and we present some numerical simulations
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2

Coulon, Anne-Charline. "Propagation in reaction-diffusion equations with fractional diffusion." Doctoral thesis, Universitat Politècnica de Catalunya, 2014. http://hdl.handle.net/10803/277576.

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This thesis focuses on the long time behaviour of solutions to Fisher-KPP reaction-diffusion equations involving fractional diffusion. This type of equation arises, for example, in spatial propagation or spreading of biological species (rats, insects,...). In population dynamics, the quantity under study stands for the density of the population. It is well-known that, under some specific assumptions, the solution tends to a stable state of the evolution problem, as time goes to infinity. In other words, the population invades the medium, which corresponds to the survival of the species, and we want to understand at which speed this invasion takes place. To answer this question, we set up a new method to study the speed of propagation when fractional diffusion is at stake and apply it on three different problems. Part I of the thesis is devoted to an analysis of the asymptotic location of the level sets of the solution to two different problems : Fisher-KPP models in periodic media and cooperative systems, both including fractional diffusion. On the first model, we prove that, under some assumptions on the periodic medium, the solution spreads exponentially fast in time and we find the precise exponent that appears in this exponential speed of propagation. We also carry out numerical simulations to investigate the dependence of the speed of propagation on the initial condition. On the second model, we prove that the speed of propagation is once again exponential in time, with an exponent depending on the smallest index of the fractional Laplacians at stake and on the reaction term. Part II of the thesis deals with a two dimensional environment, where reproduction of Fisher-KPP type and usual diffusion occur, except on a line of the plane, on which fractional diffusion takes place. The plane is referred to as 'the field' and the line to 'the road', as a reference to the biological situations we have in mind. Indeed, it has long been known that fast diffusion on roads can have a driving effect on the spread of epidemics. We prove that the speed of propagation is exponential in time on the road, whereas it depends linearly on time in the field. Contrary to the precise asymptotics obtained in Part I, for this model, we are not able to give a sharp location of the level sets on the road and in the field. The expansion shape of the level sets in the field is investigated through numerical simulations.<br>Esta tesis se centra en el comportamiento en tiempos grandes de las soluciones de la ecuación de Fisher- KPP de reacción-difusión con difusión fraccionaria. Este tipo de ecuación surge, por ejemplo, en la propagación espacial o en la propagación de especies biológicas (ratas, insectos,...). En la dinámica de poblaciones, la cantidad que se estudia representa la densidad de la población. Es conocido que, bajo algunas hipótesis específicas, la solución tiende a un estado estable del problema de evolución, cuando el tiempo tiende a infinito. En otras palabras, la población invade el medio, lo que corresponde a la supervivencia de la especie, y nosotros queremos entender con qué velocidad se lleva a cabo esta invasión. Para responder a esta pregunta, hemos creado un nuevo método para estudiar la velocidad de propagación cuando se consideran difusiones fraccionarias, además hemos aplicado este método en tres problemas diferentes. La Parte I de la tesis está dedicada al análisis de la ubicación asintótica de los conjuntos de nivel de la solución de dos problemas diferentes: modelos de Fisher- KPP en medios periódicos y sistemas cooperativos, ambos consideran difusión fraccionaria. En el primer modelo, se prueba que, bajo ciertas hipótesis sobre el medio periódico, la solución se propaga exponencialmente rápido en el tiempo, además encontramos el exponente exacto que aparece en esta velocidad de propagación exponencial. También llevamos a cabo simulaciones numéricas para investigar la dependencia de la velocidad de propagación con la condición inicial. En el segundo modelo, se prueba que la velocidad de propagación es nuevamente exponencial en el tiempo, con un exponente que depende del índice más pequeño de los Laplacianos fraccionarios y también del término de reacción. La Parte II de la tesis ocurre en un entorno de dos dimensiones, donde se reproduce un tipo ecuación de Fisher- KPP con difusión estándar, excepto en una línea del plano, en el que la difusión fraccionada aparece. El plano será llamado "campo" y la línea "camino", como una referencia a las situaciones biológicas que tenemos en mente. De hecho, desde hace tiempo se sabe que la difusión rápida en los caminos puede causar un efecto en la propagación de epidemias. Probamos que la velocidad de propagación es exponencial en el tiempo en el camino, mientras que depende linealmente del tiempo en el campo. Contrariamente a los precisos exponentes obtenidos en la Parte I, para este modelo, no fuimos capaces de dar una localización exacta de los conjuntos de nivel en la carretera y en el campo. La forma de propagación de los conjuntos de nivel en el campo se investiga a través de simulaciones numéricas
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3

Prehl, Janett. "Diffusion on fractals and space-fractional diffusion equations." Doctoral thesis, Universitätsbibliothek Chemnitz, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001068.

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Ziel dieser Arbeit ist die Untersuchung der Sub- und Superdiffusion in fraktalen Strukturen. Der Fokus liegt auf zwei separaten Ansätzen, die entsprechend des Diffusionbereiches gewählt und variiert werden. Dadurch erhält man ein tieferes Verständnis und eine bessere Beschreibungsweise für beide Bereiche. Im ersten Teil betrachten wir subdiffusive Prozesse, die vor allem bei Transportvorgängen, z. B. in lebenden Geweben, eine grundlegende Rolle spielen. Hierbei modellieren wir den fraktalen Zustandsraum durch endliche Sierpinski Teppiche mit absorbierenden Randbedingungen und lösen dann die Mastergleichung zur Berechnung der Zeitentwicklung der Wahrscheinlichkeitsverteilung. Zur Charakterisierung der Diffusion auf regelmäßigen und zufälligen Teppichen bestimmen wir die Abfallzeit der Wahrscheinlichkeitsverteilung, die mittlere Austrittszeit und die Random Walk Dimension. Somit können wir den Einfluss zufälliger Strukturen auf die Diffusion aufzeigen. Superdiffusive Prozesse werden im zweiten Teil der Arbeit mit Hilfe der Diffusionsgleichung untersucht. Deren zweite Ableitung im Ort erweitern wir auf nichtganzzahlige Ordnungen, um die fraktalen Eigenschaften der Umgebung darzustellen. Die resultierende raum-fraktionale Diffusionsgleichung spannt ein Übergangsregime von der irreversiblen Diffusionsgleichung zur reversiblen Wellengleichung auf. Deren Lösungen untersuchen wir mittels verschiedener Entropien, wie Shannon, Tsallis oder Rényi Entropien, und deren Entropieproduktionsraten, welche natürliche Maße für die Irreversibilität sind. Das dabei gefundene Entropieproduktions-Paradoxon, d. h. ein unerwarteter Anstieg der Entropieproduktionsrate bei sinkender Irreversibilität des Prozesses, können wir nach geeigneter Reskalierung der Entropien auflösen<br>The aim of this thesis is the examination of sub- and superdiffusive processes in fractal structures. The focus of the work concentrates on two separate approaches that are chosen and varied according to the corresponding regime. Thus, we obtain new insights about the underlying mechanisms and a more appropriate way of description for both regimes. In the first part subdiffusion is considered, which plays a crucial role for transport processes, as in living tissues. First, we model the fractal state space via finite Sierpinski carpets with absorbing boundary conditions and we solve the master equation to compute the time development of the probability distribution. To characterize the diffusion on regular as well as random carpets we determine the longest decay time of the probability distribution, the mean exit time and the Random walk dimension. Thus, we can verify the influence of random structures on the diffusive dynamics. In the second part of this thesis superdiffusive processes are studied by means of the diffusion equation. Its second order space derivative is extended to fractional order, which represents the fractal properties of the surrounding media. The resulting space-fractional diffusion equations span a linking regime from the irreversible diffusion equation to the reversible (half) wave equation. The corresponding solutions are analyzed by different entropies, as the Shannon, Tsallis or Rényi entropies and their entropy production rates, which are natural measures of irreversibility. We find an entropy production paradox, i. e. an unexpected increase of the entropy production rate by decreasing irreversibility of the processes. Due to an appropriate rescaling of the entropy we are able to resolve the paradox
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4

Fei, Ning Fei. "Studies in reaction-diffusion equations." Thesis, Heriot-Watt University, 2003. http://hdl.handle.net/10399/310.

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5

Grant, Koryn. "Symmetries and reaction-diffusion equations." Thesis, University of Kent, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.264601.

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6

Ninomiya, Hirokazu. "Separatrices of competition-diffusion equations." 京都大学 (Kyoto University), 1995. http://hdl.handle.net/2433/187159.

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本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものである.<br>Kyoto Journal of Mathematics, vol35(3), pp.539-567, 1995, http://projecteuclid.org/euclid.kjm/1250518709<br>Kyoto University (京都大学)<br>0048<br>新制・課程博士<br>博士(理学)<br>甲第5884号<br>理博第1591号<br>新制||理||889(附属図書館)<br>UT51-95-D203<br>京都大学大学院工学研究科数学専攻<br>(主査)教授 西田 孝明, 教授 渡辺 信三, 教授 岩崎 敷久<br>学位規則第4条第1項該当
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7

Coulon, Chalmin Anne-Charline. "Fast propagation in reaction-diffusion equations with fractional diffusion." Toulouse 3, 2014. http://thesesups.ups-tlse.fr/2427/.

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Cette thèse est consacrée à l'étude du comportement en temps long, et plus précisément de phénomènes de propagation rapide, des équations de réaction-diffusion de type Kisher-KPP avec diffusion fractionnaire. Ces équations modélisent, par exemple, la propagation d'espèces biologiques. Sous certaines hypothèses, la population envahit le milieu et nous voulons comprendre à quelle vitesse cette invasion a lieu. Pour répondre à cette question, nous avons mis en place une nouvelle méthode et nous l'appliquons à différents modèles. Dans une première partie, nous étudions deux problèmes d'évolution comprenant une diffusion fractionnaire : un modèle de type Fisher-KPP en milieu périodique et un système coopératif. Dans les deux cas, nous montrons, sous certaines conditions, que la vitesse de propagation est exponentielle en temps, et nous donnons une expression précise de l'exposant de propagation. Nous menons des simulations numériques pour étudier la dépendance de cette vitesse de propagation en la donnée initiale. Dans une seconde partie, nous traitons un environnement bidimensionnel, dans lequel le terme de reproduction est de type Fisher-KPP et le terme diffusif est donné par un laplacien standard, excepté sur une ligne du plan où une diffusion fractionnaire intervient. Le plan est nommé "le champ" et la ligne "la route", en référence aux situations biologiques que nous voulons modéliser. Nous prouvons que la vitesse de propagation est exponentielle en temps sur la route, alors qu'elle dépend linéairement du temps dans le champ. La forme des lignes de niveau dans le champ est étudiée au travers de simulations numériques<br>This thesis focuses on the long time behaviour, and more precisely on fast propagation, in Fisher-KPP reaction diffusion equations involving fractional diffusion. This type of equation arises, for example, in spreading of biological species. Under some specific assumptions, the population invades the medium and we want to understand at which speed this invasion takes place when fractional diffusion is at stake. To answer this question, we set up a new method and apply it on different models. In a first part, we study two different problems, both including fractional diffusion : Fisher-KPP models in periodic media and cooperative systems. In both cases, we prove, under additional assumptions, that the solution spreads exponentially fast in time and we find the precise exponent of propagation. We also carry out numerical simulations to investigate the dependence of the speed of propagation on the initial condition. In a second part, we deal with a two dimensional environment, where reproduction of Fisher-KPP type and usual diffusion occur, except on a line of the plane, on which fractional diffusion takes place. The plane is referred to as "the field" and the line to "the road", as a reference to the biological situations we have in mind. We prove that the speed of propagation is exponential in time on the road, whereas it depends linearly on time in the field. The expansion shape of the level sets in the field is investigated through numerical simulations
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8

Knaub, Karl R. "On the asymptotic behavior of internal layer solutions of advection-diffusion-reaction equations /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/6772.

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9

Coville, Jerome. "Equations de reaction diffusion non-locale." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2003. http://tel.archives-ouvertes.fr/tel-00004313.

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Cette thèse est consacrée à l'étude des équations de réaction diffusion non-locale du type $u_(t)-(\int_(\R)J(x-y)[u(y)-u(x)]dy)=f(u)$. Ces équations non-linéaires apparaissent naturellement en physique et en biologie. On s'intéresse plus particulièrement aux propriétés (existence, unicité, monotonie) des solutions du type front progressif. Trois classes de non-linéarités $f$ (bistable, ignition, monostable) sont étudiées. L'existence dans les cas bistable et ignition est obtenue via une technique d'homotopie. Le cas monostable nécessite une autre approche. L'existence est obtenue via une approximation des équations sur des semi-intervales infinis $(-r,+\infty)$. L'unicité et la monotonie des solutions sont quand elles obtenues par méthode de glissement. Le comportement asymptotique ainsi que des formules pour les vitesses sont aussi établis.
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10

Cifani, Simone. "On nonlinear fractional convection - diffusion equations." Doctoral thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-15192.

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11

Lunney, Michael E. "Numerical dynamics of reaction-diffusion equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ61659.pdf.

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12

Bradshaw-Hajek, Bronwyn. "Reaction-diffusion equations for population genetics." Access electronically, 2004. http://www.library.uow.edu.au/adt-NWU/public/adt-NWU20041221.160902/index.html.

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13

Parvin, S. "Diffusion-convection problems in parabolic equations." Thesis, University of Manchester, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382761.

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14

Coville, Jérôme. "Equations de réaction-diffusion non-locale." Paris 6, 2003. https://tel.archives-ouvertes.fr/tel-00004313.

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15

Cinti, Eleonora <1982&gt. "Bistable elliptic equations with fractional diffusion." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2010. http://amsdottorato.unibo.it/3073/1/Cinti-Eleonora-Tesi.pdf.

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This work concerns the study of bounded solutions to elliptic nonlinear equations with fractional diffusion. More precisely, the aim of this thesis is to investigate some open questions related to a conjecture of De Giorgi about the one-dimensional symmetry of bounded monotone solutions in all space, at least up to dimension 8. This property on 1-D symmetry of monotone solutions for fractional equations was known in dimension n=2. The question remained open for n>2. In this work we establish new sharp energy estimates and one-dimensional symmetry property in dimension 3 for certain solutions of fractional equations. Moreover we study a particular type of solutions, called saddle-shaped solutions, which are the candidates to be global minimizers not one-dimensional in dimensions bigger or equal than 8. This is an open problem and it is expected to be true from the classical theory of minimal surfaces.
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Cinti, Eleonora <1982&gt. "Bistable elliptic equations with fractional diffusion." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2010. http://amsdottorato.unibo.it/3073/.

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This work concerns the study of bounded solutions to elliptic nonlinear equations with fractional diffusion. More precisely, the aim of this thesis is to investigate some open questions related to a conjecture of De Giorgi about the one-dimensional symmetry of bounded monotone solutions in all space, at least up to dimension 8. This property on 1-D symmetry of monotone solutions for fractional equations was known in dimension n=2. The question remained open for n>2. In this work we establish new sharp energy estimates and one-dimensional symmetry property in dimension 3 for certain solutions of fractional equations. Moreover we study a particular type of solutions, called saddle-shaped solutions, which are the candidates to be global minimizers not one-dimensional in dimensions bigger or equal than 8. This is an open problem and it is expected to be true from the classical theory of minimal surfaces.
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17

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\&apos;evy measures. In the end, this leads to an existence proof for initial data in $L^{\infty}(\mathbb{R}^d)$
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18

Sun, Xiaodi. "Metastable dynamics of convection-diffusion-reaction equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0002/NQ34630.pdf.

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19

Davidson, Fordyce A. "Bifurcation in systems of reaction-diffusion equations." Thesis, Heriot-Watt University, 1993. http://hdl.handle.net/10399/1444.

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20

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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21

Yu, Weiming. "Identification of Coefficients in Reaction-Diffusion Equations." University of Cincinnati / OhioLINK, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1076186036.

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22

Al-Ofl, Abdalaziz Saleem. "Analysis of complex nonlinear reaction-diffusion equations." Thesis, Durham University, 2008. http://etheses.dur.ac.uk/2422/.

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A mathematical analysis has been carried out for some nonlinear reaction- diffusion equations on open bounded convex domains Ω C R(^d)(d < 3) with Robin boundary conditions- Existence, uniqueness and continuous dependence on initial data of weak and strong solutions are proved. A numerical analysis has also been undertaken for these nonlinear reaction- diffusion equations on the above domains. A fully practical piecewise linear finite element approximation is proposed for which existence and uniqueness of the numerical solution are proved. Semi-discrete and fully discrete error estimates are given. A practical algorithm for computing the numerical solution is given and its convergence is proved. Finally, some numerical simulations in one-dimensional space are exhibited.
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23

Cardanobile, Stefano. "Diffusion systems and heat equations on networks." [S.l. : s.n.], 2008. http://nbn-resolving.de/urn:nbn:de:bsz:289-vts-64278.

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24

Hagberg, Aric Arild. "Fronts and patterns in reaction-diffusion equations." Diss., The University of Arizona, 1994. http://hdl.handle.net/10150/186901.

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This is a study of fronts and patterns formed in reaction-diffusion systems. A doubly-diffusive version of the two component FitzHugh-Nagumo equations with bistable reaction dynamics is investigated as an abstract model for the study of pattern phenomenologies found in many different physical systems. Front solutions connecting the two stable uniform states are found to be key building blocks for understanding extended patterns such as stationary domains and traveling pulses in one dimension, and labyrinthine structures, splitting spots, and spiral wave turbulence in two dimensions. The number and type of front solutions is controlled by a bifurcation that we derive both analytically and numerically. At this bifurcation, called the nonequilibrium Ising-Bloch (NIB) bifurcation, a single stationary Ising front loses stability to a pair of counterpropagating Bloch fronts. In two dimensions, we derive a boundary where extended fronts become unstable to transverse perturbations. In addition, near the NIB bifurcation, we discover a multivalued relation between the front speed and general perturbations such as curvature or an external convective field. This multivalued form allows perturbations to induce transitions that reverse the direction of front propagation. When occurring locally along an extended front, these transitions nucleate spiral-vortex pairs. The NIB bifurcation and transverse instability boundaries divide parameter space into regions of different pattern behaviors. Before the bifurcation, the system may form transient patterns or stationary domains consisting of pairs of Ising fronts. Above the transverse instability boundary, two-dimensional planar fronts destabilize, grow, and finger to form a space-filling labyrinthine, or lamellar, pattern. Beyond the bifurcation the multiplicity of Bloch front solutions allows for the formation of persistent traveling pulses and spiral waves. Near the NIB bifurcation there is an intermediate region where new unexpected patterns are found. One-dimensional stationary domains become unstable to oscillating or breathing domains. In two dimensions, the transverse instability and local front transitions are the mechanisms behind spot splitting and the development of spiral wave turbulence. Similar patterns have been observed recently in the ferrocyanide-iodate-sulfite reaction.
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25

Büger, Matthias. "Systems of reaction-diffusion equations and their attractors." Giessen : Selbstverlag des Mathematischen Instituts, 2005. http://catalog.hathitrust.org/api/volumes/oclc/62216537.html.

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26

Meiler, Maria. "Analytic advances in difference equations of diffusion processes /." Göttingen : Sierke, 2009. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=017611057&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.

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27

Lin, Xue Lei. "Separable preconditioner for time-space fractional diffusion equations." Thesis, University of Macau, 2017. http://umaclib3.umac.mo/record=b3691377.

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28

Sionoid, Peadar N. "Nonlinear wave equations with diffusion, diffraction and dispersion." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319935.

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29

Xu, Lu. "Large deviations technique on stochastic reaction-diffusion equations." Thesis, University of Warwick, 2008. http://wrap.warwick.ac.uk/2736/.

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There are two different problems studied in this thesis. The first one is a travelling wave problem. We will improve the result proved in [4] to derive the ergodic property of the travelling wave behind the wavefront. The second problem is a large deviation problem concerning solutions to certain kind stochastic partial differential equations. We will first briefly introduce some basics about SPDE in chapter 2. In chapter 3, we will prove a large deviation principle for super-Brownian motion when it is considered as a solution to an SPDE, using the LDP for super-Brownian motion when it is considered as a measure-valued branching process as solution to a martingale problem. In chapter 4, we will prove another LDP result for solutions of a stochastic reaction-diffusion equation with degenerate noise term. Finally in chapter 5, we will explore some applications of those LDP results proved previously.
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30

Meiler, Maria. "Analytic advances in difference equations of diffusion processes." Göttingen Sierke, 2008. http://d-nb.info/992791685/04.

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31

Ferguson, R. C. "Numerical techniques for the drift-diffusion semiconductor equations." Thesis, University of Bath, 1996. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362239.

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32

Carter, David, Boguslaw Kruczek, and F. Handan Tezel. "Application of Maxwell Stefan equations to characterize silicalite membranes." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-198056.

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33

Ding, Weiwei. "Propagation phenomena of integro-difference equations and bistable reaction-diffusion equations in periodic habitats." Thesis, Aix-Marseille, 2014. http://www.theses.fr/2014AIXM4737.

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Cette thèse concerne les phénomènes de propagation de certaines équations d'évolution dans des habitats périodiques. Dans la première partie, nous étudions les phénomènes d'expansion de certaines équations d'intégro-différence spatialement périodiques. Tout d'abord, nous établissons une théorie générale sur l'existence des vitesses de propagation pour des systèmes d'évolution noncompacts, sous l'hypothèse que les systèmes linéarisés ont des valeurs propres principales. Ensuite, nous introduisons la notion d'irréductibilité uniforme des mesures de Radon finies sur le cercle. On démontre que tout opérateur de convolution généré par une telle mesure admet une valeur propre principale. Enfin, nous prouvons l'existence de vitesses de propagation pour certains équations d'intégro-différence avec des noyaux de dispersion uniformément irréductibles. Dans la deuxième partie, nous étudions les phénomènes de propagation de front pour des équations de réaction-diffusion spatialement périodiques avec des non-linéarités bistables. Nous nous concentrons d'abord sur les solutions de type fronts pulsatoires. Sous diverses hypothèses, il est prouvé que les fronts pulsatoires existent lorsque la période spatiale est petite ou grande. Nous caractérisons aussi le signe des vitesses et nous montrons la stabilité exponentielle globale des fronts pulsatoires de vitesse non nulle. Nous étudions ensuite les solutions de type fronts de transition. Sous des hypothèses convenables, on prouve que les fronts de transition se ramènent aux fronts pulsatoires avec une vitesse non nulle. Mais nous montrons aussi l'existence de nouveaux types de fronts de transition qui ne sont pas des fronts pulsatoires<br>This dissertation is concerned with propagation phenomena of some evolution equations in periodic habitats. The main results consist of the following two parts. In the first part, we investigate the spatial spreading phenomena of some spatially periodic integro-difference equations. Firstly, we establish a general theory on the existence of spreading speeds for noncompact evolution systems, under the hypothesis that the linearized systems have principal eigenvalues. Secondly, we introduce the notion of uniform irreducibility for finite Radon measures on the circle. It is shown that, any generalized convolution operator generated by such a measure admits a principal eigenvalue. Finally, applying the above general theories, we prove the existence of spreading speeds for some integro-difference equations with uniformly irreducible dispersal kernels. In the second part, we study the front propagation phenomena of spatially periodic reaction-diffusion equations with bistable nonlinearities. Firstly, we focus on the propagation solutions in the class of pulsating fronts. It is proved that, under various assumptions on the reaction terms, pulsating fronts exist when the spatial period is small or large. We also characterize the sign of the front speeds and we show the global exponential stability of the pulsating fronts with nonzero speed. Secondly, we investigate the propagation solutions in the larger class of transition fronts. It is shown that, under suitable assumptions, transition fronts are reduced to pulsating fronts with nonzero speed. But we also prove the existence of new types of transition fronts which are not pulsating fronts
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34

Wei, Hui Qin. "Preconditioners for solving fractional diffusion equations with discontinuous coefficients." Thesis, University of Macau, 2017. http://umaclib3.umac.mo/record=b3691375.

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35

Filho, Sergio Muniz Oliva. "Reaction-diffusion systems on domains with thin channels." Diss., Georgia Institute of Technology, 1993. http://hdl.handle.net/1853/28837.

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36

Meral, Gulnihal. "Numerical Solution Of Nonlinear Reaction-diffusion And Wave Equations." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/3/12610568/index.pdf.

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In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quadrature method (DQM) is used for the spatial discretization of IBVPs and Cauchy problems defined by the nonlinear reaction-diffusion and wave equations. The DRBEM and DQM applications result in first and second order system of ordinary differential equations in time. These systems are solved with three different time integration methods, the finite difference method (FDM), the least squares method (LSM) and the finite element method (FEM) and comparisons among the methods are made. In the FDM a relaxation parameter is used to smooth the solution between the consecutive time levels. It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure with exponential and rational radial basis functions is found suitable for exterior wave problem. The same result is also valid when DQM is used for space discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear reaction-diffusion and wave equations.
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37

Ryan, John Maurice-Car. "Global existence of reaction-diffusion equations over multiple domains." Texas A&M University, 2004. http://hdl.handle.net/1969.1/3312.

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Systems of semilinear parabolic differential equations arise in the modelling of many chemical and biological systems. We consider m component systems of the form ut = D&#916;u + f (t, x, u) &#8706;uk/&#8706;&#951; =0 k =1, ...m where u(t, x)=(uk(t, x))mk=1 is an unknown vector valued function and each u0k is zero outside &#937;&#963;(k), D = diag(dk)is an m × m positive de&#64257;nite diagonal matrix, f : R × Rn× Rm &#8594; Rm, u0 is a componentwise nonnegative function, and each &#937;i is a bounded domain in Rn where &#8706;&#937;i is a C2+&#945;manifold such that &#937;i lies locally on one side of &#8706;&#937;i and has unit outward normal &#951;. Most physical processes give rise to systems for which f =(fk) is locally Lipschitz in u uniformly for (x, t) &#8712; &#937; × [0,T ] and f (·, ·, ·) &#8712; L&#8734;(&#937; × [0,T ) × U ) for bounded U and the initial data u0 is continuous and nonnegative on &#937;. The primary results of this dissertation are three-fold. The work began with a proof of the well posedness for the system . Then we obtained a global existence result if f is polynomially bounded, quaipositive and satisfies a linearly intermediate sums condition. Finally, we show that systems of reaction-diffusion equations with large diffusion coeffcients exist globally with relatively weak assumptions on the vector field f.
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38

Wang, Shuyu. "Reaction-diffusion equations and the Laplacian in Hilbert space." Thesis, University of Ottawa (Canada), 1990. http://hdl.handle.net/10393/5772.

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This dissertation consists of two parts. First, we study some problems associated with reaction-diffusion equations with variables in finite-dimensional space. We investigate the positivity of solutions, the existence of positive invariant regions, and we also make some stability analysis. In part II, we study the Levy-Laplacian in infinite-dimensional space. We explore some properties of this Laplacian and solve some boundary value problems.
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39

Baugh, James Emory. "Group analysis of a system of reaction-diffusion equations." Thesis, Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/28554.

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40

Hill, Adrian T. "Attractors for convection-diffusion equations and their numerical approximation." Thesis, University of Oxford, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.314907.

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41

Qu, Lei, and 瞿磊. "Multiplicity and stability of two-dimensional reaction-diffusion equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2001. http://hub.hku.hk/bib/B31226656.

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42

Fullwood, Timothy Brent. "Pattern formation and travelling waves in reaction-diffusion equations." Thesis, University of Warwick, 1995. http://wrap.warwick.ac.uk/4251/.

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This thesis is about pattern formation in reaction - diffusion equations, particularly Turing patterns and travelling waves. In chapter one we concentrate on Turing patterns. We give the classical approach to proving the existence of these patterns, and then our own, which uses the reversibility of the associated travelling wave equations when the wave speed is zero. We use a Lyapunov - Schmidt reduction to prove the existence of periodic solutions when there is a purely imaginary eigenvalue. We pay particular attention to the bifurcation point where these patterns arise, the 1: 1 resonance. We prove the existence of steady patterns near a Hopf bifurcation and then include a similar result for dynamics close to a Takens - Bogdanov point. Chapter two concentrates on travelling waves and looks for the existence of such in three different ways. Firstly we prove the conditions that are needed for the travelling wave equations to go through a Hopf bifurcation. Secondly, we look for the existence of travelling waves as the wave speed is perturbed from zero and prove when this occurs, again, using a Lyapunov - Schmidt reduction. Thirdly we describe a result proving the existence of periodic travelling waves when the wave speed is perturbed from infinity. In the last part of chapter two we prove the stability of such waves for A-w systems. In chapter three we discuss computer simulations of the work done in the earlier chapters. We present the mappings used and prove that their behaviour is similar to the original partial differential equations. The two specific examples we give are a predator prey model and the complex Ginzburg - Landau equations.
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43

Kay, Alison Lindsey. "Travelling fronts and wave-trains in reaction-diffusion equations." Thesis, University of Warwick, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.342513.

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44

Vafadari, Cyrus. "Monte Carlo methods for parallel processing of diffusion equations." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/82451.

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Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Nuclear Science and Engineering, 2013.<br>"June 2013." Cataloged from PDF version of thesis.<br>Includes bibliographical references (page 14).<br>A Monte Carlo algorithm for solving simple linear systems using a random walk is demonstrated and analyzed. The described algorithm solves for each element in the solution vector independently. Furthermore, it is demonstrated that this algorithm is easily parallelized. To reduce error, each processor can compute data for an independent element of the solution, or part of the data for a given element for the solution, allowing for larger samples to decrease stochastic error. In addition to parallelization, it is also shown that a probabilistic chain termination can decrease the runtime of the algorithm while maintaining accuracy. Thirdly, a tighter lower bound for the required number of chains given a desired error is determined.<br>by Cyrus Vafadari.<br>S.B.
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45

Zimmermann, Nils E. R., Timm J. Zabel, and Frerich J. Keil. "Transport into zeolite nanosheets: diffusion equations put to test." Diffusion fundamentals 20 (2013 ) 53, S. 1-2, 2013. https://ul.qucosa.de/id/qucosa%3A13629.

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46

Nadin, Grégoire. "Equations de réaction-diffusion et propagation en milieu hétérogène." Paris 6, 2008. http://www.theses.fr/2008PA066491.

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Cette thèse est consacrée à l'étude d'une équation de réaction-diffusion de type monostable en milieu hétérogène. Dans une première partie nous étudions les valeurs propres principales généralisées associées à une linéarisation de cette équation en milieu périodique en temps et en espace. Puis, nous donnons des propriétés d'existence et d'unicité des solutions entières de l'équation. Dans une seconde partie, nous prouvons l'existence de fronts pulsatoires en milieu périodique en temps et en espace. Une caractérisation de la vitesse de ces fronts est utilisée pour étudier la dépendance entre les coefficients de l'équation et cette vitesse. La troisième partie est consacrée à l'étude de phénomènes de propagation dans des milieux plus généraux. Nous prouvons l'existence de fronts pulsatoires dans des milieux presque périodiques. Pour des milieux hétérogènes généraux, nous montrons qu'il existe une vitesse positive d'expansion. L’existence de fronts de propagation est prouvée pour deux équations non-locales dans la quatrième partie : d'abord l'équation de Fisher-Keller-Segel, puis l’équation de Fisher avec un terme de saturation non-local.
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47

Trojan, Alice von. "Finite difference methods for advection and diffusion." Title page, abstract and contents only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phv948.pdf.

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Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.
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48

Howard, Martin. "Non-equilibrium dynamics of reaction-diffusion systems." Thesis, University of Oxford, 1996. http://ora.ox.ac.uk/objects/uuid:4485a178-6262-4487-b40f-7c7ec790d687.

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Fluctuations are known to radically alter the behaviour of reaction-diffusion systems. Below a certain upper critical dimension d<sub>c</sub> , this effect results in the breakdown of traditional approaches, such as mean field rate equations. In this thesis we tackle this fluctuation problem by employing systematic field theoretic/renormalisation group methods, which enable perturbative calculations to be made below d<sub>c</sub>. We first consider a steady state reaction front formed in the two species irreversible reaction A + B → Ø. In one dimension we demonstrate that there are two components to the front - one an intrinsic width, and one caused by the ability of the centre of the front to wander. We make theoretical predictions for the shapes of these components, which are found to be in good agreement with our one dimensional simulations. In higher dimensions, where the intrinsic component dominates, we also make calculations for its asymptotic profile. Furthermore, fluctuation effects lead to a prediction of asymptotic power law tails in the intrinsic front in all dimensions. This effect causes high enough order spatial moments of a time dependent reaction front to exhibit multiscaling. The second system we consider is a time dependent multispecies reaction-diffusion system with three competing reactions A+A → Ø, B + B → Ø, and A + B → Ø, starting with homogeneous initial conditions. Using our field theoretic formalism we calculate the asymptotic density decay rates for the two species for d ≤ d<sub>c</sub>. These calculations are compared with other approximate methods, such as the Smoluchowski approach, and also with previous simulations and exact results.
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49

Manay, Siddharth. "Applications of anti-geometric diffusion of computer vision : thresholding, segmentation, and distance functions." Diss., Georgia Institute of Technology, 2003. http://hdl.handle.net/1853/33626.

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50

Carter, David, Boguslaw Kruczek, and F. Handan Tezel. "Application of Maxwell Stefan equations to characterize silicalite membranes." Diffusion fundamentals 24 (2015) 8, S. 1, 2015. https://ul.qucosa.de/id/qucosa%3A14522.

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