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1
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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2
Hellander, Stefan. "Stochastic Simulation of Reaction-Diffusion Processes." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-198522.
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Numerical simulation methods have become an important tool in the study of chemical reaction networks in living cells. Many systems can, with high accuracy, be modeled by deterministic ordinary differential equations, but other systems require a more detailed level of modeling. Stochastic models at either the mesoscopic level or the microscopic level can be used for cases when molecules are present in low copy numbers. In this thesis we develop efficient and flexible algorithms for simulating systems at the microscopic level. We propose an improvement to the Green's function reaction dynamics algorithm, an efficient microscale method. Furthermore, we describe how to simulate interactions with complex internal structures such as membranes and dynamic fibers. The mesoscopic level is related to the microscopic level through the reaction rates at the respective scale. We derive that relation in both two dimensions and three dimensions and show that the mesoscopic model breaks down if the discretization of space becomes too fine. For a simple model problem we can show exactly when this breakdown occurs. We show how to couple the microscopic scale with the mesoscopic scale in a hybrid method. Using the fact that some systems only display microscale behaviour in parts of the system, we can gain computational time by restricting the fine-grained microscopic simulations to only a part of the system. Finally, we have developed a mesoscopic method that couples simulations in three dimensions with simulations on general embedded lines. The accuracy of the method has been verified by comparing the results with purely microscopic simulations as well as with theoretical predictions.eSSENCE
3
Smith, Stephen. "Stochastic reaction-diffusion models in biology." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/33142.
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Every cell contains several millions of diffusing and reacting biological molecules. The interactions between these molecules ultimately manifest themselves in all aspects of life, from the smallest bacterium to the largest whale. One of the greatest open scientific challenges is to understand how the microscopic chemistry determines the macroscopic biology. Key to this challenge is the development of mathematical and computational models of biochemistry with molecule-level detail, but which are sufficiently coarse to enable the study of large systems at the cell or organism scale. Two such models are in common usage: the reaction-diffusion master equation, and Brownian dynamics. These models are utterly different in both their history and in their approaches to chemical reactions and diffusion, but they both seek to address the same reaction-diffusion question. Here we make an in-depth study into the physical validity of these models under various biological conditions, determining when they can reliably be used. Taking each model in turn, we propose modifications to the models to better model the realities of the cellular environment, and to enable more efficient computational implementations. We use the models to make predictions about how and why cells behave the way they do, from mechanisms of self-organisation to noise reduction. We conclude that both models are extremely powerful tools for clarifying the details of the mysterious relationship between chemistry and biology.
4
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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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.
6
Larsson, Stig. "On reaction-diffusion equation and their approximation by finite element methods /." Göteborg : Chalmers tekniska högskola, Dept. of Mathematics, 1985. http://bibpurl.oclc.org/web/32831.
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7
Kieri, Emil. "Accuracy aspects of the reaction-diffusion master equation on unstructured meshes." Thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-145978.
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The reaction-diffusion master equation (RDME) is a stochastic model for spatially heterogeneous chemical systems. Stochastic models have proved to be useful for problems from molecular biology since copy numbers of participating chemical species often are small, which gives a stochastic behaviour. The RDME is a discrete space model, in contrast to spatially continuous models based on Brownian motion. In this thesis two accuracy issues of the RDME on unstructured meshes are studied. The first concerns the rates of diffusion events. Errors due to previously used rates are evaluated, and a second order accurate finite volume method, not previously used in this context, is implemented. The new discretisation improves the accuracy considerably, but unfortunately it puts constraints on the mesh, limiting its current usability. The second issue concerns the rates of bimolecular reactions. Using the macroscopic reaction coefficients these rates become too low when the spatial resolution is high. Recently, two methods to overcome this problem by calculating mesoscopic reaction rates for Cartesian meshes have been proposed. The methods are compared and evaluated, and are found to work remarkably well. Their possible extension to unstructured meshes is discussed.
8
Lee, Isobel Micheline. "The existance of multiple steady-state solutions of a reaction-diffusion equation." Thesis, University of Oxford, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329934.
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Gibbs, Simon Paul. "Solutions of the reaction-diffusion eikonal equation on closed two-dimensional manifolds." Thesis, Glasgow Caledonian University, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357134.
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10
Josien, Marc. "Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux." Thesis, Paris Est, 2018. http://www.theses.fr/2018PESC1120/document.
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Le travail de cette thèse a porté sur l'étude mathématique et numérique de quelques modèles multi-échelles issus de la physique des matériaux. La première partie de ce travail est consacrée à l'homogénéisation mathématique d'un problème elliptique avec une petite échelle. Nous étudions le cas particulier d'un matériau présentant une structure périodique avec un défaut. En adaptant la théorie classique d'Avellaneda et Lin pour les milieux périodiques, on démontre qu'on peut approximer finement la solution d'un tel problème, notamment à l'échelle microscopique. Nous obtenons des taux de convergence dépendant de l'étalement du défaut. On démontre aussi quelques propriétés des fonctions de Green d'un problème elliptique périodique avec conditions de bord périodiques. Les dislocations sont des lignes de défaut de la matière responsables du phénomène de plasticité. Les deuxième et troisième parties de ce mémoire portent sur la simulation de dislocations, d'abord en régime stationnaire puis en régime dynamique. Nous utilisons le modèle de Peierls, qui couple échelle atomique et échelle mésoscopique. Dans le cadre stationnaire, on obtient une équation intégrodifférentielle non-linéaire avec un laplacien fractionnaire: l'équation de Weertman. Nous en étudions les propriétés mathématiques et proposons un schéma numérique pour en approximer la solution. Dans le cadre dynamique, on obtient une équation intégrodifférentielle à la fois en temps et en espace. Nous en faisons une brève étude mathématique, et comparons différents algorithmes pour la simuler. Enfin, dans la quatrième partie, nous étudions la limite macroscopique d'une chaîne d'atomes soumis à la loi de Newton. Des arguments formels suggèrent que celle-ci devrait être décrite par une équation des ondes non-linéaires. Or, nous démontrons --sous certaines hypothèses-- qu'il n'en est rien lorsque des chocs apparaissentIn this thesis we study mathematically and numerically some multi-scale models from materials science. First, we investigate an homogenization problem for an oscillating elliptic equation. The material under consideration is described by a periodic structure with a defect at the microscopic scale. By adapting Avellaneda and Lin's theory for periodic structures, we prove that the solution of the oscillating equation can be approximated at a fine scale. The rates of convergence depend upon the integrability of the defect. We also study some properties of the Green function of periodic materials with periodic boundary conditions. Dislocations are lines of defects inside materials, which induce plasticity. The second part and the third part of this manuscript are concerned with simulation of dislocations, first in the stationnary regime then in the dynamical regime. We use the Peierls model, which couples atomistic and mesoscopic scales and involves integrodifferential equations. In the stationary regime, dislocations are described by the so-called Weertman equation, which is nonlinear and involves a fractional Laplacian. We study some mathematical properties of this equation and propose a numerical scheme for approximating its solution. In the dynamical regime, dislocations are described by an equation which is integrodifferential in time and space. We compare some numerical methods for recovering its solution. In the last chapter, we investigate the macroscopic limit of a simple chain of atoms governed by the Newton equation. Surprisingly enough, under technical assumptions, we show that it is not described by a nonlinear wave equation when shocks occur
11
Couture, Chad. "Steady States and Stability of the Bistable Reaction-Diffusion Equation on Bounded Intervals." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/37110.
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Reaction-diffusion equations have been used to study various phenomena across different fields. These equations can be posed on the whole real line, or on a subinterval, depending on the situation being studied. For finite intervals, we also impose diverse boundary conditions on the system. In the present thesis, we solely focus on the bistable reaction-diffusion equation while working on a bounded interval of the form $[0,L]$ ($L>0$). Furthermore, we consider both mixed and no-flux boundary conditions, where we extend the former to Dirichlet boundary conditions once our analysis of that system is complete. We first use phase-plane analysis to set up our initial investigation of both systems. This gives us an integral describing the transit time of orbits within the phase-plane. This allows us to determine the bifurcation diagram of both systems. We then transform the integral to ease numerical calculations. Finally, we determine the stability of the steady states of each system.
12
Roy, Christian. "The Origin of Wave Blocking for a Bistable Reaction-Diffusion Equation : A General Approach." Thèse, Université d'Ottawa / University of Ottawa, 2012. http://hdl.handle.net/10393/22704.
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Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.
13
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 pulsatoiresThis 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
14
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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Kunert, Gerd. "Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes." Universitätsbibliothek Chemnitz, 2000. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200000867.
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We consider a singularly perturbed reaction-diffusion problem and
derive and rigorously analyse an a posteriori residual error
estimator that can be applied to anisotropic finite element meshes.
The quotient of the upper and lower error bounds is the so-called
matching function which depends on the anisotropy (of the
mesh and the solution) but not on the small perturbation parameter.
This matching function measures how well the anisotropic finite
element mesh corresponds to the anisotropic problem.
Provided this correspondence is sufficiently good, the matching
function is O(1).
Hence one obtains tight error bounds, i.e. the error estimator
is reliable and efficient as well as robust with respect to the
small perturbation parameter.
A numerical example supports the anisotropic error analysis.
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Grosman, Serguei. "The robustness of the hierarchical a posteriori error estimator for reaction-diffusion equation on anisotropic meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601418.
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Singularly perturbed reaction-diffusion problems
exhibit in general solutions with anisotropic
features, e.g. strong boundary and/or interior
layers. This anisotropy is reflected in the
discretization by using meshes with anisotropic
elements. The quality of the numerical solution
rests on the robustness of the a posteriori error
estimator with respect to both the perturbation
parameters of the problem and the anisotropy of the
mesh. The simplest local error estimator from the
implementation point of view is the so-called
hierarchical error estimator. The reliability
proof is usually based on two prerequisites:
the saturation assumption and the strengthened
Cauchy-Schwarz inequality. The proofs of these
facts are extended in the present work for the
case of the singularly perturbed reaction-diffusion
equation and of the meshes with anisotropic elements.
It is shown that the constants in the corresponding
estimates do neither depend on the aspect ratio
of the elements, nor on the perturbation parameters.
Utilizing the above arguments the concluding
reliability proof is provided as well as the
efficiency proof of the estimator, both
independent of the aspect ratio and perturbation
parameters.
17
Seymen, Zahire. "Solving Optimal Control Time-dependent Diffusion-convection-reaction Equations By Space Time Discretizations." Phd thesis, METU, 2013. http://etd.lib.metu.edu.tr/upload/12615399/index.pdf.
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Optimal control problems (OCPs) governed by convection dominated diffusion-convection-reaction
equations arise in many science and engineering applications such as shape optimization of the technological
devices, identification of parameters in environmental processes and flow control problems.
A characteristic feature of convection dominated optimization problems is the presence of sharp layers.
In this case, the Galerkin finite element method performs poorly and leads to oscillatory solutions.
Hence, these problems require stabilization techniques to resolve boundary and interior layers accurately.
The Streamline Upwind Petrov-Galerkin (SUPG) method is one of the most popular stabilization
technique for solving convection dominated OCPs.
The focus of this thesis is the application and analysis of the SUPG method for distributed and
boundary OCPs governed by evolutionary diffusion-convection-reaction equations. There are two approaches
for solving these problems: optimize-then-discretize and discretize-then-optimize. For the
optimize-then-discretize method, the time-dependent OCPs is transformed to a biharmonic equation,
where space and time are treated equally. The resulting optimality system is solved by the finite
element package COMSOL. For the discretize-then-optimize approach, we have used the so called allv
at-once method, where the fully discrete optimality system is solved as a saddle point problem at once
for all time steps. A priori error bounds are derived for the state, adjoint, and controls by applying
linear finite element discretization with SUPG method in space and using backward Euler, Crank-
Nicolson and semi-implicit methods in time. The stabilization parameter is chosen for the convection
dominated problem so that the error bounds are balanced to obtain L2 error estimates. Numerical examples
with and without control constraints for distributed and boundary control problems confirm the
effectiveness of both approaches and confirm a priori error estimates for the discretize-then-optimize
approach.
18
Mbroh, Nana Adjoah. "On the method of lines for singularly perturbed partial differential equations." University of the Western Cape, 2017. http://hdl.handle.net/11394/5679.
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Magister Scientiae - MScMany chemical and physical problems are mathematically described by partial differential equations (PDEs). These PDEs are often highly nonlinear and therefore have no closed form solutions. Thus, it is necessary to recourse to numerical approaches to determine suitable approximations to the solution of such equations. For solutions possessing sharp spatial transitions (such as boundary or interior layers), standard numerical methods have shown limitations as they fail to capture large gradients. The method of lines (MOL) is one of the numerical methods used to solve PDEs. It proceeds by the discretization of all but one dimension leading to systems of ordinary di erential equations. In the case of time-dependent PDEs, the MOL consists of discretizing the spatial derivatives only leaving the time variable continuous. The process results in a system to which a numerical method for initial value problems can be applied. In this project we consider various types of singularly perturbed time-dependent PDEs. For each type, using the MOL, the spatial dimensions will be discretized in many different ways following fitted numerical approaches. Each discretisation will be analysed for stability and convergence. Extensive experiments will be conducted to confirm the analyses.
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Meinecke, Lina. "Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-284085.
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Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. Analytical solutions can be derived only for the simplest models and numerical simulations are necessary in most cases to evaluate the models and their properties and to compare them with measured data. This thesis focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution. The second part of this thesis investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment.
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Wang, Xiuquan. "Parameter Estimation in the Advection Diffusion Reaction Model With Mean Occupancy Time and Boundary Flux Approaches." OpenSIUC, 2014. https://opensiuc.lib.siu.edu/dissertations/976.
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In this dissertation, we examine an advection diffusion model for insects inhabiting a spatially heterogeneous environment and moving toward a more favorable environment. We first study the effects of adding a term describing drift or advection toward a favorable environment to diffusion models for population dynamics. The diffusion model is a basic linear two-dimensional diffusion equation describing local dispersal of species. The mathematical advection terms are taken to be Fickian and describe directed movement of the population toward the favorable environment. For this model, the landscape is composed of one homogeneous habitat patch embedded in a spatially heterogeneous environment and the boundary of the habitat inhabited by the population acts as a lethal edge. We also derived the mean occupancy time and the boundary flux of the habitat patch. The diffusion rate and advection parameters of the advection diffusion model are estimated based on mean occupancy time and boundary flux. We then introduce two methods for the identification of these coefficients in the model as well as the capture rate. These two new methods have some advantages over other methods of estimating those parameters, including reduced computational cost and ease of use in the field. We further examine the statistical properties of new methods through simulation, and discuss how mean occupancy time and boundary flux could be estimated in field experiments.
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Galbally, David. "Nonlinear model reduction for uncertainty quantification in large-scale inverse problems : application to nonlinear convection-diffusion-reaction equation." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/43079.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008.Includes bibliographical references (p. 147-152).
There are multiple instances in science and engineering where quantities of interest are evaluated by solving one or several nonlinear partial differential equations (PDEs) that are parametrized in terms of a set of inputs. Even though well-established numerical techniques exist for solving these problems, their computational cost often precludes their use in cases where the outputs of interest must be evaluated repeatedly for different values of the input parameters such as probabilistic analysis applications. In this thesis we present a model reduction methodology that combines efficient representation of the nonlinearities in the governing PDE with an efficient model-constrained, greedy algorithm for sampling the input parameter space. The nonlinearities in the PDE are represented using a coefficient-function approximation that enables the development of an efficient offline-online computational procedure where the online computational cost is independent of the size of the original high-fidelity model. The input space sampling algorithm used for generating the reduced space basis adaptively improves the quality of the reduced order approximation by solving a PDE-constrained continuous optimization problem that targets the output error between the reduced and full order models in order to determine the optimal sampling point at every greedy cycle. The resulting model reduction methodology is applied to a highly nonlinear combustion problem governed by a convection-diffusion-reaction PDE with up to 3 input parameters. The reduced basis approximation developed for this problem is up to 50, 000 times faster to solve than the original high-fidelity finite element model with an average relative error in prediction of outputs of interest of 2.5 - 10-6 over the input parameter space. The reduced order model developed in this thesis is used in a novel probabilistic methodology for solving inverse problems.
(cont) The extreme computational cost of the Bayesian framework approach for inferring the values of the inputs that generated a given set of empirically measured outputs often precludes its use in practical applications. In this thesis we show that using a reduced order model for running the Markov
by David Galbally.
S.M.
22
Hellander, Andreas. "Multiscale Stochastic Simulation of Reaction-Transport Processes : Applications in Molecular Systems Biology." Doctoral thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-152098.
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Quantitative descriptions of reaction kinetics formulated at the stochastic mesoscopic level are frequently used to study various aspects of regulation and control in models of cellular control systems. For this type of systems, numerical simulation offers a variety of challenges caused by the high dimensionality of the problem and the multiscale properties often displayed by the biochemical model. In this thesis I have studied several aspects of stochastic simulation of both well-stirred and spatially heterogenous systems. In the well-stirred case, a hybrid method is proposed that reduces the dimension and stiffness of a model. We also demonstrate how both a high performance implementation and a variance reduction technique based on quasi-Monte Carlo can reduce the computational cost to estimate the probability density of the system. In the spatially dependent case, the use of unstructured, tetrahedral meshes to sample realizations of the stochastic process is proposed. Using such meshes, we then extend the reaction-diffusion framework to incorporate active transport of cellular cargo in a seamless manner. Finally, two multilevel methods for spatial stochastic simulation are considered. One of them is a space-time adaptive method combining exact stochastic, approximate stochastic and macroscopic modeling levels to reduce the simualation cost. The other method blends together mesoscale and microscale simulation methods to locally increase modeling resolution.eSSENCE
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BACCOLI, ANTONELLO. "Boundary control and observation of coupled parabolic PDEs." Doctoral thesis, Università degli Studi di Cagliari, 2016. http://hdl.handle.net/11584/266880.
Full textAbstract:
Reaction-diffusion equations are parabolic Partial Differential Equations (PDEs) which
often occur in practice, e.g., to model the concentration of one or more substances, distributed
in space, under the in
uence of different phenomena such as local chemical reactions,
in which the substances are transformed into each other, and diffusion, which causes
the substances to spread out over a surface in space. Certainly, reaction-diffusion PDEs
are not confined to chemical applications but they also describe dynamical processes of
non-chemical nature, with examples being found in thermodynamics, biology, geology,
physics, ecology, etc.
Problems such as parabolic Partial Differential Equations (PDEs) and many others
require the user to have a considerable background in PDEs and functional analysis before
one can study the control design methods for these systems, particularly boundary control
design.
Control and observation of coupled parabolic PDEs comes in roughly two settingsdepending
on where the actuators and sensors are located \in domain" control, where
the actuation penetrates inside the domain of the PDE system or is evenly distributed
everywhere in the domain and \boundary" control, where the actuation and sensing are
applied only through the boundary conditions.
Boundary control is generally considered to be physically more realistic because actuation
and sensing are nonintrusive but is also generally considered to be the harder problem,
because the \input operator" and the "output operator" are unbounded operators.
The method that this thesis develops for control of PDEs is the so-called backstepping
control method. Backstepping is a particular approach to stabilization of dynamic
systems and is particularly successful in the area of nonlinear control. The backstepping
method achieves Lyapunov stabilization, which is often achieved by collectively shifting
all the eigenvalues in a favorable direction in the complex plane, rather than by assigning
individual eigenvalues. As the reader will soon learn, this task can be achieved in a rather
elegant way, where the control gains are easy to compute symbolically, numerically, and
in some cases even explicitly.
In addition to presenting the methods for boundary control design, we present the dual
methods for observer design using boundary sensing. Virtually every one of our control
designs for full state stabilization has an observer counterpart. The observer gains are
easy to compute symbolically or even explicitly in some cases. They are designed in
such a way that the observer error system is exponentially stabilized. As in the case of
finite-dimensional observer-based control, a separation principle holds in the sense that a
closed-loop system remains stable after a full state stabilizing feedback is replaced by a
feedback that employs the observer state instead of the plant state.
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Kunert, Gerd. "A posteriori H^1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100730.
Full textAbstract:
The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes.
A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably.
Furthermore three modifications of these estimators are introduced and discussed.
Numerical experiments for all estimators complement and confirm the theoretical results.
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Driver, David Philip. "An optimisation-based approach to FKPP-type equations." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/277769.
Full textAbstract:
In this thesis, we study a class of reaction-diffusion equations of the form $\frac{\partial u}{\partial t} = \mathcal{L}u + \phi u - \tfrac{1}{k} u^{k+1}$ where $\mathcal{L}$ is the stochastic generator of a Markov process, $\phi$ is a function of the space variables and $k\in \mathbb{R}\backslash\{0\}$. An important example, in the case when $k > 0$, is equations of the FKPP-type. We also give an example from the theory of utility maximisation problems when such equations arise and in this case $k < 0$. We introduce a new representation, for the solution of the equation, as the optimal value of an optimal control problem. We also give a second representation which can be seen as a dual problem to the first optimisation problem. We note that this is a new type of dual problem and we compare it to the standard Lagrangian dual formulation. By choosing controls in the optimisation problems we obtain upper and lower bounds on the solution to the PDE. We use these bounds to study the speed of the wave front of the PDE in the case when $\mathcal{L}$ is the generator of a suitable Lévy process.
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Nadukandi, Prashanth. "Stabilized finite element methods for convection-diffusion-reaction, helmholtz and stokes problems." Doctoral thesis, Universitat Politècnica de Catalunya, 2011. http://hdl.handle.net/10803/109155.
Full textAbstract:
We present three new stabilized finite element (FE) based Petrov-Galerkin methods for the convection-diffusionreaction
(CDR), the Helmholtz and the Stokes problems, respectively. The work embarks upon a priori analysis of a
consistency recovery procedure for some stabilization methods belonging to the Petrov- Galerkin framework. It was
ound that the use of some standard practices (e.g. M-Matrices theory) for the design of essentially non-oscillatory
numerical methods is not appropriate when consistency recovery methods are employed. Hence, with respect to
convective stabilization, such recovery methods are not preferred. Next, we present the design of a high-resolution
Petrov-Galerkin (HRPG) method for the CDR problem. The structure of the method in 1 D is identical to the consistent
approximate upwind (CAU) Petrov-Galerkin method [doi: 10.1016/0045-7825(88)90108-9] except for the definitions of
he stabilization parameters. Such a structure may also be attained via the Finite Calculus (FIC) procedure [doi:
10.1 016/S0045-7825(97)00119-9] by an appropriate definition of the characteristic length. The prefix high-resolution is
used here in the sense popularized by Harten, i.e. second order accuracy for smooth/regular regimes and good
shock-capturing in non-regular re9jmes. The design procedure in 1 D embarks on the problem of circumventing the
Gibbs phenomenon observed in L projections. Next, we study the conditions on the stabilization parameters to
ircumvent the global oscillations due to the convective term. A conjuncture of the two results is made to deal with the
problem at hand that is usually plagued by Gibbs, global and dispersive oscillations in the numerical solution. A multi
dimensional extension of the HRPG method using multi-linear block finite elements is also presented.
Next, we propose a higher-order compact scheme (involving two parameters) on structured meshes for the Helmholtz
equation. Making the parameters equal, we recover the alpha-interpolation of the Galerkin finite element method
(FEM) and the classical central finite difference method. In 1 D this scheme is identical to the alpha-interpolation
method [doi: 10.1 016/0771 -050X(82)90002-X] and in 2D choosing the value 0.5 for both the parameters, we recover
he generalized fourth-order compact Pade approximation [doi: 10.1 006/jcph.1995.1134, doi: 10.1016/S0045-
7825(98)00023-1] (therein using the parameter V = 2). We follow [doi: 10.1 016/0045-7825(95)00890-X] for the
analysis of this scheme and its performance on square meshes is compared with that of the quasi-stabilized FEM [doi:
10.1016/0045-7825(95)00890-X]. Generic expressions for the parameters are given that guarantees a dispersion
accuracy of sixth-order should the parameters be distinct and fourth-order should they be equal. In the later case, an
expression for the parameter is given that minimizes the maximum relative phase error in 2D. A Petrov-Galerkin
ormulation that yields the aforesaid scheme on structured meshes is also presented. Convergence studies of the
error in the L2 norm, the H1 semi-norm and the I ~ Euclidean norm is done and the pollution effect is found to be small.Presentamos tres nuevos metodos estabilizados de tipo Petrov- Galerkin basado en elementos finitos (FE) para los problemas de convecci6n-difusi6n- reacci6n (CDR), de Helmholtz y de Stokes, respectivamente. El trabajo comienza con un analisis a priori de un metodo de recuperaci6n de la consistencia de algunos metodos de estabilizaci6n que pertenecen al marco de Petrov-Galerkin. Hallamos que el uso de algunas de las practicas estandar (por ejemplo, la eoria de Matriz-M) para el diserio de metodos numericos esencialmente no oscilatorios no es apropiado cuando utilizamos los metodos de recu eraci6n de la consistencia. Por 10 tanto, con res ecto a la estabilizaci6n de conveccion, no preferimos tales metodos de recuperacion . A continuacion, presentamos el diser'io de un metodo de Petrov-Galerkin de alta-resolucion (HRPG) para el problema CDR. La estructura del metodo en 10 es identico al metodo CAU [doi: 10.1016/0045-7825(88)90108-9] excepto en la definicion de los parametros de estabilizacion. Esta estructura tambien se puede obtener a traves de la formulacion del calculo finito (FIC) [doi: 10.1 016/S0045- 7825(97)00119-9] usando una definicion adecuada de la longitud caracteristica. El prefijo de "alta-resolucion" se utiliza aqui en el sentido popularizado por Harten, es decir, tener una solucion con una precision de segundo orden en los regimenes suaves y ser esencialmente no oscilatoria en los regimenes no regulares. El diser'io en 10 se embarca en el problema de eludir el fenomeno de Gibbs observado en las proyecciones de tipo L2. A continuacion, estudiamos las condiciones de los parametros de estabilizacion para evitar las oscilaciones globales debido al ermino convectivo. Combinamos los dos resultados (una conjetura) para tratar el problema COR, cuya solucion numerica sufre de oscilaciones numericas del tipo global, Gibbs y dispersiva. Tambien presentamos una extension multidimensional del metodo HRPG utilizando los elementos finitos multi-lineales. fa. continuacion, proponemos un esquema compacto de orden superior (que incluye dos parametros) en mallas estructuradas para la ecuacion de Helmholtz. Haciendo igual ambos parametros, se recupera la interpolacion lineal del metodo de elementos finitos (FEM) de tipo Galerkin y el clasico metodo de diferencias finitas centradas. En 10 este esquema es identico al metodo AIM [doi: 10.1 016/0771 -050X(82)90002-X] y en 20 eligiendo el valor de 0,5 para ambos parametros, se recupera el esquema compacto de cuarto orden de Pade generalizada en [doi: 10.1 006/jcph.1 995.1134, doi: 10.1 016/S0045-7825(98)00023-1] (con el parametro V = 2). Seguimos [doi: 10.1 016/0045-7825(95)00890-X] para el analisis de este esquema y comparamos su rendimiento en las mallas uniformes con el de "FEM cuasi-estabilizado" (QSFEM) [doi: 10.1016/0045-7825 (95) 00890-X]. Presentamos expresiones genericas de los para metros que garantiza una precision dispersiva de sexto orden si ambos parametros son distintos y de cuarto orden en caso de ser iguales. En este ultimo caso, presentamos la expresion del parametro que minimiza el error maxima de fase relativa en 20. Tambien proponemos una formulacion de tipo Petrov-Galerkin ~ue recupera los esquemas antes mencionados en mallas estructuradas. Presentamos estudios de convergencia del error en la norma de tipo L2, la semi-norma de tipo H1 y la norma Euclidiana tipo I~ y mostramos que la perdida de estabilidad del operador de Helmholtz ("pollution effect") es incluso pequer'ia para grandes numeros de onda. Por ultimo, presentamos una coleccion de metodos FE estabilizado para el problema de Stokes desarrollados a raves del metodo FIC de primer orden y de segundo orden. Mostramos que varios metodos FE de estabilizacion existentes y conocidos como el metodo de penalizacion, el metodo de Galerkin de minimos cuadrados (GLS) [doi: 10.1016/0045-7825(86)90025-3], el metodo PGP (estabilizado a traves de la proyeccion del gradiente de presion) [doi: 10.1 016/S0045-7825(96)01154-1] Y el metodo OSS (estabilizado a traves de las sub-escalas ortogonales) [doi: 10.1016/S0045-7825(00)00254-1] se recuperan del marco general de FIC. Oesarrollamos una nueva familia de metodos FE, en adelante denominado como PLS (estabilizado a traves del Laplaciano de presion) con las formas no lineales y consistentes de los parametros de estabilizacion. Una caracteristica distintiva de la familia de los metodos PLS es que son no lineales y basados en el residuo, es decir, los terminos de estabilizacion dependera de los residuos discretos del momento y/o las ecuaciones de incompresibilidad. Oiscutimos las ventajas y desventajas de estas tecnicas de estabilizaci6n y presentamos varios ejemplos de aplicacion
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Kunert, Gerd. "A note on the energy norm for a singularly perturbed model problem." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100062.
Full textAbstract:
A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.
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Drissi, Zellaji Mourad. "Méthodes d’agrégation et méthode des familles : analyse théorique et numérique pour des systèmes de réactions-diffusion associés à certains modèles aéronomiques." Besançon, 1994. http://www.theses.fr/1994BESA2065.
Full textAbstract:
Dans ce travail, nous abordons une etude theorique et numerique de modeles mathematiques de type reaction-diffusion-convection issus de l'aeronomie (etude des atmospheres planetaires). Dans les cadres techniques semi-groupe d'une part, formulation variationnelle d'autre part on obtient des proprietes de stabilite uniforme, sur un horizon infini en temps, pour la norme uniforme, a l'aide de fonctions de lyapounov lineaires scalaires ou vectorielles obtenues comme solution d'un petit systeme de reaction-diffusion. On s'est egalement interesse a des criteres de positivite stricte des concentrations etudies, a des proprietes complementaires de stabilite asymptotique pour des systemes plus specifiques (complex-balanced selon horn et jackson). Au plan analyse numerique: l'objectif principal est d'etudier une methode classique de l'aeronomie, la methode des familles, economique en cout de calcul. Nous avons pu replacer cette methode dans un cadre mathematique de methodes multiniveaux du type methodes d'agregations (bien connues en particulier en macro-economie). Ce rapprochement nous permet de generaliser la methode des familles et de la justifier en la reliant au tres classique schema implicite. L'obtention pour ce dernier de la propriete de stabilite inconditionnelle (qui entraine sa convergence) s'effectue justement grace aux fonctions de lyapounov agregees par familles qui assurent dans cette situation de semi-discretisation en temps un role analogue a celui qu'elles jouent precedemment dans le cas continue pour l'etude de la stabilite uniforme. Les resultats numeriques traitent d'un systeme de reaction-diffusion du a turco et witten par la methode des familles et la generalisation que nous proposons
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Patout, Florian. "Analyse asymptotique d'équations intégro-différentielles : modèles d'évolution et de dynamique des populations." Thesis, Lyon, 2019. http://www.theses.fr/2019LYSEN044/document.
Full textAbstract:
Cette thèse est consacrée à l’étude de phénomènes de propagation et de concentration dans des modèles d’équations intégro-différentielles venant de la écologie. On étudie certaines équations de réaction-diffusion non locales apparaissant en dynamique de populations, ainsi que des modèles représentant l’évolution Darwinienne avec un mode de reproduction sexué.Dans une première partie, nous étudions la propagation spatiale pour une équation de réaction-diffusion ou la dispersion opère via un noyau de convolution à queue lourde. Nous mesurons de manière précise l’accélération du front de propagation de la solution. Nous proposons également une échelle adaptée pour mesurer les «petites» mutations. Dans les deux cas nous utilisons le formalisme des équations de Hamilton-Jacobi.Dans un second temps nous étudions un modèle de génétique quantitative, avec un mode de reproduction sexuée. Un petit paramètre mesure la déviation entre le trait des descendants est la moyenne des traits des parents. Dans le régime où ce paramètre est petit nous étudions l’existence de solutions stationnaires, puis le problème de Cauchy lié à ce modèle. Les solutions se concentrent autour des optima de sélection, sous la forme de perturbations de distributions Gaussiennes avec petite variance fixée par le paramètre. Notre analyse généralise le cas linéaire de la reproduction asexuée en utilisant des outils d’analyse perturbative. Enfin dans une dernière partie nous fournissons des simulations numériques et des méthodes mathématiques pour étudier la dynamique interne des équilibres dans le régime de petite variance, pour les deux modes de reproduction : asexué et sexuéThis manuscript tackles propagation and concentration phenomena in different integro-differential equations with a background in ecology. We study non local reaction-diffusion equations from population dynamics, and models for Darwinian evolution with a sexual or asexual mode of reproduction, with a preference for the former.In a first part, we study spatial propagation for a reaction diffusion equation where dispersion acts through a fat tailed kernel. We measure accurately the acceleration of the propagation front of the population. We propose as well a scaling well adapted to “small mutations” when we consider the model in the context of adaptative dynamics. This scaling is very natural following the previous spatial investigation. In both cases we look at the long time behavior and we use the Hamilton-Jacobi framework. Then we turn our attention towards a quantitative genetics model, with a sexual mode of reproduction, imposed by the “infinitesimal operator”. In this non-linear setting, a small parameter tunes the deviation between the phenotypic trait of the offspring and the mean of the traits of the parents. In the regime where this parameter is small, we prove existence of stationary solutions, and their local uniqueness. We also provide an example of non-uniqueness in the case where the selection function admits several extrema. We prove that the solution concentrates around the points of minimum of the selection function. The analysis is carried by the small perturbations of special profiles : Gaussian distributions with small variance fixed by the parameter.We then study the stability of the Cauchy problem associated to the previous model. This time we prove that at all times, for a well prepared initial data, the solutions is arbitrary close to a Gaussian distribution with small variance. The proof follows the framework of the previous : we use perturbative analysis tools, but this time an even more precise description of the correctors is needed and we linearize the equation to obtain it. In a final part we show numerical simulations and different mathematical approaches to study inside dynamics of phenotypic lineages in the regime of small variance, with a moving environement
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Dufourd, Claire Chantal. "Spatio-temporal mathematical models of insect trapping : analysis, parameter estimation and applications to control." Thesis, University of Pretoria, 2016. http://hdl.handle.net/2263/58471.
Full textAbstract:
This thesis provides a mathematical framework for the development of efficient control strategies that satisfy the charters of Integrated Pest Management (IPM) which aims to maintain pest population at a low impact level. This mathematical framework is based on a dynamical system approach and comprises the construction of mathematical models, their theoretical study, the development of adequate schemes for numerical solutions and reliable procedures for parameter identification. The first output of this thesis is the construction of trap-insect spatio-temporal models formulated via advection-diffusion-reaction processes. These models were used to simulate numerically trapping to compare with field data. As a result, practical protocols were identified to estimate pest-population size and distribution as well as its dispersal capacity and parameter values related to the attractiveness of the traps. The second major output of this thesis is the prediction of the impact of a specific control method: mating disruption using a female pheromone and trapping. A compartmental model, formulated via a system of ordinary differential equations, was built based on biological and mating behaviour knowledge of the pest. The theoretical analysis of the model yields threshold values for the dosage of the pheromone above which extinction of the population is ensured. The practical relevance of the results obtained in this thesis shows that mathematical modelling is an essential supplement to experiments in optimizing control strategies.Thesis (PhD)--University of Pretoria, 2016.
Mathematics and Applied Mathematics
PhD
Unrestricted
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Ambrosio, Benjamin. "Propagation d'ondes dans un milieu excitable : simulations numériques et approche analytique." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2009. http://tel.archives-ouvertes.fr/tel-00437402.
Full textAbstract:
Dans cette thèse, on s'intéresse à la compréhension qualitative de systèmes d'EDP de type FitzHugh Nagumo. Elle est basée sur les propriétés excitable et oscillante du système d'EDO de type FitzHugh Nagumo lorsqu'on varie la valeur d'un paramètre. Après avoir analysé les propriétés du système d'EDO, on contruit des systèmes d'EDP par couplage de Réaction Diffusion ou opérateur des ondes. La simulation numérique des systèmes montre l'émergence de patterns complexes pertinents en biologie et physiologie. D'un point de vue mathématique, cela correspond à des attracteurs non triviaux, et divers théorèmes y sont montrés.
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Ali, Olivier. "Etude théorique de la transduction mécano-chimique dans l'adhérence cellulaire." Grenoble 1, 2010. http://www.theses.fr/2010GRENY022.
Full textAbstract:
Les systèmes complexes propres à la biologie moléculaire sont des sujets d'investigations privilégiés pour la physique statistique hors équilibre. En particulier la dynamique des systèmes d'adhérents qui a déjà été l'objet de description théorique. Ces descriptions sont restreintes au comportement des plaques d'adhérence focales matures, dont la durée caractéristique est la dizaine de minutes et où beaucoup d'acteurs moléculaires différents interviennent, notamment le cortex d'actine. Cependant, la question des mécanismes moléculaires précoces, précédant la mise en place de ces structures, reste entière et ouverte. L'objectif de cette thèse est de proposer un modèle de transduction mécano-chimique bidirectionnelle — de l'intérieur de la cellule vers l'extérieur et inversement — en se basant sur le caractère allostérique de l'interaction entre les intégrines (sensibles aux propriétés des matrices extracellulaires) et un partenaire cytoplasmique activable, la taline. Ce travail se divise en trois parties : i) une modélisation du bord cellulaire qui repose sur le calcul du potentiel chimique du partenaire activable et de son cycle d'activation, ii) la résolution numérique et analytique des équations précédemment définies et iii) une évolution du précédent modèle où les intégrines sont laissés libres de diffuser et qui vont dans ce cas là se regrouper dans les zones de fortes contraintesComplex systems observed in molecular biology are subject of major interest for statistical physic out of equilibrium. This is the case in particular of the focal adhesions which has been the center of theoretical investigation. The descriptions of this system have been limited to the understanding of mature focal adhesions that have a life-span superior to ten minutes. However, the understanding of what happen at the early stage remains fully open. The goal of this PhD is to propose a model of bidirectional mecano-chemical transduction process, from the Inside to the outside and reciprocally, based on the allosteric character of the interaction between integrins (sensing the extracellular matrix) and an activable cytoplasmic Partner, talin. This work is divided in three parts : i) a modelisation of the cell border based on the calculation of the chemical potential of the activable Partner and on its cycle of activation, ii) the numerical and analytical resolution of the equations proposed before and iii) an update of the previous model where the integrins have been allowed to diffuse and present in this case the ability to cluster in high constraints regions
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Ghazaryan, Anna R. "Nonlinear convective instability of fronts a case study /." Connect to resource, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1117552079.
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Thesis (Ph.D.)--Ohio State University, 2005.Title from first page of PDF file. Document formatted into pages; contains ix, 176 p.; also includes graphics. Includes bibliographical references (p. 172-176). Available online via OhioLINK's ETD Center
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Kunert, Gerd. "Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100011.
Full textAbstract:
Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation.
To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter.
The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well.
A numerical example supports the analysis of our anisotropic error estimator.
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Contri, Benjamin. "Equations de réaction-diffusion dans un environnement périodique en temps - Applications en médecine." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4711/document.
Full textAbstract:
Cette thèse est consacrée à l'étude d'équations de réaction-diffusion dans un environnement périodique en temps. Ces équations modélisent l'évolution d'une tumeur cancéreuse en présence d'un traitement qui correspond à une immunothérapie dans la première partie du manuscrit, et à une chimiothérapie cytotoxique dans la suite.On considère dans un premier temps des nonlinéarités périodiques en temps pour lesquelles 0 et 1 sont des états d'équilibre linéairement stables. On étudie l'unicité, la monotonie et la stabilité de fronts pulsatoires. On exhibe également des cas d'existence et de non-existence de telles solutions. Dans la deuxième partie de la thèse, on commence par travailler sur des nonlinéarités périodiques en temps qui sont la somme d'une fonction positive traduisant la croissance de la tumeur et d'un terme de mort de cellules cancéreuses du au traitement. On s'intéresse aux états d'équilibres de telles nonlinéarités, et on va déduire de cette étude des propriétés de propagation de perturbations et l'existence de fronts pulsatoires. On raffine ensuite le modèle en considérant des nonlinéarités qui sont la somme d'une fonction asymptotiquement périodique en temps et d'un terme perturbatif. On prouve notamment que les propriétés relatives à la propagation de perturbations restent valables dans ce cadre là. Pour finir, on s'intéresse à l'influence du protocole de traitementThis phD thesis investigates reaction-diffusion equations in a time periodic environment. These equations model the evolution of a cancerous tumor in the presence of a treatment that corresponds to an immunotherapy in the firs part of the manuscript, and to a cytotoxic chemotherapy after. We begin by considering time-periodic nonlinearities for which 0 and 1 are linearly stable equilibrium states. We study uniqueness, monotonicity and stability of pulsating fronts. We also provide some conditions for the existence and non-existence of such solutions.In the second part of the manuscript, we begin by working on time-periodic nonlinearities which are the sum of a positive function which stands for the growth of the tumor in the absence of treatment and of a death term of cancerous cells due to treatment. We are interested in equilibrium states of such nonlinearities, and we will infer from this study spreading properties and existence of pulsating fronts. We then refine the model by considering nonlinearities which are the sum of an asymptotic periodic nonlinearity and of a small perturbation. In particular we prove that the spreading properties remain valid in this case. To finish, we are interested in the influence of the protocol of the treatment
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Fei, Ning Fei. "Studies in reaction-diffusion equations." Thesis, Heriot-Watt University, 2003. http://hdl.handle.net/10399/310.
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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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Kunert, Gerd. "A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes." Doctoral thesis, [S.l. : s.n.], 1999. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10324701.
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Deolmi, Giulia. "Computational Parabolic Inverse Problems." Doctoral thesis, Università degli studi di Padova, 2012. http://hdl.handle.net/11577/3423351.
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This thesis presents a general approach to solve numerically parabolic Inverse Problems, whose underlying mathematical model is discretized using the Finite Element method. The proposed solution is based upon an adaptive parametrization and it is applied specically to a geometric conduction inverse problem of corrosion estimation and to a boundary convection inverse problem of pollution rate estimation.In questa tesi viene presentato un approccio numerico volto alla risoluzione di problemi inversi parabolici, basato sull'utilizzo di una parametrizzazione adattativa. L'algoritmo risolutivo viene descritto per due specici problemi: mentre il primo consiste nella stima della corrosione di una faccia incognita del dominio, il secondo ha come scopo la quanticazione di inquinante immesso in un fiume.
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Jiménez, Oviedo Byron. "Processus d’exclusion avec des sauts longs en contact avec des réservoirs." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4000/document.
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Sy, Mouhamadou. "Etude par microscopie optique des comportements spatio-temporels thermo- et photo-induits et de l’auto-organisation dans les monocristaux à transition de spin." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLV032/document.
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Ce travail de thèse est dédié à la visualisation par microscopie optique des transitions de phases, thermo- et photo-induites dans des monocristaux à transition de spin. L’étude des cristaux du composé [{Fe(NCSe)(py)2}2(m-bpypz)] a permis de montrer la possibilité de contrôler la dynamique de l’interface HS/BS (haut spin/bas spin) par une irradiation lumineuse appliquée sur toute la surface du cristal ou de manière localisée. Les investigations expérimentales menées sur l’effet de l’intensité de la lumière sur la température de transition ont mis en évidence d’une part l’importance du couplage entre le cristal et le bain thermique, et d’autre part le rôle de la diffusion de la chaleur dans le monocristal. En parallèle, un modèle basé sur une description de type Ginzburg-Landau, a permis de mettre sur pied une description de type réaction diffusion des effets spatio-temporels accompagnant la transition de spin dans un monocristal. Celui-ci a permis d’identifier et de comprendre le rôle des paramètres pertinents entrant en jeu dans le contrôle du mouvement de l’interface HS/BS. Les résultats obtenus sont très encourageants et reproduisent avec une grande fidélité les données expérimentales. Cependant l’origine de l’orientation de l’interface HS/BS observée par microscopie optique dans les cristaux du composé [{Fe(NCSe)(py)2}2(m-bpypz)] était restée mystérieuse. Pour résoudre cette question, nous avons développé un modèle électro-élastique qui tient compte du changement de volume au cours de la transition de spin. Ce dernier nous a conduits à analyser l’effet de la symétrie du réseau cristallin et de la forme du cristal sur l’orientation de l’interface élastique. En l’appliquant au composé [{Fe(NCSe)(py)2}2(m-bpypz)], en tenant compte du caractère anisotrope du changement de la maille élémentaire lors du passage HSBS, nous avons réussi à retrouver quantitativement l’orientation du front observée expérimentalement en microscopie optique. Ceci confirme bien le rôle primordial de l’élasticité dans le comportement des matériaux à transition de spin. Des études sous lumière à très basse température nous ont donné la possibilité de suivre en temps réel, l’effet LIESST (Light Induced Excited Spin State Trapping), la re-laxation coopérative du cristal ainsi que l’instabilité photo-induite LITH (Light Induced Thermal Hysteresis). Un monde fascinant est apparu autour de cette dernière, avec la présence de comportements totalement inédits. Ainsi, et pour la première fois, nous avons mis en évidence l’existence de phénomènes d’auto-organisation et de comportements autocatalytiques du front de transition. Cette physique non-linéaire dénote un comportement actif du cristal, par suite d’une subtile préparation autour d’un état instable. Ces comportements rappellent les structures dissipatives de Turing et ouvrent des perspectives fascinantes pour cette thématique, tant sur le plan expérimental que théoriqueThis thesis work is devoted to visualization by optical microscopy of thermo- and photo-induced phase transitions, in switchable spin transition single crystals. The study of crystals of the compound [{Fe (NCSe) (py) 2} 2 (m-bpypz)] showed the possibility to control reversibly the dynamics of the HS/LS interface through a photo-thermal effect generated by an irradiation of the whole crystal or using a spatially localized light spot on the crystal surface. The investigations of the effect of the light intensity on the transition temperature have highlighted the importance of the coupling between the crystal and the thermal bath in these experiments. Concomitantly, we developped a reaction diffusion model allowing to describe and iden-tify the relevant physical parameters involved in the control of the movement of HS/LS interface. The obtained results are very encouraging and reproduce the main features of the experimental data. However the origin of the interface orientation observed by the optical microscopy in the crystal of the compound [{Fe (NCSe) (py) 2} 2 (m-bpypz)] re-mained mysterious, and needed an elastic approach to be handled. At this end, an electro-elastic model including the volume change at the spin transition was developed. By taking into account for the anisotropy of the unit cell deformation at the transition, we were able to reproduce quantitatively the experimental HS/LS interface orientation. This result confirms the crucial role of the lattice symmetry and its elastic properties in the emergence of a stable interface orientation. The last part of the thesis is devoted to the investigation of photo-induced effects at very low temperatures (~10K). There, we visualized for the first time the real time transformation of a single crystal under LIESST (Light Induced Excited Spin State Trapping) effect as well as its subsequent relaxation at higher temperatures. We have also studied the light induced instabilities through investigation on the LITH (Light Induced Thermal Hysteresis) loops. Around the latter, a fascinating world made of nonlinear effects, and patterns formation emerged, recalled the well known Turing structures. These results lead to new horizons that will give access to new theories and original experimental observations that will enrich the topics opening the new avenues to study of nonlinear phenomena in spin crossover solids
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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.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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Maach, Fatna. "Existence pour des systèmes de réaction-diffusion ou quasi linéaires avec loi de balance." Nancy 1, 1994. http://www.theses.fr/1994NAN10121.
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Notre étude concerne des problèmes d'existence (ou de non-existence) pour des systèmes de réaction-diffusion elliptiques quasi linéaires présentant deux propriétés essentielles et fréquentes dans les applications, à savoir: 1) les solutions (éventuelles) sont positives; 2) la masse totale des composants est a priori contrôlée: ceci correspond à une propriété structurelle des termes non linéaires, par exemple que leur somme est négative ou nulle. Pour les systèmes semi-linéaires deux fois deux, c'est-à-dire lorsque les termes non linéaires sont indépendants des gradients et dans le cas ou l'un des composants est de plus a priori contrôlé, nous faisons une étude complète. Nous analysons en particulier l'influence des données au bord relativement à l'existence ou la non-existence des solutions. Nous montrons ainsi, moyennant certaines hypothèses, que pour la plupart des combinaisons de données au bord, on a existence. Des résultats négatifs sont donnés pour les autres types de données au bord. Quand les termes non linéaires dépendent des gradients et quand cette dépendance est sous-quadratique, nous obtenons l'existence de solutions classiques. Nous donnons également un résultat d'existence lorsque les données sont très peu régulières. Nous étudions enfin le cas de croissance quadratique ou sur-quadratique et nous montrons l'existence de solutions classiques si les operateurs de diffusions sont proportionnels
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Akman, Tugba. "Discontinuous Galerkin Methods For Time-dependent Convection Dominated Optimal Control Problems." Master's thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613394/index.pdf.
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Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensional problem. For one-dimensional unconstrained problem, Newton-Conjugate Gradient method with Armijo line-search. For two-dimensional control constrained problem, active-set method is applied. A priori error estimates are derived for full discretized optimal control problem. Numerical results for one and two-dimensional distributed optimal control problems for diffusion convection equations with boundary layers confirm the predicted orders derived by a priori error estimates.
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Högele, Michael Anton. "Metastability of the Chafee-Infante equation with small heavy-tailed Lévy Noise." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2011. http://dx.doi.org/10.18452/16299.
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Wird der Äquator-Pol-Energietransfer als Wärmediffusion berücksichtigt, so gehen Energiebilanzmodelle in Reaktions-Diffusionsgleichungen über, deren Modellfall die (deterministische) Chafee-Infante-Gleichung darstellt. Ihre Lösung besitzt zwei stabile Zustände und mehrere instabile auf der separierenden Mannigfaltigkeit (Separatrix) der stabilen Anziehungsgebiete. Es wird bewiesen, dass die Lösung auf geeignet verkleinerten Anziehungsgebieten mit Minimalabstand zur Separatrix innerhalb von Zeitskalen relaxiert, die höchstens logarithmisch darin anwachsen. Motiviert durch statistische Belege aus grönländischen Zeitreihen wird diese partielle Differentialgleichung unter Störung mit unendlichdimensionalem, Hilbertraum-wertigen, regulär variierenden Lévy''schen reinen Sprungrauschen mit index alpha und Intensität epsilon untersucht. Ein kanonisches Beispiel dieses Rauschens ist alpha-stabiles Rauschen im Hilbertraum. Durch Erweiterung einer Methode von Imkeller und Pavlyukevich auf stochastische partielle Differentialgleichungen wird unter milden Bedingungen bewiesen, dass im Gegensatz zu Gauß''schem Rauschen die erwarteten Austritts- und übertrittszeiten zwischen Anziehungsgebieten polynomiell mit Ordnung in der inversen Intensität für kleine Rauschintensität anwachsen. In Kapitel 6 wird eine zusätzliche natürliche “Separatrixhypothese” über das Sprungmaß, eingeführt, die eine obere Schranke für die Austrittszeiten aus einer Umgebung der Separatrix impliziert. Dies ermöglicht den Nachweis einer oberen Schranke für die Austrittszeiten, welche gleichmäßig für Anfangsbedingungen in dem ganzen Anziehungsgebiet gilt. Es folgen zwei Lokalisierungsergebnisse. Schließlich wird gezeigt, dass die Lösung metastabiles Verhalten aufweist. Unter der “Separatrixhypothese” wird dies auf ein Ergebnis erweitert, welches gleichmäßig im Raum gilt.If equator-to-pole energy transfer by heat diffusion is taken into account, Energy Balance Models turn into reaction-diffusion equations, whose prototype is the (deterministic) Chafee-Infante equation. Its solution has two stable states and several unstable ones on the separating manifold (separatrix) of the stable domains of attraction. We show, that on appropriately reduced domains of attraction of a minimal distance to the separatrix the solution relaxes in time scales increasing only logarithmically in it. Motivated by the statistical evidence from Greenland ice core time series, we consider this partial differential equation perturbed by an infinite-dimensional Hilbert space-valued regularly varying (pure jump) Lévy noise of index alpha and intensity epsilon. A proto-type of this noise is alpha-stable noise in the Hilbert space. Extending a method developed by Imkeller and Pavlyukevich to the SPDE setting we prove under mild conditions that in contrast to Gaussian perturbations the expected exit and transition times between the domains of attraction increase polynomially in the inverse intensity. In Chapter 6 we introduce an additional natural separatrix hypothesis on the jump measure that implies an upper bound on the exit time of a neighborhood of the separatrix. This allows to obtain an upper bound for the asymptotic exit time uniform for the initial positions inside the entire domain of attraction. It is followed by two localization results. Finally we prove that the solution exhibits metastable behavior. Under the separatrix hypothesis we can extend this to a result that holds uniformly in space.
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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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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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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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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 numériquesThis 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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Alqawasmeh, Yousef. "Models for Persistence and Spread of Structured Populations in Patchy Landscapes." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/36845.
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In this dissertation, we are interested in the dynamics of spatially distributed populations. In particular, we focus on persistence conditions and minimal traveling
periodic wave speeds for stage-structured populations in heterogeneous landscapes.
The model includes structured populations of two age groups, juveniles and adults,
in patchy landscapes. First, we present a stage-structured population model, where we divide the population into pre-reproductive and reproductive stages. We assume that all parameters of the two age groups are piecewise constant functions in space. We derive explicit formulas for population persistence in a single-patch landscape and in heterogeneous habitats. We find the critical size of a single patch surrounded by a non-lethal matrix habitat. We derive the dispersion relation for the juveniles-adults model in homogeneous and heterogeneous landscapes. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest. We apply sensitivity and elasticity formulas to the critical size of a single-patch landscape and to the minimal traveling wave speed in a homogeneous landscape.
Secondly, we use asymptotic techniques to find an explicit formula for the traveling
periodic wave speed and to calculate the spread rates for structured populations in
heterogeneous landscapes. We illustrate the power of the homogenization method by comparing the dispersion relation and the resulting minimal wave speeds for the
approximation and the exact expression. We find an excellent agreement between
the fully heterogeneous speed and the homogenized speed, even though the landscape period is on the same order as the diffusion coefficients and not as small as the formal derivation requires. We also generalize this work to the case of structured populations of n age groups.
Lastly, we use a finite difference method to explore the numerical solutions for the
juveniles-adults model. We compare numerical solutions to analytic solutions and
explore population dynamics in non-linear models, where the numerical solution for
the time-dependent problem converges to a steady state. We apply our theory to
study various aspects of marine protected areas (MPAs). We develop a model of
two age groups, juveniles and adults, in which only adults can be harvested and
only outside MPAs, and recruitment is density dependent and local inside MPAs and
fishing grounds. We include diffusion coefficients in density matching conditions at
interfaces between MPAs and fishing grounds, and examine the effect of fish mobility
and bias movement on yield and fish abundance. We find that when the bias towards
MPAs is strong or the difference in diffusion coefficients is large enough, the relative
density of adults inside versus outside MPAs increases with adult mobility. This
observation agrees with findings from empirical studies.