Spelling suggestions: "subject:"reactiondiffusion equation"" "subject:"reaktionsdiffusion equation""
1 |
Nonclassical symmetry reductions and conservation laws for reaction-diffusion equations with application to population dynamicsLouw, Kirsten 29 May 2015 (has links)
A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 2015. / This dissertation analyses the reaction-di usion equations, in particular the modi ed
Huxley model, arising in population dynamics. The focus is on determining the classical
Lie point symmetries, and the construction of the conservation laws and group-invariant
solutions for reaction-di usion equations. The invariance criterion for determination
of classical Lie point symmetries results in a system of linear determining equations
which can be solved analytically. Furthermore, the Lie point symmetries associated
with the conservation laws are determined. Reductions by associated Lie point symmetries
are carried out. Nonclassical symmetry techniques are also employed. Here
the invariance criterion for symmetry determination results in a system of nonlinear
determining equations which may be solved albeit di cult. Nonclassical symmetries
results in exact solutions which may not be constructed by classical Lie point symmetries.
The highlight in construction of exact solution using nonclassical symmetries
is the introduction of the modi ed Hopf-Cole transformation. In this dissertation, the
di usion term and the coe cient of the source term are given as quadratic functions
of space variable in one case, and the coe cient as the generalised power law in the
other. These equations admit a number of classical Lie point symmetries. The genuine
nonclassical symmetries are admitted when the source term of the reaction-di usion
equation is a cubic.
|
2 |
Identification of Coefficients in Reaction-Diffusion EquationsYu, Weiming 31 March 2004 (has links)
No description available.
|
3 |
Propagation Failure in Discrete Inhomogeneous Medium Using a Caricature of the CubicLydon, Elizabeth 01 January 2015 (has links)
Spatially discrete Nagumo equations have widespread physical applications, including modeling electrical impulses traveling through a demyelinated axon, an environment typical in multiple scle- rosis. We construct steady-state, single front solutions by employing a piecewise linear reaction term. Using a combination of Jacobi-Operator theory and the Sherman-Morrison formula we de- rive exact solutions in the cases of homogeneous and inhomogeneous diffusion. Solutions exist only under certain conditions outlined in their construction. The range of parameter values that satisfy these conditions constitutes the interval of propagation failure, determining under what circumstances a front becomes pinned in the media. Our exact solutions represent a very specific solution to the spatially discrete Nagumo equation. For example, we only consider inhomogeneous media with one defect present. We created an original script in MATLAB which algorithmically solves more general cases of the equation, including the case for multiple defects. The algorithmic solutions are then compared to known exact solutions to determine their validity.
|
4 |
Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux / Mathematical and numerical study of some multi-scale models from materials scienceJosien, Marc 20 November 2018 (has links)
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 apparaissent / In 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
|
5 |
The Origin of Wave Blocking for a Bistable Reaction-Diffusion Equation : A General ApproachRoy, Christian 12 April 2012 (has links)
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.
|
6 |
The Origin of Wave Blocking for a Bistable Reaction-Diffusion Equation : A General ApproachRoy, Christian 12 April 2012 (has links)
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.
|
7 |
Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshesKunert, Gerd 09 November 2000 (has links) (PDF)
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.
|
8 |
A note on the energy norm for a singularly perturbed model problemKunert, Gerd 16 January 2001 (has links) (PDF)
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.
|
9 |
The Effects of Mixing, Reaction Rate and Stoichiometry on Yield for Mixing Sensitive ReactionsShah, Syed Imran A. Unknown Date
No description available.
|
10 |
The Origin of Wave Blocking for a Bistable Reaction-Diffusion Equation : A General ApproachRoy, Christian 12 April 2012 (has links)
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.
|
Page generated in 0.696 seconds