Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 / In this study we present the numerical solution o fboundary value problems for the reaction-diffusion equations in 1-d and 2-d that model phenomena such as kinetics and population dynamics.These differential equations are solved nu- merically using the finite element method (FEM).The FEM was chosen due to several desirable properties it possesses and the many advantages it has over other numerical methods.Some of its advantages include its ability to handle complex geometries very well and that it is built on well established Mathemat- ical theory,and that this method solves a wider class of problems than most numerical methods.The Lax-Milgram lemma will be used to prove the existence and uniqueness of the finite element solutions.These solutions are compared with the exact solutions,whenever they exist,in order to examine the accuracy of this method.The adaptive finite element method will be used as a tool for validating the accuracy of theFEM.The convergence of the FEM will be proven only on the real line.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 93-99).

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 50-51). Also available in electronic version. Access restricted to campus users.

Thesis (M. Sc.)--University of Stellenbosch, 2008. / Includes bibliographical references. Also available via the Internet.

Thesis (Ph. D.)--Ohio State University, 2006. / Available online via OhioLINK's ETD Center; full text release delayed at author's request until 2007 Mar 21

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 122-126).

Thesis (Ph. D.)--University of Sydney, 2008. / Includes graphs. Title from title screen (viewed November 28, 2008) Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliographical references. Also available in print form.

Numerous macroscale models of arteries have been developed, comprised of populations of discrete coupled Endothelial Cells (EC) and Smooth Muscle Cells (SMC) cells, an example of which is the model of Shaikh et al. (2012), which simulates the complex biochemical processes responsible for the observed propagating waves of Ca2+ observed in experiments. In a 'homogenised' model however, the length scale of each cell is assumed infinitely small while the population of cells are assumed infinitely large, so that the microscopic spatial dynamics of individual cells are unaccounted for. We wish to show in our study, our hypothesis that the homogenised modelling approach for a particular system can be used to replicate observations of the discrete modelling approach for the same system. We may do this by deriving a homogenised model based on Goldbeter et al. (1990), the simplest possible physiological system, and comparing its results with those of the discrete Shaikh et al. (2012), which have already been validated with experimental findings. We will then analyse the mathematical dynamics of our homogenised model to gain a better understanding of how its system parameters influence the behaviour of its solutions. All our homogenised models are essentially formulated as partial differential equations (PDE), specifically they are of type reaction diffusion PDEs. Therefore before we begin developing the homogenised Goldbeter et al. (1990), we will first analyse the Brusselator PDE with the goal that it will help us to understand reaction diffusion systems better. The Brusselator is a suitable preliminary study as it shares two common properties with reaction diffusion equations: oscillatory solutions and a diffusion term.

A fibrilação, atrial ou ventricular, é caracterizada por uma desorganização da atividade elétrica do músculo. O coração, que normalmente contrai-se globalmente, em uníssono e uniforme, durante a fibrilação contrai-se localmente em várias regiões, de modo descoordenado. Para estudar qualitativamente este fenômeno, é aqui proposto um novo modelo matemático, mais simples do que os demais existentes e que, principalmente, admite uma representação singela na forma de circuito elétrico equivalente. O modelo foi desenvolvido empiricamente, após estudo crítico dos modelos conhecidos, e após uma série de sucessivas tentativas, ajustes e correções. O modelo mostra-se eficaz na simulação dos fenômenos, que se traduzem em padrões espaciais e temporais das ondas de excitação normais e patológicas, propagando-se em uma grade de pontos que representa o tecido muscular. O trabalho aqui desenvolvido é a parte básica e essencial de um projeto em andamento no Departamento de Engenharia Elétrica da EESC-USP, que é a elaboração de uma rede elétrica ativa, tal que possa ser estudada utilizando recursos computacionais de simuladores usualmente aplicados em projetos de circuitos integrados / Atrial and ventricular fibrillation are characterized by a disorganized electrical activity of the cardiac muscle. While normal heart contracts uniformly as a whole, during fibrillation several small regions of the muscle contracts locally and uncoordinatedly. The present work introduces a new mathematical model for the qualitative study of fibrillation. The proposed model is simpler than other known models and, more importantly, it leads to a very simple electrical equivalent circuit of the excitable cell membrane. The final form of the model equations was established after a long process of trial runs and modifications. Simulation results using the new model are in accordance with those obtained using other (more complex) models found in the related literature. As usual, simulations are performed on a two-dimensional grid of points (representing a piece of heart tissue) where normal or pathological spatial and temporal wave patterns are produced. As a future work, the proposed model will be used as the building block of a large active electrical network representing the muscle tissue, in an integrated circuit simulator

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.