Automatic Qualitative Analysis of Ordinary Differential Equations Using Piecewise Linear Approximations

This paper explores automating the qualitative analysis of physical systems. It describes a program, called PLR, that takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values. PLR approximates intractable nonlinear systems with piecewise linear ones, analyzes the approximations, and draws conclusions about the original systems. It chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes, yet simple enough to be analyzed completely and efficiently. It derives additional properties, such as boundedness or periodicity, by theoretical methods. I demonstrate PLR on several common nonlinear systems and on published examples from mechanical engineering.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

A case study of pre-midlife crisis: an extraordinary transformational episode in an ordinary life

This case study explored a single in-depth narrative of an episode of crisis. The participant, an English Jewish man in his late thirties (Guy), was selected using a ‘random purposeful’ design from a sample who had previously participated in a study on the experience of crisis in pre-midlife adulthood. From a subgroup of participants chosen for giving full accounts of both inner and outer dimensions of crisis, the individual was selected randomly. Data collection comprised two interviews followed by an email discussion. The crisis occurred in Guy’s late thirties, just before the midlife transition, and so can be considered a ‘pre-midlife’ crisis. It subsumed the period surrounding leaving a high-profile banking career and a dysfunctional marriage, and the ensuing attempts to rebuild life after this difficult and emotional period. Qualitative analysis found four trajectories of personal transformation over the course of the episode: Firstly there was a shift away from the use of a conventional persona to a more spontaneous and ‘authentic’ expression of self; secondly there was a move away from materialistic values toward relational values; thirdly a developing capacity to reflect on himself and his actions; fourthly an emerging feminine component of his personality. The case study portrays an extraordinary event in the life of an ordinary man approaching middle age. It illustrates the transformative nature of crisis in ordinary lives, the dramatic nature of narrative surrounding crisis, and also illustrates existing theory about the nature of adult crises.

Some results on solvability of ordinary linear differential equations in locally convex spaces

Let $\Gamma$ be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem $\dot x = Ax$, $x(0) = x_0$ with respect to functions $x: R\to E$. It is proved that if $E\in \Gamma$, then $E\times R^A$ is-an-element-of $\Gamma$ for an arbitrary set $A$. It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to $\omega$, does not belong to $\Gamma$.