This thesis investigates the convergence of numerical schemes for the computation of equilibrium solutions. These are solutions of evolutionary PDEs that arise from (bounded, non-decaying) boundary forcing after the dissipation of any (initial data dependent) transients. A rigorous definition of the term 'equilibrium solution' is given. Classes of evolutionary PDEs for which equilibrium solutions exist uniquely are identified. The uniform well-posedness of equilibrium problems is also investigated. Equilibrium solutions may be approximated by evolutionary initialization: that is, by finding the solution of an initial boundary value problem, with arbitrary initial data, over a period of time t ϵ [0,T]. If T is chosen large enough, the analytic transient will be small, and the analytic solution over t ϵ [T, T + T₀] will be a good approximation to the analytic equilibrium solution. However, in numerical computations, T must be chosen so that the analytical transient is small in comparison with the numerical error E_h, which depends on the fineness of the grid h. Thus T = T_h, and, in general, T_h→∞ as h→0. Convergence is required over t ϵ [T_h,T_h + T₀]. The existing Lax-Richtmyer and GKS convergence theories cannot ensure convergence over such increasing periods of time. Furthermore, neither of these theories apply when the forcing does not decay. Consequently, these theories are of little help in predicting the convergence of finite difference methods for the computation of equilibrium solutions. For these reasons, a new definition of stability - uniform stability — is proposed. Uniformly stable, consistent, finite difference schemes, for uniformly well posed problems, converge uniformly over t ≥ 0. Uniformly convergent schemes converge for bounded and nondecaying forcing. Finite difference schemes for hyperbolic PDEs may admit waves of zero group velocity, even when the underlying analytic problem does not. Such schemes may be GKS convergent, provided that the boundary conditions exclude these waves. The deficiency of the GKS theory for equilibrium computations is traced to this fact. However, uniform stability finds schemes that admit waves of zero group velocity to be (weakly) unstable, regardless of the boundary conditions. It is also shown that weak uniform instabilities are the result of time-dependent analogues of the 'spurious modes' that occur in steady-state calculations. In addition, uniform stability theory sheds new light on the phenomenon of spurious modes.

The author studies two coupled reaction-diffusion equations of 'λ-w' type, on an open, bounded, convex domain Ω C R(^d) (d ≤ 3), with a boundary of class C², and homogeneous Neumann boundary conditions. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics, and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions and compactness arguments. The work provides a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations. The author also undertook the numerical analysis of the reaction-diffusion system. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case. Semi-discrete and fully-discrete error estimates are proved after establishing a priori bounds for various norms of the approximate solutions. Finally, the theoretical results are illustrated and verified via the numerical simulation of periodic plane waves in one space dimension, and preliminary results representing target patterns and spiral solutions presented in two space dimensions.

This thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has significant influence on the analytical prop¬erties of an equation. However, to date, analytical methods are of only limited practical use. Therefore this thesis focuses on the development of a reliable new method, based on finite order approximations of the underlying infinite dimen¬sional problem, which can detect small solutions. Decisions (concerning the existence, or otherwise, of small solutions) based on our visualisation technique require an understanding of the underlying methodol¬ogy behind our approach. Removing this need would be attractive. The method we have developed can be automated, and at the end of the thesis we present a prototype Matlab code for the automatic detection of small solutions to delay differential equations.

This thesis is concerned with an investigation and evaluation of collocation algorithms for solving two-point boundary value problems for systems of ordinary differential equations. An emphasis is on developing reliable and efficient adaptive mesh selection algorithms in piecewise collocation methods. General background materials including basic concepts and descriptions of the method as well as some functional analysis tools needed in developing some error estimates are given at the beginning. A brief review of some developments in the methods to be used is provided for later referencing. By utilising the special structure of the collocation matrices, a more compact block matrix structure is introduced and an algorithm for generating and solving the matrix is proposed. Some practical aspects and computational considerations of matrices involved in the collocation process such as analysis of arithmetic operations and amount of memory spaces needed are considered. An examination of scaling process to reduce the condition number is also presented. A numerical evaluation of some error estimates developed by considering the differential operator, the related matrices and the residual is carried out. These estimates are used to develop adaptive mesh selection algorithms, in particular as a cheap criterion for terminating the computation process. Following a discussion on mesh selection strategies, a criterion function for use in adaptive algorithms is introduced and a numerical scheme to equidistributing values of the criterion function is proposed. An adaptive algorithm based on this criterion is developed and the results of numerical experiments are compared with those using some well known criterion functions. The various examples are chosen in such a way that they include problems with interior or boundary layers. In addition, an algorithm has been developed to predict the necessary number of subintervals for a given tolerance, with the aim of improving the efficiency of the whole process. Using a good initial mesh in adaptive algorithms would be expected to provide some further improvement in the algorithms. This leads to the idea of locating the layer regions and determining suitable break points in such regions before the numerical process. Based on examining the eigenvalues of the coefficient matrix in the differential equation in the specified interval, using their magnitudes and rates of change, the algorithms for predicting possible layer regions and estimating the number of break points needed in such regions are constructed. The effectiveness of these algorithms is evaluated by carrying out a number of numerical experiments. The final chapter gives some concluding remarks of the work and comment on results of numerical experiments. Certain possible improvements and extensions for further research are also briefly given.

Thesis (Ph. D.)--University of Hawaii at Manoa, 1996. / Includes bibliographical references (leaves 175). / Microfiche. / x, 175 leaves, bound ill. 29 cm

[Typescript] / Includes bibliography. / 170 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematics, 1967

vi, 133 leaves / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1972

Thesis (Ph.D.) -- McMaster University, 2001. / Includes bibliographical references. Also available via World Wide Web.

Lic.-avh. (sammanfattning) Luleå : Luleå tekniska univ., 2005. / Härtill 4 uppsatser.

Thesis (M. Sc.)--University of Adelaide, Dept. of Applied Mathematics, 1992. / Includes bibliographical references (leaves 79-80).