Series expressed in terms of Chebyshev polynomials are applied using Lie series to the iterative solution of ordinary differential equations. After a discussion of initial value problems, the method is then used to solve two-point boundary value problems and an improved method of shooting type is derived and tested.

The understanding of complex wave phenomenon, such as multiple scattering in heterogeneous media, is often hindered by lack of equations modelling the exact physics. Use of approximate numerical methods, such as Finite Element Method (FEM) and Boundary Element Method (BEM), is therefore needed to understand these complex wave problems. FEM is known for its ability to accurately model the physics of the problem but requires truncating the computational domain. On the other hand, BEM can accurately model waves in unbounded region but is suitable for homogeneous media only. Coupling FEM and BEM therefore is a natural way to solve problems involving a relatively small heterogeneity (to be modelled with FEM) surrounded by an unbounded homogeneous medium (to be modelled with BEM). The use of a classical FEM-BEM coupling can become computationally demanding due to high mesh density requirement at high frequencies. Secondly, BEM is an integral equation based technique and suffers from the problem of non-uniqueness. To overcome the requirement of high mesh density for high frequencies, a technique known as the ‘Partition of Unity’ (PU) method has been developed by previous researchers. The work presented in this thesis extends the concept of PU to BEM (PUBEM) while effectively treating the problem of non-uniqueness. Two of the well-known methods, namely CHIEF and Burton-Miller approaches, to overcome the non-uniqueness problem, are compared for PUBEM. It is shown that the CHIEF method is relatively easy to implement and results in at least one order of magnitude of improvement in the accuracy. A modified ‘PU’ concept is presented to solve the heterogeneous problems with the PU based FEM (PUFEM). It is shown that use of PUFEM results in close to two orders of magnitude improvement over FEM despite using a much coarser mesh. The two methods, namely PUBEM and PUFEM, are then coupled to solve the heterogeneous wave problems in unbounded media. Compared to PUFEM, the coupled PUFEM-PUBEM apporach is shown to result between 30-40% savings in the total degress of freedom required to achieve similar accuracy.

Type systems were invented in the early 1900s to provide foundations for Mathematics where types were used to avoid paradoxes. Type systems have then been developed and extended throughout the years to serve different purposes such as efficiency or expressiveness. The λ-calculus is used in programming languages, logic, mathematics, and linguistics. Intersection types are a kind of types used for building semantic models of the λ-calculus and for static analysis of computer programs. The confluence property was used to prove the λ-calculus’ consistency and the uniqueness of normal forms. Confluence is useful to show that logics are sensibly designed, and to make equality decision procedures for use in theorem provers. Some proofs of the λ-calculus’ confluence are based on syntactic concepts (reduction relations and λ-term sets) and some on semantic concepts (type interpretations). Part I of this thesis presents an original syntactic proof that is a simplification of a semantic proof based on a sound type interpretation w.r.t. an intersection type system. Our proof can be seen as bridging some semantic and syntactic proofs. Expansion is an operation on typings (pairs of type environments and result types) in type systems for the λ-calculus. It was introduced to prove that the principal typing property (i.e., that every typable term has a strongest typing) holds in intersection type systems. Expansion variables were introduced to simplify the expansion mechanism. Part II of this thesis presents a complete realisability semantics w.r.t. an intersection type system with infinitely many expansion variables. This represents the first study on semantics of expansion. Providing sound (and complete) realisability semantics allows one to study the algorithmic behaviour of typed λ-terms through their types w.r.t. a type system. We believe such semantics will cast some light on the not yet well understood expansion operation. Intersection types were used in a type error slicer for the SML programming language. Existing compilers for many languages have confusing type error messages. Type error slicing (TES) helps the programmer by isolating the part of a program contributing to a type error (a slice). TES was initially done for a tiny toy language (the λ-calculus with polymorphic let-expressions). Extending TES to a full language is extremely challenging, and for SML we needed a number of innovations. Some issues would be faced for any language, and some are SML-specific but representative of the complexity of language-specific issues likely to be faced for other languages. Part III of this thesis solves both kinds of issues and presents an original, simple, and general constraint system for providing type error slices for ill-typed programs. We believe TES helps demystify language features known to confuse users.

This thesis consists of two distinct parts which deal with two-point boundary value problems and parabolic problems, respectively. In Section 1 we examine the numerical solution of a two-point boundary value problem by a collocation method based on the consistency relationship of regular splines. An existence and convergence result is established which generalises the 0(h^2) convergence result of the cubic spline collocation scheme for the problem in question. Contrary to most previously documented finite element schemes this method employs splines that may be non-linear in structure. Consequently, by a judicious choice of regular spline, the dominant terms of the true solution may be imitated more accurately than by the conventional polynomial based splines. The scheme is implemented by an algorithm that examines the suitability of various classes of regular splines and determines the subsequent deployment of them. The second section investigates semi-discrete finite element schemes for approximating the linear parabolic equation. A standard finite element discretization is employed for the space variable whilst an A0-stable, linear multistep, multiderivative discretization scheme, (L.M.S.D.) is used in time. We consider both the homogeneous and the nonhomogeneous linear parabolic equations and derive optimal convergence results for the above schemes. The convergence results achieved with a k-step L.M.S.D. scheme, incorporating the first m derivatives, generalise and extend the studies of several authors who concentrate on the particular cases of linear multistep formulae, m-l, and one-step schemes, k=1. Ao-stable L.M.S.D. 's are constructed and their implementation procedures examined. The suitability of selecting a L.M.S.D. method, with m, k>1, in a semi-discrete Galerkin scheme is discussed, and its superiority over semi-discrete Galerkin schemes, that incorporate linear multistep or one-step formulae, is confirmed in several aspects. Finally, a class of quasi-linear parabolic equations is solved by a semi-discrete Galerkin scheme that is third order accurate in time. This method is based on a particular third order L.M.S.D. scheme and requires the solution of linearly algebraic systems of equations at each time level. Thus, we improve on all the previously documented linearised schemes as they are only second order accurate in time. All the schemes described in Section 2 are unconditionally stable.

In this thesis we have proposed the meshless adaptive method by radial basis functions (RBFs) for the solution of the time-dependent partial differential equations (PDEs) where the approximate solution is obtained by the multiquadrics (MQ) and the local scattered data reconstruction has been done by polyharmonic splines. We choose MQ because of its exponential convergence for sufficiently smooth functions. The solution of partial differential equations arising in science and engineering, frequently have large variations occurring over small portion of the physical domain, the challenge then is to resolve the solution behaviour there. For the sake of efficiency we require a finer grid in those parts of the physical domain whereas a much coarser grid can be used otherwise. During our journey, we come up with different ideas and have found many interesting results but the main motivation for the one-dimensional case was the Korteweg-de Vries (KdV) equation rather than the common test problems. The KdV equation is a nonlinear hyperbolic equation with smooth solutions at all times. Furthermore the methods available in the literature for solving this problem are rather fully implicit or limited literature can be found using explicit and semi-explicit methods. Our approach is to adaptively select the nodes, using the radial basis function interpolation. We aimed in, the extension of our method in solving two-dimensional partial differential equations, however to get an insight of the method we developed the algorithms for one-dimensional PDEs and two-dimensional interpolation problem. The experiments show that the method is able to track the developing features of the profile of the solution. Furthermore this work is based on computations and not on proofs.

Theoretical solutions to a time optimal problem for a charging process using two heat sources and a minimal system model have been obtained. These solutions have included different heat source ratios and have been applied to a system with simple external heat exchanger and also to a perfectly mixed system. The optimal solutions have been obtained by the application of Pontryagin's maximum principle. It is shown that in using minimal system models the charging policy may show time reductions of up to 10.2 per cent over the best constant charging rate policy. A dual computer controller/process-Simula tor system has been developed, enabling comprehensive simulation studies, under wide ranging system conditions, to be made. The controller was developed in such a manner as to enable it to be transferred to the rig without further modification. The controller used a 'search' and 'control' strategy in order to operate the process at a maximum Hamiltonian function value. However, because of the step-wise nature of the approximation to the charge rate, which was used to implement the 'control' and 'search' policy, the overall control must be regarded as sub-optimal. Results obtained from an experimental rig showed that although all implementations of the control strategy were an improvement over the best constant charge rate policy, the 'optimal' results were not reproducable owing to difficulties in achieving tight control over the performance of the long tubular external heat exchanger. Results from the simulations and practical implementations are discussed and comparisons with other work have been made. Suggestions for further studies are also given.

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Using suitable coupled Navier-Stokes Equations for an incompressible Newtonian fluid we investigate the linear and non-linear steady state solutions for both a homogeneously and a laterally heated fluid with finite Prandtl Number (Pr=7) in the vertical orientation of the channel. Both models are studied within the Large Aspect Ratio narrow-gap and under constant flux conditions with the channel closed. We use direct numerics to identify the linear stability criterion in parametric terms as a function of Grashof Number (Gr) and streamwise infinitesimal perturbation wavenumber (making use of the generalised Squire’s Theorem). We find higher harmonic solutions at lower wavenumbers with a resonance of 1:3exist, for both of the heating models considered. We proceed to identify 2D secondary steady state solutions, which bifurcate from the laminar state. Our studies show that 2D solutions are found not to exist in certain regions of the pure manifold, where we find that 1:3 resonant mode 2D solutions exist, for low wavenumber perturbations. For the homogeneously heated fluid, we notice a jump phenomenon existing between the pure and resonant mode secondary solutions for very specific wavenumbers .We attempt to verify whether mixed mode solutions are present for this model by considering the laterally heated model with the same geometry. We find mixed mode solutions for the laterally heated model showing that a bridge exists between the pure and 1:3 resonant mode 2D solutions, of which some are stationary and some travelling. Further, we show that for the homogeneously heated fluid that the 2D solutions bifurcate in hopf bifurcations and there exists a manifold where the 2D solutions are stable to Eckhaus criterion, within this manifold we proceed to identify 3D tertiary solutions and find that the stability for said 3D bifurcations is not phase locked to the 2D state. For the homogeneously heated model we identify a closed loop within the neutral stability curve for higher perturbation wavenumubers and analyse the nature of the multiple 2D bifurcations around this loop for identical wavenumber and find that a temperature inversion occurs within this loop. We conclude that for a homogeneously heated fluid it is possible to have abrup ttransitions between the pure and resonant 2D solutions, and that for the laterally heated model there exist a transient bifurcation via mixed mode solutions.

The goal of ecology is to investigate the interactions among organisms and their environment. However, ecological systems often exhibit complex dynamics. The application of mathematics to ecological problems has made tremendous progress over the years and many mathematical methods and tools have been developed for the exploration, whether analytical or numerical, of these dynamics. Mathematicians often study ecological systems by modelling them with partial differential equations (PDEs). Calculating the stability of solutions to these PDE systems is a classical question. This thesis first explores the concept of stability in the context of predator-prey invasions. Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised. Vegetation pattern formation in water-limited environments is another topic where stability theory plays a key role; indeed in mathematical models, these patterns are often the results of the dynamics that arise from perturbations to an unstable homogeneous steady state. Vegetation patterns are widespread in semi-deserts and aerial photographs of arid and semi-arid ecosystems have shown several kilometers square of these patterns. On hillsides in particular, vegetation is organised into banded spatial patterns. We first choose a domain in parameter space and calculate the boundary of the region in parameter space where pattern solutions exist. Finally we conclude with investigating how changes in the mean annual rainfall affect the properties of pattern solutions. Our work also highlights the importance of research on the calculation of the absolute spectrum for non-constant solutions.