With demand for aircraft travel set to double in the next twenty years, targets are in place to reduce noise levels and emissions. For example, one target is that the effective perceived noise from aircraft in 2020 should be half of the 2000 level. One of the key noise components is the aeroengine. Building and designing an aeroengine costs millions of pounds and furthermore, to prove the aeroengine is safe, it has to be tested to destruction. Engineers and mathematicians are employed to design aeroengines that will not only be quieter but more fuel efficient and produce fewer harmful emissions while maintaining or improving performance. The main topic of this thesis is investigating rotor-stator interaction which occurs when the turbulent, swirling air produced by the rotor hits the stator and generates noise. We do this in two distinct ways, firstly we calculate the Green’s function for pressure in a turbofan duct with swirling mean flow and secondly we investigate the effect of turbulence hitting an isolated aerofoil. The Green’s function allows engineers to calculate the noise from rotor-stator interaction in simple cases and can be used in beamforming to analyse noise sources in the aeroengine. We consider an infinite duct, and use the Euler equations to derive a sixth order partial differential equation for pressure in the duct. We then find a Green’s function of this equation, which can be done numerically or analytically using high-frequency asymptotics. Our main interest is the analytic Green’s function, which we compare to numerical results. We begin by assuming the base flow has shear and swirling components in a constantly lined duct, and our analytic Green’s function is a new result. We then calculate the Green’s function for a base flow with variable entropy and a lining that varies with circumferential position. To consider flow-blade interaction we simulate the turbulent wake of the rotor hitting a single stator blade. Tests in wind tunnels have shown that, depending on the parameters, introducing a serration on the leading edge of the aerofoil can reduce the noise significantly. We build an analytical model to investigate the effect of the serrated edge, which again involves solving a differential equation by using a Green’s function. It also requires modelling the turbulence, which we do by using either deterministic eddies or stochastic eddies. We show it is possible to reduce the noise by using a serrated leading edge, but it is hard to predict the correct choice of serration to minimise the noise.

In this thesis we present certain spaces of analytic functions on the complex half-plane, including the Hardy, the Bergman spaces, and their generalisation: Zen spaces. We use the latter to construct a new type of spaces, which include the Dirichlet and the Hardy-Sobolev spaces. We show that the Laplace transform defines an isometric map from the weighted L^2(0, ∞) spaces into these newly-constructed spaces. These spaces are reproducing kernel Hilbert spaces, and we employ their reproducing kernels to investigate their features. We compare corresponding spaces on the disk and on the half-plane. We present the notions of Carleson embeddings and Carleson measures and characterise them for the spaces introduced earlier, using the reproducing kernels, Carleson squares and Whitney decomposition of the half-plane into an abstract tree. We also study multiplication operators for these spaces. We show how the Carleson measures can be used to test the boundedness of these operators. We show that if a Hilbert space of complex valued functions is also a Banach algebra with respect to the pointwise multiplication, then it must be a reproducing kernel Hilbert space and its kernels are uniformly bounded. We provide examples of such spaces. We examine spectra and character spaces corresponding to multiplication operators. We study weighted composition operators and, using the concept of causality, we link the boundedness of such operators on Zen spaces to Bergman kernels and weighted Bergman spaces. We use this to show that a composition operator on a Zen space is bounded only if it has a finite angular derivative at infinity. We also prove that no such operator can be compact. We present an application of spaces of analytic functions on the half-plane in the study of linear evolution equations, linking the admissibility criterion for control and observation operators to the boundedness of Laplace-Carleson embeddings.

No description available.

The analysis of the p- and hp-versions of the finite element methods has been studied in much detail for the Hilbert spaces W1,2 (omega). The following work extends the previous approximation theory to that of general Sobolev spaces W1,q(Q), q 1, oo . This extension is essential when considering the use of the p and hp methods to the non-linear a-Laplacian problem. Firstly, approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces W1,q(Q) are given. This analysis shows that the traditional view of avoiding the use of high order polynomial finite element methods is incorrect, and that the rate of convergence of the p version is always at least that of the h version (measured in terms of number of degrees of freedom). It is also shown that, if the solution has certain types of singularity, the rate of convergence of the p version is twice that of the h version. Numerical results are given, confirming the results given by the approximation theory. The p-version approximation theory is then used to obtain the hp approximation theory. The results obtained allow both non-uniform p refinements to be used, and the h refinements only have to be locally quasiuniform. It is then shown that even when the solution has singularities, exponential rates of convergence can be achieved when using the /ip-version, which would not be possible for the h- and p-versions.

In this thesis we develop and implement a new algorithm for finding the solutions of linear Hamiltonian systems arising from ordinary differential equation (ODE) eigenproblems; a large source of these systems is Sturm-Liouville problems, and these will provide the angle of approach. The convergence properties of the algorithm will be analysed, as will the performance of the algorithm for large values of eigenparameter. An algorithm is also proposed to find high-index eigenvalues of general Sturm-Liouville problems.

Elementary properties of continued fractions are derived from sets of three-term recurrence relations and approximation methods are developed from this simple approach. First, a well-known method for numerical inversion of Laplace transforms is modified in two different ways to obtain exponential approximations. Differential-difference equations arising from certain Markov processes are solved by direct application of continued fractions and practical error estimates are obtained. Approximations of a slightly different form are then derived for certain generalised hypergeometric functions using those hypergeometric functions that satisfy three-term recurrence relations and have simple continued fraction expansions. Error estimates are also given in this case. The class of corresponding sequence algorithms is then described for the transformation of power series into continued fraction form. These algorithms are shown to have very general application and only break down if the required continued fraction does not exist. A continued fraction in two variables is then shown to exist and its correspondence with suitable double power series made feasible by the generalisation of the corresponding sequence method. A convergence theorem, due to Van Vleck, is adapted for use with this type of continued fraction and a comparison is made with Chisholm rational approximants in two variables. Some of these ideas are further generalised to the multivariate case. Such corresponding fractions are closely related to other fractions that may be used for point-wise bivariate or multivariate interpolation to function values known on a mesh of points. Interpolation algorithms are described and advantages and limitations discussed. The work presented forms a basis for a wide range of further research and some possible applications in numerical mathematics are indicated.

The work in this thesis is concerned with three subclasses of the string algebras: domestic string algebras, gentle algebras and derived-discrete algebras (of non-Dynkin type). The various questions we answer are linked by the theme of the Krull-Gabriel dimension of categories of functors. We calculate the Cantor-Bendixson rank of the Ziegler spectrum of the category of modules over a domestic string algebra. Since there is no superdecomposable module over a domestic string algebra, this is also the value of the Krull-Gabriel dimension of the category of finitely presented functors from the category of finitely presented modules to the category of abelian groups. We also give a description of a basis for the spaces of homomorphisms between pairs of indecomposable complexes in the bounded derived category of a gentle algebra. We then use this basis to describe the Hom-hammocks involving (possibly infinite) string objects in the homotopy category of complexes of projective modules over a derived-discrete algebra. Using this description, we prove that the Krull-Gabriel dimension of the category of coherent functors from a derived-discrete algebra (of non-Dynkin type) is equal to 2. Since the Krull-Gabriel dimension is finite, it is equal to the Cantor-Bendixson rank of the Ziegler spectrum of the homotopy category and we use this to identify the points of the Ziegler spectrum. In particular, we prove that the indecomposable pure-injective complexes in the homotopy category are exactly the string complexes. Finally, we prove that every indecomposable complex in the homotopy category is pure-injective, and hence is a string complex.

This thesis examines the patterns in one spatial dimension that arise from the study of two predator-prey reaction diffusion systems that includes nonlocal terms. The first model considered was one proposed in Gourley & Britton (1996). However, this model was shown to have serious flaws and consequently a second, improved model was proposed. It is this second model that is the main focus of this thesis. A spatially uniform solution with constant, nonzero numbers of predator and prey exists. A linear stability study of this solution shows that both Hopf and Turing bifurcations can occur along with several types of codimension two bifurcation points. Weakly nonlinear analysis of the Hopf and Turing bifurcations are performed and the results compared with computations. An investigation of the of the codimension two points is also made: one is a Takens-Bogdanov point where both Hopf and steady-state bifurcations meet and the second where Hopf and steady-state bifurcations onset with wavenumber in the ratio 1:2.

No description available.

Here we consider a range of Laplacian-based dynamics on graphs such as dynamical invariance and coarse-graining, and node-specific properties such as convergence, observability and consensus-value prediction. Firstly, using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterise convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions. We show that the EEP with respect to a node can reveal nodes in the graph with increased rate of asymptotic convergence to the consensus value as characterised by the second smallest eigenvalue of the quotient Laplacian. Secondly, we extend this characterisation of the relationship between the EEP and Laplacian based dynamics to study the synchronisation of coupled oscillator dynamics on networks. We show that the existence of a non-trivial EEP describes partial synchronisation dynamics for nodes within cells of the partition. Considering linearised stability analysis, the existence of a nontrivial EEP with respect to an individual node can imply an increased rate of asymptotic convergence to the synchronisation manifold, or a decreased rate of de-synchronisation, analogous to the linear consensus case. We show that high degree 'hub' nodes in large complex networks such as Erdös-Rényi, scale free and entangled graphs are more likely to exhibit such dynamical heterogeneity under both linear consensus and non-linear coupled oscillator dynamics. Finally, we consider a separate but related problem concerning the ability of a node to compute the final value for discrete consensus dynamics given only a finite number of its own state values. We develop an algorithm to compute an approximation to the consensus value by individual nodes that is ε close to the true consensus value, and show that in most cases this is possible for substantially less steps than required for true convergence of the system dynamics. Again considering a variety of complex networks we show that, on average, high degree nodes, and nodes belonging to graphs with fast asymptotic convergence, approximate the consensus value employing fewer steps.