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Contributions to the theory of generalised hypergeometric seriesAgarwala, R. P. January 1953 (has links)
This thesis deals with various aspects of development in the field of both ordinary and basic hypergeometric functions. It comprises six chapters. The first chapter gives a brief survey of some of the recents developments in this field including the work done in the present thesis. The second chapter gives a systematic study of the transformations connected with the partial sums of generalised hypergeometric series, both ordinary and basic. The main theorem proved in the chapter gives the most general relation of its type. The third chapter is concerned with the development of the transformation theory of bilateral cognate trigonometrical series and generalises all the known results in that field. The fourth chapter gives the integral analogues of some of the transformations of basic series analogous to those for the ordinary series in Chapter VI of Bailey's Cambridge Tract. The fifth chapter deals with a systematic classification and study of two and three term relations between special kinds of well-poised series of the type and gives a new method, by integrals, of deducing these transformations easily. The last and the sixth chapter gives a number of identities involving basic analogues of Appell's hypergeometric functions of two variables and some associated functions.
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Some new approximations for the solution of differential equationsMason, J. C. January 1965 (has links)
No description available.
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Reducing the index of differential-algebraic equations by exploiting underlying structuresMcKenzie, Ross January 2016 (has links)
Differential-algebraic equations arise from the equation based modelling of physical systems, such as those found for example in engineering or physics. This thesis is concerned with square, sufficiently smooth, potentially non-linear differential-algebraic equations. Differential-algebraic equations can be classified by their index. This is a measure of how far a differential-algebraic equation is from an equivalent ordinary differential equation. To solve a differential-algebraic equation one usually transforms the problem to an ordinary differential equation, or something close to one, via an index reduction algorithm. This thesis examines how the index reduction (using dummy derivatives) of differential-algebraic equations can be improved via structural analysis, specifically the Signature Matrix method. Improved and alternative algorithms for finding dummy derivatives are presented and then a new algorithm for finding globally valid universal dummy derivatives is presented. It is also shown that the structural index of a differential-algebraic equation is invariant under order reduction.
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Analysis of differential-delay equations for biologyEzeofor, Victory S. January 2017 (has links)
In this thesis, we investigate the role of time delay in several differential-delay equation focusing on the negative autogenous regulation. We study these models for little or no delay to when the model has a very large delay parameter. We start with the logistic differential-delay equation applying techniques that would be used in subsequent chapters for other models being studied. A key goal of this research is to identify where the structure of the system does change. First, we investigate these models for critical point and study their behaviour close to these points. Of keen interest is the Hopf bifurcation points where we analyse the parameter associated with the Hopf point. The weakly nonlinear analysis carried out using the method of multiple time scale is used to give more insight to these model. The centre manifold method is shown to support the result derived using the multiple time scale. Then the second study carried out is the study of the transition from a sinelike wave to a square wave. This is analysed and a scale deduced at which this transition gradually takes place. One of the key areas we focused on in the large delay is to solve for a certain constant a' associated with the period of oscillation. The effect of the delayed parameter is shown throughout this thesis as a major contributor to the properties of both the logistic delay and the negative autogenous regulation.
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The theory of rational integral functions of several sets of variables and associated linear transformationsWallace, Andrew Hugh January 1949 (has links)
The theme of this paper is the unification of two theories which arose and were developed independently of one another in the latter part of the 19th century and the beginning of the 20th, namely the theory of series expansion of rational integral functions of several sets of variables, homogeneous in the variables of each set, that is the series expansion of algebraic forms in several sets of variables, and the theory of induces linear transformations, or invariant matrices. I have divided the work into five chapters of which the first and third are purely historical; Chapter I is an account of various methods, devised before the introduction of the ideas of standard order and standard tableaux, of forming series expansions of algebraic forms, while Chapter III is mainly occupied by an account of Schnur's work on invariant matrices. Chapters II, IV and V establish the link between the two theories and, at the expense of one or two points of repetition of definitions, are self-contained and may be read consecutively, more or less without reference to the other two chapters.
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Inertial effects on thin-film wave structures with imposed surface shearSivapuratharasu, Mithilan January 2017 (has links)
This thesis provides a depth-averaged analytical and numerical approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to gravity-driven flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio e. Approximations are made assuming a Reynolds number, Re ~O (1/e) and a depth-averaged approach used to simplify the governing Navier-Stokes equations. A classical, parallel, Stokes flow is expected in the absence of any wave disturbance based on a local quadratic profile; in this work a generalised approach, which includes inertial effects, is solved. Flow structures are identified and compared with studies for Stokes flow in the limit of negligible inertial effects. Both two-tier and three-tier wave disturbances are constructed to study film profile evolution subject to shear at the free surface. An evaluation of film profiles is given from a paramet- ric study for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken which identifies new profiles not previously predicted when inertial effects are neglected. A revised integral boundary layer model incorporating a more general cubic velocity profile is also introduced, to better capture fluid re- circulation associated with a capillary region, and is developed to provide a better understanding of the internal flow dynamics within the thin-film layer. Notably, the wavelength and amplitude of the capillary ripples are analysed. The effect of the boundary conditions between the fluid and the plane is undertaken to simulate slip properties of various substrates over which the fluid may flow. A Navier slip condition is proposed at this boundary and its effect on the wave structure is examined both with and without the inclusion of inertia. The corresponding film dynamics are analysed with increased slip at the fluid-plane boundary and the effect on the wave structures formed are discussed. In a subsequent chapter solitary wave structures are investigated through a study of gravity-driven flow structures as associated with an oscillating inlet. The effects of increasing the film Reynolds number of these flows is evaluated together with an investigation of the stability characteristics relevant to inlet frequency and inertial effects. The effect of surface shear on solitary waves is examined, both as a stabilising and a destabilising factor on perturbations introduced at the inlet. A final section provides an overview of the outcomes from this study.
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On an equation related to Stokes wavesPichler-Tennenberg, Alex K. January 2002 (has links)
No description available.
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Assimilating data into mathematical modelsSanz-Alonso, Daniel January 2016 (has links)
Chapter 1 is a brief overview of the Bayesian approach to blending mathematical models with data. For this introductory chapter, I do not claim any originality in the material itself, but only in the presentation, and in the choice of contents. Chapters 2, 3 and 4 are transcripts of published and submitted papers, with minimal cosmetic modifications. I now detail my contributions to each of these papers. Chapter 2 is a transcript of the published paper Long-time Asymptotics of the Filtering Distribution for Partially Observed Chaotic Dynamical Systems" [Sanz-Alonso and Stuart, 2015] written in collaboration with Andrew Stuart. The idea of building a unified framework for studying filtering of chaotic dissipative dynamical systems is from Andrew. My ideas include the truncation of the 3DVAR algorithm that allows for unbounded observation noise, using the squeezing property as the unifying arch across all models, and most of the links with control theory. I stated and proved all the results of the paper. I also wrote the first version of the paper, which was subsequently much improved with Andrew's input. Chapter 3 is a transcript of the published paper \Filter Accuracy for the Lorenz 96 Model: Fixed Versus Adaptive Observation Operators" [Law et al., 2016], written in collaboration with Kody Law, Abhishek Shukla, and Andrew Stuart. My contribution to this paper was in proving most of the theoretical results. I did not contribute to the numerical experiments. The idea of using adaptive observation operators is from Abhishek. Chapter 4 is a transcript of the submitted paper\Importance Sampling: Computational Complexity and Intrinsic Dimension" [Agapiou et al., 2015], written in collaboration with Sergios Agapiou, Omiros Papaspiliopoulos, and Andrew Stuart. The idea of relating the two notions of intrinsic dimension described in the paper is from Omiros. Sergios stated and proved Theorem 4.2.3. Andrew's input was fundamental in making the paper well structured, and in the overall writing style. The paper was written very collaboratively among the four of us, and some of the results were the fruit of many discussions involving different subsets of authors. Some of my inputs include: the idea of using metrics between probability measures to study the performance of importance sampling, establishing connections to tempering, the analysis of singular limits both for inverse problems and filtering, most of the filtering section and in particular the use of the theory of inverse problems to analyze different proposals in the filtering set-up, the proof of Theorem 4.2.1, and substantial input in the proof of all the results of the paper not mentioned before. This paper aims to bring cohesion and new insights into a topic with a vast literature, and I helped towards this goal by doing most of the literature review involved.
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Asymptotic and numerical solutions of a two-component reaction diffusion systemBarwari Bala, Farhad January 2016 (has links)
In this thesis, we study a two-component reaction diffusion system in one and two spatial dimensions, both numerically and asymptotically. The system is related to a nonlocal reaction diffusion equation which has been proposed as a model for a single species that competes with itself for a common resource. In one spatial dimension, we find that this system admits traveling wave solutions that connect the two homogeneous steady states. We also analyse the long-time behaviour of the solutions. Although there exists a lower bound on the speed of travelling wave solutions, we observe that numerical solutions in some regions of parameter space exhibit travelling waves that propagate for an asymptotically long time with speeds below this lower bound. We use asymptotic methods to both verify these numerical results and explain the dynamics of the problem, which include steady, unsteady, spike-periodic travelling and gap-periodic travelling waves. In two spatial dimensions, the numerical solutions of the axisymmetric form of the system are presented. We also establish the existence of a steady axisymmetric solution which takes a form of a circular patch. We then carry out a linear stability analysis of the system. Finally, we perform bifurcation analysis of these patch solutions via a numerical continuation technique and show how these solutions change with respect to variation of one bifurcation parameter.
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Methods of proving uniqueness of stationary distributions for stochastic PDEsHorridge, Paul Robert January 2001 (has links)
In this thesis, we consider solutions u = u(t, x) for t > 0 and x e R, in time and one space dimension, of stochastic PDEs of the form dtu = Au + a(u) + b(u) dW where W is space-time white noise. The area is surveyed in Pardoux and an introduction to the concepts of white noise and SPDE solutions can be found in Walsh.
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