The solutions of nonlinear ordinary or partial differential equations are important in the study of fluid flow and heat transfer. In this thesis we apply the Homotopy Analysis Method (HAM) and obtain solutions for several fluid flow and heat transfer problems. In chapter 1, a brief introduction to the history of homotopies and embeddings, along with some examples, are given. The application of homotopies and an introduction to the solutions procedure of differential equations (used in the thesis) are provided. In the chapters that follow, we apply HAM to a variety of problems to highlight its use and versatility in solving a range of nonlinear problems arising in fluid flow. In chapter 2, a viscous fluid flow problem is considered to illustrate the application of HAM. In chapter 3, we explore the solution of a non-Newtonian fluid flow and provide a proof for the existence of solutions. In addition, chapter 3 sheds light on the versatility and the ease of the application of the Homotopy Analysis Method, and its capability in handling non-linearity (of rational powers). In chapter 4, we apply HAM to the case in which the fluid is flowing along stretching surfaces by taking into the effects of "slip" and suction or injection at the surface. In chapter 5 we apply HAM to a Magneto-hydrodynamic fluid (MHD) flow in two dimensions. Here we allow for the fluid to flow between two plates which are allowed to move together or apart. Also, by considering the effects of suction or injection at the surface, we investigate the effects of changes in the fluid density on the velocity field. Furthermore, the effect of the magnetic field is considered. Chapter 6 deals with MHD fluid flow over a sphere. This problem gave us the first opportunity to apply HAM to a coupled system of nonlinear differential equations. In chapter 7, we study the fluid flow between two infinite stretching disks. Here we solve a fourth order nonlinear ordinary differential equation. In chapter 8, we apply HAM to a nonlinear system of coupled partial differential equations known as the Drinfeld Sokolov equations and bring out the effects of the physical parameters on the traveling wave solutions. Finally, in chapter 9, we present prospects for future work.

This text provides an overview of problems in the field of data assimilation. We explore the possibility of recreating unknown data by continuously inserting known data into certain dynamical systems, under certain regularity assumptions. Additionally, we discuss an alternative statistical approach to data assimilation and investigate the utilization of the Ensemble Kalman Filter for assimilating data into dynamical models. A key challenge in numerical weather prediction is incorporating convective precipitation into an idealized setting for numerical computations. To answer this question we examine the modified rotating shallow water equations, a nonlinear coupled system of partial differential equations and further assess if this primitive model accurately mimics phenomena observed in operational numerical weather prediction models. Numerical experiments conducted using a Deterministic Ensemble Kalman Filter algorithm support its applicability for convective-scale data assimilation. Furthermore, we analyze the frequency spectrum of numerical forecasts using the Wavelet transform. Our frequency analysis suggests that, under certain experimental settings, there are similarities in the initialization of operational models, which can aid in understanding the problem of intialization of numerical weather prediction models.

This paper presents a method for interactive design by means of extending the PDE based approach for surface generation. The governing partial differential equation is generalized to arbitrary order allowing complex shapes to be designed as single patch PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating the geometry of shape that satisfying an arbitrary set of boundary conditions. Both the boundary conditions which are defined as curves in 3-space and the spine of the corresponding PDE are utilized as interactive design tools for creating and manipulating geometry intuitively. In order to facilitate interactive design in real time, a compact analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme even in the case of general boundary conditions satisfies exactly the boundary conditions where the resulting surface has an closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples.

Yes / This paper extends the PDE method of surface generation. The governing partial differential equation is generalised to sixth order to increase its flexibility. The PDE is solved analytically, even in the case of general boundary conditions, making the method fast. The boundary conditions, which control the surface shape, are specified interactively, allowing intuitive manipulation of generic shapes. A compact user interface is presented which makes use of direct manipulation and other techniques for 3D interaction.

No / This work aims to develop a personalised avatar based virtual environment for motivating weight loss and weight management. Obesity is a worldwide epidemic which has not only enormous resource impact for the healthcare systems but also has substantial health as well as a psychological effect among the individuals who are affected. We propose to tackle this issue via the development of a personalised avatar, the form of which can be adjusted to show the present and the future self of the individual. For the avatar design and development phase, we utilise a parametric based mathematical formulation derived from the solutions of a chosen elliptic partial differential equation. This method not only enables us to generate a parameterised avatar model, but it also allows us to quickly and efficiently create various avatar shapes corresponding to different body weights and even to different body postures. / This research was funded by the NIHR Research for Patient Benefit Programme (project reference PB-PG-1215-20016).

This thesis discusses a novel method for modelling facial action units. It presents facial action units model based on boundary value problems for accurate representation of human facial expression in three-dimensions. In particular, a solution to a fourth order elliptic Partial Differential Equation (PDE) subject to suitable boundary conditions is utilized, where the chosen boundary curves are based on muscles movement defined by Facial Action Coding System (FACS). This study involved three stages: modelling faces, manipulating faces and application to simple facial animation. In the first stage, PDE method is used in modelling and generating a smooth 3D face. The PDE formulation using small sets of parameters contributes to the efficiency of human face representation. In the manipulation stage, a generic PDE face of neutral expression is manipulated to a face with expression using PDE descriptors that uniquely represents an action unit. A combination of the PDE descriptor results in a generic PDE face having an expression, which successfully modelled four basic expressions: happy, sad, fear and disgust. An example of application is given using simple animation technique called blendshapes. This technique uses generic PDE face in animating basic expressions. / Ministry of Higher Education, Malaysia and Universiti Malaysia Terengganu

This thesis is concerned with a solution theory for quasilinear singular stochastic partial differential equations. We approach the theory of regularity structures, a tool to tackle singular stochastic PDEs, from a new perspective which is well suited for, but not restricted to, quasilinear equations. In the first part of this thesis, we revisit the algebraic aspects of the theory of regularity structures. Although we approach regularity structures from a different perspective than originally done, we show that the same (Hopf-) algebraic structure is underlying. Trees do not play any role in our construction, hence the Hopf algebras underlying rough paths and regularity structures are not at our disposal. Instead, our alternative point of view gives a new (Lie-) geometric interpretation of the structure group, arising from simple actions on the nonlinearity of the equation and a parametrization of the solution manifold. In the second part of this thesis, we revisit the probabilistic aspects of the theory of regularity structures. We construct and stochastically estimate the centered model, which captures the local behaviour of the solution manifold. This is carried out under a spectral gap assumption on the driving noise, and based on a novel application of Malliavin calculus in regularity structures. In deriving the renormalized equation we are guided by symmetries, so that natural invariances of the model are built in. In the third part of this thesis, we make again use of the Malliavin derivative to obtain a robust characterization of the model, which persists for rough noise even as a mollification is removed. This allows for a simple derivation of invariances of the model that are not present at the level of approximations. Furthermore, we give a convergence result of models, which together with the characterization establishes a universality result in the class of noise ensembles satisfying uniformly a spectral gap assumption.

As a perturbation method, averaging is a mathematical tool for dynamic analysis of time-periodic and space-periodic dynamical systems, including those governed by partial differential equations. The control design procedure presented in this work uses averaging techniques, the well-developed linear control strategies, and finite element methods. The controller is designed based on the linear averaged dynamics of a time- or space-periodic system. The controller is then used for trajectory tracking or stabilization of the periodic system. The applicability and performance of the suggested method depend on different physical parameters of the periodic system and the control parameters of the controller. The effects of these parameters are discussed in this work. Numerical simulations show acceptable performance of the proposed control design strategy for two linear and nonlinear time- and space-periodic systems, namely, the one-dimensional heat equation and the Chafee-Infante equation with periodic coefficients. / M.S. / Dynamic analysis and control of dynamical systems with varying parameters is a challenging task. It is always of great help if one can perform the analyses for an approximate system with constant parameters and use the results to study and control the original system with varying parameters. Averaging is a mathematical tool that is used to approximate a system with periodic parameters with a ``simpler'' system with constant parameters. In this research averaging is used for design of controllers for systems with periodic parameters. First, an approximate system with constant parameters, called the averaged system, is determined. The averaged system is used for design of a controller which can be then be used for the original system with periodic parameters.

Finite difference approximations are methods for solving differential equations by approximating derivatives. This work will begin with how to solve a partial differential equation (PDE) called the advection equation, ut + cux = 0. Both analytically, and approximately with three different finite difference methods for the spatial part of the equation: • Central in space, • First order upwind in space, • Beam-Warming in space, and forward Euler for the temporal part. We then use the theoretical approximations considered for the advection equation and apply it on Maxwell’s equations for electromagnetism in 1D. This is a system of advection equations that describes how electromagnetic waves propagate through a dielectric material. In the end of this work we will model this electromagnetic wave, or wave of light moving through materials with different refraction indexes.

<p>A model with nonlinear Schrödinger (NLS) equation used for describing pulse propagations in photopolymers is considered. We focus on a case in which change of refractive index is proportional to the square of amplitude of the electric field and consider 2-dimensional spatial domain. After formal derivation of the NLS approximation from the wave-Maxwell equation, we establish well-posedness and perform rigorous justification analysis to show smallness of error terms for appropriately small time intervals. We conclude by numerical simulation to illustrate the results in one-dimensional case.</p> / Master of Science (MSc)