M.Sc. / In this work we concentrate on two generalizations of Lie symmetries namely conditional symmetries in the form of Q-symmetries and approximate symmetries. The theorems and definitions presented can be used to obtain exact and approximate solutions for nonlinear partial differential equations. These are then applied to various nonlinear heat and wave equations and many interesting solutions are given. Chapters 1 and 2 gives an introduction to the classical Lie approach. Chapters 3, 4 and 5 deals with conditional -, approximate -, and approximate conditional symmetries respectively. In chapter 6 we give a review of symbolic algebra computer packages available to aid in the search for symmetries, as well as useful REDUCE programs which were written to obtain the results given in chapters 2 to 5.

This thesis focuses on the mathematical analysis of two partial differential equation systems. Consistent improvement of mathematical computation allows more and more questions to be addressed in the form of numerical simulations. At the same time, novel materials arising from advances in physics and material sciences are creating new problems which must be addressed. This thesis is divided into two parts based on analysis of two such materials: organic semiconductors and graphene. In part one we derive a generalized reaction-drift-diffusion model for organic photovoltaic devices -- solar cells based on organic semiconductors. After formulating an appropriate self-consistent model (based largely on generalizing partly contradictory previous models), we study the operation of the device in several specific asymptotic regimes. Furthermore, we simulate such devices using a customized 2D hybrid discontinuous Galerkin finite element scheme and compare the numerical results to our asymptotics. Next, we use specialized asymptotic regimes applicable to a broad range of device parameters to justify several assumptions used in the formulation of simplified models which have already been discussed in the literature. We then discuss the potential applicability of the simulations to real devices by discussing which parameters will be the most important for a functioning device. We then give further generic 2D numerical results and discuss the limitations of the model in this regime. Finally, we give several perspectives on proving existence and uniqueness of the model. In part two we derive a second-order finite difference numerical scheme for simulation of the 2D Dirac equation and prove that the method converges in the electromagnetically static case. Of particular interest is the application to electrons in graphene. We demonstrate this convergence numerically with several examples for which explicit solutions are known and discuss the manner in which errors appear and propagate. We furthermore extend the Dirac system with Poisson's equation to investigate interesting electronic effects. In particular, we show that our numerical scheme can successfully simulate a beam-splitter and Veselago lens, both of which have been predicted analytically to appear in graphene.

In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.

This thesis explores higher order numerical methods for solving fractional differential equations.

This thesis introduces and analyses a numerical method for solving time-dependent partial differential equations (PDEs) on surfaces. This method is based on the closest point method, and solves the surface PDE by solving a suitably chosen equation in a band surrounding the surface. As it uses an implicit closest point representation of the surface, the method has the advantages of being simple to implement for very general surfaces, and amenable to discretization with a broad class of numerical schemes. The method proposed in this work introduces a new equation in the embedding space, which satisfies a key consistency property with the surface PDE. Rather than alternating between explicit time-steps and re-extensions of the surface function as in the original closest point method, we investigate an alternative approach, in which a single equation can be solved throughout the embedding space, without separate extension steps. This is achieved by creating a modified embedding equation with a penalty term, which enforces a constraint on the solution. The resulting equation admits a method of lines discretization, and can therefore be discretized with implicit or explicit time-stepping schemes, and analysed with standard techniques. The method can be formulated in a straightforward way for a large class of problems, including equations featuring variable coefficients, higher-order terms or nonlinearities. The effectiveness of the method is demonstrated with a range of examples, drawing from applications involving curvature-dependent diffusion and systems of reaction-diffusion equations, as well as equations arising in PDE-based image processing on surfaces.

M.Sc. / There have been an extensive study on solutions of differential equations modeling physical phenomena that blows up in finite time. The blow-up time often represents an important change in the properties of such models and hence it is very important to compute it as accurate as possible. In this work, an adaptive in time numerical method for computing blow-up solutions for nonlinear ODEs is introduced. The method is named implicit midpoint-implicit Euler method (IMIE) and is based on the implicit Euler and the implicit midpoint method. The method is used to compute blow-up time for different examples of ODEs, PDEs and VIDEs. The PDEs studied are reaction-diffusion equations whereby the method of lines is first used to discretize the equation in space to obtain a system of ODEs. Quadrature rules are used to approximate the integral in the VIDE to get a system of ODEs. The IMIE method is then used then to solve the system of ODEs. The results are compared to results obtained by the PECEIE method and Matlab solvers ode45 and ode15s. The results show that the IMIE method gives better results than the PECE-IE and ode15s and compares quite remarkably with the 4th order ode45 yet it is of order 1 with order 2 superconvergence at the mesh points.

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Philosophiae Doctor - PhD / In this thesis, we design and implement a class of numerical methods (based on integral transforms) to solve PDEs for pricing a variety of financial derivatives. Our approach is based on spectral discretization of the spatial (asset) derivatives and the use of inverse Laplace transforms to solve the resulting problem in time. The conventional spectral methods are further modified by using piecewise high order rational interpolants on the Chebyshev mesh within each sub-domain with the boundary domain placed at the strike price where the discontinuity is located. The resulting system is then solved by applying Laplace transform method through deformation of a contour integral. Firstly, we use this approach to price plain vanilla options and then extend it to price options described by a jump-diffusion model, barrier options and the Heston’s volatility model. To approximate the integral part in the jump-diffusion model, we use the Gauss-Legendre quadrature method. Finally, we carry out extensive numerical simulations to value these options and associated Greeks (the measures of sensitivity). The results presented in this thesis demonstrate the spectral accuracy and efficiency of our approach, which can therefore be considered as an alternative approach to price these class of options.

Philosophiae Doctor - PhD / The purpose of this study was to investigate the role of visualization in the conceptualisation and solution of problems in multivariate calculus and dynamical systems. The theoretical basis, and the visual and analytical aspects of evaluating multiple integrals, and the stability analysis of dynamical systems, were established. To address the research questions, a teaching experiment with activities to facilitate visualization of 3D objects and phase portraits of non-linear dynamic systems was conducted with an experimental class (n = 24) which received six activity sessions in the computer Laboratory in addition to traditional lectures. The control class (n = 26) received traditional lectures and tutorial instruction. Both groups were lectured by the researcher using the same set of class notes, assignments, worksheets and tutorials. Additional support materials were posted on the Blackboard on Web-City. The activities included tasks in the computer laboratory that reinforced visualization and spatial ability factors such as surface features, nets, projections, cross-sections and rotation of 3D objects as well as phase portraits of systems of differential equations. The students were tested at several time points, and over both the short and long term to assess the impact on their visual and analytical solutions to problems in the two study domains. The pre-test on prior knowledge indicated no significant differences between the means of the experimental and control groups. Results indicate that there were no significant differences between the achievement of the two groups in Test 1 and Test 2 while the activities were ongoing, but towards the end of the semester significant differences in favour of the experimental group were recorded. A multiple linear regression analysis confirmed that in addition to prior knowledge as measured by the pretest, two of the spatial factors were significant predictors of achievement for the domains under investigation. Students had difficulties in visualising 3D regions of integration and in switching the order of triple integrals. Very few (18%) recognised the need for split integrals to span the required area or volume. While students could find analytical solutions to systems of differential equations and describe the stability of individual equilibrium points using eigenvalues, they struggled with translating rates of change into slopes on the phase portraits, with the interpretation of the solutions and in describing the global behaviour of the system. Students had difficulties in visualizing the region of integration in R³, the stability of equilibrium points in the phase portraits, and in coordinating the treatments and conversions between the geometric, numerical, symbolic and algebraic registers. The tendency to work in the algebraic register to determine the limits of the integral was noted, and students opted to use analytic methods in conducting a stability analysis of the given dynamic system rather than the geometric method. This study adds to research on visualization in mathematics by examining how exposure to technologically enhanced representations complement and promote the conceptualisation of solutions to problems involving multiple integrals and systems of differential equations.

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