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.

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.

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.

M.Sc. (Mathematics) / Please refer to full text to view abstract

Differential equations, in particular partial differential equations, are used to mathematically describe many physical phenomenon. The importance of being able to solve these types of equations can therefore not be overstated. This thesis is going to elucidate one method, the method of characteristics, which can in some cases be used to solve partial differential equations. To further the reader’s understanding on the method this paper will provide some important insights on differential equations as well as show examples on how the method of characteristics can be used to solve partial differential equations of various complexity. We will also in this paper present some important geometric complications for linear partial differential equations which one might have to take into consideration when using the method.

Stochastic partial differential equations (SPDEs) are a central object of study in the field of stochastic analysis. Their study involves a number of different tools coming from probability theory, functional analysis, harmonic analysis, statistical mechanics, and dynamical systems. Conversely SPDEs are an extremely useful paradigm to study scaling limit phenomena encountered throughout many other areas of mathematics and physics. The present thesis is concerned mainly with one particular SPDE called the Kardar-Parisi-Zhang (KPZ) equation, which appears universally as a fluctuation limit of height profiles of microscopic models such as interacting particle systems, directed polymers, and corner growth models. Such limit results are deemed instances of ``weak KPZ universality," a field born from the seminal paper of Bertini and Giacomin. We extend results on weak KPZ universality in a number of different directions. In one direction, we prove a version of Bertini-Giacomin's result in a half-space by adapting their methods to this setting, thus extending a result of Corwin and Shen and completing the final step towards the proof of a conjecture about fluctuation behavior of half-space KPZ. In another direction, we also prove a result for the free energy for directed polymers in an octant converging to the KPZ equation in a half-space with a nontrivial normalization at the boundary. In a third direction, we return to the whole-space regime and extend the Bertini-Giacomin result to the case of several different initial data coupled together, proving joint convergence of ASEP with its basic coupling to KPZ driven by the same realization of its noise. Finally we prove a ``nonlinear" version of the law of the iterated logarithm for the KPZ equation in a weak-noise but strong-nonlinearity regime. Beyond their intrinsic purpose, one application of all these extensions and generalizations is to take limits of known results and identities for discrete systems and pass them to the limit to obtain nontrivial information about the KPZ equation itself, which is a well-known methodology launched by I. Corwin and coauthors.

In my Dissertation, I consider the system of thermoelasticity endowed with poly- convex energy. I will present the equations in their mathematical and physical con- text, and I will explain the relevant research in the area and the contributions of my work. First, I embed the equations of polyconvex thermoviscoelasticity into an aug- mented, symmetrizable, hyperbolic system which possesses a convex entropy. Using the relative entropy method in the extended variables, I show convergence from ther- moviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. In addition, I establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions. Then, I prove a measure-valued versus strong uniqueness result for adiabatic poly- convex thermoelasticity in a suitable class of measure-valued solutions, de ned by means of generalized Young measures that describe both oscillatory and concentra- tion e ects. Instead of working directly with the extended variables, I will look at the parent system in the original variables utilizing the weak stability properties of certain transport-stretching identities, which allow to carry out the calculations by placing minimal regularity assumptions in the energy framework. Next, I construct a variational scheme for isentropic processes of adiabatic polyconvex thermoelasticity. I establish existence of minimizers which converge to a measure-valued solution that dissipates the total energy. Also, I prove that the scheme converges when the limit- ing solution is smooth. Finally, for completeness and for the reader's convenience, I present the well-established theory for local existence of classical solutions and how it applies to the equations at hand.

We study a two-dimensional Fermi-Pasta-Ulam lattice in the long-amplitude, small-wavelength limit. The one-dimensional lattice has been thoroughly studied in this limit, where it has been established that the dynamics of the lattice is well-approximated by the Korteweg–De Vries (KdV) equation for timescales of the order ε^−3. Further it has been shown that solitary wave solutions of the FPU lattice in the one dimensional case are well approximated by solitary wave solutions of the KdV equation. A two-dimensional analogue of the KdV equation, the Kadomtsev–Petviashvili (KP-II) equation, is known to be a good approximation of certain two-dimensional FPU lattices for similar timescales, although no proof exists. In this thesis we present a rigorous justification that the KP-II equation is the long-amplitude, small-wavelength limit of a two-dimensional FPU model we introduce, analogous to the one-dimensional FPU system with quadratic nonlinearity. We also prove that the cubic KP-II equation is the limit of a model analogous to a one-dimensional FPU system with cubic nonlinearity. Further we study whether stability of line solitons in the KP-II equation extends to stability of one-dimensional FPU solitary waves in the two-dimensional lattices. / Thesis / Doctor of Philosophy (PhD)

Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD. / Master of Science / The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

This thesis continues the work started with my previous supervisor, Dr. Shaohua Chen. In [15], the authors developed some tools that showed the boundedness or blowup of solutions to a semilinear parabolic system with homogeneous Neumann boundary conditions. This system, the so called ’Activator-Inhibitor Model’, is of interest as it is used to model biological processes and pattern formation. Similar tools were later adapted to deal with the same parabolic system in [3], in which the authors prove global boundedness of solutions under homogeneous Dirichlet conditions. This new problem is of mathematical interest as the solutions may grow singular near the boundary. Shortly after, a different system was considered in [4], where the authors proved global boundedness of solutions to a system featuring similar singular reaction terms. The goal of this thesis is twofold: first, the tools developed that allow us to tackle these sorts of problems will be demonstrated in detail to showcase its utility; the second is to then use these tools to generalize some of these previous results to a larger class of singular parabolic systems. In doing so, we expand the classical literature found in [14] and other notable works, where nonsingular equations are extensively treated. The motivation for the first should be clear. While there are numerous bodies of text treating nonsingular problems, there are no collections available dealing with these types of singularities exclusively. This is of practical use to other mathematicians studying partial differential equations. The motivation for the second is, perhaps, more practical. There are a growing number of models found in physics, chemistry and biology that may be generalized to a singular type system. Through allowing those individuals to treat these problems, we may gain valuable insights into the real world and how these processes behave. / Thesis / Master of Science (MSc)