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This thesis deals with the analysis and numerical simulation of anisotropic nonlinear partial differential equations (PDEs) and dynamical systems in biology. It is divided into two parts, motivated by the simulation of fingerprint patterns and the modelling of biological transport networks. The first part of this thesis deals with a class of interacting particle models with anisotropic repulsive-attractive interaction forces and their continuum counterpart. These models are motivated by the simulation of fingerprint databases, which are required in forensic science and biometric applications. In existing interacting particle models, the forces are isotropic and the continuum limits of these particle models are given by nonlocal aggregation equations with radially symmetric potentials. The central novelty in the models we consider is an anisotropy induced by an underlying tensor field. This innovation does not only lead to the ability to describe real-world phenomena more accurately, but also renders their analysis significantly harder compared to their isotropic counterparts. We discuss the role of anisotropic interaction, study the steady states and present a stability analysis of line patterns. We also show numerical results for the simulation of fingerprints, based on discrete and continuum modelling approaches. The second part of this thesis focuses on a new dynamic modeling approach on a graph for biological transportation networks which are ubiquitous in living systems such as leaf venation in plants, blood circulatory systems, and neural networks. We study the existence of solutions to this model and propose an adaptation so that a macroscopic system can be obtained as its formal continuum limit. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. We also show the global existence of weak solutions of the macroscopic gradient flow. Results of numerical simulations of the discrete gradient flow illustrate the convergence to steady states, their non-uniqueness as well as their dependence on initial data and model parameters. Based on this model we propose an adapted model in the cellular context for leaf venation, investigate the model analytically and show numerically that it can produce branching vein patterns.

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 deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.

This thesis deals with the analysis and numerical simulation of different partial differential equation models arising in socioeconomic sciences. It is divided into two parts: The first part deals with a mean-field price formation model introduced by Lasry andLions in 2007. This model describes the dynamic behaviour of the price of a good being traded between a group of buyers and a group of vendors. Existence (locally in time) of smooth solutions is established, and obstructions to proving a global existence result are examined. Also, properties of a regularised version of the model are explored and numerical examples are shown. Furthermore, the possibility of reconstructing the initial datum given a number of observations, regarding the price and the transaction rate, is considered. Using a variational approach, the problem can be expressed as a non-linear constrained minimization problem. We show that the initial datum is uniquely determined by the price (identifiability). Furthermore, a numerical scheme is implemented and a variety of examples are presented. The second part of this thesis treats two different models describing the motion of (large) human crowds. For the first model, introduced by R.L. Hughes in 2002, several regularised versions are considered. Existence and uniqueness of entropy solutions are proven using the technique of vanishing viscosity. In one space dimension, the dynamic behaviour of solutions of the original model is explored for some special cases. These results are compared to numerical simulations. Moreover, we consider a discrete cellular automaton model introduced by A. Kirchner and A. Schadschneider in 2002.By (formally) passing to the continuum limit, we obtain a system of partial differential equations. Some analytical properties, such as linear stability of stationary states, areexamined and extensive numerical simulations show capabilities and limitations of the model in both the discrete and continuous setting.