Die Berechnung des 1912 von Birkhoff eingeführten chromatischen Polynoms eines Graphen stellt bekanntlich ein NP-vollständiges Problem dar. Dieses gilt somit erst recht für die Verallgemeinerung des chromatischen Polynoms zum bivariaten chromatischen Polynom nach Dohmen, Pönitz und Tittmann aus dem Jahre 2003. Eine von Averbouch, Godlin und Makowsky 2008 vorgestellte Rekursionsformel verursacht durch wiederholte Anwendung im Allgemeinen einen exponentiellen Rechenaufwand. Daher war das Ziel der vorliegenden Dissertation, ...

In this thesis, a non-linear system of partial differential equations is studied, describing the motions of an elastic structure which is immersed into an incompressible viscous fluid. The displacement of the elastic structure is modelled by a Lamé system and the fluid velocity as well as the fluid pressure are described by the Navier-Stokes equations. The structure and the fluid are coupled via two boundary conditions at the interface which correspond to continuity of velocities and forces. As the elasticity is ...

The retinal disease age-related macular degeneration is the most common cause of vision loss in industrialized countries. In this thesis, motivated by the drug (antibody) treatment of this disease, we designed long-term three dimensional Finite Element simulations of the drug distribution in the healthy human eye. The underlying model consists of a time-dependent convection-diffusion equation coupled to a stationary Darcy equation describing the flow of the aqueous humor through the vitreous medium. We replaced the ...

In this work, we will first review the Quantum Mechanics theory to derive the main equations. Next, we will analyze these equations by Functional Analysis methods to find conditions for existence, uniqueness, multiplicity, and other properties as positivity. Next, we will review and develop some numerical methods for solving the nonlinear Schrödinger equation, its time version, generalizations with rotational terms, and systems of NLSE (NLSS). We notice that the main problem to run numerical methods is the memory ...

This dissertation is dedicated to the analysis of the Navier-Stokes equations in a timeperiodic framework in the so-called layer domain Π = R2 × (0, 1), described by: ∂tu − νΔu + (u · ∇)u + ∇p = f in [0, T] × Π, div u = 0 in [0, T] × Π, u|∂Π = a for all t ∈ [0, T] , u|t=0 = u|t=T in Π. The velocity field u and the pressure p are unknowns, while the external force f is prescribed. Challenges arise due to unboundedness of the layer Π and from introduction of a nonhomogeneous boundary condition a. The investigated ...

We give in this work some results on twin basic quantum calculus. The (p,q)-derivative, the (p,q)-integral and some (p,q)-Taylor formula are discussed. Next, we introduce and provide several results about (p,q)-Gamma and (p,q)-Beta functions. Some (p,q)-Sturm Liouville problems are introduced and some (p,q)-hypergeometric solutions are obtained. The work ends with the introduction (p,q)-Appell polynomials and some (p,q)-Laplace transforms.

The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are allowed to depend on the solution and the step size. As a result of this, we also refer to them as non-standard additive RK (NSARK) methods. We motivate and introduce modified Patankar (MP) schemes as a subclass of NSARK methods and emphasize their importance. The first major part ...

Energy-aware scheduling is crucial in the current economy and green production initiatives. However, with the rise of renewable energy sources, power production is subject to uncertain weather conditions. Hence, within a network, some balancing group managers must maintain a balance between production and demand, which requires precise energy orders for specific times. Therefore, those managers must prioritize accurate energy orders to manage this complex problem. The primary aim of this thesis is to manage one ...

In this thesis, we investigate the existence and regularity results of a ferroelectric phase-field model, which is a state-of-the-art model arising in recent years from the engineering area for the ferroelectric study.

LIMA steht für "LehrInnovation in der Studieneingangsphase 'Mathematik im Lehramtsstudium' – Hochschuldidaktische Grundlagen, Implementierung und Evaluation". Hinter dem Akronym verbirgt sich ein interdisziplinäres Gemeinschaftsprojekt der Universitäten Paderborn und Kassel. Es begann im Januar 2009 und endete im August 2012. Gefördert wurde LIMA vom Bundesministerium für Bildung und Forschung (BMBF) im Rahmen der „Zukunftswerkstatt Hochschullehre“. Das übergeordnete Ziel des Vorhabens war, den Studienbeginn für ...