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978-3-8439-2215-9, Reihe Mathematik
Modelling and Control of Balance Laws with Applications to Networks
150 Seiten, Dissertation Universität Mannheim (2015), Softcover, A5
In this thesis essentially two types of partial differential equations and their applications to networks are considered. The first system is of hyperbolic type with an additional source term. We consider the control of boundary value problems and investigate stability properties. The tool to analyse stability and exponential decay is Lyapunov functions. The theoretical results are transferred to specific applications, such as electric transmission lines and road networks.
The second partial differential equation is of parabolic type and describes the movement of an organism under the influence of certain substances. In literature, this process is known as chemotaxis. The chemotaxis model is extended to networks by introducing coupling conditions which ensure the conservation of mass. We prove existence and uniqueness from a numerical point of view. To ensure stability and positivity for the numerical scheme, a network CFL condition is derived by a detailed computation. Finally, several numerical examples are presented.