In this work, we address two different topics, which are discussed in the main chapters. The first part of the master thesis deals with the limits of validity of certain mathematical models from the area of spacebased acquisition, pointing, and tracking (APT) laser systems, written as systems of ordinary differential equations (ODEs). The interest in the analysis and numerical solution of these models was strongly motivated from the international cooperation with Professor Jose Maria Gambi from the Charles III University of Madrid, Spain. Our main goal was to demonstrate the considerable improvement that can be achieved by appropriately correcting the standard equations. This investigation was carried out for three different types of satellites and a number of different combinations of parameters. For the numerical solution of the respective model equations, the standard MATLAB code ode45 was used. In the second part of the work, we considered a discretisation of the stationary epsilondependent WignerEquation, which for epsilon=0 turns out to be an Index2 differential algebraic equation (DAE). This activity was developed within the cooperation with Professor Anton Arnold from the Vienna University of Technology, Vienna, Austria. Main aim here was to find information about the solutions behaviour, in the limit epsilon to 0, to support the analysis of the original, continuous problem. To numerically simulate this class of problems, we used a special version of the collocation method implemented in the MATLAB code bvpsuite and embedded into the leastsquares minimization algorithm.
