The space tether system under consideration consists of two rigid bodies with significantly different ballistic coefficients. Because of this difference one of the bodies acts as a stabilizer for the main body – a spacecraft – during the motion of the tether system in the atmosphere. The investigations are focused on the stability of motion of the tether system in the atmosphere. During its motion in the atmosphere the tether system makes use of torques from aerodynamic forces to maintain a desired orientation. This aerodynamic method of stabilization is passive and does not require energy expenses. Such a tether system can be used to stabilize the motion before landing onto the surface of Earth or other planets with atmospheres. The aerodynamic tether system is helpful for returning payloads from outer space, especially using small landing modules. It is also possible to utilize in the removal of space debris by reducing the altitude of their orbits. By achieving the spacecraft motion stability during descent the tether system enables a reduction in the target landing area at the final stage of the descent. The modelling of motion of the tether system includes two parts – (i) the deployment of the tether system, and (ii) the descent of deployed tether system through the dense layers of the atmosphere. The motion of the deployed tether system is investigated with regard to the terms of its stability. The tether system can be in stable motion even if either or both bodies are statically unstable. The stability of the system is assessed relative to the parameters – the mass, the geometrical dimensions of the bodies and the length of the tether. It is found that increasing the length of the tether, as a controlled part of the deployment process during descent, can provide an additional stabilizing factor for the tether system. The model of the deployment process, based on the model of an elastic tether, represents the tether as a set of nodes with mass and with elastic connections. The control of the deployment is based on the length and the rate of change of the length of the tether. The aerodynamic resistance of the tether and its mass characteristics are both taken into consideration during modelling of the deployment. The described and numerically realized mathematical models allows the parameters for the space tether system motion in the atmosphere to be determined.

This dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the time-fractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²^-α), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integer-order case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a non-polynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semi-discrete ordinary differential equations in the initial value variable is integrated in time with a non-polynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a non-polynomial approximation in the first sub-interval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with non-polynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.