Wieslaw Marszalek
In this thesis, we analyze infinite dimensional differential algebraic equations (DAEs). First, we extend the notion of index to partial DAEs. Three different type of indices (modal, perturbation and algebraic) are defined and compared with each other. The comparison with finite dimensional DAEs is also done. It is shown that infinite dimensional DAEs exhibit richer behavior than finite dimensional DAEs, since the former may have solutions (and indices) which depend not only on the forcing functions, data (initial and boundary conditions), but also on the region of interest and method of approximation. In particular, the method of lines (MOL) with finite difference and finite element approximation is analysed.
Then, we analyse the traveling wave solutions in both linear and nonlinear infinite dimensional systems. The solutions of dissipative systems of conservation laws in gas dynamics and magnetohydrodynamics (MHD) naturally lead to DAEs when one looks for a special solution in the form of a traveling wave between the left and right equilibria. The structure of the DAE (semi-explicit or conservative) depends on the dissipative mechanism involved.
Next, we analyse the singularity induced bifurcation in MHD, when an equilibrium is placed at the singularity of a MHD DAE. It is shown that one may be able to integrate through the singularity to connect two equilibria lying on the opposite sides of that singularity. In some cases we reach and leave the equilibrium at the singularity in finite time.
Our analysis is illustrated by many numerical examples. We also present a few related research topics for further research.