Rukhsana Kausar

• Various physical phenomenons with sudden transients that results into structrual changes can be modeled via switched nonlinear differential algebraic equations (DAEs) of the type \[ E_{\sigma}\dot{x}=A_{\sigma}x+f_{\sigma}+g_{\sigma}(x). \tag{DAE} \] where \(E_p,A_p \in \mathbb{R}^{n\times n}, x\mapsto g_p(x),\) is a mapping, \(p \in \{1,\cdots,P\}, P\in \mathbb{N} f \in \mathbb{R} \rightarrow \mathbb{R}^n , \sigma: \mathbb{R} \rightarrow \{1,\cdots, P\}\). Two related common tasks are: Task 1: Investigate if above (DAE) has a solution and if it is unique. Task 2: Find a connection among a solution of above (DAE) and solutions of related partial differential equations. In the linear case \(g(x) \equiv 0\) the task 1 has been tackeled already in a distributional solution framework. A main goal of the dissertation is to give contribution to task 1 for the nonlinear case \(g(x) \not \equiv 0\) ; also contributions to the task 2 are given for switched nonlinear DAEs arising while modeling sudden transients in water distribution networks. In addition, this thesis contains the following further contributions: The notion of structured switched nonlinear DAEs has been introduced, allowing also non regular distributions as solutions. This extend a previous framework that allowed only piecewise smooth functions as solutions. Further six mild conditions were given to ensure existence and uniqueness of the solution within the space of piecewise smooth distribution. The main condition, namely the regularity of the matrix pair \((E,A)\), is interpreted geometrically for those switched nonlinear DAEs arising from water network graphs. Another contribution is the introduction of these switched nonlinear DAEs as a simplication of the PDE model used classically for modeling water networks. Finally, with the support of numerical simulations of the PDE model it has been illustrated that this switched nonlinear DAE model is a good approximation for the PDE model in case of a small compressibility coefficient.