In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than the absolute dimension of the described process, and the actual lower-dimensional configuration space of the system is deeply implicit. For linear higher-index differential-algebraic equations, we develop an appropriate numerical method based on properties of canonical subspaces and on the so-called geometric reduction. Taking into account the fact that higher-index differential-algebraic equations lead to ill-posed problems in naturally given norms, we modify this approach to serve as transfer conditions from one time-window to the next in a time stepping procedure and combine it with window-wise overdetermined least-squares collocation to construct the first fully numerical solvers for higher-index initial-value problems.

翻译：与常规常微分方程不同，精确设定初始条件的问题仅在微分代数方程的背景下出现，其中系统的动态自由度小于所描述过程的绝对维度，且系统实际低维构型空间具有深度隐式特性。针对线性高阶指标微分代数方程，我们基于典型子空间特性和所谓几何约简方法，开发了相应的数值计算方法。考虑到高阶指标微分代数方程在自然给定范数下会导致不适定问题，我们将该方法改进为时间步进过程中时间窗口间的传递条件，并结合窗口式超定最小二乘配置法，构建了首个适用于高阶指标初值问题的全数值求解器。