In this chapter and the next we study differential-algebraic equations (DAEs), already introduced in Section 1.3. Here we consider the mathematical structure of such system and some essential analytical transformations. Numerical approaches and discretizations are discussed in the next chapter. But here, too, our motivation remains finding practical computer solutions. Compared to Chapters 2 and 6, this chapter is unusually long. One reason is that DAE theory is much more recent than ordianry differential equation (ODE) theory. As a result, DAE theory is more in a state of flux, and good expositions are scarce. More importantly, understanding the principles highlighted here is essential for, and will get you a long way towards, constructing good numerical algorithms.
To get a taste of the similarity and the difference between DAEs and ODEs, consider two functions y( t) and z( t) which are related on some interval [0, b] by
and the task of recovering one of these functions from the other. To recover z from y one needs to differentiate y( t) an automatic process familiar to us from a first calculus course. To recover y from z one needs to integrate z( t) a less automatic process necessitating an additional side condition (such as the value of y(0)).
This would suggest that differentiation is a simpler, more straightforward process than integration. On the...
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