16.2 Example: chemical kinetics
As an example of a system of DAEs, consider following chemical kinetics problem(Robertson 1966). The nondimensionalized DAE consists of two differential equations and one algebraic constraint. The differential equations describe the reactions from reactants \(y_1\) and \(y_2\) to the product \(y_3\), and the algebraic equation describes the mass conservation. (Serban and Hindmarsh 2021).
The state equations implicitly defines the state \((y_1(t), y_2(t), y_3(t))\) at future times as a function of an initial state and the system parameters, in this example the reaction rate coefficients \((\alpha, \beta, \gamma)\).
Unlike solving ODEs, solving DAEs requires a consistent initial condition. That is, one must specify both \(y(t_0)\) and \(y'(t_0)\) so that residual function becomes zero at initial time \(t_0\) \[ r(y'(t_0), y(t_0), t_0) = 0 \]