## 16.2 Example: chemical kinetics

As an example of a system of DAEs, consider following chemical kinetics problem. 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. .

$\frac{dy_1}{dt} + \alpha y_1 - \beta y_2 y_3 = 0 \frac{dy_2}{dt} - \alpha y_1 + \beta y_2 y_3 + \gamma y_2^2 = 0 y_1 + y_2 + y_3 - 1.0 = 0$

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 $$t_0$$ $r(y'(t_0), y(t_0), t_0) = 0$

### References

Robertson, H. H. 1966. “The Solution of a Set of Reaction Rate Equations.” In Numerical Analysis, an Introduction, 178–82. Lodon; New York: Academic Press.
Serban, Radu, and Alan C. Hindmarsh. 2021. “Example Programs for IDAS.” LLNL-TR-437091. Lawrence Livermore National Laboratory.