The chemical reaction system is missing

[mbaudin47]

From (Saltelli, Chan, Scott, 2000), p.44:

A + A -> products

with rate constant k = X1 * exp(-X2 / T) where T=300 K is the absolute temperature.

The concentration of A is:

dY/dt = -2kY^2

where Y(0)=X3 is the initial condition.

(Saltelli, Chan, Scott, 2000) use

• X1 ~ U(8.97e6, 3.59e7)
• X2 ~ U(0,1000)
• X3 ~ U(1.0,1.2)

The solution is:

Y(t) = [2 * X1 * exp(-X2 * t / T) + 1 / X3]^{-1}

The time interval is [1.e-10, 1.e-3], in base 10 logarithmic scale.

The expectation of Y(t) decreases from 1.1 to 0. The standard deviation increases and then decreases: the maximum is reached at 1.e-7 approximately. The table 2.20 shows the expected value and the variance depending on time for 14 time values in the [1.e-10, 1.e-3] interval.

The same model is used in (Saltelli, Chan, Scott, 2000), p.160 for a reliability analysis.

References:

• "Sensitivity analysis", A.Saltelli, K.Chan, E.M.Scott, Wiley (2000).
  p.44 Model 11 : "A nonlinear test model: the chemical reaction system"

• Tilden, J.W., Constanza, V., McRae, G.J. and Seinfeld, J.H. (1980). Sensitivity analysis of chemically reacting systems. Chemical Physics, Springer Series, 18,69-91.