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Simulating ODE Measurement Values

Forward Analytic Solution
The forward problem has the following closed form analytic solution  \[
\begin{array}{rcl}
     y_0 (t , a) & = & a_0 * \exp( - a_1 * t )
     \\
     y_1 (t , a) & = & 
     a_0 * a_1 * \frac{\exp( - a_2 * t ) - \exp( -a_1 * t )}{ a_1 - a_2 }
\end{array}
\] 


Simulation Parameter Values
 \bar{a}_0 = 1   initial value of  y_0 (t, a)
 \bar{a}_1 = 2   transfer rate from compartment zero to compartment one
 \bar{a}_2 = 1   transfer rate from compartment one to outside world
 \sigma = 0   standard deviation of measurement noise
 e_k = 0   simulated measurement noise,  k = 1 , 2 , 3, 4
 s_k = k * .5   time corresponding to the k-th measurement,  k = 1 , 2 , 3, 4

Measurement Values
The simulated measurement values are given by the equation  \[
\begin{array}{rcl}
z_k 
& = &  y_1 ( s_k , \bar{a} ) + e_k
\\
& = & 
\bar{a}_0 * \bar{a}_1 * 
     \frac{\exp( - \bar{a}_2 * s_k ) - \exp( -\bar{a}_1 * s_k )}
          { \bar{a}_1 - \bar{a}_2 }
\end{array}
\] 
for  k = 1, 2, 3, 4 .
Input File: omh/ipopt_cppad_ode2.omh