estimating parameters in a differential equation with mathematica 6

• To: mathgroup at smc.vnet.net
• Subject: [mg89041] estimating parameters in a differential equation with mathematica 6
• Date: Sat, 24 May 2008 03:55:08 -0400 (EDT)

```I need a help
I have bee trying to solve a differential equation to fit experimental data with mathematica, but something is always wrong. can somebody kindly help me!
this is the experiemtal data
data = {{0.0000001, -0.000001}, {1, -1.386}, {2, -1.933}, {4, \
-3.767}, {5, -5.009}, {6, -6.322}, {7, -7.177}, {7.5, -7.679}};
and the formula for least square
Plus @@ Apply[(f[#1] - #2)^2 &, data, {1}]
then the differential equation is as follow
{f'[t] = -n Log[
1 + \[ExponentialE]^(
k (-22.46` + x/(0.00225`+ 0.04` x)))] (-f[t]/
Log[1 + \[ExponentialE]^(
k (-22.46` + x/(0.00225`+ 0.04` x)))])^(-1 + n)/n}
where "n" and "k" are to be estimated by the FindMinimum operation. so the expression I have used is this one
sse[n_?NumberQ, k_?NumberQ] :=
Block[{sol, f},
sol = NDSolve[{f'[
x] = -n Log[
1 + \[ExponentialE]^(
k (-22.46` + x/(0.00225`+ 0.04` x)))] (-f[x]/
Log[1 + \[ExponentialE]^(
k (-22.46` + x/(0.00225`+ 0.04` x)))])^(((-1) + n)/n),
f[0.0000001] == -0.000001}, f, {x, 0.00000001, 7.5}][[1]];
Plus @@ Apply[(f[#1] - #2)^2 &, data, {1}] /.sol]
followed by this:
FindMinimum[sse[n, k], {n, 0.00001, 100}, {k, 0.00001, 100}]

but i can get any solution.this one of the comments i receive
ReplaceAll::reps: {-(0.00001 \
Log[1.+<<18>>^(0.00001<<1>><<1>>)])/(-(1. \
<<1>>)/(Log<<1>><<1>>]))^99999.,f[1.*10^-7]==-1.*10^-6} is neither a \
list of replacement rules nor a valid dispatch table, and so cannot \
be used for replacing. >>

NDSolve::deqn: Equation or list of equations expected instead of \
-(0.00001 Log[1+\[ExponentialE]^(0.00001 \
<<1>>)])/(-f[x]/Log[<<1>>])^99999. in the first argument {-(0.00001 \
Log[1+\[ExponentialE]^(0.00001 \
<<1>>)])/(-f[x]/(Log<<1>><<1>>]))^99999.,f[1.*10^-7]==-1.*10^-6}. >>

it seems there is a syntax problem somewhere.
Thanks in advance for any help,
Sylvain

```

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