Fit several functions with lmfit with shared parameters, no datasets- in Python

Question

I would like to find the parameters E_u, tau_max, and G from these 3 functions.

The functions are the following:

Function 1: 0=0.009000900090009*E_u*(0.000103939092728486*exp(1500000.0/tau_max) + 0.000157703794137242*exp(2999000.0/tau_max) + 0.00017784012*exp(4500000.0/tau_max) + 0.00025534696*exp(6000000.0/tau_max) + 0.00027086158*exp(7500000.0/tau_max) + 0.000280826592271819*exp(9000000.0/tau_max) + 0.0004132622*exp(10501000.0/tau_max))*exp(-10501000.0/tau_max) + 1000000.0*G*(0.000467438377626028*exp(2999000.0/tau_max) + 0.00117770839577636*exp(4500000.0/tau_max) + 0.00197826966391473*exp(6000000.0/tau_max) + 0.00312798328672298*exp(7500000.0/tau_max) + 0.00434787369844519*exp(9000000.0/tau_max) + 0.00561383708066149*exp(10501000.0/tau_max))*exp(-10501000.0/tau_max)/tau_max

Function 2: 1.13624775718=0.09000900090009*E_u*(0.000103939092728486*exp(15000.0/tau_max) + 0.000157703794137242*exp(29990.0/tau_max) + 0.00017784012*exp(45000.0/tau_max) + 0.00025534696*exp(60000.0/tau_max) + 0.00027086158*exp(75000.0/tau_max) + 0.000280826592271819*exp(90000.0/tau_max) + 0.0004132622*exp(105010.0/tau_max))*exp(-105010.0/tau_max) + 10000.0*G*(0.000467438377626028*exp(29990.0/tau_max) + 0.00117770839577636*exp(45000.0/tau_max) + 0.00197826966391473*exp(60000.0/tau_max) + 0.00312798328672298*exp(75000.0/tau_max) + 0.00434787369844519*exp(90000.0/tau_max) + 0.00561383708066149*exp(105010.0/tau_max))*exp(-105010.0/tau_max)/tau_max

Function 3: 1.13106678093=0.9000900090009*E_u*(0.000103939092728486*exp(150.0/tau_max) + 0.000157703794137242*exp(299.9/tau_max) + 0.00017784012*exp(450.0/tau_max) + 0.00025534696*exp(600.0/tau_max) + 0.00027086158*exp(750.0/tau_max) + 0.000280826592271819*exp(900.0/tau_max) + 0.0004132622*exp(1050.1/tau_max))*exp(-1050.1/tau_max) + 100.0*G*(0.000467438377626028*exp(299.9/tau_max) + 0.00117770839577636*exp(450.0/tau_max) + 0.00197826966391473*exp(600.0/tau_max) + 0.00312798328672298*exp(750.0/tau_max) + 0.00434787369844519*exp(900.0/tau_max) + 0.00561383708066149*exp(1050.1/tau_max))*exp(-1050.1/tau_max)/tau_max

Answer 1

You have 3 non-linear equations and 3 unknowns in a set of transcendental equations. There will not be a closed-form solution, but you can get numerical values for the parameters. I know this is in the python section, but you should look into Mathematica for this. There is a good example here: https://mathematica.stackexchange.com/questions/9875/numerically-solving-two-dependent-transcendental-equations