Solving Multiple First Order Differential Equations for Constants

Hi,

I hope someone can help me.

I am looking to solve these 6 first order differential equations for the unknowns k1, k2, k3, k4, k5 and k6, with known concentrations in the square brackets (any number for them will do just now to get a working script!). Could someone help me obtain a script to do this?

Many thanks in advance.

Answer 1

[Torsten]

Replace the time derivatives by difference quotients.

Then you get a linear system of equations

A*[k1;k2;k3;k4;k5;k6] = b

where the number of rows in A is equal to (6*number of measurement times).

This can subsequently be solved using

k = A\b.

Best wishes

Torsten.

[HK]

Thank you very much!

I am still slightly confused though. Is there any chance you could type out some of the script? I think that would help me understand a lot better.

Thanks again

[Torsten]

Which data from measurements do you have ?

[HK]

I know all the values in moles or mass for the quantities in the square brackets, eg. [MeOH] = 1.412 moles.

Thanks again so much

[Torsten]

Ok. Then for [TG], e.g., you can use the equations

[TG]_(i+1) - [TG]_(i) = dt * (-k1*[TG]_(i+1)*[MeOH]_(i+1) + k2*[DG]_(i+1)*[ME]_(i+1))

for i=1,2,...,N.

Same for the other components.

You see that you get a linear system of equations in the unknowns k1,k2,...,k6, as stated above.

This can be solved using the mldivide - operator.

[HK]

Slowly but surely I think I am getting somewhere! (Sorry I am inexperienced with Matlab and trying to improve).

Can you identify the errors that I still have? Am I solving for k in the command window correctly? All of the equations for 'TG(i+1) - TG(i)' etc are in correctly but there are errors around the equals signs saying 'Parse error at '=': usage might be invalid MATLAB syntax'.

[Torsten]

To get approximate values for k1,...,k6, you need measurement data for the concentrations [TG], [DG],,...,[MeOH] at different times t1,...,tn, not only at t=0. Do you have such data ?

[HK]

Yes, I have concentration data at the end of the reaction (t=60 mins/3600s) and also at two intermediate times (t = 2 mins/120s and t = 12 mins/720s). Should I state all of these values at the top of the script like I did for the values at t=0? Should the script then run or do additional changes need to be made in its layout?

Thanks again!

[Torsten]

Since your data are taken at times with such a great temporal distance, you should try the method described here:

https://de.mathworks.com/matlabcentral/answers/43439-monod-kinetics-and-curve-fitting

Best wishes

Torsten.