# Finding best parametric function estimation for ODE of first order

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Hello to every one.

Until now I've solved differential equations numerically under given formulae for their coefficients and given initial / boundary conditions. But recently I've been dealt with the inverse problem, namely: find the best fit to the solution of the ODE regarding some unknown paramateres. The hard part here is due to the fact that we need to determine not a single value but the values of a whole parametric function.

Consider the following initial value problem for ODE of first order:

Solving the equation by separation of the variables we obtain the exact solution

Let's consider however this problem statement: given the Cauchy's problem

whose solution is known over the grid Find the estimated values of the function fitting the solution , i.e. establish that they're eventually located close enough to the curve .

I read a lot about the topic but I didn't find what I searched for. Could you give me some short explanations or guidelines on how to perform this idea in general and later using MATLAB toolboxes?

Kind regards,

Lyudmil Yovkov, PhD

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### Accepted Answer

Torsten
on 18 Jul 2021

Edited: Torsten
on 18 Jul 2021

Search for a book on optimal control of ODEs and DAEs, e.g.

Matthias Gerdts:

Optimal control of ODEs and DAEs

De Gruyter 2011

Or start with an introductory script:

silo.tips/download/optimal-control-of-odes-introductory-lecture

Finally, you will have to use Matlab's lsqcurvefit or fmincon to determine the coefficients c_i on the grid x_i that minimize

sum_{i=0}^{i=n} (u_i - u_i(c_i))^2

where u_i(c_i) are the values for u obtained from the ODE by using the c_i as coefficients in front of du/dx.

##### 11 Comments

Torsten
on 21 Jul 2021

### More Answers (2)

Alan Stevens
on 18 Jul 2021

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