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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

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.

Torsten
on 21 Jul 2021

Alan Stevens
on 18 Jul 2021

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