Nonlinear least-squares fitting of curve described by PDEs

Hi people. I would like to fit a curve described by a system of two 2nd degree partial differential equations (PDEs) using lsqnonlin. While it is simple to write your anonymous function when you have a single equation for your model, how can you do it when you have a system of PDEs, which do not have an analytic solution for the parameter of interest (the one to be fitted on the experimental data)? The PDEs have a number of free variables for which I would like to get the values that best fit the data. I could also try a better method than lsqnonlin, if there is one. Thanks

Answer 1

[Torsten]

Although the description here is for ODEs, it can easily be adopted for PDEs:

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

Best wishes

Torsten.

[Xen]

Hi Torsten. The link was helpful, but I am not sure how to code the first and second derivatives of variables that depend both on x and t. Lets say I have these equations

dS/dt = a * dS/dx + b * d2S/dx2

dX/dt = c * dX/dx + d * d2X/dx2

How would you code the DifEq(t, x) function? And can the ode45 function solve this system? According to Matlab documentation you can only pass tspan, but not, say, xspan. Thanks.

[Torsten]

Use "pdepe".

Best wishes

Torsten.

[Xen]

Despite that I can solve the PDEs for a random set of parameter values, I can't get it to work with lsqcurvefit. How can I pass my initial parameter estimates in the anonymous PDE function? I am thinking of just solve it for random parameters' combinations, compare with my experimental data and manually find the best solution...

[Torsten]

The "Anonymous Function" approach is the most flexible way to pass extra parameters to the PDE functions:

https://de.mathworks.com/help/optim/ug/passing-extra-parameters.html

Best wishes

Torsten.