I have some data represented as a surface below with time and distance axes. I would like to fit this to all possible solutions for the diffusion equation that is:
dS/dt = D.d2S/dx2
where S is the signal representing concentration here, t is time and x is distance.
I think the pdepe in MATLAB can do this but I'm not sure how to implement it? I'm familiar with fitting data to a function which often is the analytical solution to diff. eqns, but am not sure how to solve and fit at the same time. Could someone explain/demonstrate this to me?
Example matrix data attached. All data(:,1) are on the distance axis and data(1,:) are on the time axis.
Thanks!
Hi.
You didn't give any specifics (initial, boundary, etc.) so I apologize if this is a bit general. I'm assuming you have a fixed diffusion coefficient and want to find it from the data. If not, I hope this may still point you in the right direction.
First, I agree that you have to go with a numerical solution. Any sort of Fourier series or LaPlace transform will result in an infinite number of terms, which will be really difficult to work with.
Next, I would suggest an approach like this:
1) Set up your PDE in the MATLAB solver with appropriate IC and BC and meshing for x and t
2) Assume a "guess" for D, and integrate the equation
3) Compare the numerical solution at selected points to your data, then adjust D, integrate again, compare and adjust D, and repeat until you have a satisfactory solution.
The compare-and-adjust can be done in MATLAB using fmincon, which will perform a nonlinear regression type optimization. To do this, minimize the sum of the squared residuals (S-data - S-calc)^2 over all the compared points.
A word of caution... sometimes fmincon is stubborn and keeps going to the same local minimum over a range of initial guesses. In this case, you should consider bounds on your guess for D. If that does not work out, you may need to search for a global solution-- you can write your own Multistart routine in MATLAB (Multistart is described on the web) or use the Global Optimization Toolbox from MATLAB.
An alternative way to do the search for D is by using MATLAB with Microsoft Excel via the Spreadsheet Link (available on the MATLAB site). With this, you call the MATLAB (as a function from Excel) to integrate the PDE, passing the guess for D and the initial conditions into MATLAB, which will integrate and pass the S points for comparison back to Excel. Using a function in Excel like SUMXMY2 to calculate the sum of the squared residuals, you can then use the SOLVER in Excel to keep repeating that sequence until the sum of squares is minimized by adjusting D. This has the advantage that the global minimization can be accomplished using Multistart in the SOLVER, and you may have a shorter learning curve than with the MATLAB toolboxes.
Either method will require some work and deciding how many points (in x and t) to compare... I don't have any general rules for that but you can figure that you by some trial and error, and some educated guessing lol
By the way, If you have other parameters instead of D, you can do the same by adjusting those parameters after choosing what you want to go into the sum of squares to be minimized,
Hope this helps. Good luck!
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