MATLAB: Data fitting by Integration with variable limit with a unknown parameter.

curve fittingfinding unknown parametersintegration

I want to fit experimental data with equation:
y = C(1) * 74.826 * (x/T(1))^3 * integration (t^4*exp(t)/(exp(t)-1)^2), 0, T(1)/x) + C(2) * 24.942 * (T(2)/x)^2 * exp(T(2)/x)/(exp(T(2)/x)-1)^2 + C(3) * 24.942 * (T(3)/x)^2 * *exp(T(3)/x)/(exp(T(3)/x)-1)^2
with six unknown parameters: C(1), T(1), C(2), T(2), C(3), T(3).
The integral part is giving error all times. How to write the programm please help.
Thank you.

Best Answer

You can do something like this:
y_fcn = @(CT,x) CT(1) * 74.826 * (x./CT(4)).^3 * integral(t.^4.*exp(t)./(exp(t)-1).^2), 0, CT(4)./x) + ...
                CT(2) * 24.942 * (CT(5)./x).^2  .* exp(CT(5)./x)./(exp(CT(5)./x)-1).^2 +  CT(3)*24.942*(CT(6)./x).^2.*exp(CT(6)./x)./(exp(CT(6)./x)-1).^2;
err_fcn = @(CT,x,y) sum((y-y_fcn(CT,x)).^2);
CT0 = [1 2 3 4 5 7]; % Some sensible initial guess for C and T, that we concatenate together as CT = [C,T]
CT_best = fminsearch(@(CT) err_fcn(CT,x_obs,y_obs),CT0)
You most likely have to use element-wise operations in the equation for y (I might have missed one or two multiplictions or power-operations). Then it should be fine to use ordinary least-square fitting of the C and T parameters (provided you have enough data-points, and they have the same standard deviation - but I expect you to know this already).
HTH
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