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How to solve exponential equation to find unknown constants

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Nidhi
Nidhi on 8 Aug 2021
Commented: Nidhi on 8 Aug 2021
I am trying to find the values of the constants a, b and c from the exponential equation. The data points of x and t needs to be exported from an excel file(given) . I have tried solving it using lsqcurvefit but I am facing problem in finding the initial values of the parameters to solve the problem.
The equation is
Here, x and t are variables. The value of y is constant and equal to 6.

Accepted Answer

J. Alex Lee
J. Alex Lee on 8 Aug 2021
data = readtable("Book1.xlsx");
Alternatively, based on my reparameterization above you can as a first pass say everything after looks like saturation, take the average, and declare
mask = data.t < 4;
F = mean(data.x(~mask))
F = 157.1525
Now you can linearize the problem and say
To visualize
z = log(F-data.x);
plot(data.t,z,'.')
Warning: Imaginary parts of complex X and/or Y arguments ignored.
hold on
As expected, the plot is pretty linear for , so you can do a linear fit and visualize
lf = polyfit(data.t(mask),z(mask),1)
lf = 1×2
-1.3886 1.7279
plot(data.t,polyval(lf,data.t),'-')
Then convert the linear fit parameters back to
H = -lf(1)
H = 1.3886
G = exp(lf(2))
G = 5.6287
And then you're back to the algebra to get back to . This procedure, which now only relies on observing the saturation behavior, results in
figure
plot(data.t,data.x,'.')
hold on
plot(data.t,F - G*exp(-H*data.t),'-')
  3 Comments
Nidhi
Nidhi on 8 Aug 2021
@J. Alex Lee I understood how to find the values now. Thank you so much once again!

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More Answers (1)

J. Alex Lee
J. Alex Lee on 8 Aug 2021
Do you have an intuition of what the graph of looks like? Hint: it looks like your plot, scaled and shifted roughly only in the y direction, so roughly whatever appears in front of t in the exponential is roughly unity. Next step, simplify your parameterization a bit so you can intuitively scale and shift...say
That makes it even easier to see the link between your data. At , and at , so the scale factor and shift factor is .
Now it's an algebra problem to get from to .

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