How to solve exponential equation to find unknown constants

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.

Answer 1

[J. Alex Lee]

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),'-')

[Nidhi]

@J. Alex Lee Thank you so much for your help. Can you please show me how I can find the values of the constants a,b and c?

[J. Alex Lee]

that's just algebra, sorry, no it is too tedious for me.

[Nidhi]

@J. Alex Lee I understood how to find the values now. Thank you so much once again!

Answer 2

[J. Alex Lee]

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 .