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Solving 3 Simultaneous Exponential Equations

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I am trying to solve these 3 simultaneous exponential equations for a,b and c (This is from Vogels Viscosity Equation):
159.2543 = a*exp(b/(291.15-c))
117.2699 = a*exp(b/(293.15-c))
63.8384 = a*exp(b/(299.15-c))
I would really appriciate it if someone could show me how to write the code to solve them please!
Thank you in advance!

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Accepted Answer

Star Strider
Star Strider on 20 Dec 2019
Try this:
x = [291.15; 293.15; 299.15];
y = [159.2543; 117.2699; 63.8384];
% % % MAPPING: a = b(1), b = b(2), c = b(3)
objfcn = @(b,x) b(1).*exp(b(2)./(x - b(3)));
B0 = [0.07; 450; 230];
[B,normresid] = fminsearch(@(b) norm(y - objfcn(b,x)), B0)
xv = linspace(min(x), max(x));
figure
plot(x, y, 'p')
hold on
plot(xv, objfcn(B,xv), '-r')
hold off
grid
Values of:
a = 0.970
b = 180
c = 255
give a reasonable fit to the data.

  2 Comments

Rory Thornton
Rory Thornton on 20 Dec 2019
Great, fits the rest of my data really well too.
Thank you so much!

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More Answers (1)

David Goodmanson
David Goodmanson on 21 Dec 2019
HI Rory,
just for completeness
y1 = 159.2543;
y2 = 117.2699;
y3 = 63.8384;
x1 = 291.15;
x2 = 293.15;
x3 = 299.15;
A1 = log(y1/y2)/log(y2/y3);
A2 = (x2-x1)/(x3-x2);
c = (x1*A1-x3*A2)/(A1-A2);
% back substitute
b = log(y1/y2)*(x1-c)*(x2-c)/(x2-x1);
a = y1/exp(b/(x1-c));
a
b
c
a =
10.6205
b =
42.5004
c =
275.4539
% these should be small
y1 - a*exp(b/(x1-c))
y2 - a*exp(b/(x2-c))
y3 - a*exp(b/(x3-c))
ans =
0
ans =
-4.2633e-14
ans =
-7.1054e-14
The c result is fairly close to Star Strider's, but for some reason a and b differ from that result by quite a bit. The checks here show agreement at all three y points, but the best fit isn't necessarily the one that goes through all three points exactly.

  1 Comment

Rory Thornton
Rory Thornton on 22 Dec 2019
Thank you! It's just a line of best fit, there will be many viable solutions which may have very different a/b/c values. This fits the data well too.

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