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I am working for a mathematical model in which I intend to fit the model itself to statistical data to determine the values that my parameters must have. I am dealing with equations of the sort:

$$0.001=\frac{e^be^{-c}}{1-\int_0^9{e^{at}}dt}$$ and I wish to find all the combinations of a, b c greater or equal to $0$ that may satisfy this condition (I know that there should only be one). I have thought of a brute force approach, but it would be unrealistic to solve this problem in that way.

There has to exist some sort of function to solve this. I have been reading on data annealing, but it doesnt do the trick, and normal data fitting doesn't really make sense, since i don't have "points" to which I must find an equation, but rather i just need to find those three parameters.

Nick Heumann
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  • Are you sure the upper limit of the integral should be $t$? Also, if you know that $a,b,c$ satisfy the equation $a, b+d,c+d$ also satisfy the equation for any $d$. – podiki Nov 17 '21 at 02:06
  • my bad, the upper limit is a constant T that is known, should've specified – Nick Heumann Nov 17 '21 at 09:57

1 Answers1

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As @podiki states, the equation only depends on the difference of $b$ and $c$, let's call this $d$. Then the equation is equivalent to:

$$0.001 = \frac{e^d}{1 - \int_0^9 e^{at}dt }$$

You can easily evaluate the integral:

$$\int_0^9 e^{at}dt = \frac 1 a (e^{9a} - 1).$$

Therefore the equation is equivalent to:

$$e^d = \frac{0.001}{a} (e^{9a} - 1).$$

Its unique solution is:

$$d = \ln \left( \frac{0.001}{a} (e^{9a} - 1) \right),$$

which is well definied, if and only if $a \ne 0$.

d8411BT
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