Imagine I have a differential equation and I have some data and the model is supposed to fit the data. If I now rescale the time in the range 0 to 1, do I need to adjust the parameters of the equations? How would one adjust them?
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1$\begingroup$ I’m voting to close this question because it's off topic for computational science but might be appropriate in math.stackexchange. $\endgroup$– Brian BorchersNov 18, 2020 at 0:56
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3$\begingroup$ Actually, if we could see the structure of the equation it would be clearer if this question belongs more to math or to computational science $\endgroup$– Maxim UmanskyNov 18, 2020 at 3:03
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Yes. Let's assume you have an ODE of the form $$ x'(t) = kx(t) $$ and that your coefficient is $k=42$. If the physical units of $x$ are meters and of $t$ seconds, then what that really means is that $k=42 \frac{1}{s}$.
So now if you rescale time -- say, you want to measure time in minutes, you still have $k=42 \frac{1}{s}$ but you want to express this also in minutes, which would make it $$ k=42 \frac{1}{s} = 42 \frac{1}{\frac{1}{60}\text{min}} = 42\times 60 \frac{1}{\text{min}} = 2520 \frac{1}{\text{min}}. $$ That means, the coefficient is exactly the same as before, but it has a different numeric value that would have to be input into a program.
answered Nov 18, 2020 at 18:21