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I am trying to take the Laplace transform of floor(f(t)) in order to solve the differential equation f'(t)=floor(f(t)).

I know that http://functions.wolfram.com/IntegerFunctions/Floor/22/04/" [Broken] gives L(floor(t)) = (e^(-s))/(s(1-e^(-s))) instead -- are these equivalent?) and L(f(t)) = F(t) (of course), but I realized that I have no idea how to take the Laplace transform of a composition of functions, and no table I have been able to find contains L(floor(f(t))) or rules about compositions of functions. There is plenty of information on convolutions, but that isn't what I'm looking for unless this problem can be reformulated as one.

My question, at its essence, is this: given functions f and g, how do I determine L(f(g(t))? More specifically I want to know L(floor(f(t))), but any insight on the general case (if the problem does generalize) would be much appreciated.

In terms of background for the problem, I'm trying to model some cells for my cancer research, and my system of equations based upon the simple case f'=a*f(t) overestimates growth rates by effectively allowing hypothetical "fractions of cells" to divide. The discrete case based on f'=a*floor(f(t)) should solve this problem, but I do not know how to obtain an analytical solution for it.

I apologize if I am missing something obvious here, as well as for not knowing LaTeX.

Thank you!

Edit: I had originally posted a version of this thread in the Calculus and Analysis section https://www.physicsforums.com/showthread.php?p=2524246" since it was more concerned with the Laplace transform itself than solving the differential equation once I had it, but I realized that my query was still more germane to the Differential Equations forum.

Edit2: [tex]\mathcal{L}(\text{floor}(t))=\frac{e^{-s}}{s(1-e^{-s})}[/tex] or perhaps [tex]\mathcal{L}(\text{floor}(t))=\frac{1}{s(e^{s}-1)}[/tex] -- included for clarity now that I have learned some LaTeX.

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# Laplace Transform of Composition of Functions: Floor(f(t))

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