I'm trying to find a function, $G(t)$, that satisfies the relation
$$G(t+1) - G(t) = f(t)$$
for some known function, $f(t)$, for $t \in R$. I have been able to come up with solutions for a few instances of $f(t)$ by trial and error and inspection, namely:
$$\begin{align}
f(t) & = m t + f_0 & & => & & G(t) = (m/2) t^2 + (f_0 - m/2) t
\\
f(t) & = f_0 (1 + r)^t & & => & & G(t) = (f_0 / r) (1 + r)^t
\end{align}$$
where $f_0$, $r$, and $m$ are constants. But I can't seem to identify any method which can be applied generally.
I'm having a tough time even naming the class of problem - I wrote "continuous difference equation," though that doesn't seem to be a commonly used term. My current thinking is that I might be able to write this as an integral equation in some way, since the $G(t+1) - G(t)$ is reminiscent of the Fundamental Theorem of Calculus, and the solutions I've found so far are kind of similar to the integrals of $f(t)$, though just a bit off. If anyone has seen anything like this before or knows even what to call this type of problem it would be greatly appreciated!
Answer
That's exactly the subject of what is called Indefinite Sum
$$
\Delta G(t) = G(t + 1) - G(t) = f(t)\quad \Rightarrow \quad G(t) = \Delta ^{\, - 1} f(t) = \sum\nolimits_{\,t} {f(t)}
$$
that is to find the function $G(t)$ whose finite difference is a given function $f(t)$.
The topic has many similarities with the (definite / indefinite) integrals, and the solution methods are much alike:
- basic theory and identities;
- table of basic integrals;
- lot of expertise and intuition to put up a chain combining the above.
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