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I often see papers which analytically solve a set of ODEs taking into account the different timescales over which each of the variables change. For example, one might have a set of ODEs

$$\frac{dA}{dt} = \text{constant} $$ $$\frac{dB}{dt} = f(B,C) $$ $$ \frac{dC}{dt} = g(B,C)$$

where $f(B,C)$, $g(B,C)$ are just arbitrary functions of $B,C$.

Now $A$ changes over a timescale $T_A$ which is very short (i.e. $dA/dt = \text{constant}$ is big) whilst $B,C$ change over the same timescale $T_{BC}$ which is long.

In this case can someone explain how the set of ODEs could be solved (perturbatively, approximately, etc.) taking into account the different timescales?

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  • $\begingroup$ Usually one takes $\Delta t$ to be the minimum of the timesteps $\endgroup$ – Kyle Kanos Nov 22 '19 at 1:48
  • $\begingroup$ You have 3 functions but one of them ($A$) is independent of the other two ($B$ and $C$). Are you looking for a proof or the general concept or examples? $\endgroup$ – DanielTuzes Dec 6 '19 at 14:02
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I think the technique you are looking for is the so-called adiabatic decoupling, used in quantum physics quite often, where they refer to it as adiabatic approximation.

Let's suppose you have 2 variables $A$ and $B$ and * $A$ changes fast in time with a characteristic time value $T_A$, but in a limited range, and much above $T_A$, an expected value of $A$ can be expressed as a function of $B(t)$ * $B$ changes slowly in time with a characteristic time value $T_B$. It's value is not necessary limited.

Your differential equations are then

\begin{gathered} \frac{{dA\left( t \right)}}{{dt}} = f\left( {A\left( t \right),B\left( t \right)} \right), \hfill \\ \frac{{dB\left( t \right)}}{{dt}} = g\left( {A\left( t \right),B\left( t \right)} \right). \hfill \\ \end{gathered}

Because $B$ changes slowly, you keep the value of $B$ constant at around $t \in {t_0} \pm \delta {T_B}$, for some small value $\delta$. You approximate $f$ with $\tilde f$ around $B(t_0)$ and go for the solution of the equation

$$\frac{{dA\left( t \right)}}{{dt}} = {{\tilde f}_{B\left( {{t_0}} \right)}}\left( {A\left( t \right)} \right).$$

The solution $A(t)$ corresponds therefore for a steady $B$ value. It can be still a function of $B$, but doesn't depend on its time derivative.

Now you convolve the 2nd equation with a good window function with a width in the order of $T_A$. You get

$$\frac{{d\left\langle {B\left( t \right)} \right\rangle }}{{dt}} = \left\langle {g\left( {A\left( t \right),B\left( t \right)} \right)} \right\rangle.$$

In a hopefully lucky case you act with the $\left\langle \bullet \right\rangle $ operator on the arguments individually resulting in an equation of $$\frac{{d\left\langle {B\left( t \right)} \right\rangle }}{{dt}} = g\left( {\left\langle {A\left( t \right)} \right\rangle ,\left\langle {B\left( t \right)} \right\rangle } \right).$$ You approximate $A$ with $\tilde A$ so that $\left\langle {A\left( t \right)} \right\rangle \approx \left\langle {{{\tilde A}_{B\left( {{t_0}} \right)}}\left( t \right)} \right\rangle = {f_2}\left( {\left\langle {B\left( t \right)} \right\rangle } \right)$. What you have at this point is a differential equation for ${\left\langle {B\left( t \right)} \right\rangle }$ that does is time independent of $A$.

Example

Balls/particles impacting the walls of a container and the container has an initial speed. What is the speed of the container?

You neglect the effect of the moving container on the balls/container and solve the differential equations for the particles (maybe in a statistical manner). Then you can and express the total force acting on the wall. The total force will have an average value if you make the average couple of times over the flight time of the balls/particles.

The net force (e.g., the weight) of the particles contributes to the total force acting on the container, but you no longer resolve the effect of each ball/particle, but use only their net effect. (The pressure can modify the shape of the container, e.g. in the case of a balloon.) You solve the differential equation for the container including the effective contribution of the particles/gas.

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