I don't think this is a really hard or advanced problem but I don't know how to describe it technically, so I am going to try to make it simple (even if it takes a long time in words).
This is a totally idealized, abstract thought experiment so please bear with me.
I have three containers: A, B, and C. Each one has an interior temperature. A is connected to B and C in such a way that increases to A's temperature should, over time, increase the temperatures of B and C. The three containers are perfectly insulated, not from each other, but there is no surrounding medium for heat to go into.
At the beginning of the story A, B, and C have the same interior temperature. At time 0, I magically set the temperature of A. As time advances, the heat from A begins diffusing into B and C.
In my imagination, A is sending a certain amount of heat into its neighbors in every time-unit. This amount of heat it is sending is proportional to how much heat it has already got (and has something to do with the surface area/conductivity of the interface between the containers, which is constant because I don't care about it).
When it sends the heat, it doesn't have it any more. So over time it is losing heat, and the others are gaining the sent heat, and correspondingly the transfers are smaller and smaller.
I want an expression which does the following: for an arbitrary given time t, tell me how much heat A has lost since time 0.
Realism isn't the point: as far as I am concerned I could be talking about the concentration of a gas diffusing through the containers, or any number of similar situations (lotto wins diffusing smoothly to creditors and extended family over time, etc.)
That is the mathematical problem I am thinking about. My personal problem is this. It is easy for me to think about this from one time step to the next. But since each change in A depends on the last change in A, I can't see a simple way to determine the state at time t except to compute and apply the difference at every time step. I know this is actually a calculus problem but I don't have the intuition to choose the right tools here. And this is really what I am trying to do by means of this silly problem: to build my intuition about how to use the calculus in a self-motivated way (not just test questions).