Leveraging Solutions with Duhamel's Theorem

There are many transient heat transfer problems that can be solved analytically for a simple boundary condition, but would be difficult or impossible to solve with a more complicated, time-dependent boundary condition. In many cases, if the original solution can be formulated, possibly through parameterization, as the response to a unit step change on the boundary, Duhamel’s theorem can be used to extend the simple solution to a more complex case.
As an illustration, imagine that this totally fictional function represents the response of a system to a unit step change in the boundary condition where the red line represents the system response, and the blue line represents the boundary condition:

Using Duhamel’s theorem, we can combine that response to several step changes of various sizes, both positive and negative, and even to continuous functions, like this: 

The boundary depicted includes an initial step up, two steps down of different sizes, an increasing ramp, a decreasing exponential, and a step up and back down again. The important point is that we didn’t have to derive any additional solution beyond what we already had in the first figure, we just had to use our solution over and over again, in the appropriately shifted, scaled, and sometimes integrated way to produce the overall solution with the very complicated boundary condition.

Duhamel's Theorem, Part 2