Suppose I'm solving
$$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?
Okay, so I don't really see the point you're trying to convey and how it relates to my question. Below, I'll write out a set steps in reasoning to see if you can pinpoint the error I'm making.
Let's only consider the case ##n = 2## i.e traveling around the ring twice.
Periodicity demands that...
I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be:
$$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...
Imagine a bubble vibrating in air. Because it vibrates, it's interfacial area increases, thus new molecules are added and removed from the surface as it vibrates.
Consider a molecule is initially at position X_0 at the interface, and over a certain amount of time molecules squeeze and disappear...