Abstract:
In this paper, we derive the fundamental solution to the heat equation with a discontinuous diffusion coefficient in the free space, with an absorbing boundary, and with a reflecting boundary. We use the fundamental solution with an absorbing boundary to make connections with the transition probability density of absorbed Skew Brownian Motion (SBM) and derive a formula for the first passage time of SBM. We also consider an oceanographic situation where the steepness of the continental shelf and slope at the shelfbreak are different. This creates a discontinuous coefficient in the Arrested Topographic Wave (ATW) equation, which is a form of the heat equation. Using the fundamental solution with a reflecting boundary naturally applies to the ATW and from this result, we can derive a formula the vertical velocity (the up- and downwelling) of ocean currents at the shelfbreak.
