This example illustrates Laplace transform solution for a parabolic partial differential equation.
Find the flow velocity in a viscous fluid being dragged along by an accelerating plate.
Try a Laplace transform in .
Transform the partial differential equation:
Transform the boundary condition:
Solve the partial differential equation:
Apply the boundary condition at that must be regular there:
Apply the given boundary condition at :
Solving for and plugging it into the solution of the ordinary differential equation,
has been found:
We need to find the original function corresponding to the transformed
We do not really know what is, just that it transforms back
to . However, we can find the other part of in the
How does times this function transform back? The product of
two functions, say
, does not transform back
to . The convolution theorem Table 6.3 # 7 is needed: