- 7.2.1 The physical problem
- 7.2.2 The mathematical problem
- 7.2.3 Transform the problem
- 7.2.4 Solve the transformed problem
- 7.2.5 Transform back

7.2 A parabolic example

This example illustrates Laplace transform solution for a parabolic partial differential equation.

7.2.1 The physical problem

Find the flow velocity in a viscous fluid being dragged along by an accelerating plate.

7.2.2 The mathematical problem

- Semi-infinite domain :
- Unknown vertical velocity
- Parabolic
- One homogeneous initial condition
- One Neumann boundary condition at and a regularity constraint at
- Constant kinematic viscosity

Try a Laplace transform in .

7.2.3 Transform the problem

Transform the partial differential equation:

Transform the boundary condition:

7.2.4 Solve the transformed problem

Solve the partial differential equation:

This is a constant coefficient ordinary differential equation in , with simply a parameter. Solve from the characteristic 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:

7.2.5 Transform back

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
tables.

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: