Subsections

### 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: