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: 