16.7. Solving PDEs with Boundary and Initial Conditions

Try these first

1. 1.

   What is an initial condition in a PDE?

   Hint

   Think about what information you need at time t = 0.

2. 2.

   Name the three types of boundary conditions.

   Hint

   Recall the DNR acronym.

3. 3.

   What do Dirichlet conditions specify?

   • The derivative at the boundary
   • The function value at the boundary
   • No flow at the boundary

   Hint

   Recall the type of condition that dictates what is happening right at the edge.

4. 4.

   True or False: A well-posed problem is one that is stable.

   • True
   • False

   Hint

   Consider what well-posed means in connection with solutions.

5. 5.

   A cylindrical rod of length L is held at fixed temperatures at both ends (Dirichlet conditions). The initial temperature distribution is given as u(x,0) = sin(πx/L). Solve the PDE for different times t > 0.

   Hint

   Consider using Fourier series to express the initial condition.

6. 6.

   A string fixed at both ends vibrates according to the wave equation with initial displacement u(x,0) = f(x) and initial velocity ∂u/∂t (x,0) = g(x). Derive the general solution of the wave equation subject to Neumann boundary conditions.

   Hint

   Focus on how the wave behavior is altered based on the boundary at the fixed ends.