18.3. General Steps in the Eigenfunction Expansion Method

Try these first

1. 1.

   Define eigenfunction in the context of PDEs.

   Hint

   Think about functions that remain unchanged apart from scaling.

2. 2.

   What step follows after separating variables in a PDE?

   Hint

   Identify what your variables relate to.

3. 3.

   What is the first step in the Eigenfunction Expansion Method?

   • Combine solutions
   • Identify and solve the spatial part
   • Express the initial condition

   Hint

   Think about how we approach separating different parts.

4. 4.

   True or False: Eigenvalues can be complex.

   • True
   • False

   Hint

   Recall the properties of eigenvalues in Sturm-Liouville problems.

5. 5.

   Given the boundary conditions for a rod held at two ends and subjected to a heat flux, derive the eigenfunctions and eigenvalues. Next, use them to find the general solution of the heat equation.

   Hint

   Focus first on applying the correct boundary conditions to obtain eigenvalues.

6. 6.

   Consider a wave equation defined on a string of length L with fixed ends. Describe how you would derive the first three eigenfunctions and show how they form a complete basis for the problem.

   Hint

   Use the knowledge of harmonic frequencies.