18. Partial Differential Equations

Try these first

1. 1.

   What is an eigenfunction?

   Hint

   Think about functions that remain proportional under transformations.

2. 2.

   Define a boundary value problem.

   Hint

   Consider where conditions are set in a physical system.

3. 3.

   What method is commonly used to solve linear PDEs?

   • Eigenfunction Expansion
   • Matrix Method
   • Gradient Descent

   Hint

   Think about the techniques discussed in class.

4. 4.

   True or False: Eigenfunctions are always orthogonal.

   • True
   • False

   Hint

   Recall how eigenfunctions relate to inner product properties.

5. 5.

   Consider a second-order PDE. Formulate a Sturm-Liouville problem that yields a set of eigenfunctions appropriate for boundary conditions $u(0)=0$ and $u(L)=0$.

   Hint

   Focus on how the boundary conditions influence the form of the solution.

6. 6.

   Using the heat equation example, derive how the coefficients in the expansion relate to the initial condition using the inner product.

   Hint

   Recall how orthogonality simplifies finding coefficients in the expansion.