15. Partial Differential Equations

Try these first

1. 1.

   Define a Fourier series in your own words.

   Hint

   Think of it as breaking down a complex signal.

2. 2.

   List the Dirichlet conditions.

   Hint

   These conditions ensure that the series converges.

3. 3.

   What does a Fourier series represent?

   • A single polynomial function
   • A sum of sine and cosine functions
   • An ordinary differential equation

   Hint

   Think about what functions are used to create a Fourier series.

4. 4.

   True or False: The Dirichlet conditions are necessary for the convergence of Fourier series.

   • True
   • False

   Hint

   Recall the essential requirements for Fourier representation.

5. 5.

   Given a non-periodic function defined on $[0, L]$, apply half-range expansions to derive a Fourier sine series representation.

   Hint

   Think of representing your function in terms of odd extensions.

6. 6.

   Using the method of separation of variables, derive the solution to the wave equation under given boundary and initial conditions.

   Hint

   Pay close attention to how the initial velocity influences the form of the sine and cosine series.