12.3. Solution of the Heat Equation by Separation of Variables

Try these first

1. 1.

   What does the assumption $u(x,t) = X(x) T(t)$ represent?

   Hint

   Think about how we can break the problem into simpler parts.

2. 2.

   Name one type of boundary condition we could apply.

   Hint

   What specifies temperature at the boundaries?

3. 3.

   What does the method of separation of variables aim to do?

   • Combine variables into one
   • Split functions into separate dependent variables
   • Eliminate PDEs altogether

   Hint

   Think about the word 'separation'.

4. 4.

   True or False: The solution to the heat equation can include both sine and cosine functions.

   • True
   • False

   Hint

   Recall how we solved the spatial part.

5. 5.

   Given a rod of length $L$ with heat starting at a linear distribution, derive the temperature at later times using separation of variables.

   Hint

   Begin with writing the equation and identify the boundaries.

6. 6.

   How would the solution change if we modified the boundary condition to Neumann conditions?

   Hint

   Consider how heat flux will reflect in the function behavior.