3.1.2. Linear PDEs

Try these first

1. 1.

   Define a linear PDE and give an example.

   Hint

   Think about equations you've seen that represent physical phenomena.

2. 2.

   What is the heat equation?

   Hint

   Consider the role of time and space in heat transfer.

3. 3.

   What characterizes a linear PDE?

   • Terms with higher powers of variables
   • Terms only to the first power
   • Mixed derivatives only

   Hint

   Consider how exponents affect linearity.

4. 4.

   True or False: Laplace's Equation is a nonlinear PDE.

   • True
   • False

   Hint

   Think about the definition of linear vs. nonlinear.

5. 5.

   Prove that if u(x,y) is a solution to $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ and v(x,y) is a solution, then au + bv is also a solution for constants a and b.

   Hint

   Break down the terms of both functions.

6. 6.

   Calculate the steady-state temperature distribution along a rod described by the heat equation with boundaries held at constant temperatures of T1 and T2.

   Hint

   Consider the boundary values set for T1 and T2.