3.2. Classification of Second-Order Linear PDEs

Try these first

1. 1.

   Define a discriminant in your own words.

   Hint

   Think about how the discriminant relates to the solutions.

2. 2.

   What is the condition for a PDE to be hyperbolic?

   Hint

   Remember the symbol >.

3. 3.

   What is the condition for a PDE to be classified as elliptic?

   • D > 0
   • D = 0
   • D < 0

   Hint

   Think about Laplace's equation.

4. 4.

   True or False: Hyperbolic PDEs can describe diffusion processes.

   • True
   • False

   Hint

   Recall what hyperbolic equations are used for.

5. 5.

   Construct a second-order linear PDE that is hyperbolic, provide the coefficients, and calculate the discriminant. Explain why it is hyperbolic.

   Hint

   Select coefficients carefully to ensure D remains positive.

6. 6.

   Given the equation $A \frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2} + C \frac{\partial^2 u}{\partial y^2} = 0$, determine conditions on A, B, and C so it becomes an elliptic PDE.

   Hint

   Manipulate A, B, and C values until D < 0 is achieved.