16.10. Canonical (Standard) Forms of Second-Order PDEs

Try these first

1. 1.

   What is the general form of a second-order PDE?

   Hint

   Look for the highest order derivatives present.

2. 2.

   Define discriminant in the context of second-order PDEs.

   Hint

   Recall how it relates to the coefficients A, B, and C.

3. 3.

   What is the canonical form of an elliptic PDE?

   • A: \\( \\frac{\\partial^2 u}{\\partial \\xi^2} + \\frac{\\partial^2 u}{\\partial \\eta^2} = 0 \\)
   • B: \\( \\frac{\\partial^2 u}{\\partial \\xi^2} = \\frac{\\partial u}{\\partial \\eta} \\)
   • C: \\( \\frac{\\partial^2 u}{\\partial \\xi \\partial \\eta} = 0 \\)

   Hint

   Recall the different forms discussed in class.

4. 4.

   If D = B^2 - 4AC < 0, which type of PDE do you have?

   • True
   • False

   Hint

   Think about how discriminants categorize the equations.

5. 5.

   Given the PDE $2 \frac{\partial^2 u}{\partial x^2} - 8 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0$, classify it and find the canonical form.

   Hint

   Pay close attention to the coefficients A, B, and C.

6. 6.

   Derive a transformation for the equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ into its canonical form.

   Hint

   Focus on recognizing how changes in variables can simplify complex equations.