16.5. Linear and Nonlinear PDEs

Try these first

1. 1.

   Define what makes a PDE linear.

   Hint

   Think about how variables appear in the equation.

2. 2.

   Provide one example of a linear PDE.

   Hint

   Look for an equation where all terms are in the first degree.

3. 3.

   Which of the following equations is a linear PDE?

   • \\left( \\frac{\\partial u}{\\partial x} \\right)^2 + \\frac{\\partial u}{\\partial y} = 0
   • \\frac{\\partial u}{\\partial t} + k \\frac{\\partial^2 u}{\\partial x^2} = 0
   • 2u^2 + \\frac{\\partial u}{\\partial x} = 0

   Hint

   Look for powers that are only one.

4. 4.

   True or False: Nonlinear PDEs can have superposition applied to their solutions.

   • True
   • False

   Hint

   Remember the characteristics of solutions in linear versus nonlinear equations.

5. 5.

   Devise a nonlinear PDE that incorporates both x and y derivatives and explain how it fits the definition of nonlinear.

   Hint

   Make sure one term is a product of derivatives or powers.

6. 6.

   Explain the significance of superposition in linear PDEs and why it doesn’t apply to nonlinear PDEs with an example.

   Hint

   Think of concrete examples showing different behaviors in solutions.