Questions: Classification of PDEs (Elliptic, Parabolic, Hyperbolic)

Question 1

For the PDE Au_xx + 2Bu_xy + Cu_yy + (lower order terms) = 0, how is the type determined?

• A. By the sign of B² - AC
• B. By the sign of A + C
• C. By the sign of AC - B²
• D. By whether A, B, C are positive

Answer: (A)

The discriminant B² - AC determines the type: positive gives hyperbolic, zero gives parabolic, and negative gives elliptic. This directly parallels the discriminant for conic sections.

Question 2

The heat equation u_t = ku_xx is classified as parabolic.

• T. True
• F. False

Answer: True

Written in standard form with x and t as the two independent variables, the heat equation has A = k, B = 0, C = 0, giving discriminant B² - AC = 0, which is the parabolic case.

Question 3

What physical behavior is typically associated with hyperbolic PDEs?

Model answer: Wave propagation with finite speed

Hyperbolic PDEs describe phenomena where disturbances propagate at finite speeds along characteristic curves, such as sound waves, electromagnetic waves, and vibrating strings.

Question 4

A PDE can change classification from one region of the domain to another.

• T. True
• F. False

Answer: True

When the coefficients A, B, C depend on the spatial variables, the discriminant B² - AC can change sign across the domain. The Tricomi equation y·u_xx + u_yy = 0 is elliptic for y > 0, parabolic at y = 0, and hyperbolic for y < 0.

Question 5

Which type of PDE requires specification of both initial conditions and boundary conditions for a well-posed problem on a bounded domain?

• A. Elliptic
• B. Parabolic
• C. Hyperbolic
• D. Both parabolic and hyperbolic

Answer: (D)

Both parabolic and hyperbolic PDEs are evolution equations that require initial conditions (specifying the state at t = 0) along with boundary conditions. Elliptic PDEs describe equilibrium states and require only boundary conditions.